university of twente
faculty of engineering technology (CTW)
department of engineering fluid dynamics
Numerical modelling of non-linear shallow water waves
Supervisor Author Prof. Dr. Ir. H.W.M. L.H. Lei BSc Hoeijmakers
August 22, 2017
Preface
An important part of the Master Mechanical Engineering at the University of Twente is an internship. This is a nice opportunity to obtain work experience and use the gained knowledge learned during all courses. For me this was an unique chance to go abroad as well. During my search for a challenging internship, I spoke to prof. Hoeijmakers to discuss the possibilities to go abroad. I was especially attracted by Scandinavia. Prof. Hoeijmakers introduced me to Mr. Johansen at SINTEF. I would like to thank both prof. Hoeijmakers and Mr. Johansen for making this internship possible.
SINTEF, headquartered in Trondheim, is the largest independent research organisation in Scandinavia. It consists of several institutes. My internship was in the SINTEF Materials and Chemistry group, where I worked in the Flow Technology department. This department has a strong competence on multiphase flow modelling of industrial processes. It focusses on flow assurance market, multiphase reactors and generic flow modelling. During my internship I worked on the SprayIce project. In this project SINTEF develops a model for generation of droplets sprays due to wave impact on marine structures. The application is marine icing. I focussed on understanding of wave propagation.
I would to thank all people of the Flow Technology department. They gave me a warm welcome and involved me in all social activities, like monthly seminars during lunch and the annual whale grilling. In special I would like to thank Sverre Gullikstad Johnson and Stein Tore Johansen for the daily supervision, willingness to answer questions and discussing progress and results!
My internship assignment was really interesting and challenging. These three months gave me a nice insight in how it is to work in fundamental research and especially in the field of numerical modelling. Beside daily work, I attended the international conference on computa- tional fluid dynamics organized in Trondheim, which showed me the wide range of applications in computational fluid dynamics. All of this made me have an awesome time in Norway!
3 Summary
Marine icing is ice accretion on offshore structures and vessels. Heavy ice accretion can be a severe threat, since it leads to safety problems or damaging of equipment. It can even lead to instability with all its fatal consequences. In SINTEF, fundamental research is done to develop models for forecasting marine icing. Marine icing is mainly caused by sea spray, which is induced by water waves impacting on vessels or offshore structures. In this internship a numerical study is done to verify the possibility of modelling of non-linear water waves using ANSYS Fluent.
Before a numerical study was done in ANSYS Fluent, a 1D Boussinesq model had been elab- orated. Boussinesq type of systems are mathematical models to describe shallow water waves. One type of shallow water waves are solitary waves, which are characterized by maintaining its shape over a long distance, while it propagates with constant velocity. Solitary waves will never merge. The 1D Boussinesq model was used to gain insight in (numerical) behaviour of these waves, which could be used as a benchmark for the numerical study in ANSYS FLe- unt. The Boussinesq model was discretised on a staggered grid using upwinding schemes. The initialization was based on the analytical expression of solitary waves. Results were validated with experiments of solitary wave interactions. In these experiments two cases were considered; head-on and overtaking collisions. For both cases, the 1D Boussinesq model was capable in reproducing the wave interaction with a relatively small error. For overtaking collisions friction was added to come to a better fit.
In ANSYS Fluent a 2D geometry was built, which was meshed on a uniform grid. Volume of fluid method was used to model the multi phases of water and air. Initial conditions were set with a user-defined functions. This user-defined function enhanced the standard the features of ANSYS Fluent. The volume fractions, velocity field and pressure distribution were initial- ized using this function. The results were again validated with experimental data. The data was really well presented by the 2D model. However, numerical dissipation was quite significant.
It can be concluded that ANSYS Fluent is a suitable CFD method to handle wave propa- gation. In the continuation of the project, geometry and user defined functions can be adjusted to make it applicable for this marine icing project at SINTEF.
4 Contents
1 Introduction 5
2 Wave Theory 7 2.1 Linear wave theory ...... 7 2.2 Non-linear wave theory ...... 9
3 1D Boussinesq model 12 3.1 Discretization Boussinesq model ...... 12 3.1.1 Mass conservation ...... 13 3.1.2 Momentum conservation ...... 14 3.1.3 Initial and boundary conditions ...... 15 3.2 Single wave propagation ...... 16 3.3 Experimental and numerical comparisons ...... 17 3.3.1 Head-on collisions ...... 19 3.3.2 Overtaking collisions ...... 19 3.3.3 Accuracy study ...... 25
4 2D Naviers-Stokes model 26 4.1 Initial conditions ...... 26 4.1.1 Initialization volume fraction ...... 27 4.1.2 Initialization velocity ...... 27 4.1.3 Initialization pressure ...... 28 4.2 Solution methods ...... 29 4.3 Single wave propagation ...... 29 4.4 Experimental and numerical comparisons ...... 30 4.4.1 Head-on collisions ...... 30 4.4.2 Overtaking collisions ...... 30 4.4.3 Accuracy study ...... 36
5 Conclusion 37
6 Recommendations 38
7 Bibliography 39
8 Appendix A1
5 1. Introduction
Marine icing is ice accretion on offshore structures and can occur in sufficiently cold and windy conditions. It can lead to severe safety problems, such as slippery decks, ladders and handrails. Equipment like valves and winches can be become useless, causing delays in operation. Radar antennas can be damaged by icing. Heavy ice accretion can increase the size of structural members, which can lead to higher wind forces. It also increases the total weight and raises the center of gravity, which can lead to instability. In figure 1.1 examples of marine icing are shown. It clearly visualises the enormous impact of marine icing.
(a) (b)
Figure 1.1: Examples of marine icing
Marine icing is mainly caused by sea spray. It can be induced by wave impacts on vessels or offshore structures. Due to wave impact, a spray of droplets can be generated, which is moved and cooled by ambient air. If the ambient air temperature is below the freezing point of the seawater, these small supercooled water droplets impacts on vessels or offshore structures and freeze. The process of spray formation is related to different complicated phenomena, including propagation of the free water surface, wave slamming, spray formation after impact and droplet break up. These droplets will eventually lead to ice formation on a surface. Figure 1.2 presents an overview of these phenomena. In the SprayIce project, fundamental research is done to develop models for spray generation, which contain all these phenomena. This gives insight in the droplet distribution and where and in which amount marice icing occurs.
6 CHAPTER 1. INTRODUCTION
Figure 1.2: Formation of sea spray
This internship assignment is part of the SprayIce project. It focusses on numerical mod- elling of wave propagation. A method is needed which generates oncoming waves to an object. It must be verified that the wave structure is not destroyed by numerical dissipation before impact. In chapter 2, an introduction to wave theory is given. Starting from the Navier-Stokes equations the linear wave theory is derived, which is expanded to the non-linear wave theory ending up with the 1D Boussinesq model. In chapter 3, the 1D Boussinesq model has been elaborated in more detail. The discretized model is given, together with initial and boundary conditions. Results based on this model are presented and compared with experimental data. A grid refinement study has been performed to check the accuracy of different grid sizes and time steps. These observations can be used as a benchmark for 2D simulations. The 2D simula- tions have been perfomed in commercial CFD software, which are discussed in chapter 4. The initialization of different flow variables is explained and similar results as for the 1D model are presented. Conclusions can be found in chapter 5. Recommendations for future work will be given in chapter 6.
7 2. Wave Theory
Surface waves are disturbances on the interface between two fluids. In this report there will be focussed on surface waves between water and air in oceans. Ocean waves can be mainly divided in two types: gravitational and capillary waves. Capillary waves can be observed as small ripples on even flat sea surfaces. This report focusses on gravitational waves. These are sustained by the gravitational force after an initial disturbance on the surface. This can occur due to tide, wind, currents, earthquakes, ships, etc. In figure 2.1 examples of both types of waves are shown.
(a) Capillary waves (b) Gravitational waves
Figure 2.1: Examples of different type of water waves
2.1 Linear wave theory
The simplest way to describe propagation of water waves is the linear wave theory. The math- ematical theory will be reviewed, which leads to expressions for the shape of the free surface and velocity field of the wave [1]. It is assumed that the ocean is incompressible and inviscid. The equations for respectively conservation of mass and momentum are then given as ∇ · u = 0 (2.1) ∂u 1 + u · ∇u = − ∇p + g (2.2) ∂t ρ The ocean can also be considered as irrotational, and hence the velocity field u can be written as the gradient of a potential function, u = ∇φ. The conservation of mass, equation (2.1), then becomes the Laplace’s equation ∇2φ = 0 (2.3) With substituting this potential in the momentum equation, equation (2.2), one can obtain Bernoulli’s equation for an incompressible, inviscid and irrotational fluid ∂φ 1 p + ∇φ · ∇φ + + gz = 0 (2.4) ∂t 2 ρ
8 CHAPTER 2. WAVE THEORY
It is assumed that the sea bottom is horizontal and located at depth z = −D. Futhermore, it is assumed that there is no flow through the sea bottom. This boundary condition can be given by
∂φ = 0 at z = −D (2.5) ∂z
To derive an expression for the free surface, η(x, t), two boundary conditions at the surface will be imposed. The first condition is based on the no-slip condition at the sea surface. This means that water particles located at the surface remain there. Its vertical velocity can either be expressed in terms of the velocity potential or it can be expressed in terms of the time derivative of the free surface vertical position. Since the fluid element may be moving in the horizontal direction as well as in the vertical, it needs to be expressed in terms of the material derivative
Dη ∂η ∂φ ∂η = + (2.6) Dt ∂t ∂x ∂x
Combining this with the velocity in terms of the potential gives
∂φ ∂η ∂φ ∂η = + at z = η(x, t) (2.7) ∂z ∂t ∂x ∂x which is known as the kinematic boundary condition. For the second boundary condition it is assumed that water pressure at the the surface is equal to the air pressure just above the surface. This is known as the dynamic boundary condition
∂φ 1 + ∇φ · ∇φ + gη = 0 at z = η(x, t) (2.8) ∂t 2
In order to obtain boundary value problem for linear water waves, it is assumed that surface waves are sufficiently small compared with its wavelength. This means that the boundary condition can be evaluated at z = 0 instead of z = η. Furthermore, higher order terms can be neglected and so the linearised boundary conditions are given by
∂φ ∂η = at z = 0 (2.9) ∂z ∂t and ∂φ + gη = 0 at z = 0 (2.10) ∂t By solving η in equation (2.10) and inserting in equation (2.9), the combined free surface condition can be found ∂2φ ∂φ + g = 0 at z = 0 (2.11) ∂t2 ∂z which can be used for obtaining an expression for the potential. It is assumed that the potential can be expressed as a product of different functions depending on a single variable
φ(x, z, t) = X(x)Z(z)T (t) (2.12)
Inserting this into Laplace’s equation, equation (2.3) and make use of the boundary conditions in equation (2.5) and (2.11), leads to sinusoidal functions for the velocity potential and free surface.
9 CHAPTER 2. WAVE THEORY
2.2 Non-linear wave theory
As described in the previous section, the boundary conditions are linearised and some higher order terms are neglected. This is only allowed when the wave height is sufficiently small compared to the wave length. Otherwise non-linearity should be included. In figure 2.2, an overview is given for the validity of several wave theories as a function of dimensionless water depth and wave height. The regime the Sprayice project focusses on is in the shallow water regime. As can be seen, the linear theory is not valid here. One of the best and most used approximations to model shallow water waves is the Boussinesq wave theory. This non-linear theory can be applied for ’long waves’, which means that the water depth is much smaller that the wave length. The essential idea of the Boussinesq approximation is to eliminate the vertical coordinate from the flow equations, while retaining some of the influence of the vertical flow under water waves. This first derivation was done by Joseph Boussinesq in 1871 [3]. It is based on a truncated Taylor expansion of the velocity potential evaluated at the sea bed z = −D. Subsequently, this Taylor expansion of the velocity field can be substituted in the boundary conditions, given in equation (2.7) and (2.8) which finally leads to the following set of equations ∂η ∂ 1 ∂3u + [(η + D)u ] ≈ D3 b (2.13) ∂t ∂x b 6 ∂x3 ∂u ∂u ∂η 1 ∂3u b + u b + g ≈ D2 b (2.14) ∂t b ∂x ∂x 2 ∂t∂2x
Figure 2.2: Validity of several theories for water waves [2]
10 CHAPTER 2. WAVE THEORY
where ub is the velocity at the bottom [4]. The complete derivation can be found in the original paper by Boussinesq [3]. If equation (2.13) is differentiated to time and equation (2.14) multiplied by D/3 and differentiated to x, it can be shown that both equations can be combined to one single, fourth-order differential equation for the free surface
∂2η ∂2η ∂2 3 η2 d2 ∂2η − gd − gD + ≈ 0 (2.15) ∂t2 ∂x2 ∂x2 2 d 3 ∂x2
The Boussinesq equations can handle wave propagation in arbitrary directions. A simplified model is the Korteweg-de Vries (KDV) equations, which are only capable in describing uni- directional waves. The advantage of the KDV equations is the existence of an analytical solution. The KDV equation can be written as
∂η 3η ∂η D2 ∂3η + U 1 + + U ≈ 0 (2.16) ∂t 2D ∂x 6 ∂x3
A solution of this equation is the cnoidal wave solution. Cnoidal waves solutions are in terms of the Jacobi elliptic function and are valid for any wave length larger than approximately 10 times the water depth. These type of waves will not be elaborated in this report. In case of an infinite wave length, the cnoidal waves solution reduces to a simpler form, which are knowns as solitary wave solutions. This wave phenomena was first described by John Scott Russell in 1834, who observed a solitary wave solution in the Union Canal in Scotland. He described his observation as follows [5]:
I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped − not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles. I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.
The key properties of a soliton is that it maintains its original shape over a long distance, while it propagates with a constant velocity. In contrast to normal waves, solitons will never merge. The free surface of a soliton can be given as
x − ct η(x, t) = A sech2 (2.17) ∆ where r 4D ∆ = D (2.18) 3A and wave velocity c = pg(D + A) (2.19) In figure 2.3, an example of a soliton wave propagating is shown for selected time steps.
11 12 CHAPTER 2. WAVE THEORY
Figure 2.3: Wave propagation of a soliton, (D = 1m, A = 0.3m) 3. 1D Boussinesq model
In chapter 2 the original Boussinesq equations have been derived. However, nowadays there are potentially many different but formally equivalent Boussinesq systems. The reason for this is due to the fact that lower-order relations can be used systematically to alter the higher- order terms without disturbing the formal level of approximation and the considerable choice of dependent variables available for the description of the motion. The general type family of Boussinesq type systems can be formulated in the following non-dimensional, unscaled variables [6].
∂η ∂u ∂(ηu) ∂3u ∂3η + + + a − b = 0 (3.1) ∂t ∂x ∂x ∂x3 ∂t∂x2 ∂u ∂η ∂u ∂3η ∂3u + + u + c − d = 0 (3.2) ∂t ∂x ∂x ∂x3 ∂t∂2x where a, b, c, d are real coefficients and η = η(x, t), u = u(x, t) real functions. By choosing these coefficients and functions, different types of Boussinesq systems can be obtained [7]. A scaled and dimensional Boussinesq system is derived by Peregrine [8]. Peregrine derived the following system ∂η ∂ + [(D + η)u] = 0 (3.3) ∂t ∂x ∂u ∂u ∂η 1 ∂3u + u + g = D2 (3.4) ∂t ∂x ∂t 3 ∂t∂2x In case of a flat bottom, i.e. the water depth does not depend on time and position, the equations can also be written as: ∂ ∂ (D + η) + [(D + η)u] = 0 (3.5) ∂t ∂x ∂u ∂u ∂ 1 ∂3u + u + g (D + η) = D2 (3.6) ∂t ∂x ∂x 3 ∂t∂2x Substituting H = D + η, ∂H ∂(Hu) + = 0 (3.7) ∂t ∂x ∂u ∂u ∂H 1 ∂3u + u + g = H2 (3.8) ∂t ∂x ∂x 3 ∂t∂2x Equation (3.7) is based on mass conservation, equation (3.8) is on momentum conservation. This system can be solved numerically. In the next section the discretization will be elaborated, which leads to results for water depth and velocity.
3.1 Discretization Boussinesq model
Before a model is made in commercial CFD, a numerical 1D model has been programmed in Fortran77 by my supervisor Stein Tore Johansen. This model gives insight in the correctness
13 CHAPTER 3. 1D BOUSSINESQ MODEL of the used equations and gives understanding of the numerical behaviour. It can be used as a simple analysis to check how coarse the grid size and how large the time step could be before the scheme will show unstable or inaccurate results. The model will be validated with experi- mental data. The purpose of this numerical model is not to construct a model as accurate and efficient as possible, but it can be used as a benchmark for more advanced CFD analyses. It is mainly based on first and second order schemes. To avoid checkerboard patterns in the solution, a staggered grid is used. In figure 3.1, this principle is shown. The velocity components are placed at the cell edges, the height components are placed in the middle of a cell. Based on this staggered grid, the discretization of equation (3.7) and (3.8) can be performed.
Figure 3.1: Staggered 1D grid
3.1.1 Mass conservation The first term of mass equation is discretized with a first order explicit Euler method.
n+1 n ∂H H − Hi ≈ i (3.9) ∂t ∆t Hi The second term in equation (3.7) can be divided in two derivatives. These are both discretized with an upwind scheme. n n n n H 1 − H 1 u 1 − u 1 ∂ n i+ 2 i− 2 n i+ 2 i− 2 (Hu) ≈ u + H (3.10) ∂x i ∆x i ∆x Hi
1 1 where the i + 2 and i − 2 boundaries are not specified yet. This is done by upwinding. For the water height, it is defined as
H 1 = a 1 Huw + (1 − a 1 )Hdw (3.11) i+ 2 i+ 2 i+ 2 H 1 = a 1 Huw + (1 − a 1 )Hdw (3.12) i− 2 i− 2 i− 2 where ’a’ is the convection coefficient, which has a value between 0 and 1. This value determines the order of upwinding. Subscripts ’uw’ and ’dw’ stands respectively for upwind and downwind 1 faces. The way these are specified depends on the direction of the local velocity. For the i + 2 boundary this means, when locally the direction is upstream (ui+1 > 0), it is given as
Huw = Hi (3.13)
Hdw = Hi+1 (3.14)
In case when the direction is downstream (ui+1 < 0)
Huw = Hi+1 (3.15)
Hdw = Hi (3.16)
14 CHAPTER 3. 1D BOUSSINESQ MODEL
1 For the i − 2 boundary, the same procedure can be followed. When local velocity is positive (ui > 0), it is given as
Huw = Hi−1 (3.17)
Hdw = Hi (3.18)
When local velocity is negative (ui < 0)
Huw = Hi (3.19)
Hdw = Hi−1 (3.20)
For the velocity a similar derivation can be done
u 1 = a 1 uuw + (1 − a 1 )udw (3.21) i+ 2 i+ 2 i+ 2 u 1 = a 1 uuw + (1 − a 1 )udw (3.22) i− 2 i− 2 i− 2 Since the local velocities at these faces are not directly known, it is obtained by local averaging. 1 1 The plus and minus sign denote respectively the i + 2 and i − 2 boundary. u + u u¯+ = i+1 i (3.23) i 2 u + u u¯− = i i−1 (3.24) i 2
From here, the local stream direction can be determined. Then the variables uuw and udw are defined in the same way as for the water height. In case of first order upwind (a = 1) and a upstream velocity, the expression in equation (3.10) reduces to
n n n n ∂ Hi ui+1 − Hi−1ui (Hu) ≈ (3.25) ∂x ∆x Hi
Substituting the discretized terms of equation (3.9) and (3.25) in the mass equation, (3.7) and multiplying with ∆t gives an expression for the water height at the next time step
∆t Hn+1 = Hn + Hnun − Hn un (3.26) i i ∆x i i+1 i−1 i
3.1.2 Momentum conservation A similar procedure can be done for the momentum equation 3.8. The first term is again discretized using a first order explicit Euler method:
n+1 n ∂u u − ui ≈ i (3.27) ∂t ∆t ui For the second term a upwind scheme is used.
n n u 1 − u 1 ∂u n i+ 2 i− 2 u ≈ u (3.28) ∂x i ∆x ui
This can be written in a more conservative form [9]
∂u ∂u2 ∂u u = − u (3.29) ∂x ∂x ∂x
15 CHAPTER 3. 1D BOUSSINESQ MODEL
Discretizing equation (3.29) with a upwind scheme gives the following expression
+ n − n u¯i u 1 − u¯i u 1 + − ∂u i+ 2 i− 2 n u¯ − u¯ u ≈ − u i i (3.30) ∂x ∆x i ∆x ui un − un u¯+(un − un) − u¯−(un − un) i+ 1 i− 1 i i+ 1 i i i− 1 i un 2 2 ≈ 2 2 (3.31) i ∆x ∆x The third term is discretized with a first order upwind scheme. n n ∂H Hi − Hi−1 g ≈ g (3.32) ∂x ∆x ui The last term is a combination of spatial and time derivatives. For the spatial derivative, a standard second order central difference scheme is used. The time derivative is discretized with a first order explicit Euler method. Due to staggering the local water height is calculated by local averaging.
3 n n 2 n+1 n+1 n+1 n n n ! 1 2 ∂ u 1 Hi − Hi−1 ui+1 + ui−1 − 2ui ui+1 + ui−1 − 2ui H ≈ − (3.33) 3 ∂t∂2x 3∆t 2 (∆x)2 (∆x)2 ui Combining terms of equation (3.27), (3.28), (3.32) and (3.33) leads to the following expression n n n+1 n u − u n n u − u i+ 1 i− 1 H − H i i + un 2 2 +g i i−1 ≈ ∆t i ∆x ∆x ! 1 Hn − Hn 2 un+1 + un+1 − 2un+1 un + un − 2un i i−1 i+1 i−1 i − i+1 i−1 i (3.34) 3∆t 2 (∆x)2 (∆x)2
After rearranging terms and multiplying with ∆t an experssion for ui at the next time step can be found " # " # " # 1 Hn − Hn 2 1 Hn − Hn 2 1 Hn − Hn 2 un+1 1 + 2 i i−1 − un+1 i i−1 − un+1 i i−1 i 3(∆x)2 2 i+1 3(∆x)2 2 i−1 3(∆x)2 2 n n 2 n ∆t n n n ∆t n n 1 Hi − Hi−1 n n n ≈ ui − g Hi − Hi−1 − ui (ui+ 1 − ui− 1 ) − 2 (ui+1 + ui−1 − 2ui ) ∆x ∆x 2 2 3(∆x) 2 (3.35) The system of equations can be written in matrix-vector notation and subsequently solved with n+1 a tridiagonal matrix algorithm to obtain results for ui .
3.1.3 Initial and boundary conditions The initial conditions for height and velocity are described by the analytical solution of a solitary wave. The height at t0 is described by x − ct H(x, t ) = D + A sech2 0 (3.36) 0 ∆
The velocity at t0 is given by following approximation [4, p. 390] η u(x, t ) = c (3.37) 0 D Due to the staggered grid, the velocity and height are not located at the same point. This is taken into account for the velocity initialization by taking the local average of η η¯ u(x, t ) = c (3.38) 0 D
16 CHAPTER 3. 1D BOUSSINESQ MODEL with η(x − ∆x, t ) + η(x, t ) η¯ = 0 0 (3.39) 2 At the boundaries velocity is set to zero. This leads to reflection of waves at the boundary. In the model grid size and the CFL value, C, can be chosen. Based on these input values, the time step is determined as follows C∆x ∆t = (3.40) c Since time is solved explicitly, maximum CFL value should be one.
3.2 Single wave propagation
Solving equation (3.26) and (3.35), with the initialization given in equation (3.36) and (3.38), one could obtain results for the water height and wave velocity. It is modelled with a grid size ∆x = 0.01 and a time step based on CFL = 0.5. In figure 3.2, the water height is shown at several time steps. The wave propagates to the right with a wave speed as in equation (2.19). After 5 seconds (figure 4.5b), the wave encounters the wall, where after it is reflected. Next, the wave propagates to the left (figure 4.5c). It is clearly visible that it behaves as a solitary wave. It propagates at constant velocity, while it maintains its shape. The same simulation is performed with a lower water depth, D = 0.05 m. The results are presented in figure 3.3. It can be clearly seen that wave speed is significantly lower and the width is narrower than in comparison with the higher water depth. This is in accordance with the literature. Another effect which can be seen is the formation of oscillatory tail waves, which propagates to the left. As explained in chapter 2, solitary waves are characterized by maintaining its shape over a long distance. Apparently, the solution of the used model together with these wave properties is not an exact solitary wave solution. This can be explained by the Ursell number, which indicates the non-linearity of long surface waves. This dimensionless number is given as follows
(a) t = 0. (b) t = 2.
(c) t = 5. (d) t = 6.
Figure 3.2: Single wave propagation at several time steps, length of channel L = 10m, water depth D = 0.2m, wave height A = 0.01m
17 CHAPTER 3. 1D BOUSSINESQ MODEL
(a) t = 0. (b) t = 0.5.
(c) t = 1. (d) t = 2.
Figure 3.3: Single wave propagation at several time steps, length of channel L = 10m, water depth D = 0.05m, wave height A = 0.01m and distinguishes three different long wave cases [4]. O(1) linear shallow water waves AL2 U = = O(1) cnoidal and solitary waves (3.41) D3 O(1) nonlinear shallow water waves where L is the characteristic horizontal scale (e.g. wavelength). From a certain ratio between water depth and wave amplitude initialization based on solitary waves is no longer valid. This is visualized in figure 3.4, which shows waves with different amplitude together with an indication of the corresponding Ursell number. For a wave with sufficiently high Ursell number, a tail wave arises. The formation of tail waves is mainly caused by an overpredicted initial velocity field. It leads to disturbances in a wave, which are restored by forming these tail waves. A way to diminish these waves is to reduce the initial velocity by a specific factor. The velocity initialization, equation (3.38), changes to η¯ u(x, t ) = f · c (3.42) 0 D In figure 3.5, results for both with and without reduction factor are shown. It can be seen that tail waves are partly diminished by this reduction factor, but it still does not behave as a solitary wave. A drawback of this reduction factor is a slight decrease in amplitude. An optimum for this reduction factor could be found. To get completely rid of these tail waves, a more proper velocity initialization must be found. In this report, it has not been investigated in more detail.
3.3 Experimental and numerical comparisons
For the Boussinesq model, results can be presented demonstrating the accuracy of the used numerical schemes. Wave propagation and interaction of solitary waves are studied and com- pared with experimental data. The experiments have been conducted at the W.G. Prichard
18 CHAPTER 3. 1D BOUSSINESQ MODEL
(a) A = 0.004 m, U ≈ 60 (b) A = 0.003 m, U ≈ 45
(c) A = 0.002 m, U ≈ 30 (d) A = 0.001 m, U ≈ 15
Figure 3.4: Single wave propagation for different wave amplitudes at t = 0.5 s, water depth D = 0.05m
Figure 3.5: Comparison single wave propagation at t = 2 with reduction factor f = 0.85: original initialization (orange), initialization with reduction factor (blue).
Fluid Mechanics Laboratory of Penn State University, in a wave channel of length 13 m [11]. The quiescent water depth was set to 0.05 m. Solitary waves were generated by a horizontal, piston-like motion of a paddle, driven by a linear motor which allowed the generation of highly accurate and repeatable wave profiles. The measurements of the water surface were done by a bottom-mounted pressure tranducer and by four non-contacting wave gages. In particular, two cases are considered; head-on collisions and overtaking collisions.
19 CHAPTER 3. 1D BOUSSINESQ MODEL
3.3.1 Head-on collisions
In case of head-on collision two counter-propagating solitary waves with different wave ampli- tude have been considered. The initial conditions are chosen such that it matches the first experimental data set as accurate as possible. This is done by superposition of two single soli- tary wave solutions, equation (3.36), where the second wave has a negative velocity. The initial time t0 for both waves is based on the wave peaks
x t = max (3.43) 0 c where xmax is x-coordinate of the wave peak. Figure 3.6 records the wavetank measurements of the experiments of this collision at eight time steps. Solutions are obtained using ∆x = 0.005 m and a time step based on a CFL = 0.5. The reduction factor for velocity initialization is empirically set to f = 0.85. This diminishes tail waves most effectively. During collision (figure 3.6c) it can be seen that the waves merge and the solution rises to an amplitude larger than the sum of the amplitudes of the two incident solitary waves. After collision, the two waves emerge and separate from each other. As a result of this collision, the amplitudes of the two resulting solitary waves are slightly smaller than the incident amplitudes. The details of the interaction are relatively well represented by the numerical model. It reproduces the measured profile with a relatively small error.
3.3.2 Overtaking collisions
In this case, two co-propagating solitary waves have been considered. Both waves have the same direction of velocity, but are ordered so that initially the larger amplitude wave trails the smaller one. The collision consists of the larger solitary wave catching up the smaller wave, passing on and subsequently separating from it. It is modelled with same grid size and time step as for the case with head-on collisions. Again, the velocity initialization is reduced by a factor f = 0.85. Figure 3.7 shows eight measurements of wave profiles, together with the numerical results. Again, the initial conditions are set such that it fits the first experimental data set most accurately, (figure 4.7a). Since the interaction occurs with wave velocities of the same sign, it takes place over a long distance. Therefore, it is necessary in the experimental case to have the measurement device move in a reference frame adapted to the mean velocity of both waves, which leads to measurements in a local frame. For comparing with the numerical model, which is in the absolute frame, the experimental data has been shifted according to the distance travelled based on the mean velocity of the waves. Overall, the shape of the experimental data and the numerical simulations are well correlated. However, two types of errors are observed. First, tail waves are still present in the numerical model. The velocity could be reduced even more, but it will affect wave amplitude as well. The second error is in overshooting the measured amplitude. As can be seen, the measured amplitude decays in time, while in the numerical simulation it maintains its initial amplitude. This is mainly due to dissipative processes in the experiments, like wall friction. These effects are more significant for overtaking collisions as for head-on collisions, since it extends over a longer time interval. Due to higher amplitude wave speed will increase compared to the experimental data, according to equation (2.19). This can be particularly observed in the larger wave, which propagates with a higher velocity than in the measurements. To include dissipative effects, a simple friction source is added in the momentum equation, equation (3.8). The friction term is based on laminar flow
2uµ τ = (3.44) w H
20 CHAPTER 3. 1D BOUSSINESQ MODEL
The friction source becomes |τ |A S = f · w (3.45) f ρ∆V |τ | w∆x = f · w (3.46) ρ w∆xH 2ν = f · u (3.47) H2 where f is the friction factor. This term can be determined empirically. Which is discretized as follows 2ν S = f · un+1 (3.48) fi n n 2 i Hi +Hi−1 2 and subsequently added to the term of the left hand side of the momentum discretization, equation (3.35). The results are plotted in figure 3.7 (orange lines). The friction term is set to f = 0.6. The effect of the added friction is clearly visible and gives an indication of the influence of friction. For 1D Boussinesq model, friction has not been investigated in more detail. Based on results for both head-on and overtaking collisions it can be concluded that the 1D Boussinesq model is capable in generating correct waves and gives useful information for the 2D Navier-Stokes model.
21 CHAPTER 3. 1D BOUSSINESQ MODEL
(a) t = 0 s
(b) t = 0.50074 s
(c) t = 0.75264 s
(d) t = 0.8018 s
22 3.3. EXPERIMENTAL AND NUMERICAL COMPARISONS 23
(e) t = 0.85095 s
(f) t = 0.89396 s
(g) t = 1.02912 s
(h) t = 1.20116 s
Figure 3.6: Head-on collision of two solitary waves of height A1 = 0.01217 m and A2 = 0.01063 m at selected time steps: numerical results (solid line), experimental results (dots) CHAPTER 3. 1D BOUSSINESQ MODEL
(a) t = 0 s
(b) t = 2.5990 s
(c) t = 3.5021 s
(d) t = 4.1472 s
24 3.3. EXPERIMENTAL AND NUMERICAL COMPARISONS 25
(e) t = 4.697 s
(f) t = 5.5972 s
(g) t = 6.6017 s
(h) t = 8.3989 s
Figure 3.7: Overtaking collision of two solitary waves of heights A1 = 0.02295 m and A2 = 0.00730 m at selected time steps: numerical results (blue line), experimental results (dots), numerical results with friction (orange line) CHAPTER 3. 1D BOUSSINESQ MODEL
3.3.3 Accuracy study In the previous section the numerical model is validated with experimental data and it is observed that model presents well accurate results. This accuracy can be analysed in more detail by performing a grid refinement study and decreasing time step. The purpose of this test is to observe how coarse the grid could be, before inaccurate results are obtained. A single wave is considered. In figure 3.8, dependency of grid size can be seen. It is clearly visible that grid sizes of ∆x = 0.16 m and ∆x = 0.08 m lead to poor results. As the grid size decreases, the waves shift rightwards and the amplitudes increase slightly. From a grid size of ∆x = 0.02 m acceptable results are obtained. For even smaller grid sizes, a really small shift is observed. It can be concluded that a grid size of ∆x = 0.02 m gives sufficiently accurate results. Sensitivity of time step this can be obtained as well. The same case is considered. Several CLF values have been evaluated, which are plotted in figure 3.9. It is observed that for increasing CFL, the wave shift rightwards. Differences between CFL values below one are not significant. As expected, for CFL values larger than 1 amplitude is overshooted and the solution becomes unstable.
Figure 3.8: Water height at t = 1 for different grid sizes ∆x: 0.16 m (orange) 0.08 m (blue), 0.04m (red), 0.02 m (green), 0.01 m (black), 0.005 m (yellow); time step based on CFL = 0.5
Figure 3.9: Water height at t = 1 for different CFL values: 0.1 (orange), 0.5 (blue), 1 (green), 2 (red), 5 (violet); grid size ∆x = 0.01 m
26 4. 2D Naviers-Stokes model
In chapter 3 it has been shown that the 1D Boussinesq model is capable of modelling realistic waves. In this chapter, the commercial CFD software ANSYS Fluent is used to model solitary waves. This software solves the general mass and momemtum equations as in equation (2.1) and (2.2). With proper initial and boundary conditions waves can be modelled. The geometry is based on a simple 2D channel, shown in figure 4.1. It is meshed on a uniform grid using rectangular elements. Since it is multiphase flow of water and air, the volume of fluid method is used. Viscous terms are neglected in this simulation. Left and right boundaries are modelled as walls, which means that incoming waves will be reflected. The bottom is also modelled as a wall. The top boundary is a pressure outlet with constant pressure. To enable Fluent generating waves, initialization is the most crucial part.
Figure 4.1: Geometry channel
4.1 Initial conditions
For initialization in ANSYS Fluent, a user-defined function has been used. An user defined function, or UDF, is a C function that can be loaded in ANSYS Fluent to enhance the standard features. In this UDF, most important variables have been initialized. The initialized variables are the volume fraction of both water and air, the velocity in x- and y-direction and the pressure distribution. The complete code of the UDF can be found in the appendix, chapter 8.
27 CHAPTER 4. 2D NAVIERS-STOKES MODEL
4.1.1 Initialization volume fraction The volume fraction for water and air is based on the analytical solution of a solitary wave, equation (3.36). The water level is given by
x − ct H(x, t ) = D + A sech2 0 (4.1) 0 ∆
If a specific cell is fully above the water level, the volume fraction of air is set to one, when it is fully below the water level it is set to zero. At the interface, a fraction between one and zero is obtained. Mathematically, this is done as follows
Yti − min(max(Hi,Ybi ),Yti ) αairi = (4.2) Yti − Ybi where Yti and Ybi are the y-coordinates of respectively top and bottom face of cell i. When in a cell volume fraction of air is known, volume fraction of water can simply be obtained by:
αwateri = 1 − αairi (4.3)
Figure 4.2 shows the contour plot of the volume fraction for both water and air. The red part corresponds with the air fraction, the blue part with water fraction. Between both fractions, an interface can be seen.
Figure 4.2: Initialization volume fraction
4.1.2 Initialization velocity For initialization of the velocity, distinction must be made between water and air. The water velocity in x-direction is chosen similar to the 1D Boussinesq model
A x − ct u(x, t ) = c sech2 0 (4.4) 0 D ∆
As can be seen, the velocity does not depend on y-position. This is not completely realistic, but for the initial case it is sufficient. The water velocity in y-direction is derived based on mass conservation, equation (2.1).
∂u ∂v + = 0 (4.5) ∂x ∂y ∂v ∂u = − (4.6) ∂y ∂x Z y ∂u v = − dy (4.7) 0 ∂x The expression in equation (4.4) can be substituted in equation (4.7). Workig out this integral, the following expression for the velocity in y-direction is found.
2cA x − ct x − ct v(x, y, t ) = · tanh 0 sech2 0 · y (4.8) 0 ∆H ∆ ∆
28 CHAPTER 4. 2D NAVIERS-STOKES MODEL
The velocity of air is set to zero. Again, this is not completely correct. The initial velocity of water causes a positive momentum. For conservation of momentum, there should be an initial opposite velocity of air. Due to the huge difference in density between water and air (factor of 1000), this contribution can be neglected and therefore it is allowed to set air velocity to zero. In figure 4.3 contour plots are presented for the velocity initialization.
(a) velocity x-direction
(b) velocity y-direction
(c) velocity magnitude
Figure 4.3: Initialization velocity
4.1.3 Initialization pressure The pressure is initialized from the dominant terms of the momentum equation, equation (2.2). The dominant terms are ∂p = ρg (4.9) ∂y y Integrating gives Z y ∂p Z y dy = ρgydy (4.10) y0 ∂y y0 where y0 is the coordinate of the top boundary. At this boundary the pressure is known, since it measured as a pressure outlet. Due to multiphase problem, the integrals must be split into two parts
∗ ∗ Z y ∂p Z y ∂p Z y Z y dy + dy = ρairgydy + ρwatergydy (4.11) ∗ ∗ y0 ∂y y ∂y y0 y The coordinate y∗ is at the interface between water and air. It is assumed that the air pressure is constant and equal to the pressure at the top boundary. The first term on the left hand side is therefore equal to zero. The first term of the right hand side can be neglected. Since the huge difference in density of air and water, this term hardly contributes. Equation (4.11) simplifies to Z y ∂p Z y dy = ρwatergydy (4.12) y∗ ∂y y∗ Solving this equation gives an expression for the water pressure distribution
∗ ∗ p(x, y) = ρwatergy(y − y ) + p(y ) (4.13)
29 CHAPTER 4. 2D NAVIERS-STOKES MODEL
Where p(y∗) is equal to atmospheric pressure. Equation (4.1) can be substituted for y∗. The contour plot of the pressure distribution is figured in figure 4.4, where the highest pressure occurs at the bottom of the domain.
Figure 4.4: Initiliazitation pressure distribution
4.2 Solution methods
The solution methods that have been used to discretize mass and momentum equations are briefly discussed. A pressure-based solver is chosen, which is common used for low-speed flow problems. It obtains velocity field from the momentum equation. The pressure field is ex- tracted by solving a pressure equation which can be obtained by manipulating continuity and momentum equations. The spatial discretization is done by upwind schemes. Several upwind schemes are available in ANSYS Fluent, all differing in accuracy. A second-order upwind scheme is used. Upwind schemes are quite diffusive and therefore generally unsuitable for multiphase flows. Additional schemes are needed for tracking sharp interfaces between the both phases. A compressive method is chosen. Time derivatives are discretized implicit in time with second- order accuracy. For coupling between velocity and pressure, the SIMPLE algorithm is used. A more detailed description of discretization schemes and mathematical expressions can be found in the theory guide of ANSYS Fluent [12].
4.3 Single wave propagation
Based on the initialization and the used solution methods, results can be obtained for single wave propagation. figure 4.5 presents contour plots of the air phase. It clearly shows the wave propagating to the right, after which it hits the wall. The wave reflects and propagates to the left, while it maintains its shape. The results corresponds with the 1D Boussinesq model in figure 3.2.
(a) t = 1.
(b) t = 2.5.
(c) t = 4.
Figure 4.5: Single wave propagation at several time steps, length of channel L = 5m, water depth D = 0.2m, wave height A = 0.01m, grid size ∆x = 0.005m, time step ∆t = 0.007s
30 CHAPTER 4. 2D NAVIERS-STOKES MODEL
4.4 Experimental and numerical comparisons
As for the 1D model, the accuracy can be demonstrated using experimental data of solitary wave interactions. Again, head-on collisions and overtaking collisions have been considered.
4.4.1 Head-on collisions The initial conditions have been derived in a similar way as for the 1D model. Figure 4.6 presents the results of the numerical model together with experimental measurements at selected time steps. The free surface of the 2D model can be found by obtaining all points where volume fraction of air is exactly equal to 0.5. This is the interface between water and air. To diminish the tail waves, the velocity initialization is reduced by factor f = 0.85. The grid size is 0.005 m and the time step is set to 0.007 s. A very good agreement with the experimental data is observed. The maximum height of the wave is predicted really well by the numerical model, just as the wave speed. During the emerge, figure 4.6c, the numerical model shows a slightly narrower wave. When waves are separated, figure 4.6h, the free surface between both waves is lightly undershooted in the numerical model.
4.4.2 Overtaking collisions The overtaking case is modelled with same grid size and time step as the previous case. The velocity initialization is reduced by a factor f = 0.85. Results are plotted in figure 4.7. A good match is observed between the numerical model and experimental data. The shape is well correlated and in contrast to the 1D Boussinesq model the amplitude of the 2D model decreases over time. Despite good agreement of the 2D model, the question that arises is how this decay is induced. As mentioned above, viscous effects have not been taking into account. Hence dissipation by wall friction can not be the reason. Another indication of a dissipative effect is the shift of experimental data. As explained in chapter 3, in case of overtaking collision the experimental data is shifted with the distance based on the mean velocity of both waves. However, for the 2D model mean velocity is reduced by 4% (e.g. 0.02 m/s) to achieve a good fit. Numerical dissipation could be an explanation for this decay. This is supported by observations given in figure 4.8, where the evolution of a single soliton is shown. In figure 4.8a the solution based on initialization and solution methods, as explained in respectively section 4.1 and section 4.2, are given. It is observed that after 14 seconds amplitude has decreased with almost 40% of its initial amplitude. This is the main difference between the 1D model and the 2D model. The 1D model is derived for solitary wave types of solutions and therefore there should be almost no dissipation, which is indeed observed in the results for head-on and overtaking collisions, figure 3.6 and figure 3.7. The 2D model makes use of general mass and momentum equations and to model correct solitary waves initialization and solution methods are crucial. This is shown in Figure 4.8b, where velocity is initialized without velocity reduction factor. The decay after 14 seconds is in this case 30% of its initial amplitude and the wave peak has slightly moved rightwards, indicating a higher wave speed. To verify differences in solutions methods, a simulation is performed where the spatial discretization is done with the QUICK scheme instead second-order upwind. The QUICK sheme will be typically more accurate on structured meshes aligned with the flow direction. The evolution is presented in figure 4.8c. However, no significant differences are seen. Numerical dissipation has not been investigated in more detail. Based on the observations for the overtaking collision, it seems that numerical dissipation is in same order of magnitude as friction in the experimental case.
31 32 CHAPTER 4. 2D NAVIERS-STOKES MODEL
(a) t = 0 s
(b) t = 0.50074 s
(c) t = 0.75264 s
(d) t = 0.8018 s 4.4. EXPERIMENTAL AND NUMERICAL COMPARISONS 33
(e) t = 0.85095 s
(f) t = 0.89396 s
(g) t = 1.02912 s
(h) t = 1.20116 s
Figure 4.6: Head-on collision of two solitary waves of height A1 = 0.01217 m and A2 = 0.01063 m at selected time steps: numerical results (solid line), experimental results (dots) CHAPTER 4. 2D NAVIERS-STOKES MODEL
(a) t = 0 s
(b) t = 2.5990 s
(c) t = 3.5021 s
(d) t = 4.1472 s
34 4.4. EXPERIMENTAL AND NUMERICAL COMPARISONS 35
(e) t = 4.697 s
(f) t = 5.5972 s
(g) t = 6.6017 s
(h) t = 8.3989 s
Figure 4.7: Overtaking collision of two solitary waves of heights A1 = 0.02295 m and A2 = 0.00730 m at selected time steps: numerical results (solid line), experimental results (dots) 36 CHAPTER 4. 2D NAVIERS-STOKES MODEL
(a) Original solution method with velocity reduction
(b) Original solution method without velocity reduction
(c) Adapted solution method without velocity reduction
Figure 4.8: Single wave evolution for different solution methods and initialization, at selected time steps t = 0.7 s, t = 7.0 s and t = 14.0 s CHAPTER 4. 2D NAVIERS-STOKES MODEL
4.4.3 Accuracy study The accuracy of the 2D model has been tested to verify dependency of grid size and time step. Especially for 2D and 3D simulations computational time becomes more important. The objective in the SprayIce project is to model correct waves with a grid size as coarse as possible. Figure 4.9 shows the results for different grid size for a single wave. It can be seen that the x-coordinate of the wave peak is hardly affected by grid size. It is observed that grid sizes ∆x = 0.04 m and ∆x = 0.02 are too coarse. The wave becomes wider and decreases in amplitude. From a grid size ∆x = 0.01 m a sufficiently accurate solution is obtained, only a small shift leftwards is seen. The predicted amplitude does not change for these grid sizes. On a grid size ∆x = 0.01 m simulations with different time steps have been performed, which are presented in figure 4.10. The largest possible time step is around ∆t = 0.02s. Larger time steps does not lead to a converging solution. For smaller time steps peak shifts leftwards, which is in accordance with the 1D model. However, a remarkable difference is observed. For decreasing time step the size of the shift increases. For even smaller time steps than ∆t = 0.001s a shift can no longer be seen.
Figure 4.9: Water height at t = 1 for different grid sizes ∆x: 0.04 m (blue) 0.02 m (red), 0.01m (green), 0.005 m (yellow), 0.0025 m (purple); time step ∆t = 0.005 s.
Figure 4.10: Water height at t = 1 for different time step ∆t: 0.02s (green), 0.01s (yellow), 0.005s (purple), 0.0025s (brown), 0.001s (red); grid size ∆x = 0.01 m
37 5. Conclusion
In the SprayIce project, a CFD method was needed that can handle wave propagation without severe numerical dissipation. It must be verified that wave structures are not destroyed by numerical dissipation before impact on vessel or offshore structures.
The 1D Boussinesq model gives insight in the behaviour of solitary waves. Initialization has been done based on the analytical expression of a solitary waves. The model is able in gen- erating waves with the correct characteristics of a soliton. For some cases it is observed that oscillatory tail waves arise, which can be explained by the Ursell number. Due to overprediction of initial velocity, disturbances occur and tail waves are formated. A way to diminish tail waves is by reducing initial velocity. Accuracy of the used model is tested by presenting results of experimental data. Two cases are considered, head-on and overtaking collisions. For head-on collision, the details of the interaction are well presented. Wave height and speed are in good accordance with the experiments. For overtaking collisions, some errors can be observed. De- spite reduction in initial velocity, tail waves are still present. Besides, wave height is slightly overestimated. Since wave speed is related to its amplitude, wave speed is overshooted as well. Overshoot in amplitude could be explained by absence of friction in the numerical model. The 1D Boussinesq model maintains its initial amplitude, while in the experimental case amplitude decays over time. Friction effects become more significant in overtaking collision than head-on collision, since it runs over a longer time interval. This has been tested by implementing a simple friction term, which was based on laminar flow. It is shown that with a friction term, wave height and speed are predicted more accurately. Results of the 1D Boussinesq model give useful insight in single solitary waves and interaction of solitary waves, which can be used for the 2D Navier-Stokes model as a benchmark.
A similar process is done for a 2D Navier-Stokes model. Commercial CFD software, ANSYS Fluent, has been used. For generating correct waves, it is observed that initialization is crucial. With an user-defined function important flow variables can be initialized. Results for single wave propagation are in accordance with the Boussinesq model. Characteristics of solitons are well presented. Again, results are verified with experimental data of solitary wave interaction. The results for head-on collisions are in really good agreement. Interaction of overtaking collisions is modelled quite accurate. In contrast to the Boussinesq model, the amplitudes in the 2D model decreases over time. Since viscous effects are not included, friction can not be the reason for this decay. It is shown that the 2D model generates numerical dissipation. It can concluded that nu- merical dissipation in the model is in the same order of magnitude as friction in the experiments.
Eventually, it can be concluded that commercial CFD can be used to model water waves. In the continuation of the SprayIce project, the user-defined function can be changed to the de- sired wave case. This can used as input for modelling of wave-vesssel impact, droplet break-up and spray formation.
38 6. Recommendations
As mentioned in this report, the 1D Boussinesq system was mainly used to gain useful informa- tion about solitary waves. There are many different but equivalent types of Boussinesq systems. In this report only one type of Boussinesq system have been handled. The original Boussinesq system has been extended to improve for example non-linear behaviour or frequency dispersion, or to include wave breaking or variable water depth. Several Boussinesq systems should be used to gain insight in similarities and differences. This increases knowledge of modelling solitary waves, which can subsequently be used as a more accurate benchmark.
The 1D model has been modelled with own written code. The advantage is that one has fully control of the discretization. A more detailed study should be performed into the numerical behaviour of this 1D model. Different discretization schemes should be used to check differences in accuracy. The effect of writing terms in (non)-conservative form can be analyzed either. Since the 2D model is performed using commercial software, one may not have fully control of the discretization. Observations in different numerical schemes for the 1D model should be used to change the numerical methods in the 2D model. It should be checked if different schemes lead to less numerical dissipation.
Friction effects can be investigated in more detail. It is shown that for the 1D model in overtak- ing collisions friction plays a role. In the 2D simulation, flow is assumed as inviscid, so friction is not included. However, it is observed that numerical dissipation is in the same order as friction in the experiments. If the simulation is carried out with viscous flow, even more dissipation is expected. If numerical dissipation can be decreased, one may change to viscous flow.
As can be seen in single wave propagation, tail waves arise in some cases. It is expected that it is a physical phenomenon, because these waves are not an exact soliton. More research should be done to investigate the fundamental cause of these waves. Initialization and numerics may also have influence on the origin of tail waves.
Simulations are performed for small water waves and wave height yet. Eventually, it must be scaled up to a higher dimensions, such that it is applicable to for the SprayIce project. For scaling up it should be checked if the Ursell number is still in the solitary wave regime. Since droplet break-up and spray formation is a 3D problem, modelling of wave propagation should be extended to a 3D case. Therefore, geometry and the UDF should be adjusted.
39 7. Bibliography
[1] J. Billingham, A.C. King Wave Motion, Cambridge University Press, (75-86), 2006.
[2] B. le M´ehaut´e, An introduction to hydrodynamics and water waves, Springer, ISBN 0-387- 07232-2, 1976.
[3] J. Boussinesq, Th´eriedes ondes et des remous qui se propagent le long d’un canal rectangu- laire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond J. Math´ematiquesPures et Appliqu´ees,(55-108), 1872.
[4] A. Svendsen, Introduction to nearshore hydrodynamics World Scientific, (381-423), 2005.
[5] J.S. Russel Report on waves, Fourteenth meeting of the British Association for the Advance- ment of Science, 1844.
[6] J.L. Bona, M. Chen, J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory, J. Nonlinear Sci. 12, (283-318), 2002.
[7] Denys Dutykh, Theodoros Katsanounis, Dimitros Mitsotakis, Finite volum schemes for dis- persive wave propagation and runup J. Computational Physics 230 (3035-3061), 2011.
[8] D.H. Peregrine Long waves on a beach, J. Fluid Mech. 27, (815-827), 1967.
[9] G.S. Stelling, M. Zijlema, An accurate and efficient finite-difference algorithm for non- hydrostatic free-surface flow with application to wave propagation Int. J. for numerical meth- ods in fluids, (1-23), 2003.
[10] G. Stelling, S.P.A Duinmeijer, A staggered conservative scheme for every Froude number in rapidly varied shallw water flows Int. J. for numerical methods in fluids, (1329-1354), 2003.
[11] W. Craig, Solitary water wave interactions 2005.
[12] ANSYS Inc, ANSYS Fluent Theory Guide
40 8. Appendix
This C code generates the initialization for important flow variables (volume fraction, velocity and pressure distribution). First, it defines general water and wave properties. Functions are given which contain mathematical expressions for initialization of all variables, as explained in section 4.1. In the function DEFINE INIT loops are performed over all subdomains. Each subdomain corresponds to a specific phase (eg. water or air). In a subdomain there is looped over all cells and with C CENTROID coordinates of a certain cell can be obtained. In the cell center, flow variables can be defined. A cell can consist of multiple phases and therefore coordinates of cell faces are needed as well to obtain an fraction. These are found with F CENTROID. This C code can be compiled, after which it can be hooked to ANSYS Fluent.
1 /******************************************************* UDF for initializing flow flied variables 3 *******************************************************/
5 # include"udf.h"
7 #define PRIMARYPHASE_ID 2/*air_phase*/ #define SECONDARYPHASE_ID 3/*water_phase*/ 9 #define WATER_DEPTH 0.05/*[m]*/ #define AMP_WAVE 0.01/*[m]*/ 11 #define T0 1.5/*[s]*/ #define G 9.81/*[m/s^2]*/ 13 #define RHO_AIR 1.225/*[kg/m^3]*/ #define RH0_WATER 998.2/*[kg/m^3]*/ 15 #define P_REF 0/*[Pa(g)]*/
17
19 /*Calculate the volume fraction of the air phase*/ real primary_phase_vol_frac(real ybot,real ytop, real xpos) 21 { real delta = WATER_DEPTH*pow(4. * WATER_DEPTH/(3. * AMP_WAVE),0.5); 23 real c = pow(G*(WATER_DEPTH + AMP_WAVE),0.5); real eta = AMP_WAVE*pow(1. /cosh((xpos-c*T0)/delta),2.0);/*analytical solution of soliton wave*/ 25 real water_level = WATER_DEPTH + eta; real vol_frac; 27 vol_frac = (ytop-fmin(fmax(water_level,ybot),ytop))/(ytop-ybot); 29 return vol_frac; } 31 /*Calculate the volume fraction of the water phase, knowing that water_phase= 1-air_phase*/ 33 real secondary_phase_vol_frac(real ybot,real ytop, real xpos) { 35 return 1. - primary_phase_vol_frac(ybot,ytop,xpos); } 37 /*Calculate the velocity inx-direction of the water phase*/
A1 CHAPTER 8. APPENDIX
39 real secondary_phase_u_vel(real xpos, real vol_frac) { 41 real delta = WATER_DEPTH* pow(4. * WATER_DEPTH/(3. * AMP_WAVE),0.5); real c = pow(G*(WATER_DEPTH + AMP_WAVE),0.5); 43 real eta = AMP_WAVE*pow(1. /cosh((xpos-c*T0)/delta),2.0); real vel_u ; 45 if (vol_frac > 0.01) { 47 vel_u = c*eta/WATER_DEPTH; } 49 else { 51 vel_u = 0; } 53 return vel_u; } 55 /*Calculate the velocity iny-direction of the water phase*/ 57 real secondary_phase_v_vel(real xpos, real ypos, real vol_frac) { 59 real delta =WATER_DEPTH* pow(4. * WATER_DEPTH/(3. * AMP_WAVE),0.5); real c = pow(G*(WATER_DEPTH + AMP_WAVE),0.5); 61 real vel_v ; if (vol_frac>0.01) 63 { vel_v = 2. *AMP_WAVE* c * tanh((xpos - c * T0) / delta)*pow(1. / cosh((xpos - c * T0) / delta),2.) * ypos / (delta*WATER_DEPTH); 65 } else 67 { vel_v = 0; 69 } return vel_v; 71 }
73 /*Calculate the pressure distribution in the water phase, based on the dominant terms of the momentum equation*/ real primary_phase_press(real vol_frac) 75 { real press = vol_frac*P_REF; 77 return press; } 79 real secondary_phase_press(real xpos, real ypos, real vol_frac) 81 { real delta = WATER_DEPTH*pow(4. * WATER_DEPTH/(3. * AMP_WAVE),0.5); 83 real c = pow(G*(WATER_DEPTH + AMP_WAVE),0.5); real eta = AMP_WAVE*pow(1. /cosh((xpos-c*T0)/delta),2.0); 85 real water_level = WATER_DEPTH + eta; real press = (-RH0_WATER * G*(ypos - water_level) + P_REF)*vol_frac; 87 return press; } 89
91 DEFINE_INIT(wave_initialization, mixture_domain) { 93 int phase_domain_index; cell_t c; 95 Thread *t; face_t f; 97 Thread *tf; Domain *subdomain; 99 real x[ND_ND];
A2 CHAPTER 8. APPENDIX
real y_b ;/*Initial value bottom_face*/ 101 real y_t ;/*Initial valua top_face*/ real fac ; 103 int n;
105 /*loop over all subdomains(phase) in the superdomain(mixture)*/ sub_domain_loop(subdomain,mixture_domain,phase_domain_index) 107 { /*loop if primary phase*/ 109 if (DOMAIN_ID(subdomain) == PRIMARYPHASE_ID) { 111 /* loop over all cell threads in the domain*/ thread_loop_c(t,subdomain) 113 { /*loop over all cells*/ 115 begin_c_loop_all(c,t) { 117 y_b = 1. e99 ;/*Initial value bottom_face*/ y_t = -1.e99;/*Initial valua top_face*/ 119 /*loop over all faces ofa cell*/ 121 c_face_loop(c,t,n) { 123 f = C_FACE(c,t,n); tf = C_FACE_THREAD(c,t,n); 125 /*F_CENTROID gives the coordinates of the faces, the if statements determines the top and bottom face ofa specific cell*/ F_CENTROID(x,f,tf); 127 if(x[1]
A3 CHAPTER 8. APPENDIX
tf = C_FACE_THREAD(c,t,n); 163 F_CENTROID(x,f,tf); if(x[1]
A4