Simulating Unsteady Conduit Flows with Smoothed Particle Hydrodynamics

Simulating unsteady conduit flows with smoothed particle hydrodynamics

Citation for published version (APA): Hou, Q. (2012). Simulating unsteady conduit flows with smoothed particle hydrodynamics. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR733420

DOI: 10.6100/IR733420

Document status and date: Published: 01/01/2012

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Download date: 23. Sep. 2021 Simulating Unsteady Conduit Flows with Smoothed Particle Hydrodynamics Copyright ⃝c by Qingzhi Hou, Eindhoven, The Netherlands. All rights are reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the author.

A catalogue record is available from the Eindhoven University of Technology Library Proefschrift. - ISBN: 978-90-386-3167-7

NUR 919 Subject headings: initial boundary value problems; moving boundaries; channel flows; two-phase flows; waterhammer; pipe filling and emptying; isolated slug; smoothed particle hydrodynamics 2010 Mathematics Subject Classification: 65D07, 65D10, 65N15, 65N35, 76B10, 76D05, 76D50, 76M28, 76T10

The work described in this thesis has been ﬁnancially supported by the China Scholarship Council (CSC). Simulating Unsteady Conduit Flows with Smoothed Particle Hydrodynamics

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magniﬁcus, prof.dr.ir. C.J. van Duijn, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op maandag 25 juni 2012 om 16.00 uur

door

Qingzhi Hou

geboren te Shandong, China Dit proefschrift is goedgekeurd door de promotor: prof.dr. R.M.M. Mattheij

Copromotor: dr.ir. A.S. Tijsseling Summary

Pipelines are widely used for transport and cooling in industries such as oil and gas, chemical, water supply and sewerage, and hydro, fossil-fuel and nuclear power plants. Unsteady pipe flows with large pressure variations may cause a range of problems such as pipe rupture, support failure, pipe movement, vibration and noise. The unsteady flow is generally caused by flow velocity changes due to valve or pump operation. Water hammer is the best known and extensively studied phenomenon in this respect. Fast transients may also occur in rapid pipe filling and emptying processes. Due to high driving heads, the advancing liquid column may achieve a high velocity. When this high-velocity column is blocked or restricted in its flow, high water-hammer pressures may result. Another scenario is that of slug flow, which arguably is the most dangerous type of two-phase pipe flow. Heavy isolated liquid slugs travelling at high speed be- have like cannonballs. Damage is likely to happen when these slugs impact on barriers such as pumps, bends and partially closed valves. Advancing liquid columns occurring in rapid pipe filling and emptying can be seen as a special case of isolated slugs.

In this thesis, we present a Lagrangian particle method for solving the Euler equations with application to water hammer, rapid pipe ﬁlling and emptying, and isolated slugs travelling in an empty pipeline. As a meshfree method, the smoothed particle hydrodynamics (SPH) used herein is suitable for problems encompass- ing moving boundaries and impact events, which are the common features of the concerned topics.

We ﬁrst present the kernel and particle approximation concepts, which are two essential steps in SPH. Based on numerical approximation rules, the SPH discrete form of the Euler and Navier-Stokes equations are derived. To treat various boundary conditions, we apply several types of image particles that are particularly designed to complete the kernels truncated by system boundaries. The global conservation of mass and linear momentum is then demonstrated. The SPH errors in the integral approximation and summation approximation are analysed based on given particle distribution patterns. Other problems such as particle clustering, tensile instability, particle boundary layer and lacking of polynomial reproducing abilities (incompleteness) are also discussed together with possible remedies. vi Summary

Before applying the implemented particle solver to the thesis topics, we first thoroughly test it against a selection of two-dimensionale benchmarks, which have a close relationship with the concerned problems. They include dam-break, jet impinging onto an inclined plane, emerging jet under gravity, free overfall and flow separation at bends. Good agreements with analytical and numerical solutions in literature are found. The convergence rate of SPH is shown to be of first order, which is consistent with the theoretical analysis.

For the rapid pipe filling problem, we apply the 1D SPH solver to the laboratory experiment of Liou & Hunt [116]. The velocity head at the inlet has to be taken into account to obtain a good agreement with the experiment. Water elasticity does not play a role and the friction formulation for steady state flows can be used. Head transition analysis provides deeper insight into the hydrodynamic behaviour of the filling process. As a special case of pipe filling, water hammer due to liquid impact at partially and fully closed valves is studied. The results agree well with standard MOC solutions. Similar observations are made for the rapid emptying process.

For the isolated slug travelling in a voided pipeline and impacting on a bend, we apply the 1D and 2D SPH solvers to the laboratory experiments of Bozkus [24]. To obtain the arrival velocity of the slug at the elbow, a 1D model including mass loss at the slug tail is used. In the slug impact, flow separation at the bend plays a vital role, which is typical 2D flow behaviour at a geometrical discontinuity. With a flow contraction coefficient obtained from 2D SPH solutions, the improved 1D model gives good results for the reaction force, not only in magnitude but also its duration and shape.

Finally, to study the evolution of air/water interfaces and its possible effect on filling and emptying processes, a new experimental study is performed in a large- scale pipeline. It is found that in filling the water front tends to split into two fronts propagating with different velocities. This results in air intrusion on top of a water platform. In emptying, flow stratification occurs at the water tail. Consequently, the validated 1D assumption of vertical air/water interfaces for small-scale systems with relatively high driving head may not be applicable to large-scale systems. The interface evolution does not play an important role in filling, the overall behaviour of which can be well predicted with 1D SPH solutions. However, flow stratification largely prolongs the overall draining process. Samenvatting

Pijpleidingen worden gebruikt voor transport en koeling in de olie- en gasin- dustrie, in de chemische industrie, voor watervoorziening en riolering, en in en- ergiecentrales werkende op waterkracht, fossiele brandstof of kernreacties. In- stationaire buisstromingen met grote drukvariaties kunnen een scala aan problemen veroorzaken, zoals lei-dingbreuk, schade aan verankering en ophanging, verplaatsing van buizen, trillingen en lawaai. De instationaire stroming wordt meestal veroorzaakt door veranderingen van de vloeistofsnelheid ten gevolge van het manipuleren van kleppen of pompen. Waterslag is het bekendste en meest bestudeerde verschijnsel op dit gebied. Waterslag kan ook optreden bij het snel vullen en legen van leidingen. Bij een hoge aandrij-vende druk kan de bewegende vloeistofmassa een grote snelheid bereiken. Wanneer zo’n hoge- snelheidsmassa botst op een blokkade of lokale weerstand, kan dit resulteren in waterslag en de bijbehorende hoge drukvariaties. Een ander scenario is dat van propstroming, mogelijk de meest gevaarlijke vorm van twee-fasenstroming. De zware vloeistofproppen die zich verplaatsen met hoge snelheid zijn te vergeli- jken met kanons-kogels. Schade is bijna onvermijdelijk wanneer een dergelijke prop botst op een pomp, een bochtstuk of een gedeeltelijk gesloten kleplichaam. De bewegende vloeistofmassa’s bij het vullen en ledigen van leidingen mogen beschouwd worden als een speciaal geval van de ge¨ısoleerde prop. Dit proefschrift beschrijft een Lagrangiaanse deeltjesmethode voor het oplossen van de Euler vergelijkingen met toepassing op het gebied van waterslag, het snel vullen en legen van leidingen en de beweging van ge¨ısoleerde vloeistofproppen in een verder lege buis. De gebruikte roostervrije methode ”smoothed particle hydrodynamics” (SPH) is geschikt voor problemen met bewegende randen en botsingen met randen, zoals van belang voor de beschouwde onderwerpen. We introduceren eerst de begrippen kernen deeltjesbenadering, twee essentiele¨ stappen in de SPH methode. Gebaseerd op numerieke benaderingsregels wordt de discrete SPH vorm van de Euler en Navier-Stokes vergelijkingen afgeleid. Bij de behandeling van de gestelde randvoorwaarden passen we verschillende types gespiegelde deeltjes toe die speciaal zijn geconstrueerd om de door randen afge- broken kernen weer volledig te maken. Het globale behoud van massa en impuls wordt dan aangetoond. De fouten gemaakt in de SPH integraal- en sommatiebe- naderingen worden geanalyseerd voor gegeven verdelingen van de deeltjes. An- dere problemen, zoals het samenklonteren van deeltjes, treksterkte instabiliteit, viii Samenvatting randlagen van deeltjes en de onmogelijkheid om polynomen te reproduceren (on- volledigheid) worden besproken samen met mogelijke oplossingen. Voordat de ge¨ımplementeerde deeltjes-oplosser wordt toegepast op de onderwerpen van het proefschrift, testen we deze eerst en grondig voor een aantal tweed- imensionale standaardproblemen die sterk gerelateerd zijn aan de beschouwde onderwerpen. Dit zijn: dambreuk, waterstraal botsend op een scheve plaat, fontein, waterkering en loslating van de stroming bij een bocht. De overeenkom- sten met analytische en numerieke resultaten uit de literatuur zijn goed. De eerste-orde convergentiegraad van SPH wordt aangetoond, wat in overeenstem- ming is met de theoretische analyse. Voor het probleem van het snel vullen van een leiding, passen we de 1D SPH methode toe op het laboratoriumexperiment van Liou en Hunt [116]. Het snel- heidsafhankelijke drukverlies bij de buisinlaat wordt in rekening gebracht om goede overeenkomst met het experiment te krijgen. De elasticiteit van het water speelt zoals verwacht geen rol en een stationaire wrijvingswet kan worden gebruikt. Een analyse van energieomzettingen leidt tot dieper inzicht in het hydro- dynamisch gedrag van het vulproces. Als een speciaal aspect van vullen, is waterslag ten gevolge van de botsende vloeistofkolom op een gedeeltelijk of volledig gesloten klep bestudeerd. De resultaten berekend met SPH komen goed overeen met de conventionele oplossing verkregen met de karakteristiekenmethode. Voor het snelle ledigen van leidingen zijn vergelijkbare constateringen gedaan. Voor de ge¨ısoleerde prop zich voortbewegend in een lege leiding en botsend op een haaks bochtstuk, passen we de 1D en 2D SPH methodes toe op de laborato- riumexperimenten van Bozkus [24]. Om de snelheid waarmee de prop de bocht raakt te bepalen is een 1D model met massaverlies aan de achterkant van de prop gebruikt. Bij de botsing van de prop met het bochtstuk speelt loslating van de stroming een cruciale rol, wat kenmerkend is voor een 2D stroming rond een ge- ometrische discontinu¨ıteit. Met de invoering van een contractiecoefﬁci¨ ent¨ waar- van de waarde bepaald is met behulp van 2D SPH simulaties, geeft het verbeterde 1D model goede resultaten voor de kracht op de bocht, niet alleen wat betreft de grootte, maar ook wat betreft duur en vorm. Uiteindelijk, om de evolutie van een lucht-water front en de invloed daarvan op het vullen en legen te bestuderen, is er een nieuwe experimentele studie op grote schaal uitgevoerd in een lange pijpleiding. Bij het vullen is waargenomen dat het water-front zich splitst in twee fronten die zich met verschillende snelheden verplaatsen. Dit resulteerde in het indringen van lucht boven een min of meer vlakke laag water. Bij het legen treedt deze stratiﬁcatie op aan de achterkant van de waterkolom. Het gevolg is dat de 1D aanname van verticale lucht-water fronten, zoals gevalideerd in experimenten op kleine schaal met relatief hoge aan- drijvende druk, niet toepasbaar is voor grotere leidingen. De vervorming van het lucht-water front speelt geen belangrijke rol bij het vullen van leidingen: het globale gedrag kon goed worden voorspeld met de 1D SPH oplossing. Daarente- gen verlengt het ontstaan van een horizontale water-lucht laag wel de tijdsduur van het legen van leidingen. Contents

Summary v

Samenvatting vii

1 Introduction 1 1.1 Motivation for the study ...... 1 1.2 Problem statement and method justiﬁcation ...... 3 1.3 Outline of the thesis ...... 5

2 Mathematical Modelling 7 2.1 Introduction ...... 7 2.2 Basic concepts and assumptions ...... 7 2.2.1 Infinitesimal fluid element moving with the flow ...... 8 2.2.2 The material derivative ...... 8 2.2.3 The divergence of the velocity ...... 9 2.3 The continuity equation ...... 10 2.4 The momentum equation ...... 10 2.5 Summary of the governing equations ...... 13 2.5.1 The Navier-Stokes equations for viscous flow ...... 13 2.5.2 The Euler equations for inviscid flow ...... 13 2.5.3 Equation of state ...... 14 2.6 Initial and boundary conditions ...... 14 2.7 Moving fluid domain ...... 16

3 Smoothed Particle Hydrodynamics (SPH) 17 3.1 Introduction ...... 17 3.2 Literature review on SPH ...... 19 3.3 SPH approximations ...... 20 3.3.1 Function approximation ...... 20 3.3.2 Gradient and divergence ...... 23 3.3.3 Second derivatives ...... 26 3.3.4 Kernels and their properties ...... 27 3.4 SPH ﬂuid dynamics ...... 31 3.4.1 Field equations ...... 31 x Contents

3.4.2 SPH continuity equation ...... 31 3.4.3 SPH momentum equation ...... 32 3.4.4 SPH viscosity ...... 33 3.4.5 SPH equation of state ...... 35 3.4.6 Particle movement and time integration ...... 36 3.4.7 Particle search ...... 38 3.4.8 SPH boundary conditions ...... 38 3.4.9 SPH conservation properties ...... 43 3.4.10 Particle clustering and tensile instability ...... 44 3.4.11 Particle boundary layer ...... 46 3.5 Error, incompleteness and improvements ...... 47 3.5.1 Error estimation ...... 47 3.5.2 Incompleteness ...... 55 3.5.3 Improvements ...... 56

4 Selected Test Problems 59 4.1 Introduction ...... 59 4.2 Dam-break ...... 61 4.2.1 Basic theory ...... 61 4.2.2 Test problem and SPH setup ...... 62 4.2.3 Numerical results ...... 63 4.2.4 Convergence behaviour ...... 65 4.2.5 Summary ...... 67 4.3 Impinging jet ...... 68 4.3.1 Basic theory ...... 68 4.3.2 Test problem and SPH setup ...... 69 4.3.3 Numerical results ...... 70 4.3.4 Parametric study ...... 73 4.3.5 Summary ...... 76 4.4 Jet ﬂow under gravity ...... 76 4.4.1 Test problem and SPH setup ...... 77 4.4.2 Numerical results ...... 78 4.5 Flow separation at bends ...... 82 4.5.1 Test problem and SPH setup ...... 84 4.5.2 Numerical results ...... 85 4.6 Quasi-3D model ...... 90

5 Rapid Filling and Draining of Pipelines 93 5.1 Introduction ...... 93 5.2 Mathematical modelling ...... 95 5.2.1 Pipe ﬁlling ...... 96 5.2.2 Pipe emptying ...... 99 5.3 Discrete SPH dynamic equations ...... 102 5.3.1 SPH for 1D water hammer ...... 102 Contents xi

5.3.2 Treatment of the boundaries ...... 103 5.4 Numerical results ...... 104 5.4.1 Pipe ﬁlling ...... 104 5.4.2 Rigid column versus water hammer ...... 111 5.4.3 Pipe draining ...... 114

6 Slug Flow in a Voided Pipeline 119 6.1 Introduction ...... 119 6.2 State of the art ...... 121 6.2.1 Laboratory tests ...... 121 6.2.2 Reaction forces ...... 126 6.3 Governing equations ...... 127 6.3.1 Slug motion in an empty pipe ...... 127 6.3.2 Driving air pressure ...... 129 6.3.3 Elbow pressure and reaction force ...... 129 6.4 Results and discussion ...... 131 6.4.1 1D SPH simulations ...... 131 6.4.2 2D SPH simulations ...... 133 6.4.3 Comparison of peak pressures at the elbow ...... 136 6.4.4 Parameter variation ...... 138

7 Filling and Emptying of a Large-scale Pipeline: Experiments and Simu- lation 141 7.1 Experimental apparatus ...... 141 7.1.1 System origin and coordinates ...... 142 7.1.2 Tanks and pipes ...... 143 7.1.3 Supports and connections ...... 145 7.1.4 Valves ...... 147 7.1.5 Instruments, uncertainty and data acquisition ...... 148 7.2 Experimental variables and procedure ...... 150 7.2.1 Experimental variables ...... 150 7.2.2 Experimental procedure ...... 151 7.3 Experimental results ...... 153 7.3.1 Steady-state water flow ...... 153 7.3.2 Pipe filling ...... 153 7.3.3 Pipe emptying ...... 162 7.4 Numerical simulation of pipe filling ...... 170

8 Conclusions and Recommendations 173 8.1 Concluding remarks ...... 173 8.2 Recommendations ...... 174

Appendix A: SPH Corrections 177 A.1 Incompleteness ...... 177 A.2 Restoring completeness ...... 178 xii Contents

A.2.1 MSPM and SSPM ...... 178 A.2.2 RKPM ...... 179 A.3 Comparison and discussion ...... 181 A.3.1 MSPM vs SPH ...... 182 A.3.2 SSPM vs MSPM ...... 183 A.3.3 SSPM vs RKPM ...... 183

Bibliography 185

Index 199

Acknowledgements 203

Curriculum Vitae 207 Chapter 1

Introduction

1.1 Motivation for the study

Piping systems are widely used for the transportation of ﬂuids in different applications such as power stations, water supply and sewerage works, oil and gas industry, and chemical plants. In these systems, many accidents and incidents occur [108]. For example, in 2009 a terrible accident happened in the Sayano- Shushenskaya hydropower plant, the largest power plant in Russia, and led to grave consequences. The turbine hall and engine room were ﬂooded due to pipe rupture, the ceiling of the turbine hall collapsed, 9 of 10 turbines were damaged or destroyed, and 75 people were killed [189]. Frequent incidents include displaced and broken pipes, failure of supports and hydraulic machinery, noise, vibration and fatigue.

What are the reasons for these accidents and incidents happening around us? Except for external excitations, such as earthquakes, wind loads and vehicle impacts, the most common reason is unsteady ﬂuid dynamics within the pipeline. There are three special cases that deserve attention due to their violent appear- ance:

Water hammer. When fluid flowing in a pipe is suddenly stopped due to valve operation or pump failure, a large pressure rise results and excites the system. The pressure waves in the system impact the valve or pump like a hammer [230]. For illustration, we refer to a personal experience. During the pipe filling and draining experiments, described in Chapter 7, the simultaneous closure of control valves caused severe water-hammer in the experimental test rig [22]. Several joints close to an elbow and to a U-bend were disconnected (see Figs. 1.1 and 1.2b) and pipe anchors screwed into the floor were lifted up (see Fig. 1.2a). Five steel bars supporting the U-bend were displaced and one of them broke off and 2 Introduction was knocked down to the ground floor 5 metres below (see Fig. 1.2b).

Figure 1.1: Joint failure in the PVC pipeline apparatus [22].

Figure 1.2: Support failures in the PVC pipeline apparatus [22].

Rapid filling and emptying of a pipeline. Take a sewage system as an example. During rain periods, water from roofs and roads flows through manholes to the underground sewer system. When the advancing water flow is blocked somewhere, a regime transition from open channel flow to pressurized flow takes place. Damage is easily induced [236, 237], because sewage systems are not designed to take any overpressure. Surface and basement flooding due to function loss may result. In addition, when a new pipe system is taken into operation, the 1.2 Problem statement and method justification 3

filling (system start-up) has to be done very carefully. The same holds for system cleaning. In rapid pipe filling and emptying, violent unsteady flows may occur.

Isolated slugs in an empty line. The scenario of this problem is for example as follows. When steam lines in power plants are stopped for maintenance, condensed steam accumulates in lower locations and forms water slugs. When the system is reactivated, the water slugs propelled by highly compressed air or steam may achieve very high velocities and propagate like ”bullets” in the pipeline. When such ”bullets” impact on an elbow, pump or partially closed valve, damage may easily be the result [170].

These three violent cases make up the topics in this thesis.

1.2 Problem statement and method justiﬁcation

Water hammer, rapid filling and emptying, and travelling slugs in pipelines are sketched in Fig. 1.3. These are simplified physical models or laboratory test rigs. The three cases have several physical features in common. First, since the fluid velocity is generally high, the flows are turbulent with large Reynolds number. Second, as two components – air and water – are involved in pipe filling and emptying, slug flow and sometimes water hammer, they are two-phase flows with possible flow-regime transitions. Third, they are unsteady flows with moving boundaries confined within slender structures. Mathematically they can be modelled as initial boundary value problems governed by the Navier-Stokes (N-S) equations. What makes them special are the moving boundaries and contact discontinu- ities due to fluid-solid impact. Special attention is needed to understand local flow behaviour, such as moving and deforming water fronts and flow separation at bends.

For flows in slender pipes and channels, one-dimensional models, such as rigid- column models and elastic models, are preferred because of low computational cost and acceptable accuracy. Rigid-column models have been applied to rapid pipe filling [27, 116, 183] and emptying [105], and travelling slugs [24–26, 98, 170, 231]. When the flow impacts a solid or is suddenly accelerated, an elastic model has to be used [136, 137, 219].

One-dimensional models cannot adequately describe flow stratification, flow separation, impact on a bend and water front evolution as encountered in the considered problems. Two- and three-dimensional models are highly needed. However, flows in two and three dimensions with moving and deforming free boundaries are difficult to simulate using traditional mesh-based methods with either Eule- rian or Lagrangian grid. The fluid-solid impacts are hard to tackle. Therefore, our approach is based on moving ”particles”. In particular, we use the smoothed particle hydrodynamics (SPH) numerical method. SPH is a proper choice here 4 Introduction

Figure 1.3: Sketches for the problems considered in this thesis: (a) water hammer [211], (b) rapid pipe ﬁlling, (c) rapid pipe draining and (d) isolated slug in a void pipe [24]. because of its Lagrangian, mesh-less and inherently elastic features. The extension from one-dimensional SPH to two and three dimensions is straightforward, but expensive. SPH is a good method for impact simulations too [93, 117]. 1.3 Outline of the thesis 5

Using the SPH method, we model the water-hammer problem, rapid pipe filling and emptying, and isolated slug motion in a pipe and its impact on a bend. Even in 1D setting, the moving particle model is already remarkably versatile and gives satisfactory results when compared to physical experiments. To improve existing 1D models, 2D SPH simulations give crucial insight into local flow behaviour, such as free-surface evolution, flow separation, contraction and stratification, and velocity and pressure distribution. Flow stratification and deformation of water front and tail occur in rapid pipe filling and draining. This is confirmed by the novel large-scale laboratory measurements reported herein. Flow separation at a sharp bend turns out to be vital to accurately predict slug impact (magnitude and duration) in both the 2D and improved 1D models.

Important basic topics include mathematically deriving and evaluating the SPH method, applying it to the N-S equations to establish the SPH ﬂuid dynamics equations, investigating and extending the SPH treatment of various boundary conditions, estimating the accuracy of the SPH approximations (both the function and its derivatives), analyzing the incompleteness of SPH and ﬁnding possible improvements.

1.3 Outline of the thesis

The outline of this thesis is as follows. In Chapter 2, we present the mathematical modelling of the concerned problems, concentrating on the Navier-Stokes (N-S) equations in Lagrangian form and the formulations of various boundary conditions. The smoothed particle hydrodynamic (SPH) method is introduced in Chapter 3 for solving the N-S equations. Different approximation rules are evaluated and a formulation fulfilling the conservation properties is derived. To properly complete the kernels associated with particles close to the boundary, the implementation of the image particle approach is detailed. Based on given particle distributions, the numerical error in SPH is estimated. New algorithms to improve the completeness condition are proposed. Also, in Chapter 3, essential features of the numerical implementation for the particle solver are presented. In Chapter 4, the coded SPH solver is verified against several selected 2D benchmarks that are closely related to the defined problems. The procedure to set up a numerical experiment is introduced, with focus on how to choose the key parameters. The SPH numerical convergence rate is checked and compared with the theory. The application of the 1D SPH solver to rapid pipe filling and emptying is worked out in Chapter 5. The SPH results are validated against experiments of Liou & Hunt [116] and numerical solutions found in literature. The rigid-column model is established and validated. Fluid dynamics in the filling and emptying of conduits is examined through numerical experiments. The SPH results for the water-hammer problem are validated against the exact solution by the method of characteristics. Chapter 6 presents the application of the 1D and 2D SPH solver to the isolated slug experiment of Bozkus [24]. Pressure loss and 6 Introduction reaction force at the bend are estimated from the flow separation results obtained in Chapter 4. Chapter 7 deals with the large-scale laboratory experiments performed at Deltares, Delft. The water-hammer experiments have been published in [23, 99] and a general description of the filling and emptying experiments has been reported in [105]. The focus of Chapter 7 is on the water-air and air-water interface evolutions in the filling and emptying processes. The 1D SPH model and the rigid-column model developed in Chapter 5 are used to interpret the new experiments. The considered problems, models and methods, and the related topics are so many and diverse that an overall literature review for all of them is not possible. The review of a specific problem is given where it is studied. Chapter 2

Mathematical Modelling

2.1 Introduction

For the flow phenomena described in Chapter 1, the continuum mechanics principle can still be applied and the Navier-Stokes equations are valid. Before the derivation of the equations is carried out, several physical and mathematical concepts are introduced in Section 2.2. With the model of an infinitesimally small fluid element moving with the flow, the continuity equation and momentum equation are derived in Sections 2.3 and 2.4, respectively. The equations for viscous and inviscid flow are summarized in Section 2.5 in the notation used in SPH literature. The physical boundary conditions and the mathematical representa- tions are described in Section 2.6. Section 2.7 describes the treatment of a moving domain.

2.2 Basic concepts and assumptions

In obtaining the basic equations governing the ﬂuid motion, the following phi- losophy is adopted:

• Choose the appropriate fundamental principles from the laws of physics. • Apply these physical principles to a suitable model of the ﬂuid ﬂow. • Extract the mathematical equations which state the physical principles.

From the physical principles, the mass and momentum conservation laws will be used to derive the governing equations. Four alternative models of the flow 8 Mathematical Modelling are constructed in [3]. Here the model of an infinitesimally small element moving with the fluid will be used, because this is in close relation with the basics of SPH. This will be further explored in Chapter 3.

2.2.1 Infinitesimal fluid element moving with the flow

In mathematical physics, a physical model is needed ﬁrst, from which a corresponding mathematical model can be derived. In this section, we discuss a model of the ﬂow which is in particular useful to understand the ”particle” concept that will be elaborated in Chapter 3.

Figure 2.1: A ﬂow ﬁeld represented by streamlines.

Consider a flow field represented by streamlines as shown in Fig. 2.1. An infinitesimal fluid element moving with the flow will be extracted and examined. The fluid element is very small but large enough to contain a huge number of molecules so that the physical principles for continuum mechanics are applicable. The fluid element has a fixed mass, but its volume will be time-dependent. Since the flow field is described by fluid elements, the physical principles will be applied to one infinitesimal fluid element instead of the whole flow field. The fundamental partial differential equations (PDEs) will then be derived in what is generally called the non-conservation form.

2.2.2 The material derivative

The material derivative is an important notation and concept used in SPH. It has an important physical meaning that will be helpful to understand the basic 2.2 Basic concepts and assumptions 9 ideas of SPH. To understand its physical meaning, an inﬁnitesimally small ﬂuid element moving on a pathline in a three-dimensional domain is shown in Fig. 2.2. Then the material derivative is expressed in vector notation as:

D ∂ := + v · ∇, (2.1) Dt ∂t

∇ ∂ ∂ ∂ where t is time and := i ∂x + j ∂y + k ∂z in a Cartesian coordinate system.

Figure 2.2: Fluid element moving in a ﬂowing ﬂuid – sketch for the material derivative.

From the definition (2.1), the material derivative D/Dt is the time rate of change in a moving fluid element. The first term on the right hand side (RHS) of (2.1) is the local derivative, which is the time rate of change at a fixed point. The second term is the convective derivative, which is the spatial rate of change due to the movement of the fluid element in the flow field. The material derivative can be applied to any scalar field variable in the flow, such as density, velocity components, pressure, etc. The material derivative is essentially the same as the total derivative with respect to time [3] in calculus.

2.2.3 The divergence of the velocity

The physical meaning of the divergence of the velocity given in this section is helpful to understand the mass conservation property of the SPH equations.

Assume that we are dealing with a velocity ﬁeld v in a three-dimensional space in a Cartesian coordinate system. The symbol ∇ · v is deﬁned by

∇ · v = ∂u/∂x + ∂v/∂y + ∂w/∂z, 10 Mathematical Modelling where v := {u, v, w}T. Equivalently, it can be expressed as [3]:

1 D(δV) ∇ · v = (2.2) δV Dt where δV is the volume of the infinitesimal fluid element as sketched in Fig. 2.2. The relation (2.2) clearly demonstrates the physical meaning of ∇ · v. The divergence of the velocity is the time rate of change of the volume of a moving fluid element, per unit volume.

2.3 The continuity equation

The mass conservation law will be applied to the infinitesimally small fluid element moving with the flow, resulting in the continuity equation.

Consider the flow model sketched in Fig. 2.2. Assume that the fluid element has a fixed mass δm and a time-dependent volume δV. Then

δm = ρδV, (2.3) in which ρ is the mass density of the ﬂuid. The mass conservation law is stated as the rate of change of δm being zero. According to the physical meaning of the material derivative introduced in Section 2.2.2, we have D(δm) = 0. (2.4) Dt Replacing mass by density and volume gives [ ] Dρ 1 D(δV) + ρ = 0. (2.5) Dt δV Dt

The term in brackets in Eq. (2.5) is equal to ∇ · v given by Eq. (2.2). Hence, combining Eqs. (2.2) and (2.5), we obtain the continuity equation Dρ + ρ∇ · v = 0. (2.6) Dt

2.4 The momentum equation

Newton’s second law can be applied to the infinitesimal fluid element moving with the flow. The momentum equation is then derived. The moving fluid element with forces exerted on it is sketched in Fig. 2.3. Under action of these forces the fluid element will change its position, shape and volume. When applied to the moving fluid element in Fig. 2.3, Newton’s second law states that the net force 2.4 The momentum equation 11 on the fluid element equals its mass times its acceleration. Although it is a vector relation, only the x component of Newton’s second law is considered, i.e.

Fx = δmax, (2.7) where Fx is the x component of the net force, δm is the mass of the ﬂuid element, and ax is the x component of the acceleration.

Figure 2.3: Forces on the moving ﬂuid element used in the derivation of the x component of the momentum equation.

The forces acting on the ﬂuid element include body forces and surface forces. Assume that the body force per unit mass in x-direction is fx. The surface forces in the x-direction exerted on the ﬂuid element are depicted in Fig. 2.3. The symbol τij denotes a stress in the j direction exerted on a plane perpendicular to the i axis. The total net force in the x-direction is written as [ ] ∂p ∂τ ∂τ ∂τ F = − + xx + yx + zx dx dy dz + f ρ dx dy dz. (2.8) x ∂x ∂x ∂y ∂z x

The ﬁxed mass of the ﬂuid element is δm = ρdxdydz. Since ax is the material rate of change of the velocity u, we have ax := Du/Dt due to the Lagrangian approach. By substituting Fx, δm and ax into Eq. (2.7), we obtain Du ∂p ∂τ ∂τ ∂τ ρ = − + xx + yx + zx + ρf . (2.9) Dt ∂x ∂x ∂y ∂z x

Equation (2.9) is the x component of the momentum equation. Similarly, the y 12 Mathematical Modelling and z components of the momentum equation read Dv ∂p ∂τ ∂τ ∂τ ρ = − + xy + yy + zy + ρf (2.10) Dt ∂y ∂x ∂y ∂z y and Dw ∂p ∂τ ∂τ ∂τ ρ = − + xz + yz + zz + ρf . (2.11) Dt ∂z ∂x ∂y ∂z z

Newtonian fluids are defined as fluids where the shear stress is proportional to the time rate of strain, i.e., the spatial derivative of velocity. The fluids considered in this thesis are assumed to be Newtonian fluids. For such fluids, we have ∂u τ = λ∇ · v + 2µ , (2.12) xx ∂x ∂v τ = λ∇ · v + 2µ , (2.13) yy ∂y ∂w τzz = λ∇ · v + 2µ , (2.14) ( ∂z ) ∂v ∂u τ = τ = µ + , (2.15) xy yx ∂x ∂y ( ) ∂w ∂u τ = τ = µ + , (2.16) xz zx ∂x ∂z ( ) ∂w ∂v τ = τ = µ + , (2.17) yz zy ∂y ∂z where µ is the dynamic viscosity coefficient of the fluid and λ is the second viscosity coefficient. The dynamic viscosity coefficient is a measure of the internal molecular friction of the fluid (momentum diffusion). The following hypothesis (Stokes) is generally used for λ: 2 λ = − µ. 3

Substituting all the shear forces into (2.9) – (2.11), we obtain the three momentum equations ( ) [ ( )] Du ∂p ∂ ∂u ∂ ∂v ∂u ρ = − + λ∇ · v + 2µ + µ + Dt ∂x ∂x ∂x ∂y ∂x ∂y [ ( )] (2.18) ∂ ∂w ∂u + µ + + ρf , ∂z ∂x ∂z x

[ ( )] [ ] Dv ∂p ∂ ∂v ∂u ∂ ∂v ρ = − + µ + + λ∇ · v + 2µ Dt ∂y ∂x ∂x ∂y ∂y ∂y [ ( )] (2.19) ∂ ∂w ∂v + µ + + ρf ∂z ∂y ∂z y 2.5 Summary of the governing equations 13 and [ ( )] [ ( )] Dw ∂p ∂ ∂w ∂u ∂ ∂v ∂w ρ = − + µ + + µ + Dt ∂z ∂x ∂x ∂z ∂y ∂z ∂y ( ) (2.20) ∂ ∂w + λ∇ · v + 2µ + ρf . ∂z ∂z z

2.5 Summary of the governing equations

By applying two physical conservation laws to a moving ﬂuid element, the continuity and momentum equations have been derived in Sections 2.3 and 2.4, respectively. The equations for viscous and inviscid ﬂow are summarized in this section in the notation used in SPH literature.

2.5.1 The Navier-Stokes equations for viscous ﬂow

For viscous ﬂow the governing equations are the Navier-Stokes equations. The Greek superscripts α, β and ϑ are introduced to denote the coordinate directions. The summation in the equations is indicated by repeated indices. Then the governing equations as used for SPH in Chapter 3 are:

Continuity equation Dρ ∂Vα = −ρ , (2.21) Dt ∂xα Momentum equation DVα 1 ∂σαβ = , (2.22) Dt ρ ∂xβ where Vα is the α-th component of the ﬂuid velocity v, and σ is the total stress tensor consisting of the isotropic pressure p and the viscous stress τ,

σαβ = −pδαβ + ταβ, (2.23) where δαβ is the Kronecker delta and ταβ = µεαβ with ∂Vα ∂Vβ 2 ∂Vϑ εαβ = + − δαβ. (2.24) ∂xβ ∂xα 3 ∂xϑ

2.5.2 The Euler equations for inviscid ﬂow

The governing equations for inviscid ﬂow are known as the Euler equations. By dropping all terms involving viscosity from the Navier-Stokes equations, the Eu- ler equations are obtained as 14 Mathematical Modelling

Continuity equation Dρ ∂Vα = −ρ , (2.25) Dt ∂xα Momentum equation DVα 1 ∂p = − . (2.26) Dt ρ ∂xα

2.5.3 Equation of state

To close the system of equations for either viscous or inviscid ﬂow, additional information is provided by the equation of state. The Tait-Murnaghan equation of state [162] relating pressure p and density ρ is given by (( ) ) 2 γ ρ0c0 ρ p − p0 = − 1 , (2.27) γ ρ0 where p0 is the reference pressure, ρ0 is the reference density at reference pressure, c0 is the speed of sound, and γ is a positive constant. In this thesis atmospheric pressure is taken as the reference pressure p0.

2.6 Initial and boundary conditions

The equations governing the ﬂow have been presented and explained in the previous sections. To obtain a particular solution to these equations, initial and boundary conditions are needed. In this section, the physical initial and boundary conditions used in later chapters are described.

Initial conditions

Initially (at t = 0), we impose a given velocity v0 and pressure to the ﬂow, i.e.

v = v0, p = p(t = 0). (2.28)

The initial density ρ(t = 0) is calculated from Eq. (2.27).

Boundary conditions

Several types of boundaries are considered and four of them are described here for free-surface flows. They are the wall boundary, free surface boundary, inflow boundary and outflow boundary. Other boundaries such as supply reservoir, conduit bend, partially closed valve, etc will be described when they are encountered in specific problems. In order to efficiently formulate the boundary conditions, let Vn denote the velocity component orthogonal to the boundary (n 2.6 Initial and boundary conditions 15

is the exterior normal direction),Vt the velocity component parallel to the boundary (t is the tangential direction), and ∂Vn/∂n and ∂Vt/∂n their derivatives in the normal direction.

(i) Wall boundary

With respect to the fixed wall boundary, the condition is different for viscous and inviscid flows. For a viscous flow, the boundary condition is the no-slip condition, i.e. fluid does not penetrate through the boundary and the fluid at the wall is at rest, i.e. Vn = 0 and Vt = 0. (2.29) For an inviscid flow, there is no skin friction to induce its ’sticking’ to the surface. Hence the flow velocity at the wall is a finite, non-zero value. Moreover, there cannot be mass flow into or out of a non-permeable wall. The flow velocity vector immediately adjacent to the wall must therefore be tangent to the wall. Conse- quently, the wall boundary condition for an inviscid flow is given by

Vn = 0 and ∂Vt/∂n = 0. (2.30) The first condition of (2.30) implies that there is no flow penetrating through the boundaries, whilst the second implies that there are no frictional losses at the boundary. Apart from being enforced for the frictionless wall boundary, the free- slip condition is often imposed along a line or plane of symmetry, thereby reducing the size of the domain where the flow needs to be calculated by a half.

(ii) Free surface boundary

On a free-surface a kinematic and a dynamic condition are required. The kinematic boundary condition is written as

∂Vt/∂n = 0, (2.31) which is the same as the second condition of (2.30). The difference is that n is the exterior normal direction to the free-surface which is generally changes with time before a steady flow is reached. This condition mathematically states that in the exterior normal direction the velocity component Vt has a maximum value at the free surface. Physically, it means that the particles of air at the surface basically move at the same speed as the particles of water at the surface, because there is no shear on the surface. It also implies that a fluid particle initially on the surface will remains on it during the evolution of the surface. Since surface tension is not taken into account and the atmospheric pressure is taken as the reference pressure p0, the dynamic boundary condition is p = 0. (2.32) (iii) Inflow boundary

For the inﬂow boundaries considered herein, the velocity is prescribed at the inlet. For an inﬂow boundary with a given velocity, the condition is

Vn = Vn0, and Vt = Vt0. (2.33) 16 Mathematical Modelling

A constant and uniform pressure is also described at the inﬂow section.

(iv) Outﬂow boundary

At the outlet, the velocity is assumed not to change in the direction normal to the boundary. Accordingly, the outﬂow conditions are given by

∂Vn/∂n = 0, and ∂Vt/∂n = 0. (2.34)

2.7 Moving ﬂuid domain

For free-surface flows the domain occupied by the fluid may change with time. In addition to the pressure and the velocity, the domain Ω(t), which the fluid occupies at time t, is also to be determined. Besides the governing equations and the initial and boundary conditions, the problem description also requires the initial configuration Ω0. Starting from it, the fluid domain evolves with time according to the current velocities at ∂Ω(t) (boundary of the domain). Chapter 3

Smoothed Particle Hydrodynamics (SPH)

In this chapter, the SPH method and its variations are described. The errors in the SPH approximations of a function and its derivatives are estimated. A detailed outline is as follows: In Section 3.1, a brief introduction to meshfree methods is given. As one of the earliest meshfree methods, SPH and its variations are reviewed in Section 3.2. The basic concepts and formulations of SPH are described in Section 3.3. In Section 3.4 the SPH equations for ﬂuid ﬂows are derived and their conservation properties are shown. A numerical implementation is also derived with much attention to the treatment of boundary conditions. The numerical errors in the SPH integral approximation and in the summation approximation are estimated in Section 3.5, where corrective methods aiming at the improvement of the consistency in SPH approximations are presented.

3.1 Introduction

Fluid flow can be descried by both Eulerian and Lagrangian approaches [117]. Conventional numerical methods such as the finite difference method (FDM), finite volume method (FVM) and finite element method (FEM) partition the region of interest into a finite number of (small) cells, associate degrees of freedom (DOFs) with each cell and lead to a (non)linear system in these DOFs. In an Eulerian approach the collection of cells and associated DOFs (called mesh) is static in space. In a Lagrangian approach the numerical mesh follows the material motion. For Eulerian grids, various difficulties appear in modelling complex geometries, free surface flows, moving interfaces, deformable boundaries, etc. For Lagrangian grids, difficulties occur in dealing with problems with large deformation, pene- 18 Smoothed Particle Hydrodynamics (SPH) tration and crack propagation, as mesh connectivity and topology changes. For problems with complex geometry, the generation of a quality mesh and proper mesh refinements are computationally expensive. Moreover, mesh-based methods are not suitable for simulations of explosions, high velocity impacts and discrete particle systems.

Over the past decades, a new class of computational methods has been developed, which approximate the governing equations only based on a set of uncon- nected nodes without the need for an additional mesh. They are usually referred to as meshfree or meshless methods and have been a major research focus starting from the middle of the 1990s. For the basics of the mathematical theory and numerical implementation of different meshfree methods, see the review papers by, among others, Monaghan (1992) [151], (2005) [156], Belytschko et al. (1996) [16], (1998) [17], Randles & Libersky (1996) [182], Li & Liu (2002) [111], Babuskaˇ et al. (2003) [7], (2004) [8], Fries & Matthies (2004) [63], Nguyen et al. (2008) [164], and Liu & Liu (2010) [121]. The work of Monaghan [151, 156] and Liu & Liu [121] is about smoothed particle hydrodynamics (SPH) and its applications, something that we focus on herein. Randles & Libersky [182] summarized the use of SPH in solid mechanics and recent developments in this ﬁeld are reviewed by Liu & Liu [121]. Fries & Matthies [63] classiﬁed the meshfree methods and described them from their different origins and viewpoints. For meshfree methods based on weak forms, a numerical implementation similar to FEM is used. Babuskaˇ and his co-workers [7] discussed meshfree methods (weak form) from the mathematical point of view. The core technique of meshfree methods (weak form) is the partition of unity of the shape functions.

Two regular international workshops on meshfree methods have been established. One is the annual workshop SPHERIC [197] which started in 2006. It is mainly on SPH and its variations (strong form). The other is the biennial workshop Meshfree Methods for PDEs [144] which started in 2003 and focuses on the rig- orous mathematical background of meshfree methods typically based on weak forms. The SPH method has been implemented in LS-DYNA [131] as a commercial toolbox for high-velocity impact (HVI) problems. There are at least three open source codes available around the world. The first one can be found via the link ’http://www.liugr.org’. This code is in Fortran 77 and developed by Liu & Liu accompanying their textbook [117]. The code is clear and useful for beginners, but less efficient because of an ineffective searching algorithm. The second code named HYDRA can be found from ’http://www.pearcef.org’. The code has been developed by Couchman et al. accompanying the paper [45] for applications in astrophysics. It is written in Fortran 95. For the application of SPH in ocean engineering with large-scale free surfaces, the code ”SPHERIC” has been developed by the SPHERIC committee and this can be found via ’http://www.spheric.org’. All three codes have their own features and specific application targets. Since none of them is proper for the problems considered in this thesis, we developed our own 1D codes and worked on the 2D codes developed by Dr. Kruisbrink and Dr. Pearce (one of the developers of HYDRA [45, 135, 172, 209]) of the University of Nottingham. It is an interaction between Matlab 2010b and Fortran 95. 3.2 Literature review on SPH 19

3.2 Literature review on SPH

SPH methodology. SPH is a meshless, Lagrangian, particle method that uses an approximation technique to calculate field variables like velocity, pressure, position, etc. In SPH the governing PDEs for fluid dynamics are directly trans- formed into ordinary differential equations (ODEs) by constructing their integral forms with a kernel (or smoothing) function and its gradient. Unlike FDM, FVM and FEM, the SPH method uses a set of particles without predefined connectivity to represent a continuum system and thus it is easy to handle problems with complex geometries. It does not suffer from mesh distortion and refinement problems that limit the usage of FEM for large deformation problems, HVIs in solid mechanics (where solids are treated fluid-like but with material strength), and hydrodynamic problems with free surfaces and moving boundaries. As a Lagrangian method, SPH naturally tracks material history information due to movement of the particles. It is an ideal alternative for attacking fluid dynamics problems. It has the strong ability to incorporate complex physics into the SPH formulations. It is easy to work with and acceptable accuracy is usually obtained. The main disadvantage is that it is more time consuming than conventional methods and parallel computation techniques are needed if many particles are used. SPH differs from other meshfree methods in two aspects. First, ”particles” (nodes) are moving with the fluid, thus representing a fluid flow. They offer nice visualization (animation) possibilities. The particles satisfy the Navier- Stokes equations, so that SPH is a proper method for fluids. In other meshfree methods [63, 164], the weak form of a governing equation and a Galerkin procedure of FEM are generally applied. In these other methods, fixed points without priori connectivity are treated as nodes as in FEM. They are mainly used in solid mechanics with small deformations [164]. Second, SPH approximation techniques are directly used to discretize the governing equations (Euler or Navier- Stokes equations) and all field information such as position, velocity and density are carried by particles. However, the calculated nodal values in other meshfree methods for solids are not the required displacements and a back substitution is usually needed, which is a typical FEM procedure.

SPH history. SPH was invented independently by Lucy [133] and Gingold & Monaghan [69] for three-dimensional astrophysical problems, such as the formation and evolution of proto-stars and galaxies. Since the motion of astrophysical particles satisfies the same conservation laws as gas flows, it was modelled by gas dynamics. SPH is a truly meshfree Lagrangian method based on strong formulations. During its early development (1980s), it was mainly applied to compressible astrophysical and cosmological flows. Monaghan [151] presented an el- egant review for applications in this field. Today, SPH is being used to model the collapse and formation of galaxies [20, 45, 173], coalescence of black holes with neutron stars, detonations in white dwarfs and even the evolution of the uni- verse. Recently Rosswog [185] presented a detailed review of the SPH method with particular focus on its astrophysical applications. At the beginning of the 1990s, Libersky & Petschek [114] extended SPH to solid dynamics, especially to 20 Smoothed Particle Hydrodynamics (SPH)

HVI problems. Monaghan (1994) [152] developed a concept of artificial compressibility and used an equation of state for liquids with artificially low speed of sound to model incompressible free-surface flows. After that, SPH was used to model a vast range of fluid dynamics problems. The applications of SPH in the fluid dynamics field include low Reynolds number flows [159], viscous and heat conducting flows [41, 158], multiphase flows [87, 88, 153], coastal hydrodynamics including water wave impact and sloshing [71] and many more [121]. Throughout the development of the SPH method, several benchmarks have been systematically investigated such as Sod’s shock tube, blast waves and dam break.

SPH drawbacks. Regardless of a great amount of success achieved with SPH, several technical drawbacks had to be overcome before it was fully developed. Among the shortcomings, four aspects, namely, boundary deficiency, nodal incompleteness, zero–energy mode and tensile instability are well known. These drawbacks often result in a loss of accuracy and stability, and a failure of convergence in the SPH calculations. To address these problems, a number of useful techniques have been developed, such as symmetrising [150], normalization [93] and regularization [182] of the first-derivative approximations without changing the kernels, employment of ghost-particles [205], use of extra stress points [56], and introducing artificial pressure terms [72, 155]. Interested readers are referred to Belyschko et al. [17] and Liu et al. [119, 120] for boundary deficiency and nodal completeness, Liu & Liu [117] and Vignjevic [223] for zero–energy mode, and Monaghan [155] and Liu & Liu [121] for tensile instability.

3.3 SPH approximations

3.3.1 Function approximation

As a meshfree method, SPH is built on a set of distributed particles in a domain without a grid or mesh. It consists of two steps of approximations. One is the kernel approximation and the other is the particle approximation.

A continuous scalar ﬁeld f : Ω ⊂ R3 → R can be written in integral form as ∫ ′ ′ ′ f(x) = f(x )δ(x − x )dx , ∀ x ∈ Ω, (3.1) Ω where x → δ(x − x ′) is the Dirac delta distribution. The domain Ω has boundary ∂Ω. The Dirac delta distribution is difﬁcult to use as a function in a collocation process [112]. As a main technical ingredient of SPH, a continuous function W(x− x ′, h) known as kernel is chosen to mimic the Dirac delta, such that for h → 0, W(x − x ′, h) → δ(x − x ′). The kernel W is assumed to be radial symmetric in its support domain Γ, which is totally within Ω (Γ ⊂ Ω) and has boundary ∂Γ. The so-called smoothing length h is a dilation parameter that determines the support 3.3 SPH approximations 21 size of the kernel W. The approximated function is then written as ∫ ′ ′ ′ ⟨f⟩ (x) := f(x )W(x − x , h)dx , (3.2) Ω where ⟨·⟩ denotes an approximation operator. For a vector function f = [f1, . . . , fn], the approximation is componentwise and we have ⟨f⟩ = [⟨f1⟩ ,..., ⟨fn⟩]. This step is known as kernel approximation or integral approximation.

Suppose that space∪ Ω is partitioned∩ into N parts and each part has a volume Ωb. ∅ ̸ We have Ω = b Ωb and Ωa Ωb = , a = b. From (3.2) we get ∑ ⟨ ⟩ . b f (x) = f(xb)W(x − xb, h)Ωb =: f(x). (3.3) b

Note that ”volume” Ωb is an area segment in two-dimensions and a line segment in one-dimension. In much of the SPH literature, the ’hat’ in the approximated b function f(x) is often deleted without further explanation, but herein we distinguish them for the convenience of error estimation in Section 3.5.1.

Assume that the continuum has a mass density distribution ρ(x). A point ’b’ is chosen to represent one part, which is referred to as a particle hereafter. Then the mass of particle b is mb = ρbΩb ⇔ Ωb = mb/ρb. (3.4) where ρb = ρ(xb). Using (3.4) and collocation, (3.3) is rewritten as ∑ ∑ b mb fa = fbWabΩb = fbWab, (3.5) ρb b b b b where fa := f(xa), fb := f(xb) and Wab := W(xa − xb, h). This step is called particle approximation or summation approximation.

The idea of the SPH particle approximation for a 2D function is sketched in Fig. 3.1. Particles b within domain Γ are referred to as neighbours of particle a. One particle and its neighbour are interacting particles and form an interaction pair. Remark 3.1. If we have ∫ ′ ′ W(x − x , h)dx = 1, (3.6) Ω then (3.2) always holds for constant functions. Condition (3.6) is referred to as normalization condition of the kernel.

Remark 3.2. The factor W(x−xb, h)Ωb is a weight function in the approximation (3.3) of the function values at scattered points, i.e. f(xb).

With respect to the kernel approximation, it is easy to prove that the operator ⟨·⟩ is linear. If f1 and f2 are smooth functions and C is a constant, then

⟨f1 + f2⟩ = ⟨f1⟩ + ⟨f2⟩ , (3.7) 22 Smoothed Particle Hydrodynamics (SPH)

Figure 3.1: Sketch for SPH particle approximation: (a) side view and (b) top view.

⟨Cf2⟩ = C ⟨f2⟩ . (3.8)

The operator b· is also linear.

Theorem 3.1. The SPH kernel approximation of the product of two smooth functions f1 and f2 is generally different from the product of the kernel approximation of these two functions 1, i.e.

⟨f1f2⟩ ̸= ⟨f1⟩⟨f2⟩ . (3.9)

1Two sides of (3.9) are said to be equal in [117], which is not correct. 3.3 SPH approximations 23

Proof. Apparently when h → 0 or any one of two functions f1 and f2 is constant, two sides of (3.9) will be exactly equal. However, when h > 0, ⟨f1f2⟩ = ⟨f1⟩⟨f2⟩ does not hold even for a linear function. Take scalar functions f1(x) = f2(x) = x as an example. As W(x − x ′, h) is radial symmetric in Ω, together with the normalization condition (3.6), we have ∫ ′ ′ ′ ⟨f1f2⟩ (0) = (f1f2)(x )W(0 − x , h)dx ∫Ω ′ ′ ′ = (x )2W(0 − x , h)dx ∫Ω ′ ′ ′ = (0 − x )2W(0 − x , h)dx > 0, Ω and ∫ ′ ′ ′ ⟨f1⟩ (0) = ⟨f2⟩ (0) = x W(0 − x , h)dx = 0. Ω Consequently, ⟨f1f2⟩ ̸= ⟨f1⟩⟨f2⟩ .

3.3.2 Gradient and divergence

Aside from satisfying the conditions such as even function and normalization, it is assumed that kernel W(x − x ′, h) has some other properties:

• It has a compact support Γ such that

′ W(x − x , h) = 0 ∀ x ∈/ Γ. (3.10)

• Its gradient satisﬁes the antisymmetry condition [112, 117] ∫ ′ ′ ∇W(x − x , h)dx = 0. (3.11) Ω

For a 3D domain ∂Γ is the surface of a sphere. Deﬁne η = x − x ′, then in componentwise, we have dη ∇ W(η, h) = ∇ W(η, h) = ∇ W(η, h), x η dx η and dη ∇ ′ W(η, h) = ∇ W(η, h) = −∇ W(η, h). x η dx ′ η Thus ′ ′ ∇xW(x − x , h) = −∇x ′ W(x − x , h). (3.12) 24 Smoothed Particle Hydrodynamics (SPH)

If (∇f)(x) is taken as a function and substituted into (3.2), utilising integration by parts, one can show that ∫ ( ) ′ ′ ′ ⟨∇f⟩ (x) = ⟨∇xf⟩ (x) = ∇x ′ f (x )W(x − x , h)dx Ω ∫

′ ′ ′ ′ ′ = ( ) ( − ) − ( )∇ ′ ( − ) f x W x x , h ∂Ω f x x W x x , h dx ∫ Ω ′ ′ ′ = f(x )∇xW(x − x , h)dx (3.12) ∑Ω . = f(xb)∇W(x − xb, h)Ωb (3.4) ∑b mb c = f(xb)∇W(x − xb, h) =: ∇f(x), (3.13) ρb b

If we differentiate both sides of (3.2), the gradient of the approximated function is obtained as ∫ ′ ′ ′ ∇ ⟨f⟩ (x) := f(x )∇W(x − x , h)dx , (3.14) Ω or in summation form ∑ ∇b mb ∇ f(x) := f(xb) W(x − xb, h). (3.15) ρb b

Comparing (3.13) and (3.14), one sees that c b ⟨∇f⟩ = ∇ ⟨f⟩ ⇒ ∇f = ∇f, (3.16) which means that the SPH approximation of the gradient is commutative. This is an important property and it is an exact relation for Γ ⊂ Ω.

Remark 3.3. The factor ∇W(x − xb, h)Ωb is a weight function in the approximation (3.13). It implies that the derivative of a function can be approximated from neighbouring function values.

SPH Rule I. A concise notation of (3.15) is ∑ c mb ∇fa = fb∇aWab, (3.17) ρb b c c where ∇fa := ∇f(xa) and ∇aWab := ∇W(xa − xb, h). Although the gradient approximation has been given by (3.17), special formulations closely related to hydrodynamics are more often used in SPH. They are given below.

SPH Rule II. With (3.11) and (3.13), we have ∫ ∫ ′ ′ ′ ′ ′ ⟨∇f⟩ (x) = f(x )∇W(x − x , h)dx − f(x) ∇W(x − x , h)dx , (3.18) Ω Ω 3.3 SPH approximations 25 which in particle approximation form is ∑ ( ) c mb ∇fa = fb − fa ∇aWab. (3.19) ρb b Remark 3.4. The SPH rule II is a partition of nullity (PN) (this means that the sum of weights of particle a and its neighbours is zero), while the original SPH gradient (3.17) is not. The SPH gradient (3.19) is exact for constant functions, but the original (3.17) is not.

Similarly, another SPH gradient is written as [112, 150] ∑ ( ) mb ∇fa = fb + fa ∇aWab. (3.20) ρb b Although this gradient approximation is symmetric, it is not a PN. When it was applied to the 2D heat equation, the numerical error was much larger than that given by its counterpart (3.19) [30].

SPH Rule III. As a primary variable, mass density is an important quantity in standard SPH. To obtain higher accuracy, the triviality [117] ( ) ρ∇f = ∇ ρf − f∇ρ (3.21) is often used. Then, applying (3.17) to the gradients in (3.21) yields ( ) ∑ ( ) d∇ = − ∇ ρ f a mb fb fa aWab. (3.22) b SPH Rule IV. When the gradient of a scalar ﬁeld is written as ( ) ∇f f f = ∇ρ + ∇ , (3.23) ρ ρ2 ρ one can construct another SPH gradient: (d) ∑ ( ) ∇f fa fb = mb + ∇aWab. (3.24) ρ a ρ2 ρ2 b a b The SPH gradient (3.24) can also be derived naturally from a variational principle as shown in [112, 177].

SPH Rule V. Many more alternative formulations of the SPH gradient are possible. A general identity for ρ∇f can be written as ( ( )) ρ∇f = ρ2−σ f∇ρσ−1 + ρ2σ−2∇ fρ1−σ , (3.25) with SPH equivalent ( ) ∑ ( ) d 2−σ mb ρ∇f = −ρ fa − fb ∇aWab, (3.26) a a ρ2−σ b b 26 Smoothed Particle Hydrodynamics (SPH) where σ is a constant. When both sides of (3.25) is divided by ρ2, we obtain ( ) ( ) ∇f f 1 1 f = ∇ + ∇ , (3.27) ρ ρσ ρ1−σ ρ2−σ ρσ−1 and its SPH equivalent is (d) ∑ ( ) ∇f fa fb = mb + ∇aWab. (3.28) ρ a ρσρ2−σ ρσρ2−σ b a b b a

With respect to σ = 2, SPH gradient (3.26) reduces to (3.22), and (3.28) reduces to (3.24). Several gradient formulations with different values of σ such as σ = 1/2 and 3/2 are highlighted for problems with large density gradients [177]. The case with σ = 1 in (3.28) is (d) ∑ ( ) ∇f fa + fb = mb ∇aWab. (3.29) ρ a ρaρb b

More generalized SPH approximations of ρ∇f and ∇f/ρ can be derived (see e.g. [178]), which is the reason why so many different variants of SPH formulations have been proposed [112, 117, 177]. These variants generally have different conservation properties and calculation efﬁciencies.

The above approximation rules hold for gradient of a scalar ﬁeld is valid for the divergence of a vector ﬁeld. From SPH rule II, we have ∑ d mb ∇ · fa = − fab · ∇aWab. (3.30) ρb b

The notation fab := fa − fb will be used for vector quantities throughout this chapter. From SPH rule III, we get ( ) ∑ \∇ · = − · ∇ ρ f a mbfab aWab. (3.31) b

Divergence approximations from other SPH rules can be similarly derived. The SPH approximation of the curl of a vector ﬁeld can be found in [178].

3.3.3 Second derivatives

A straightforward SPH approximation uses the second derivative of the kernel, but this proves to be very noisy and sensitive to particle disorder (more details can be found in e.g. [178]). It is better to approximate the second derivatives utilising only the ﬁrst derivative of the kernel [151, 178]. Various forms are proposed and used for discretisation of the Laplacian operator in an SPH context [47, 107, 151, 3.3 SPH approximations 27

158,160,178,191,224]. The Laplacian operator may be estimated using the integral approximation (see e.g. [151, 177] for the derivation) ∫ ( ) ′ r · ∇W ′ ⟨⟨∆f⟩⟩ (x) = 2 f(x) − f(x ) dx , || ||2 (3.32) Ω r giving the SPH Laplacian ∑ ( ) m ( )r · ∇ W c = b − ab a ab ∆f a 2 fa fb 2 , (3.33) ρb r b ab

′ where r := x−x , rab := xa−xb and rab := ||rab|| (Euclidean distance between two particles a and b). This formulism has been used for heat conduction problems where a second-order parabolic PDE is directly solved (see e.g. [41, 95]), but it is not needed when a ﬁrst-order system is solved as in [92].

3.3.4 Kernels and their properties

The kernel function W(x − x ′, h) determines the accuracy of function approximation (3.5), while the kernel gradient determines the approximation accuracy of the ﬁrst and second derivatives given in Sections 3.4.2 and 3.4.3. To achieve the required accuracy, the kernel and its gradient have to be chosen properly. Be- fore going further, several kernels commonly used in SPH are summarized here, because they are a key element in SPH. The properties will be demonstrated by taking the widely used cubic spline kernel as an example. It is believed that this will be helpful in understanding the SPH approximations and related topics discussed later in this chapter.

The kernel function Wab := W(xa − xb, h) can be written in a general manner as

1 W := W(r , h) = w (q) , (3.34) ab ab hd where d is the dimension of the system and q := rab/h.

Standard kernels. Since the kernel is supposed to mimic the Dirac delta distribution, the ﬁrst option that comes to one’s mind could be the Gaussian

λ 2 W(q, h) := e−q , (3.35) hd where the normalization factor λ = π−d/2. The Gaussian has the advantage that the spatial derivative is inﬁnitely smooth and therefore exhibits good stability properties. The inventors of SPH, Gingold & Monaghan (1977) [69], used the Gaussian kernel in their original work on a three-dimensional astrophysical problem. For practical applications involving boundaries, however, a Gaussian kernel is rarely used these days because it is not compactly supported (the approximation spans 28 Smoothed Particle Hydrodynamics (SPH) the entire spatial domain). In addition, the computational cost for the approximation is of O(N), where N is the total number of sampling points in the domain. For this reason, kernels which have similar shapes as the Gaussian, but that are compactly supported, are popular in SPH. Then the calculation is reduced to a sum over closely neighbouring points, which reduces the cost to O(n), where n is the number of contributing neighbours [177].

Based on the Gaussian, Monaghan & Lattanzio (1985) [149] recommended the super Gaussian for better accuracy and stability

λ 3 2 W(q, h) = ( − q2)e−q , (3.36) hd 2 √ where λ = 1/ π in d = 1 dimension. Similar to the Gaussian, it is not compactly supported. In addition, part of it is negative (when q2 > 3/2), which may induce nonphysical behaviour such as negative density [117]. To make the Gaussian and super Gaussian compactly supported, Zhang & Batra [233, 235] made some corrections. The revised (truncated and normalized) Gaussian is { 2 λ e−q − e−4, 0 6 q < 2, w(q, h) = (3.37) hd 0, 2 6 q, √ √ where λ = [ 1.04823/ π, 1.10081/π, 1.18516/(π π)] for d = [1, 2, 3], and their revised super Gaussian is { 2 λ (4 − q2)e−q , 0 6 q < 2, W(q, h) = (3.38) hd 0, 2 6 q, √ where λ = [2/(7 π)] for d = 1. Although it was demonstrated that these revised Gaussian and super Gaussian perform better in correction schemes like MSPH [233] and SSPH [235], the gradient of these two kernels is not compactly supported.

In the other original SPH work, Lucy (1977) [133] employed the quartic spline function (fourth-order polynomial) as the kernel: { λ (1 + 3q)(1 − q)3, 0 6 q < 1, W(q, h) = (3.39) hd 0, 1 6 q, where λ = [ 5/4, 5/π, 105/(16π)] for d = [1, 2, 3]. The most commonly used kernel is the cubic spline function (third-order polynomial)   2 1  − q2 + q3, 0 6 q < 1,  3 2 λ W(q, h) = 1 3 (3.40) d  ( − ) 6 h  2 q , 1 q < 2,  6 0, 2 6 q, 3.3 SPH approximations 29 where λ = [ 1, 15/(7π), 3/(2π)]. After it was ﬁrst employed by Monaghan & Lat- tanzio (1985) [149], it became the most popular kernel in the SPH community. In this thesis it is the ﬁrst choice. Its properties are summarized later in this section.

To be more like a Gaussian function but more stable, Morris [159] proposed two other high order spline kernels. Fulk & Quinn [64] analysed 20 different one- dimensional SPH kernels. Various methods of obtaining an objective measure of the quality and accuracy of the SPH kernel were addressed. The conclusion was that, in general, bell-shaped kernels perform better than other shapes. Liu & Liu [117] proposed a general approach to construct kernels, which was also presented in Price’s thesis [177]. With this approach, new kernels can be constructed to fulﬁll speciﬁc problem-dependent requirements.

Kernel derivatives. Kernel derivatives are of greater interest because they determine the approximation accuracy of the ﬁrst and second spatial derivatives given in PDEs. The ﬁrst derivatives of the kernel are written as ( ) α α α ∂Wab 1 dw xa − xb ∂ Wab := α = , (3.41) ∂x h dq rab and the second derivatives are ( )( ) ( ( )( )) 2 2 α α β β αβ α α β β ∂ Wab 1 d w xa − xb xa − xb 1 dw δ xa − xb xa − xb α β = 2 2 + − 3 , ∂x ∂x h dq rab h dq rab rab (3.42) where the Kronecker delta is { 1, α = β, δαβ := 0, α ≠ β.

Properties of the cubic spline kernel. The cubic spline function and its derivatives are shown in Fig. 3.2. Graphs of other kernels can be found in [64, 117]. The properties of the cubic spline kernel are summarized below:

• Normalization condition (3.6). This property assures that a constant function is reproduced exactly [117].

• Compact support: W(x − x ′, h) = 0, if r > 2h,(2h is the radius of the kernel support. This property works in two ways. On one hand, from a physical point of view, it means that particle b has no effect on particle a if it is beyond a certain distance. On the other hand, from a mathematical point of view, it generates sparse matrices and thus reduces the computational effort.

• Positivity: W(x − x ′, h) > 0. If this is not satisﬁed, it is possible to get nonphysical quantities such as negative energy, temperature and density. 30 Smoothed Particle Hydrodynamics (SPH)

0.5

-0.5

-1 removed hump

-1.5 w dw/dq d2w/dq 2 -2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 q

Figure 3.2: Cubic spline kernel and its derivatives.

• Radial symmetry: W(x − x ′, h) = W(x′ − x, h) = W(r). This property means that particles at the same distance to a third particle have equal inﬂuence on it.

• δ function consistency: W(x − x ′, h) → δ(x − x ′) for h → 0.

• Monotonicity: W(x − x ′, h) is decreasing with respect to r. It states that with the increase of the distance between two interacting particles, the mutual affection decreases. That is, nearer particles have a bigger effect on a given particle.

• ′ ∈ 2 Smoothness: W(x − x , h) C0(Ω).

The derivatives of the cubic spline kernel have the following properties:

• Normalization conditions (e.g. two-dimensional case): ∫ ∫ ( ) ( ) ′ ′ ′ ′ x − x ∂xWdx = 1, y − y ∂yWdy = 1. (3.43) Ω Ω

• Antisymmetry of the gradient (3.11).

• Symmetry of the second derivative. 3.4 SPH ﬂuid dynamics 31

3.4 SPH ﬂuid dynamics

3.4.1 Field equations

The ﬁeld equations for ﬂuid dynamics have been derived in Chapter 2. They are outlined below for convenience. It is assumed that there are no mass and heat sources, no chemical reactions, and no heat conduction. These assumptions are more than reasonable for the problems considered herein.

• Continuity equation: Dρ = −ρ∇ · v (3.44) Dt where ρ is the mass density, v is the velocity ﬁeld v := dx/dt.

• Momentum equation:

Dv ∇p = − + ν∆v + g, (3.45) Dt ρ

where p is the pressure, ν is the kinematic viscosity and g is the body force due to gravity.

• Constitutive equation: The system is closed by the Tait-Murnaghan equation of state (see Section 2.5.3 or [162]) (( ) ) 2 γ ρ0c0 ρ p − p0 = − 1 , (3.46) γ ρ0

where γ is a positive constant, ρ0 is the reference density and c0 is the speed of sound.

3.4.2 SPH continuity equation

There are much-used SPH continuity equations derived from different approximation rules. First, from Eq. (3.44) we get

1 Dρ = −∇ · v. (3.47) ρ Dt

Applying the divergence approximation (3.30) to the right-hand term of (3.47), we have ∑ ( ) m − ∇ · = b · ∇ v a vab aWab. ρb b 32 Smoothed Particle Hydrodynamics (SPH)

The approximation operator ’hat’ has been omitted without losing clarity. There- fore, the SPH continuity equation is written as ∑ Dρa mb = ρa vab · ∇aWab. (3.48) Dt ρb b

From the divergence approximation (3.31), we obtain ( ) ∑ ∇ · = − · ∇ ρ v a mbvab aWab. b Therefore, an alternative SPH continuity equation is ∑ Dρ a = m v · ∇ W . (3.49) Dt b ab a ab b

This is (3.48) with ρa = ρb. If the density is expressed in the form of (3.4), the smoothed density becomes ∑ ρa = mbWab. (3.50) b For a compressible gas Eq. (3.50) generally performs better than Eqs. (3.48) and (3.49) to ﬁnd the smoothed density [148,154]. One may use Eq. (3.50) to calculate density for free-surface ﬂows, then a particle number correction is needed [71]. It is preferable to approximate the rate of change of density for the problems considered herein and we use the SPH continuity equation in the form of (3.49).

3.4.3 SPH momentum equation

The SPH momentum equation without diffusion term (second term on the right hand of Eq. (3.45)) is derived ﬁrst. It is the equation of motion for inviscid ﬂow. According to the SPH approximation rules given in Section. 3.3.2, three different SPH equations of motion are derived as follows.

Form 1. Based on the gradient approximation (3.24), we have ( ) ∑ ( ) ∇p pa pb = mb + ∇aWab, ρ a ρ2 ρ2 b a b which leads to ∑ ( ) Dv p p a = − m a + b ∇ W + g. (3.51) Dt b ρ2 ρ2 a ab b a b

Form 2. Based on the gradient approximation (3.29), we have ( ) ∑ ( ) ∇p pa + pb = mb ∇aWab. ρ a ρaρb b 3.4 SPH ﬂuid dynamics 33

Therefore, we obtain ∑ ( ) Dva pa + pb = − mb ∇aWab + g. (3.52) Dt ρaρb b

Form 3. From Eq. (3.45) (without the diffusion term), we get

Dv ρ = −∇p + ρg, (3.53) Dt Applying the gradient approximation (3.19) to the term ∇p, we have ∑ ( ) m ( ) ∇ = b − ∇ p a pb pa aWab, ρb b which leads to ∑ ( ) Dva pb − pa = − mb ∇aWab + g. (3.54) Dt ρaρb b

The ﬁrst two SPH momentum equations are symmetric with respect to pa and pb, whilst the third SPH momentum equation is not, although it fulﬁls a PN. This leads to serious consequences for momentum conservation as will be shown in Section 3.4.8. We prefer Eq. (3.51), although simulations with either Eq. (3.51) or (3.52) do not give noticeable differences [117].

3.4.4 SPH viscosity

Artificial viscosity. When the SPH method is used to simulate compressible flows, due to fluid elasticity, spurious oscillations may occur in both the velocity field and the pressure field, which may ruin the simulation completely [112]. Such numerical instabilities are either due to a discontinuity in velocity field or improper approximation of a hyperbolic system. Various analyses have been de- voted to this topic in the context of FDMs and FEMs, and many measures have been taken to prevent such numerical instabilities. A common technique used in FDMs is to add artificial diffusion terms to the discrete momentum equation (also to the energy equation if it is involved) to suppress or to ”smear out” undesirable oscillations. As an approximation of the strong form of a PDE, SPH is similar to FDMs. Therefore, adding artificial diffusion term into SPH momentum equation is recommended to treat the numerical instabilities [151].

Two types of artiﬁcial viscosities are commonly used in CFD to stabilize numerical computations [112, 117]. One is the von Neumann-Richtmyer viscous pressure { ( ) 2 α(∆x)2ρ ∇ · v , ∇ · v < 0, Π1 := (3.55) 0, ∇ · v > 0, 34 Smoothed Particle Hydrodynamics (SPH) where α is a constant and ∆x is the grid size. It is proportional to the square of the velocity divergence. The other one is proportional to the velocity divergence and speed of sound c, { ( ) α∆xρc ∇ · v , ∇ · v < 0, Π2 := (3.56) 0, ∇ · v > 0.

Applying the SPH rules given in Section 3.3.2, the approximations of (3.55) and (3.56) can be derived (see e.g. [112]).

However, the standard artificial viscosities do not work very well in SPH shock simulation [9, 62, 148]. To suit the SPH character, Monaghan & Gingold [148] proposed the following artificial viscosity. The corresponding SPH momentum equation with artificial viscosity is ∑ ( ) Dv p p a = − m a + b + Π ∇ W + g. (3.57) Dt b ρ2 ρ2 ab a ab b a b

The effect of the artificial viscous force is reflected in Πab, which is given as [148]   2  −αcµab + βµab , vab · rab < 0, Π = ρ¯ab (3.58) ab  0, vab · rab > 0, with hv · r µ = ab ab , ab 2 2 (3.59) rab + 0.01h where β is a constant,ρ ¯ab = (ρa + ρb)/2 and h is the smoothing length. When time-dependent h and/or space-dependent c are used for individual particles, parameters h and c also need to be averaged like the density [87,148,156,177]. The viscous term induces a shear and bulk viscosity. The linear µab term corresponds to bulk viscosity, and the quadratic µab term corresponds to von Neumann- 2 2 Richtmyer viscosity [112, 117]. The constant 0.01h added to rab is to avoid a zero denominator. Remark 3.5. The artificial viscosity (3.58) is better than the conventional artificial viscosity (3.55) and (3.56) used in FDMs, because a particle spacing term, rab, is incorpo- rated in the viscous force, reflecting the length scale of the particle distribution. Remark 3.6. For most problems considered in this thesis we take β as zero, as its effect can be neglected in most incompressible flow problems as suggested by Monaghan [152]. The constant α is problem dependent. For gas dynamics α = 0.5 ∼ 1 is generally used. For free surface flows, a smaller magnitude should be applied [146, 152]. In most calculations in this thesis we take α = 0.1 and its effect will be discussed later in the numerical tests.

Real viscosity. When ﬂuid viscosity is included in the governing equations, the SPH momentum equation has the same form as (3.57), but with different Πab term. Two SPH formulations for the term ν∆v are listed below. 3.4 SPH ﬂuid dynamics 35

Form 1. The widely used SPH viscosity for viscous ﬂows is [151] ν + ν v · r Π = −8 a b ab ab . ab 2 2 (3.60) ρa + ρb rab + 0.01h Another commonly used SPH viscosity for low Reynolds ﬂows [160] is similar to (3.60), but about 4 times smaller.

Form 2. Recently Monaghan [158] suggested a better viscosity to be

Kvsigvab · rab Πab = − , (3.61) ρabrab where K is a constant, typically 0.5 for shocks. The signal velocity is vsig = ca+cb. This viscous term respresents a shear viscosity and a bulk viscosity. However, the effect of the bulk viscosity is negligible for nearly incompressible ﬂuid. The kinematic shear viscosity is 15 ν = Kv h, (3.62) 112 sig which is deduced using the integral expressions given in [156].

3.4.5 SPH equation of state

The SPH pressure can be obtained straightforwardly from (3.46) as (( ) ) 2 γ ρ0c0 ρa pa = − 1 . (3.63) γ ρ0

Pressure pa is relative to the reference pressure p0 (atmospheric pressure). The physical meaning of this equation and its parameters has been discussed in Sec- tion 2.5.3. Here the focus is on how to choose the parameters ρ0, γ and c0, which are practically used in SPH simulations.

The reference density is the fluid density under reference pressure, e.g. 1000 kg/m3 for water. The choice of γ = 7 is valid for liquid water [152, 162], and is now widely used in standard SPH [121]. For free-surface water flows, different choices of γ (γ = 1 ∼ 7) generally give a very small influence on the results [146, 160]. For example, when γ = 1, one obtains

2 pa = c0(ρa − ρ0), (3.64) which is the equitation of state widely used in liquid acoustics [230]. The choice of γ = 1 is used throughout the simulations unless otherwise mentioned.

The choice of c0 needs some effort as shown below, because artificial values will be used for the flows. The density variation δρ is given by [152] δρ VL = , (3.65) ρ c2τ 36 Smoothed Particle Hydrodynamics (SPH) where L is a typical length scale of the flow, τ is a typical time scale and V is a typical velocity. For the problems we are considering (weakly compressible flows with one or multi free surface), V = L/τ, so that the relative fluctuation 2 in density is proportional to Ma , where Ma := V/c0 is the Mach number [152]. 3 For fluids like water with c0 ∼ 10 m/s, due to low convective velocities (three orders of magnitude smaller than c0, i.e. V ∼ 1 m/s), the density variation is extremely small (δρ/ρ = Ma2 ∼ 10−6), so that it is customary to approximate the weakly compressible fluid by an ideally incompressible fluid (ρ is constant and c0 = ∞). The approach in SPH is different. The real weakly compressible fluid is approximated by an artificial fluid which is more compressible (δρ/ρ ∼ 10−2). The artificial fluid has a speed of sound (c ∼ 10 m/s) that is much smaller than the real speed of sound c0. Since c is still one order of magnitude larger than the convective velocity (V ∼ 1 m/s), the fluid has small density fluctuations (δρ/ρ ∼ 1%). To make the behaviour of the artificially made more compressible fluid sufficiently close to the real fluid, we control the density variation within 4% for simulations (free- surface flows) performed in this thesis. After the typical velocity V is chosen, e.g. the maximum velocity in the flow Vmax = max(||v||), a suitable choice of c can be made to achieve the desired density variation.

3.4.6 Particle movement and time integration

Particle movement. Since SPH particles move with the ﬂow, the particle position change is determined by dx a = v . (3.66) dt a This is a Lagrangian feature of SPH. It is also the reason why free surfaces or interfaces do not need to be tracked explicitly in SPH.

Time integration. The spatial operators in the ﬁeld equations have been written in SPH particle summation form resulting in a set of ODEs in time. To integrate the discrete SPH equations, one has to use proper numerical integration schemes. Implicit methods permit larger time steps to be taken while maintaining stability. However, large time steps cannot be used in SPH because the numerical results are highly related to particle position change within one time step. In other words, a small time step has always to be taken to reduce possible numerical error due to large particle motion, no matter whether implicit or explicit methods are applied. If implicit methods are used, the solution of a linear or even nonlinear system of equations is required. That may be very expensive, especially when many particles are used in the simulation. Consequently, the majority of SPH codes use explicit time integration algorithms. Here the Euler forward method is used because it is the cheapest in data storage comparing with other explicit time- integration methods such as modiﬁed Euler, Runga-Kutta and leapfrog [185]. It is outlined below.

In SPH flow motion is driven by small density changes. Therefore, the sequence 3.4 SPH fluid dynamics 37 of the SPH algorithm is given by the following equations. The density is first calculated by ( ) Dρ n ρn+1 = ρn − ∆t a , (3.67) a a Dt where superscripts n and n + 1 indicate previous and present time steps, respectively, and ∆t is the numerical time step. The rate of change of density is calculated from (3.49). With the updated density, the pressure is then obtained from (3.64). The velocity at the present time step is calculated by ( ) Dv n+1 vn+1 = vn − ∆t a , (3.68) a a Dt where the particle acceleration is calculated from (3.57) with the obtained pressure and density. Finally, the position is updated for the next time step calculations by n+1 n n+1 xa = xa + va ∆t. (3.69) The scheme is first-order accurate, fully explicit in time and conditionally stable. A recommended time step size, due to the Courant-Friedrichs-Lewy (CFL) condition, is h ∆t ≤ C , (3.70) CFL c + V where V is the particle velocity (or maximum velocity among all particles), and CCFL is a constant between 0 and 1. The smoothing length is generally taken as h = ηd0 with η = 1.1 ∼ 1.33 as shown in Section 3.5.1. Here d0 is the initial particle spacing. Due to Ma = V/c, the CFL condition (3.70) can be rewritten as

d ∆t ≤ λ 0 , (3.71) c where λ = ηCCFL/(1 + Ma). Apart from the CFL condition, another time step control is used in SPH to limit the acceleration [149]. It reads √ ≤ h ∆t 0.01minb | | , (3.72) a b | | | | where a b = Dvb/Dt is the magnitude of particle acceleration, equivalent to force per unit mass; the ’min’ is over all particles. For viscous ﬂow, an extra viscous force stability condition needs to be satisﬁed [107, 191] √ h2 ∆t ≤ 0.125minb . (3.73) νb

The time step ∆t in SPH is chosen as the minimum of the above three conditions. More sophisticated time step constraints can be found in [158]. For the simulations herein the ﬁrst two time step controls are mostly used because the ﬂow is inviscid in most cases. 38 Smoothed Particle Hydrodynamics (SPH)

3.4.7 Particle search

In SPH a priori connectivity between the particles is not needed. This explains why SPH is referred to as a meshfree method. However, one needs to find the contributing neighbours b (time and space dependent), when a field quantity is calculated for a specific particle a. To efficiently access neighbouring particles at each time step, the summations in the SPH equations can be evaluated using the link-list algorithm (see [117, 149]). The computational complexity of the link-list algorithm is O(N), where N is the total number of particles. The list generation dominates the computation time in large problems involving many particles.

The link-list algorithm used in SPH originates from Monaghan & Lattanzio [149]. A grid of bookkeeping square cells with size 2h is used and a list of particles belonging to each cell is created. Only particles in neighbouring cells can then contribute to the state-change of particles in a given cell. The list generation algorithm with an optimized cell size used in [45] is employed herein to decrease the number of operations. Various other optimization procedures have been proposed, such as those employed in the SPMHD [178] and the MPS method [101]. When the smoothing length h is time-dependent (for problems with large density variation, see e.g. [117, 158, 177]), the link-list algorithm is not as efﬁcient as the tree algorithm [80]. The smoothing length is constant (propositional to the initial particle spacing) for all problems considered herein.

3.4.8 SPH boundary conditions

Truncated kernel. Before boundary conditions are numerically treated, we first introduce the concept of truncated kernel. For the SPH equations derived above, we assumed that the kernel support Γ for a particle is fully within the problem domain Ω. However, if a particle is close to the system boundary ∂Ω, kernel support associated with this particle will be truncated as sketched in Fig. 3.3. This is called ′ ′ that the kernel support is not fully filled. Then the term f(x )W(x − x , h)|∂Ω in Eq. (3.13) is not zero. This is known as boundary deficiency that is a drawback of ′ ′ SPH. One may numerically evaluate the non-zero term f(x )W(x − x , h)|∂Ω, but we use the image particle approach to complete the truncated kernel herein. This technique slightly enlarges the problem domain and is commonly used in SPH and FDMs (ghost grid points). Its efficiency and accuracy have been underlined in recent reviews of SPH (e.g. [121, 156]). Therefore, apart from the enforcement of boundary conditions, the essential step is to make kernel supports for all fluid particles fully filled. For convenience, a particle located within the computational domain is referred to as a fluid particle and other particle types are defined where they are created.

Rigid wall boundary

Free-slip boundary. A common way to enforce the free-slip boundary condition in 3.4 SPH ﬂuid dynamics 39

Figure 3.3: Boundary deﬁciency due to truncated kernel by the boundary of the system domain.

SPH is to use a local mirroring of the flow to the other side of the solid boundary as shown in Fig. 3.4. To ensure that the kernel associated with a fluid particle close to the wall is fully filled, the thickness of the imaged flow should be larger than the kernel radius (2h for the cubic spline kernel). It is taken herein as 2.5h as shown in Fig. 3.4. To exactly maintain a free-slip condition on the boundary, the m sign of the tangent velocity component is the same as its image, i.e. Vt = Vt, m whilst the sign of the normal velocity component is reversed, i.e. Vn = −Vn. Consequently, the velocity components of a mirror particle for a wall with inclined angle β anti-clockwise from the horizontal (see Fig. 3.4) are { m Vx = Vx cos (2β) + Vy sin (2β), m (3.74) Vy = Vx sin (2β) − Vy cos (2β).

Particle pressure is mirrored as well to make the mass ﬂux vanish on the surface of the boundary. Due to the movement of the ﬂuid particles, the mirror particles change their positions at each time step too.

The mirror particle approach works very well for straight walls. A curved wall can be replaced by several straight walls. Alternatively, the continuous wall concept [103] can be used. It is similar to another commonly used approach – wall particles with repulsive forces (wall particle approach) [152]. The idea of them is that when the distance of a ﬂuid particle to the wall is less than a critical value (a function of h), a repulsive force will be exerted on the particle. The continuous wall is tested in the dam-break in Section 4.2. Although the wall particle approach is easier to be implemented and useful for irregular boundaries, we will not use it as unphysical shear stresses and the ”toothpaste jet” phenomenon [222] may be induced. In addition, it requires an ad hoc coefﬁcient in the repulsive force and some tuning to choose a proper wall particle distance. 40 Smoothed Particle Hydrodynamics (SPH)

Figure 3.4: Mirror particle approach for the enforcement of the free-slip condition: (a) ﬂuid particles and the mirror images; (b) velocity determination of a mirror particle.

In the simulations carried out in later chapters, two straight free-slip walls often join at some point and hence form a corner. To complete the support of the kernel associated with a particle close to the joint, the corner need to be treated as well. Otherwise, ﬂuid particles may leave the computational domain at some time (unphysical particle penetration), and the ﬁnal results may be affected. To the author’s knowledge, this problem has not been treated well in SPH. Here we apply a similar idea as the mirror particle approach to deal with it.

As shown in Fig. 3.5, when a fluid particle is in flow regions that need to be mirrored and its distance to the joint is less than 2.5h, a mirror particle is created and called corner mirror particle. The symmetry of the fluid particle with its corner image about the joint gives the position of the corner mirror. Its velocity is anti- symmetric about the joint (see Fig. 3.5). This concept is only used for sharp and blunt angles because the mirror particle approach is sufficient for a right angle.

No-slip boundary. Many different ways have been followed to impose a no-slip boundary condition in SPH, such as fixed wall particles with repulsive forces [152], mirror particles [160] and fixed ghost particles [224]. The disadvantages of the first approach has been discussed above, and only the latter two approaches are examined here. The idea of the mirror particle approach is similar to the enforcement of a free-slip boundary condition. The difference is that, in order to maintain a zero velocity, the sign of both tangent velocity component and normal velocity component need to be reversed, i.e.

m m Vt = −Vt, and Vn = −Vn. Another possible choice is

m m Vt = 0, and Vn = 0. In the third approach ghost particles are ﬁxed behind the wall with a certain dis- 3.4 SPH ﬂuid dynamics 41

Figure 3.5: Mirror particle approach for the treatment of the corner: (a) corner mirror particles; (b) velocity determination of a corner mirror particle. tribution (e.g. evenly distributed) in a layer, the thickness of which needs to be larger than 2h. The situation of two orthogonal walls forming a 90 degrees corner is shown in Fig. 3.6 as an example. The zero velocity condition (2.29) is imposed by the non-movement of the ghost particles. The algorithm for pressure change of the ghost particles is as follows. When a ghost particle b contributes to the pressure evolution of a ﬂuid particle a, the same term is added to the pressure evolution of particle b. Thus, the pressure forces are kept symmetric with respect to the wall, ensuring a homogenous Neumann condition for the pressure.

Figure 3.6: Fixed ghost particle approach for the enforcement of the no-slip boundary condition at an inner corner.

Free surface

For the problems considered herein, we are actually modelling free surface flow (water and void) instead of two-phase water-air flow. This is the physical statement of the dynamic condition (2.32). The kinematic condition (2.31) and the 42 Smoothed Particle Hydrodynamics (SPH) dynamic condition (2.32) for the free surface boundary (see Section 2.6) are automatically satisfied in SPH and can be visualized by setting monitoring particles on the surface. The kinematic condition (2.31) implies that a particle originally on the surface will remain on it. This is satisfied by the particle movement Eq. (3.66). Note that the density change of a free surface particle may not be exactly zero, but very close to it, because it has very similar velocity as its neighbours. Although this is an approximation, the error is generally small and will reduce when more particles are used.

Different from the treatment of solid boundaries given above, the dynamic condition (2.32), i.e. p = 0, at the free surface does not need to be treated explicitly, as it has been implicitly implemented in the SPH momentum equation. This is a critical step to understand the advantage of SPH for problems with a free surface and is explained below. Apparently, the support of the kernel associated with a particle close or on the surface is not full, as there is no particle outside the free surface. To satisfy the requirement of the full kernel support, assume that some dummy particles with index ” b′ ” exist outside the water adjacent to the free surface. Due to the dynamic condition (2.32), we have pb ′ = 0. Then from the momentum equation (3.59) (without gravity and viscosity), we obtain ∑ ( ) ∑ Dva pa pb pb ′ ∇ ′ ∇ ′ = − mb 2 + 2 aWab − mb 2 aWab . (3.75) Dt ρ ρ ρ ′ b a b b ′ b The second term on the right hand side of (3.75) is zero and the dummy particles have no contribution to the velocity change of a free surface particle. Conse- quently, we do not need to complete the incomplete kernel supports with imaged particles. In fact, the dynamic condition (2.32) automatically results in a zero boundary integral in (3.13). It is consistent with that the boundary integral is not needed when the SPH continuity equation is used to calculate the density change, as concluded in [222].

Inlet / Outlet

To impose the velocity boundary condition (2.33) and more importantly to fill the kernel associated with a fluid particle close to the inlet, a block of particles with identical velocity is placed ahead of the inlet and indexed as inlet particles (see Fig. 3.7a). The width of this inlet block needs to be larger than the kernel radius. If the support of the kernel associated with a fluid particle is truncated by the inlet, inlet particles contribute to the density and velocity calculation of this fluid particle. The velocity of the inlet particle itself does not need to be calculated as it is explicitly provided as a boundary condition. The pressure of the inlet particles can be constant or time-dependent depending on the problem. An inlet particle moves with the given velocity. After crossing the inlet and entering the computational domain, it becomes a fluid particle and its velocity and pressure will evolve from next time step on. A new inlet particle is generated at the upstream end of the inlet block.

Similarly, to ensure that the fluid flows out of the computational domain freely, 3.4 SPH fluid dynamics 43

Figure 3.7: Enforcement of the (a) inlet boundary condition with inlet particles and (b) outlet boundary condition with outlet particles. an outflow block is placed behind the outlet (see Fig. 3.7b). When a fluid particle leaves the computational domain and enters the outlet block, it becomes an outlet particles whose properties (velocity and pressure) will not change. It will be deleted after leaving the outflow block at its downstream end.

3.4.9 SPH conservation properties

The conservative properties of the discrete SPH equations have been extensively studied [69, 112, 133, 151, 156, 222].

Mass conservation. In traditional SPH for astrophysics, the summation approximation of density, i.e. (3.50), is often used to compute density changes instead of the SPH continuity equation (3.49). To show the mass conservation, the density approximation (3.50) can be taken over all the particles. In the original papers on SPH [69, 133], it was asserted that using Eq. (3.50) for density calculation is equivalent to conservation of mass, but more correctly it is equivalent to global conservation of mass [222].

When the SPH continuity equation (3.49) is used for density calculation (in this thesis), the global mass conservation is shown below. Due to the symmetry of the kernel and antisymmetry of its gradient, we have Wab = Wba and ∇aWab = −∇bWba. Then from (3.49) and noticing ma = mb = C, we obtain

m˙ b→a = mbvab · ∇aWab = m˙ a→b, (3.76) wherem ˙ b→a is the transferred mass in unit time from particle b to particle a. Thus the global conservation of mass is veriﬁed. The conservation of mass can be deduced from (3.50) as shown in [112], or from basic calculus as shown in [222]. 44 Smoothed Particle Hydrodynamics (SPH)

A posteriori check for mass conservation may be conveniently performed by counting the number of particles. For a closed system (without inlet and outlet), the mass of the system is ∑N M = ma, (3.77) a=1 which is a constant (conserved) because the particle mass ma is constant and the total particle number N is fixed. For a system with inlet and outlet, the total number of fluid particles changes with how many particles are entering (from inlet) and leaving (from outlet) the system. When the flow is close to steady state, the total number of fluid particles will be close to constant. This accounts for the global mass conservation, allows for a posteriori check and can be a flag to determinate an SPH simulation.

Momentum conservation. For rigid body motion of an inviscid flow, i.e. the velocity is uniform, it is seen that the viscosity term Πab is zero for both the artificial viscosity (3.58) and the real viscosity (3.60) – (3.61). Together with the preserved mass, the momentum is conserved. For general flows, global conservation of linear momentum (without viscosity and external force) can be seen from ∑ ∑ ∑ ∑ ( ) D Dv p p m v = m a = − m m a + b ∇ W = 0, Dt a a a Dt a b ρ2 ρ2 a ab a a a b a b (3.78) where the double summation is zero because of the antisymmetry of the kernel gradient and symmetry of the rest. Apparently, the SPH momentum equation (3.52) also conserves linear momentum, but Eq. (3.54) does not. In fact, linear momentum conservation can be shown by noticing the following action-reaction law of Newton for a pair of interacting particles a and b ( ) pa pb → ∇ → Fb a = −mamb 2 + 2 aWab = −Fa b, (3.79) ρa ρb where Fb→a is the force exerted by particle b on particle a.

The conservation of angular momentum and energy by the SPH discrete equations can be veriﬁed as well [111, 112, 151, 156, 178, 222]. In fact, as proved by Price [177, 178] and Li & Liu [112], using the variational principle, the SPH equations (3.49) and (3.57) can be directly derived from the density approximation (3.50). Thus the global conservation of ﬂuid quantities is guaranteed.

3.4.10 Particle clustering and tensile instability

Particle clustering. A”pairing instability” may occur when an improper smoothing length h = ηd0 is used, where d0 is the initial particle spacing. This is due to the shape of the kernel gradient as shown in Fig. 3.2, and this is a consequence of the fact that this kernel is designed to give good function approximation, rather 3.4 SPH ﬂuid dynamics 45 than necessarily being the best choice for calculating gradients [178]. Particu- larly, the kernel gradient contains a minimum value at q = 2/3 and is zero at the origin (see Fig. 3.2). This characteristic means that the mutual repulsive force Fb→a (3.79) decreases for neighbouring particles within the ”hump” of the kernel gradient. The net result is that two particles located within the ”hump” form an attracting ”pair”, eventually falling on top of each other. This explains the origin of the name ”pairing instability” [178] and manifests itself by particles clustering unrealistically close.

In gas dynamics complete merging of particles occurs when η & 1.5, corresponding to the placement of the nearest neighbour inside the ”hump”. There is also an intermediate regime 1.225 . η . 1.5 where the pairs form but do not completely merge. A sensible choice of η = 1.2 is recommended by Price [178], with which the pairing instability is avoided. Although the pairing instability is quite benign, it is the main reason one cannot simply ”stretch” the cubic spline to large neighbour numbers in order to obtain convergence [178]. For free surface ﬂows, η ∼ 1.3 has been often used by Monaghan [152,158] and other researchers [5,146,222,224].

In fact, to make the simulation more smooth a large smoothing length (more neighbouring particles) needs to be chosen. To avoid the pairing instability, ﬁxes have been proposed for the above ”hump” effect by Couchman et al. [45]. They suggested to modify the gradient of the cubic spline kernel as follows

  − 1, 0 6 q 6 2/3,   2 ′ − 3q + 2.25q , 2/3 6 q < 1, w (q) := G  (3.80)  − 0.75(2 − q)2, 1 6 q < 2,  0, 2 6 q,

with w itself and G = λ/hd unchanged. That is, the ”hump” is removed by simply making the kernel gradient ﬂat within q < 2/3 as shown in Fig. 3.2. Two different smoothing lengths are used in this thesis. For 1D simulations, h is taken as 1.2d0 and the gradient of (3.40) is used. For 2D simulations, h = 1.33d0 and the kernel gradient with moved ”hump” (3.80) is used.

Tensile instability. There is an inherent drawback of SPH for stress wave propagation in materials with strength (solids) known as ”Tensile instability”. A com- prehensive analysis that discusses the roots of tension instability has been made by Swegle et al. [200]. The tensile instability may not only be a problem for solids, but also occurs in ﬂuids [155]. The origin of tensile instability is brieﬂy discussed herein. When particles are under expansion at a given time, and the particle distance becomes so large that particle a has no neighbours, i.e. rab > 2h, then the forces between particles given by (3.79) become zero. If particle b is still depart- ing from particle a at this time, unphysical material breaking may happen. The tensile instability manifests itself by particles separating unrealistically far. 46 Smoothed Particle Hydrodynamics (SPH)

3.4.11 Particle boundary layer

In this section a numerical boundary layer problem is presented. As discussed in Section 3.4.9, ﬁxed ghost particles are widely used for the treatment of the no-slip boundary. To complete the kernel supports, the layer thickness of the ﬁxed ghost particles should be larger than 2h. This setup may cause unphysical boundary layer problems, which is also closely related to ”particle crystallization” [222].

Figure 3.8: Particle boundary layer in the simulation of viscous axisymmetric pipe ﬂow.

Suppose a fluid is flowing in an axisymmetric pipe (or a channel) with diameter D as shown in Fig. 3.8. If the number of fluid particle layers is not many, e.g. 10, 60% of the fluid particles will be directly affected by the boundary ghost particles. If the real boundary layer, i.e. the flow region significantly affected by the boundary, is thin (turbulent flow), spurious phenomena may result. Therefore, to avoid unphysical boundary layer behaviour, the number of fluid particle layers should be much larger than that of fixed ghost particles (six in Fig. 3.8). This requirement may cause difficulties to the modelling of viscous flows in slender structures such as pipes and channels. Suppose that to overcome the possible boundary layer problem, the number of fluid particle layers should be at least 10 times that of the ghost particle layers (100 times were practically used in the simulation of flow over a backward-facing step in [91]). Then we will have 60 layers of fluid particles in the transverse direction. For a slender pipe, the ratio of its length and diameter can be larger than 1000. Then at least (60 + 6) × 60 × 1000 = 3.96 × 106 particles are to be used in a simulation. As a small time step has to be used in SPH, the computation will be very costly if the simulation time is not short. This explains why parallel computation is highly needed in the context of SPH. For simulations carried out in this thesis, the maximum number of particles is limited to about 20000 to obtain reasonable computation times on a standard PC. Note that this limitation occurs not only in SPH, but also in conventional mesh- based methods. The large L/D values also explain why 1D models are mostly used for pipe and channel flows. 3.5 Error, incompleteness and improvements 47

3.5 Error, incompleteness and improvements

3.5.1 Error estimation

From the description in Section 3.3, it is evident that there are two major steps in SPH. The ﬁrst step is the integral approximation and the second is the summation approximation. Consequently, in order to estimate the numerical error involved in SPH, one needs to analyse the errors in these two steps. Deﬁne

′ ′ ′ ϵ1 := f − ⟨f⟩ and ϵ := f − ⟨f ⟩ (3.81) ∞ 1 ∞ as the error in the integral approximation of the function and its ﬁrst-order directive, respectively. Deﬁne

b ′ ′ b′ ϵ2 := ⟨f⟩ − f and ϵ := ⟨f ⟩ − f (3.82) ∞ 2 ∞ as the error in the summation approximation of the function and its ﬁrst-order directive, respectively. Deﬁne

b ′ ′ b′ ϵ := f − f and ϵ := f − f (3.83) ∞ ∞ as the total error in the SPH approximation of the function and its ﬁrst-order directive, respectively.

Without loss of generality, the analysis is restricted to a single dimension. Again, we assume that the compact support of the kernel is fully within the problem domain, i.e. Γ ⊂ Ω (see Fig. 3.1). For convenience we use η to replace x ′.

Integral approximation errors. Assume that kernel W(x − η, h) has compact support Γ = {η |x−η| ≤ 2h}, in which the kernel∫ is symmetric and positive. It satisﬁes the normalization condition Eq. (3.6), i.e. Γ W(x − η, h)dη = 1. For a function f ∈ C3(Ω),Γ ⊂ Ω, we assume that its second- and third-order derivatives in Ω ′′ ′′′ are bounded, i.e. C1 = sup{|f (x)|} and C2 = sup{|f (x)|} exist. Then we have x∈Ω x∈Ω the following Lemma to estimate the error in the integral approximation of the function f.

Lemma 3.1. The SPH approximation of an arbitrary function f ∈ C3(Ω) is at least of second-order accuracy, if the kernel W(x − η, h) is symmetric and positive in its compact support Γ ⊂ Ω and fulﬁlls the normalization condition (3.6). 48 Smoothed Particle Hydrodynamics (SPH)

Proof. Applying the Taylor series expansion to (3.2) yields ∫ ⟨f⟩ (x) = f(η)W(x − η, h)dη ∫Ω [ ] ( ) ( ) 1 2 = f(x) + f′(x) η − x + f′′(ξ) η − x W(x − η, h)dη Ω ∫ 2 ∫ ( ) (3.84) = f(x) W(x − η, h)dη + f′(x) η − x W(x − η, h)dη ∫ Ω Ω ( ) 1 2 + f′′(ξ) η − x W(x − η, h)dη, 2 Ω in which Lagrange’s form for the∫ remainder( ) has been used so that ξ lies in Γ. Symmetry of the kernel results in Ω x − η W(x − η, h)dη = 0. Together with |x − η| ≤ 2h and the normalization condition (3.6), we have

ϵ1 = f(x) − ⟨f⟩ (x) ∫ ∞ ( ) 1 2 = f′′(ξ) η − x W(x − η, h)dη 2 ∞ ∫Ω 1 ≤ |f′′(ξ)| |η − x|2W(x − η, h)dη (3.85) 2 ∞ Ω ∫ 2 ≤ 2C1h W(x − η, h)dη ∞ Ω 2 = 2C1h .

Remark 3.7. Lemma 3.1 also holds for multiple dimensions due to the radial symmetry of the kernel in Γ ⊂ Ω. The Taylor series expansion for a multivariate function is ( ) ( ) ( ) T f(η) = f(x) + ∇f(x) η − x + η − x Hf(ξ) η − x , (3.86) where ∇ is the gradient operator and H is the Hessian matrix.

Deﬁnition 3.1. The integral( of the) product of the compactly supported kernel W(x − k η, h) and the k-th monomial x−η over the support Γ ⊂ Ω is called the k-th moment of the kernel, and denoted by Mk, i.e. ∫ ( ) k Mk = x − η W(x − η, h)dη. (3.87) Ω

With this deﬁnition, Lemma 3.1 can be generalized to the following Lemma: Lemma 3.2. The SPH approximation of an arbitrary function f ∈ Ck+1(Ω) is at least of kth-order accuracy, if the kernel satisfy the following moment conditions { 1, i = 0, Mi = (3.88) 0, i ≠ 0, i ≤ k. 3.5 Error, incompleteness and improvements 49

Note that M0 = 1 is the normalization condition (3.6) and M2i+1 (i ∈ N) is always zero (in one-dimension only) because of the symmetry of the kernel.

We have the following Lemma to estimate the numerical error in the integral approximation of the ﬁrst-order derivative.

Lemma 3.3. The SPH kernel approximation of the ﬁrst-order derivative of the function f ∈ C3(Ω) is at least of second-order accuracy, if the kernel W(x − η, h) is symmetric and positive in its compact support Γ ⊂ Ω and fulﬁlls the normalization condition (3.6).

Proof. From the ﬁrst derivative approximation (3.13), we know ∫ ′ ′ ⟨f ⟩ (x) = f(η)W (x − η, h)dη. (3.89) Ω Consequently, the error in the integral approximation of the ﬁrst-order derivative is

′ ′ ⟨ ′⟩ ϵ1 = f (x) − f (x) ∫ ∞ ∫

= f ′(x) − f′(η)W(x − η, h)dη + f′(η)W(x − η, h)dη − ⟨f′⟩ (x) ∞ ∫Ω Ω ∫

≤ f′(x) − f ′(η)W(x − η, h)dη + f ′(η)W(x − η, h)dη − ⟨f′⟩ (x) ∞ Ω ∫ Ω ( ) ′ 2 = 2C2h + f(η)W(x − η, h) dη (3.85) ∞ Ω 2 = 2C2h + f(η)W(x − η, h) ∂Ω ∞ 2 = 2C2h . (3.90)

Remark 3.8. When the function f in the kernel approximation is replaced by its ﬁrst- order derivative, by Lemma 3.1 one also obtains

′ ′ ′ 2 ϵ = f − ⟨f ⟩ ≤ 2C2h . 1 ∞

Remark 3.9. For the derivative approximation of a scalar function, the accuracy is third- order because M2 = 0 due to the antisymmetry of the kernel gradient (3.11) ∫ W ′(x − η, h)dη = 0 Ω and the normalization condition of the kernel derivative (3.43) ∫ ( ) ′ η − x W (x − η, h)dη = 1. Ω 50 Smoothed Particle Hydrodynamics (SPH)

This is clear if we substitute the following Taylor series expansion into (3.89), ( ) ′′ ( ) ′′′ ( ) (4) ( ) ′ f (x) 2 f (x) 3 f (ξ) 4 f(η) = f(x)+f (x) η−x + η−x + η−x + η−x . (3.91) 2! 3! 4! For multi-dimensional cases, since the second order moments involving the mixed derivatives are not zero, the SPH kernel approximation of the ﬁrst-order derivative is second- order accuracy.

Similar to the generalization of the error estimate of the function approximation, the following lemma for the error estimation of the derivative approximation is proposed. Lemma 3.4. The SPH kernel approximation of the ﬁrst-order derivative of the function f ∈ Ck+1(Ω) is at least of kth-order accuracy if the kernel is compactly supported and satisﬁes the moment conditions (3.88).

Based on Lemmas 3.2 and 3.4, the following theorem for the accuracy of the integral approximation of a function and its derivatives is obtained. Theorem 3.2. The SPH kernel approximations of both the function f ∈ Ck+1(Ω) and its ﬁrst-order derivative are at least of kth-order accuracy for a compactly supported kernel, which satisﬁes the moment conditions (3.88).

With Theorem 3.2, it is possible to construct kernels such that the second moment M2 is also zero (super Gaussian given by (3.36)), resulting in error ϵ1 of O(h4) (see e.g. [156]). However, the disadvantage of such high-order kernels (k > 2) is that they become negative in some part of the support, which may result in non-physical behaviour such as negative density, temperature and energy [117, 156, 177].

Total and summation approximation errors. Consider a 2n + 1 evenly distributed particles in the domain Γ = {xb|−2h ≤ xa −xb ≤ 2h} as shown in Fig. 3.9. We have xb = xa + d0 (b = −n, . . . , n), where Ωb = d0 = O(h) is the distance between two particles. For the cubic spline kernel (3.39), we know Wab = Wa,−b ≥ 0, −1 ′ ′ ′ −2 Wab = O(h ), Wab < 0 (b > 0) and Wab = −Wa,−b = O(h ). Deﬁne ′ αb := Wabd0, then we have αb = α−b and αb = O(1). Deﬁne βb := Wabd0, −1 then we have β0 = 0, β−b = −βb and βb = O(h ).

The normalization condition of the kernel (3.6) in summation form is ∑n ∑n ∑n Wabd0 = αb = α0 + 2 αb = 1, (3.92) b=−n b=−n b=1 which is called partition of unity (PU). The antisymmetry of the kernel derivative (3.11) in summation form is ∑n ∑n ′ Wabd0 = βb = 0, (3.93) b=−n b=−n 3.5 Error, incompleteness and improvements 51

Figure 3.9: Sketch for evenly spaced particles in a 1D domain.

which is called partition of nullity (PN). The normalization condition of the kernel derivative (3.43) in summation form is

∑n ( ) ∑n ∑n ′ xb − xa Wabd0 = bd0βb = 2 bd0βb = 1. (3.94) b=−n b=−n b=1

Assume that function f(x) is sufﬁciently smooth. Expanding f(xb) = f(xa + bd0) around xa by the Taylor series yields

( ) ( ) 2 3 ′ bd0 ′′ bd0 ′′′ f(xb) = f(xa) + f (xa)bd0 + f (xa) + f (xa) ( ) 2! 3! 4 (3.95) bd0 + f(4)(x ) + O(h5). 4! a

Theorem 3.3. For the SPH approximation of a sufﬁciently smooth function with evenly distributed particles and a cubic spline kernel, the total approximation error ϵ is of order 2.

Proof. From summation approximation of the function (3.3) and with αb = O(1), we have 52 Smoothed Particle Hydrodynamics (SPH)

∑n b f(xa) = αbf(xb) b=−n ( ) ∑n ∑4 k bd0 (k) 5 = αb f (xa) + O(h ) (3.95) k! b=−n k=0 ( ) ( ∑n ) ∑n 2 bd0 (3.96) = α + 2 α f(x ) + 2 α f′′(x ) + O(h4) 0 b a b 2! a b=1 ( ) b=1 ∑n 2 bd0 ′′ 4 = f(xa) + 2 αb f (xa) + O(h ) (3.92) 2! b=1 2 = f(xa) + O(h ). Thus we obtain the total error in the function approximation as

b 2 ϵ = f(xa) − f(xa) = O(h ). (3.97) ∞

Corollary 3.1. For the SPH approximation of a sufﬁciently smooth function with evenly distributed particles and a cubic spline kernel, the summation approximation error ϵ2 is of order 2.

Proof. With the deﬁnitions of the errors given above, we know

b ϵ = f(xa) − f(xa) ∞ b = f(xa) − ⟨f⟩ (xa) + ⟨f⟩ (xa) − f(xa) ∞ (3.98) b ≤ f(xa) − ⟨f⟩ (xa) + ⟨f⟩ (xa) − f(xa) ∞ ∞

= ϵ1 + ϵ2.

Due to Theorem 3.3, ϵ is of order 2, while ϵ1 is at least of order 2 because of 2 Lemma 3.1. Therefore, we have ϵ2 = O(h ).

−d Remark 3.10. For d-dimensions (d = 2, 3), we have Wab = O(h ) and Ωd = d O(h ). Therefore, αb = WabΩb is still of order O(1). Theorem 3.3 and corollary 3.1 hold for multiple dimensions. ′ Remark 3.11. When particles become disordered, the f (xa) term will not drop out from 2 (3.96) due to the losing of symmetry, and hence ϵ2 is in between O(h) and O(h ). If (3.92) is largely violated due to disordered particle distribution, the accuracy will be even lower or totally lost. Theorem 3.4. For the SPH approximation of the ﬁrst-order derivative with evenly distributed particles and the cubic spline kernel, the total approximation error ϵ′ is of order 2. 3.5 Error, incompleteness and improvements 53

Proof. From summation approximation of the derivative (3.13) and with βb = O(h−1), we have

∑n b′ f (xa) = βbf(xb) b=−n ( ) ∑n ∑4 k bd0 (k) 4 = βb f (xa) + O(h ) (3.95) k! b=−n k=0 ( ) ∑n ∑n ( ) ∑n 3 ′ bd0 ′′′ = β f(x ) + 2 β bd f (x ) + 2 β f (x ) + O(h4) b a b 0 a b 3! a b=−n b=1 b=1 ( ) ∑n ( ) ∑n 3 ′ bd0 ′′′ 4 = 2 βb bd0 f (xa) + 2 βb f (xa) + O(h ) (3.93) 3! b=1 ( ) b=1 ∑n 3 ′ bd0 ′′′ 4 = f (xa) + 2 βb f (xa) + O(h ) (3.94) 3! b=1 ′ 2 = f (xa) + O(h ). (3.99) Thus we obtain the total error in the ﬁrst-order derivative approximation as

′ ′ b′ 2 ϵ = f (xa) − f (xa) = O(h ). (3.100)

Corollary 3.2. For the SPH approximation of a sufﬁciently smooth function with evenly ′ distributed particles and a cubic spline kernel, the summation approximation error ϵ2 is of order 2.

Proof. With the deﬁnitions of the error in the integral, summation and total approximation given above, we have

′ ′ b′ ϵ = f (xa) − f (xa) ∞ ′ ′ ′ b′ = f (xa) − ⟨f ⟩ (xa) + ⟨f ⟩ (xa) − f (xa) ∞ (3.101) ′ ′ ′ b′ ≤ f (xa) − ⟨f ⟩ (xa) + ⟨f ⟩ (xa) − f (xa) ∞ ∞ ′ ′ = ϵ1 + ϵ2.

′ ′ Due to Theorem 3.4, ϵ is of order 2, while ϵ1 is at least of order 2 because of ′ 2 Lemma 3.3. Therefore, we have ϵ2 = O(h ).

′ −d−1 Remark 3.12. For d-dimensions (d = 2, 3), we have Wab = O(h ) and Ωd = d ′ −1 O(h ). Therefore, βb = WabΩb is still of order O(h ). Theorem 3.4 and corollary 3.2 hold for multiple dimensions. 54 Smoothed Particle Hydrodynamics (SPH)

′′ Remark 3.13. When particles become disordered, the f (xa) term will not drop out from ′ 2 (3.99) due to the losing of symmetry, and hence ϵ2 is in between O(h) and O(h ). If (3.93) and (3.94) are largely violated due to disordered particle distribution, the accuracy will be even lower or totally lost.

Remark 3.14. For the cubic spline kernel we have M0 = 1 and M1 = 0, but M2 ≠ 0. Consequently, the accuracy in the above SPH ﬂuid dynamics (multiple dimensions) is at most second order. Furthermore, for kernels with similar shapes as the cubic spline kernel (e.g. kernels in [64]), Theorems 3.3 and 3.4 hold. Corollaries 3.1 and 3.2 hold too.

Although an equi-spaced particle distribution can be established for initial setup, the particles will become disordered during the course of the SPH calculations. The exact pattern of the disorder depends on the specific problem. In one of the first SPH computations [133], the disorder was described by a probability distribution and hence the errors could be estimated in the same way as a Monte −1/2 Carlo estimate, i.e. ϵ2 = O(NP ) (NP is the number of points in the support) [165]. However, as noticed in another original SPH work [69], the errors were much smaller that the Monte Carlo estimate would suggest. It was attributed to that the probability estimates allow fluctuations that are inconsistent with the physics. As argued by Monaghan [156], the SPH particles are disordered but in an orderly way. When particles are disordered but evenly distributed, i.e. the well-known disordered equi-distribution in random number generation, the error −1 d−1 ϵ2 = O(NP (log NP) ) according to the quasi-Monte Carlo estimation [165].

Now we will examine the relationship between N, NP and h. Take a 2D domain with unit area as the example. The area associated with one point is Ωb = 1/N. If we suppose that the average particle distance is d0 and Ωb is a square, we get 2 ∝ ∝ −2 another expression Ωb = d0. Since h d0, we have N h . The number of points within the support of the kernel can be estimated from NP = C Nh2, where C is a constant depending on the area of the total domain and the support. More generally, one can show that N ∝ h−n and NP ∝ CNhn, where n is the number of dimensions. Consequently, in a 2D domain we have

ϵ2 = O(h) (3.102) for a random particle distribution, and 2 ϵ2 = O(h log h) (3.103) for a disordered equi-distribution. Since the particle distribution is neither totally random nor equal, the approximation error is something in between (3.102) and (3.103).

The error in the summation approximation ∫ ∑ b ϵ2 = ⟨f⟩ − f = f(η)W(x − η, h)dη − f(xb)W(x − xb, h)Ωb ∞ ∞ Ω b highly depends on the particle distribution pattern. The right hand term is often called nodal integration [112, 165], and determines the summation approximation 3.5 Error, incompleteness and improvements 55 accuracy. Quinlan et al. [179] used the Euler-MacLaurin formula to estimate the summation approximation error for a regular 1D particle distribution. The same technique is extended for the approximation of a 3D function [2] and its ﬁrst- order derivatives [1]. Several other papers have also been dedicated to the error estimation in the summation approximation (see e.g. [147, 156, 222]), but before more work has been done examining the particle distributions that occur most frequently in SPH, it is not clear what exactly error bounds we could use.

From the above error analysis we notice that the errors in the summation approximation are generally larger than that in the integral approximation. Hence ′ ′ the total errors ϵ and ϵ in the SPH calculations are dominated by ϵ2 and ϵ2, respectively. This is the reason why only ϵ2 is mostly discussed in literature [1,2, 179, 222].

3.5.2 Incompleteness

Ascertaining whether a meshfree method is consistent for unstructured grid of nodes is quite difficult, as compared to checking whether a finite-difference scheme for a uniform grid is consistent [17]. Therefore, in Section 3.5.1 we estimated the approximation errors in SPH based on given particle distributions, such as even distribution or disordered equi-distribution. For meshfree methods, a common way is to examine the reproducing conditions, i.e. completeness, instead. The completeness states the function reproducing ability of an approximation algorithm and is equivalent to the consistency condition in FDMs [17, 111, 112]. We have the following definition for completeness [17] or nodal completeness [112,122]. b Definition 3.2. An approximation f is complete to order k (kth-order completeness) if any polynomial up to order k can be represented exactly.

b Thus if f is given by ∑ b f(x) = ϕb(x)fb, (3.104) b where ϕb(x) are the approximating functions (shape functions in FEM) and fb := f(xb) are the nodal values, then if fb are given by a polynomial of order k, the ap- b proximation f(x) should reproduce the polynomial exactly if the approximation is complete to order k. These are known as reproducing conditions in the literature on wavelets.

The discrete form of the kernel approximation to a function (3.3) is similar to shape function approximation in FEM. For example, (3.3) can be rewritten as (3.104), where the approximation functions ϕb(x) given by

ϕb(x) = W(x − xb, h)Ωb (3.105) 56 Smoothed Particle Hydrodynamics (SPH) play the same role as the shape functions in FEM. Based on collocation methods, the SPH approximations are constructed only at the nodes (particles), ∑ b fa = ϕabfb, (3.106) b where ϕab := W(xa − xb, h)Ωb. It is another form of the summation approximation (3.5). This approximation is not interpolant, i.e.

W(xa − xb, h)Ωb ≠ δab. (3.107) b b So the nodal variables fb do not correspond to f(xb), i.e. fb ≠ f(xb). This is consistent with the analysis in Section 3.3.1. The cubic spline kernel satisﬁes the normalization condition (3.6) (i.e. M0 = 1), but its discrete counterpart does not necessarily reproduce constant functions for a disordered set of particles, i.e. ∑ W(x − xb, h)Ωb ≠ 1. (3.108) b does not hold. Thus the standard SPH summation approximation cannot exactly reproduce even a constant function, i.e. it lacks zeroth-order completeness.

Similarly, the discrete forms (summation forms) of the antisymmetry of the kernel derivative (3.11) and its normalization (3.43), i.e. Eqs. (3.93) and (3.94) (in one dimension) do not hold. So the lack of reproducing conditions in the summation approximation to the derivatives can be accordingly shown. That is, the standard SPH summation approximations to the ﬁrst-order directives cannot exactly reproduce linear functions. For details on completeness analysis, one is referred to [17, 112].

3.5.3 Improvements

An approximation widely used in ﬁtting data is [192] ∑ / ∑ b fa = ϕabfb ϕab, (3.109) b b which is known as Shepard functions. They can reproduce constant functions and their gradients [17]. By applying Taylor series expansion to the SPH kernel approximation of the function, (3.109) is derived by Chen et al. [29, 30, 32].

To restore the completeness in SPH approximations, a number of techniques have been summarized by Belytschko et al. (1998) [17]. The following gives a brief review on the recent developments. The key idea is to fulﬁll the moment conditions (3.88) (in discrete form) by using modiﬁed kernel and its directives. Inspired by the ideas presented in [30, 120, 235], two new correction schemes are proposed. Since they are not further explored herein, they are presented in the Appendix A. 3.5 Error, incompleteness and improvements 57

To overcome the incompleteness of SPH, a reproducing kernel particle method (RKPM) has been proposed by Liu and his co-workers [122, 123, 126] with the kernel modified via a correction function. It was proved that RKPM can restore nth-order nodal completeness for a polynomial function and its derivatives. It was also found that the tensile instability problem was removed and numerical oscillations at sharp wave fronts were apparently absent without using an artificial viscosity [123]. Based on the framework of RKPM, by using the technique of multilevel decomposition, Liu and his co-workers [124,125] proposed a new class of multiscale reproducing kernel particle methods; by introducing the moving least square technique, Li et al. [127] presented a moving least square reproducing kernel particle method; by combing a partition of unity and the associated wavelet functions, Li & Liu [110] developed a synchronized reproducing kernel approximation. Recently by incorporating Galerkin weak forms into RKPM, a new method named reproducing kernel element method (RKEM) was presented by Liu and his co-workers [113, 128, 132, 194]. The central idea of RKEM is to combine the strengths of both finite element methods and the meshfree feature of RKPM. By combining the moving least square technique with the kernel estimate in conventional SPH, the moving least square particle method (MLSPH) was invented by Dilts [52, 53]. This method allows for direct enforcement of boundary conditions without using ghost particles and improves dynamic impact simulations, but it is much more time consuming (a factor 7 to 8 times) than the standard SPH as pointed out by Monaghan [155]. Chen and his co-workers [30, 32] proposed a corrective kernel particle method (CSPM) by applying a Taylor series expansion to the kernel estimate. The boundary deficiency was addressed and the accuracy in the interior domain was enhanced. The tensile instability could be removed, but artificial viscosity had to be used. Based on the framework of CSPM, by retaining higher-order derivative terms in the Taylor series expansions, Zhang & Betra [11–13, 234] further developed a modified smoothed particle hydrodynamics (MSPH) method. MSPH is an improved version of CSPM with better accuracy and stability, but lower computational efficiency. Based on the same idea as MSPH, Liu et al. [59,119,120] presented a finite particle method (FPM). By applying the synchronized reproducing particle approximation [110] to the corrected kernel estimates [30, 233], a symmetric smoothed particle hydrodynamics (SSPH) was developed by Batra & Zhang [14, 235]. To find kernel estimates of the derivatives, the SSPH method does not use the kernel derivatives while the aforementioned methods do. Therefore, a larger class of kernels than standard is admitted in SSPH. The above meshfree particle methods are mainly used for solids, where particles are taken fixed.

Chapter 4

Selected Test Problems

The SPH equations with the initial and boundary conditions have been introduced and discussed in Chapter 3. They have been applied to a number of typical test problems, four of which are presented in this chapter. They are the dam- break, impinging jet, jet flow under gravity and flow separation at bends. A common feature of these problems is that there is at least one moving free surface, so that the liquid is assumed to be incompressible and inviscid, and hence governed by the Euler equations. Apart from validating the developed code, they are used to verify the SPH equations given in Chapter 3. The test problems are closely related to the rapid pipe filling/draining and the isolated slug travelling in a void line studied in later chapters. This chapter is organized as follows. Numerical modelling of free-surface flows is briefly discussed in Section 4.1. The four test examples are presented in the subsequent four sections. For each test problem a brief literature review is first given together with the specific objectives. Then the setup of the numerical experiments is described with the focus on how to choose the problem geometry and the proper SPH parameters. To validate the developed SPH solver, the obtained solutions are compared with theoretical, experimental and/or numerical results from literature.

4.1 Introduction

A notoriously difficult theoretical problem is the free-surface flow in the presence or absence of body forces, e.g. gravity. To obtain a meaningful solution, delicate transformations, simplifying assumptions and singular points need special attention in the potential flow theory. Nevertheless, the theoretical solutions often have discrepancies with the experimental results because of improper assumptions. Consequently, a numerical simulation of free-surface flows is often the only option. For traditional mesh-based methods, difficulties mainly arise from 60 Selected Test Problems the use of grids (fixed or moving). To overcome the meshing problems it is possible to eliminate the grid, in part or completely [190]. The numerical simulation of moving surfaces was not possible until the development of the particle-in-cell (PIC) or marker-and-cell (MAC) methods [76,77]. Some other particle approaches such as molecular dynamics (MD), Monte Carlo (MC), lattice Bolztmann method (LBM), dissipative particle dynamics (DPD), vortex particle methods, etc (see the reviews on particle methods, e.g. [102,117,190]), may not be sufficient for the free surface flow problems considered herein because of the large scale, the lack of computational efficiency and the too high complexity.

The MAC method is based on an Eulerian framework and explicit free-surface tracking techniques. In the original MAC method, massless marker particles are used to locate the surface while finite-difference schemes are used to solve the hydrodynamic equations. The pressure is calculated from the pressure Poisson equation. To project the field quantities of a particle onto the fixed background mesh, numerical mapping (interpolation) is needed at every time step. Many modifications to the original MAC have been proposed by adopting better numerical techniques to solve the pressure and more efficient algorithms to track the free-surface, such as volume of fluid (VOF) [81] and level set methods (LSM) [73]. The MAC and its modifications retain coupling between particles and a mesh, and hence only eliminate the grids partially [190]. Although the MAC method is flexible and robust for the simulation of free-surface flows, it has some disadvantages. First, since tedious surface tracking is necessary in MAC, the numerical implementation is often complicated [73]. Second, due to forward and backward information transition between the particles and the background mesh, the numerical accuracy is generally low [73]. Third, extension of the simulations from 2D to 3D is never an easy task. Lastly, the enforcement of boundary conditions for an irregular domain is difficult. A decent review of the MAC method was given by McKee et al. recently [142], where the details on several new techniques can be found. The core ideas of MAC and the numerical implementation are summarized by Griebel et al. [73], and a C++ code is included.

In 1994 another efficient macroscopic particle approach, the SPH method, based on a Lagrangian framework and originally developed for astrophysical problems (gas dynamics without boundaries), was used to model free-surface flows by its inventor, Monaghan [152]. In this method, the problem domain is covered by a large number of particles, with or without direct physical meaning, as explained in Chapter 3. Since spatial derivatives are approximated using the moving particles, which serve as interpolation points, there is no need to use any background mesh (grid) and hence any problem caused by the grid is resolved. Compar- ing with the MAC method, in SPH the free surface does not need to be tracked explicitly because its motion is naturally captured by the particles representing it. Numerical mapping of the information onto a grid is not necessary due to its Lagrangian feature. These features simplify (conceptually) the simulation of free-surface flows to a great extent. 4.2 Dam-break 61

4.2 Dam-break

A dam is a barrier that obstructs or directs fluid flow, often creating a reservoir behind it. The 2D reservoir considered herein is a rectangular column of fluid supported by a horizontal bottom and confined by two vertical walls as sketched in Fig. 4.1a. When the right wall is instantaneously removed, the fluid is free to flow along the horizontal bed (see Fig. 4.1b).

Figure 4.1: Deﬁnition sketch of the two-dimensional dam-break problem: (a) initial equilibrium state; (b) ﬂow evolution after instantaneously removing of the right wall.

The dam-break problem has been widely investigated numerically and experimentally (see e.g. [77, 81, 140, 152, 167, 176, 191, 207]). It is a proper test case to verify methods dealing with free-surface ﬂows with large movements. Prin- ciples of shallow water theory cannot be used to simplify the analysis of this two-dimensional problem because the streamline curvature is big and a non- hydrostatic pressure distribution exists. Due to the non-linearity of the free- surface condition, computational difﬁculties occur in traditional methods. Fur- thermore, the free surface is changing in time. This explains why the only re- course to solving dam-break problems (especially in 3D) had been experiment [140] until the development of marker-and-cell (MAC) method by Harlow & Welch [77], and the mesh-free Lagrangian SPH particle method [152].

The two-dimensional dam-break problem is simulated and the results are compared with both numerical and experimental results from literature. The main objective of the present study is to validate our code for hydraulic engineering problems involving a single free surface encountered, e.g., in pipe ﬁlling. The numerical convergence behaviour of SPH is studied to verify the error analysis presented in Section 3.5.

4.2.1 Basic theory

An idealization of a dam-break problem is shown in Fig. 4.2. From a frictionless, semi-inﬁnite horizontal channel with constant depth of water H on one side of 62 Selected Test Problems the gate (dam) and no water on the other side, the gate is suddenly removed. If vertical accelerations are neglected, the velocity of a point on the surface V at y depth is √ √ V = 2 gH − 3 gy. (4.1) The moving free-surface proﬁle is described by √ √ x = (2 gH − 3 gy)t. (4.2)

It is a parabola with vertex at the leading edge and concave upward. At x = 0, the position of the removed dam, the water depth√ remains constant at y = 4H/9. The leading edge moves downstream at V = 2 gH. For further details one is referred to [199], but it is remarked here that the basic theory is of limited value because of the simplifying assumptions.

Figure 4.2: Dam-break velocities and free-surface proﬁle.

4.2.2 Test problem and SPH setup

The speciﬁc case studied in [152] and [191] is reproduced for comparison. The setup for the numerical solution of the dam-break problem is shown in Fig. 4.3. In this ﬁgure, the problem domain is taken as (0, 0.4) × (0, 0.225) m2, with the origin of the 2D coordinate system at the left corner. The horizontal bed coincides with the x-axis and the left wall coincides with the y-axis. The depth of the water column is H = 0.2 m and the width is W = 0.1 m.

The initial distribution of SPH particles comprises a square pattern with particle distance d0 = 0.005 m, which is the same as used in [191]. The total number of ﬂuid particles is N = 20 × 40. Fluid particles are initially under hydrostatic pressure with zero velocity. The initial particle densities are calculated through the equation of state (3.63). Mirror particles are used to impose the free-slip boundary conditions at the bottom√ and left wall. According to Eq. (4.1), the maximum horizontal velocity is 2 gH = 2.8 m/s, so that the SPH sound speed is taken as c = 30 m/s based on the requirement of a low Mach number. The smoothing −5 length is h = 1.33d0 and the minimum time step is ∆tmin = λd0/c = 3.33 × 10 s, where λ = 0.2. The simulation time is Ts = 0.2 s, during which the water front 4.2 Dam-break 63

Dam break (gravity) − Case 1 (α = 0.1, c = 30 m/s, V = 0 m/s) 0 0.25 1800 left wall 0.2 1600

1400 0.15 computational domain 1200

0.1 1000 y [m] fluid particles 800 0.05 600 horizontal bed 400 0 mirror particles 200 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 x [m]

Figure 4.3: SPH setup of the two-dimensional dam-break problem including the problem domain, coordinate system, boundaries (left wall and horizontal bed), ﬂuid particles, mirror particles and initial conditions. Hydrostatic pressure in Pascal indicated by colourbar. Mirror particles behind the left wall are not shown. remains within the computational domain. The√ following non-dimensional variables are used: X = x/H, Y = y/H and T = t g/H, for the leading edge, the vertical height and the time, respectively.

We solve the two-dimensional ﬂow problem to obtain (i) the water surface proﬁle at given time, (ii) the time history of the leading edge, and (iii) the entire velocity distribution.

4.2.3 Numerical results

The free-surface profiles and particle distributions for the calculated flow at four different times are presented in Fig. 4.4. A wedged-shape water front is formed at the lower right corner of the free surface. Numerical results calculated by ISPH in [191] are given for comparison. A good overall agreement is observed. The flow pattern is initially in qualitative agreement with theory given in Section 4.2.1. The water height at the dam site remains about 4/9 times the initial height H and the free surface profile initially is a parabola. The differences are mainly due to two aspects: (1) in the theory the reservoir size is infinite (W = ∞), but it is finite (W = 0.1) in the numerical simulation; (2) the vertical acceleration of the flow is ignored in the theory.

Figure 4.5 depicts the movement of the leading edge (water-front). The SPH results are compared with the experiments [140] and computations by MAC [81] 64 Selected Test Problems

(a) (b) 0.2 0.2

0.18 0.18 t = 0.05 s t = 0.1 s 0.16 0.16

0.14 0.14

0.12 0.12

0.1 0.1

0.08 0.08

Vertical distance (m) 0.06 Vertical distance (m) 0.06

0.04 0.04

0.02 0.02

0 0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 Horizontal distance (m) Horizontal distance (m)

0.18 0.18 t = 0.15 s t = 0.18 s 0.16 0.16

0.14 0.14

0.12 0.12

0.1 0.1

0.08 0.08

Vertical distance (m) 0.06 Vertical distance (m) 0.06

0.04 0.04

0.02 0.02

0 0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 Horizontal distance (m) Horizontal distance (m)

Figure 4.4: Water distribution at four time levels: open circles – present SPH; ﬁlled circles – Shao and Lo [191].

and ISPH [191]. All the numerical results are in good agreement and have a similar trend as the experiments but the leading edge is faster.

Particle distributions with associated velocities at four time levels are shown in Fig. 4.6. The velocities on the free surface have larger values while the ﬂuid at large distance from the surface (left lower corner) remains almost stationary. The computed velocity of the leading edge at time t = 0.2 s is roughly 2 m/s from Fig. 4.6, which√ agrees with the result in [191], but differs from the theoretical solution 2 gH = 2.8 m/s. This implies that the basic theory in Section 4.2.2 is not valid for the numerical simulation.

The continuous wall concept [103] for the enforcement of the free-slip boundary condition at the bottom and left wall is tested and similar results are obtained as shown in Fig. 4.7. Compared with mirror particles, the continuous wall concept is easier to implement, especially for corners and sharply curved wall. 4.2 Dam-break 65

1.5 SPH MAC+VOF ISPH Experiment

0.5 Non−dimensional leading edge X=x/H

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Non−dimensional time T=t(g/H)1/2

Figure 4.5: Two-dimensional dam-break problem: movement of the leading edge. T = 0 indicates the removal of the right wall and X is the position of the leading edge. The experimental data is from [140].

4.2.4 Convergence behaviour

To investigate the numerical convergence of the SPH method, four simulations with different particle numbers were performed. These runs have particle numbers of N = 10 × 20, N = 20 × 40, N = 40 × 80 and N = 80 × 160, corresponding to particle spacing, d0 of 0.01, 0.005, 0.0025 and 0.00125 m, respectively. The smoothing lengths are correspondingly h = 1.33d0. To be consistent, the corresponding minimum time steps were two times, one time, a half and a quarter of the original 3.33 × 10−5 s. The non-dimensional positions of the leading edge for the four runs are plotted in Fig. 4.8.

Convergence is observed as the particles become ﬁner, i.e. the initial particle spacing becomes smaller. To estimate the convergence rate, it is assumed that the numerical error Ed0 of any run with particle spacing d0 is proportional to n d0 , where n is the order of convergence. Let E10×20, E20×40, E40×80 and E80×160 denote the four numerical errors, thus the relationship between the errors and particle spacing is n n EN×2N − E2N×4N . (d0,N×2N) − (d0,2N×4N) = n n , (N = 10, 20). (4.3) E2N×4N − E4N×8N (d0,2N×4N) − (d0,4N×8N) Since by deﬁnition we have

d0,10×20 = 2d0,20×40 = 4d0,40×80 = 8d0,80×160, (4.4) relationship (4.3) can be simpliﬁed to

EN×20 − E2N×4N . = 2n, (N = 10, 20). (4.5) E2N×4N − E4N×8N 66 Selected Test Problems

0.2 0.2

0.18 0.18

0.16 0.16 t = 0.05 s t = 0.1 s 0.14 0.14

0.12 0.12

0.1 0.1

0.08 0.08

Vertical distance (m) 0.06 Vertical distance (m) 0.06

0.04 0.04

0.02 0.02

0 0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 Horizontal distance (m) Horizontal distance (m)

0.2 0.2

0.18 0.18

0.16 0.16 t = 0.15 s t = 0.2 s 0.14 0.14

0.12 0.12

0.1 0.1

0.08 0.08 Vertical distance (m) Vertical distance (m) 0.06 0.06

0.04 0.04

0.02 0.02

0 0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 Horizontal distance (m) Horizontal distance (m)

Figure 4.6: Velocity distributions after dam break at four time levels.

Consequently, n can be determined. It is assumed that the error differences between the runs at any time can be represented by the differences in the corresponding locations of the leading edge shown in Fig. 4.8, e.g.

E10×20 − E20×40 = |X10×20 − X20×40|. (4.6)

With this assumption and the relationship (4.5), the index n can be evaluated at any time of the run. The averaged value of n over the simulation time is 0.97 for the three coarser runs (4.3) (N = 10) and 0.99 for the three finer runs (4.3) (N = 20). Thus, the current SPH numerical method is estimated to be first-order accurate, which confirms the theoretical analysis in Section 3.5.1. This is consistent with a second-order error in the derivative approximation for an ordered particle distribution, which reduces to first-order when the particles become disordered. It is necessary to mention that n is varying when the averaging is taken over different time intervals. When the average is stopped earlier, smaller convergence rates (about 0.8) are obtained; conversely, larger convergence rates (about 1.15) are obtained when the averaging is started from T = 0.7 onwards. The reason is that the differences between the leading edge locations are very small and thus subjected to round-off errors (cancellation). 4.2 Dam-break 67

1.5 SPH+mirror SPH+contiuous wall

0.5 Non−dimensional leading edge X=x/H

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Non−dimensional time T=t(g/H)1/2

Figure 4.7: Time evolution of the leading edge with different numerical treat- ments of the bottom and left wall boundary conditions.

1.4 N = 10×20 N = 20×40 1.2 N = 40×80 N = 80×160 1

0.8

0.6

0.4

0.2 Non−dimensional leading edge X=x/H

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Non−dimensional time T=t(g/H)1/2

Figure 4.8: Non-dimensional position of the leading edge for four different particle numbers (initial spacings) to check convergence.

4.2.5 Summary

The two-dimensional dam-break problem has been simulated with the SPH particle method. The numerical results, including free-surface profiles, positions of the leading edge and velocity distributions, are compared to basic theory, experiments and numerical simulations from literature. Good agreement with the numerical results from literature is obtained. There are several aspects – not pur- 68 Selected Test Problems sued herein – accounting for the difference between the numerical results and the experiments. For the velocity of the leading edge, the difference with basic theory is due to the finite size of the upstream reservoir and the non-zero vertical acceleration. The numerical convergence behaviour is found to be first-order.

4.3 Impinging jet

Jet flows occur in many real world situations, such as fluid mixing, fuel filling, combustors and ejectors, fire-fighting, weir flows, waterfalls, the flow past the bow of a ship, etc. Being one kind of free surface flows in hydrodynamics, jet flows are of great practical importance, but the simulation is difficult. In the case of steady jet flows, the free surfaces are previously unknown and the objective is to locate them, which has been the focus when using potential flow theory. In unsteady jet flows, the location of the free surface evolves with time and the dynamic boundary conditions bounding the jet are difficult to satisfy. When a jet impinges on a solid surface (e.g. the tip of an ocean wave impacts on the bow of a ship or an offshore structure), a large impact force may result. The prediction of hydrodynamic impact loads plays a vital role in the design of many structures, particularly for marine, coastal and offshore engineering. It is also of importance in accessing the integrity of piping systems.

A jet impinging on an inclined ﬁxed plane is chosen as test example herein because (i) it has two free surfaces, a situation for which SPH has not been fully exploited, and (ii) analytical solutions for steady state jets based on a control volume approach [199] and the potential ﬂow assumption [174] are available for comparison, and hence it is possible to do a parametric study for the SPH solver. Important SPH parameters are systematically assessed.

4.3.1 Basic theory

Figure 4.9 sketches the top view of a 2D jet emerging from a channel and impacting on a ﬁxed planar wall with an inclination angle β. The planar wall is smooth and there is no gravity involved. The velocity of the jet is V0, the channel width is D0, and the range of β is 0 < β ≤ 90o. For 90o < β < 180o the solution is symmetric and will not be considered herein.

For the steady-state situation, the magnitude of the average velocity of the flows parallel to the wall (V1 and V2) is the same as that of the incoming flow. This follows from Bernoulli’s equation if losses in the impact region are neglected. The division of flow can be computed from the momentum and volume conservation 4.3 Impinging jet 69

Figure 4.9: Two-dimensional jet impinging on an inclined planar wall: (a) initial state, (b) steady state.

[199], and the widths of the leaving ﬂows D1 and D2 are given by

D D D = 0 (1 + cos β),D = 0 (1 − cos β). (4.7) 1 2 2 2

The steady force per unit length exerted on the plane is

2 F = ρV0 D0 sin β. (4.8)

4.3.2 Test problem and SPH setup

In the setup of the numerical tests, the channel width is D0 = 0.1 m. The symmetry line of the channel is taken as the x−axis with its origin at the inflow section, which is fixed for most of the test cases. The length of the inclined wall is 0.25 m and its centre is fixed at (0.25, 0) m. The computational domain, location of the outflow section, and the initial length of the fluid block depend on the inclination angle β. The particles are initially uniformly distributed with spacing d0 = 0.005 m and the smoothing length is taken as h = 1.33d0. The initial pressure is the same as the reference pressure. The velocity components are V0x = 1 m/s and V0y = 0 m/s. The speed of sound is taken as c = 10 m/s. The time step is ∆t = 0.2d0/c, and the simulation time is taken as Ts = 2.5 s, after which a steady state is practically reached. The initial setup for the case with β = 45o is shown in Fig. 4.10. A monitoring particle is set up (see Fig. 4.10) to check the kinematic free surface boundary condition, i.e. a particle initially on the surface remains on it during the movement of the particle. The following non-dimensional variables X := x/D0, Y := y/D0 scaled by the channel width are used. 70 Selected Test Problems

Jet flow − Case 1 (α = 0.1, c = 10 m/s, V = 1 m/s, β = 45o) 0

0.25

0.2 computational domain 0.15

0.1 inlet particles 0.05

0 fixed wall y [m] −0.05

−0.1 fluid particles −0.15 monitoring particle −0.2

−0.25 0 0.1 0.2 0.3 0.4 0.5 x [m]

Figure 4.10: Case setup for the two-dimensional jet ﬂow impinging on an inclined planar wall (β = 45o case).

4.3.3 Numerical results

The first case study is for the jet impacting on a vertical wall, i.e. β = 90o. Due to symmetry only half of the flow is simulated. The symmetry line is a free- slip boundary that is modelled by mirror particles. From the theoretical analysis it is known that this is the most violent case giving the highest jet force. The particle distribution at steady state is shown in Fig. 4.11a, where the shape of the free surface is clearly shown by the outer layer of particles. The numerical prediction agrees well with the potential flow solution [174]. The difference at the most curved part is mainly due to the coarse SPH model and can be reduced by using higher resolution with d0 = 0.0025 m as shown in Fig. 4.11b. The results interpreted below are from this finer model.

The trajectory of the monitored particle, initially located at (0, 0.0475) m (left upper corner of the fluid block in Fig. 4.11), has been checked. It remains part of the free surface during its movement until it leaves the computational domain. This posteriori verification confirms the correct numerical enforcement of the kinematic free surface boundary condition as described in Section 3.4.8.

The velocity distributions at four different time levels are shown in Fig. 4.12. The initial particle distribution has a wedge-shaped front as shown in Fig. 4.12a. If the flow front is steep (as in Fig. 4.10), a high impact pressure pulse is generated when the fluid hits the vertical wall. When this pressure pulse arrives at the inlet, the particles close to the inlet will be distorted. To attenuate the resulting particle distortion, longer simulation times are necessary. As shown in Fig. 4.12c, the jet flow at time t = 1 s is close to its final steady state. 4.3 Impinging jet 71

(a) (b) 2.5 2.5

2 2

1.5 1.5

1 1 Nondimensional distance Y Nondimensional distance Y 0.5 0.5

0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Nondimensional distance X Nondimensional distance X

Figure 4.11: Solution for the two-dimensional jet impinging normally onto a smooth wall with initial particle distance (a) D0/d0 = 10 and (b) D0/d0 = 20. Dots – SPH particles; solid line – potential ﬂow theory [174].

(a) (b) 2.5 2.5

t = 0 s t = 0.5 s

2 2

1.5 1.5

1 1 Nondimensional distance Y Nondimensional distance Y 0.5 0.5

0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Nondimensional distance X Nondimensional distance X

t = 1.0 s t = 2.0 s

2 2

1.5 1.5

1 1 Nondimensional distance Y Nondimensional distance Y 0.5 0.5

0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Nondimensional distance X Nondimensional distance X

Figure 4.12: Velocity distribution of the jet impinging an orthogonal plane at different time levels.

The velocity along the vertical wall is plotted in Fig. 4.13a with comparison to the theoretical solution given in [146]. The velocity at a sample point on the wall 72 Selected Test Problems is interpolated from the particles within a small disc having the same radius as the kernel. The interpolation algorithm is the MLS approximation with linear basis functions [16], which is widely used in meshfree methods [111, 112]. The SPH solution has good agreement with theory. The pressure distribution in the entire steady flow is shown in Fig. 4.13b. The pressure at the stagnation point is very 2 close to the theoretical value ρV0 /2 = 500 Pa, with a deviation of 25 Pa. When the pressure is calculated from the SPH velocity field according to the Bernoulli theorem, the deviation is reduced to 15 Pa. This means that although both of them have reasonable accuracy, the pressure that obtained from the velocity field has better agreement with the theory than that directly obtained from the pressure (density) field. This is consistent with the conclusion that the pressure field generally suffers from a numerical fluctuation (known as pressure noise) in the standard SPH (see e.g. [107, 146]). Integration of the pressure along the vertical wall gives a force of 514 N/m, which is almost half of the analytical value (1000 N/m) determined by Eq. (4.8). The relative error is only 3 percent. This again indicates a good agreement between the SPH calculation and theory.

(a) (b) 1 2.5 SPH 0 0.2 0.4 0.9 Theory p/ρV2 0 0.8 2

0.7

0.6 1.5

0.5

0.4 1 Velocity (m/s)

0.3 Nondimensional distance Y 0.2 0.5

0.1

0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 Nondimensional distance Y Nondimensional distance X

Figure 4.13: Two-dimensional jet impinging onto an orthogonal plane: (a) velocity distribution along the vertical wall with comparison to theory and (b) dimensionless pressure distribution.

Results for the cases β = 45o and β = 60o are shown in Fig. 4.14a and 4.14b, respectively. The theoretical free-surface profiles are not given in [174], but the theoretical width determined from Eq. (4.7) is plotted in Fig. 4.14 for the sake of validation. The flow width at the upper outflow section (large discharge) agrees better with the theory than that at the lower outflow section (small discharge). More particle layers remain orderly in the large discharge section, but particles in the small discharge section hardly stay in ordered layers. Although the difference between the numeric and theoretical predictions at the lower outflow section is not significant, some small fluctuations exist on the free surface, especially for the β = 45o case.

Other inclined angles are also tested and the results together with the above three cases are shown in Fig. 4.15, where the theoretical solutions given by Eq. (4.7) are also presented for comparison. The symbols are for the SPH results, and the solid 4.3 Impinging jet 73

(a) (b) 2.5 2.5 SPH SPH 2 Theory 2 Theory

1.5 1.5

1 1

0.5 0.5

0 0

−0.5 −0.5

−1 −1 Nondimensional distance Y Nondimensional distance Y −1.5 −1.5

−2 −2

−2.5 −2.5 0 1 2 3 4 5 0 1 2 3 4 5 Nondimensional distance X Nondimensional distance X

Figure 4.14: Two-dimensional jet ﬂow impinging on a ﬁxed wall with inclined angle (a) β = 45o and (b) β = 60o. and dash lines are for the theoretical results. For all cases examined the agreement at the large discharge section (D1) is better than that at the small discharge section (D2). The maximum error in the large discharge section is only 1.1 percent, while that in the small discharge section is 6 percent.

D1/D0 1 D2/D0

0.8

0.6

0.4 Normalized discharge 0.2

0 10 20 30 40 50 60 70 80 90 Inclined angle β (o)

Figure 4.15: Normalized discharges of the outﬂow sections as a function of β. The symbols denote SPH results and the lines show the theoretical solution.

4.3.4 Parametric study

There are several key factors in SPH, such as the kernel function, smoothing length, speed of sound (or Mach number) and artificial viscosity, that may affect the numerical results for free-surface flows. The smoothing length and kernel (gradient) have been discussed in Sections 3.4.10. They are also systematically in- 74 Selected Test Problems vestigated in the literature (see e.g. [64,151,159,177]). Here we focus on the other two factors. From (3.63), it can be seen that the artificial speed of sound plays an important role in SPH for free-surface flows. It is commonly determined from the requirement on the Mach number Ma := Vmax/c0 ∼ 0.1 (see Section 3.4.5). The artificial viscosity term Πab in Eq. (3.57) is originally added to the momentum equation to suppress possible numerical oscillations at sharp wave fronts in gas dynamics [151]. For free surface flows, the term involving α introduces both shear and bulk viscosity into incompressible flows. With negligible changes in the density, the viscosity is almost entirely shear viscosity [152]. In this section we investigate the effect of these two factors on the SPH results, and the β = 90o case with d0 = 0.0025 m is taken as the test example.

Mach number. Five different values of the Mach number, Ma = 0.5, 0.2, 0.05, 0.02 and 0.01 are tested. The corresponding values of the speed of sound are c = 2, 5, 20, 50 and 100 m/s respectively, because Vmax = V0 = 1 m/s. For the value of Ma = 0.5 (c = 2 m/s), particles penetrate through the wall at the early stage of the simulation. This is because with a low speed sound, the pressure force from mirror particles is not sufficient to stop the fluid particles moving towards the wall during the initial unsteadiness. The results for the remaining four values of Mach number are shown in Fig. 4.16, where only the particles at the free-surface are plotted. The previous solution for c = 10 m/s (Ma = 0.1) is also presented for the sake of comparison. The number of fluid particles varied from to 4130 (Ma = 0.2) to 4600 (Ma = 0.01). It is seen that the results for Mach numbers in the range [0.05, 0.2] (c corresponds to the range [5, 25] m/s) are close to each other. Closer examination indicates that the calculation with c = 5 m/s (Ma = 0.2) gives slightly better relation fit to the theory [174]. For Mach numbers smaller than 0.02 (c0 > 50 m/s), the SPH solution departs from the theory and the free surface becomes wavy.

2.5 Ma = 0.2 Ma = 0.1 Ma = 0.05 2 Ma = 0.02 Ma = 0.01

1.5

1 Nondimensional distance Y 0.5

0 0 0.5 1 1.5 2 2.5 Nondimensional distance X

Figure 4.16: The effect of the Mach number on the free surface proﬁle in the 2D jet impinging onto an orthogonal plate.

Recently, Gomez-Gesteira´ et al. [71] concluded that a real speed of sound can be 4.3 Impinging jet 75 used in SPH, but a much lower c is practically employed to allow for a larger time step as limited by the CFL condition. To verify the effectiveness of this conclusion, a value of c0 = 1000 m/s (Ma = 0.001), which is of the order of the real speed of sound, was tested. The particles become highly disordered and the free surface proﬁle is largely different from the theory. Consequently, a real speed of sound cannot be used in this case with the inlet velocity V0 = 1 m/s, because the Mach number (Ma = 0.001) has a large departure from the above range of Ma. It is not just because of the time restriction as argued by Gomez-Gesteira´ et al. [71]. Similarly, when a speed of sound c = 1000 m/s was used in [222], a ’weird’ phenomenon referred to as ’crystallized state’ was observed, which is one of the spurious transport simulations in SPH [130]. This is because in that case the maximum velocity is about 2 m/s, and hence the Mach number is 0.002. Naturally, the following question is asked: If the Mach number requirement is satis- ﬁed, does SPH allow for a high speed of sound? To answer it, numerical experiments with c = 1000 m/s and different jet velocities V0 are carried out. Three different values, namely, V0 = 50, 100 and 200 m/s are tested, which are in the range of Ma ∈ [0.05, 0.2], and the results are shown in Fig. 4.17. All the results are in good agreement with the theory and a closer examination demonstrates that the SPH solution with Ma = 0.2 agrees slightly better with the theory.

2.5 V = 50 0 V = 100 0 V = 200 2 0

1.5

1 Nondimensional distance Y 0.5

0 0 0.5 1 1.5 2 2.5 Nondimensional distance X

Figure 4.17: SPH solutions for the problem of 2D jet impinging onto an orthogonal plate with a real speed of sound c = 1000 m/s. The number of ﬂuid particles is about 4130 for all three cases.

Based on the numerical experiments, it is concluded that the choice of a speed of sound is limited by a practical time step, but more importantly it should satisfy the restriction on the Mach number Ma = 0.05 ∼ 0.2.

Coefficient of artificial viscosity. The approach used for the enforcement of a free-slip boundary condition affects the choice of the coefficient α in the Πab term (see Eq. (3.57)). In the wall particle approach, a typical value of α is 0.01 (see e.g. [152,191]). In the mirror particle approach, different values of α from 0.1 to 0.3 have been used (see e.g. [5,146,156]). As discussed in Section 3.4.8, the mir- 76 Selected Test Problems ror particle approach has advantages over the wall particle approach, and it has become a standard method to treat free-slip boundaries. To find a proper choice of α in the mirror particle approach, numerical experiments with different α values were performed. It was found that in the range of α ∈ [0.05, 0.2], the SPH solutions have good agreement with the theory and no significant difference between SPH solutions is observed. Hence the typical α value in the mirror particle approach is one order of magnitude higher than that in wall particle approach. This may be attributed to that wall particles not only create a repulsive force, but also induce a shear force. In contrast, mirror particles can model the free-slip boundary very well without creating artificial shear.

4.3.5 Summary

In this section, the problem of a jet impinging onto a fixed plane at various inclination angles was studied using the Lagrangian particle SPH method. Through the numerical experiments, the following conclusions were obtained: 1) the calculated free surface profile showed a good agreement with the theoretical prediction; 2) the velocity distribution and the pressure distribution are consistent with the theoretical results; 3) the SPH solution is not sensitive to the Mach number (speed of sound) and artificial viscosity coefficient α as long as they are within predefined ranges.

4.4 Jet ﬂow under gravity

Steady state jets emerging from a nozzle or channel and falling under gravity are often modelled by potential ﬂow theory. For a channel with a given inclination angle, the shape√ of each jet depends upon a single parameter, the Froude number Fr = V0/ gH, which ranges from zero to inﬁnity. Here V0 is the emerging velocity of the jet, g is the acceleration of gravity and H is the channel height. For Fr → ∞ the jet is slender and of parabolic shape. It becomes thicker as Fr decreases, and reaches a limiting form at Fr ≈ 0.

A brief literature review is presented here. Based on potential flow theory, several studies were on thin waterfalls (see e.g. [38, 40, 196]) and slender jets (see e.g. [67, 214, 216, 218]). Based on the assumption that one wall of the nozzle is infinitely long, a rising stream from an inclined nozzle which falls down as a single jet was investigated in [51]. Two limit cases are obtained when the nozzle is horizontal [70] and vertical [218]. For the horizontal case, a limit configuration is the free overfall problem (see the review [49]). An extension of the jet from a uniform nozzle is the flow of a stream emerging from a simple orifice in a vertical wall and falling under gravity (see e.g. [143, 171, 215]) and streams emerging from a two-dimensional curved nozzle [213]. A more complicated case that a stream 4.4 Jet flow under gravity 77 emerging from a vertical nozzle, hitting a horizontal plate of finite extent and then falling under gravity was recently studied in [37]. The methods used in these papers can be summarized as follows. From the potential flow theory, with the use of conformal mapping or hodograph transformations, an integral-differential equation for the free surface(s) of the jet is formulated. The resulting equation is then discretised via partition of the integration interval [70,196,216] or truncated power series [37, 51, 218] to yield a finite set of nonlinear equations which are solved numerically by Newton’s method.

The test case considered here is a two-dimensional thick jet emerging from a horizontal channel and falling under gravity. Two horizontal channel walls allowing for different offsets are used to model the effect of the nozzle geometry, a limiting case of which is the free overfall problem. Numerical SPH solutions are compared with published results from potential ﬂow theory, experiments and alternative numerical researches. The main objective is to validate the SPH code for analysis of hydrodynamic problems involving two free surfaces and gravity. This problem is closely related to the pipe ﬁlling and emptying problem studied in Chapter 5.

4.4.1 Test problem and SPH setup

The flow of interest sketched in a Cartesian coordinate system is shown in Fig. 4.18. Upstream there is a uniform flow in a horizontal channel of height H. The channel ends at x = 0 after which the stream of water falls freely under gravity. The lower and upper walls of the channel may end at different values of x. Let the upper and lower walls end at (l, H) and (0, 0) respectively, where l is the amount of overhang relative to the lower wall, which may be positive, negative (see Fig. 4.18), or zero. It is√ convenient to parameterize the steady state jet by the Froude number Fr = V0/ gH. One needs to find the free-surface profiles for given Fr and the dimensionless length L = l/H.

Figure 4.18: Deﬁnition sketch of the stream emerging from a horizontal slot and falling under gravity.

In the numerical simulations the computational domain is taken as (−0.1, 0.4) × (−0.1, 0.1) m2. The inlet section is between the points (−0.1, 0) and (0.1, 0.1). De- pending on Fr, the flow leaves the computational domain from the bottom side and/or right side (see Figs. 4.19 – 4.21), where the outlet section is defined. The length of the lower channel wall is 0.1 m, and the channel width is 0.1 m. The channel is initially filled with uniformly distributed fluid particles having veloc- 78 Selected Test Problems

ity V0 and zero pressure. Ahead of the inlet section, there are speciﬁed layers of inlet particles. For inlet particles, the uniform horizontal velocity is determined by the given Froude number, and the vertical velocity is zero. The pressure is assumed to be hydrostatic and the density is calculated from the equation of state (3.63). Mirror particles are used for the free-slip boundary condition at the channel walls. The initial particle distance is d0 = 2.5 mm. The speed of sound c depends on the given Froude number and satisﬁes the condition on Mach number. The minimum time step is ∆tmin = 0.2d0/c. The simulation time to reach a steady-state depends on the Froude number. The criterion to stop the simulation is that the change of the total kinematic energy is less than a given tolerance.

We simulate this two-dimensional inviscid ﬂow problem from a given initial state until a steady state is reached. For various values of Fr and l, the calculated free surface proﬁles are compared with the predictions from literature. The geometry is scaled by the channel height W: X = x/H, Y = y/H and L = l/H.

4.4.2 Numerical results

For a given dimensionless length L, the minimum Froude number at which the ﬂow detaches from the downstream end of the upper wall is designated as Frmin. That is to say, for a ﬂow with Fr < Frmin, the detachment will occur earlier.

Emerging jet. The jets for L = −0.5 (underhang) and various Fr are shown in Fig. 4.19 with theoretical results from literature. For this case Frmin = 0.517 [70]. As the Froude number decreases to Frmin, the simulated jet falls more rapidly and the curvature of the free surfaces increases. For large Froude number Fr = 3.162 (Fig. 4.19d), the numerical solutions of both upper and lower surfaces agree very well with potential flow theory. For the medium values Fr = 1.414 (Fig. 4.19c) and 1 (Fig. 4.19b), the lower free-surface profile matches the theory, but the upper one is different from the theoretical prediction. For Frmin = 0.517 (Fig. 4.19a), both lower and upper profiles are different from the theory, although the SPH flow nicely detaches from the separation point.

The discrepancy may be attributed to two possible reasons. The first one is the assumption for the inlet pressure distribution. Since the pressure at the inlet is not known, we assumed that it is hydrostatic with atmospheric pressure at the top wall. For high Froude number, the flow is dominated by inertia, and SPH works well. With the decrease of Fr, hydrostatic pressure and gravity play a more important role. An improper setup of the pressure distribution at the inlet section will result in inaccurate results. The second possible reason is the selected geometry of the computational domain. It determines the location of the inlet section (theoretically it is assumed to be infinitely far [70]) and outlet section.

When L = 0, the channel is a slot and the results are shown in Fig. 4.20. For the overhang case L = 0.5, the results are shown in Fig. 4.21. Again the SPH results 4.4 Jet ﬂow under gravity 79

(a) (b)

1 1

0.5 0.5 Y 0 Y 0

−0.5 −0.5

−1 −1 −1 0 1 2 3 4 −1 0 1 2 3 4 X X

1 1

0.5 0.5

Y 0 Y 0

−0.5 −0.5

−1 −1 −1 0 1 2 3 4 −1 0 1 2 3 4 X X

Figure 4.19: Underhang free-surface proﬁles for various Froude numbers: (a) Frmin = 0.517, (b) Fr = 1, (c) Fr = 1.414, (d) Fr = 3.162. Open circles – [70], dots – SPH.

have good agreement with the theory for large Froude numbers, but show big discrepancies for low Froude numbers, especially for the upper free surface. As exhibited in Figs. 4.19a, 4.21a and 4.21b, the particle layers close to the upper separation point become sparse, because SPH slightly suffers from the tensile instability problem. This is consistent with the ﬁnding that tensile instability is not only restricted to solids, but may also occur in ﬂuids [155]. The pressure correction term as used in [155] for the collision of two solid rings may be applied to resolve the problem.

From the above twelve cases, it is concluded that good agreement is only found for Froude numbers larger than one. For small Froude numbers, the effect of gravity is signiﬁcantly underestimated in the theoretical solutions. Experimental validation is needed here.

Free overfall. If the dimensionless length L is smaller (negative) than a certain value, a special case – the free waterfall or overfall – will result, in which the upper wall has no effect on the resulting jet. This certain value depends on Froude number. In the numerical simulations of this problem, three different Froude numbers resulting in subcritical, critical and supercritical ﬂows are considered. There is an important hydraulic parameter in the overfall problem – the brink depth, deﬁned as the water depth at x = 0, which is often used for discharge estimation [15, 61]. 80 Selected Test Problems

(a) (b)

1 1

0.5 0.5

Y 0 Y 0

−0.5 −0.5

−1 −1 −1 0 1 2 3 4 −1 0 1 2 3 4 X X

1 1

0.5 0.5

Y 0 Y 0

−0.5 −0.5

−1 −1 −1 0 1 2 3 4 −1 0 1 2 3 4 X X

Figure 4.20: Slot free-surface proﬁles for various Froude numbers: (a) Fr = 1, (b) Fr = 1.414, (c) Fr = 2.5, (d) Fr = 4.472. Open circles – [70], dots – SPH.

(a) (b)

1 1

0.5 0.5

Y 0 Y 0

−0.5 −0.5

−1 −1 −1 0 1 2 3 4 −1 0 1 2 3 4 X X

1 1

0.5 0.5

Y 0 Y 0

−0.5 −0.5

−1 −1 −1 0 1 2 3 4 −1 0 1 2 3 4 X X

Figure 4.21: Overhang free-surface proﬁles for various Froude numbers: (a) Fr = 1.414, (b) Fr = 2, (c) Fr = 3.162, (d) Fr = 4. Open circles – [70], dots – SPH.

For the subcritical flow case, SPH is applied to the experimental setup of Ra- jaratnam [180]. The inflow depth and discharge are equal to 0.132 m and 0.143 m3s−1m−1. The Froude number is Fr = 0.953, which is close to critical. In Fig. 4.22, the computed free-surface profiles are compared to experimental [180], theoretical [100] and numerical [145] results. The present SPH results agree bet- 4.4 Jet flow under gravity 81 ter with the theoretical [100] and experimental [180] results than the numerical solution by MAC with VOF [145]. The brink depth is predicted very well with an error of only 2 percent.

0.8

0.6

0.4

0.2

-0.2

-0.4

NondimensionalY distance -0.6

-0.8

-1 -4 -3 -2 -1 0 1 Nondimensional distance X

Figure 4.22: Free surface profiles in a subcritical free overfall (Fr = 0.953). Dots – SPH, filled circles – [180], filled squares – [100], filled triangles – [145].

For the critical flow case, we consider the problem presented in [70, 196]. The inflow depth and discharge are equal to 0.1 m and 0.99 m3s−1m−1, respectively. The computed free-surface profiles are compared with the theory [70, 196] as shown in Fig. 4.23. The computed free-surface profiles close to the brink point (x = 0) match well with the theoretical results. The agreement becomes less far downstream, where the upper surface becomes slightly wavy in SPH. The possible reason is the setup of the outflow section. Although potential flow theory was used in both [70] and [196], different solutions were obtained due to different numerical methods. The SPH results have better overall agreement with the predictions in [196].

0.8

0.6

0.4

0.2

-0.2

-0.4

NondimensionalY distance -0.6

-0.8

-1 -4 -3 -2 -1 0 1 2 3 Nondimensional distance X

Figure 4.23: Free surface profiles in a critical free overfall (Fr = 1). Dots – SPH, filled circles – [70], filled squares – [196]. 82 Selected Test Problems

The theoretical solution for the free overfall with a supercritical ﬂow is taken from [226]. The Froude number is 2. As shown in Fig. 4.24, the agreement of the surface proﬁles between the numerical simulation and theory is good and this is consistent with the other results for Fr > 1.

0.8

0.6

0.4

0.2

-0.2

-0.4

NondimensionalY distance -0.6

-0.8

-1 -4 -3 -2 -1 0 1 2 3 4 Nondimensional distance X

Figure 4.24: Free surface proﬁles in a supercritical free overfall (Fr = 2). Dots – SPH, ﬁlled squares – [226].

4.5 Flow separation at bends

Flow separation occurring in piping systems has received much attention because it determines the energy losses and forces on components such as pipes, valves, tees, elbows and bends. This phenomenon is also significant to flow in open channels and around turbomachine blades. When a fluid passes a bend, it is likely to separate from the inner corner (see Fig. 4.25). The size of the separation void depends on the Reynolds number of the flow and the geometry of the bend. To model the flow separation problem, there are two possible ways to go. One is using potential flow theory and the other is to solve the full Navier-Stokes equations. Applying different separation models, potential flow theory has long been used to model fluids. Although the energy losses resulting from separation cannot be directly predicted by potential flow theory, a good estimate of the size of the separation region, the pressure gradient and velocity distributions can be obtained. The potential flow solution usually describes well high Reynolds flows. One can predict the energy dissipation resulting from flow separation through solving the Navier-Stokes equations. Such solutions can be quite difficult to find because the separation streamline is not known in advance. Potential flow theory applied to flow separation in 2D bends is briefly reviewed herein.

Using conformal mapping and Roshko’s free streamline theory [184], Lichtarowicz & Markland [115] solved the potential ﬂow round a right-angled elbow for two different ratios of the channel width Rb := s/b (see Fig. 4.25). In Roshko’s model, 4.5 Flow separation at bends 83

Figure 4.25: Deﬁnition sketch of a separated ﬂow in a right-angled bend.

separation is represented through a surface of discontinuity that separates the flow into two regions: 1) the main flow where the velocity is continuous and pos- sesses a potential, and 2) a secondary region extending theoretically to infinity and bounded by the surface of discontinuity EE′ (undisturbed flow). The veloc- ′ ity at the separation point (Vs) is greater than that of the flow after point E (Vd) (undisturbed flow) and remains constant along the separated free streamline CE, which curves gradually until its direction is restored to that of the undisturbed flow. The direction then remains constant, but the velocity decreases to that of the downstream value Vd. By mapping the physical plane to a hodograph domain, Mankbadi & Zaki [139] studied the flow pattern for symmetric and asymmetric bends with various turning angles. Different from [115], Kirchhoff’s free streamline theory has been used in [139]. In Kirchhoff’s theory, the velocity on the whole surface CE is assumed to remain constant and is equal to that of the downstream flow at infinity, i.e. Vs = Vd. For a right-angled elbow, analytical solution in [139] is different from that in [115]. To understand the discrepancy of these two theoretical solutions, Chu [39] recently studied separated flow in symmetric bends of arbitrary turning angles (see Fig. 4.26), using Kirchhoff’s free streamline theory and a similar numerical method. The results of Chu [39] are in good agreement with the solution in [115]. The same discrepancy was encountered between the solution in [39] and that in [139], and hence possible numerical errors may have happened in [139] as pointed out by Chu [39]. The hodograph transformation method used in [39, 139] is efficient for two-dimensional and axisymmetric problems [39, 78]. Its results should be a good prediction of the actual physical flow [79], if the fluid is at very high Reynolds numbers and the region adjacent to the free-streamline is gas. However, the solution procedure of the indirect hodograph method is rather restrictive [39], and it is extremely difficult to impose the boundary conditions as assumed in the theory [79]. 84 Selected Test Problems

Figure 4.26: Deﬁnition sketch of a separated ﬂow in a symmetric bend with turning angle β.

The flow separation problem is studied herein to investigate the fluid dynamics at bends including flow contraction and pressure distribution. Better understanding of this problem gives insight to slug hydrodynamic behaviour at the bend as studied in Chapter 6.

4.5.1 Test problem and SPH setup

There are two series of flow separation problems are tested. A right-angled elbow is considered first (see Fig. 4.25), where the chosen coordinate system is also shown. The inner walls B ′C and CD ′, and the outer walls BA and AD form the fixed boundaries. At point C the flow separates and is bounded downstream by the curved free streamline CE′. The width of upstream channel is fixed at b = 1 m, whilst the downstream channel width s is varying and related to b by the ratio ′ ′ of channel width Rb. The length of the inner walls B C and CD is 2 m. For outer walls, |AD| = 3 m and |BA| depends on the given Rb. The second series of tests are the flow separation at a symmetric bend (Rb = 1) with different turning angles (see Fig. 4.26). The length of B ′C and CD ′ and the channel width b are the same as the first case. For outer walls, their length depends on the given turning angle β. The computational domain is chosen to include the whole bend and hence changes with the given values of Rb and β. The fluid flows into the bend from the inlet section BB ′ and flows out from the outlet section DD ′. Boundary conditions including the free surface, free-slip walls, inlet, outlet and the formed corner at point A are treated using the methods described in Section 3.4.8.

Initially there are some uniformly distributed ﬂuid particles in the upstream bend and inlet particles in the inlet block with particle distance d0 = 0.05 m. The velocity is V∞ = Vx = 1 m/s and Vy = 0. The smoothing length h is taken as 4.5 Flow separation at bends 85

1.33d0. The reference density is taken as the initial density, which implies initial zero pressure. Since the contraction coefficient Cc := d/b is β dependent, the maximum velocity varies with Rd and β due to mass conservation. The speed of sound is chosen to fulfill the requirement of the Mach number. The time step for all cases considered herein is fixed at 0.0001, which is small enough to satisfy the stability conditions. When the kinematic energy of all particles in the computational domain arrives at a constant (relative difference between two time step is less than a given tolerance), the simulation is stopped and assumed to reach a steady state. The number of particles at ’steady state’ depends on the value of Rb and the turning angle β. Note that exact steady state does not exist in SPH. The total number of fluid particles in the computational domain at ’steady state’ may slightly change (0.1%). The following variables scaled by the upstream channel width are used: X = x/b and Y = y/b.

4.5.2 Numerical results

Symmetric right-angled bend. Simulations have been carried out for several cases with different ratio of the channel widths Rb and turning angles β. A right-angled bend with Rb = 1 was considered ﬁrst (see Fig. 4.25). The SPH solution is shown in Fig. 4.27, where the outer particle layer represents the free surface.

1.5

0.5

0 Nondimensional distance Y −0.5

−1

−2 −1.5 −1 −0.5 0 0.5 1 Nondimensional distance X

Figure 4.27: Particle paths and theoretical free-streamline in a right-angled bend. Dots – SPH particles, open squares – L & M [115], ﬁlled circles – Chu [39].

The computed free streamline matches the theoretical solutions very well. The agreement of the current results with the solution in [115] is slightly better than with that in [39]. Some error may be present in the results extracted from [39], in which the coordinate information is incomplete. The averaged outlet velocity at Y = 2 is Vd = 1.89 m/s, which is slightly smaller than the prediction (Vd = 1.90 m/s) of Chu [39]. 86 Selected Test Problems

The velocity distribution along wall AD is shown in Fig. 4.28, which has been scaled by Vd. The present prediction is consistent with the theoretical solution of [39]. The particle velocity along the free streamline CE ′ is constant at a value close to Vd. This is consistent with Kirchhoff’s free streamline theory.

0.9

0.8

0.7

0.6

0.5

0.4

0.3 Nondimensional velocity 0.2

0.1

0 −1 −0.5 0 0.5 1 1.5 Nondimensional distance Y

Figure 4.28: Velocity distribution along wall AD. Solid line – SPH, open circles – Chu [39].

2 The pressure coefficient is defined as Cp := p/(ρVd/2), in which p is the pressure along the outer wall AD. There are two ways to determine pressure p at steady state. One way is to use the calculated velocity distribution through the Bernoulli theorem. The other way is to directly interpolate from the pressure field. The distribution of Cp along AD determined using the first approach is shown in Fig. 4.29, where also presented are the results in [115, 139]. The predicted Cp by SPH has better agreement with the solutions of in [115] than those in [139]. This is consistent with the conclusion of Chu [39], that possible numerical errors may have happened in the method of Mankbadi & Zaki. The directly interpolated pressure distribution has less agreement with the theoretical solutions because of the noise in the pressure field (not shown).

Asymmetric right-angled bend. For the asymmetric case (Rb ≠ 1), several values of Rd are tested and four steady-state flow fields are displayed in Fig. 4.30. For the free-streamline profiles, there are no results available for comparison, but the computed results can be verified to some extent through the contraction coefficient Cc := d/b as shown in Table 4.1. Due to particle fluctuations, the flow width DE′ at the outflow section has a small variation in time. Consequently, the evaluation of Cc involves averaging over a certain time interval at steady - but fluctuating - state. The calculated contraction coefficients agree very well with the theoretical ones. The maximum relative error is not more than 1.5 percent. As shown in Table 4.1, the contraction coefficient Cc decreases as Rb increases. When Rb becomes so large that the separation point C has no effect on the downstream flow, the current setup will be half of the jet emerging from a channel and impinging on an orthogonal plane as shown in Fig. 4.11. Clearly, following the 4.5 Flow separation at bends 87

0.9

0.8

0.7

0.6

0.5 Cp

0.4

0.3

0.2

0.1

0 −1 −0.5 0 0.5 1 1.5 Nondimensional distance Y

Figure 4.29: Pressure coefﬁcient along wall AD. Solid line – SPH, open squares – L & M [115], open circles – Chu [39].

same definition, the contraction coefficient for that case is 1. That is, at Rb = 2.5 (the SPH setup of the jet impinging problem; see Fig. 4.11), Cc is equal to 1, which can also be found in e.g. [146, 174]. As a consequence, there will be a critical Rb which results in a minimum Cc. On the other hand, when Rb → 0, the contraction coefficient approaches another constant 0.611 [115].

Table 4.1: Values of contraction coefﬁcients Cc at Y = 2 for different ratios of channel width Rb.

Rb (Ratio) 0.5 0.6 0.7 0.8 0.9 1.0 1.2 2.5 L & M [115] 0.584 0.573 0.560 0.551 0.537 0.526 0.500 1.0† Present 0.59 0.58 0.56 0.55 0.53 0.53 0.50 0.99‡

† Theory and ‡ SPH solution of the jet impinging on an orthogonal wall.

Symmetric acute and obtuse bend. For the flow fields in bends with other than right turning angles, several typical results are shown in Fig. 4.31 together with the results from potential flow theory. For the first three cases β = 15o, β = 30o and β = 45o, the theoretical profiles of the free-streamlines are not given in [39, 139]. For the other three cases β = 60o, β = 120o and β = 150o, the numerical results agree very well with the theoretical solutions. The particle number in the computational domain increases with turning angles β, approximately from 1800 (β = 15o) to 2950 (β = 150o). To achieve a steady state, the simulation times varied from 4 seconds (β = 15o) to 5 seconds (β = 150o). That is, after about 45000 time-steps, the plotted final states are achieved. All the calculations were performed on a normal PC, and the computation time varied between 15 and 25 minutes for one complete case. By using more particle layers at the inlet section, the obtained agreement can even be improved at the expense of compu- 88 Selected Test Problems

(a) (b) 2 2

1.5 1.5

1 1

0.5 0.5

0 0 Nondimensional distance Y Nondimensional distance Y −0.5 −0.5

−1 −1 −2 −1.5 −1 −0.5 0 0.5 −2 −1.5 −1 −0.5 0 0.5 Nondimensional distance X Nondimensional distance X

1.5 1.5

1 1

0.5 0.5

0 0 Nondimensional distance Y Nondimensional distance Y −0.5 −0.5

−1 −1 −2 −1.5 −1 −0.5 0 0.5 −2 −1.5 −1 −0.5 0 0.5 1 Nondimensional distance X Nondimensional distance X

Figure 4.30: Flow fields in a right-angled bend with different ratio of bend width: (a) Rb=0.6, (b) Rb=0.8, (c) Rb=0.9 and (d) Rb=1.2. tational time. As shown in Fig. 4.31, with the increase of β, a larger portion of the flow is affected by the outer corner of the bend, and less particles stay in layers when rounding the bend. When the turning angle is larger than 90o, some fluid is trapped at the corner as shown in Figs. 4.31e and 4.31f. The coefficient of contraction for various turning angles is shown in Table 4.2. The numerical results are consistent with the theoretical predictions. The relative error is less than 1 percent.

Table 4.2: Values of contraction coefﬁcients Cc for different turning angles β.

β (o) 15 30 45 60 90 120 150 Chu [39] 0.893 0.792 0.701 0.625 0.528 0.467 0.434 Present 0.89 0.79 0.70 0.63 0.53 0.47 0.43

From Table 4.2 the estimated maximum velocity at steady state for the cases considered is Vmax = V∞/Cc < 2.5 m/s. Thus a speed of sound c = 25 m/s should 4.5 Flow separation at bends 89

(a) (b)

0.5 1 0.8

0.6

0.4 0 0.2

−0.2 −0.5 −0.4

Nondimensional distance Y Nondimensional distance Y −0.6

−0.8

−1 −1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Nondimensional distance X Nondimensional distance X

1.5

0.5 0.5

0 0

Nondimensional distance Y −0.5 Nondimensional distance Y −0.5

−1 −1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Nondimensional distance X Nondimensional distance X

(e) (f)

1.5 1.5

1 1

0.5 0.5

0 0 Nondimensional distance Y Nondimensional distance Y −0.5 −0.5

−1 −1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 −2 −1 0 1 2 3 Nondimensional distance X Nondimensional distance X

Figure 4.31: Flow fields in symmetric bends with various turning angles: (a) β = 15o, (b) β = 30o, (c) β = 45o, (d) β = 60o, (e) β = 120o and (f) β = 150o. Dots – SPH particles, filled circles – Chu [39]. guarantee the SPH requirement of low Mach number for all test cases. How- ever, when c = 25 m/s is used, during early unsteady stage, particles penetrate through the outer corner of bends with large turning angles (e.g. β = 120o and 150o). This is because the maximum velocity during the unsteady parts of the numerical simulation can be much higher than 2.5 m/s (see below). For a flow 90 Selected Test Problems starting with a vertical front (see Fig. 4.32a), a large portion of it turns downwards to the outer corner due to the constraining walls BA and AD (see Fig. 4.32b). Due to mass conservation, the flow velocity increases and may reach a high value before it arrives at the outer corner, e.g. a velocity of 9.5 m/s in the β = 150o case. Pressure forces exerted from mirror particles behind the wall are not high enough to fully stop the high velocity flow going to the corner, and some particles penetrate through the corner. This happens mainly during the early unsteady phase of the simulation, and disappears in steady state. To avoid early stage particle penetration, a possible way is to use a larger c, e.g. c = 95 m/s, to fulfill the requirement of the Mach number. However, when c = 95 m/s is used, the Mach number at steady state will not be in a correct range Ma = 0.05 ∼ 0.2 as given in Section 4.3.4. Hence time-dependent speed of sound has to be used and one more equation needs to be solved (see e.g. [5] for this new concept). This com- plicates the numerical simulation. In fact, there is a simple way to avoid early stage particle penetration without using a larger or time-dependent c than practically desired. The initial inlet flow is set up with a wedge front (the angle of which is larger than β) as shown in Fig. 4.32c. The maximum velocity during the unsteady state is now reduced to 1.4 m/s and the particle distribution is less disordered (see Fig. 4.32d). Although the ultimate free-streamline profiles at steady state show no significant change, the unsteady simulation becomes smoother, and the steady state is achieved earlier.

Another practical situation is the flow in branched channels as systematically studied in [79]. To demonstrate the capability of the present method to simulate flow separation in branched channels, the first example in [79] consisting of two inlets with identical velocity is examined here, and the results are shown in Fig. 4.33. It is seen that for both the free streamline and the contact line, the computed solutions are comparable with the analytical solutions. The differences are mainly due to the current coarse model, and can possibly be improved by using more particles.

4.6 Quasi-3D model

For typical 2D problems involving free surfaces have been investigated above. For viscous pipe flows, the dimension is three and this cannot be reduced to two because of the presence of gravity. As mentioned in Chapter 3, for flows in (networks of) slender structures such as pipes, channels and conduits, 2D modelling is time-consuming, not to mention 3D. However, for inviscid flows with negligible lateral interaction, the 3D pipe flow can be treated as a combination of 2D flows. This is explained as follows.

Since we are interested in the water front evolution during filling and emptying, let Fig. 4.34a sketch the situation of the flow at a certain moment. There is no flow interaction in z−direction, because the fluid is inviscid. To determine the evolu- 4.6 Quasi-3D model 91

(a) (b)

1.5 1.5

1 1

0.5 0.5

0 0 Nondimensional distance Y Nondimensional distance Y −0.5 −0.5

−1 −1 −2 −1 0 1 2 3 −2 −1 0 1 2 3 Nondimensional distance X Nondimensional distance X

1.5 1.5

1 1

0.5 0.5

0 0 Nondimensional distance Y Nondimensional distance Y −0.5 −0.5

−1 −1 −2 −1 0 1 2 3 −2 −1 0 1 2 3 Nondimensional distance X Nondimensional distance X

Figure 4.32: Sketch of possible particle penetration at the outer corner when improper c0 is used: (a) a vertical front (b) velocity distribution, and possible reme- diation, (c) a wedge front and (d) velocity distribution. tion of the 3D water front (see Fig. 4.34a), the pipe is sliced into N vertical planes. Then the ﬂow is represented by a set of 2D domains (channels), one of which is shown in Fig. 4.34b. After the 2D simulations with the same inlet information but with different channel widths Dj, (j = 1, ··· ,N) have been completed, the 3D water front can be constructed from the obtained 2D results. Although 2D simulations can be carried out as shown in this chapter, the length of the problem domain should be within a reasonable range relative to D for computational efﬁciency. Otherwise, 1D models should be used with losing of the water front deformation. 92 Selected Test Problems

2 SPH particle Free−streamline Contact−line 1.5

0.5

0 Nondimensional distance Y −0.5

−1 −2 −1 0 1 2 3 Nondimensional distance X

Figure 4.33: Flow field with free-streamline, contact-line and boundary walls in a branched channel. Dots – SPH particles, filled circles and squares – Hassenpflug [79].

Figure 4.34: Sketch of a moving water front in a circular pipe with the quasi-3D treatment. Chapter 5

Rapid Filling and Draining of Pipelines

This chapter reports the development and application of an SPH based simulation of rapid filling and draining of pipelines. The water-hammer equations with a moving boundary are used herein to model the filling and emptying processes, and the SPH method is employed to solve the governing equations. To assign various boundary conditions, the SPH pressure boundary concept proposed recently in literature is used and extended. Except for imposing boundary conditions, this concept more importantly ensures completeness of the kernels associated with particles close to the boundaries. As a consequence, the boundary deficiency problem encountered in conventional SPH is remedied. The employed particle method with the SPH pressure boundary concept aims to predict the unsteady flow during rapid pipe filling and emptying. It is validated against laboratory tests, rigid-column solutions and numerical results from literature. Results obtained with the present approach show better agreement with the test data than those from rigid-column theory and the elastic model solved by either the box scheme or the method of characteristics (MOC). It is concluded that SPH is a promising tool for the simulation of rapid filling and emptying of pipelines with undulating elevation profiles.

5.1 Introduction

Fast transient flow in piping systems is generally caused by rapid changes in flow conditions due to sudden valve operation, pump start up or shut down, power failure, etc. The phenomenon is generally called pressure surge or water hammer, and it may damage hydraulic machinery, piping and supports. If possible, 94 Rapid Filling and Draining of Pipelines it should be anticipated in the design process and prevented in practice. Fast transients may also occur in rapid pipe filling and emptying processes. Rapid pipe filling and emptying occur in various hydraulic applications, such as water- distribution networks, storm-water and sewage systems, fire-fighting systems, oil transport pipelines and pipeline cleaning. With respect to rapid filling of an empty pipeline, while the water column is driven by a high head, air is expelled by the advancing water column. If the generated air flow is not blocked by valves, the water column grows with little adverse pressure and attains a high velocity. For emptying of a pipeline initially filled with water, while the air is blown into the pipeline, water is expelled out of the system. If the driving air pressure is high and the resistance from pipe components is low, the water column shortens with high acceleration and a high velocity as a result. When the advancing column is suddenly stopped (fully or partially), severe pressure changes occur in the system [75, 141, 237]. Also, for a pipeline with a large undulating elevation profile, water column separation may occur at high elevation points. This changes the hydraulics significantly and may cause pressure surges more harmful than the initial water hammer when the separated columns rejoin [21, 116]. A reliable model that can predict the magnitude of the water column velocities, the possible occurrence of column separation and the induced overpressure in the system is highly desirable.

Pipe filling. For the 1D modelling of the rapid filling of pipelines, the rigid- column theory based on a set of ODEs is commonly used. The rigid-column filling model for pipes in series was formulated in [116]. The model describes the unsteady motion of a lengthening water column filling empty pipelines with an undulating elevation profile. The model was validated against laboratory tests, but only the early stage of the filling process matched. As shown in [6], by decoupling the momentum equation from the pressure head at the pipe segment junctions, the filling process can be more efficiently modelled. The rigid-column model with the decoupling technique was recently extended to represent a branched system with undulating pipe segments [183]. By coupling the rigid-column model with an entrapped air model, Cabrera et al. [27] addressed filling pipes initially with air entrapped between water columns. The rigid-column model gives good results as long as the flow remains axially uniform. When the water column is disturbed somewhere in the system, pressure oscillations along its length or even column separation may occur and the rigid-column model will fail. The elastic model based on a set of PDEs for unsteady flow in conduits [230] is capable of dealing with potential fast transients in rapid pipe filling. However, the elastic model with a moving boundary is difficult to solve using traditional mesh-based methods. A recent attempt is the fully implicit box or Preissmann finite-difference scheme in [136]. This method uses a fixed spatial grid and a flexible temporal grid, and hence the Courant number is time dependent. The obtained results gave acceptable agreement with the laboratory tests in [116]. However, a serious and unsolved numerical convergence problem occurred due to an uncontrol- lable large Courant number. To solve this problem, the method of characteristics (MOC) was applied [137]. Since both the spatial and temporal grid are fixed in the MOC, the Courant number is constant and an interpolation has to be used to 5.2 Mathematical modelling 95 deal with the increasing water-column length.

Pipe emptying. The pipe emptying problem did not receive much attention in literature because it often occurs together with other hydraulic phenomena. Dif- ferent from pipe filling, the air is under pressure and the interaction between air and water plays an important role in the draining process. In the case of pipeline emptying with compressed air supplied from the upstream end as studied in [105], the moving water column is pressurized and its tail becomes free surface flow, i.e. mixed flow situations are possible. Pipe emptying experiments for a large scale pipeline were recently performed by Laanearu et al. [104, 105] to study two-phase flow transitions. To explore how simple models can be used to explain the phenomena observed, a control volume approach was proposed by in [105]. The biggest disadvantage was that the coefficients in the model had to be calibrated from the specific experimental results by means of curve fitting.

Water hammer. For the simulation of severe transient flows in pipes, several methods are available. Before the digital computer era, graphical techniques were used. The accuracy of this method could be low, since friction was not properly taken into account. The solution procedure was very elaborate and is rarely used nowadays. For details on the graphical method, one is referred to the summary [134]. Nowadays, the most commonly used approach is the computerised MOC, which has been employed numerically to simulate transient flow in complex pipe systems since the 1960s. Later, water-hammer models were extended with associated phenomena such as gas release, column separation, unsteady friction, pipe wall viscoelasticity and fluid-structure interaction. Details on MOC and its practice can be found in [230] and the recent review papers [21, 68]. Other methods for the numerical solution of the transient flow equations include the explicit [28] and implicit [161,206] FDMs, the FEM [193,201], Godunov method [74,90], lattice- Boltzmann method (LBM) [35, 36] and an explicit central-difference scheme with TVD [225]. A direct finite-difference methodology was described in [188], but no results were presented. SPH is used – for the first time – herein.

5.2 Mathematical modelling

The pipe flows considered in this chapter are governed by Navier-Stokes equations as deduced in Chapter 2. The corresponding discrete SPH dynamic equations developed in Chapter 3 can be directly used. However, to efficiently simulate flows in slender structures like a pipeline, 1D models are mostly used. Two of them are established here: one is the elastic model and the other is the rigid- column model. 96 Rapid Filling and Draining of Pipelines

5.2.1 Pipe ﬁlling

Elastic model. Consider a pipeline system equipped with a reservoir, a valve and a number of pipes as sketched in Fig. 5.1. The downstream end of the system is open to air. Pipes with different slopes represent an undulating elevation proﬁle. For the i-th pipe, its length, diameter and inclination angle are Li, Di and θi. Here θi is from the horizontal taken in the clockwise manner. The centre line of the pipes is taken as the x axis with its origin at the inlet. The valve is located at a distance L0 from the inlet. After the valve is opened, the water will advance into the empty pipe due to the reservoir pressure.

Figure 5.1: Sketch of the ﬁlling of a pipeline with undulating elevation proﬁle.

Since the ﬂow is treated as one-dimensional, the continuity equation (2.25) reduces to Dρ ∂V = −ρ , (5.1) Dt ∂x where capital notation V has been used for the 1D velocity for clarity. Similarly, capital notation P will be used for the 1D pressure.

With γ = 1 and taking the total derivative on both sides of Eq. (2.27), we obtain DP Dρ = c2 . (5.2) Dt 0 Dt Substituting Eq. (5.2) into Eq. (5.1) yields the continuity equation in term of pressure, DP ∂V = −ρc2 . (5.3) Dt 0 ∂x The liquid compressibility is taken into account through the wave speed, while the density in (5.3) is taken constant. Assume that filled pipes remain full and a well-defined liquid front exists. Then the momentum equation (2.26) with Darcy- Weisbach friction becomes DV 1 ∂P fV|V| = − + g sin θ − , (5.4) Dt ρ ∂x 2D 5.2 Mathematical modelling 97 where g is the gravitational acceleration, f the friction factor and D the pipe diameter. Return flow may occur, so V|V| instead of V2 is used in (5.4). Equations (5.3) and (5.4) are the classical water-hammer equations. A direct deduction of this 1D first-order hyperbolic system can be found for instance in [230].

With regard to the friction factor f, the Darcy-Weisbach friction law developed for steady turbulent flow is used. Although the validity of the traditional assumption of using steady friction has been challenged, it should not be an issue for the inertia driven problems studied herein. Different from [116], where a constant friction factor f was used, the friction factor used herein depends on Reynolds number Re, which changes with the flow velocity. Its instantaneous value is calculated from   1 (fake value), Re = 0,  64/Re, 0 < Re ≤ 2500, f = [ ( )] (5.5)  −2  ε/D 5.74  0.25 log + , Re > 2500, 3.7 Re0.9 where ε/D is the relative roughness and Re := VD/ν with ν the kinematic viscosity. For turbulent flow (Re > 2500), the above expression is known as the Swamee– Jain formula. The relative pipe roughness can be directly measured, and it is often provided by the manufacture. It may also be found from handbooks. The kinematic viscosity ν = 10−6 m2/s is used herein for water.

Assume that the reference pressure P0 in (2.27) is the atmospheric pressure. Then the initial conditions are

V(x, 0) = 0, P(x, 0) = ρgHR + x sin θ1, (0 ≤ x ≤ L0), (5.6) where HR is the reservoir head. Assume that: (1) air in the empty pipe can ﬂow out with negligible resistance, and consequently it has no effect on the motion of the water column; (2) the resistance of the open valve and the pipe connections is negligible. Then the upstream boundary condition is ( ) V2 V2 P(0, t) = ρg H − i − K i , (5.7) R 2g 2g

2 where Vi is the velocity at the inlet, Vi /(2g) is the velocity head and K the entrance head loss coefﬁcient. This condition is deduced by applying the Bernoulli equation at the point on the reservoir surface and the point after the vena contracta [199]. The downstream boundary condition is

P(L(t), t) = 0, (5.8) where L(t) is the length of the water column. Its change is taken into account by ∫ t L = L0 + Vfdt, (5.9) 0 in which Vf is the velocity of the water front. 98 Rapid Filling and Draining of Pipelines

Rigid-column model. Based on the developments in [6, 116], a mathematical model is established and implemented. This model includes the velocity head at the inlet and the time-dependent friction factor, items that were not included in the original developments. Four assumptions made in [116] are applied in the current model. These assumptions state that the pipe remains full during ﬁlling, that the pressure at the front is atmospheric, that the water-pipe system is incompressible (i.e. rigid water column) and that steady friction may be used. Deﬁne Hi as the head at the downstream end of pipe i. Suppose that the water front is travelling in the (i + 1)-th pipe as shown in Fig. 5.1. Applying Newton’s law of motion to the advancing water column in the (i + 1)-th pipe yields

2 l(t) dQ fi+1l(t) Q = Hi − 2 + l(t) sin θi+1, (5.10) gAi+1 dt Di+1 2gAi+1 where l(t) is the water column length in the partially filled pipe i + 1 being filled, A = πD2/4 is the pipe cross-sectional area and Q = VA is the uniform flow rate. Similarly, for the fully-filled j-th (j = 2, ··· , i) pipe, we have

2 Lj dQ fjLj Q ··· = Hj−1 − Hj − 2 + Lj sin θj, j = 2, , i, (5.11) gAj dt Dj 2gAj

Due to entrance head loss and velocity head at the inlet, the equation for the ﬁrst pipe is slightly different and reads ( ) 2 L1 dQ f1L1 Q = HR − H1 − K + 1 + 2 + L1 sin θ1. (5.12) gA1 dt D1 2gA1

By adding the Eqs. (5.10), (5.11) and (5.12), all interior heads Hj (j = 1, ··· , i) are cancelled, and one equation is obtained for the discharge Q of the water column     ∑i ∑i 2 1  Lj l(t)  dQ  fjLj fi+1l(t) K + 1 Q + = HR − 2 + 2 + 2 g Aj Ai+1 dt D A D A A 2g j=1 j=1 j j i+1 i+1 1 ∑i + Lj sin θj + l(t) sin θi+1. (5.13) j=1

The water column length in pipe segment i + 1, i.e. l(t), is determined by

∑i L(t) = Lj + l(t), (5.14) j=1 where L(t) is the total water-column length. Substituting (5.14) into (5.13) gives

1 dQ Q2 (C L(t) + C ) = H − (C L(t) + C + C ) + C L(t) + C , (5.15) g 1 2 dt R 3 4 7 2g 5 6 5.2 Mathematical modelling 99

where the coefﬁcients Ck (k = 1, ··· , 7) are deﬁned as 1 C1 = , Ai+1 ∑i ∑i Lj C2 = − C1 Lj, Aj j=1 j=1

fi+1 C3 = 2 , Di+1Ai+1 ∑i ∑i f L C = j j − C L , 4 D A2 3 j j=1 j j j=1

C5 = sin θi+1, ∑i ∑i C6 = Lj sin θj − C5 Lj, j=1 j=1 K + 1 C7 = 2 . A1

The friction factors fn(n = 1, ··· , i + 1) are calculated from Eq. (5.5) with initial or previous ﬂow discharge Q. The total water column length is determined by ∫ t Q L = L0 + dt. (5.16) 0 Ai+1

In Eqs. (5.15) and (5.16), i starts from 0. When L(t) becomes larger than L1, i becomes 1. When L(t) becomes larger than L1 + L2, i becomes to 2. So on and so forth.

The initial conditions are

Q(0) = 0 and L(0) = L0. (5.17)

With conditions (5.17), Eqs. (5.15) and (5.16) are applied between the supply reservoir and the pipe, in which the water-column front is located. After determining dQ/dt, the head at a junction (i.e. H1, ··· ,Hi) is determined by using the momentum equations (5.12) and (5.11) for each ﬁlled pipe.

5.2.2 Pipe emptying

Elastic model. For the emptying of a pipeline with an undulating proﬁle, water initially contained in the pipes is driven out by high pressure air. Consider a pipeline equipped with two valves and upstream an air tank as sketched in Fig. 5.2. Water is initially ﬁlled in the pipe between two valves. The upstream valve is open and the downstream valve is closed. This is an illustration of the pipe draining experiments studied in Chapter 7. 100 Rapid Filling and Draining of Pipelines

Figure 5.2: Sketch of the draining of a pipeline with undulating elevation proﬁle.

The elastic model developed in Section 5.2.1 for pipe filling can be directly used to model the pipe emptying problem, but boundary conditions need to be changed accordingly. The downstream valve is taken as the origin of the axial coordinate system. Then the moving boundary condition for the shortening water column after valve opening is ( ) P L(0) − L(t), t = Pair, (5.18) where Pair is the pressure of the air, L(0) is the initial water-column length, and L(t) is the column length at time t, which is calculated by ∫ t L = L(0) − Vfdt, (5.19) 0 where Vf is the air front (air-water interface) velocity. Air flow is not modelled herein, in the sense that the upstream driving air pressure Pair is a given function of time, which may be constant. In this chapter it is taken as a constant. In Chapter 7, it is the time-dependent measured pressure. The boundary condition at the outlet is ρV2 P(0, t) = K o , (5.20) v 2 where Kv is the head loss coefficient accounting for valve resistance and Vo is the flow velocity at the valve (outlet). The initial conditions are the same as (5.6), except that all inclination angles need to be included.

Rigid-column model. Deﬁne Hi as the head at the downstream end of pipe i. Suppose that the (i+1)-th pipe is being emptied (see Fig. 5.2). Note that compared with Fig. 5.1, the numbering of pipes has been reversed. Applying Newton’s law of motion to the shortening water column in the (i + 1)-th pipe yields

2 l(t) dQ fi+1l(t) Q = Hair − Hi+1 − 2 + l(t) sin θi+1, (5.21) gAi+1 dt Di+1 2gAi+1 where l(t) is the water column length in pipe i + 1 and Hair = Pair/ρg is the head at the air-water interface. For pipe j (j = 1, ··· , i), which is still fully ﬁlled 5.2 Mathematical modelling 101 with water, we have

2 Lj dQ fjLj Q ··· = Hj+1 − Hj − 2 + Lj sin θj, j = 1, , i. (5.22) gAj dt Dj 2gAj

The head at the outlet is 2 2 KvQ KvV1 H1 = 2 = . (5.23) 2gA1 2g

By adding the Eqs. (5.21), (5.22) and (5.23), all interior heads, Hj (j = 2, ··· , i + 1) are cancelled, and one equation is obtained for the discharge Q of the water column     ∑i ∑i 2 1  Lj l(t)  dQ  fjLj fi+1l(t) Kv  Q + = Hu − 2 + 2 + 2 g Aj Ai+1 dt D A D A A 2g j=1 j=1 j j i+1 i+1 1 ∑i + Lj sin θj + l(t) sin θi+1. (5.24) j=1

Again, we have ∑i l(t) = L(t) − Lj, (5.25) j=1 where L(t) is the total water-column length. Substituting (5.25) into (5.24) gives

1 dQ Q2 (C L(t) + C ) = H − (C L(t) + C + C ) + C L(t) + C , (5.26) g 1 2 dt R 3 4 7 2g 5 6 where the constant coefﬁcients Ck, (k = 1, ··· , 6) are the same as in Eq. (5.15). The coefﬁcient C7 is Kv C7 = 2 . A1 The water column length is determined by ∫ t Q L(t) = L(0) − dt. (5.27) 0 Ai+1

In Eqs. (5.26) and (5.27), i starts from N − 1 where N is the total number of pipes. When L(t) becomes smaller than LN−1 + ··· + L2 + L1, i becomes N − 2. When L(t) becomes smaller than LN−2 + ··· + L2 + L1, i becomes N − 3, and so on. The initial conditions are ∑N Q(0) = 0 and L(0) = Lj. (5.28) j=1 102 Rapid Filling and Draining of Pipelines

5.3 Discrete SPH dynamic equations

5.3.1 SPH for 1D water hammer

When density variations are replaced by pressure variations and the remaining densities are taken constant, the SPH gradient I and II (see Section.3.3.2) are the same. For the 1D case, they degenerate to ∑ ∂ψa 1 dWab = mb(ψb − ψa) , (5.29) ∂x ρ dxa b where the function has been notated by ψ to distinguish it from the friction factor f. The derivative of the cubic spline kernel (3.40) is given by  2  − 2q + 1.5q , 0 6 q < 1, dW sign(x − x ) ab = a b 2 6 2  − 0.5(2 − q) , 1 q < 2, (5.30) dxa h  0, 2 6 q, in which q := rab/h with rab := |xa − xb| the distance between the particles. Replacing the spatial derivatives in Eqs. (5.3) and (5.4) with the derivative approximation (5.29), one obtains the SPH water-hammer equations ∑ DPa 2 dWab = c0 mb(Va − Vb) , (5.31) dt dxa b ∑ ( ) DVa 1 dWab faVa|Va| = − 2 mb Pa − Pb + Πab + g sin θa − , (5.32) dt ρ dxa 2Da b where the subscript a denotes ﬁeld quantities and pipe geometry information at position xa. The artiﬁcial viscosity term is ( ) −c h (V − V )(x − x ) Π = 0 a b a b ; 0 . ab min 2 2 (5.33) ρ rab + 0.01h

The rate of change of particle position is

dxa = Va. (5.34) dt The length of the water column is determined by the movement of the particle at the boundary (free surface for pipe ﬁlling and air-water interface for pipe emptying) as shown below. The forward Euler method is used for time marching.

The Eqs. (5.31) and (5.32) are the SPH discretisation of the water-hammer equations. This system of equations will be applied to the pipe ﬁlling and draining problems in this chapter and Chapter 7, and to the slug ﬂow in Chapter 6. 5.3 Discrete SPH dynamic equations 103

5.3.2 Treatment of the boundaries

The enforcement of the boundary conditions (5.7), (5.18) and (5.20) has not been described in Chapter 3. The SPH pressure boundary concept proposed in [103] is employed to impose these conditions. Note again that the essential ingredient is to properly complete the truncated kernel supports using image particles.

The enforcement of the boundary condition (5.7) is sketched in Fig. 5.3a. Assume that at time t ﬂuid particle ”i” (inlet) is the one closest to the reservoir and that its velocity is Vi (taken from previous time step or initial value). To complete the kernel support associated with particle ”i”, a set of particles with spacing d0 (d0 is the initial ﬂuid particle spacing) is placed inside the reservoir. Their velocity is set equal to Vi and their pressure is ( ) V2 V2 P = ρg H − i − K i . (5.35) i R 2g 2g

The enforcement of the boundary condition (5.20) is similar to that of condition (5.7) and is sketched in Fig. 5.3b. Suppose that at time t ﬂuid particle ”o” (outlet) is the one closest to the downstream end and that its velocity is Vo (taken from previous time step or initial value). To complete the kernel support associated with particle ”o”, a set of particles with spacing d0 is placed outside the pipe end. Their velocity is taken as Vo and their pressure is

ρV2 P = K o . (5.36) o v 2 The moving air-water front condition (5.18) is imposed in a similar way. The illustration is shown in Fig. 5.3c. Suppose that at time t ﬂuid particle ”f” (front) represents the moving air-water interface and its velocity is Vf (taken from previous time step or initial value). To complete the kernel associated with particle ”f”, a set of particles with spacing d0 is placed upstream of particle ”f”. Their velocity is Vf and their pressure is Pf = Pair. (5.37) In pipe ﬁlling, the moving water front (5.18) can be regarded as the 1D case of the free surface (zero pressure) boundary condition. As discussed in Section 3.4.8, the free surface boundary condition does not need to be explicitly imposed, although the kernel associated with the particle at the water front is not complete. How- ever, if we can properly complete the truncated kernel supports, the boundary condition at the water-air interface will be imposed more accurately. More importantly, when the water front reaches an obstacle, e.g. a partially or fully closed valve (water-hammer generation), the enforcement of the free surface boundary condition has to be adapted to the no-slip condition as discussed later in this chapter. Suppose that at time t particle ”f” (front) is located at the water front and that its velocity is Vf (see Fig. 5.3d). To complete the truncated kernel, a set of particles is placed downstream of particle ”f”. The velocity of these particles is taken as Vf, and their pressure is zero. 104 Rapid Filling and Draining of Pipelines

Figure 5.3: Illustration of pressure particles for (a) reservoir inlet, (b) valve outlet, (c) water tail and (d) water front.

All the introduced image particles are referred to as pressure particles as they are used to impose pressure conditions. The number of pressure particles for each boundary, Npp, depends on the smoothing length h, because the radius of the kernel is 2h. To meet the requirement that the kernel associated with any a fluid particle needs to be fully supported, an integer Npp > 2h/d0 must be taken. Prac- tically, Npp = 4 is used, which satisfies the above requirement for all 1D cases considered in this thesis. When a pressure particle enters the computational domain (herein the pipeline), it becomes a fluid particle and a new pressure particle is generated the most upstream point. On the other hand, when a fluid particle leaves the computational domain, it becomes an outlet particle and the most downstream particle is deleted.

5.4 Numerical results

5.4.1 Pipe ﬁlling

The developed SPH algorithm for water hammer (elastic column) is first applied to the pipe filling laboratory experiments carried out in [116]. Their test apparatus is shown in Fig. 5.4. It consists of a constant-head water supply tank, a quarter- turn ball valve and two PVC pipes. The length of the pipes are L1 = 3.55 m and L2 = 3.11 m. They have the same diameter of 22.9 mm. Pipe 1 penetrates 195 mm into the water tank from a side wall near the tank bottom. Its inclination o angle is θ1 = 2.66 downward. The downstream end of pipe 2 is open to the o atmosphere. Its inclination angle is θ2 = 2.25 downward. The total gravity head is L1 sin θ1 + L2 sin θ2 = 287 mm. The valve is fitted in pipe 1 at distance 438.2 mm from the inlet. The constant inlet submergence (from the tank water surface 5.4 Numerical results 105

to the inlet centreline) is HR = 354 mm, and the length of the initial hydrostatic water column is L0 = 427 mm (from inlet to the valve). From steady-state tests, the hydraulic grade lines were established by eleven manometers. The average value of the friction factor f was 0.0247, the entrance-loss coefﬁcient K was 0.8 and the average of the relative roughness (estimated from f) was ε/D = 0.00015.

Figure 5.4: Sketch of the test rig in Liou & Hunt’s laboratory experiments [116].

Velocity and pressure. The initial particle spacing is d0 = ∆x = 10 mm, the speed of sound is c0 = 1000 m/s and the time step is ∆t = ∆x/c0. The simulation is terminated when the ﬁrst ﬂuid particle arrives at the downstream end. There are about 660 particles when the pipe is full at t = 4.6 s. Figure 5.5 depicts the predicted velocities against water column length (measured from the inlet). For comparison, Fig. 5.5 also shows the measurement of Liou & Hunt [116], their rigid-column results and the solution of the elastic model (water hammer equations with a moving boundary) solved by the box scheme [136] and MOC [137].

0 2 10D 20D Rigid−column model

1.5

SPH

1 Laboratory tests (20 data sets) Velocity (m/s)

Box scheme 0.5

MOC

0 0 1 2 3 4 5 6 7 Column length (m)

Figure 5.5: Velocity vs. column length in ﬁlling for laboratory tests [116], rigid- column model [116], and elastic model solved by the box scheme (open squares) [136], MOC (open circles) [137] and the present SPH method (solid line) [83].

The variations in the measured average velocities are shown by the horizontal error bars indicating one standard deviation above and below the mean values. The positions of the error bars indicate the locations of timing sections. Among the results from the different models and methods, the SPH solution agrees the best 106 Rapid Filling and Draining of Pipelines with the measurement, although the measured maximum velocity is not fully reached. The MOC solution [137] is very similar to the present calculation. The slight difference is attributed to a constant friction coefﬁcient used in the MOC solution [137], which becomes more clear as shown later when the friction formulae is examined. The solution by the box scheme [136] matches the late phase of the ﬁlling process well, but under-predicts the velocity in the early phase.

The pressure distribution along the advancing water column at four time levels is shown in ﬁg. 5.6. At a given time, the pressure is linearly distributed indicating a constant pressure gradient. This means that the water column moves as a rigid column and the water elasticity does not play a role (close look shows small oscillations). There is an inﬂexion point at x = 3.55 m due to the inclination angle change, but it is not evident as the slop change is very small. With the lengthening of the water column, the pressure gradient becomes smaller. The pressure difference at x = 0 is due to the velocity head and entrance head loss in Eq. (5.7). Without these two terms, the four curves will joint at one point at x = 0.

2200 t = 1.0 s 2000 t = 2.0 s 1800 t = 3.0 s t = 4.0 s 1600

1400

1200

1000

800 Pressure (Pa) 600

400

200

0 0 1 2 3 4 5 6 Column length (m)

Figure 5.6: Pressure distribution along the water column in pipe ﬁlling at four time levels.

Velocity head. The rigid-column model (labelled 0D in Fig. 5.5) largely over- estimates the velocities. Liou & Hunt explained the difference by arguing that, during the filling process, a local acceleration of reservoir water approaching the pipe inlet decreases the available head driving the flow in the pipe downstream of the valve. Since it was not possible to measure the three-dimensional velocity field in the reservoir, a virtual pipe segment having the same diameter as the pipe was assumed to exist ahead of the pipe inlet. The labels 10D and 20D in Fig. 5.5 indicate the length of a virtual pipe segment ahead of the inlet [116]. After this consideration, the rigid-column solution still has big discrepancy with the laboratory measurements (see Fig. 5.5). As explained in [137], the argument of Liou & Hunt [116] seems suspect since the particle velocity in the reservoir does not seem to be significantly large except in the inlet itself. The associated acceleration head cannot absorb much driving head. Instead, without adding a virtual pipe, a better solution is obtained by including the velocity head in [116]. To test this 5.4 Numerical results 107 idea, SPH results obtained without the velocity head term in (5.7) are presented in Fig. 5.7, where the corresponding MOC solution of [137] is also shown.

1.8

1.6

1.4

1.2

0.8 Velocity (m/s) 0.6

0.4

0.2

0 0 1 2 3 4 5 6 7 Column length (m)

Figure 5.7: Velocity vs. column length in pipe ﬁlling for rigid-column model of Liou & Hunt (open squares) [116] and the elastic model solved by MOC (open circles) [137] and SPH (solid line) without velocity head. The dashed line is the present rigid-column model.

The close similarity between the elastic model and the rigid-column model demonstrates that the main culprit for the discrepancy in Fig. 5.5 must be the missing velocity head. Theoretically, the rigid-column model should not be different from the elastic model, because water elasticity does not play a role in this case of pipe ﬁlling. The water column has a much greater tendency to occupy the empty pipe than to get compressed. Some error might exist in Liou & Hunt’s rigid-column implementation. To validate this assertion, the rigid-column model developed is Section 5.2.1 is tested with f = 0.0247 and the solution is presented in Fig. 5.7 (dashed line). The good agreement between the present rigid-column model and the elastic model (solved by both SPH and MOC) signiﬁes that there is indeed some error in Liou & Hunt’s rigid-column implementation.

Friction formulae. The variation of f is shown in Fig. 5.8. It first decreases rapidly with time and then gradually increases (see Fig. 5.8a). Most of the time it is between 0.022 and 0.025. The transition from laminar flow to turbulent flow takes place at the very beginning (see Fig. 5.8b). For starting laminar flow, the friction factor is often multiplied by an amplification coefficient 4/3. However, the effect of the amplification coefficient is insignificant, as the laminar flow occurs only at the very beginning.

To check the validity of using a constant friction factor for unsteady ﬂow, the SPH results with f = 0.0247 are shown in Fig. 5.9 and compared with the solution using the instantaneous factor given by (5.5). The MOC solution with constant friction factor [137] is also presented. The difference between the SPH results with different friction formulae is insigniﬁcant, because friction itself is unimportant in 108 Rapid Filling and Draining of Pipelines

(a) (b) 0.05 0.16

0.14 0.045

0.12 0.04

0.1 0.035 0.08 Friction factor Friction factor 0.03 0.06

0.025 0.04

0.02 0.02 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.02 0.04 0.06 0.08 0.1 Time (s) Time (s)

Figure 5.8: History of the friction factor in pipe ﬁlling: (a) whole process and (b) early stage. the early stage where the changes in f are the largest, but where inertia dominates. The SPH solution with constant friction factor agrees very well with the MOC solution [137]. Hereafter, the interpreted results are obtained with the constant friction factor f = 0.0247.

1.8

1.6

1.4

1.2

0.8 Velocity (m/s) 0.6

0.4

0.2

0 0 1 2 3 4 5 6 7 Column length (m)

Figure 5.9: Velocity vs. column length in pipe ﬁlling for SPH solutions with instantaneous (dashed line) and constant (solid line) friction factor, and the MOC solution (open circles) in [137].

Head transition. The understanding of the dynamics of the ﬂow seems to be a challenge, because the contribution of the driving and resistive forces is different in the different stages of the ﬁlling process [137]. To examine how these forces come into action, consider the friction head loss

L(t) V(t)2 h (t) = f , (5.38) f D 2g 5.4 Numerical results 109 and the acceleration head L(t)a(t) h = , (5.39) a g where V and a are the averaged (among all ﬂuid particles) velocity and acceleration. The driving pressure head obtained from (5.7) is

V2 h = H − (1 + K) , (5.40) d R 2g

Together with the gravity potential (elevation), the contribution of the acceleration head, the friction head loss and the driving pressure head are shown in Fig. 5.10. The head in the system is conserved at any moment, i.e. ha = hd + hg − hf. Quick head changes occur in the early stage of the ﬁlling. When ha = 0, the driving head reaches its minimum value and the water column obtains its maximum velocity. Increasing friction causes a nearly constant deceleration.

0.6 Friction head loss Acceleration head 0.5 Driving head Gravity head 0.4

0.3

0.2

0.1 Acceleration head (m)

−0.1 0 1 2 3 4 5 6 7 Column length (m)

Figure 5.10: Contribution of the active heads during the ﬁlling process.

Maximum velocity. For a ﬁxed pipe length, the velocity history of the water column depends on the reservoir head (HR) and the initial water column length (L0). Apparently, the maximum velocity increases with the decrease of L0 and the increase of HR. Both of them are nonlinear relationships as shown in Fig. 5.11. The maximum velocity in the experiment is about 1.71 m/s [116]. The corresponding L0 in Fig. 5.11a is 0.2 m. It implies that the length of vena contracta at the inlet is 0.43 − 0.2 = 0.23 m, which is approximately ten times of the pipe diameter D. If this effect is included, the measured maximum velocity is Fig. 5.5 will be reached.

Numerical aspects. In the box scheme for the elastic model [136], the spatial step size ∆x is ﬁxed. The temporal step size ∆t is ﬂoating because the water-column front is assumed to advance one cell during each time step. Hence the Courant number Cr = c∆t/∆x = c/V may achieve very high values, which causes a stability problem. In addition, if transients are fast, numerical dissipation cannot remain within an acceptable range resulting in distortion of the solution [136]. To 110 Rapid Filling and Draining of Pipelines

(a) (b) 1.85 2.6

1.8 2.4

1.75 2.2

1.7 2 1.65 1.8 1.6

1.6 Maximum velocity (m/s) Maximum velocity (m/s) 1.55

1.5 1.4

0 0.2 0.4 0.6 0.8 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Initial column length L (m) Reservoir head H (m) 0 R

Figure 5.11: Effect of (a) L0 and (b) HR on the maximum velocity in the ﬁlling process.

overcome these problems, the MOC method with ﬁxed ∆x and ∆t was used [137]. Interpolation was used to track the moving boundary. In contrast to the box scheme and the MOC, the temporal grids are ﬁxed in the present SPH algorithm. The spatial ”grid” (particle distances) changes due to particle movement are negligible. Although the Courant number is not constant, its change is very small (about 10−6), because the particle distribution due to movement hardly becomes random, for which tremendous energy is needed [156]. This ensures the stability and accuracy of the present SPH algorithm.

In the above simulations, a realistic speed of sound c = 1000 m/s was used. Due to this, the time step was very small to fulfill the CFL condition, and hence the simulation was time-consuming. In the 2D simulations of free-surface flows presented in Chapter 4, to allow a large time step, the weakly compressible fluid was artificially made more compressible with a much lower value of c. For the filling problem considered herein, a free surface (treated as a moving boundary) also exists. Therefore, a small c can be used. To verify this, SPH simulations with different c values are presented in Fig. 5.12a. The CPU times used for the simulations are shown in Fig. 5.12b.

The difference between simulations with small Mach number (Ma = V/c < 0.1) is insigniﬁcant. The relative error between simulations with c = 1000 m/s and c = 25 m/s is less than 1%. For large Mach number, Ma > 0.2 (c = 5 m/s), oscillations occur when the water column reaches its maximum velocity. As expected, the numerical efﬁciency is dramatically enhanced when c is decreased. The CPU time is more or less a linear function of c. The speed of sound in the box scheme was not given [136], so it is not sure if the numerical divergence problem can be resolved when a smaller c0 is used. 5.4 Numerical results 111

(a) (b) 1.8 160

1.6 140

1.4 120 1.2 100 1 80 0.8

Velocity(m/s) c = 1000 60 0.6 0 CPU(min) time c = 100 0 40 0.4 c = 25 0 c = 10 0.2 0 20 c = 5 0 0 0 0 1 2 3 4 5 6 7 0 100 200 300 400 500 600 700 800 900 1000 Column length (m) Speed of sound (m/s)

Figure 5.12: Effect of the speed of sound on (a) velocity proﬁle and (b) CPU time.

5.4.2 Rigid column versus water hammer

Pipe filling is a slow process where the water elasticity has negligible effect on the fluid dynamics. This explains why both the rigid-column model and the elastic model provided good results for this problem. It is also the reason why an artificially low speed of sound can be used in the elastic model.

Pipe filling with valve impact. To test if SPH is capable of capturing water- hammer pressures, the SPH was applied to the reservoir-pipe-valve system in [137] (see Fig. 5.13). The system is initially empty and the downstream valve is partially closed with head loss coefficient Kv = 20. The entrance head loss coefficient is 0.8.

Figure 5.13: Reservoir-pipe-valve system with a partially closed valve.

The water-column velocity versus its length is shown in Fig. 5.14. The SPH solution agrees well with the MOC results in [137], except at the early phase of the filling process. This is because a fixed spatial cell size had to be used in MOC, while there is no such restriction in SPH. The effect of the slope change is more clear compared with Fig. 5.5. The velocity of the water column (averaged value) and its front are shown in Fig. 5.15. Before the system is full at t = 12 s, the water front velocity matches the water-column velocity very well. After the water column reaches the valve, waterhammer is generated because of the sudden change 112 Rapid Filling and Draining of Pipelines of velocity. The corresponding pressure surges at the inlet and partially closed valve are shown in Fig. 5.16. The driving pressure is negative during the filling process because of the pipe elevation change (siphon). Since it is still above the vapour pressure of water, column separation does not result. The water hammer effect is captured and quickly damped out. The fluid dynamics is mainly due to momentum transfer (see Eq. (5.36)) at the valve.

4 Velocity (m/s)

0 0 20 40 60 80 100 Column length (m)

Figure 5.14: Velocity vs. water column length in pipe ﬁlling for the elastic model solved by SPH (solid line) and MOC (open circle) [137].

4 Velocity (m/s)

0 0 5 10 15 20 Time (s)

Figure 5.15: Velocity history of the water column (averaged value) (solid line) and the water front (open circles) in pipe ﬁlling.

The rigid-column model was also tested and its solution coincides with the solid line in Fig. 5.15. Although the rigid-column model still captures the averaged 5.4 Numerical results 113

5 x 10 7 Upstream Downstream 6

Pressure (Pa) 2

−1 0 5 10 15 20 Time (s)

Figure 5.16: Pressure history of the flow at the inlet (driving pressure) and the downstream valve in pipe filling. behaviour of the flow, it is unable to predict acoustic pressure and velocity variations (see Fig. 5.16). This is more clear from the next example.

Water hammer. For the veriﬁcation of SPH for severe transients, the water-hammer case studied in [211] is considered. The sketch of the problem is shown in Fig. 5.17. The length of the horizontal steel pipe is L = 20 m, its diameter is D = 797 mm and the friction factor is f = 0.02. The constant reservoir head is HR = P0/ρg = 1 × 106/1000/9.8 = 102 m. The speed of sound is 1025 m/s. When the water column arrives at the downstream valve with a uniform velocity of 1 m/s, the valve is suddenly closed. In SPH, the initial particle distance d0 = ∆x is taken as 0.05 m and the smoothing length is h = 1.2∆x. The time increment is ∆t = 5 × 10−5 s and the simulation time is 0.3 seconds. The rigid-column model will not work for this case because the water elasticity is of high importance. The entrance head loss at the inlet is neglected with respect to HR.

Figure 5.17: Reservoir-pipe-valve system for severe transients (water hammer).

The SPH velocity history at the inlet is shown in Fig. 5.18, where solutions of MOC and corrective smoothed particle method (CSPM) are also depicted. The application of CSPM to the water-hammer problem has been systematically studied [84], where all kinds of SPH parameters were varied including the smoothing 114 Rapid Filling and Draining of Pipelines length h, the artiﬁcial viscosity coefﬁcient α, the kernel W, etc.

1.5

0.5

0 Velocity (m/s) −0.5

−1

−1.5 0 0.05 0.1 0.15 0.2 0.25 0.3 Time (s)

Figure 5.18: Velocity history of water hammer at inlet for solutions of SPH (open circles), CSPM (open squares) and MOC (solid line).

Both the SPH and CSPM results agree well with the MOC solution. The steep wave front can be reasonably reproduced. By using more particles, the numerical solution can be improved as shown in [84]. Comparing with CSPM, the SPH solution suffers from a slight phase change at the late stage of the simulation due to numerical dissipation. This is consistent with the ﬁnding of Price [177]. He studied a sinusoidal wave travelling in a pipe with periodic conditions and a large phase change was found. To improve the results, a varying smoothing length was used. Another way is to use a corrected kernel as in CSPM. The pressure magnitude at the valve is well predicted as shown in Fig. 5.19. The effect of friction is not noticeable due to the short duration of the event. If there is no friction, an exact solution exists and it can be calculated by MOC [211, 230].

5.4.3 Pipe draining

To verify the SPH model for pipe emptying, we examine a similar setup as in the ﬁlling case. Differently, the water tank is replaced by an air tank, the upstream valve is located at the inlet and the downstream end is equipped with a valve (see Fig. 5.2 with i = 1, 2). The two pipes are initially full of water, which is at rest and under pressure. The upstream valve is open and the downstream valve is closed. The air tank has a constant head of HR = 0.2 m of water. Other parameters such as the pipe length, diameter and slope, are unchanged as given in Section 5.4.1. After the downstream valve open instantaneously, water is driven out of the system by the contant air pressure and gravity. The resistance of the downstream valve is neglected, i.e. Kv = 0. 5.4 Numerical results 115

6 x 10 2.5

1.5

1 Pressure (Pa) 0.5

−0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 Time (s)

Figure 5.19: Pressure history of water hammer at valve for solutions of SPH (open circles), CSPM (open squares) and MOC (solid line).

Velocity. In SPH the particle spacing is d0 = ∆x = 10 mm, and hence there are initially 666 particles in the system. The speed of sound is c = 300 m/s. The time step is ∆t = ∆x/c and the simulation is terminated after the system is empty (about 6 seconds). Again, the constant friction factor f = 0.0247 is used. The predicted variation of the column velocity is shown in Fig. 5.20. The water column experiences rapid velocity changes at the early and late stages. The early rapid change is because of the low friction, and the late rapid change is because of the small column length. The rigid-column solution agrees very well with the SPH results. The water elasticity effect is invisible because the driving air pressure is low and the pipe length is short. The variation of the column length is shown in Fig. 5.21. It shortens slowly at the early stage and rapidly at the very late stage.

Different from the filling case, the velocity increases with the shortening of the column length and experiences two rapid changes with large acceleration as shown in Fig. 5.22. At the early stage, inertia dominates and the water column acceler- 2 ates at (HR + L1 sin θ1 + L2 sin θ2)g/(L1 + L2) = 0.72 m/s . When the velocity increases, friction becomes important and decreases the acceleration. After the column length is smaller than 1.5 m, the friction force becomes unimportant again and the column experiences another rapid acceleration because of its small mass. This is also clear from the acceleration history shown in Fig. 5.23. An inflection point exists in the velocity profile in Fig. 5.20, which corresponds to the minimum acceleration exhibited in Fig. 5.23.

Head transition. To better understand the dynamics of the ﬂow during emptying, the contribution of the driving pressure head, acceleration head, friction head loss and gravity head are shown in Fig. 5.24. Since the downstream valve resistance is zero, 116 Rapid Filling and Draining of Pipelines

3.5

2.5

Velocity (m/s) 1.5

0.5

0 0 1 2 3 4 5 6 Time (s)

Figure 5.20: Velocity history of the water column in pipe draining for rigid- column model (solid line) and elastic model solved by SPH (open circles).

3 Column length (m) 2

0 0 1 2 3 4 5 6 Time (s)

Figure 5.21: Water column length variation in pipe draining for rigid-column model (solid line) and elastic model solved by SPH (open circles).

the driving head is equal to upstream constant pressure head hu = pair/(ρg). Again, the head transition and the equilibrium of the water column are veriﬁed.

Controlled pipe emptying. For controlled pipe emptying through partially opening the downstream valve, the head loss coefficient Kv significantly affects the pipe emptying process (see Fig. 5.25). The valve resistance largely affects the emptying time and maximum velocity, but its effect on the early stage of emptying is insignificant because of the low flow velocities. There is a point (t = 1.25 5.4 Numerical results 117

3.5

2.5

Velocity (m/s) 1.5

0.5

0 0 1 2 3 4 5 6 7 Column length (m) t = 0

Figure 5.22: Velocity vs. water column length in pipe draining for rigid-column model (solid line) and elastic model solved by SPH (open circles).

4 ) 2

2 Acceleration (m/s 1

−1 0 1 2 3 4 5 6 Time (s)

Figure 5.23: Acceleration history of the water column in pipe draining for rigid- column model (solid line) and elastic model solved by SPH (open circles). s) in Fig. 5.25, where the velocity histories separates. This is because the ﬂow velocity in the early stage is relatively small, and hence the head loss due to valve resistance is unimportant relative to the inertia force. 118 Rapid Filling and Draining of Pipelines

0.5 Friction head loss 0.45 Acceleration head Driving head 0.4 Gravity head

0.35

0.3

0.25

Head (m) 0.2

0.15

0.1

0.05

0 0 1 2 3 4 5 6 7 Column length (m) t = 0

Figure 5.24: Contribution of the active heads during the emptying process.

3.5 Kv = 0 Kv = 0.5 3 Kv = 1 Kv = 1.5

2.5

1.5 Velocity (m/s)

0.5

0 0 1 2 3 4 5 6 7 Time (s)

Figure 5.25: Effect of downstream valve resistance on water-column velocity in the emptying process. Chapter 6

Slug Flow in a Voided Pipeline

6.1 Introduction

For gas-liquid flow in a horizontal pipe, different steady flow regimes have been classified as shown in Fig. 6.1. The classification is based on the relative velocity between the gas and the liquid (see Fig. 6.2).

Figure 6.1: Two-phase ﬂow patterns in a horizontal pipe [138, 202].

Among these flow regimes the slug flow has been thoroughly investigated due to its violent behaviour. Much of the work published on slug flow is related to power plants and oil recovery industries. Dukler and his co-authors proposed physical models to predict the development of the slug flow [54], the slug frequency [203], the minimum stable slug length [55] and flow regime transition [202, 204]. A review on the modelling of slug flow can be found in [58]. The behaviour of the slug in terms of velocity, pressure drop, void fraction and holdup, 120 Slug Flow in a Voided Pipeline

Figure 6.2: Flow regime map. Comparison of theory and experiment. Water-air, 25oC, 1 atm., 2.5 cm diameter, horizontal pipe. Solid lines: theory; hatched lines: measurements [138]. has a significant effect on chemical reactions in the gas-liquid mixture. In addition, pressure surges and impact forces are unavoidable when the slug encounters obstructions such as valves, bends, flow meters and branches. For a water slug emerging from the end of a horizontal pipe and impacting on a vertical plate, Sakaguchi et al. [187] measured and modelled the generated force. The dynamics of the slug in nearly horizontal pipes [232], a vertical pipe with a bend [208] and an ’S’ shaped riser [48] was measured and modelled. In these studies, the slug was assumed to be continuous (coherent liquid with finite length) and steady. The impact force of the steady slug is that of a jet Eq. (4.8)

2 F = ρsAVs , (6.1) where ρs is the liquid density, Vs the slug velocity and A the pipe cross-sectional area. The velocity range of steady slug ﬂow is 0.1 to 5 m/s (see Fig. 6.2), such that the generated impact pressure is small (less than 0.25 bar) and no damage is to be expected [48, 187, 208].

Steady slug ﬂow is less interesting for us. We are interested in a single slug accelerated along a void pipeline by high-pressure gas or steam. The focus is on the hydrodynamic behaviour of the slug and its impact on obstructions. The scenario is one where water has accumulated behind a valve in a steam line, and the valve is suddenly opened so that the liquid slug is forced to move along the pipe like a ”bullet” in a gun (Owen & Hussein [170]). This scenario frequently occurs and causes troubles in the power industry where energy is produced by steam turbines. When the high velocity slug impacts on an obstruction such as elbow [24,60] or oriﬁce plate [170], excessive hydrodynamic loads are generated, 6.2 State of the art 121 which may lead to severe damages. Over 50 occurrences have been attributed to this mechanism and an incident was reported in [231]. Therefore, it is of value to consider this scenario. Apart from the geometry of the impact target, the most important factor is the impinging slug itself (its velocity, length and shape) and the driving pressure behind it. As indicated by Bozkus et al. [26], the most needed liquid slug data are those for large diameter pipes used in practical power-plant piping systems.

A proper understanding of the hydrodynamics of an individual slug in a voided line, including its development and impact mechanism, is important to evaluate the damage risk and hence take corresponding measures to mitigate it. The objectives of this chapter are to review the experimental and mathematical modelling of the isolated slug motion and impact, recognize and validate the assumptions in the model, apply the SPH method and compare the results with experiments and simulations found in literature.

6.2 State of the art

Steady slug flow has been intensively studied and the focus was on the kinematic behaviour of the slug. However, for a liquid slug in an initially empty line with an obstruction, there is not a wide range of studies. The first study was by Fenton & Griffith (1990) [60]. They experimentally and analytically studied the peak forces at an elbow due to clearing a liquid slug initially trapped upstream. Similarly, Neumann [163] studied the forces on a pipe bend resulting from clearing a pool of liquid. After these two attempts, the isolated slug problem gained more attention. Bozkus and his co-authors (Bozkus 1991 [24]; Bozkus & Wiggert 1997 [25]; Bozkus et al. 2004 [26]; Kayhan & Bozkus 2011 [98]) made great contributions to this topic. The hydrodynamic behaviour of the slug (motion and impact) has also been investigated by Owen & Hussein (1994) [170] and Yang & Wiggert (1998) [231]. Among these studies, the works of Fenton & Griffith [60], Neumann [163], Bozkus [24], Owen & Hussein [170] and Bozkus et al. [26] need to be underlined due to their original experimental frameworks.

6.2.1 Laboratory tests

Fenton & Griffith (1990). As said, Fenton & Griffith [60] investigated the forces experienced by a pipe bend, when a trapped upstream liquid slug was cleared by a high pressure gas flow. The experimental test rig is shown in Fig. 6.3. The 2.44 m long pipe with inner diameter of 25 mm was slightly inclined upward and could be filled with water in its lower section, so that liquid slugs with different lengths were obtained. An elbow open to the atmosphere was placed at the end of the pipe. When the ball valve was suddenly opened, the trapped water was forced 122 Slug Flow in a Voided Pipeline to move rapidly and hit the elbow. To reduce the effect of structural vibration on the measured impact forces, the elbow was not rigidly fixed to the pipe, but fixed to a support behind it, on which a strain gauge force transducer was installed. Thus, the axial impact forces at the elbow were measured when the slug passed through it. The experiments were varied by changing the initial volume of the entrapped water, the distance of the trapped water to the bend and the driving air pressure in the vessel (414, 552 and 690 kPa). The estimated arrival velocity of the slug at the elbow was varying from 18 to 23 m/s. It was found that the force on the bend was mainly due to quasi-steady momentum transfer in changing the fluid flow direction around the bend. The waterhammer phenomenon was not observed. Once the slug had travelled six or more times its initial length, it was dispersed so much that the magnitude of the forces experienced at the bend dramatically dropped.

Figure 6.3: Experimental setup of Fenton & Grifﬁth [60].

Neumann (1991). Neumann [163] investigated the forces on a pipe bend resulting from clearing a pool of liquid and the effect of pipe diameter change in a similar setup as that of Fenton & Griffith (see Fig. 6.3). The difference is that the initial trapped water was replaced by a water pool of varying depth. It was found that the generated forces were negligible as long as a transition from stratified flow to slug flow did not occur, which was true when the liquid fraction in the pool was less than 20%. When the transition indeed occurred, to calculate the effective slug length it was assumed that, no matter how shallow the pool, the water was concentrated in the middle of the pool as a slug, whose length was the water volume divided by pipe cross-sectional area. The estimated slug arrival velocity was from 17 to 36 m/s. To characterize the hydrodynamics, a dimensionless length ∗ D = L/L0, referred to as dispersion distance, was used, where L0 is the initial slug length and L is the distance measured from the tail of the slug to the bend. It was found that for D∗ ∼ 5 the force at the bend was greatly reduced, which is consistent with the findings in [60]. When D∗ > 6 the force was even negligible. When the 1 inch pipe connected to the bend (see Fig. 6.3) was reduced to 0.75 inch, the experimental results had negligible change. 6.2 State of the art 123

Bozkus (1991). The experimental setup of Bozkus [24] is depicted in Fig. 6.4 and consists of a 9.45 m long and 50 mm inside diameter PVC pipe with a ball valve upstream and a 90o elbow downstream. Upstream of the valve there was a pipe section whose length could be changed to generate slugs of various lengths. This pipe connected to an air tank that supplied the pressurized air to drive the slugs. When the valve was suddenly opened, the slug accelerated due to the pressurized air behind it, hit the elbow and released into the atmosphere. Two pressure transducers were installed at the elbow to record the pressure history (see Fig. 6.4). Slugs with lengths ranging from 1.22 to 3.35 m were propelled into the pipe under reservoir pressures ranging from 69 to 275 kPa.

Figure 6.4: Experimental setup used by Bozkus in [24]; SGP = slug generate pipe.

The peak pressures measured at the elbow varied signiﬁcantly (largely scattered data), even though nearly identical initial and boundary conditions were imposed. Thus, each individual test run was repeated 8-10 times. Bozkus attributed the large scattering of the data to the hand-operated valve, because different opening manoeuvres may have affected the slug dynamics. Comparing Figs. 6.3 and 6.4, the differences between the two test rigs, such as pipe material, diameter, inclination angle, force measurement, etc., are evident. In addition, the elbow in Fig. 6.4 was rigidly connected to the pipe.

The slug velocity at the elbow was not measured. The estimated arrival velocity varied from 15 to 30 m/s. Two general trends were observed. For relatively short slugs (Ls = 1.22 and 1.52 m), the response exhibited a single peak followed by a rapid pressure decay. The effect of air entrainment was so large that the short slugs had practically broken up. For relatively long slugs (Ls = 2.13, 2.74 and 3.35 m), a double-peaked response was observed. The double-peak phenomenon was correlated with the arrival of two lumps of masses – instead of one – at the elbow, because the single slug broke up. In his analytical model, he attributed the double-peak phenomenon to a short-lived water hammer taking place along the slug body directly after valve opening.

Owen & Hussein (1994). The test rig of Owen & Hussein [170] is shown in Fig. 6.5. Upstream of the valve is a pressure source and downstream is an isolated slug of water being propelled into an empty pipe (D = 50 mm) and impinging on an 124 Slug Flow in a Voided Pipeline orifice with an inner diameter ranging from 25 to 35.4 mm. Air, with pressures up to 11 bar (absolute) and of ambient temperature was used as the driving gas. The volume of the air reservoir was sufficiently large for the air pressure drop to be less than 3% in each test. This is different from Bozkus’ experiment [24], where a noticeable drop of the tank pressure was present. A butterfly valve was located between the reservoir and the steel pipe to enable the rapid air expulsion. The water slug was held between the closed valve and a thin polythene sheet sand- wiched between flanges downstream of the valve. By using two different pipes (0.99 and 2.16 m long) to hold the slug, it was possible to obtain three different slug lengths (L0 = 0.99, 2.16 and 3.15 m), whilst maintaining the overall length of the rig constant at 13.0 m. To measure the slug velocity at the orifice plate, two conductivity probes 0.215 m apart were used. To measure the impact force at the orifice plate, a pressure transducer was located close to its upstream face. For each slug length, different air pressures were used. To propel the slug down the pipe, the valve was opened quickly by hand so that the sudden rise in pressure behind the water forced it through the polythene sheet.

Figure 6.5: Experimental setup of Owen & Hussein [170].

Comparing with the test rig of Bozkus [24] (see Fig. 6.4), the biggest advantage of this setup is that the velocity of the slug arriving at the impact target (oriﬁce plate therein) was measured. As shown below, the slug velocity is the most important ingredient in determining the impact force at the target. In addition, the driving air pressure used in [170] had a much larger range than in [24].

To confirm the repeatability of the experiment, a number of valve openings were carried out and no discernible difference in the slug velocity was detected. This means that no dependence on the hand-operated valve was observed, whilst Bozkus attributed his large scattering data to it [24, 25]. This suggests that the measurements in [170] might be more accurate and reliable than those in [24]. To test if the orifice plate had an effect on the slug motion, slug velocities with and without the orifice plate were measured. No distinguishable difference between the velocity measurements was observed. This suggests that the air being compressed between the orifice plate and the slug has little effect on the velocity of the slug as it approaches the plate. This is consistent with the conclusion of Mar- tino et al. [141] that when the orifice ratio d/D (d being the orifice diameter and D the pipe diameter) is larger than 0.2, the presence of the orifice will not affect the arriving velocity of the advancing front at the pipe end. 6.2 State of the art 125

For the tests with the 2.16 and 3.15 m long slugs, the traces recorded from the conductivity probes showed a step-change as the front of the slug passed by. For the 0.99 m slug, however, the recorded trace was extremely erratic, indicating that the slug had broken up. This was conﬁrmed by the trace from the pressure transducer, which showed a much lower impact pressure. This observation is consistent with that in [24,25,60,163]. The measured arrival velocities were varying from 32 to 58 m/s for the 2.16 m long slug, and from 27 to 42 m/s for the 3.15 m long slug. Only the measured peak pressures (without the pressure traces) were presented in [170].

Bozkus et al. (2004). As indicated by Bozkus & Wiggert [25], the pipe diameter used in [60] was relatively small compared with the pipe diameters in real piping systems. The weakness of the experimental setup in [24] was that the operation of the valve might have affected the slug dynamics. To avoid these disadvantages, a new experimental setup was built (see Fig. 6.6), which has the same conﬁguration as that in Fig. 6.3. The main differences were that the diameter of the steel pipe was 100 mm instead of 25 mm and that the elbow was rigidly connected to the pipe. Similar as in [60], the disadvantage is the use of an inclined pipe for slug initialisation. The slanted shape of the slug front with respect to the pipe cross- section has an increased effect on the occurrence of Taylor instability resulting in more air entrainment in the slug body, especially for short slugs [26,48,97]. Under different driving air pressures (2, 3, 4, and 5 bar (gauge)), slugs with four initial lengths, L0 = 3, 4, 5, and 6 m, were ﬁred. The arrival velocity of the slug at the elbow was not measured. The estimated slug velocities were varying from 15 to 30 ∗ m/s. Since D = L/L0 < 5 for all the slugs (long slugs), no evident slug breaking up was observed. Moreover, double-peak phenomena were not observed at all in this experiment. The reason for the inconsistent results in Bozkus’ two experiments in [24] and [26] is not entirely clear.

Figure 6.6: Experimental setup of Bozkus et al. [26]. 126 Slug Flow in a Voided Pipeline

6.2.2 Reaction forces

When a liquid slug passes through an elbow, dynamic forces are imparted on it. The impact force used in [60, 163] is the same as Eq. (6.1). In [24, 25], Bozkus included the effect of the driving air Pair, i.e.

2 F = PairA + ρsAVs . (6.2) The experimental peak pressures were underestimated by (6.2). For improvement, a ﬁctitious surge tank was assumed to exist at the elbow [26], and the impact force was 2 F = PairA + ρsAVs + ρsgz, (6.3) where z is the head rise in the hypothetical tank. The impact forces of Bozkus [24–26] are suspicious because the derivation of (6.2) and (6.3) might be incorrect.

The impact force largely depends on the slug velocity Vs, for the calculation of which different propagation models have been proposed [24–26, 60, 170]. In [60, 163] the slug was assumed to instantaneously accelerate to the velocity of the driving gas, which was referred to as ”nominal velocity” [60]. The force calculated by Eq. (6.1) with the nominal velocity underestimated the measured peak force ∗ in the worst case by a factor 2.5 for D = L/L0 < 4, and largely overestimated it for D∗ > 5.

The analytical study of Bozkus [24] was almost the same as the advanced model in [60]. The difference was that the mass loss due to shearing effects and gravity as the slug moves in the pipe was considered. For short slugs, the calculated peak pressures were much higher (about 2 times) than the experimentally obtained ones. For medium and long slugs, the calculation reasonably matched the experimental peaks, but the shape of the simulated pressure history was completely different from the measurements. In another work of Bozkus et al. [26], the mass loss from the main slug was neglected. The computed impact pressures underestimated the measured peaks.

A quasi two-dimensional model was developed in [231], taking air entrainment into account. The water slug was modelled as a number of concentric cylinders sliding through each other. Since the inner cylinders moved faster than the outer ones, the air penetrated into the slug and the tail of the outer cylinders was considered as holdup. At the elbow, the concentric cylinders kept on sliding into each other, until the void between the cylinders reached the bend. To test the model, the experimental data of Bozkus [25] were used. Again, the numerical solution underestimated the experimental outcome. The advantage of the model is that it takes into account air entrainment. The disadvantages relate to two aspects. First, since gravity was not considered, the computed ﬂow was symmetric. This is inconsistent with the experimental visualization in [24], which is similar as shown in Fig. 6.7. Second, the effect of a laminar boundary layer was so large that the arrival velocity of the slug at the elbow was dramatically underestimated resulting in an underestimation of the impact forces. 6.3 Governing equations 127

Recently, Kayhan [97] developed a new model, in which the holdup coefficient was not a constant (5%) [24], but a coordinate-dependent function fitted to the experimental data [24]. To predict the impact force at the elbow, the effect of the vertical branch of the elbow was taken into account. Based on many simplifications and complicated 3D coordinate transformations, a 1D numerical approach along a curved line was established. The calculated transient forces at the elbow were compared with those obtained experimentally by Bozkus [24] and numerically by previous researchers [25, 231]. The new method predicted the peak pressures with higher accuracy than the previous methods [24,231]. Although the pressure traces were also improved, they still largely disagreed with the measurements. The contribution of this new model is that it takes the effect of the elbow flow into account. However, the model is based on a fully developed quasi-steady flow without separation at the elbow, which seems unrealistic.

The model of Owen & Hussein [170] is different from the aforementioned models. First, instead of numerically solving the gas dynamics equations, the driving air pressure upstream of the slug was directly obtained from a gas expansion wave. Second, the effect of air ahead of the slug was considered, the pressure of which was given by a gas compression wave. Third, the change of the tank pressure was neglected, because the volume of the tank was sufﬁciently large such that the drop in pressure as the air was expelled was less than 3%. Fourth, the mass loss of the slug during motion was neglected. As stated by Owen & Hussein [170]: ”It was appreciated that as the slug moves through the pipe, it is shedding liquid behind it. However, the area of the slug upon which the gas is acting will still be that of the pipe cross-section, and the total mass of the liquid being accelerated is that of the whole slug.” The calculated velocities for the 2.16 and 3.15 m slugs (not breaking up) showed good agreement with the experiments.

6.3 Governing equations

In this section, the model for slug motion in an empty line and the determination of the driving air pressure developed by Bozkus [24] and used in [25, 26, 97, 98] is brieﬂy revisited. This model is used in this thesis to determine the conditions for the slug impact at the elbow. They include the arriving time of the slug, the slug length and velocity, and the driving air pressure. With these conditions, the SPH model developed in Section 5.3 is used to predict the slug impact force at the elbow. A new calculation of reaction force based on ﬂow separation is derived.

6.3.1 Slug motion in an empty pipe

A moving liquid slug that losses its mass is sketched in Fig. 6.7. In developing the model the following assumptions were made in [24] 128 Slug Flow in a Voided Pipeline

• The slug flow is one-dimensional, incompressible and with planar front. • The rigid pipe is fixed in space, hence no fluid-structure interaction occurs. • The pressure ahead of the water front is atmospheric. • No gas (air) entrainment occurs into the liquid (water), (i.e. no mixture).

• Vr1 ≈ Vr2 = Vs (see Fig. 6.7). • The shear resistance to the slug motion is assumed to be that of a quasi- steady flow. • The moving slug loses its mass at a constant rate due to shearing and gravity effects. The mass loss is referred to as ”holdup” in steady slug flows [54, 55, 202, 204] and is accounted for in the equations by a holdup coefficient denoted by αs.

Figure 6.7: Control volume moving with the liquid slug in an empty pipe adapted from [24].

Applying the Reynolds transport theorem to the moving and shortening control volume shown in Fig. 6.7, the following governing equations were derived in [24]:

Momentum equation [ ( )] dVs fs 2 1 − αs 2 Pair − Patm + − Vs = , (6.4) dt 2D L αs ρsL

Continuity equation ( ) dL 1 = − − 1 Vs, (6.5) dt αs and the equation for slug position dx s = V , (6.6) dt s 6.3 Governing equations 129

where αs = A3/A2 = 5% herein, Patm is the atmospheric pressure and xs is the position of the advancing slug tail. The initial conditions are

Vs(0) = 0, L(0) = L0 and xs(0) = 0. (6.7)

6.3.2 Driving air pressure

When gas acoustics is used to model the gas behind the slug, the following differential equations for one-dimensional, unsteady, nonuniform ﬂow of an isothermal compressible gas were used in [24]:

Conservation of mass ∂P ∂P ∂V a + V a + ρ c2 a = 0, (6.8) ∂t a ∂x a a ∂x Conservation of momentum 2 ∂Va ∂Va 1 ∂Pa Pdry Va + Va + = −fa , (6.9) ∂t ∂x ρa ∂x 4A 2 Equation of state dP dρ a = c2 a , (6.10) dt a dt with wave speed √ ca = RT, (6.11) where the subscript ’a’ stands for air, ca is the isothermal speed of sound, Pdry is the dry perimeter of the cross-section (air), R is the universal gas constant and T is the gas temperature in Kelvin. Note that the pressures in Eqs. (6.4), (6.8) – (6.10) are also in absolute scale. The initial conditions are

Va(x, 0) = 0, Pa(x, 0) = Ptank(t = 0) = P0, x < 0. (6.12) and the boundary conditions are

Pa(−Ltank, t) = Ptank(t),Pa(xs, t) = Pair,Va(xs, t) = Vs, (6.13) where Ltank is the distance from the tank to the slug tail at t = 0. The boundary conditions at the slug tail couple the gas dynamics equations (6.8) – (6.11) to the Eqs. (6.4) – (6.6) for the slug.

6.3.3 Elbow pressure and reaction force

From the study of ﬂow separation at sharp bends in Section 4.5 it is evident that the assumption made in [24–26] that the leaving liquid occupies the full cross- section of the vertical pipe is not entirely reasonable. To develop a new model considering the effect of ﬂow separation, the control volume in Fig. 6.8 is utilized. 130 Slug Flow in a Voided Pipeline

Figure 6.8: Control volume for the liquid slug at the elbow considering ﬂow separation.

Applying the Bernoulli equation to points 1 and 2 in Fig. 6.8, we have

ρ V2 ρ V2 ρ V2 P + s s1 = P + s s2 + K s s1 , (6.14) 1 2 2 2 e 2 where Ke is the coefﬁcient of minor loss at the elbow. The unsteady Bernoulli equation may be used too, then an extra term LedVs/dt (Le is the elbow length) exists at the right hand side of (6.14). It is negligible comparing with other terms. To satisfy mass conservation, the leaving ﬂow velocity is

Vs2 = Vs/Cc, (6.15) where Cc is the contraction coefficient due to flow separation. Consequently, together with Vs1 = Vs and P2 = 0, one obtains ρ V2 P = (1/C2 − 1 + K ) s s . (6.16) 1 c e 2 To compare with the measurements, the pressure at the elbow (transducers) needs to be estimated. Since the two pressure transducers are close to the corner, we assume that the flow velocity at the transducers is close to zero. Furthermore, we assume that the head loss occurs mainly in x-direction. Then we have

2 2 2 ρsVs ρsVs 1 ρsVs Pe = P1 + − Ke = 2 . (6.17) 2 2 Cc 2

The pressure at the elbow can be calculated from point 2 using Vs2, the same formula as (6.17) is obtained.

Applying the linear momentum equation to the control volume, the reaction forces are obtained. The reaction force in x-direction is the same as Eq. (6.1) and the force in y-direction is

2 2 Ry = ρsA2Vs2 = ρsAVs /Cc. (6.18) 6.4 Results and discussion 131

6.4 Results and discussion

The experiment of Bozkus [24] (see Fig. 6.4) is simulated in this section. The short slugs are not important for us, because they break up and give relative small impact forces as experimentally observed in [24, 60, 163, 170]. We focus on the medium and long slugs. To predict slug motion in an empty pipe, the 1D model of Bozkus presented in Sections 6.3.1 and 6.3.2 is used. When the slug hits the elbow, the arrival time, the slug length and velocity, and the driving air pressure are known from the model. With these conditions, the 1D SPH model developed in Section 5.3 and 2D model developed in Section 3.4 are used to determine the slug dynamics after impacting. The speed of sound in the liquid is 440 m/s, which was determined from spectral analysis of the pressure history recorded by the transducer installed at the SGP (slug generation pipe, see Fig. 6.4) [24].

6.4.1 1D SPH simulations

In the SPH water-hammer equations, pressure Eq. (6.18) is used as the downstream boundary condition and the measured driving air pressure (Fig. 5.1 in [24]) is used as the upstream boundary condition. With these two conditions, the velocity history of the slug Vs(t) is solved from the model. The pressure at the transducers is then obtained from (6.17) and the reaction forces in two directions are calculated from Eqs. (6.1) and (6.18). For a right angled bend, the theoretical contraction coefficient Cc is equal to 0.5255 (see Table 4.1). This value is not used herein because the practical used bend was not right angled but a curved elbow. The contraction coefficient for the elbow in Fig. 4.4 [24] is Cc = 0.51, which is obtained from 2D SPH simulations. The head loss coefficient Ke for a standard elbow varies from 0.7 to 0.9 [46, 199]. Herein we take Ke = 0.9. The contribution of flow separation to P1 is about 3 times of the minor loss contribution. The initial particle spacing is ∆x = 0.01 m and the smoothing length is h = 1.2∆x. The time step is taken as ∆t = ∆x/c0 with c0 = 440 m/s. The simulation is terminated when all fluid particles are expelled out of the 9.45 m pipe. Since the reaction forces are not measured, only the pressure history at the elbow is presented.

For initial slug length L0 = 2.74 m and initial tank pressure P0 = 137.82 kPa (20 psi), the pressure obtained from Eq. (6.17) is shown in Fig. 6.9 together with numerical solutions from other models. Although there is a mismatch in arrival times and the double peak behaviour is not captured, the maximum pressure is well predicted. In addition, the simulated trend has good agreement with the measurements after the second pressure peak. Double peak pressure behaviour was indeed predicted in [24, 25, 231], but the predicted trend completely contra- dicted with the experimental observations.

In the simulations of Bozkus [24–26], the pressure at the elbow is computed from Eq. (6.2) or (6.3) (divided by A). Since ﬂow separation and head loss were not con- 132 Slug Flow in a Voided Pipeline

180

160

140

120

100

Dynamic pressure (psig) 40

0.7 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 Time (s)

Figure 6.9: Pressure history at the elbow. Comparison between experiments [24] (solid line with circles) and numerical predictions by Bozkus & Wiggert [25] (solid line with squares), Yang & Wiggert [231] (solid line with diamonds) and present 1D SPH (solid line).

sidered, pressure P1 is equal to zero. Then under driving air pressure, the slug passes the elbow with acceleration and hence increasing velocity. This is inconsistent with the experiment. In the present model, pressure P1 given by Eq. (6.16) reduces slug velocity and hence impact force and pressure. If only flow separation is considered, the pressure peak will be underestimated. Consequently, to accurately capture the slug hydrodynamics, sufficient information of the flow behaviour at the elbow, such as flow separation and head loss, is highly needed.

Based on the pressure history at the elbow, Bozkus [24] determined the impulse of the slug at the elbow by computing the product of the area under the pressure- time curve and the pipe cross-sectional area, i.e. ∫ t2 I = A P(t)dt, (6.19) t1 where P(t) is the pressure history at the elbow, and the integral limits t1 and t2 are the start and end points on the time axis. To accurately predict the impulse, not only the peak pressure, but also the time of duration plays an important role. The duration is the time from the slug hitting the bend to the instant when pressure decreases to tank pressure. The present model not only predicts the maximum pressure very well (Fig. 6.9), but also gives an accurate prediction of the duration and hence the impulse. This statement is more apparent if the calculated arrival time at the elbow is shifted to that of the experiment, as shown in Fig. 6.10e.

For other slugs with different initial lengths and driving air pressures, the results are shown in Fig. 6.10 together with numerical solutions from other researchers 6.4 Results and discussion 133

[24, 25, 98, 231]. Similar conclusions as drawn above for the case of L0 = 2.74 m and P0 = 20 psi hold. The decreasing pressure trend was predicted in [98], but the pressure peak magnitude and duration were underestimated to a large extent. When the slugs exited the elbow, the pressures were highly overestimated too.

The single pressure peak in present study and in [98] may be attributed to two aspects. First, the deformation of the slug’s front is disregarded in the present and in Kayhan’s model. However, the initially vertical front indeed became wedged shape as visualized in the experiments [24], although the shape change was relatively small compared with the slug tail deformation. In the new experiments of Bozkus [26] with an initial wedge-shaped front, only one gradually developing pressure peak was observed. More importantly, air entrainment at the advancing leading edge was not considered in the models. This is inconsistent with the observation in [166] for a long slug moving in a pipeline. Air entrainment reduces the liquid density to a large extent and hence the impact force. This phenomenon was discussed in [98, 231], but not modelled. Since the depth of intrusion is limited, air entrainment can only affect the early stage of the slug impact. This explains the complex slug behaviour occurring before the second peak.

6.4.2 2D SPH simulations

Since the slug velocity is high (more than 13 m/s) and the impact duration is short (less than 0.1 s), gravity and viscosity do not play an important role and are therefore neglected. With the idea presented in Section 4.5, the pipe with the elbow is represented by a vertical plane through the central axis, so that a two-dimensional simulation becomes possible. The driving air pressure is taken constant. The 2D SPH setup for the case of L0 = 2.74 m and P0 = 137.82 kPa (20 psi) is shown in Fig. 6.11a. The curved outer wall is modelled as three connected straight sections as shown in Fig. 6.11b. The initial particle distribution is hexa- hedral (see Fig. 6.11b) with particle spacing d0 = 0.0058 m. The smoothing length is h = 1.33d0. The time step is ∆t = 0.2d0/c0. The simulation is stopped when all particles have left the computational domain.

The pressure history of the slug impact at point B (see Fig. 6.11b) is shown in Fig. 6.12. The 2D SPH solution has similarity with the 1D solution, but suffers from pressure noise. After numerical smoothing using the ”smoothn” scheme [65], the magnitude of the pressure is comparable with the 1D solution. The slightly lower pressure magnitude may be attributed to the curved bend (Fig. 4.4 in [24]) instead of the mitre bend (see Fig. 6.11b), and the minor loss (no loss in 2D SPH). The water front shape may also have some effect.

Flow fields at two different times after slug impact are shown in Fig. 6.13. Indeed, quasi-steady flow separation has been established after only 0.01 seconds. That is about one tenth of the duration of the slug impact. This verifies that the flow separation theory used to obtain the pressure and reaction forces in Section 6.3 is 134 Slug Flow in a Voided Pipeline

(a) (b) 220

120 P = 10 psi, L = 2.13 m 200 P = 20 psi, L = 2.13 m 0 0 0 0 180 100 160

80 140 120

60 100

80 40 60 Dynamic pressure (psig) Dynamic pressure (psig) 20 40

20 0 0

0.88 0.89 0.9 0.91 0.92 0.93 0.94 0.65 0.66 0.67 0.68 0.69 0.7 Time (s) Time (s)

P = 40 psi, L = 2.13 m P = 10 psi, L = 2.74 m 0 0 70 0 0 250 60

200 50

40 150

30 100 20 Dynamic pressure (psig) Dynamic pressure (psig) 50 10

0 0 −10 0.48 0.49 0.5 0.51 0.52 0.53 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 Time (s) Time (s)

(e) (f) 180 120 P = 20 psi, L = 2.74 m P = 20 psi, L = 3.35 m 160 0 0 0 0

140 100

120 80 100 60 80

60 40 Dynamic pressure (psig) Dynamic pressure (psig) 40 20 20

0 0

0.72 0.74 0.76 0.78 0.8 0.82 0.84 0.85 0.86 0.87 0.88 0.89 0.9 0.91 0.92 Time (s) Time (s)

(g) (h) 160 P = 30 psi, L = 3.35 m P = 40 psi, L = 3.35 m 0 0 0 0 140 200

120 150 100

80 100 60

40 Dynamic pressure (psig) Dynamic pressure (psig) 50

0 0 0.69 0.7 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.59 0.6 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 Time (s) Time (s)

Figure 6.10: Pressure history at the elbow. Comparison between experiments [24] (solid line with circles) and numerical predictions by Bozkus & Wiggert [25] (solid line with squares), Yang & Wiggert [231] (solid line with diamonds), Kayhan & Bozkus [98] (solid line with triangles) and present SPH (solid line). 6.4 Results and discussion 135

slug impact - case 1 ( = 0.1, V = 22.4 m/s) 0 0.1 0

Y [m] -0.1 0 0.5 1 1.5 2 X [m]

0.06 A B 0.04

C 0.02

0 D y [m]

-0.02

-0.04

-0.06

-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 x [m]

Figure 6.11: SPH setup of two-dimensional slug impact at elbow for the case of L0 = 2.74 m and P0 = 137.82 kPa (20 psi): (a) overview and (b) details at the elbow.

180 Original signal (2D SPH) 160 Smoothed signal (2D SPH) 140 Experiment 1D SPH 120

100

40 Dynamic pressure (psig) Dynamicpressure 20

-20 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.8 0.81 0.82 Time (s)

Figure 6.12: Pressure history after slug impact at the elbow (point B) for the case of L0 = 2.74 m and P0 = 137.82 kPa (20 psi). 136 Slug Flow in a Voided Pipeline

0.04 0.04

0.02 0.02

0 0 y [m] y y [m] y

-0.02 -0.02 t = 0.005 s t = 0.01 s

-0.04 -0.04

-0.06 -0.06 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0. 1 x [m] x [m]

(a) (b)

Figure 6.13: Flow separation at the elbow at two different times. a step in the correct direction.

The two essential features of the slug impact problem, i.e. the pressure magnitude and the impact duration, have been well predicted by the 1D and 2D SPH simulations. To capture the details of the complex slug behaviour at the bend, a 3D SPH simulation is needed.

6.4.3 Comparison of peak pressures at the elbow

Tabulated data for the Pmax values are given in Table 6.1. In this table, together with the experimental data for different initial slug lengths and initial tank pressures, the maximum pressures predicted by previous researchers and by the present study are summarized.

A comparison between experimental and numerical maximum pressures Pmax is shown in Fig. 6.14. The predictions in the present study provide a better approximation to the experimental values measured in [24]. The 1D SPH solutions have better agreement with the experiments than the 2D solutions. The predictions of previous researchers highly underestimate the measurements (see Table 6.1). This underestimation was attributed to the inadequateness of the momentum transfer theory. As stated by Yang & Wiggert [231]: ”There are some differences between experiment and prediction of the pressure-time histories of the slug at the elbow. This indicates that the simplified incompressible momentum transfer theory may be inadequate to predict the impulse. Based on analysis of the data and an estimation using simplified one-dimensional waterhammer theory for two-component flow, it is suggested that an acoustic response takes place when the high-speed slug impacts on the elbow. The waveform of the pressure-time histories is mainly dependent on the acoustic propagation of the wave. Therefore, the propagation of that wave in the liquid slug is an important issue that requires further study.” The cited waterhammer phenomenon has been checked in [26, 97, 231]. To match the waterhammer pressure to the experiment, 6.4 Results and discussion 137 89 55 40 81 174 251 292 115 178 247 121 162 69.70 66.24 107.00 177.88 213.66 289.44 138.35 196.10 262.49 100.59 152.27 201.00 99.71 59.26 89.17 32.49 62.34 90.82 138.34 168.29 220.62 129.59 164.57 120.78 Computed Computed Computed – – – – 96.16 35.24 61.49 27.41 65.12 115.01 152.14 173.46 Computed 48.17 30.96 68.96 22.30 47.18 73.30 104.89 160.57 219.03 106.29 146.10 101.51 of Bozkus [24] of Y& W [231] of Kayhan [97] of 1D SPH of 2D SPH 5 8 18 41 9 31 6 7 12 16 – – [24] 79 71 65 131 124 3527 151 222 32 264 4 4240 171 207 17 11 16 14 – – [24] 38 96 56 78 63 139 135 173 104 126 Table 6.1: Comparison of predicted and measured peak pressures (psi (gauge)). 30 20 40 20 30 40 20 30 40 slug tank experiment experiment peak values peak values peak values peak values peak values (7 ft) (9 ft) (11 ft) Initial Initial First peak Second peak Computed length pressure 2.13 m 10 2.74 m 10 3.35 m 10 138 Slug Flow in a Voided Pipeline

300

250

200 (psig) max 150

100 Calculated P

0 0 50 100 150 200 250 300 Experimental P (psig) max

Figure 6.14: Comparison between experimental and calculated maximum pressures: straight line with error bars – experiment of Bozkus [24], circles – 1D SPH, squares – 2D SPH, diamonds – Kayhan [97].

many assumptions on the water-air mixture density ρm and the speed of sound c0 have been made. These assumptions may be reasonable and useful, but a care- ful validation is needed.

The above argument seems improper for us because the fluid at the elbow is more likely to occupy the empty space than to be compressed and hence causing a water hammer. This is similar to the observation in pipe filling and emptying studied in Chapter 5. The current study indicates that the momentum transfer theory is adequate to predict the slug impact at the elbow, but that the effect of flow separation and pressure loss must be taken into account.

6.4.4 Parameter variation

The maximum impact pressure at the elbow is needed for pipe design purposes. Apart from the driving air pressure and water elasticity (speed of sound), the slug impact force depends on the slug length and velocity. To study the effect of these factors, 1D numerical experiments were carried out by taking the case of L0 = 2.74 m and P0 = 137.82 kPa (20 psi) as an example.

It was found that the driving air pressure slightly affects the pressure history as shown in Fig. 6.15. The length of the slug arriving at the elbow (Lse) only affects the time of duration of the impact (see Fig. 6.16). According to Eq. (6.17), the pressure magnitude depends on the arriving velocity of the slug at the elbow (Vse). It signiﬁcantly affects the pressure magnitudes and impulse as shown in Fig. 6.17. The effect of water elasticity is also checked by varying the speed of 6.4 Results and discussion 139

140 With driving air pressure No driving air pressure 120

100

40 Dynamic pressure (psig)

0 0 0.02 0.04 0.06 0.08 0.1 Time (s)

Figure 6.15: Effect of the driving air pressure on the pressure history at the elbow.

sound c0. Similar as the pipe ﬁlling and emptying, its effect is negligible because the water slug is more likely to change its momentum than to be compressed. In practical applications, the pressure (6.17) and reaction forces Eqs. (6.1) and (6.18) should be used to predict the slug impact at a sharp bend.

150 L = 1.5 m se L = 1.8 m se L = 2.2 m se L = 2.5 m se 100

50 Dynamic pressure (psig)

0 0 0.02 0.04 0.06 0.08 0.1 Time (s)

Figure 6.16: Effect of the impacting slug length Lse on the pressure history at the elbow. 140 Slug Flow in a Voided Pipeline

V = 15 m/s 250 se V = 20 m/s se V = 22.4 m/s se V = 30 m/s 200 se

150

100 Dynamic pressure (psig)

0 0 0.02 0.04 0.06 0.08 0.1 Time (s)

Figure 6.17: Effect of the impacting slug velocity Vse on the pressure history at the elbow. Chapter 7

Filling and Emptying of a Large-scale Pipeline: Experiments and Simulation

This chapter describes the experimental apparatus designed and built at Deltares, Delft, The Netherlands, as part of the EC Hydralab III project, to investigate the ﬁlling and emptying of a large-scale pipeline. Different from other laboratory studies, the scale of this experiment is close to the practical situation in industrial plants. It includes a variety of components and the operation procedure is rather complex. All these features make the current study particularly useful as a test case for real situations. To determine the dynamic characteristics of the viscoelas- tic PVC pipeline, water-hammer tests were performed and numerically validated. These are out of the range of the thesis and are reported elsewhere; see Bergant et al. [22, 23]. The experimental setup is presented in Section 7.1 with a detailed description of different components and measurement devices. The experimental variables and procedure are described in Section 7.2. The measured data are presented in Section 7.3. Section 7.4 shows preliminary numerical simulations with their discussion and comparison to the experimental results.

7.1 Experimental apparatus

The piping system used in the experiments is illustrated in Fig. 7.1. The experimental apparatus consisted of a water tank, an air tank, steel supply pipelines (for water and air), a PVC inlet pipe, a pipe bridge, a horizontal long PVC pipeline (the test section), an outlet steel pipeline and a basement reservoir. The detailed 142 Filling and Emptying of a Large-scale Pipeline: Experiments and Simulation information is given below, and part of it can be found in [104, 105].

Figure 7.1: Sketch of the large-scale PVC pipeline apparatus at Deltares, Delft, The Netherlands [23].

7.1.1 System origin and coordinates

The downstream end of the PVC pipe bridge was deﬁned as the origin of the coordinate system. It is the starting point of the test section. The x-coordinate follows the central axis of the pipeline, the y-coordinate is not used herein and the z- coordinate is the vertical elevation. Coordinates of measuring instruments and 7.1 Experimental apparatus 143 other important components are shown in Fig. 7.2.

Figure 7.2: System dimensions and coordinates of the instruments [23].

7.1.2 Tanks and pipes

A water tank (see Fig. 7.3) with a constant 25 m head relative to the centre line of the inlet (see Fig. 7.4a) was used to supply water in the ﬁlling experiment and an air tank with a 70 m3 volume was used to supply air in the emptying experiment. 144 Filling and Emptying of a Large-scale Pipeline: Experiments and Simulation

The water supply steel pipe, from the T-junction (x = −27.2 m, in practice it is a Y-junction, but it is not important in view of the large-scale) to the upstream steel-PVC connection (x = −14 m), was 13.2 m long. The air supply steel pipe, from the check valve (x = −43.1 m) to the T-junction, was 15.9 m long. The vertical leg of the T-junction was 3.6 m long. The inner diameter of the steel pipes was 206 mm, with a wall thickness of 5.9 mm. The PVC pipe was 275.2 m long and its diameter was 250 mm with an average wall thickness of 7.3 mm. It consisted of two parts. The first part included a PVC inlet pipe and a pipe bridge. It was from the upstream steel-PVC connection to the selected starting point of the test section (x = 0 m) and its length was 14 m. With the aid of a piezometer tube, the bridge was used to set up the initial water column for pipe filling. The second part was the horizontal PVC pipe of length 261.2 m (from x = 0 m to the downstream PVC-steel connection). Most of the measurements took place in this section. The outlet steel pipe connected the downstream end of the PVC pipe to the basement reservoir. It contained two ”segments” of different diameter connected by a reducer. The reducer had a length of 0.3 m and located 0.3 m upstream of the outlet flow meter. The first segment was 8.8 m long and the diameter was 250 mm with a wall thickness of 7 mm. The second segment was 2 m long and the diameter was 200 mm with a wall thickness of 5 mm. The upstream inlet and downstream outlet steel pipe segments are shown in Fig. 7.4, and the corresponding CAD drawing is depicted in Fig. 7.5.

Figure 7.3: The water tank with a constant 25 m head at Deltares, Delft, The Netherlands. It was brought down in 2011. 7.1 Experimental apparatus 145

Figure 7.4: Laboratory view: (a) upstream and (b) downstream steel pipes.

Figure 7.5: Upstream and downstream steel pipe segments and valves: CAD drawing.

7.1.3 Supports and connections

To suppress pipe motion and associated FSI effects [22, 23, 99, 210, 229], it was attempted to structurally restrain the pipe system as much as possible. The PVC pipeline was fixed to the concrete floor by metal anchors and supported with wooden blocks to reduce sagging (see Fig. 7.6a). The pipe bridge – elevated 1.3 146 Filling and Emptying of a Large-scale Pipeline: Experiments and Simulation m above the main pipeline axis – was supported by a tube-frame (see Fig. 7.6a). However, it appeared hard to fix the most downstream elbow; at this point a very heavy mass was attached with a rope to reduce its vertical movement (see Figs. 7.4b and 7.5). The PVC pipeline segments were attached to each other by bolted connections and flanges (see Fig. 7.6a). Wherever the PVC pipe needed to turn its direction, a large radius bend (R = 5DPVC) was used. There were four 90-degree bends in the test section as shown in Fig. 7.2. A long bend was used at the 180 degree turning point as exhibited in Figs. 7.2 and 7.6b.

Figure 7.6: Laboratory view: (a) PVC pipe bridge with the supporting tube-frame and pipe anchors and (b) long turn. 7.1 Experimental apparatus 147

7.1.4 Valves

There were nine valves in the system (see Fig. 7.1). Several small air venting valves are not shown; these mainly located upstream of the pipe bridge. The check valve (x = −43.1 m) was used to prevent entrance of water into the air system. The water inlet valve (x = −34.6 m, unlabelled) connected to the water tank remained open during all experiments. The remaining seven valves numbered from V0 to V6 were actively operated manually or automatically in the experiments (see Figs. 7.1, 7.2 and 7.5). Three of them (valves V0, V2 and V6) were used in the filling process. The upstream service valve V0 (DN200) was operated manually to supply water. The automatic control valve V2 (DN150) was used for flow regulation. A small-size on/off valve V6 in a transparent stand pipe located at the upstream end of the pipe bridge (see Figs. 7.1 and 7.7a) was used for monitoring the initial front of the water column. Four valves (V1,V3,V4 and V5) were used in the emptying process. The manually operated valve V1 (DN300) was used to supply air into the system. The automatically operated valve V3 (DN250) at five diameters distance from V1 was used to regulate the air flow. The downstream manual valve V4 (DN200) was used to regulate the outflow. Its orifice was maxi- mally open at 0 degree position and fully closed at 90 degree position (with ruler as shown in Fig. 7.7b). The relative positions from 9/9-opening to 0/9-opening are henceforth used to characterize the outflow conditions. The manually operated on/off valve V5 (DN200), mounted three diameters downstream from V4, was used to start the emptying process. All valves were butterfly valves except valve V3, which was a cage valve.

Figure 7.7: Laboratory view: (a) transparent stand tube at pipe bridge with small on/off valve and (b) manually operated outﬂow control valve with ruler. 148 Filling and Emptying of a Large-scale Pipeline: Experiments and Simulation

7.1.5 Instruments, uncertainty and data acquisition

A maximum of 28 instruments were installed along the whole system as sketched in Fig. 7.8. There were twelve pressure transducers, six water level meters, four thermometers, three void fraction meters and three flow meters. The coordinates of the pressure transducers and flow meters are shown in Fig. 7.2. The coordinates of the other measuring instruments are listed in Table 7.1, where the type, output range, position (within pipe cross-section) and other detailed information is provided. The three transparent sections had lengths of 0.7 metres with transparent windows 0.5 metres long. A high-speed camera set up at these sections (see Fig. 7.8) recorded the water-air (filling) and air-water (emptying) interface shapes and the air-water mixing process. A removable accelerometer was used to measure pipe vibration amplitude and frequency caused by impacting liquid slugs. The measuring sections were numbered in sequence from upstream to downstream and the instruments were accordingly labelled as indicated in Ta- ble 7.1. It was found that the pressure transducer pdv at x = 269.5 m and the vertex flow meter at x = −47.5 m did not work properly. Hence their measurements are not used in the following data analysis.

Figure 7.8: Layout of measuring instruments in the large-scale PVC pipeline apparatus.

According to nominal values provided by the manufactures, the estimated uncertainty for steady-state conditions was ± 1.0% in the flow-rate measurements, ± 0.1% in the pressure measurements, ± 0.8 oC in the temperature measurements and ± 15 mm in the water-level measurements. All pressure transducers were of the strain-gauge type with a natural frequency of 10 kHz. They were all installed flush-mounted as good as possible. 7.1 Experimental apparatus 149 5 6 7 2 3 4 8 9 1 24 25 26 27 28 29 23 10 11 12 13 14 15 16 17 18 19 20 21 22 data column – type vertex strain-gauge strain-gauge strain-gauge strain-gauge strain-gauge strain-gauge strain-gauge strain-gauge strain-gauge strain-gauge strain-gauge strain-gauge conductivity conductivity conductivity conductivity conductivity conductivity conductivity conductivity conductivity electromagnetic electromagnetic platinum resistance – – – – – top top top bottom left-sideleft-side platinum resistance left-side left-side left-side platinum resistance position right-side right-side right-side right-side right-side right-side right-side C o C C right-side platinum resistance C o o o range 0 – 5 bar 0 – 5 bar 0 – 5 bar 0 – 5 bar 0 – 5 bar 0 – 5 bar 0 – 5 bar 0 – 50 0 – 50 0 – 50 0 – 100 % 0 – 100 %0 – 100 % right-side 0 – 10 bar 0 – 10 bar 0 – 500 L/s 0 – 200000 s -100 – 300 12.163 mm/V12.135 mm/V top-bottom 12.318 mm/V top-bottom 11.932 mm/V top-bottom 12.246 mm/V top-bottom 12.178 mm/V top-bottom top-bottom 9 1 6 9 3 1 3 3 5 7 8 9 1 3 5 7 8 9 1 – section air tank air inlet air inlet air inlet inlet steel pipe 0 – 500 L/s inlet steel pipe 0 – 5 bar inlet steel pipe 0 – 5 bar outlet steel pipe 0 – 5 bar outlet steel pipe 0 – 500 L/s – – 0.5 1.6 1.7 1.6 46.4 46.6 46.6 46.4 -14.3 -27.7 -14.0 -47.5 -46.3 -46.5 141.9 251.7 269.5 252.8 270.3 111.7 183.7 206.8 252.8 111.7 183.7 206.8 252.9 1 3 5 7 8 9 1 6 9 DAQ channel setting and instruments with coordinates and section labelling. u t 1 5 7 8 9 3 1 2 u dv 3t 3b dv – uv air air air T T T p p p p p p p Q p p p VF VF VF p T Q p WL WL WL WL WL WL Q Table 7.1: time pressure pressure pressure pressure pressure pressure pressure pressure pressure pressure pressure pressure flow rate flow rate flow rate water-level water-level water-level water-level water-level water-level temperature temperature temperature temperature void fraction void fraction void fraction 4 5 6 1 2 3 7 8 9 0 – 23 24 25 28 29 22 10 11 12 13 14 15 16 17 18 19 20 21 DAQ channel output mark coordinate (m) 150 Filling and Emptying of a Large-scale Pipeline: Experiments and Simulation

Apart from the uncertainties in the measurements, there were few uncertainties in the filling and emptying process itself. First, the response time of the measuring instruments is different. For example, the difference between the response time of pu (x = −14 m) and upstream electromagnetic flow meter (EMF, x = −14.3 m) was about 1 second. The response time of upstream EMF was taken as t = 0 in pipe filling, and the response time of downstream EMF was taken as t = 0 in pipe emptying. Second, the PVC pipeline was assumed initially empty in each pipe filling run. However, it was found that some water remained between the sections 3 and 8 after a filling and subsequent emptying run. The amount of un- cleared water varied per test. When valve V4 was fully open and high pressure air (more than 1 bar) was used to empty the pipe, the level of remaining water was about 30 mm on average. The water level was up to 60 mm when low pressure air (less than 1 bar) was used for draining. The remaining water is mainly attributed to pipe skin friction and turbulence. Third, before a pipe filling run, the air entrapped in the steel pipe between the T-connection and the check valve was not released. It is not clear what is its effect on the filling process.

Deltares’ 32-channel data acquisition system DAQ was used for synchronized recording of flow rate (inflow Qu, outflow Qdv, air flow Qa), pressure (p1, p3t, p3b, p5, p7, p8, p9), temperature (T1, T3, T9), water level (WL1, WL3, WL5, WL7, WL9) and void fraction (VF1, VF3, VF9); see Fig. 7.8. Video camera and accelerometer recordings were not electronically synchronized with the data acquisition system DAQ recordings. The used channels of the DAQ system are listed in Table 7.1. A sampling rate of 100 Hz was used to record the experimental quantities: discharge, gauge pressure, water level, void fraction and temperature.

7.2 Experimental variables and procedure

7.2.1 Experimental variables

All pipe filling experiments were carried out with the downstream valves V4 and V5 fully open and a constant driving head of 21.4 m (x = −34.6 m, relative to z = 0 m). In the pipe emptying tests, two different experimental variables were systematically investigated. The first one was the upstream driving air pressure, chosen because it highly affects the air-water interface movement. Five different air pressures were used in this investigation. They were 2, 1.5, 1, 0.5 and 0 barg. The last case implies that the draining is due to gravity only (driven by the downstream vertical pipe segment). The second experimental variable was the degree of opening of the outlet control valve V4 (0 degrees – fully open; 90 degree – fully closed). By changing the closing position of valve V4 different outflow rates are obtained and the corresponding movement of the air-water interface is largely affected. Five different valve openings were tested. They were 9/9, 8/9, 6/9, 4/9 and 2/9 open. Not all the test options were combined and the thirteen test conditions used are listed in Table 7.2. 7.2 Experimental variables and procedure 151

Table 7.2: Variables settings in the emptying experiments (valve V4 setting: 9/9 fully open, 0/9 fully closed).

Case 1 2 3 4 5 6 7 8 9 10 11 12 13 † pa 2.0 1.5 1.0 0.5 0.0 2.0 2.0 2.0 2.0 1.0 1.0 1.0 1.0 ϕ‡ 9/9 9/9 9/9 9/9 9/9 8/9 6/9 4/9 2/9 8/9 6/9 4/9 2/9

† tank air pressure (barg); ‡ dimensionless valve opening.

7.2.2 Experimental procedure

To describe the procedure for pipe filling and emptying experiments, Fig. 7.5 is used. In the pipe filling experiments, air at atmospheric pressure initially present in the system is replaced by water. Both downstream valves V4 and V5 are initially open. First, the upstream valve V0 was opened manually. Then the automatic valve V2 was opened from 0% to 15% until the height of the water in the pipe bridge reached a level of 0.4 metre (0.04 barg reading from pressure transducer pu at x = −14 m as shown in Fig. 7.2). After closing valve V2 the water level gradually approached the desired height (1 m) due to a small leakage of V2. When the required water level of 1 m (x = −6.5 m) was reached, as visually observed from the small transparent stand pipe (see Fig. 7.7a), the small-size on/off valve on it was closed, and valve V2 was fully opened immediately. Then the filling process started at t = 0. After some time (about three minutes) a steady state was reached, i.e. the inlet and outlet flow discharges were equal and constant. Then the outlet control valve V4 was closed slowly from 0 to 75 degrees to avoid possible pressure surges. After the gradual closure of valves V5 and V2 the filling process was completed.

Water entirely filling the pipeline was driven out by compressed air from the upstream high-pressure tank (pressure initially fixed between 0 barg and 2 barg). The initial conditions for each emptying run were established by the completed filling process. Air entrapped in the high-elevation air supply line (elevation 1.2 m) was ventilated through the small-size air-venting valve mounted five diameters downstream of the check valve. Similarly, air entrapped in the pipe bridge was released by a ventilating hose connected to its top. Water supply valve V0 was then manually closed, so that the unwanted leakage of valve V2 in closed position was eliminated. Then valve V4 was set to the desired degree of opening for a controlled emptying process. After valve V1 was opened (valve V3 was always open at 15% for flow regulation), the static water-column in the system was pressurized by the high-pressure air. When the induced pressure oscillations had become small enough, valve V5 was opened manually as quickly as possible and then the emptying process started at t = 0. After the main air-water interface arrived at the pipe end and all water slugs were driven out of the system, valve V1 was closed and the emptying process was considered completed. 152 Filling and Emptying of a Large-scale Pipeline: Experiments and Simulation 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Step tprecordings Stop V1 possible manually as Close quickly as V5 manually Open 15% to opened always is V3 and V1 manually Open degree desired to V4 manually Set V0 manually bridge down- Close pipe pipe and valve supply check of air stream including degrees system 75 Ventilate to (gradually) V2 degree automatically Close 0 from (gradually) V4 manually % 100 Close to 0% from V2 automatically Open % 0 to 15% from V2 automatically Open % 15 to 0% from V2 automatically Open V0 manually Open recordings Start Operation al 7.3: Table xeietlpoeuefracmlt iefiln n mtigcycle. emptying and filling pipe complete a for procedure Experimental eoecoigV atutlalsusaeot(visual out are slugs inspection) all audible and until wait V1 closing Before emptying during drops pressure Air stabi- for seconds 60 lization for wait and system Pressurize pressure air than less pressure System (gradually) V5 manually close Also seconds 60 started for closing wait V2 and Also steady is system until Fill bridge) pipe in pressure until Wait filling) pipe pressure until Wait V5 and V4 or valves downstream before Open value desired filling to during pressurization tank Air Description p p u u ie pt . a wtrlevel (water bar 0.1 to up rises ie pt .4br(vertical bar 0.04 to up rises mtigends Emptying starts Emptying emptying for Initialization ends Filling starts Filling filling for Initialization Note 7.3 Experimental results 153

In fact, the filling and emptying experiments were continuously performed one after the other. The whole procedure including initialization for filling, pipe filling, steady-state water flow, initialization for emptying (air ventilation, valve V4 adjustment and water-column pressurization) and pipe emptying is detailed in Table 7.3. For every test, the runs were repeated at least five times for nominally the same initial and boundary conditions to assess the repeatability of the unsteady two-phase flow in the pipeline and to enable statistical and error analysis.

7.3 Experimental results

7.3.1 Steady-state water ﬂow

Seventy eight steady-state flow measurements have been carried out in between the filling and emptying experiments, and eight of them were used to determine the head losses due to skin friction and due to the 180 degree long turn (see Fig. 7.2) [85, 105]. The velocity of the steady flow is about 4 m/s. The Reynolds number is about 950000. The calibrated friction factor is f = 0.0136. The head loss due to the 180 degree smooth turn is negligible because the minor loss coefficient klb = 0.06 is small [46]. The pipe relative roughness is about 0.00011, which is slightly smaller than that of the PVC pipe (0.00015) used by Liou & Hunt [116].

The hydraulic grade line for the PVC pipe is straight as shown in Fig. 7.9. This is consistent with the above calculated negligible head loss due to the 180 degree long turn. The pressure head at x = −27.7 m is 13.5 m. Including entrance loss and velocity head (0.75V2/g = 1.2 m), the large head loss 21.4 − 3.1 − 13.5 − 1.2 = 3.6 m at the very beginning is because of the head losses mainly due to valves V0 and V2 in Fig. 7.8. The head loss coefﬁcient of one fully opened valve is about 1.8 Kv = 42/(2×9.8) = 2.2, which is high comparing to normal fully open butterﬂy valves [46].

7.3.2 Pipe ﬁlling

Inflow rate. The flow rates measured by the upstream and downstream flow meters are shown in Fig. 7.10. Without losing information, the early part of the outflow measurements is left out for clarity. Good repeatability is exhibited by three representative runs. The arriving time of the water front at the downstream flow meter (x = 270.3 m) is 54 ± 0.8 s. The area covered by the inflow rate curve between t = 0 and t = 54 s and the x-axis is the amount of water that has entered the pipeline when the water front arrives at the downstream flow meter. The averaged value of this area is 11150 litres. It is less than the pipe volume ∀ = 2 × × 3 AL = πDPVC/4 (270.3 + 6.5) 10 = 12047 litres. The volume discrepancy is equivalent to a 20.6 m long PVC pipe. This implies that when the water column 154 Filling and Emptying of a Large-scale Pipeline: Experiments and Simulation

10 Head (m)

−5 −50 0 50 100 150 200 250 Locations (m)

Figure 7.9: Hydraulic grade line for the entire pipeline. arrives at the downstream ﬂow meter, not all air is expelled out of the system. This matter is further examined in next section.

Run 1 250 Run 2 Run 3

200

150

inflow outflow 100 Flow rate (L/s)

0 20 40 60 80 100 Time (s)

Figure 7.10: Measured inflow and outflow rates in three typical pipe filling runs.

Now we focus on the measurements by the upstream flow meter. As shown in Fig. 7.10, the flow rate first rises to its maximum value of 270 ± 3 L/s in about 4 seconds time. Then it experiences a rapid decreasing from 270 L/s to 250 L/s in 2 seconds time. After that, the flow gradually decelerates until a steady state is reached. The inflow velocity V – determined from the averaged flow rate of 9 runs – is shown in Fig. 7.11 by the solid line. The trend of the curve is the same as 7.3 Experimental results 155 that in the small-scale experiments of Liou & Hunt [116]. The filling time is much longer (about 57 seconds) because of the large-scale. The measured maximum velocities are high (about 6.2 m/s) due to the 25 m high (relative to the inlet at x = −34.6 m) water tank driving the flow. The friction head loss at steady state in the whole system is about L V2 34.6 + 271 42 h = f = 0.0136 × × = 13.6 m. f D 2g 0.2354 2 × 9.8

6 V 1 5 V

3 Velocity (m/s) 2 V 2

0 0 20 40 60 80 100 Time (s)

Figure 7.11: Velocity history of the inﬂow and the water column fronts in the ﬁlling experiments. Solid line – adapted from upstream EMF measurements; symbols – indirect measurements (squares – V1 – leading front, diamonds – V2 – secondary front).

Water-air interface. Since the initial water front (water-air interface) splits into two water fronts during the filling process, the sketch in Fig. 7.12 is introduced for the sake of clearness. To determine the water front velocities V1 and V2, the measuring method of [116] is used. Two different groups of instruments are used as the front timing sections. The first group consists of the six water level meters (WL1, WL3, WL5, WL7, WL8 and WL9) located at x = 1.7, 46.4, 111.7, 183.7, 206.8 and 252.9 m. The second group consists of the pressure transducers (p1, p3b, p5, p7, p8 and p9) located at x = 1.6, 46.6, 111.7, 183.7, 206.8 and 252.8 m. Except for pressure transducer p3b located at the pipe bottom, the other five transducers are in a horizontal plane with the pipeline axis. The water level meters are in a vertical plane with the pipeline axis. The arrival times of the advancing water fronts at the timing sections are captured by the two groups of instruments. The average speeds of the fronts are then computed from the distances between the timing sections and the travel times.

The determined water front velocities are shown in Fig. 7.11 by symbols; the squares indicate V1 and the diamonds V2. Note that the velocities V1 and V2 156 Filling and Emptying of a Large-scale Pipeline: Experiments and Simulation

Figure 7.12: Water front evolution in the filling process – one original water front splitting into two fronts: (a) initial state, (b) early stage of filling and (c) forming of two water fronts. are velocities averaged over the trajectory between two subsequent sections. The location of the symbols indicate the arrival times of the water fronts at the timing sections. Only velocities determined from water level meters are presented in Fig. 7.11. This is because the pressure transducers gave results close to that from the water level measurements for the leading front 1, whilst they could not accurately sense the arrival of the secondary front 2. The reason why the pressure transducers do not work well for detecting front 2 is the small pressure rise due to the arriving of water front 2. The water front starts from x = −6.5 m and has splitted into two fronts when it arrives at section 1 (x = 1.7 m). The overall trend of the velocities of both water fronts follows the inflow measurement.

The timing sections, arrival times and water front velocities are listed in Tables 7.4 and 7.5. When water front 1 is travelling between section 3 (x = 46.4 m) and section 9 (x = 252.9 m), the velocity difference between V and V1 is roughly 0.4 m/s, except at section 8 (x = 206.8 m) where it decreases to 0.24 m/s. Similarly, before water front 2 arrives at section 9, the velocity difference between V and V2 is about 1 m/s except at section 8 (x = 206.8 m), where the difference increases to 1.25 m/s. The velocity variation at section 8 is because of a hydraulic jump occurred between sections 8 and 9 as shown below from the water level meter measurements.

The water level meter measurements for three repeated runs are shown in Fig. 7.13. A good repeatability is seen again. The splitting of the water front is evident. The water levels in one typical run are shown in Fig. 7.14. As shown in Figs. 7.13 and 7.14, the pipeline is not entirely empty before a ﬁlling test. Water is mainly present between section 3 (x = 46.6 m) and section 8 (x = 206.8 m), and has the highest level 35 mm at section 5 (x = 111.7 m). This is because of the incomplete emptying before the ﬁlling test. The shown results are from the tests using 2 bar air pressure for emptying. When lower air pressures were used, more water 7.3 Experimental results 157

Table 7.4: Propagation of water front 1 in the ﬁlling process. The unit for x, t and V is m, s and m/s, respectively.

Section x t1 V1 V V1 − V 1 1.7 1.61± 0.07 5.09 2.61 2.48 3 46.4 9.12± 0.16 5.96 5.66 0.30 5 111.7 21.09± 0.36 5.46 5.05 0.41 7 183.7 35.28± 0.33 5.07 4.50 0.43 8 206.8 40.20± 0.35 4.70 4.36 0.24 9 252.9 50.34± 0.44 4.55 4.10 0.45

Table 7.5: Propagation of water front 2 in the ﬁlling process. The unit for x, t and V is m, s and m/s, respectively.

Section x t2 V2 V V − V2 1 1.7 5.44± 0.18 1.38 5.84 4.46 3 46.4 14.51± 0.25 4.92 5.30 0.38 5 111.7 32.48± 0.40 3.63 4.56 0.96 7 183.7 56.17± 0.59 3.04 4.00 0.96 8 206.8 64.39± 0.36 2.81 4.06 1.25 9 252.9 80.42± 0.41 2.87 4.03 1.16

stayed in the system. The evolution of the water front then becomes much more complex and the repeatability of the tests is much less.

Figure 7.14 indicates that the shape of water front 1 does not change much until it arrives at section 9 (x = 252.9 m). Its ﬁnal length is about 3 metres as estimated from the rise time (0.5 – 0.75 s) and the measured velocity. The average height of water front 1 is more or less constant (193 ± 8 mm) until it arrives at section 9 (x = 252.9 m) where it increases to 230 mm. Water front 2 reaches section 1 (x = 1.7 m) at about t = 5.4 s. Its shape slightly changes with time. The average height of water front 2 is approximately constant until it arrives at section 8 (x = 206.8 m). It decreases to 6 mm at section 9 (x = 252.9 m). The dramatic change of the height of the two water fronts indicates that a large ﬂow regime transition (hydraulic jump) must have occurred between sections 8 (x = 206.8 m) and 9 (x = 252.9 m). The distance between the two water fronts lengthens with time. It is about 20 m when water front 1 arrives at x = 1.7 m (t = 1.6 s), and it increases to 75 m when water front 1 arrives at x = 206.8 m (t = 40.2 s). According to these observations, a quantitative analysis of the water-air interface evolution is sketched in Fig. 7.15. 158 Filling and Emptying of a Large-scale Pipeline: Experiments and Simulation

(a) WL1 (b) WL3 250 250

200 200

150 150

100 100 Water leverl (mm) Water leverl (mm)

50 50 Run 1 Run 1 Run 2 Run 2 Run 3 Run 3 0 0 1 2 3 4 5 6 9 10 11 12 13 14 15 Time (s) Time (s)

200 200

150 150

100 100 Water leverl (mm) Water leverl (mm)

50 50 Run 1 Run 1 Run 2 Run 2 Run 3 Run 3 0 0 20 22 24 26 28 30 32 34 35 40 45 50 55 Time (s) Time (s)

(e) WL8 (f) WL9 250 250

200 200

150 150

100 100 Water leverl (mm) Water leverl (mm)

50 50 Run 1 Run 1 Run 2 Run 2 Run 3 Run 3 0 0 40 45 50 55 60 65 50 55 60 65 70 75 80 85 Time (s) Time (s)

Figure 7.13: Water levels at six different locations in three repeated ﬁlling runs: (a) WL1, (b) WL3, (c) WL5, (d) WL7, (e) WL8 and (f) WL9. See Tables 7.1 and 7.4 for locations.

The evolution of the water-air interface indicates the intrusion of air. When water front 1 has reached the pipeline end, the pipe at section 7 (x = 183.7 m) is still not full (see Fig. 7.14). The thickness of the air layer between the two water fronts 7.3 Experimental results 159

250

200

150

100 Water leverl (mm) WL1 WL3 50 WL5 WL7 WL8 WL9 0 0 10 20 30 40 50 60 70 80 90 Time (s)

Figure 7.14: Water level changes at six different locations in one typical ﬁlling run.

Figure 7.15: Illustration of air intrusion in the filling experiments (unit: m). The intruded air lengthens while its thickness does not change in stages (b), (c), (d) and (e). A hydraulic jump occurred between stages (e) and (f), after which entrapped air is gradually expelled from the system. is more or less constant before the occurrence of the hydraulic jump. The length of the air layer increases with time. That is, with the advancement of the water fronts, more air enters the water column from its top. This is consistent with 160 Filling and Emptying of a Large-scale Pipeline: Experiments and Simulation the conclusion drawn from the flow rate measurement that the system is not full when the leading front arrives at the downstream flow meter.

Based on the heights of the water fronts shown in Figs. 7.13 and 7.15 and together with Fig. 7.11 and Tables 7.4 and 7.5, the conservation of volume at the moment (t = 21 s) that water front 1 arrives at section 5 (x = 111.7 m) is checked. The 3 inﬂow rate is Qin = VA = 5.05 × 0.0435 = 0.22 m /s. The volume change at the 3 water fronts is Qout = V1A1 + V2A2 = 5.46 × 0.0381 + 4.5 × 0.0054 = 0.23 m /s. The areas A1 and A2 are obtained from the water level 192 mm. Velocity V1 is taken from Table 7.4 and V2 is interpolated from Table 7.5. Approximately Qin equals Qout. Thus the conservation of volume is veriﬁed.

The volume of the intruded air is also checked. When water front 1 arrives at x = 206.8 m (t = 40.2 s), the volume of the filled water (covering area of the flow rate curve between t = 0 and t = 40.2 s) is ∀1 = 8700 litres. The initial 3 voided pipe volume is ∀2 = LA = (206.8 + 6.5) × 0.0435 × 10 = 9275 litres. Hence the volume of the intruded air is ∀air = ∀2 − ∀1 = 575 litres. We used the measured water level (191 mm) to calculate ∀air too. The air volume on top of the stratified water platform is 0.0057 × 103 × 75 = 430 litres. The air volume on top of the wedge-shaped water front is about 100 litres (estimated from the length and the height of the wedge-shaped front). Consequently, the intruded air volumes calculated using the two approaches are more or less the same. The discrepancy is because of the oscillation of the stratified free surface (spikes in Fig. 7.13e). This verification confirms the water level measurements.

Pressure. Pressure histories at different locations along the PVC test pipeline are shown in Fig. 7.16. When the water column arrives at the downstream bend (x = 267 m) at t = 54 s, the recorded pressures show some oscillations due to impact. The magnitude of the oscillation at section 9 (x = 252.9 m) is 0.1 barg, 2 × × 2 5 which is approximately equal to 0.5ρV1 = 0.5 1000 4.5 /10 = 0.1 barg.

The pressure histories of transducers pu (x = −14 m) and puv (x = −27.7 m) are shown in Fig. 7.17 together with p1 (x = 1.6 m). At the beginning of ﬁlling, pu and puv experience a rapid change (short lived peak for 3 seconds) and then gradually increase as p1 does. The rapid change of pu and puv may be attributed to the entrapped air between the T-junction (x = −27.2 m) and the check valve (x = −43.1 m). The peak in driving pressure can be part of the reason for the short-lived ﬂow rate peak (see Fig. 7.11).

The pressure distribution (hydraulic grade line) at three time levels are shown in Fig. 7.18. The chosen times are the instants when water front 1 arrives at the transducers p5 (x = 111.7 m), p7 (x = 183.7 m) and p9 (x = 252.8 m). The linear pressure distribution along the water column means a uniform pressure gradient decreasing in time. This is consistent with the observation in the small- scale ﬁlling experiment studied in Chapter 5 (see Fig. 5.6). It implies that the advancing water column behaves like a rigid column, although air intrusion takes place. In addition, it means that the intruded air is not under atmosphere. This 7.3 Experimental results 161

p 1.2 1 p 3b p 1 5 p 7 p 8 0.8 p 9

0.6

Pressure (barg) 0.4

0.2

0 10 20 30 40 50 60 70 Time (s)

Figure 7.16: Pressure history at six different locations along the PVC test section in one typical run. See Table 7.1 for locations.

1.5 p uv p u p 1

0.5 Pressure (barg)

0 10 20 30 40 50 60 70 80 Time (s)

Figure 7.17: Pressure history at x = −27.7 m (puv) and −14 m (pu) in a typical ﬁlling run.

can be clariﬁed as follows. When water front 1 arrives at section 9 (x = 252.8 m), water front 2 does not arrive at section 7 (x = 183.7 m) (see Fig. 7.14). If the air is under atmospheric pressure, the pressure distribution between sections 7 and 9 will be ﬂat and close to zero. This is apparently not the case shown in Fig. 7.18. 162 Filling and Emptying of a Large-scale Pipeline: Experiments and Simulation

1.2 t = 21.1 s t = 35.4 s t = 50.6 s 1

0.8

0.6

Pressure (barg) 0.4

0.2

0 0 50 100 150 200 250 Location (m)

Figure 7.18: Pressure distribution along water column in the ﬁlling process at three time levels when water front 1 arrives at section 5 (x = 111.7 m) – circles, section 7 (x = 183.7 m) – squares, and section 9 (x = 252.9 m) – diamonds. The x-coordinates of the symbols are the locations of the pressure transducers.

7.3.3 Pipe emptying

Outflow rate. The outflow rates measured in case 1 (see Table 7.2) are depicted in Fig. 7.19. The shown measurements are from rest to the instant when the leading air front (air-water interface) arrives at the outlet flow meter. Good repeatability is verified by three typical runs. Three cases where the repeatability is not well established are the cases 5, 9 and 13. They represent the more extreme conditions (case 5: draining under gravity only; cases 9 and 13: 2/9 opening of the downstream butterfly valve V4) and for the time being will not be further processed due to their lack of repeatability. The outflow rates for other cases can be found in [85].

The water column acceleration changed roughly in three stages. After the sudden opening of the downstream valve V5, the flow accelerated rapidly with water hammer from rest. Then followed a slowly increasing discharge (more or less linear). Before the air-water front visibly and audible flushed out from the outlet, a second rather rapid flow acceleration occurred. The profile is very similar to the numerical simulation of the small-scale pipe emptying studied in Section 5.4.3. Similar trends are observed in cases 2, 3, 4, 6, 7, 10 and 11. In cases 8 and 12 the second rapid acceleration is absent (see Fig. 7.20). The detailed information for the evolution of the leading air front is listed in Table 7.6. The maximum difference of the emptying time is about 2.5 seconds, and the maximum flow rate difference is about 20 L/s relative to 300 L/s outflow rate. The maximum value of the standard deviation, determined for all valid cases, is less than 3.4 7.3 Experimental results 163

400 Run 1 Run 2 350 Run 3

300

250

200

150 Flow rate (L/s)

100

0 0 10 20 30 40 50 Time (s)

Figure 7.19: Outﬂow rate in the emptying experiment. Case 1: 2.0 barg initial air-circuit pressure and 9/9 opening of valve V4.

percent. The arrival time is the instant that the leading air front arrives at the downstream flow meter. It is designated as td (drainage time). Similar to pipe filling, the area covered by the outflow rate curve from t = 0 until t = td and the x-axis is the volume Vd of water driven out of the system. The volume of the initial water in the pipeline between the check valve (x = −43.1 m) and the downstream flow meter (x = 270.3 m) is about ∀ = L1A1 + L2A2 + L3A3 = (29.1 × 0.0296 + 275.2 × 0.0435 + 9.1 × 0.0437) × 103 = 13230 litres, which is larger than the integrated drainage ∀d in Table 7.6. When valve V4 is fully open, the drainage at td increases with the driving air pressure. For a given driving air pressure, the drainage decreases with reducing of the opening of valve V4. The averaged volume Vd of the expelled water is 10309±498 L. Apparently, not all the water is driven out of the system at time td, which implies that the initial planar air-water interface has stratified during draining. This becomes more clear from the water level meter measurements examined below. With respect to case 1, the volume of the remaining water is equivalent to a 55 m long PVC pipe. When the remaining water is assumed to locate between section 3 (x = 46.4 m) and the downstream bend (x = 267 m), its level is about 70 mm. It is higher than the water levels observed in Fig. 7.14, because the emptying (after td) continues with some water slugs (see water level measurements in next section).

To show the effect of the driving air pressure pair and the resistance of valve V4, ten typical flow-rate curves for the ten cases are presented in Fig. 7.20. Intuitively, flow with 1.0 barg driving pressure and 4/9-opening of valve V4 (case 12) should have the lowest flow rate and thus the slowest emptying process, while flow with the highest driving air pressure (2 barg) and the largest opening of valve V4 (9/9- 164 Filling and Emptying of a Large-scale Pipeline: Experiments and Simulation

Table 7.6: Evolution of the leading air front in the emptying process. The unit for pair, td, Qmax and ∀d is barg, s, L/s and L, respectively.

Case pair ϕ td Qmax ∀d 1 2.0 9/9 47.24± 0.59 361.24 ± 1.08 10807 2 1.5 9/9 51.96± 1.22 330.07 ± 3.90 10597 3 1.0 9/9 59.21± 0.57 285.99 ± 0.96 10321 4 0.5 9/9 71.96± 1.19 230.90 ± 4.02 10027 6 2.0 8/9 46.56± 1.10 372.64 ± 4.17 10783 7 2.0 6/9 50.28± 0.60 307.18 ± 6.50 10680 8 2.0 4/9 65.66± 0.33 183.25 ± 1.38 10349 10 1.0 8/9 59.11± 1.01 293.75 ± 9.74 10381 11 1.0 6/9 63.19± 0.50 239.54 ± 3.17 10317 12 1.0 4/9 83.43± 0.48 137.16 ± 1.75 9811

opening) (case 1) should result in the highest flow rates and consequently the most rapid emptying process. The first guess is obviously correct as shown in Table 7.6 and Fig. 7.20. But next to case 1 the flow in case 6 (pa = 2 barg, 8/9- opening of valve V4) has the largest flow rates and shortest emptying time. This may be attributed to the resistance characteristics of the downstream butterfly valve V4, i.e. 8/9 is more or less fully open, as can also be seen from cases 3 and 10. The flow rate increases more quickly with a higher driving pressure with respect to a given opening of valve V4. With a given driving pressure, the flow rate decreases with the reducing of the valve opening. They are very similar as the numerical simulations shown in Fig. 5.25.

For convenience the valid 10 test cases are classified into three groups according to the magnitudes of the outflow rates. The first group includes cases 1, 2, 6 and 7, of which the magnitudes are higher than 300 L/s (velocity is higher than 6.9 m/s). The second group includes cases 3, 4, 10 and 11, of which the final magnitudes are higher than 200 L/s (velocity is higher than 4.6 m/s). The last group includes cases 8 and 12, of which the flow rates are lower than 200 L/s. The three groups are referred to as violent, intermediate and gentle emptying, respectively.

The measurements of the upstream EMF for the different test cases are also examined. It is found that there is a time difference between the start time of the two EMFs (see Fig. 7.21). The time difference is approximately the water hammer wave travelling time (L/c) in the water column. Before the leading air front passes the upstream EMF, the measured ﬂow-rate curve shows the same trend as that of the downstream EMF (more or less parallel but without ﬂuctuations).

Air-water interface. During emptying, the original vertical front (air-water interface) becomes stratiﬁed, forming a main air front and a ”hold-up”. For clearness, the sketch for the air front evolution during draining is shown in Fig. 7.22. It is 7.3 Experimental results 165

400 Case 1 Case 2 350 Case 3 Case 4 300 Case 6 Case 7 Case 8 250 Case 10 Case 11 Case 12 200

150 Flow rate (L/s)

100

0 0 10 20 30 40 50 60 70 80 90 Time (s)

Figure 7.20: Outﬂow rates in the emptying experiment for 10 different conditions (Table 7.2. The time duration of the extracted data is from the opening of downstream valve at t = 0 s to the instant td when the main air front passes the downstream ﬂow meter.

300 Run 1 Run 2 Run 3 250 downstream EMF

200

150 Flow rate (L/s) 100 upstream EMF

0 0 10 20 30 40 50 60 Time (s)

Figure 7.21: Flow rate measurements of the upstream and downstream ﬂow meters in the emptying experiment (case 10).

different from the evolution of the water-air interface in pipe ﬁlling (see Fig. 7.15). 166 Filling and Emptying of a Large-scale Pipeline: Experiments and Simulation

Figure 7.22: Illustration of air intrusion in the emptying experiments (unit: m). The air forms a main front and water hold-up behind it.

To determine the velocity of the main air front, the water level meters are used again as the timing points. Pressure transducers are not used because it is difﬁcult to accurately determine the arriving time of the air due to much noise. The results for case 6 are shown in Fig. 7.23. The location of the dots indicate the arrival times of the leading air front at the timing sections. They have the same trend as the outﬂow-rate curves.

4 Velocity (m/s) 3

0 0 10 20 30 40 50 Time (s)

Figure 7.23: Velocity history of the outﬂow (solid line) and the main air front (dots) in the emptying experiments (case 6).

The velocity differences between the outﬂow and the main air front are listed in 7.3 Experimental results 167

Table 7.7. The difference during large acceleration periods – air front arrives at section 1 (x = 1.7 m) and section 8 (x = 206.8 m) – is about 1 m/s for the examined six cases. During the ”linear” acceleration period, the velocity difference is more or less constant (about 0.75 m/s). At the late stage of emptying (the air front is at section 9 (x = 252.8 m)), the averaged velocity difference is 0.83 m/s.

Table 7.7: Velocity differences between the outﬂow and the leading air front in the emptying process. It is also the air intrusion velocity relative to the outﬂow (Unit: m/s).

Case \ x (m) 1.7 46.4 111.7 183.7 206.8 252.9 1 0.98 0.74 0.78 0.74 1.01 0.82 2 1.05 0.74 0.76 0.71 1.02 0.81 6 1.00 0.71 0.74 0.74 1.04 0.84 7 0.99 0.77 0.73 0.71 0.98 0.79 10 1.03 0.75 0.76 0.77 1.05 0.87 11 1.01 0.74 0.78 0.76 0.98 0.82

The water level meter measurements in case 6 (most violent flow in group 1) are shown in Fig. 7.24. For clearness the time interval between two points is 0.25 s. Behind the main air front, a water ”tail” known as ”hold-up” in normal slug flow (see Chapter 6) is formed. That is, with compressed air flowing in from upstream, not all water initially filling in the system is immediately flushed out. This is consistent with the observation from Table 7.6 and Fig. 7.23. After the passing of the air front, the ”tail” at section 1 (x = 1.7 m) is small (about 10 mm deep) and the air-water interface is almost planar. At the other sections, the ”tail” is thicker, e.g. about 100 mm at x = 46.6 m. The other three cases in group 1 confirmed this flow regime transition (stratification). The development of the water ”tail” is because of gravity, turbulence and pipe skin friction. The water level change of the ”hold-up” after t = 45 s is because of the formulation of slugs. The water level increase at section 1 after t = 45 s may be because of the flow returning from the 180 degree long bend. Another possible contribution is the flushing of the water remaining ahead of the pipe bridge. Its thickness is about 20 mm. For the intermediate and gentle cases, similar trends as in the violent case are observed. The only difference is the ”tail” thickness [85].

By taking case 6 as an example, the volume of the remaining water is checked. When the leading air front arrives at section 8 (x = 206.8 m) at t = 40.2 s (see Fig. 7.23), the volume of the expelled water (area integral of the outflow rate curve between t = 0 and t = 40.2 s) is ∀1 = 8585 litres. The volume of the initially filled 3 water is ∀2 = L1A1 + L2A2 = [29.1 × 0.0296 + (14 + 206.8) × 0.0435] × 10 = 10466 litres. Hence the volume of the remaining water is ∀rw = ∀2 − ∀1 = 1881 litres. We can also calculate ∀rw using the measured water levels at t = 40.2 s. From Fig. 7.24, the water levels at sections 3, 5, 7 and 8 is close to a constant 65 mm. The water level is 6 mm at section 1. We assume that the stratified flow starts from the 168 Filling and Emptying of a Large-scale Pipeline: Experiments and Simulation

250 WL1 WL3 WL5 200 main WL7 air font WL8 WL9 150

100 Water leverl (mm)

50 hold−up

0 0 10 20 30 40 50 60 70 80 Time (s)

Figure 7.24: Water levels at six different locations along the PVC pipeline in the emptying experiments for the most violent case (case 6).

middle of sections 1 and 3, i.e. x = 22.4 m with level of 65 mm. Then the volume of the remaining water is (206.8−22.4)×0.0098×103 = 1808 litres. The volumes of the remaining water calculated using the two approaches are approximately the same (relative difference is 4 percent). This conﬁrms the accuracy and consistency of the measurements in pipe emptying tests.

The conservation of volume at the moment (t = 37 s) when the leading air front arrives at section 7 (x = 183.7 m) is also checked. The outflow velocity reading from Fig. 7.23 is 6.45 m/s. The velocity of the leading air front is 7.19 m/s. The level of the water tail (stratified flow) is about 58 mm which is averaged from the water levels at sections (see Fig. 7.24) and hence the cross-sectional area of the tail 2 3 is 0.0083 m . The outflow rate is Qout = VA = 6.45 × 0.0435 = 0.281 m /s. The air flow rate at the leading water front is Qa = VaAa = 7.19 × 0.0352 = 0.253 3 m /s. The difference between Qout and Qa is attributed to the movement of the water tail.

Pressure. The pressure measurements for case 6 are shown in Fig. 7.25. The results for cases 10 and 12 can be found in [85]. The pressure oscillations within the ﬁrst 20 seconds are due to water hammer caused by the fast opening of the valve. For the violent and intermediate cases (cases 6 and 10), the water-pressure variations are characterized by a concave-up increase, until air arrives at the measurement sections. For the gentle case (case 12), the curvature is insigniﬁcant and the water-pressure increases more or less linearly. After some time, when the air front propagates through the downstream end, impact of water slugs at the downstream bend causes pressure surges from the end of the pipeline up to section 7 7.3 Experimental results 169

(x = 183.7 m). Due to the large forces exerted by the water slugs, the downstream bend had large visible movements. The number of slugs varied from 2 to 4 for the three different cases. After the air front passes a section, the pressure at that place equals the driving air pressure. Consequently, the driving air pressure decreases more or less linearly in time during emptying. This is conﬁrmed from the recordings of pair at x = −46.5 m. The pressure change has similar shape as the velocity variation shown in Fig. 7.19.

first arrival of air at different locations p 2 1 p 5 p 7 p 1.5 8 p 9

0.5 Pressure (barg)

0 slug impact

−0.5 −20 0 20 40 60 80 100 120 Time (s)

Figure 7.25: Pressure history at ﬁve different locations along the PVC pipeline in the emptying experiments for the most violent case (case 6).

From the pressure history at section 9 (see Fig. 7.26), the Joukowsky pressure drop, i.e. ∆P = ρc∆V, is checked. The initial pressure decrease is 2.5 bar. The corresponding velocity change is 0.75 m/s as shown in Fig. 7.23. Accordingly, the pressure variation due to the velocity change is ∆P = ρc∆V = 1000 × 348 × 0.75 = 2.6 bar. The speed of sound c = 348 m/s is obtained from the water hammer tests [23]. This agrees well with the observed pressure change (the difference is about 4 percent). With the shortening of the water column, the pressure amplitudes becomes smaller and smaller. After about 12 seconds, the water-hammer event has been damped out. The pressure then smoothly increases.

The pressure distributions along the moving water column at the instant when the air front arrives at section 5 and section 7 are shown in Fig. 7.27. The straight ”hydraulic grade line” means a constant pressure gradient and a rigid column. With the shorting of the water column, pressure at section 9 (x = 252.9 m) increases. 170 Filling and Emptying of a Large-scale Pipeline: Experiments and Simulation

1.5

0.5 Pressure (barg)

−0.5 0 5 10 15 20 Time (s)

Figure 7.26: Pressure history at section 9 (x = 252.8 m) in case 6.

1.8 t = 26.1 s t = 36.8 s 1.6

1.4

1.2

0.8 Pressure (barg)

0.6

0.4

0.2 100 120 140 160 180 200 220 240 260 Location (m)

Figure 7.27: Pressure distribution along the water column in the emptying process (case 6) at two time levels: t = 26.1 s when the air front arrives at section 5 (x = 111.7 m) – circles and t = 36.8 s when it arrives at section 7 (x = 183.7 m) – squares. The x-coordinates of the symbols are the locations of the pressure transducers. 7.4 Numerical simulation of pipe ﬁlling

The rigid-column model developed in Section 5.2.1 is applied to the current large- scale experiments. Only the pipe ﬁlling process is simulated herein. One is re- 7.4 Numerical simulation of pipe ﬁlling 171 ferred to our recent paper [105] for the large-scale pipe emptying problem.

The location of the pressure transducer pu (x = −14 m) is taken as the upstream boundary and the measured pressure (see Fig. 7.17) is applied as the driving pressure. The initial water column length is L0 = 7.5 m. The friction factor is f = 0.0136. The fourth-order Runge-Kutta method is used for the time integration of Eqs. (5.15) and (5.16).

The rigid-column solution is compared with the laboratory test in Fig. 7.28. The predicted flow rate history has the same trend as the experiment. The captured short-lived peak at t = 4 s is because of the initial peak in the driving pressure pu. The smaller peak is attributed to the 3D effect of the pipe bridge (complex overflow). Uncertainties concerning the measured pipe bridge geometry and the response time of the pressure transducer may have some effect too. It was found that the numerical peak flow rate is sensitive to the length of the upstream branch of the bridge. As mentioned in Section 7.1.5, the response time of the pressure transducer is approximately 1 second earlier than the flow meter. This gives some flexibility in choosing the starting time of the driving pressure.

Laboratory test 250 Rigid−colum model

200

150

Flow rate (L/s) 100

0 0 10 20 30 40 50 60 Time (s)

Figure 7.28: Flow rate history for pipe ﬁlling test 4015 (solid line) and 1D numerical simulation (dashed line).

Chapter 8

Conclusions and Recommendations

In this thesis we presented a Lagrangian particle method for solving the 2D Euler equations and 1D Navier-Stokes equations with applications to water hammer, rapid pipe ﬁlling and emptying, and isolated slug propagation and impact in an empty pipeline. The implemented particle solver has been thoroughly tested against a selection of 2D examples. New large-scale laboratory tests have been performed, simulated and analysed.

8.1 Concluding remarks

• The Lagrangian meshless SPH method is a proper approach for inertial driven flows with free and moving interfaces. It globally conserves mass and momentum. The applied image particle approach efficiently overcomes the boundary deficiency problem in SPH and its implementation is straightforward. The incompleteness of SPH due to discretisation is remedied by using modified kernels. The convergence rate of SPH is first order. Due to particle movement neighbour search is needed, leading to higher computational costs compared to traditional mesh-based methods.

• Starting from a non-equilibrium initial state, SPH was able to solve a range of 2D steady ﬂow problems very well.

• The 1D SPH results for rapid pipe ﬁlling agree with the experimental data of Liou & Hunt [116].

• For the water-hammer problem, the 1D SPH solver and its variations (e.g. 174 Conclusions and Recommendations

CSPM) give results consistent with the exact solution by the method of characteristics.

• For the slug impact force on a bend, the 1D and 2D SPH results are consistent with each other and they show better agreement with available measurements than solutions found in literature, because of improved modelling. The ﬂow separation at the bend has a major effect on the local pressure magnitude and distribution. New formulas for elbow pressure and reaction force are derived and used to predict slug impact at a sharp bend.

• Different from previous experiments [116,219,221], in the large-scale filling tests, the original water front splits into two fronts travelling at different speeds. The stratified flow between the two water fronts has a nearly constant height (the air on top has a constant thickness) until a hydraulic jump takes place.

• Although the applied 1D model cannot capture the observed flow regime transition from pressurized pipe flow to stratified open channel flow in pipe filling, it gives satisfactory results for the overall behaviour of the advancing water column.

8.2 Recommendations

• The considered problems are high Reynolds pipe flows and they are treated either as 2D inviscid flows or 1D turbulent flows. Consequently, the enforcement of the no-slip boundary condition treated in Section 3.4.8 and the SPH diffusion term introduced in Section 3.4.4 have not been tested. They can be validated against the benchmarks of Couette flow, Poiseuille flow and driven cavity flow [117]. It is also an ideal case to test the effect of viscosity in various unsteady laminar flows.

• The weakly compressible SPH methodology used in this thesis slightly suffers from pressure noise. Many improvements have been proposed, such as incompressible SPH [101,107,191], adding density diffusion to the continuity equation [146], particle redistribution [43], variable smoothing length [117,156,177,178], variable speed of sound [5], etc. A thorough comparison and evaluation of these techniques is recommended.

• The boundary deﬁciency problem in SPH has been overcome mainly by the image particle approach in this thesis. The new algorithms proposed in Appendix A deal with the same problem but have not been fully examined. Their application to ﬂuid dynamic problems should be further explored.

• SPH encompasses much potential strategies to model two-phase flows [43, 87,88,156] with surface tension [160] and fluid-structure interaction [4]. Af- ter these additions are introduced in the current SPH solver, the entire process of pipe filling and emptying (2D or even 3D) can be simulated. The 8.2 Recommendations 175

isolated slug and steady two-phase slug ﬂow can be modelled in such a way too.

• All ingredients needed for 2D SPH simulations have been derived in Chap- ter 3. Extensive numerical simulations are feasible with the ever increasing computing power. The extension from 2D to 3D is straightforward (but expensive) and the implemented code is already organised for a 3D computation. Since SPH itself is highly suitable for parallel computing, parallelisa- tion using MPI or GPU will speed up the computation to a large extent.

• A 2D or 3D SPH simulation of the pipe bridge used in the Delft experiment would give vital insight in the formation of the two water fronts in the pipe ﬁlling. In addition, 2D and 3D simulations should clarify the impact of liquid slugs on bends.

• For the ﬁlling and emptying of long pipelines, 1D and 2D/3D SPH simulations should be combined. The 1D solutions describe the global behaviour, whereas the 2D/3D solutions describe the evolution of water fronts and tails. A full 2D/3D simulation would require too many particles and lead to unacceptable computation times.

Appendix A: SPH Corrections

A.1 Incompleteness

To have a second-order accurate function approximation, the moment conditions of the kernel have to be satisﬁed (see Lemma 3.2) ∫ W(x − η, h)dη = 1, Ω ∫ ( ) k x − η W(x − η, h)dη = 0, k = 1, 2. Ω

Similarly, to have a second-order accurate derivative approximation, the kernel gradient must satisfy (see Lemma 3.4) ∫ ( ) x − η W ′(x − η, h)dη = 1, Ω ∫ ( ) k x − η W ′(x − η, h)dη = 0, k = 0, 2. Ω For the cubic spline kernel with a full support, these conditions only hold in continuous form. After having been approximated by a summation form, they are generally not valid because of the error in the summation approximation. That is to say, the following discrete moment conditions for the kernel and its gradient do not hold ∑ W(x − xb, h)Ωb = 1, (A.1) b ∑ ( ) k x − xb W(x − xb, h)Ωb = 0, k = 1, 2, (A.2) b ∑ ( ) ′ x − xb W (x − xb, h)Ωb = 1, (A.3) b ∑ ( ) k ′ x − xb W (x − xb, h)Ωb = 0, k = 0, 2. (A.4) b 178 Appendix A: SPH Corrections

Condition (A.1) is generally called partition of unity (PU) [7, 112] and plays an essential role in meshfree methods [7, 8, 63]. Conditions (A.1) – (A.4) are viewed as the consistency or completeness conditions.

A.2 Restoring completeness

There is no analytical function W that can satisfy all the completeness conditions (A.1) – (A.4). To restore the completeness, a modified kernel that takes into account the information of particle distribution has to be constructed. Here we first present three algorithms, two of which are new. Then they are compared to the original SPH and other popular modifications.

A.2.1 MSPM and SSPM

By Taylor expansion we have ( ) ( ) . ′ 1 ′′ 2 f(η) − f(x) = f (x) η − x + f (x) η − x , (A.5) 2 where higher derivatives have been omitted. Multiplying both sides of (A.5) with compactly supported functions ϕk(x − η, h)(k = 1, 2) and integrating over the support of ϕk yields ∫ ∫ ∫ ∫ . ′ ′′ 1 f(η)ϕ dη−f(x) ϕ dη = f (x) (η−x)ϕ dη+f (x) (η−x)2ϕ dη, k = 1, 2. k k k 2 k (A.6) Writing the integrals in (A.6) in summation form yields ( ∑ ∑ ) ( ) ′ ∑b fbϕ1Ωb − f(x) ∑b ϕ1Ωb . f (x) = K ′′ , (A.7) b fbϕ2Ωb − f(x) b ϕ2Ωb f (x) where the moment matrix K is ( ) ∑ 1 2 (xb − x)ϕ1 2 (xb − x) ϕ1 K := Ωb 1 2 . (A.8) (xb − x)ϕ2 (xb − x) ϕ2 b 2 Suppose that matrix K is non-singular, then we obtain the approximated derivatives of function f(x) at point xa as ( ) ( ∑ ∑ ) b′ fa . −1 ∑b fbϕ1Ωb − fa ∑b ϕ1Ωb b′′ = Ka (A.9) fa b fbϕ2Ωb − fa b ϕ2Ωb with ( ∑ ∑ ) 1 2 −1 −1 ∑b Ωb(xb − xa)ϕ1 ∑b Ωb 2 (xb − xa) ϕ1 Ka := 1 2 . (A.10) b Ωb(xb − xa)ϕ2 b Ωb 2 (xb − xa) ϕ2 A.2 Restoring completeness 179

′ ′ ′′ ′′ If we take ϕ1 = Wab := W (xa − xb, h) and ϕ2 = Wab := W (xa − xb, h), we obtain ( ) ( ∑ ( ) ) b′ ′ fa . −1 ∑b (fb − fa)WabΩb b′′ = Ka ′′ , (A.11) fa b fb − fa WabΩb with ( ∑ ∑ ) ′ 1 2 ′ ∑b(xb − xa)WabΩb ∑b 2 (xb − xa) WabΩb Ka = ′′ 1 2 ′′ . (A.12) b(xb − xa)WabΩb b 2 (xb − xa) WabΩb We refer to the above approximation as the modiﬁed smoothed particle method (MSPM). It has been successfully applied to the 1D and 2D transient heat conduction [86].

2 If we choose the functions ϕ1 = (xb −xa)Wab and ϕ2 = (xb −xa) Wab according to Lemma 3.2, we obtain ( ) ( ∑ ( )( ) ) b′ f − f x − x W Ω fa −1 ∑ b( b a)( b a) ab b 1 b′′ = Ka 2 (A.13) 2 fa b fb − fa xb − xa WabΩb with ( ( ) ( ) ) ∑ 2 ∑ 3 b (xb − xa) WabΩb b (xb − xa) WabΩb Ka := ∑ 3 ∑ 4 . (A.14) b xb − xa WabΩb b xb − xa WabΩb

Comparing with MSPM (A.11), the moment matrix Ka is symmetric and consequently this new approximation is referred to as symmetric smoothed particle method (SSPM). Note that the factor 1/2 has been placed on the left-hand side of (A.13).

A.2.2 RKPM

The above approximations directly correct the derivatives. Since the constant- reproducing condition in the derivative approximation is equivalent to the linear- reproducing condition in the function approximation [112], some of the approximations directly correct the approximation of the function instead of its derivatives. As an example, a widely used technique, the reproducing kernel particle method (RKPM) [122] is described here.

The SPH kernel approximation of a function f is ∫ ⟨f⟩ (x) = f(η)W(x − η, h)dη. Ω It does not even have zeroth-order consistency (completeness) (see Section 3.5.2). In the RKPM [122], this is remedied by modifying the kernel function to f W(x − η, h) := C(x − η)W(x − η, h), (A.15) 180 Appendix A: SPH Corrections

f where W is the corrected kernel and C(x − η) is the so-called correction function. It is assumed to have the following form,

C(x − η) := pTb, (A.16) where ( ) T p := 1, x − η, (x − η)2 (A.17) is a polynomial basis and ( ) T b := b0(x), b1(x), b2(x) (A.18) contains the unknown coefﬁcients. Only up to second-order monomials are included in p for simplicity. Then the kth order (k = 0, 1, 2) moment of the corrected kernel is ∫ f k f Mk = (x − η) Wdη ∫Ω = (x − η)kpTbWdη Ω (A.19) ∑2 = bn(x)Mk+n(x)(k = 0, 1, 2), n=0 where Mk+n(x) is the (k + n)th (k, n = 0, 1, 2) moment of the original kernel W.

f By replacing W by W and expanding f(η) about point x using a Taylor series (retaining up to second derivatives), from (3.2) we obtain ∫ f ⟨f⟩ (x) := f(η)W(x − η, h)dη ∫ Ω[ ] . ′ 1 ′′ = f(x) + f (x)(x − η) + f (x)(x − η)2 Wdηf (A.20) Ω 2 1 = f(x)Mf (x) + f′(x)Mf (x) + f′′(x)Mf (x). 0 1 2 2 To reproduce the function, the following moment conditions of the corrected kernel need to be satisﬁed f f M0(x) = 1 and Mk(x) = 0 (k = 1, 2). (A.21)

From (A.19) and (A.21), we get

Mb = (1, 0, 0)T, (A.22) where the moment matrix M is   M0(x) M1(x) M2(x)   M := M1(x) M2(x) M3(x) . (A.23) M2(x) M3(x) M4(x) Then we have b(x) = M−1(1, 0, 0)T, (A.24) A.3 Comparison and discussion 181 with which we get the approximated function (in integral form) as ∫ f ⟨f⟩ (x) := f(η)W(x − η, h)dη ∫ Ω (A.25) = f(η)pTbWdη. Ω

The ﬁrst and second directive approximations are obtained as ∫ ( ) ′ ′ ⟨f ⟩ (x) := f(η) pTbW dη, ∫Ω ( ) ′′ (A.26) ′′ ⟨f ⟩ (x) := f(η) pTbW dη, Ω where the derivatives of coefﬁcients b are calculated by

′ ′ b (x) = −M−1(x)M (x)b(x), [ ] ′′ ′′ ′ ′ (A.27) b (x) = −M−1(x) M (x)b(x) + 2M (x)b (x) .

Since the moment conditions of the corrected kernel determine the capabilities of f reproducing monomials, W is often called reproducing kernel, which accounts for the origin of the name ’RKPM’. After replacing the integrals in (A.25) and (A.26) by Riemann sums, the RKPM approximations are obtained. When terms up to order m are retained in the Taylor series expansion of the function, the kernel approximations of the function, its first and second derivatives are consistent to order m, m − 1 and m − 2, respectively [112, 235]. For efficiency reasons only the first-order monomials are retained in RKPM, thus reproducing linear functions [122, 123].

A.3 Comparison and discussion

If the discrete moment conditions of the kernel and its derivatives, (A.1) – (A.4), are satisﬁed (as assumed in the standard SPH method), the moment matrices in MSPM, SSPM and RKPM are identity matrices. Then they all degenerate to the standard SPH method. Unfortunately, this assumption is not true. There is not such a kernel that can satisfy all the discrete moment conditions. In addition, when kernels are truncated by boundaries, the correction methods such as MSPM, SSPM, RKPM and moving least square (MLS) [16, 52] have the advantage that it is not necessary to complete the truncated kernel with image particles (mirror, ghost, inlet, outlet, etc). The correction algorithms are generally more expensive than standard SPH because a series of small systems without (e.g. CSPM [30], MSPM and SSPM) or with extra operations (e.g. RKPM and MLSPH [52, 53]) are needed. Furthermore, invertibility of the moment matrix (K and M) in these corrective methods needs special attention, which restricts 182 Appendix A: SPH Corrections the possible particle (point) distribution patterns. These methods are therefore mainly used in solids where particles are not moving or when their distribution is controllable. For these and other meshfree methods, the details can be found in [16, 82, 111, 112, 118, 164].

Now we review the relationship between MSPM and the original SPH and several other correction schemes given in the literature. The focus is on the better completeness of MSPM. Then we show the advantages of SSPM over MSPM and RKPM.

A.3.1 MSPM vs SPH

′ Suppose that the ﬁrst- and second-order moments of function ϕ1 = Wab satisfy the following conditions ∑ ∑ ′ 2 ′ Ka11 = (xb − xa)WabΩb = 1 and Ka12 = (xb − xa) WabΩb = 0, b b then from (A.11) we get ∑ ( ) b′ ′ fa = fb − fa WabΩb. (A.28) b

One can see that expression (A.28) is exactly the 1D case of∑ (3.19). If we further ′ ′ require the zeroth-order moment of Wab to be zero, i.e. b WabΩb = 0, we obtain ∑ b′ ′ fa = fbWabΩb, (A.29) b which is the 1D case of (3.17). Expression (A.28) is a symmetrisation of (A.29) and gives a better derivative approximation. This improvement was ﬁrst proposed by Monaghan in [150].

Similarly, to get the following second-derivative approximation ∑ ( ) b′′ ′′ f (xa) = fb − fa WabΩb, (A.30) b

′′ two moment conditions for ϕ2 = Wab including ∑ ∑ ′′ 1 ′′ K = (x − x )W Ω = 0 and K = (x − x )2W Ω = 1, a21 b a ab b a22 2 b a ab b b b need to be fulﬁlled. These are much harder to be satisﬁed (if not impossible) than the moment conditions of W ′, because W ′′ is less smooth than W ′ (see Fig. 3.2). Thus the approximation given by (A.30) will be poor. This explains mathematically why second-order derivatives are not approximated directly, as mentioned A.3 Comparison and discussion 183 in Section 3.3.3. The extra number of moment conditions in SPH will be 25 and 81 for 2D and 3D problems, respectively.

If only condition Ka12 = 0 is satisﬁed, we obtain the Johnson-Beissel correction used in their work on simulating penetration and fragmentation of solids [93], ∑ ( ) f − f W ′ Ω b′ ∑ b ( b a ) ab b fa = ′ . (A.31) b xb − xa WabΩb

This correction is equal to the Randles-Libersky correction [182].

In [93,182] only ﬁrst-order derivatives were corrected, but the Chen-Beraun correction [29] corrects even second-order derivatives by ∑ ( ) ∑ ( ) ′ b′ ′′ b′′ b fb − fa WabΩb − fa b xb − xa WabΩb f = ∑ ( ) , (A.32) a 1 2 ′′ b 2 xb − xa WabΩb

b′ where the term fa in the numerator is obtained from (A.31). The approximation schemes (A.31) and (A.32) determine the corrective smoothed particle method (CSPM). Comparing to the MSPM algorithm, the CSPM algorithm needs 1, 10 and 36 extra conditions for 1D, 2D and 3D problems, respectively. The effect of particle distribution pattern and smoothing length on the singularity of the moment matrix in 2D CSPM is studied in [82].

A.3.2 SSPM vs MSPM

Comparing with MSPM, the symmetry property of SSPM saves computational time and memory to a large extent, especially in 2D and 3D problems. In addition, there are no derivatives of the kernel in SSPM, so that the requirement for continuity of the kernel has been reduced and more choices for kernels can be made.

When a uniform particle volume is assumed, i.e. Ωb is the same for all particles, SSPM is the same as the generalized ﬁnite difference method (GFDM), the derivation of which follows an MLS procedure; see e.g. [18, 66, 175] for the details.

If we make certain assumptions on the elements of matrix Ka as in the MSPM algorithm, many other and simpler SPH corrections can be obtained.

A.3.3 SSPM vs RKPM

Comparing with RKPM, more choices for kernels can be made in SSPM. In addition, two advantages of SSPM over RKPM are noticed. First, the SSPM moment 184 Appendix A: SPH Corrections matrix K is one block of matrix M in RKPM, because a correction is made to the function approximation in RKPM, but not directly to the derivatives as in SSPM. If we simultaneously perform function approximation in SSPM, the moment matrix K will be exactly the same as matrix M in RKPM. This work has been done by Zhang & Batra [235], resulting in the symmetric smoothed particle hydrodynamics (SSPH) method [14, 235]. When the strong form of a PDE is solved, we prefer a direct correction to the derivatives. This is because for initial-boundary value problems (considered in this thesis), only derivatives in the governing equations need to be approximated and function values are obtained from either the initial condition or a previous time step. When the weak form of a PDE is solved using a Galerkin method, direct correction to the function approximation is often used as in the element-free Galerkin (EFG) method [16], which is one of the most popular meshfree methods for solids. The basis functions (shape functions) in EFG are constructed using the MLS approximation. The similarity between MLS and RKPM is discussed in e.g. [111,112,118,235]. Second, the computational efﬁciency of SSPM is much higher than RKPM, because the corrections to the kernel and its derivatives are performed simultaneously in SSPM, whereas much extra work is needed in RKPM to obtain the corrected kernel derivatives. Bibliography

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acceleration head, 109, 115 corner mirror particles, 40 action-reaction law of Newton, 44 corrective smoothed particle method, 113, air entrainment, 123, 125, 126, 133 183 air front, 100 Courant number, 109 air tank, 123, 143 Courant-Friedrichs-Lewy, 37 air-water interface, 100, 103, 163, 164, critical flow, 81 167 cubic spline, 28 antisymmetry of the kernel gradient, 23, 49, 50 dam-break, 61 approximating functions, 55 Darcy-Weisbach, 96 arrival velocity, 122, 123, 125 degrees of freedom, 17 artificial fluid, 36 density variation, 35 artificial viscosity, 73, 75, 102 detachment, 78 asymmetric right-angled bend, 86 Dirac delta distribution, 20 discharge, 80 Bernoulli theorem, 68, 72, 86, 97, 130 disordered equi-distribution, 54, 55 body force, 11 dispersion distance, 122 boundary deficiency, 38 divergence of the velocity, 10 boundary integral, 42 driving pressure head, 109, 115 box scheme, 105 dummy particles, 42 branched channels, 90 duration, 132 brink depth, 79 dynamic boundary condition, 15, 42 dynamic viscosity, 12 CFL condition, 37 channel, 68, 76 elastic model, 96, 99 Chen-Beraun correction, 183 element-free Galerkin, 184 commutative, 24 emerging jet, 78 compact support, 23, 29, 47 entrance head loss, 97 completeness, 55, 178 Euler equations, 13 compression wave, 127 Euler-MacLaurin, 55 conduit, 90 Eulerian, 17 contact line, 90 evenly distributed particles, 50 continuous wall, 39 expansion wave, 127 contraction coefficient, 85, 130, 131 control volume, 128, 129 field variables, 19 controlled pipe emptying, 116 flow front, 70 convective derivative, 9 flow pattern, 63, 83, 119 convective velocity, 36 flow rate, 98 200 Index

flow regime transition, 119, 157, 167 interaction pair, 21 flow regimes, 119 interpolant, 56 flow separation, 129 inviscid flow, 13 fluid particles, 38, 104 isolated slug, 3, 123 fluid-solid impact, 3 free overfall, 76, 79 Jet flows, 68 free streamline, 83, 84, 86, 90 Johnson-Beissel correction, 183 free surface boundary, 15, 41 joint failure, 2 free-slip boundary, 38, 62, 75 Joukowsky, 169 free-slip condition, 15 free-surface profile, 62, 72, 77, 80, 81 kernel, 20, 27 friction head, 115 kernel approximation, 20, 21 friction head loss, 108, 155 kernel derivatives, 29 Froude number, 76 kernel radius, 29, 39 kernel support, 29 gas acoustics, 129 kinematic boundary condition, 15, 41, gas dynamics equations, 129 69 Gaussian, 27 kinematic viscosity, 97 generalized finite difference method, 183 Kirchhoff’s free streamline, 83, 86 ghost particles, 40, 46 Kronecker delta, 13, 29 global mass conservation, 43 Lagrangian, 17 head transition, 108, 115 leading air front, 162, 164 Hessian matrix, 48 leading edge, 62, 133 hodograph transformation, 83 link-list algorithm, 38 holdup, 127, 128 liquid slug, 121 holdup coefficient, 128 local derivative, 9 HYDRA, 18 hydraulic grade line, 160, 169 MAC, 60 hydraulic jump, 159 Mach number, 36, 62, 74, 75, 89, 110 marker particles, 60 image particle approach, 38 mass conservation law, 10 impact force, 120, 124, 126 mass density, 10, 21 impinging jet, 68 mass loss, 126, 127 impulse, 132 material derivative, 9 inclination angle, 68, 96, 106 mesh, 17 incompleteness, 55 mesh distortion, 19 incompressible, 36 mesh-based methods, 18, 59 infinitesimal fluid element, 8 meshfree methods, 18, 72 inflection point, 115 Meshfree Methods for PDEs, 18 inflow boundary, 15 meshless methods, 18 inlet block, 42 minimum Froude number, 78 inlet boundary, 42 minor loss, 130, 153 inlet particles, 42 mirror particle approach, 39 integral approximation, 21 mirror particles, 39, 62 integral approximation errors, 47 MLS approximation, 72 interacting particles, 21, 44 modified kernel, 178 Index 201 modified smoothed particle method, 179 particle spacing, 62, 84, 131, 133 moment, 48 particle-in-cell, 60 moment conditions, 48 particles clustering, 45 moment matrix, 178 particles separating, 45 momentum transfer, 112, 122, 136 partition of nullity, 25, 51 monitoring particle, 69 partition of unity, 50, 178 monotonicity, 30 pipe bridge, 142, 171 Monte Carlo, 54 pipe emptying, 162 moving boundaries, 3 pipe filling, 94, 153 moving fluid domain, 16 pressure, 13 moving fluid element, 9 pressure coefficient, 86 moving least square, 181 pressure distribution, 72, 78, 106, 169 pressure noise, 72, 133 Navier-Stokes equations, 13 pressure particles, 104 neighbour, 21, 25 pressure transducers, 123 neighbouring particles, 38, 45 net force, 11 quartic spline, 28 Newton’s second law, 10 quasi-3D model, 90 Newtonian fluids, 12 quasi-Monte Carlo, 54 no-slip boundary, 40, 46 quasi-steady flow, 133 no-slip condition, 15 nodal completeness, 55 radial symmetry, 30 nodal integration, 54 Randles-Libersky correction, 183 nominal velocity, 126 rapid pipe filling and emptying, 2 non-permeable wall, 15 ratio of channel width, 84 normalization factor, 27 reaction force, 126, 129 normalization of the kernel, 21, 29, 50 reference density, 14 normalization of the kernel derivative, reference pressure, 14 30, 49, 51 relative roughness, 97, 153 reproducing conditions, 55 orifice plate, 124 reproducing kernel particle method, 179 outflow boundary, 16 repulsive force, 39 outlet block, 43 reservoir head, 97, 109 outlet boundary, 42 revised Gaussian, 28 outlet particles, 43, 104 revised super Gaussian, 28 Reynolds number, 97, 153 pairing instability, 44 Reynolds transport theorem, 128 particle, 19, 21 rigid wall boundary, 38 particle acceleration, 37 rigid-column model, 98, 100, 112 particle approximation, 21 Roshko’s free streamline, 82 particle boundary layer, 46 particle crystallization, 46, 75 separation point, 78, 83, 86 particle disorder, 26 shape functions, 56 particle distribution, 63, 90, 178 Shepard, 56 particle penetration, 90 skin friction, 15, 153, 167 particle position, 36 slender jets, 76 particle search, 38 slender structures, 46, 90, 95 202 Index smoothing length, 20 tree algorithm, 38 speed of sound, 14, 73, 131, 169 truncated kernel, 38 SPH, 19 turbulent flow, 97 SPH artificial viscosity, 33 turning angles, 84, 87 SPH continuity equation, 31, 32 two-phase flows, 3 SPH Laplacian, 27 SPH mass conservation, 43 undisturbed flow, 83 SPH momentum conservation, 44 undulating elevation profile, 96 SPH momentum equation, 32 SPH pressure boundary, 103 valve resistance, 100, 115, 163 SPH real viscosity, 34 variational principle, 25, 44 SPH Rule I, 24 velocity, 13 SPH Rule II, 24 velocity distribution, 70, 86 SPH Rule III, 25 velocity head, 97 SPH Rule IV, 25 vena contracta, 97, 109 SPH Rule V, 25 viscous flow, 13 SPH water-hammer equations, 102, 131 viscous stress, 13 SPHERIC, 18 von Neumann-Richtmyer, 33 stagnation point, 72 wall boundary, 15 steady slug flow, 119, 120 wall particles, 39 stratified flow, 122 water front, 103, 155 subcritical flow, 80 water front evolution, 156 summation approximation, 21 water hammer, 1, 93, 113, 168 summation approximation error, 50 water level, 158 super Gaussian, 28 water slugs, 3, 168 supercritical flow, 82 water tail, 168 support domain, 20 water tank, 143 support failures, 2 water-air interface, 155 surface forces, 11 water-hammer equations, 97 surface tracking, 60 wedge-shaped front, 70, 133, 160 Swamee–Jain, 97 symmetric acute and blunt bends, 87 zeroth-order completeness, 56 symmetric bend, 84 symmetric right-angled bend, 85 symmetric smoothed particle method, 179 symmetrisation, 182

Tait-Murnaghan equation of state, 14, 31 Taylor instability, 125 Taylor series expansion, 48, 50 tensile instability, 45, 79 thin waterfalls, 76 total approximation error, 50 total stress tensor, 13 travelling slugs, 3 Acknowledgements

Before finishing my Masters degree in Engineering Mechanics in the Northwest- ern Polytechnical University in 2007, I was planning to do a PhD at Tongji Uni- versity and then go abroad as a postdoctoral researcher. The China Scholarship Council (CSC) happened to start a 5-year programme in 2007 to financially support 5000 Chinese students per year to study abroad. I fortunately became one of the 410 students who got the opportunity to do their PhD abroad. This made my dream of overseas study come true in advance. After some struggling for places and directions, I came to Eindhoven University of Technology in the Netherlands and joined CASA as a visiting PhD student in Applied Mathematics. Although it was an adventure, I was not disappointed as I learned a lot and enjoyed the geographic and scientific journey very much. I would not have been able to complete this trip without the contribution from countless people over the past four and half years. I would like to take the opportunity to express my gratitude to some of them.

In the first place, I am sincerely grateful to my promotor, Prof. Dr. Bob Mattheij for offering me the precious opportunity to work on this PhD project. I would like to thank him for much financial support including the payment for the first 18 months accommodation, first year health insurance, registration fee for many training courses and the support to attend the 16th European Conference on Mathematics in Industry. Then I would like to express my gratitude to my copromotor Dr. Arris Tijsseling for the great support that he has offered me in working on this thesis. I benefited a lot from his expertise in pipe flows and tremendous daily discussions. Due to his encouragement, I had the great chance to participate in the EC-HYDRALAB III projects ”Unsteady friction in pipes and ducts” and ”Transient vapourous and gaseous cavitation in pipelines”. It was him who brought me to the computational and experimental hydraulic engineering field. Besides my two supervisors, my defense committee is completed by Prof. Dr. Wil Schilders, Dr. Frazer Pearce, Dr. Jos van ’t Westende, Dr. Jacques Dam and Dr. Anton Bergant. I would like to thank them all for the invested time and their valuable comments and suggestions.

I would like to give my appreciation to some colleagues who made a direct contribution to this research. I am grateful to Dr. Jos Maubach for the clariﬁcation 204 Acknowledgements of mathematical notations and many fruitful discussions on the error analysis in Chapter 3. I thank Dr. Bas van der Linden for several computer programming related questions.

I am indebted to Prof. Alan Vardy (University of Dundee) and Dr. Anton Bergant (Litostroj Power d.o.o., Ljubljana) for offering me the opportunity to participate in the EC-HYDRALAB III project ”Unsteady friction in pipes and ducts”. I thank Dr. Janek Laanearu (TU Tallinn) for inviting me to join the EC-HYDRALAB III project ”Transient vapourous and gaseous cavitation in pipelines”. I would like to thank the staff at Deltares, Delft, where the EC projects were carried out: Dr. Hugo Hartmann and Dr. Jos van ’t Westende for their help in the preparation of the project HYIII-Delft-4. I would like to thank Richard Tuin, Martin Boele and Theo Ammerlaan for their expert technical advices during the measurement periods. As project leaders, Dr. Anton Bergant and Dr. Janek Laanearu made great and continuous efforts. I also thank Anton for his continuous help and advice in the experimental data analysis. Having been involved in these two projects brought me a lot of valuable experimental experience. Through out these projects, I had the great opportunity to work with many experts in hydraulics and collaborate with them on the scientiﬁc publications [22, 23, 105].

I would like to thank Prof. Andrew Cliffe for inviting me to be at the University Technology Centre (UTC) for Mechanical Transmissions Systems (Rolls & Royce), The University of Nottingham, for a half year project cooperation. I am grateful to Dr. Arno Kruisbrink and Dr. Frazer Pearce for allowing me to use and extend their code ”HYDRA”. The many discussions with them and with Thomas Yue gave me useful insight into the SPH method. I would like to thank all people in UTC, in particular Dr. Wang Chen and Dr. Li Ran, for providing the enjoyable working environment during my period in Nottingham.

I would like to thank Prof. Lixiang Zhang (Kunming University of Science and Technology) for presenting our work [83] at the International Conference on Ad- vances in Computational Modeling and Simulation, December, 2011, Kunming, China. I am indebted to Dr. Zafer Bozkus (Middle East Technical University) for many useful discussions on the isolated slug problem.

Being part of CASA has been an unforgettable experience. I would like to thank my colleagues and friends for the pleasant and stimulating working environment. It is great to have had Hans, Godwin, Zoran, Maria, Agnieszka, Mirela, Roxana, Nico, Neda, Corien, Akshy, Lucia, Xiulei as my ofﬁce mates. Many thanks to Ali, Andriy, Antonino, David, Erwin, Evgeniya, Iason, Jan-Willem, Kundan, Laura, Maria, Mark, Martien, Maxim, Michel, Oleg, Patricio, Peter, Shona, Shruti, Sinatra, Sudhir, Tasnim, Valeriu, Volha, Yabin, Yves and many more. In particular, I thank Hans, Iason, Patricio, Oleg and Yabin for several scientiﬁc discussions. I also enjoyed many interesting discussions with Alireza Kermat and Wu Wenyan when they were visiting CASA. I am very thankful to Enna van Dijk for all the help on various administrative matters from the very beginning until the late end of my study. 205

I am also thankful to my Chinese friends who made my personal life in the Netherlands more colourful. They are Hua Yicun, Wang Xijian, Ye Jianhong, Huang Zhengxing, Wu Zhen, Lv Yixin, Liu Lei, Liu Jie, Yu Qixiao, Qin Lu, Liu Shoumin and Li Di, Liu Zhipeng and Wang Yanru, Ma Zhe and Liu Liyuan, Xu Wei and Xing Nana. Thank you all for the delicious Chinese food and enjoyable games. Particularly, I thank Shoumin for having been a great housemate for almost three years.

I am deeply grateful to my family and friends at home for their continuous support over the years, without which there would have been no chance for me to have reached this stage. I would like to express my sincere appreciation to my parents, my brother and sisters for their patience and support. Finally, I appreci- ate my lovely Li Hui for her patience, care, understanding and love.

Darcy Q. Hou

Eindhoven, June 2012. 206 Acknowledgements Curriculum Vitae

Qingzhi Hou was born on April 16th 1978 in Yutai, Shandong Province, China. After finishing his pre-university education at Yutai and Yucai high-school in Jin- ing in 2000, he started his Bachelor studies in Civil Engineering at the Northwest- ern Polytechnical University, Xi’an, China. He received his Bachelor’s degree cum laude in 2004 and won a scholarship to continue his study and obtain a Mas- ter degree in Engineering Mechanics at the same university. He was a part-time engineering auditor at the Audit Department of Northwestern Polytechnical Uni- versity from September 2004 to September 2005. At the end of 2006, Qingzhi did an internship at Guangdong Daya Bay nuclear power plant, Shenzhen, China. In this three-month internship, he focused on the thermodynamic stress and strain analysis of piping systems. During the Master studies, he was involved in two research projects funded by the National Natural Science Foundations of China (No. 50475147 and No. 10202020), which resulted in five published papers. The Master thesis was written under the supervision of Prof. J. Ren and was entitled ”Dynamic characteristics and the influencing factors of buried fluid-conveying pipelines”. After finishing his Master study at the beginning of 2007, he started a research project at the Wind Engineering group of Prof. M. Gu, Tongji University, Shanghai, China. In this project he focused on wind tunnel tests and analysis of tall buildings.

From February 2008 till June 2012 Qingzhi has been working as a visiting PhD researcher at the Eindhoven University of Technology within the Centre for Analy- sis, Scientiﬁc Computing and Applications (CASA) under the supervision of Prof. R.M.M. Mattheij and Dr. A.S. Tijsseling, where he is better known as Darcy. The PhD project was sponsored by the China Scholarship Council (CSC) and the results are presented in this dissertation. During his PhD studies he participated in two EC-HYDRALAB III projects both under contract No. 022441 (RII3). From October 2010 to March 2011, he worked as a visiting scholar at the University Technology Centre for Mechanical Transmissions Systems, The University of Not- tingham, UK.