Evaluation of Drag Estimation Methods for Ship Hulls

Home , Capesize, Ship measurements, Waterline, Waterline length

DEGREE PROJECT IN ENGINEERING PHYSICS, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2020

Evaluation of drag estimation methods for ship hulls

HAMPUS TOBER

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

Abstract

This study aims to evaluate to which degree CFD can complement traditional towing-tank experiments, and to develop a computationally cheap and robust methodology for these type of simulations. Two radically diﬀerent surface ship hulls were chosen for the trials: a capesize bulk carrier and a fast displacement hull. A bare submarine hull was also used to benchmark the software in the early stages of the study. All simulations were Reynolds–Averaged Navier–Stokes (RANS) simulations using the k-ω-SST turbulence model. The chosen software was OpenFOAM 5.x and foam-extend 4.1 coupled with the commercial expansion Naval Hydro Pack, which is supposed to handle high Courant numbers well.

Producing a high-quality mesh which is able to capture both the free surface and the boundary layers proved to be of utmost importance and the meshing procedure is thus discussed in detail. A majority of the surface ship simulations suffered from a phenomenon known as numerical ventilation. The effect seemed to be much worse for the fast-displacement hull and the drag estimates for this hull deviated around 16.1% from experimental data. The bulk carrier was less affected and the drag estimates for this hull only deviated around 6.3% from experimental data. In order to reduce the computational cost, all surface ship simulations were run with a maximum Courant number of 25 and some consequences of this are also discussed.

Acknowledgements

I would like to express my greatest appreciation to Marcus Winroth and Mattias Liefvendahl for their patient guidance and unparalleled support throughout this project. I would like to thank everyone from the departments of Underwater Technology and Naval Systems for welcoming me to FOI and for making me feel like part of the group. I would also like to thank Ricardo Vinuesa for all the good feedback and comments I’ve gotten throughout the project. Finally I want to thank Director Nobuyuki Hirata of the National Maritime Research Institute in Japan for providing me with data and answering my questions about his papers.

Contents

1 Introduction 1 1.1 Submarines ...... 1 1.2 Surface Ships ...... 3 1.3 Ship terminology ...... 4 1.4 Semi-Empirical Methods ...... 5 1.5 Fluid Mechanics ...... 6 1.6 Pressure Gradient Flows ...... 6 1.7 Turbulence ...... 6

2 Cases 7 2.1 Case 1: Joubert Bare Hull ...... 7 2.2 Case 2: Japanese Bulk Carrier ...... 10 2.3 Case 3: MARIN Systematic Series FDS-5 ...... 12

3 Method - Computational Fluid Mechanics 14 3.1 RANS ...... 14 3.2 Eddy viscosity models ...... 15 3.3 Solving algorithm ...... 16 3.4 Wall functions ...... 18 3.5 Two-phase ﬂows ...... 18 3.6 Rigid Body Motion ...... 19

4 Computational Setup 20 4.1 OpenFOAM ...... 20 4.2 Meshing ...... 20 4.3 Case 1: Joubert Bare Hull ...... 21 4.4 Case 2: Japanese Bulk Carrier ...... 24 4.5 Case 3: MARIN Systematic Series FDS-5 ...... 29

5 Results 32 5.1 Case 1: Joubert Bare Hull ...... 32 5.2 Case 2: Japanese Bulk Carrier ...... 35 5.3 Case 3: MARIN Systematic Series FDS-5 ...... 45

6 Conclusions 50 6.1 Case 1: Joubert Bare Hull ...... 50 6.2 Surface Ships ...... 50 6.3 Summary ...... 52 6.4 Outlook ...... 52

Appendices 55

A Meshes 55

B Residuals 63 1 Introduction

Traditional towing-tank tests are notoriously complex to perform. First an accurate scale model of the ship hull needs to be manufactured, where parameters such as displacement, weight distribution and surface geometry will be sources of error. The next step is to design a measurement campaign where the model is tested under the right conditions. Here the measurement devices, and possibly the testing facilities will introduce errors to the measurements. Finally the data needs to be scaled up to give estimates of the performance of the full scale ship and this process will both introduce new errors and possibly scale up the previously mentioned errors. For these reasons, towing tank tests can easily become very expensive and are usually only performed near the end of the design campaign.

A numerical towing tank based on Computational Fluid Dynamics (CFD) could instead allow evaluations of designs multiple times during the design process. See [1] for a relatively recent summary of state of the art numerical ship hydrodynamics. Simulations of full-scale ships would most likely be too computationally expensive due to the high Reynolds numbers and large meshes, but it would be possible to run simulations of models of different scales to further investigate how the final results can be scaled up as accurately as possible. When developing the simulation methodology emphasis was put on keeping down computational costs to allow entire series of hull forms to be tested within a reasonable time frame. In order to ensure that the methodology would work for many different types of ships three vastly different hull types were tested, one high speed displacement hull, one bulk carrier hull belonging to the largest class of bulk carriers and one bare submarine hull. The software chosen as basis for the numerical towing tank was the open source finite volume based software OpenFOAM.

1.1 Submarines The idea of submarines has been around for hundreds of years but it wasn’t until the end of the 18th century that working prototypes were built, the ﬁrst being Bushnell’s Turtle followed by Fulton’s Nautilus. These early submarines were man-powered military vessels designed for the purpose of planting explosive charges on the underside of warships. Although the vessels were technologically advanced for their time, they had little to no success.

Major strides in submarine technology were made over the course of the 1800s and by the end of the century the Holland-class submarine was designed for the British Royal Navy by John Philip Holland. The Holland-class had many features of modern submarines and was powered by a petrol engine coupled with an electric motor.

However it was during the 20th century that the development of submarines really took oﬀ, mainly due to the two world wars. Many of the early military submarines were however optimized for surface operation and little attention was given to underwater performance.

1 Figure 1: Top: German Type VII-C U-boat (1930s) Bottom: German Type XII U-boat (1940s)

Towards the end of the Second World War German designers started to realize the improvements in underwater performance that could be gained by streamlining the hull and removing appendages. The diﬀerence in design philosophy can be seen by comparing the earlier Type VII with the later Type XII [2] (see Figure 1).

After the war both the British and American navies obtained Type XII’s for evaluation and were stunned by the technological advances that the Germans had made. Their subsequent designs, the British Porpoise-class and American Tang-class, were heavily inﬂuenced by the German submarine [3] [4]. Modern day submarines are even more streamlined and are usually based around an axisymmetric base hull, such as the Joubert Bare Hull which is discussed in Chapter 2.

1.1.1 Resistance The resistance acting on a body moving through a 1-phase ﬂuid (constant density) is usually divided into two components, pressure resistance (or form resistance), Rp, and frictional resistance,Rf .

Pressure resistance: Since all ﬂuids are viscous to some degree there will be a boundary layer on all surfaces of the body in contact with the ﬂuid. This disrupts the fore-aft symmetry of the pressure acting on the body, resulting in a net drag force known as the pressure resistance.

Frictional resistance: The no-slip condition at the surface of the submarine means that the ﬂuid counteracts the movement of the body by exerting tangential shear forces on the surface as it moves forward. The resulting drag coming from these shear forces is referred to as the frictional resistance

These resistances are often scaled with the dynamic pressure acting on the body to produce the non-dimensional coeﬃcients Cp and Cf . The scaling is done according to

Rp C = (1) p 1 2 2 ρu∞S and similarly for Cf , where ρ is the density of the fluid, u∞ is the inflow velocity and S is a characteristic area which in this study will be taken as the wet area of the submarine. We want to emphasize that Cp, in this study, refers to the streamwise (x) component of the pressure coefficient. Furthermore, when the coefficients are written with capital letters such as Cp this refers to the integral of the local coefficient values over the entire body, according to Z Cp = cp dSh , (2) Sh where Sh is the area of the hull and cp is the local coefficient.

2 1.2 Surface Ships Unlike submarines, surface ships have been around for thousands of years and even the earliest designs showed some understanding of ship hydrodynamics. Up until the invention of propeller driven ships it was however not as important to estimate the hydrodynamic drag on the hull. In order to properly dimension the propulsion system of a ship it is crucial to have a good estimate of its resistance at design speed, so when the ﬁrst steamships were developed in the mid 1800s this immediately became a problem and many of the early propeller driven ships thus had badly dimensioned propulsive systems.

In the 1870s William Froude started developing methods for estimating the resistance of full scale ships. He experimented with scale models and performed towed resistance tests using planks of diﬀerent length and surface texture. A few of Froude’s ideas are discussed in more detail in Section 1.4

1.2.1 Resistance For bodies moving close to or on the free surface of the ﬂuid another type of resistance becomes important, namely the resistance associated with the production of waves. With this in mind it is possible to make another division of the total resistance, RT , into viscous resistance,Rv, and residual resistance,RR. Using equation (1) this can be written in terms of non-dimensional coeﬃcients as

CT = Cv + CR. (3)

For surface ships the area under water will be used in the scaling of the coeﬃcients since the drag from the water is vastly dominating. This is discussed further in Chapter 4.

Viscous resistance: The component of the total resistance here referred to as viscous resistance is the sum of the frictional resistance, Rf , and the pressure resistance Rp. The reason for this grouping is that the pressure resistance can be argued to be the result of viscous effects, much like the frictional resistance. The pressure resistance is sometimes also referred to as viscous pressure resistance. This viscosity dependence means that the viscous resistance will scale with the Reynolds number of the flow, defined as

ρuL uL Inertial forces Re = = = , (4) µ ν Viscous forces where ρ is the density of water, L is a characteristic length which in most naval applications is taken as the length of the ship, µ is the dynamic viscosity of water and ν is the kinematic viscosity of water.

Residual resistance: The residual resistance is what is left of the total resistance once the viscous resistance has been subtracted. This component is usually dominated by the so called wave-making resistance and is even denoted as such in some references dealing with large displacement ships such as bulk carriers or container ships. When considering high speed crafts, other factors such as wave-breaking and spray can also become important.

All these components being related to surface waves, which are governed by gravity, means that the residual resistance will scale with the Froude number which is deﬁned as u Inertial forces Fn = √ = , (5) Lwl g Gravitational forces where Lwl is the length of the vessel at the waterline and g is the gravitational acceleration. [5]

3 1.3 Ship terminology For clariﬁcation a few of the terms speciﬁc to the ship building community are here explained since they are used both throughout this report as well as in many of the references.

Figure 2: Some ship measurements on a bulk carrier

Length between perpendiculars (Lpp) is the distance between the forward and aft perpendiculars. On wooden ships the stem was usually taken as the forward perpendicular and the stern post as the rear perpendicular, however on many modern ships it is not as simple. For instance on the bulk carrier in Fig (2) the forward perpendicular is right above the bulb on the bow and the aft perpendicular is the planned axis of rotation of the rudder.

Waterline length (Lwl) is the length of the ship measured at the waterline, at design load.

Beam at waterline (Bwl) is the widest width of the ship at the waterline, at design load.

Draft (T) is the distance from the lowest point of the hull to the surface.

Wet area (S) is the area of the hull which is under water.

Displacement (∇) is the volume of water being displaced by the presence of the hull. It is directly proportional to the weight of the ship and the density of the water.

Block coeﬃcient (CB) is the ratio of the displacement ∇ to a box with the dimensions Lwl x Bwl x T.

Amidships is the midpoint between the two perpendiculars.

Sinkage is the diﬀerence in draft experienced when the ship is moving relative to when it is stationary. Where along the hull the sinkage is measured can vary, but in this study it is deﬁned as the vertical movement of the center of mass.

Trim is the diﬀerence in pitch experienced when the ship is moving relative to when it is stationary. It is sometimes also expressed as the diﬀerence in sinkage between the fore and aft perpendiculars.

Center of Gravity (CoG) is the center of mass of the ship.

Center of Buoyancy (CoB) is the center of mass of the ﬂuid displaced by the hull.

4 1.4 Semi-Empirical Methods Semi-empirical methods are a type of performance prediction methods used extensively by the ship building community. As the name implies they are based to some degree on empirical data, but also contain estimations and modeling. A common approach is to start with experimental data from scale models which can then both be extrapolated to full scale and interpolated to make estimates about similar hull forms.

One major problem when scaling up resistance data is the fact that Cv and CR scale with diﬀerent non-dimensional numbers. Since it is practically impossible to keep both the Reynolds number and Froude number constant between a model test and a full scale ship, the total resistance cannot be scaled directly. The ﬁrst systematic approach to this problem was suggested by Froude in the 1870s and consisted of running model trials at the correct Froude number, measuring the total resistance and then subtracting an estimate of the frictional resistance, leaving only the residual resistance.

CR = CT − Cf (6)

Note that Froude used the frictional resistance instead of the viscous resistance as in Section 1.2.1, simply because no theory explaining viscous boundary layers existed at that time.

This method was adopted by many ship builders, and at the International Towing Tank Conference (ITTC) in 1957 [6] a standardized formula for calculating the frictional resistance was accepted in favor of Froude’s experimental plank-towing data. The formula was originally derived from the skin friction of a ﬂat plate but has been corrected to also take into account some three-dimensional eﬀects. It can be written as 0.075 Cf = . (7) log(Re) − 22

This procedure was later developed further and now also includes a form factor (k) to take the pressure (form) resistance into account as well (see ITTC 1978 [7]). The expression for the viscous resistance then becomes

Cv = (1 + k)Cf , (8) and the expression for the total resistance becomes

CT = (1 + k)Cf + CR (9)

There are of course limitations to this approach, and the diﬀerent types of resistance will interact with each other. The creation of waves will for instance change the wetted area of the hull and thus change the frictional resistance. Nevertheless, the ITTC78 approach has today essentially become an industry standard.

5 1.5 Fluid Mechanics The ground pillar of modern fluid mechanics is undoubtedly the Navier–Stokes equations. They are a set of coupled differential equations describing the conservation of mass, momentum and energy of a moving fluid. For the applications discussed in this report, both air and water can be considered to be incompressible fluids. Under these conditions the energy equation uncouples from the mass and momentum equations and it will therefore not be considered. Conservation of mass, often referred to as the continuity equation, can be written as ∂ρ ∂(ρui) + = 0 (10) ∂t ∂xj and since water and air are Newtonian fluids the conservation of momentum can be written as 2 ∂(ρui) ∂(ρujui) ∂ ui 1 ∂p + = ν 2 − + fi , i = 1, 2, 3 , (11) ∂t ∂xj ∂xj ρ ∂xi where ρ is the density of the fluid, u is the flow velocity, p is the pressure and f is the sum of all body forces. The subscripts i and j denotes directions in the cartesian coordinate system. Solving this set of equations yields the pressure and velocity fields of the flow domain which can then be used to calculate other flow quantities, such as the skin friction and pressure drag.

1.6 Pressure Gradient Flows As soon as there is curvature in the geometry pressure gradients will form which will either decelerate or accelerate the flow, depending on the curvature. Pressure gradients which decelerates the flow are called adverse pressure gradients (APG, ∂p/∂x > 0) while pressure gradients accelerating the flow are called favorable pressure gradients (FPG, ∂p/∂x < 0).

Adverse pressure gradients will often increase the complexity of fluid mechanics problems and can in some cases lead to flow separation. This happens when the flow closest to the wall is decelerated to a standstill and starts flowing opposite to the outer flow (see Figure 3). Adverse pressure gradients can be expected to arise near the converging stern sections on both submarines and surface ships. The importance of pressure gradient effects on turbulent boundary layers is discussed in detail in two articles by Vinuesa et al. [8] [9].

Figure 3: Boundary layer with an APG

1.7 Turbulence When the inertial forces of a flow grow large enough relative to the dampening viscous forces, the flow will turn turbulent. A turbulent flow is characterized by chaotic fluctuations in both flow velocity and pressure. These fluctuations give rise to eddies in a range of different sizes, from the scale of the geometry or domain (depending on the problem) down to the smallest viscous length scales. The presence of these eddies greatly increases the mixing rate of both momentum and energy, meaning larger wall shear stresses in wall bounded flows.

Due to the high Reynolds numbers of ships and submarines, their boundary layers are almost entirely turbulent with transition occurring within the ﬁrst few percentages of the length of the vessel. When performing experiments with scale models it is important to use tripping devices to ensure that the relative location of the transition point remains the same.

6 2 Cases

In this chapter the three test cases of the study are discussed and the chosen sources for reference data are summarized. The test cases were one submarine hull without appendages, one bulk carrier and one high speed displacement hull.

2.1 Case 1: Joubert Bare Hull The DSTO Joubert Bare Hull (JBH) comes from a series of generic submarine geometries based on the work done by P.N. Joubert in the early 2000s [3] [4]. The series was developed to be representative of many modern submarines without having any full scale equivalent, meaning that the geometries can remain unclassiﬁed and be discussed in the open literature. A number of articles documenting both experimental testing and CFD of the bare hull conﬁguration have been published and some of them are summarized below.

Figure 4: Joubert Bare Hull

Since this study mainly focuses on capturing the drag and the magnitude of its components using CFD, a number of articles presenting relevant experimental data have been chosen as references. The table below summarizes what kind of data can be found in these articles as well as some information about the experiments.

References Re Exp. / Sim Data obtained Cf Cp CT Manovski et al. 2018 [10] 3.8x106 Exp. Integrated 3.0490 Clarke et al. 2016 [11] 12x106 Exp. (Raw) Integrated 2.8819 0.82246 3.7044 12x106 Exp. (Corr) Integrated 2.5954 0.63529 3.2307 Jones et al. 2013 [12] 3.5x106 Exp. Integrated 3.0941 1.6876 4.7817 5.4x106 Exp. Integrated 2.8757 1.6876 4.5632 Anderson et al. 2012 [13] 5.4x106 Exp. Measured 4.7581 5.4x106 RANS Measured 3.4821 Quick et al. 2012 [14] 5.4x106 Exp. (Trip) Measured 4.76 5.4x106 Exp. (No trip) Measured 4.03

Table 1: Summary of data available from the references for the JBH

In the articles by Anderson et al. and Quick et al. measured values of the total resistance coeﬃcient, CT , are given, but in the rest of the references there are only plots of cf and cp along the body. Since one of the main goals of the study was to capture the total resistance, the curves were digitalized using the software Engauge and a MATLAB code was developed that integrates cf and cp over the body to give an estimate of Cf and Cp. In Table 1 the column Data obtained indicates whether the drag coeﬃcients were measured in the study or integrated from curves.

7 Quick et al. 2012 The article by Quick et al. [14] documents the first phase of a larger study aiming to build a body of information about a generic submarine shape using both experimental testing and CFD. The report documents a series of wind-tunnel tests performed in the DSTO Low Speed Wind Tunnel with the purpose of gathering steady-state aerodynamic force data, as well as asses the flow characteristics around the body. In this early phase of the study only the bare hull of the submarine was considered. A model was machined from aluminum, measuring 1.35 m long with a fineness ratio (length over max diameter) of 7.3. This resulted in an estimated blockage ratio of 2.1% relative to the wind tunnel test-section. The surface roughness of the model was measured to be less than 0.8 µm. Force and moment coefficients are presented for a number of different angles of attack and sideslip.

The authors mention that tests were done to check the influence that the support pylon and pitch-arm might have had on the flow angularities and the results showed that the influence was larger than expected. However, no attempts were made to correct the force and moment data since it will be part of an "incremental database" and subsequent tests of fully appended submarines will use the same mounting. The wake effect from the model support can clearly be seen in the asymmetry in the Cx (CT in x-direction) vs. angle of attack data as well, indicating that the presented total drag coefficient might very well be too high.

Anderson et al. 2012 Anderson et al. [13] documents a joint study between Swedish FOI and Australian DSTO. Both experimental and computational results are presented. The experiments were performed in the DSTO Low Speed Wind Tunnel located in Melbourne, Australia. Two different models were used, one of the bare hull and one of the fully appended configuration. Both models were 1.35 m long and the bare hull version was the same model as the one used by Quick et al. [14]. The bare hull configuration, which is the main interest of the current study, was run at a free stream velocity of 60 m/s, meaning a Reynolds number of 5.4 million. The skin-friction coefficient, cf , was measured along the model using the Preston tube method, described in detail in the original article by Preston [15]. The authors claim that the total drag coefficient, CT was scaled using the surface area of the submarine, but when studying their data it seems more likely that the length, L, squared was used.

Since the experimental setup was the same as that used by Quick et al., the previously mentioned effect of the support pylon can be expected to be visible in these results as well. The authors mention that the effect can be seen in the PIV measurements in that the turbulence levels are higher on the lower side of the model, and the downwards divergence of the wake. It is also mentioned that the pressure coefficient is slightly asymmetric in the lower half of the near wake, which according to the authors is most likely due to the support pylon being misaligned.

Jones et al. 2013 The article by Jones et al. [12] documents a study investigating different kinds of tripping devices to be used with the Joubert generic submarine geometries. The study focuses on the bare hull configuration and the effect that the tripping devices have on the skin friction and pressure coefficients along the model. All measurements were done in the DSTO Low Speed Wind Tunnel and the bare hull model used was the same 1.35 m model used by Quick et al. [14]. The skin friction coefficient was measured using the Preston tube method [15].

The tripping devices used were wires with diameters of 0.1 mm, 0.2 mm and 0.5 mm, and also a 3 mm wide tape on which grit of an average particle diameter of 190 µm had been distributed. Curves of the skin friction coefficient, cf , and pressure coefficient, cp, along the body are presented for a range of velocities for all the different tripping devices as well as for the case without tripping. For the 0.1 mm tripping wire, measurements were only taken at 3 measurement locations since it was found that this tripping wire merely moved the transition point a bit upstream rather than all the way to the wire.

The cp curves showed good agreement with numerical results for the front half of the model, but for the rear half the experimental cp curve looks shifted and when integrating cp along the model the total value of the form drag is signiﬁcantly larger than that predicted by RANS. This might very well also be a result of the support pylon and pitch arm.

8 Clarke et al. 2016 Clarke et al. [11] is an experimental study of the JBH in the cavitation tunnel at the Australian Maritime Collage. The model used was the same 1.35 m long model described by Quick et al. [14], using a diﬀerent more streamlined mounting solution. The blockage was estimated to be around 8.5% and thus a blockage correction based on CFD-results was applied to the measured data. Skin friction measurements were done at Reynolds numbers of 6x106 and 12x106 using the Preston tube method [15] and this data was presented in terms of cf vs x/L curves. The characteristics of the cavitation tunnel, including free stream turbulence intensity, are discussed in another article by D. Butler et al. [16].

Manovski et al. 2018 Manovski et al. [10] describes an experimental study where a long-distance Particle Image Velocimetry-system (PIV) was benchmarked for the purpose of measuring mean as well as ﬂuctuating velocity components in turbulent boundary layers. The system was tested using a 2 m long carbon-ﬁber composite model of the JBH in the DSTO Low Speed Wind Tunnel. All tests were run at zero angle of incidence with a free stream velocity of 28.8 m/s, meaning a Reynolds number of roughly 3.8×106.

The PIV data was used to produce estimates of boundary layer proﬁles and skin friction along the model. As a reference the skin friction coeﬃcient, cf , was also measured along the model using the Preston tube method [15].

Conclusions After reading through all the references, the articles by Anderson et al. [13] and Clarke et al. [11] were chosen as the primary references. Even though the wind-tunnel experiments by Anderson et al. seem to have been affected by the mounting pylons to some degree, cf was measured on the top side of the model where the effect is expected to be small. Furthermore the experiments, including facilities, models and tripping devices, are very well documented which is of great value. The study by Clarke et al. was chosen since it uses the same model as Anderson et al. but a different testing facility and mounting solution. It is also good that the authors estimate the blockage and makes an attempt to correct it.

Integration of cf and cp

One problem with references using Preston tube measurements for cf is that the method relies on the boundary layer being turbulent. For this reason, cf data is only available from, at best, right after the tripping location. This means that the integration of cf misses out a signiﬁcant part of the total friction. The contribution from these neglected parts was estimated for the data from Anderson et al. and Clarke et al. This was done by cutting the cf curves from the corresponding simulations and then comparing them with the uncut curves.

In the experiments by Anderson et al. cf was sampled from x/L = 0.05 to x/L = 0.79 (measured from the nose of the model). In the experiments by Clarke et al., cf was sampled from x/L = 0.13 to x/L = 0.96 (measured from the nose of the model). The results from the comparisons are presented in the tables below.

Cf (Full) 3.1265 Cf (Full) 2.7726 Cf (Cut) 2.6605 Cf (Cut) 2.3999

Table 2: Left: Anderson et al. Right: Clarke et al.

The contribution from the neglected parts of the geometry was roughly 15.0% for the data from Anderson et al. and roughly 13.4% for the data from Clarke et al. For this reason the cf and cp curves were chosen as the main results to compare for the JBH. The integrated values are still presented as sort of low point for the total values of Cf and Cp.

9 2.2 Case 2: Japanese Bulk Carrier The Japanese Bulk Carrier (JBC) is a generic capesize bulk carrier designed through a joint program involving the National Maritime Research Institute (NMRI), Yokohama National University (YNU) and the Ship Building Research Centre of Japan (SRC), all from Japan. Capesize meaning that it is too large to transit any of the canals, and thus has to travel around the capes. Figure 5 shows the JBC model from the side and Table 3 presents a few properties of the hull.

Figure 5: Japanese Bulk Carrier

symbol Model Full Scale Length (between perpendiculars) LPP 7.0 m 280 m Length (Waterline) Lwl 7.125 m 285 m Beam at water line Bwl 1.125 m 45 m Draft T 0.4125 m 16.5 m Wet area S 12.22 m2 19556.1 m2 Displacement volume ∇ 2.787 m3 178369.9 m3 Block coeﬃcient CB 0.8580 0.8580 Form factor k 0.314 0.314 Long. pos. of CoB fwd amidship LCoB 2.5475 % of LPP 2.5475 % of LPP 1/3 Slenderness Lwl / ∇ 5.0630 5.0630

Table 3: Ship properties

10 Hino et al. 2016 Hino et al. [17] was a study with the purpose of producing benchmark data for a ship equipped with an energy saving device (ESD), to be used as validation for future CFD-analyses. The chosen ship was the JBC of which two scale models were constructed, one 7 m long at NMRI and one 3.2 m long at Osaka University.

Both towed tests and self propulsion tests were performed. Stereo-PIV was used in both cases to capture the ﬂow ﬁeld between the ESD and the propeller, and resistance data was collected for the 7 m model. The towed resistance measurements were performed in the large towing tank at NMRI, measuring 400 m long, 18 meters wide and 8 m deep. Data was collected for Froude numbers ranging from 0.12 to 0.16 meaning Reynolds numbers ranging from 6.28x106 to 8.38x106. Both the Froude numbers and Reynolds numbers were calculated using the length between perpendiculars, Lpp.

The data was presented in the form of the residual resistance and through private communications with one of the authors it was concluded that they had used equation (8) with the formfactor from Table 3 to subtract an estimate of the viscous resistance. The data is plotted in Figure 6 together with the viscous resistance estimate, as well as the sum of the two which should be the measured total resistance. This data will be used as reference data for all the simulations of the JBC.

4.5 4.38

4 4.37 3.5 4.36 3 4.35 3 3 2.5 4.34

C x10 2 R T C x10 C C 4.33 1.5 f C v 4.32 1

0.5 4.31

0 4.3 0.12 0.13 0.14 0.15 0.16 0.12 0.13 0.14 0.15 0.16 Fr Fr

Figure 6: Left CR(Cw), Cf and Cv vs Fr Right CT vs Fr

Also available in the article was the trim and sinkage at the design Froude number. These are presented in Table 4. Notice that the trim is here expressed as the diﬀerence in sinkage between the perpendiculars. The trim and sinkage will also be used to check the validity of the results, since capturing the ships position in the water is essential to get a good estimate of the resistance.

Sinkage (% of Lpp) -0.086 Trim (% of Lpp) -0.180

Table 4: Trim and sinkage at Fn = 0.142

11 2.3 Case 3: MARIN Systematic Series FDS-5 The MARIN Systematic Series is a series of high-speed displacement ship hulls which were developed in the 1980s through a joint industry program involving Delft University and the Maritime Research Institute Netherlands (MARIN) as well as the navies of the Netherlands, Australia and the US. The purpose of the program was to evaluate the seakeeping and endurance of high speed displacing mono-hull vessels in waves. The basis of the series is the parent hull called the “fast displacement ship 5” (FDS-5) which was chosen through a pre-study. The other hulls in the series are systematic iterations of the FDS-5 where the parameters being varied are length over beam ratio, beam over draft ratio and block coeﬃcient. The entire program, including all results, was summarized in a book by Kaspenberg et al. [18] published by MARIN in 2014. Figure 7 shows the FDS-5 model used and Table 5 presents a few properties of the hull.

Figure 7: MARIN Systematic Series FDS-5

symbol Model Length (between perpendiculars) LPP 5.0 m Length (Waterline) Lwl 5.0 m Beam at water line Bwl 0.625 m Draft T 0.15625 m Wet area S 2.9288 m2 Displacement volume ∇ 0.195 m3 Block coeﬃcient CB 0.396 Long. pos. of cB fwd amidship LcB -5.11% of LPP 1/3 Slenderness Lwl / ∇ 8.62

Table 5: Ship properties

12 Kaspenberg et al. 2014 As mentioned, the book by Kaspenberg et al. [18] summarizes the entire joint program including, of interest to this study, a measurement campaign where models of all the hull forms were manufactured and tested. The measurements were performed in the towing tank at MARIN in the Netherlands and consisted of both seakeeping tests in waves and calm water resistance tests. In the book there is also information about the design of the parent hull as well as instruction on how the data can be interpolated to other similar hull forms.

Tabulated data from the resistance tests is presented for all hull forms and a range of velocities. The data includes the total drag as well as the sinkage at the fore and aft perpendiculars. The data for the FDS-5 is plotted in Figure 8, where Cf was calculated using the ITTC-57 friction line and CR is simply CT minus Cf . The sinkage at the fore and aft perpendiculars was recalculated to the trim and sinkage at the CoG and this is also presented below.

3.5 5.5 C f 3 C R 5

3 3 2.5 4.5 x 10

2 T C x 10 C 4 1.5

1 3.5 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 Fn Fn

5 0

0 -0.5 ]

-5 ° -1 -10 Trim [

Sinkage [mm] -1.5 -15

-20 -2 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 Fn Fn

Figure 8: Left CR and Cf vs Fn Right CT vs Fn

All models were ﬁtted with a 25mm wide tripping-strip with sand grains with a diameter of 250 µm. It was mounted at x/L = 0.1 aft of the point where the hull cuts the water when stationary (which is also the fore perpendicular for this hull).

13 3 Method - Computational Fluid Mechanics

Developing a theory for solving the Navier–Stokes equations with turbulence analytically has proven to be notoriously difficult, so difficult in fact that a large part of modern day fluid mechanics research is instead conducted within the field of Computational Fluid Dynamics (CFD). Here the aim is to discretize the governing equations and solve them numerically using the ever increasing power of computers.

The process of discretizing the governing equations and then solving them numerically without any further simplifications is called Direct Numerical Simulation (DNS). Such simulations resolve the entire spectra of turbulent fluctuations and thus require very fine computational grids and are often extremely computationally expensive. The number of floating-point operations required also increases as Re3, meaning that this type of simulation only is feasible for low to moderate Reynolds numbers. Direct numerical simulations are almost exclusively used for research and the industry instead relies on different kinds of modeling to reduce the complexity of the problem.

The approach used in this study is known as Reynolds–averaged Navier–Stokes (RANS) and is discussed in more detail in the section below. Other crucial elements to surface ship simulations are the ability to model two phases (air and water) at the same time, as well as the ability to model the movement of the ship hull with respect to the free surface. The software used throughout this study was the open source software OpenFOAM, both version 5.X and the fork foam-extend coupled with the commercial extension Naval Hydro Pack. This is discussed in more detail in Chapter 4.

3.1 RANS As discussed resolving the entire spectra of turbulent scales is often extremely computationally expensive and in many practical applications the bulk flow is of much greater significance than the turbulent fluctuations so studying the averaged flow quantities can often be sufficient. This is done by first decomposing the flow into an averaged part and a fluctuating part in a process known as Reynolds decomposition. Using this decomposition the flow quantities can be written as 0 ui = Ui + ui (12) p = P + p0

0 0 where Ui and P are the averages and ui and p are the ﬂuctuations with zero mean. Substituting (12) into the momentum equations (11) and then averaging results in the Reynolds–averaged Navier–Stokes equations,

∂Ui ∂Ui ∂ h ∂Ui 1 0 0 i + Uj = ν − P δij − huiuji , i = 1, 2, 3 , (13) ∂t ∂xj ∂xj ∂xj ρ where h...i denotes average. These equations are almost identical to the regular Navier–Stokes momentum 0 0 equations (11) with the exception of the new term huiuji, which is a tensor containing the covariances of the velocity ﬂuctuations. This term is usually referred to as the Reynolds stress tensor even though it is technically 0 0 ρhuiuji which has the dimension of a stress. The introduction of this stress tensor means that there are now 6 additional unknowns resulting in a total of 10 unknowns and 4 equations. This leads to the so called closure problem of RANS. The two most common methods for dealing with this problem are to use either eddy viscosity models (EVMs) or Reynolds-stress models (RSMs).

Eddy viscosity models model the Reynolds stresses using the Boussinesq eddy–viscosity hypothesis which states that 2 hu0 u0 i = kδ − 2ν hS i , (14) i j 3 ij T ij where νT is the eddy viscosity and k is the turbulent kinetic energy. The hypothesis assumes that the deviatoric Reynolds stress tensor is proportional to the mean of the strain rate tensor.

14 In Reynolds-stress models individual transport equations are instead solved for each Reynolds-stress component, thus removing the need for any additional modeling. This does however also make RSMs relatively computationally expensive for being RANS models. This study focuses on the use of eddy viscosity models which are explained in more detail below.

3.2 Eddy viscosity models The purpose of eddy viscosity models is simply to determine an expression for the eddy viscosity and this ∗ ∗ expression usually consists of a velocity scale u (xi, t) and a length scale l (xi, t). The simplest EVMs, algebraic models, relates these scales to the geometry of the flow and thus require the user to know details of the ∗ ∗ flow beforehand. The much more commonly used two-equation models instead relate u (xi, t) and l (xi, t) to turbulent quantities of the flow and then solve separate transport equations for these quantities. The workings of a few such models is discussed below [19] [20].

3.2.1 k- model The k- model was first developed by B.E Launder and W.P Jones in the 1970s [21] and has since evolved and improved significantly. The model was one of the first examples of a two-equation model, a model where the eddy viscosity is expressed in terms of two turbulence quantities which are solved for using two model transport equations [20]. The k- model uses the turbulent kinetic energy, k, and the turbulence dissipation rate, .

The k-transport equation can be written as " # ∂k ∂k ∂ νT ∂k ∂ui + Uj = + huiuji − (15) ∂t ∂xj ∂xi σk ∂xi ∂xj and the -transport equation for incompressible ﬂow can be written as

" # 2 ∂ ∂ ∂ νT ∂ ∂ui + Uj = + C1 huiuji − C2 , (16) ∂t ∂xj ∂xi σ ∂xi k ∂xj k where σk, σ, C1 and C2 are model constants. The eddy viscosity is then estimated as

2 cνρk ν = , (17) T where cν is a constant which has been empirically determined to be approximately 0.09. The model is widely used today, but is known to struggle with adverse pressure gradient ﬂows for which it predicts excessively high shear stresses. [22]

3.2.2 k-ω model The k-ω model was presented by David C. Wilcox in 1988 [23] and uses the turbulent kinetic energy, k, and the specific rate of dissipation of k, ω, as its two model parameters. Wilcox wrote that ω can be thought of as“... the ratio of the turbulent dissipation rate to the turbulent mixing energy k”. It is defined as ω = , (18) β · k where β is a closure coefficient equal to 3/40. The transport equation for k is exactly the same as in the k- model and the transport equation for ω can be written as " # ∂ω ∂ ∂ νT ∂ω Cω1ω ∂ui 2 + Uj = + huiuji − Cω2ω , (19) ∂t ∂xj ∂xi σω ∂xi k ∂xj

15 where σω, Cω1 and Cω2 are model constants. Unlike the the k- model, the k-ω model has proven to be very accurate when modeling boundary layers with adverse pressure gradients. It does however come with one major setback: It requires a non-zero inﬂow condition for ω and is very sensitive to this value. According to F. R. Menter [22] the eﬀect of the inlet ω value on the magnitude of the modeled eddy viscosity can be as large as 100%. The model is also know to under-predict the spreading rate for free shear layers.

3.2.3 k-ω-SST model The k-ω shear stress transport model, k-ω–SST in short, was published by F. R. Menter in 1994 [22] and can be thought of as a merger between the the k- model and the k-ω model. The model makes use of a blending function which allows it to use the k-ω formulation in the vicinity of a wall and then gradually change to the k- formulation as the distance to the wall grows larger. The blending function takes on values between 0 and 1 and uses the wall normal distance as a switching variable.

The k-ω-SST model is able to handle strong adverse pressure gradients without the heavy reliance on free stream conditions. For this reason it was chosen as the primary turbulence model to be used in this study.

3.2.4 Turbulence at the free surface Traditional two-equation turbulence models, like the ones used in this study, are known to struggle near the free surface. The abrupt change in ﬂuid density at the surface often makes the models over-predict the production of turbulence. In reality the surface restricts the turbulent length scales, and stretches the eddies parallel to the surface which leads to increased dissipation in the direction normal to the surface. [24]

The eﬀect of the free surface on the friction force is however assumed to be negligible and since the focus here is on drag, the inability of traditional two-equation models to capture turbulence near the free surface is considered to be outside the scope of this study.

3.3 Solving algorithm Due to the coupled nature of the Navier-Stokes equations the pressure field needs to be known in order to calculate the velocity field and the velocity field needs to be known to calculate the pressure field. One way around this problem is to use a predictor-corrector algorithm, i.e. guess one of the fields, calculate the other field, go back and correct the initial guess and then repeat until convergence. Two very common algorithms of this type are the PISO and SIMPLE algorithms, which are discussed briefly below.

3.3.1 SIMPLE The Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) is an iterative algorithm that often requires multiple iterations per time step before convergence. An overview of the algorithm-steps is presented below. [25]

1) Guess the pressure ﬁeld, p, or if available take the pressure ﬁeld from the previous time step.

2) Calculate the velocity ﬁeld, u∗, from the pressure ﬁeld using a linearized version of the momentum equations.

3) Calculate a pressure correction, pcorr.

4) Use the correction to obtain a corrected pressure field, p∗, and a corrected velocity field, u∗∗, which satisfies continuity.

5) Solve any additional transport equations.

6) Check for convergence. If converged advance to next time step, otherwise repeat from step 2 with corrected pressure ﬁeld.

16 3.3.2 PISO The PISO (Pressure-Implicit with Splitting of Operators) algorithm is based on the SIMPLE algorithm but splits the solution process into an implicit predictor step and one to several explicit corrector steps where operations on the pressure field are uncoupled from the velocity. This means that the momentum equation is only solved once per time step which in turn means that any errors in the initial pressure field remain unaffected by the following iterations. [26] An overview of the steps of the PISO algorithm is presented below.

Predictor step) This step is exactly the same as step 1 and 2 in the SIMPLE algorithm although the predictor step is only performed once per time step. A first estimate of the velocity field, u∗, is calculated using the linearized momentum equation and the pressure field, p, which is either taken from the previous time step or guessed if not available.

Corrector step) The corrector step is the same as steps 3 and 4 in the SIMPLE algorithm and this step can be repeated multiple times per time step. The velocity ﬁeld calculated from the momentum equation will not be divergence free and will thus not satisfy the continuity equation. A pressure correction is therefore used to obtain a corrected pressure, p∗. This pressure is then used to calculate the corrected velocity, u∗∗. As stated earlier this corrector step can be repeated multiple times by calculating a new pressure correction from the corrected ﬁelds p∗ and u∗∗. Although convergence is often reached within just a few iterations and the algorithm is even considered by some to be non-iterative. [25]

3.3.3 PIMPLE The PIMPLE algorithm is a hybrid between SIMPLE and PISO. It basically solves a predetermined number of SIMPLE loops each time step, with a predetermined number of PISO pressure corrections within each loop. The number of SIMPLE loops to perform is ususally refered to as the number of outer correctors (nOuterCorrectors) and the number of PISO loops to perform is called simply the number of correctors (nCorrectors). The workings of the PIMPLE algorithm as well as its implementation in OpenFOAM is explained in detail in the book by T. Holzmann [27].

3.3.4 Convergence OpenFoam allows 3 solver convergence criteria to be speciﬁed and if one of them is reached the solver will stop. The criterias are maxIter which is the maximum allowed number of iterations allowed, tolerance which is the residual value considered to be suﬃciently low and relTol which is the value of the current residual divided by the initial residual at which to stop.

3.3.5 Hydrostatic Pressure Eﬀects For the two-phase simulations an alternative pressure, pd, is solved for instead of the total pressure. This alternative pressure is deﬁned as pd = p − ρgh . (20) According to the authors of the OpenFOAM Users Guide [28] this formulation is numerically convenient for cases where the hydrostatic pressure is important.

3.3.6 Unsteady RANS Even though RANS solutions are averaged it is possible to capture the time dependence of the mean ﬂow if small enough time steps are used. Such simulations are referred to as Unsteady RANS (URANS). In OpenFOAM the PISO and PIMPLE solvers (PisoFoam and PimpleFoam) are transient solvers which can be used for URANS. They are thus somewhat sensitive to the size of the time step.

The SIMPLE algorithm is integrated in OpenFOAM through the steady solver SimpleFoam. It neglects the time derivatives in the governing equations and can for that reason only be used to reach steady solutions. Many steady solvers operate much like unsteady solvers, but without the need to resolve the time history of the ﬂow much larger time steps as well as variable time steps can be used. Some solvers even use local time stepping where the time step is unique and optimized in each cell. [25]

17 3.4 Wall functions In order to properly predict the friction forces on any surface it is of great importance to capture the viscous and turbulent mechanisms of the boundary layer. The inner regions of the boundary layer adds significant complexion to the process of turbulence modeling. Here the wall both dampens wall normal components of the turbulence, rendering it anisotropic, and increases the production of turbulence through shear forces. Moreover, the profiles of U and are very steep in this region meaning that a very fine mesh is required to capture the physics. The result is that a substantial fraction of the computational cost of wall-resolved RANS simulations comes from this near wall region, and many two-equation models even fail to properly model the near wall physics. An often much computationally cheaper approach is to use wall functions. These functions add boundary conditions a small distance out from the wall as well, thus reducing the need to solve the RANS equations in this region. The boundary conditions are generated based on analytical models describing the near wall flow.

When measuring distances close to the wall one often uses a non-dimensional wall–normal distance (y+) scaled with the viscous length scale. It is defined as uτ y y+ ≡ (21) ν where uτ is the friction velocity and y is the wall normal distance. The friction velocity is a velocity scale p based on the viscous stresses at the wall and is defined as uτ ≡ τw/ρ where τw is the wall shear stress. + When using wall functions it is desirable to have the first grid point in the lower log-law region, i.e. a ∆yw somewhere between 30 and 300. Wall functions have proven to work well for parallel flows with weak or ideally zero pressure gradients making them a viable choice for many ship hull geometries.

3.5 Two-phase ﬂows Simulations of surface ships adds one extra degree of diﬃculty compared to simulations of vessels moving through a single medium (such as submarines or aircraft). In order to properly resolve the domain around a surface ship both of the two phases (air and water), as well as the free surface between them needs to be modeled accurately. The most common way to do this is to consider both phases as part of one continuous medium and model the free surface as a jump in density and viscosity, meaning that only one set of governing equations is required to describe the motion of the entire domain.

Since marine hydrodynamics problems can be considered incompressible, the density of both air and water are assumed to be constant. The diﬀerence in density across the free surface can be expressed as

[ρ] = ρ− − ρ+ , (22) where the superscripts + and - denotes an infinitesimal distance from the surface on the water and air side respectively. The velocity field is is also defined as being continuous according to

[u] = u− − u+ = 0 , (23) which basically says that the velocity of the air in the immediate vicinity of the surface must be the same as the velocity of the water just under the surface. Finally the surface tension is neglected which results in a continuous pressure ﬁeld, [p] = 0 . (24) Equations 22, 23 and 24 together make up the jump conditions which needs to be satisﬁed across the free surface. How the viscosity is handled is described in the next section.

This approach to two-phase problems adds the requirement of a method to determine which phase (i.e. which density and viscosity) to consider in each calculation. The two most widely used such methods are the Level Set method and the Volume of Fluids (VoF) method. In this study only the latter is considered and it is described in more detail in the next section.

18 3.5.1 Volume of fluids The Volume of Fluids method was first described by Hirt and Nichols in 1981 [29] and is based on an indicator function which represents the volume fraction of water (Vw) in each cell to the total volume (V) of that cell. This indicator function can be written as Vw α = , (25) V where α = 1 means water and α = 0 means air. The effective kinematic viscosity is modeled as being continuous across the surface. Using the indicator function α the effective viscosity can be written as

νe = ανe,w + (1 − α)νe,a , (26) where νe,w is the kinematic viscosity of water and νe,a is the kinematic viscosity of air. This allows the use of conventional eddy-viscosity models to model the turbulence.

The indicator function is treated as a passive scalar which is convected by the velocity ﬁeld. Its motion is described by the transport equation

∂α ∂(uiα) ∂ + + (ur,iα(1 − α)) = 0 , (27) ∂t ∂xi ∂xi where the last term is a compression term to prevent excessive smearing of the surface and ur is the relative velocity ﬁeld normal to the surface. Equation 27, together with equations 11 and 10 makes up the set of governing equations for the two-phase ﬂow cases. The compression term is described in more detail in the PhD thesis by Rusche [30], and its implementation in foam-extend is explained in the article by Vukčević et al. [31].

3.5.2 Ghost Fluid Method One feature included in the Naval Hydro Pack extension is the ghost fluid method for polyhedral meshes. The Ghost Fluid Method is a method which implicitly enforces the jump conditions over the surface. It creates ghost cells at all points in the vicinity of the free surface, meaning that all these points contains the mass, momentum and energy of both the fluid in that cell and the ghost fluid which is the fluid on the other side of the interface (i.e. the ghost fluid for "air cells" is water and vice versa). The value of the VoF-indicator function (α) is then used to determine which of the fluids to consider in each cell. The Ghost Fluid Method is described in more detail in the articles [32] and [33] by Fedkiw et al. The implementation of the Ghost Fluid Method in foam-extend is described in the article by Vukčević et al. [34]. The method was used in all simulations of surface ships in this study.

3.6 Rigid Body Motion Capturing the movement of the ship with respect to the surface is crucial when trying to estimate the drag. As the ship rotates and translates the wetted surface changes which aﬀects both the skin friction and the procution of waves. For a ship under normal operating conditions all 6 degrees of freedom (DoF) needs to be considered, but during calm water resistance test the problem can be simpliﬁed to two DoFs. These two are translation along the z-axis (sinkage) and rotation about the y-axis (trim).

In nearly all marine hydrodynamics problems the motion of the vessel is almost entirely governed by pressure forces, greatly overshadowing the viscous forces. This means that the process of solving the pressure equations and the process of solving the rigid body equations needs to be tightly coupled as well. The 6DoF code implemented in foam-extend solves the rigid body equations using a Cash-Karp scheme with variable time-step size. The equations are solved multiple times per PIMPLE/SIMPLE iteration, both before and during each PISO pressure correction, which leads to a tighter coupling than in most conventional CFD codes. According to Gatin et al. [35] this tighter coupling reduces the amount of PISO loops required to achieve convergence of the body motion equations, thus reducing the computational cost. The mesh is however only moved once per PIMPLE/SIMPLE iteration to allow proper coupling with the velocity ﬁeld and the movement of the free surface. More details of the 6DoF implementation in foam-extend can be found in the article by Gatin et al. [35].

19 4 Computational Setup

In this chapter the chosen software is brieﬂy discussed followed by a detailed discussion of the computational setup of the three cases, including meshing, boundary- and initial conditions, solvers and schemes.

4.1 OpenFOAM OpenFOAM is an open source software for solving continuum mechanics problems. Its CFD solvers are based on the Finite Volume Method meaning that the problem domain is divided into a ﬁnite number of control volumes over which the governing equations are then discretized and solved.

Naval Hydro Pack is a commercial expansion to OpenFOAM built on the fork foam-extend. The expansion offers an under-relaxed PIMPLE algorithm which is supposed to be very robust even at high Courant numbers. As previously discussed, it also offers the implementation of the ghost fluid method to better capture the free surface jump conditions. For a description of Naval Hydro Pack refer to the homepage of the company developing the expansion, Wikki Ltd.

4.2 Meshing All meshes were generated using functions included in either OpenFOAM 5.x or foam-extend 4.1.

The first step in the construction of all meshes was to make a base-mesh using blockMesh. These base-meshes were then refined and snapped to the geometry using snappyHexMesh. For the two surface ships refineMesh was also used immediately after blockMesh to make sure that the cells in the vicinity of the hull were as "cubic" as possible as this tends improve the performance of snappyHexMesh. This was needed since the base-meshes for the final surface ship meshes had vertical grid refinement near the surface. This is discussed in more detail in Section 4.4 and 4.5

As mentioned, using wall functions to model the inner part of the viscous boundary layer puts some requirements on the mesh. The first grid point should ideally be between a y+ of 30 and 300, and this is best achieved by generating a series of prism cell layers on the surface. All meshes in this study had 6 layers of rectangular prism cells closest to the surface. An expansion ratio of 1.2 was used and the thickness of the final cell layer was half of that of the first cell outside the layer. This is usually done to ensure a good transition between the prism layer and the outer mesh.

This meshing methodology was chosen since it is very ﬂexible when changing between similar geometries and still produces quality meshes relatively fast. Another positive with snappyHexMesh is that it can be run in parallel with relative ease, thus reducing the time required to generate a mesh.

All ﬁgures of meshes in this report were generated using the software ParaView. Note that the seemingly random lines appearing in some of the ﬁgures is merely an artifact that arises when cells are cut. For visualization purposes the entire domain was cut along the symmetry plane.

20 4.3 Case 1: Joubert Bare Hull A computational domain was constructed for the JBH spanning 8 L in the x-direction and 4 L in the y- and z-directions, where L is the length of the model. This resulted in a blockage ratio of roughly 0.1%. The blockage ratio is calculated as the projected frontal area of the model divided by the cross sectional area of the domain.

4.3.1 Meshing Due to the axisymmetric shape of the bare hull and the 0 degree inﬂow angle of the case, it would be possible to run half, or even a quarter of the body and then mirror the results. This was however not done since the methods developed might be used for dynamic maneuverability simulations in the future.

Initially two meshes were developed, one coarse and one ﬁner, as a ﬁrst step in a grid dependence study. A few mesh parameters from the two meshes are presented in the tables below.

Cells (Total) 830.964 Cells (Total) 1.511.314 Max Aspect Ratio 7.17 Max Aspect Ratio 9.34 Non orthogonality (Max) 64.7 Non orthogonality (Max) 61.7 Non orthogonality (Avg) 6.60 Non orthogonality (Avg) 5.61 Max skewness 1.04 Max skewness 0.79 (a) Mesh 1 (b) Mesh 2

Table 6: Mesh properties for meshes 1 and 2

Close ups of the two meshes can be found in Figures 9 and 10 and additional screenshots of both meshes can be found in Appendix A.

Figure 9: Close up of Mesh 1

21 Figure 10: Close up of Mesh 2

4.3.2 Simulation The steady-state solver simpleFoam, based on the SIMPLE-algorithm, was used to run the JBH cases. All simulations were run for 1000 iterations which proved to be more than enough for convergence (see Appendix B for residuals). As discussed in Chapter 3 the turbulence model used was k-ω-SST. Both meshes were run at Re = 5.4×106 and mesh 2 was also run at Re = 12×106. The boundary conditions are summarized in Table 7 and the initial conditions are presented in Table 8 (a) and (b).

k νt ω p u Inlet fixedValue calculated fixedValue zeroGradient fixedValue Outlet inletOutlet calculated inletOutlet fixedValue inletOutlet Sides Slip calculated slip slip slip JBH kqWallFunction nutkWallFunction omegaWallFunction zeroGradient noSlip

Table 7: Boundary conditions for the JBH

For information about the boundary conditions we refer to the oﬃcial OpenFOAM documentation [28].

22 The initial value of k was calculated as 3 k = (u I)2 (28) 0 2 ∞ and the initial value for ω was calculated as √ − 1 k ω = C 4 (29) 0 µ L where Cµ is a constant equal to 0.09 and L is the turbulent length scale of the problem, in this case taken as the length of the model (1.35 m). The turbulent intensities were taken from the two reference papers and these were 0.67% in the wind tunnel used by Jones et al. [12] and 0.5% in the cavitation tunnel used by Clarke et al. [16]. The resulting initial condition are presented in Table 8.

u∞ 60 [m/s] u∞ 134 [m/s] p0 0 p0 0 2 2 2 2 k0 0.24 [m /s ] k0 0.67 [m /s ] ω0 0.66 [1/s] ω0 1.11 [1/s] Re 5.4×106 Re 12×106 (a) Flow conditions calculated from Jones et al. [12] (b) Flow conditions calculated from Clarke et al. [11]

Table 8: Flow conditions from the two chosen JBH references

A linear interpolation scheme with gauss discretization was used for all gradient calculations. The divergences were also discretized using a Gauss scheme and the interpolation schemes used are summarized in Table 9.

u k ω Scheme linearUpwind Upwind Upwind

Table 9: The divergence schemes used in all JBH simulations.

The convergence criterias for u, k and ω were a tolerance of 10−8 and a relTol of 0.1, and the convergence criterias for p were a tolerance of 10−7 and a relTol of 0.01

23 4.4 Case 2: Japanese Bulk Carrier The computational domain for the JBC spanned 6.5 L in the x-direction, 3 L in the y-direction and 4 L in the z-direction resulting in a blockage ratio of 0.11% .

4.4.1 Meshing Much like for the JBH, two meshes were also made for the Japanese bulk carrier as a ﬁrst step of a grid dependence study. Some properties of the meshes are presented in the tables below.

Cells (Total) 1.710.752 Cells (Total) 9.510.000 Max Aspect Ratio 14.6 Max Aspect Ratio 16.3 Non orthogonality (Max) 64.8 Non orthogonality (Max) 65.0 Non orthogonality (Avg) 8.36 Non orthogonality (Avg) 5.87 Max skewness 2.09 Max skewness 8.04 ∆z at the free surface [mm] 22 ∆z at the free surface [mm] 11 (a) Mesh 1 (b) Mesh 2

Table 10: Mesh properties for meshes 1 and 2

These ﬁrst two meshes were built in a way much like the meshes for the JBH, with the exception of a set of reﬁnement boxes along the free surface to allow it to be properly resolved. Emphasis was put on having good resolution on the sides of the hull, expected to be in contact with water, and a coarser mesh was allowed along the upper surface of the model since the contribution from the air skin friction was considered to be negligible. For a bulk carrier of similar proportions, with all superstructures, the total air drag is estimated to be around 2.5% [5]. In these simulations towing tank models are used, which have no appended structures, meaning that the aerodynamic drag is considerably lower than 2.5%. Figures 11 and 12 show close ups of the two initial meshes.

Figure 11: Close up of Mesh 1

24 Figure 12: Close up of Mesh 2

A set of simulations were performed to evaluate the quality of these meshes, including one stationary test where the ship was "dropped" with the waterline at the specified design waterline, and a dynamic test where the ship was run at the design Froude number. The results are presented in Figure 13 where "Diff" indicates the difference between the values from the dynamic and the static tests. Also included in Figure 13 is reference data from Hino et al..

Sinkage Trim 0 0.12

-0.002 0.1 -0.004

0.08 -0.006 Fn = 0 - Mesh1 Fn = 0 - Mesh2 -0.008 Fn = 0.142 - Mesh1 ]

° 0.06 Fn = 0.142 - Mesh2 -0.01 Diff - Mesh1 Diff - Mesh2

Trim [ 0.04 -0.012 Fn = 0.142 - Exp. Sinkage [m]

-0.014 0.02

-0.016 0 -0.018

-0.02 -0.02 0 20 40 60 80 100 0 20 40 60 80 100 Time (s) Time (s)

Figure 13: Comparison of trim and sinkage for Mesh 1 and Mesh 2

It is clear that both meshes fail to capture the sinkage for the case at Fn = 0.142 and there is even substantial sinkage for the stationary case (Fn = 0). The reason for this was believed to be that the spatial resolution of the mesh was insuﬃcient where the hull cuts the water, meaning that the surface ended up being smeared. Therefore a new meshing campaign was started where emphasis was put on keeping a constant and much ﬁner vertical (z) resolution along the entire surface. Two new meshes were made and their properties are presented in Table 11.

25 Cells (Total) 9.560.053 Cells (Total) 14.747.319 Max Aspect Ratio 27.49 Max Aspect Ratio 49.06 Non orthogonality (Max) 64.94 Non orthogonality (Max) 64.59 Non orthogonality (Avg) 5.56 Non orthogonality (Avg) 5.13 Max skewness 2.49 Max skewness 2.52 ∆z at the free surface [mm] 9.9 ∆z at the free surface [mm] 5.4 (a) Mesh 3 (b) Mesh 4

Table 11: Mesh properties for meshes 3 and 4

The new meshes were refined using directional refinement in z around the free surface already on the blockMesh stage. A block 1 m thick in z, centered on the free surface, spanning the entire domain in x and y was constructed and then split in the z-direction, 20 times for mesh 3 and 40 times for mesh 4. In order to create a smooth transition between this free surface mesh and the outer coarser mesh, grading was added on both sides. In order for snappyHexMesh to work properly it is important to keep the cells which are to be further refined as cubical as possible. For this reason the refineMesh utility was used to refine the meshes in the xy-plane within a series of trapezoidal prism around the model until the cells in the vicinity of the hull were of roughly the same refinement level in all directions. The trapezoidal refinement regions had an angle of 19◦ to properly capture the wave patterns. The regions can be seen in Figure 15. Finally snappyHexMesh was run with refinement and layer generation enabled on the hull. The main difference between mesh 3 and 4 is the level of directional refinement in the surface region added in the initial blockMesh stage. Figures 14 and 15 show Mesh 3 from two different angles. More figures of all meshes can be found in Appendix A. Discussion around the trim and sinkage predictions of mesh 3 and 4 can be found in Chapter 5.

Figure 14: Close up of Mesh 3

26 Figure 15: Mesh 3 from above, cut along the free surface at rest.

4.4.2 Simulation The transient solver navalFoam, based on the PIMPLE-algorithm, was used to run all JBC cases. All simulations used an initial ∆t of 0.01 s and variable time stepping and meshes 1, 2 and 3 were all run for 100 s. Mesh 4 showed convergence problems in the drag force for some of the velocities so the simulations with this mesh were run for at least 300 s. This is discussed more in Chapter 5. As discussed in Chapter 3 the turbulence model used was k-ω-SST. Choosing free stream values of k and ω is not as straightforward for the surface ships as for the JBH since all experiments were performed in calm water conditions, i.e. no free stream turbulence. For this reason a few of the Naval Hydro Pack validation cases using similar geometries were studied and based on this the value of the free stream turbulence intensity was chosen to 5%. The initial conditions for k and ω were then calculated using equations (28) and (29).

Focus was put on getting good results from the case where Fn = 0.142 since this is the speciﬁed design Froude number and since this point had the most available reference data. A few more Froude numbers were also tested and the entire range is presented in the table below.

Fn 0.12 0.13 0.142 0.15 0.16 u 0.9944 1.077 1.179 1.2430 1.3259 Re 6.40x106 6.93x106 7.59x106 8.00x106 8.53x106

Table 12: Simulations performed with Mesh 3 and 4

27 The boundary conditions used in all JBC simulations are summarized in Table 13.

α k ω pd u Inlet waveAlpha fixedValue fixedValue zeroGradient waveVelocity Outlet waveAlpha inletOutlet inletOutlet zeroGradient waveVelocity Sides zeroGradient inletOutlet inletOutlet zeroGradient zeroGradient Top inletOutlet inletOutlet inletOutlet totalPressure pressureInletOutletVelocity Seabed inletOutlet kqRWallFunction omegaWallFunction zeroGradient fixedValue JBC zeroGradient kqRWallFunction omegaWallFunction zeroGradient movingWallVelocity

Table 13: Boundary conditions for the JBC

The boundary condition for u on the hull had to be set to movingWallVelocity since an ordinary no-slip boundary condition wouldn’t work with a moving geometry. waveVelocity is a boundary condition included in the Naval Hydro Pack which simply means that the velocity is set using the initWaveField command. Since no incoming waves are used it is essentially the same as using fixedValue.

As mentioned earlier a variable time-step was used, and this time step was adjusted according to a Courant number limiter. Since a part of this study consisted of evaluating the performance of Naval Hydro Pack, a quite high maximal Courant number of 25 was chosen. This was the value recommended by Wikki in their validation cases. Allowing a higher Courant number practically means allowing larger time steps, which can reduce the computational cost of the simulation. This is in line with the other goal of the study which was to develop a robust yet cheap methodology for performing this type of simulation. A test run with a maximum Courant number of 0.5 was also run using mesh 4 and the results from this simulation are discussed in Chapter 6.

Dynamic meshing and 6DoF was enabled with the model free to move in pitch (rotation about y) and heave (translation in z), and locked in the other degrees of freedom. This allows the solver to dynamically ﬁnd the correct trim and sinkage.

The implicit Euler scheme was used for all time derivatives in the JBC simulations and the gradients were calculated using a least-squares scheme. The divergences were discretized using a Gauss scheme and the interpolation schemes used are summarized in Table 14.

u α k ω Scheme linearUpwind vanLeer01DC Upwind Upwind

Table 14: The divergence schemes used in all JBC simulations. vanLeer01DC is a combination of upwind and central diﬀerencing that uses a ﬂux limiter to ensure that alpha is strictly bounded between 0 and 1. For more information about the schemes refer to the OpenFOAM Users Guide [28].

The convergence criterias for all parameters were a maxIter of 500, a tolerance of 10−8 and a relTol of 0.01. Additionally, stricter criteria were added to the ﬁnal iteration of the pressure in the PIMPLE loop. These were a maxIter of 1000 and a tolerance of 10−8 with no limit on relTol.

28 4.5 Case 3: MARIN Systematic Series FDS-5 The computational domain for the FDS-5 spanned 7.5 L in the x-direction, 3 L in the y-direction and 4 L in the z-direction resulting in a blockage ratio of roughly 0.07%.

4.5.1 Meshing Two initial meshes were also made for the FDS-5 as a ﬁrst step of a grid dependence study. Some properties of the meshes are presented in the tables below.

Cells (Total) 9.108.161 Cells (Total) 22.667.112 Max Aspect Ratio 47.49 Max Aspect Ratio 47.49 Non orthogonality (Max) 64.78 Non orthogonality (Max) 65.00 Non orthogonality (Avg) 6.49 Non orthogonality (Avg) 5.97 Max skewness 3.07 Max skewness 3.53 ∆z at the free surface [mm] 7.3 ∆z at the free surface [mm] 3.6 (a) Mesh 1 (b) Mesh 2

Table 15: Mesh properties for meshes 1 and 2

With the meshing process from the JBC in mind, having suﬃcient z-resolution around the free surface was a priority from the beginning when meshing the FDS-5 and a similar blockMesh design was therefore used. Much like for the JBC, emphasis was also put on having a good resolution around the hull and a coarser mesh was allowed along the top of the model. Trapezoidal reﬁnement regions were also used for the FDS-5 to allow the wave pattern to be properly captured while still keeping down the mesh size and these can be seen in Figure 18.

Due to the shape of the hull the layer generation was failing in some areas of mesh 1, especially around the sharp part of the bow (see Figure 16). For this reason additional reﬁnement was added in the near hull region of mesh 2 and this lead to successful prism layer generation along most of the geometry. Additional ﬁgures of both meshes are available in Appendix A.

Figure 16: Close up of the bow region of mesh 1 showing where the prism layer generation failed

29 Figure 17: Close up of mesh 1

Figure 18: Mesh 1 from above, cut along the free surface at rest.

30 4.5.2 Simulation The transient solver navalFoam, based on the PIMPLE-algorithm, was used to run all the FDS-5 cases. They all used an initial ∆t of 0.01 s and variable time stepping and were all run for 100 s. A few of the simulations using mesh 2 were stopped before they reached 100 s since they had already converged at that point and the computational cost was growing large.

As for the JBC, the values of the free stream turbulence in the towing tank was most likely close to zero since the water upstream was stationary and instead a tripping strip was used to trip the boundary layer. Since no information regarding the turbulence levels of the experiments was disclosed, the free stream turbulence levels from the Naval Hydro Pack validation cases was used here as well.

The FDS-5 does not have a design Froude number and ships of this type are usually expected to perform well in a large range of velocities. For this reason a series of different Froude numbers were tested and these are presented in the table below. Only 4 different Froude numbers were tested using mesh 4 since the relatively large size of the mesh increased the computational cost significantly.

Fn 0.356 0.500 0.571 0.642 0.786 0.858 1.000 u 2.496 3.502 4.002 4.496 5.509 6.012 7.006 Re 10.96x106 15.37x106 17.57x106 19.74x106 24.18x106 26.39x106 30.76x106 Mesh 1 X X X X X X X Mesh 2 - X X - X - X

Table 16: Simulations performed with Mesh 1 and 2. X indicates that a simulation was run at this Froude number and - indicates that no simulation was run at this Froude number.

The boundary conditions used in all the FDS-5 simulations are summarized in the table below.

α k ω pd u Inlet waveAlpha fixedValue fixedValue zeroGradient waveVelocity Outlet waveAlpha inletOutlet inletOutlet zeroGradient waveVelocity Sides zeroGradient inletOutlet inletOutlet zeroGradient zeroGradient Top inletOutlet inletOutlet inletOutlet totalPressure pressureInletOutletVelocity Seabed inletOutlet kqRWallFunction omegaWallFunction zeroGradient fixedValue FDS-5 zeroGradient kqRWallFunction omegaWallFunction zeroGradient movingWallVelocity

Table 17: Boundary conditions for the FDS-5

For the reasons discussed in Section 4.4, the Courant number limiter was set to maximum 25 for all FDS-5 simulations as well.

Dynamic meshing and 6DoF was enabled with the model free to move in pitch and heave, and locked in the other degrees of freedom in order to allow the solver to dynamically ﬁnd the correct trim and sinkage.

Like for the JBC simulations, the implicit Euler scheme was used for all time derivatives and all gradients were calculated using a least-squares scheme. The divergences were discretized using a Gauss scheme and the divergence interpolation schemes used are summarized in Table 18. u α k ω Scheme linearUpwind vanLeer01DC Upwind Upwind

Table 18: The divergence schemes used in all FDS-5 simulations.

31 5 Results

In this chapter the results from all simulations of the three cases are presented and compared with the reference data.

5.1 Case 1: Joubert Bare Hull This section contains the results from the three simulations of the JBH. Figure 19 shows an example of what the ﬂow ﬁeld can look like around the JBH.

Figure 19: The JBH with surrounding velocity ﬁled seen from the side using mesh 2 at Re = 5.4x10−6. The background is colored according to the value of the streamwise (x) component of the velocity.

Table 19 contains all the reference data from Table (1), complemented with results from the three simulations.

References Re Exp. / Sim. Data obtained Cf Cp CT Manovski et al. 2018 [10] 3.8x106 Exp. Integrated 3.0490 Clarke et al. 2016 [11] 12x106 Exp. (Raw) Integrated 2.8819 0.82246 3.7044 12x106 Exp. (Corr) Integrated 2.5954 0.63529 3.2307 Jones et al. 2013 [12] 3.5x106 Exp. Integrated 3.0941 1.6876 4.7817 5.4x106 Exp. Integrated 2.8757 1.6876 4.5632 Anderson et al. 2012 [13] 5.4x106 Exp. Measured 4.7581 5.4x106 RANS Measured 3.4821 Quick et al. 2012 [14] 5.4x106 Exp. (Trip) Measured 4.76 5.4x106 Exp. (No trip) Measured 4.03 Mesh 2 - Run 1 5.4x106 RANS Integrated 3.1265 0.22183 3.3484 Mesh 2 - Run 2 12x106 RANS Integrated 2.7726 0.27919 3.0517

Table 19: Table of resistance values from references and computations

As discussed in Chapter 2 the integrated values of the drag coeﬃcients contains large errors due to the experimental data being cut. For this reason it makes more sense to study plots of the curves if one is interested in the accuracy of the simulations.

32 5.1.1 cf curves

700 1 6 Mesh 1 Mesh 1 Mesh 1 600 0.8 Jones et al. 5 Jones et al.

500 0.6 4

400 3 + w p

y 0.4 3 c * 10 ∆ 300 f c 0.2 2 200

100 0 1

0 -0.2 0 0 0.5 1 0 0.5 1 0 0.5 1 x/L x/L x/L

+ 6 Figure 20: Plots of ∆yw , cp and cf along the side of the hull at Re = 5.4x10 using mesh 1. Reference data is from Jones et al. [12]

250 1 6 Mesh 2 Mesh 2 Mesh 2 0.8 Jones et al. 5 Jones et al. 200

0.6 4

150 3 + w p

y 0.4 3 c * 10 ∆ f

100 c 0.2 2

50 0 1

0 -0.2 0 0 0.5 1 0 0.5 1 0 0.5 1 x/L x/L x/L

+ 6 Figure 21: Plots of ∆yw , cp and cf along the side of the hull at Re = 5.4x10 using mesh 2. Reference data is from Jones et al. [12]

×10 -3 600 1 5 Mesh 2 Mesh 2 500 0.8 Clarke et al. (Raw) Clarke et al. (Raw) Clarke et al. (Corr.) 4 Clarke et al. (Corr.) 0.6 400

3 3 0.4 + w p

y 300 c * 10 ∆ 0.2 f c 2 200 0 1 100 -0.2

0 -0.4 0 0 0.5 1 0 0.5 1 0 0.5 1 x/L x/L x/L

+ 6 Figure 22: Plots of ∆yw , cp and cf along the side of the hull at Re = 12x10 using mesh 2. Reference data is from Clarke et al. [12] with and without blockage correction.

33 + When studying Figures 20, 21 and 22 the ∆yw values looks slightly too high for mesh 1 while they look reasonable for both simulations using mesh 2. The only potential problem is the spike in Figure 22 reaching slightly above 500 due to the layer addition failing near the stern of the hull. The impact of these few cells on the total drag is however expected to be minimal.

The cp curve in Figures 20 and 21 seems to diverge slightly after the model midpoint compared to the experimental data from Jones et al. [12] and the reason for this is believed to be the mounting pylon aﬀecting the experimental results, as discussed in Chapter 2. This conclusion seems even more plausible when considering the fact that no divergence from the experimental data can be seen in the cp curve in Figure 22, which also uses mesh 2.

When studying the cf curves a clear improvement can be seen for mesh 2 compared to mesh 1 in terms of matching the experimental curve. The simulation at Re = 5.4×106 using mesh 2 (Figure 21) seems to capture the overall trend of the cf curve quite well although the values diﬀer around 10%. There is also a strong peak in cf in the experimental data right after the tripping point which is not as prominent in the simulations. In order to get an even better match of the cf curve one could try to use a laminar inﬂow with numerical tripping, for instance by injecting turbulent kinetic energy in a few chosen cells. The simulation at Re = 12×106 (Figure 22) was most successful at capturing the cf curve of its corresponding experimental study. The simulation data matches the blockage corrected data well both in terms of trend and magnitude.

As mentioned the simulation at Re = 12×106 (Figure 22) shows good agreement with experimental results for both cp and cf . This is reassuring as it indicates that the simulations can in fact capture both components of the drag with good accuracy.

34 5.2 Case 2: Japanese Bulk Carrier + This section contains the results from the simulations of the JBC. The results include visualizations of ∆yw along the hull, plots of the velocity field in the wake region, wavepatterns, signs of numerical ventilation and estimates of the total drag. Figures 23 and 24 show examples of what the flow field can look like around the JBC. Note that the lines visible on the free surface in Figure 23 are merely an artifact of the visualization in ParaView.

Figure 23: The JBC with the free surface seen from above using mesh 4 at Fn = 0.16. The free surface was extracted using a contour with α = 0.5. It is colored according to the value of the z-coordinate scaled with LPP .

Figure 24: The bow region of the JBC with the free surface using mesh 4 at Fn = 0.16. The free surface was extracted using a contour with α = 0.5. It is colored according to the value of the z-coordinate scaled with LPP .

35 + As a ﬁrst check of mesh quality, ∆yw values were sampled along the center of the hull in the x-direction for + both meshes and plotted together. These curves can be seen Figure 25. The ∆yw values of the rest of the JBC simulations were also checked and proved to be of the same order of magnitude. Both meshes seem to have acceptable near wall resolution for wall-modeled RANS equations, and mesh 4 could potentially even be made a bit coarser.

Additionally cf curves were sampled along the side of the two JBC meshes at a depth of T/2 and plotted 6 together with the cf curve from the JBH simulation at Re = 5.4x10 using JBH mesh 2. The data from the JBH was included for reference since it is a very streamlined body, similar in shape to the side of the JBC.

120 7 JBC Mesh 3 JBC Mesh 3 100 JBC Mesh 4 6 JBC Mesh 4 JBH Mesh 2 5 80 3 4 + w

y 60 x 10 ∆ f 3 c 40 2

20 1

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x/L x/L

+ Figure 25: Left: ∆yw sampled along the middle of the hull in the x-direction for both meshes. Right: cf sampled along the side of the hull of the JBC at a depth of T/2 for both mesh 3 and 4 at Fn = 0.142. The cf 6 + curve from the JBH simulation at Re = 5.4x10 using JBH mesh 2 is also included for reference. Both ∆yw and cf is plotted as functions of x/L where the bow is located at x/L = 0.

Figure 25 (right) shows that the simulation using mesh 4 had signiﬁcantly lower cf along the side of the hull (x/L = 0.2 to x/L = 0.8) compared to the mesh 3 simulation. This should contribute to a lower total resistance.

36 5.2.1 Wave patterns Wave profiles were sampled for both meshes at Fn = 0.142. The profiles were sampled at two locations in the domain, the first along the side of the hull and the second in a straight line a distance from the hull. The second line was sampled at a y-distance of y = 0.1043 measured from amidship, spanning from x = -0.5 to x = Lpp Lpp Lpp 2 where 0 is the forward perpendicular and positive direction is towards the stern. The profiles were sampled in ParaView using a contour filter of constant α = 0.5. The wave profile data from both meshes is plotted together with reference data from Hino et al. [17] in Figure 26.

Waterline on hull 20 Exp. Mesh 3 3 10 Mesh 4 x 10

PP 0 z/L

-10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x/L PP y/L = 0.1043 PP 4 Exp. Mesh 3 3 2 Mesh 4

x 10 0 PP

z/L -2

-4 -0.5 0 0.5 1 1.5 2 x/L PP

Figure 26: Top: The z-coordinate of the waterline on the hull scaled with Lpp and plotted as a function of x/Lpp where 0 is the bow. Bottom: The z-coordinate of the free surface sampled in a line parallel to the hull at a distance of y/Lpp = 0.1043, scaled with Lpp and plotted as a function of x/Lpp where 0 is the bow.

When studying 26 (top) it seems like both meshes predict the waterline on the hull well although a slight improvement can be seen for mesh 4. In Figure 26 (bottom) the trend of the wave pattern is captured quite well by both meshes although the smaller, shorter wavelength waves between x/Lpp = 0.1 and x/Lpp = 0.8 are absent. This could be a sign of numerical dissipation in the mesh. It could also be related to the high Courant numbers of the simulations since a Courant number of 25 means that information can travel through 25 cells in one time-step, thus eﬀectively smearing details smaller than this.

In Figure 26 (bottom) for x < 0 a bump can be seen in the data from both simulations which does not appear in the experimental data. This is most likely related to the mesh being more refined in the area around the ship compared to further upstream. The region between the two different refinements is made up from tetrahedral cells which seems to affect the α values somewhat (see Figure 24). This could possibly be fixed by refining the mesh further upstream and having a constant refinement level along the free surface.

37 5.2.2 Mesh 4 problems An unexpected problem with mesh 4 was discovered when studying the distribution of α on the hull. For all simulations using mesh 3 as well as the simulation using mesh 4 at Fn = 0.12, the distribution of α was as expected i.e. 1 under the free surface and 0 above the free surface with a thin mixing region in between (see Figure 27). This was however not the case for all simulations using mesh 4. For all Froude numbers larger than 0.12 streaks containing air (α < 1) appeared along the hull, and the amount of air in the streaks increased with the ﬂow velocity, see Figure 28.

Figure 27: α plotted on the hull of the JBC at Fn = 0.12 using mesh 4.

Figure 28: α plotted on the hull of the JBC at Fn = 0.16 using mesh 4.

The results look very similar to a phenomena known as Numerical Ventilation (NV) in which air is "pushed" down near the bow of the hull and then transported along the hull, producing streak-like structures. Numerical ventilation is a known problem, especially in simulations of planing hulls which form an acute entrance angle with the free surface. The phenomena is discussed in more detail in an article by Stephens et al. [36]. Figure 29 show the area of the bow intersecting the free surface for the two previously discussed cases, Fn = 0.12 and Fn = 0.16.

38 Figure 29: Left: Close-up of the bow region of mesh 4 at Fn = 0.12. Right: Close-up of the bow region of mesh 4 at Fn = 0.16

When studying Figure 29 (right) it seems like air is being pushed down quite far in the innermost prism layer at Fn = 0.16. The dark red layer corresponds to roughly α = 0.85. For the case where Fn = 0.12, Figure 29 (left), no air seems to be pushed down. These conclusions agree well with Figures 27 and 28. The eﬀect of numerical ventilation is expected to be seen in the frictional resistance as well. The streaks of air along the hull will lead to the wrong viscosity being used when calculating wall shear stresses which in turn should lead to an under-prediction of the skin friction.

Numerical ventilation seems to be very dependent on the angle of the bow where it cuts the free surface, as can be seen in Figure 29, and this is also mentioned in the article by Stephens et al. [36]. According to the authors numerical diﬀusion will occur when the free surface is not aligned with the mesh, and this results in the free surface being smeared. As the angle between the bow and the surface decreases (from 90◦), the prism layer cells will become more and more misaligned with the surface and this will eventually lead to numerical ventilation. When studying Figure 29 it seems like the bow is almost perpendicular to the free surface at Fn = 0.12 and no numerical ventilation can be seen. At Fn = 0.16 there is a signiﬁcant bow wave which pushes the surface further up the hull meaning that the prism layer cells are now at an acute angle to the surface.

All simulations using this mesh also had problems with oscillations in the drag force and this was found to be caused by an oscillating group of waves in the near hull region. The oscillations seemed to make convergence slower and all mesh 4 simulations had to be run for at least 300 seconds before these oscillations had been reduced to acceptable levels. The simulation at Fn = 0.16 required roughly 1500 s of simulation time. An acceptable magnitude of the oscillations was taken to be below 1% for the total drag. Figure 30 shows oscillations in the pressure and viscous force components around their respective mean.

39 Fn = 0.142 Fn = 0.16 2 3 Pressure Pressure Viscous Viscous 1.5 2 1 1 0.5

0 0

-0.5 -1 -1 -2 Oscillation around mean [%] -1.5 Oscillation around mean [%]

-2 -3 300 350 400 450 500 1300 1350 1400 1450 1500 Simulation time [s] Simulation time [s]

Figure 30: The viscous and pressure force ﬂuctuations over the last 200 s of simulation time expressed in terms of percentage of the mean. The cases are Left: JBC mesh 4 at Fn = 0.142 and Right: JBC mesh 4 at Fn = 0.16.

It was theorized that the oscillations were related to the high Courant numbers of the simulations so this was investigated by running a simulation of mesh 4 at Fn = 0.16 with the Courant limiter set to a maximum Courant number of 0.5. Due to the massively increased computational cost this simulation was only run for 20 seconds. When studying Figure 31 the simulation with a lower maximal Courant number seems to converge signiﬁcantly faster, although the magnitude of the oscillations is still at roughly 10% after 20 seconds.

250 Max Co. = 0.5 - Pressure 200 Max Co. = 0.5 - Viscous Max Co. = 25 - Pressure 150 Max Co. = 25 - Viscous

100

-50

-100 Oscillation around mean [%] -150

-200 5 10 15 20 Simulation time [s]

Figure 31: The viscous and pressure force ﬂuctuations over the last 15 seconds expressed in terms of percentage of the mean. The simulation used JBC mesh 4 at Fn = 0.16 and a maximum allowed Courant number of 0.5

It should also be mentioned that the computational cost of the ﬁrst 20 seconds with a max Courant number of 0.5 was more than 8200 core-hours, compared to the roughly 30.000 core-hours for 1500 seconds with a maximum Courant number of 25.

40 The simulation at a lower Courant number also revealed another potential source of numerical ventilation, see Figure 32. Since the resolution of the surface is coarser further upstream the interface becomes smeared there, and if the Courant number is high this smeared interface can travel far downstream into the reﬁned regions. This is clearly seen when comparing Figure 32 (left) and (right).

Figure 32: Left: Close-up of the bow region of JBC mesh 4 at Fn = 0.16 with the Courant number limited to 25. Right: Close-up of the bow region of JBC mesh 4 at Fn = 0.16 with the Courant number limited to 0.5

As mentioned, lowering the Courant number too much would severely increase the computational cost of the simulations which is not always a feasible option. A better option could be to reﬁne the mesh around the surface upstream of the model to reduce the smearing. This is also mentioned as an eﬀective strategy against numerical ventilation in the article by Stephens et al. [36].

One reason why mesh 4 is struggling with oscillations but not mesh 3 could be that mesh 4 has a much higher resolution in the region near the hull meaning less numerical dissipation.

5.2.3 Wake plots As an additional check of the accuracy and physicality of the simulations, the wake region near the propeller mounting was investigated and compared with PIV data from Hino et al. [17].

Figure 33 show planes of the velocity distribution scaled with the free stream velocity. The ﬁrst plane was sampled at S2 which is 262.5 mm upstream of the rear perpendicular and the second plane was sampled at S4 which is 110 mm upstream of the rear perpendicular.

When studying Figure 33 the vorticies on the sides in the simulations seems to be somewhat smaller than recorded in the experiments for both meshes, but overall the simulations show good agreement with the PIV data. Mesh 4 shows a slight improvement to mesh 3 and more details in the vorticies seems to be resolved.

41 0 -0.01

-0.01

0.5 -0.02 0.6 0.6

0.4 0.7 -0.02 0.7

0.6 -0.03 0.7 0.8 -0.03 0.5 0.8 PP 0.4 PP 0.5

Z/L -0.04 Z/L 0.3 0.8

-0.04 0.4

0.4 0.2 -0.05 -0.05 0.3 0.9

-0.06 -0.06 0.9

-0.07 -0.07 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 Y/L Y/L PP PP

0 -0.01

-0.01 -0.02 0.6 0.5 0.4 0.6 0.7 -0.02 0.7 0.5

0.6 -0.03 0.7 0.8 -0.03 0.8 PP PP 0.5

0.4 0.4 Z/L -0.04 Z/L 0.8 -0.04 0.3 0.1 0.4 0.25 -0.05 0.9 -0.05 0.3

0.2 0.9

-0.06 -0.06

-0.07 -0.07 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 Y/L Y/L PP PP

Figure 33: Top to bottom: Data from Mesh 3, Mesh 4 and Experiments. Left to right: Velocity contours from plane S2 and velocity contours from plane S4. All experimental data is from Hino et al. [17]

42 5.2.4 Trim & Sinkage The trim and sinkage data from simulations using both meshes are presented in Figure 34.

-3 0.15 Exp. Exp. 0.14 -4 Mesh 3 Mesh 3 Mesh 4 Mesh 4 0.13 -5 0.12 ] ° -6 0.11

Trim [ 0.1 -7 Sinkage [mm] 0.09 -8 0.08

-9 0.07 0.12 0.13 0.14 0.15 0.16 0.12 0.13 0.14 0.15 0.16 Fn Fn

Figure 34: Left: Sinkage as a function of the Froude number for both meshes. Right: Trim angle as a function of the Froude number for both meshes. The experimental data in both ﬁgures is from Hino et al. [17].

When studying Figure 34 both meshes seem to capture the trim and sinkage quite well. A small improvement in sinkage can be seen for mesh 4 as expected since it has a higher resolution around the free surface. Numerical ventilation is known to aﬀect the trim and sinkage in simulations of planing hulls, but this does not seem to be the case for the JBC. The sinkage follows the same trend for both meshes, and when comparing the two meshes at Fn = 0.12 (no NV) and Fn = 0.16 (severe NV) the diﬀerence in sinkage is roughly the same. When it comes to the trim it is virtually unchanged when changing meshes.

43 5.2.5 Drag In this section the drag data from both JBC meshes is presented and compared with reference data from Hino et al. [17].

4.9 3.4 Exp. ITTC-57 Mesh 3 3.35 Mesh 3 4.8 Mesh 4 Mesh 4 3.3 4.7 3 3 3.25 4.6 x10 x10 f T 3.2 C C 4.5 3.15

4.4 3.1

4.3 3.05 0.12 0.13 0.14 0.15 0.16 0.12 0.13 0.14 0.15 0.16 Fn Fn

Figure 35: Left: CT as a function of the Froude number for both meshes. The experimental data is from Hino et al. [17]. Right: Cf as a function of the Froude number for both meshes compared with the ITTC-57 friction line.

In Figure 35 (left) the estimated drag is a bit high compared to the experimental data. The average discrepancy in CT was about 7.5% for mesh 3, about 5.1% for mesh 4 and about 6.3% between the meshes. In the table below are the discrepancies between all simulations and the experimental data.

Fn 0.12 0.13 0.142 0.15 0.16 Mesh 3 6.44 % 6.10 % 7.35 % 6.5 % 11.05 % Mesh 4 7.10 % 6.29 % 5.03 % 3.60 % 3.68 %

Table 20: Diﬀerence between estimated CT and experimental data for all simulations using mesh 3 and 4.

When studying Figure 35 (right) it seems like the skin friction estimate from mesh 4 is much closer to the ITTC-57 friction line than mesh 3. A large part of this decrease in friction between the meshes is however expected to be a result of the numerical ventilation and any real improvement for Fn > 0.12 is thus hard to distinguish.

The design Froude number, Fn = 0.142, is supposed to be the point of lowest CT as can be seen in the experimental data. This trend is however not captured by the simulations. For mesh 3 this seems to be caused by the frictional resistance deviating from the trend for the point Fn = 0.142. This simulation was therefore meticulously studied but no apparent cause for the deviation could be found. As has been previously shown, the wave patterns and wake ﬂow ﬁeld at this Froude number both match experimental data very well. The reason why mesh 4 fails to capture the trend could be that the numerical ventilation becomes worse as the velocity increases.

44 5.3 Case 3: MARIN Systematic Series FDS-5 + This section contains the results from the simulations of the FDS-5. The results include visualizations of ∆yw along the hull, signs of numerical ventilation and estimates of the total drag. Figures 36 and 37 show examples of what the ﬂow ﬁeld can look like around the FDS-5.

Figure 36: The FDS-5 with the free surface seen from above using mesh 2 at Fn = 1.0. The free surface was extracted using a contour with α = 0.5. It is colored according to the value of the z-coordinate scaled with LPP .

Figure 37: The stern region of the FDS-5 with the free surface using mesh 2 at Fn = 1.0. The free surface was extracted using a contour with α = 0.5. It is colored according to the value of the z-coordinate scaled with LPP .

45 + Much like for the JBC, a ﬁrst check of mesh quality was done for the FDS-5 by studying the ∆yw values along + the hull. Since there is no clear line in which to sample, the ∆yw distribution was instead plotted on the hull. The results for mesh 1 and 2 at Fn = 1.0 can be seen in Figures 38 and 39 below where the hull has been cut + at the waterline. The ∆yw values for the other Froude numbers were of similar magnitude.

+ Figure 38: ∆yw plotted along the under water part of the hull of the FDS-5. The case is mesh 1 at Fn = 1.0.

+ Figure 39: ∆yw plotted along the under water part of the hull of the FDS-5. The case is mesh 2 at Fn = 1.0.

+ Both meshes seem to be of acceptable quality for wall-modeled simulations, with average ∆yw values ranging from 100 to 200 for mesh 1 in the range Fn = 0.5 to 1.0, and from 50 to 100 for mesh 2 in the same Froude number range.

46 5.3.1 Numerical Ventilation Numerical ventilation was present in all simulations of the FDS-5 and the eﬀect was much more severe than for the JBC, as can clearly be seen in Figures 40 and 41 below.

Figure 40: α plotted on the hull of the FDS-5 at Fn = 1.0 using mesh 1

Figure 41: α plotted on the hull of the FDS-5 at Fn = 1.0 using mesh 2

The eﬀect of the numerical ventilation seems to become worse with the ﬁner mesh and when studying Figure 41 there even seems to be some zones of pure air on the hull, i.e. α = 0, for mesh 2. Figure 42 show close ups of where the bow cuts the free surface for mesh 1 and 2 at Fn = 1.0.

Figure 42: Left: Close-up of the bow region of mesh 1 at Fn = 1.0 Right: Close-up of the bow region of mesh 2 at Fn = 1.0

The conclusion that the numerical ventilation is worse for mesh 2 becomes even clearer when studying Figure 42. The innermost prism layer of mesh 2 has an α-value close to 0.5 in the hull region, meaning almost 50% air.

As discussed in Section 5.2 the angle with which the hull cuts the free surface seems to have a large eﬀect on the magnitude of the numerical ventilation. Due to its shape the FDS-5 has an acute angle to the free surface at all velocities and this is most likely why numerical ventilation was present throughout the Froude number range.

47 5.3.2 Trim & Sinkage In Figure 43 trim and sinkage data from simulations using both FDS-5 meshes is presented.

5 0 Exp. -0.2 Mesh 1 Mesh 2 0 -0.4

-0.6

-5 ] ° -0.8

-1 -10 Trim [ -1.2 Sinkage [mm] Exp. -15 Mesh 1 -1.4 Mesh 2 -1.6

-20 -1.8 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 Fn Fn

Figure 43: Left: Sinkage as a function of the Froude number for both meshes. Right: Trim angle as a function of the Froude number for both meshes. The experimental data in both ﬁgures is from Kaspenberg et al. [18].

The sinkage does not seem to be captured that well by either of the meshes. One might argue that the trend is captured in some Froude number intervals, but the relative discrepancies are still large. The trim is at least captured in terms of overall trend, but the discrepancies are large here as well. According to Stephens et al. [36] numerical ventilation is known to aﬀect the trim and sinkage in the simulations where it appears. This is because numerical ventilation will aﬀect both the buoyant forces and the hydrodynamic lift acting on the hull.

Since the numerical ventilation gets worse with increasing velocity, it was almost expected that both the trim and sinkage estimates would also get worse with increasing velocity as can be seen in Figure 43 (left).

It also seems like the FDS-5 is very sensitive to changes in Froude number around Fn = 0.5. When the Froude number increases from being less than 0.5 to 0.5 the magnitude of both the trim and the sinkage increases rapidly. If the Froude number continues to increase the trim seems to plane out while the sinkage decreases as rapidly as it increased. The rapid increase in trim seems to be captured relatively well by the simulations, while the spike in sinkage is severely underestimated. This could be due to the fact that numerical ventilation increases as the trim increases (and the bow angle becomes more acute), thus pushing more air under the hull.

48 5.3.3 Drag In this section the ﬁnal drag estimates from the FDS-5 simulations are presented and compared with reference data from Kaspenberg et al. [18].

5.5 3.6 Exp. ITTC-57 Mesh 1 3.4 Mesh 1 5 Mesh 2 Mesh 2 3.2

3 4.5 3 3 x10 x10 f T 2.8 C C 4

2.6 3.5 2.4

3 2.2 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 Fn Fn

Figure 44: Left: CT as a function of the Froude number for both meshes. The experimental data is from Kaspenberg et al. [18]. Right: Cf as a function of the Froude number for both meshes compared with the ITTC-57 friction line.

When studying Figure 44 (left) the simulations seem to underestimate the drag for the FDS-5, unlike for the JBC, and there is barely any diﬀerence between the two meshes when it comes to CT . The average discrepancy from experimental data was 15.78% for mesh 1, 16.45% for mesh 2 and 16.1 % between the two of them. Table 21 below summarizes the discrepancies for mesh 1 and 2 at all Froude numbers.

Fn 0.356 0.500 0.571 0.642 0.786 0.858 1.000 Mesh 1 10.92 % 18.67 % 16.45 % 16.57 % 15.38 % 16.30 % 16.19 % Mesh 2 - 15.36 % 17.78 % - 16.46 % - 16.20 %

Table 21: Diﬀerence between estimated CT and experimental data for all simulations using mesh 1 and 2.

The frictional force is also severely underestimated, and is even lower than the ITTC-57 friction line as can be seen in Figure 44 (right). The main reason for this is believed to be the numerical ventilation present in all simulations. This would also explain why the friction drag is even lower for mesh 2 than for mesh 1.

The simulations does however seem to capture the overall trend of CT quite well for both meshes which is surprising considering that the simulations struggled in capturing the trends in both sinkage and friction.

49 6 Conclusions

In this chapter a few conclusions are drawn from the results and possible subjects for future studies are discussed.

6.1 Case 1: Joubert Bare Hull The Joubert Bare Hull was a good first geometry for benchmarking the software. The skin friction distribution can be relatively difficult to capture with CFD, so it makes sense to start with a streamlined body and without the complexity of the two phases. The fact that the simulation using mesh 2 at Re = 12×106 showed such good agreement with the experimental data, both in terms of cp and cf , increased the confidence in the chosen methodology.

6.2 Surface Ships When comparing the simulations and results from the JBC and FDS-5 with the JBH it is obvious that simulations of surface ships brings a lot of additional problems that needs to be taken in to account.

6.2.1 Numerical Ventilation Numerical ventilation was present to some degree in almost all simulations of surface ships. One of the main reasons for this was believed to be the prism layers being misaligned with the free surface thus causing it to be smeared. [36] The coarser upstream mesh together with the high Courant numbers also seemed to be part of the problem. The combination of the two lead to the surface being smeared upstream and this smeared interface was then convected downstream to the bow. As mentioned it would be very beneﬁcial if this problem could be solved without decreasing the Courant numbers too much. A strategy for this could be to reﬁne the mesh much further upstream in order to reduce the surface smearing.

A method that could potentially salvage the results from a simulation suffering from numerical ventilation would be to recalculate the wall shear stresses using the velocity field, with a forced viscosity independent of α. This method does however rely on the errors in trim and sinkage being negligible, which is not the case for the FDS-5. It could however potentially be used for the JBC simulations. The method also assumes that the velocity field remains unaffected by the NV, which seemed to be the case in all simulations.

6.2.2 Convergence The simulations of the JBC using mesh 4 showed convergence problems for the drag force and when studying the results in ParaView there seemed to be waves oscillating in the near hull region. As mentioned it was investigated whether these waves were an eﬀect of the high Courant numbers, and the results showed that this was at least true to some degree. The oscillations could be reduced signiﬁcantly by reducing the maximal allowed Courant number, but there were still small oscillations present even at a Courant number of 0.5.

The oscillations are believed to originate from the initial bow wave being created when the simulation is initialized. Since coarse meshes tend to be more numerically dissipative these waves might have dissipated much faster on the coarser mesh 3. If this is the case it would be possible to reduce oscillations and speed up convergence by initializing the simulations using a pre-calculated velocity field. This could be done, for instance by performing a simulation of the same geometry using a coarser mesh and then interpolating the final velocity field on to the finer mesh.

It was also investigated whether the oscillations and convergence problems were somehow related to the numerical ventilation, although no correlation between the two could be found.

50 6.2.3 Computational cost Computational cost is often given in number of core-hours required to perform a simulation, however since the simulations of the JBC using mesh 4 and FDS-5 using mesh 2 were run for diﬀerent amounts of simulation time this would not be representative. Instead the computational cost is presented as the amount of core-hours required per simulation time second, see Figure 45. All simulations were run on Intel Xeon Gold 6230 nodes with 40 cores in each node. All JBC simulations were run on between 2 and 4 nodes and the FDS-5 simulations on between 3 and 5 nodes.

7.5

Mesh 3 250 7 Mesh 4 Mesh 3 Mesh 4

6.5 200

6 150 5.5

5 100

4.5 50 4 Avg. core-hours per second of sim. time Avg. core-hours per second of sim. time 3.5 0 0.12 0.13 0.14 0.15 0.16 0.4 0.5 0.6 0.7 0.8 0.9 1 Froude number Froude number

Figure 45: Computational cost presented as the average amount of core-hours required per simulation time second for Left: JBC Right: FDS-5

In general the FDS-5 seems to be more computationally expensive to simulate than the JBC. One reason for this could be that the FDS-5 was simulated at much higher velocities than the JBC, meaning much higher Reynolds numbers. The physics of the FDS-5 simulations are also quite complex with big wakes and water spraying, which could potentially impair convergence.

As mentioned, one of the JBC simulation was run with a maximal Courant number of just 0.5 and the computational cost of this simulation was over 400 core-hours per simulation time second. A simulation of the FDS-5 with a Courant number of 0.5 was also initiated, but it was later canceled since the computational cost of the ﬁrst two seconds of simulation time was over 10.000 core-hours per simulation time second.

6.2.4 Case 2: Japanese Bulk Carrier Overall the results from the JBC simulations are considered to be satisfactory. The fact that both the wave patterns and the ﬂow in the wake region show such good agreement with experimental data indicates that the ﬂow physics are at least captured to some degree.

The average discrepancy in drag was 6.1% for both meshes, and it is hard to say which is the better since they both had their own problems. 6.1% is still a bit high and this number could probably be reduced to at least below 5%.

Numerical ventilation was present in almost all simulations using mesh 4 and its impact is believed to be seen in the reduction in friction between meshes 3 and 4. A good starting point for reducing the numerical ventilation could be to improve the resolution of the free surface upstream of the hull.

There were also some oscillations in the drag force for all mesh 4 simulations and this was believed to be due to a combination of the high Courant numbers and the ﬁner mesh being being less numerically dissipative.

51 6.2.5 Case 3: MARIN Systematic Series FDS-5

The results from the FDS-5 simulations are not as satisfactory as the JBC results and a discrepancy in CT of 15% is relatively high. A large part of the dicrepancy is believed to come from the severe numerical ventilation seen in some of the cases. In the article by Stephens et al. the authors mention that many numerical studies of planing hulls report a discrepancy of 10% or less. A displacing hull should in theory be easier to simulate since it is much less sensitive to changes in trim and sinkage and also generally suﬀers less from numerical ventilation. With this in mind a discrepancy less than 10% should be achievable.

One positive result is however the fact that the trends in trim and sinkage seems to be capture quite well. The simulation even seems to predict the point of lowest sinkage, even though the magnitude of the sinkage in this point is quite wrong. This indicates that the dynamics of the hull are at least somewhat correct.

Another potential source of error was discovered when reading the book by Kaspenberg et al. [18]. The authors mention that the center of gravity (CoG) was unknown for all models and the towing point was therefore instead taken as 60% of the model height from the keel. If a model is not towed in its CoG a torque will be induced which will aﬀect the trim of the model. How large this error was is not disclosed in the book.

6.3 Summary Overall a large amount of data has been collected throughout the study, for all three test cases and this would deﬁnitely be a good basis for future studies as well as for a future numerical towing tank. Aside from this a number of potential problems and pitfalls related to surface ship simulations have also been identiﬁed which can be of great value to future studies. Due to time constraints the discrepancies in drag could not be reduced further and instead emphasis was put on collecting equal amounts of data for all three test cases.

6.4 Outlook A few interesting topics for future studies could be:

1) To further investigate numerical tripping, for instance by refining the mesh upstream and trying out different Courant numbers to find a good balance between accuracy and computaitonal cost.

2) To further optimize the meshing methodology. Even though a lot of time and work was put into the final meshing methodology of this study, there is still room for improvement. For instance, the meshes could be made more efficient by making them coarser in some areas where resolution is not needed and instead refine the surface further.

3) To perform a proper grid reﬁnement study to assess the model errors.

4) To investigate numerical tripping. Since almost all reference papers use some kind of tripping device it would be interesting to see how much can be gained in terms of cf by simulating this using numerical tripping.

5) To investigate propulsion modeling as well as the interaction between the propulsion mechanism and the ﬂow in the stern region.

52 References

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53 [18] G.K. Kapsenberg, A.B. Aalbers, A. Koops, and J.J. Blok. Fast Displacement Ships: The Marin Systematic Series. Maritime Research Institute Netherlands, 2014.

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54 Appendices

A Meshes

This appendix contains overviews of all meshes as well as close ups of some important regions.

Joubert Bare Hull Mesh 1

Figure 46: Overview of mesh 1

Overall mesh 1 looks very coarse and especially in the wake region. In Figure 47 the layer addition seems to have failed close to the tip of the stern, while the bow seems to be properly meshed.

Figure 47: Left Close up of the stern region of mesh 1 Right Close up of the bow region of mesh 1

55 Mesh 2

Figure 48: Overview of mesh 2

Mesh 2 is a significant improvement to mesh 1 and has an elongated stern-refinement region to properly resolve the wake. In Figure 49 (Left) and (Right) the layer addition also seem to have failed near the tip of the stern, much like for mesh 1. The layers looks much finer compared to those in mesh 1, which can be confirmed by comparing the y+ values between Figures 20 and 21.

Figure 49: Left Close up of the stern region of mesh 2 Right Close up of the bow region of mesh 2

56 Japanese Bulk Carrier Mesh 1

Figure 50: Overview of mesh 1

Overall mesh 1 looks rather coarse and the hull doesn’t seem to be properly resolved. The layer generation also seems to have failed in multiple areas.

Figure 51: Left Close-up of the stern region of mesh 1 Right Close-up of the bow region of mesh 1

57 Mesh 2

Figure 52: Overview of mesh 2

Mesh 2 looks like a signiﬁcant improvement to mesh 1 and the main feature missing is a good resolution around the free surface. The hull looks properly resolved and when studying Figure 53 the layer generation seems successful all over the bow. There was some problem generating layers in the area around the propeller shaft, although the eﬀect of this region on the overall drag was considered to be negligible.

Figure 53: Left Close-up of the stern region of mesh 2 Right Close-up of the bow region of mesh 2

58 Mesh 3

Figure 54: Overview of mesh 3

The quite diﬀerent meshing approach of mesh 3 and 4 can clearly be seen when comparing Figure 54 with Figures 52 and 50. The area around the free surface has a much higher resolution and the reﬁnement regions are constrained to the vicinity of the model.

Figure 55: Left Close-up of the stern region of mesh 3 Right Close-up of the bow region of mesh 3

59 Mesh 4

Figure 56: Overview of mesh 4

Mesh 4 looks very similar to mesh 3, with the exception of the even higher reﬁnement level around the free surface.

Figure 57: Left Close-up of the stern region of mesh 4 Right Close-up of the bow region of mesh 4

60 MARIN Systematic Series FDS-5 Mesh 1

Figure 58: Overview of mesh 1

Mesh 1 has a very high reﬁnement level around the free surface, but parts of the hull are still quite unresolved. The most problematic area is the sharp edge along the middle of the bow which looks choppy and the layer generation has failed completely.

Figure 59: Left Close-up of the bow region of mesh 1 from front Right Close-up of the bow region of mesh 1 from side

61 Mesh 2

Figure 60: Overview of mesh 2

Mesh 2 is similar to mesh 1 but has even higher resolution around the free surface and also seems to capture the shape of the bow much better. The layer generation seems to have failed along the sharp edge of the bow, although only in a very thin region and its impact on the total resistance was assumed to be negligible.

Figure 61: Left Close-up of the bow region of mesh 2 from front Right Close-up of the bow region of mesh 2 from front

62 B Residuals

In this Appendix the residuals from all meshes at a few key Reynolds/Froude numbers are presented.

Joubert Bare Hull

Mesh 1 - Re = 5.400.000 0 U x -2 k omega -4 p

(Residual) -6 10

Log -8

-10 0 100 200 300 400 500 600 700 800 900 1000 Time (s) Mesh 2 - Re = 5.400.000 0 U x -2 k omega -4 p

(Residual) -6 10

Log -8

-10 0 100 200 300 400 500 600 700 800 900 1000 Time (s) Mesh 2 - Re = 12.000.000 0 U x -2 k omega -4 p

(Residual) -6 10

Log -8

-10 0 100 200 300 400 500 600 700 800 900 1000 Time (s)

Figure 62: Residuals for the three JBH cases

63 Japanese Bulk Carrier Mesh 3

Fn = 0.12 -4 U x -6 k omega pd -8 (Residual) 10 -10 Log -12 0 10 20 30 40 50 60 70 80 90 100 Time (s) Fn = 0.142 -4 U x k -6 omega pd -8 (Residual) 10 -10 Log

-12 0 10 20 30 40 50 60 70 80 90 100 Time (s) Fn = 0.16 -4 U x k -6 omega pd -8 (Residual) 10 -10 Log

-12 0 10 20 30 40 50 60 70 80 90 100 Time (s)

Figure 63: Residuals for a few of the cases using Mesh 3

64 Mesh 4

Fn = 0.12 -4 U x -6 k omega pd -8 (Residual) 10 -10 Log

-12 0 50 100 150 200 250 300 Time (s) Fn = 0.142 -4 U x -6 k omega pd -8 (Residual) 10 -10 Log

-12 200 250 300 350 400 450 500 Time (s) Fn = 0.16 -4 U x -6 k omega pd -8 (Residual) 10 -10 Log

-12 0 200 400 600 800 1000 1200 Time (s)

Figure 64: Residuals for a few of the cases using Mesh 4

65 MARIN Systematic Series FDS-5 Mesh 1

Fn = 0.500 -4 U x k -6 omega pd -8 (Residual) 10 -10 Log -12 0 10 20 30 40 50 60 70 80 90 100 Time (s) Fn = 0.786 -4 U x -6 k omega pd -8 (Residual) 10 -10 Log -12 0 5 10 15 20 25 30 35 40 45 50 Time (s) Fn = 1.000 -4 U x -6 k omega pd -8 (Residual) 10 -10 Log -12 0 10 20 30 40 50 60 70 80 90 100 Time (s)

Figure 65: Residuals for a few of the cases using Mesh 1

66 Mesh 2

Figure 66: Residuals for a few of the cases using Mesh 2

67 www.kth.se