Proc IMechE Part M: Prediction of wave resistance by a ...mansim.org/articles/Prediction of wave resistance... · ments carried out at the Ata Nutku Ship Model Testing Laboratory

Review Article

Proc IMechE Part M:J Engineering for the Maritime Environment1–18� IMechE 2015Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1475090215599180pim.sagepub.com

Prediction of wave resistance by aReynolds-averaged Navier–Stokesequation–based computational fluiddynamics approach

Omer K Kinaci1, Omer F Sukas1 and Sakir Bal2

AbstractThe prediction of wave resistance in naval architecture is an important aspect especially at high Froude numbers where agreat percentage of total resistance of ships and submerged bodies is caused by waves. In addition, during hull form opti-mization, wave resistance characteristics of a ship must closely be observed. There are potential, viscous and experimen-tal methods to determine the wave resistance of a ship. Reynolds-averaged Navier–Stokes equation–based methodsusually follow the experimental method that determines the form factor first. However, it is proven in recent studiesthat the form factor changes with the Reynolds number. As the Reynolds number increases, this change in the form fac-tor is being neglected. In this study, a Reynolds-averaged Navier–Stokes equation–based prediction of wave resistance ispresented that overcomes this flaw. The methodology is validated with the benchmark problems of submerged andsurface-piercing bodies to determine the effectiveness of the proposed method. The method is also validated by experi-ments carried out at the Ata Nutku Ship Model Testing Laboratory of Istanbul Technical University for a totally sub-merged ellipsoid and the benchmark KRISO Containership. Results reveal the robustness of the present methodology.

KeywordsWave resistance, viscous resistance, Reynolds-averaged Navier–Stokes equation, KRISO Containership, form factor

Date received: 6 July 2014; accepted: 11 June 2015

Introduction

Model testing has been an indispensable part for theprediction of ship resistance due to the complex natureof flow around an underwater hull. However, thisdependency to model tests is decreasing slowly as newcalculation methods are developed by researchers anddesigners. As widely known, total resistance of a shipcould be broken down into subparts such as frictionalresistance, viscous pressure resistance and wave resis-tance. Frictional resistance of a ship can be calculatedby the International Towing Tank Conference (ITTC)formulation,1,2 but it is still a problem today to com-pute the viscous pressure resistance and the wave resis-tance empirically for ship-like bodies.

In model testing, the viscous pressure resistance iscalculated by towing the model at very low speeds gen-erally in a towing tank and implementing the methodof Prohaska or some other approaches recommendedby ITTC procedures.2 It is also possible to follow thesame procedure using the Reynolds-averaged Navier–Stokes equation (RANSE)-based computational fluid

dynamics (CFD) method to obtain the viscous pressureresistance numerically instead of conducting modeltests. The wave resistance can generally be calculatedby implementing the potential-flow-based boundaryelement methods (BEMs).3 The frictional resistancecan then be obtained by the ITTC correlation line andall these resistance components may be added to obtainthe total resistance; however, the solutions are stilldependent on whether the assumptions made or on themesh structure.

Using RANSE-based CFD, it is possible to calculatethe total resistance, but RANSE method does not

1Naval Architecture and Maritime Faculty, Yildiz Technical University,

Istanbul, Turkey2Faculty of Naval Architecture and Ocean Engineering, Istanbul Technical

University, Istanbul, Turkey

Corresponding author:

Omer K Kinaci, Faculty of Naval Architecture and Maritime, Yildiz

Technical University, Barbaros Bulvari Besiktas, Istanbul 34349, Turkey.

Email: [email protected]

by guest on August 24, 2015pim.sagepub.comDownloaded from

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directly calculate the wave resistance of a ship. Instead,it calculates the pressure resistance, which is the addi-tion of viscous pressure resistance component and waveresistance component, and the frictional resistance toobtain the total resistance. If one wants to solelyobserve the wave resistance characteristics of a ship,she or he generally consults to potential-flow-basedsolutions that ignore the viscosity of the fluid and line-arize the free-surface conditions. The main aim of thisstudy is to compute the wave resistance of ships andbodies moving under free surface by RANSE-basedCFD.

The first known wave resistance calculation startedwith the linear theory of wave resistance developed byJohn Henry Michell.4 His works were noticed a quartercentury after by other scientists. Meanwhile, in the early1900s, Havelock5 developed a potential-flow theory forcalculating the wave resistance around a hull. There area lot of works on potential-flow theory in the literature,to calculate the wave drag of a body; most of them relyon Havelock’s theory. Sclavounos6 and Sclavounos andcolleagues6–9 have also many works on ship waves. Hisstudies are mainly contributed by potential theory,ignoring viscous effects. In the work of Rigby et al.,10

the wave resistance was calculated by potential theoryusing Rankine sources. The method was first evaluatedon a sphere with infinite depth that the analytical solu-tion has been known for. Then the effects of the freesurface were introduced and the wave resistance of aWigley hull was computed and compared with theexperimental data. The proposed method gave satisfac-tory results and seemed better than the linear theorywhich failed to return accurate results at certain Froudenumbers.

The wave pattern and wave resistance of surface-piercing bodies were also predicted by Bal11 with aniterative BEM. The free surface and surface of the pier-cing body were treated separately to unite for a con-verged solution. The convergence was achieved afterseveral iterations. The effects of cavitation and finitedepth were also included in the studies.12–14 A modifiedRankine source panel was used to investigate the flowsaround Wigley and Series 60 hulls by Tarafder andSuzuki.15 In the modified method, Rankine sourceswere distributed along the surface of the geometry aswell as its image and on the free surface. They com-pared their results with the experiments and foundgood accordance in predicting the wave pattern andwave resistance. Chen16 developed a new vortex-basedpanel method involving free surface with energy dissi-pation. The influence of wave damping is implementedin the traditional potential-flow wave drag computa-tions by adding energy dissipation term into the GreenFunction. The inclusion of energy dissipation not onlycancels the singularities of the Green Function but alsoprovides better results when compared with othermethods. Chen17 improved his studies a year after,when he adopted a dissipative free-surface approach topotential-flow theory for a two-dimensional (2D)

hydrofoil moving under the free surface. Submergedand surface-piercing bodies moving at a constant speedwere also solved with the developed code in He,18 andthe wave pattern, the wave profile and the wave resis-tance were computed to validate the results.

Belibassakis et al.19 investigated the two phase flowaround a prolate spheroid, Wigley hull and Series 60ship with an iso-geometric analysis applied to the BEM.The wave resistances, pressure distributions and waveprofiles were investigated inside their study and com-parisons were made with experiments and other numer-ical results in the literature. On the other hand, thegeneral wave resistance theories, the solution methodsof wave drag models and a broad history of the devel-opments in understanding the waves generated by shipscan be found in the works of Wehausen,20 Newman21

and Tulin,22 respectively. A general evaluation of thewhole ship resistance problem can be attained fromLarsson and Raven23 and Gotman.24 A prolate spher-oid was used for a submerged body and a Wigley hullwas used to demonstrate the capability of the method.

Other than linearized potential-flow methods, fullynonlinear RANSE-based solutions have also been stud-ied broadly in the literature. In 2000, a CFD workshopheld in Gothenburg (Sweden) was a stimulating confer-ence to present the developments of in-house CFDcodes. Three different benchmark vessels were investi-gated by different institutions and universities with dif-ferent codes implementing RANSEs, and their resultswere evaluated in that workshop. A brief comparisonof all the results for these three ships generated with dif-ferent CFD codes were made in the study by Larssonet al.25 Resistance and propulsion characteristics of var-ious commercial ships were investigated by Choi et al.26

The experimental approach is implemented in RANSE-based CFD to predict the speed performance, resistanceand self-propulsion. A multiphase analysis (involvingair and water phases) of a surface effect ship (SES) withRANSEs was made by Maki et al.,27 investigating theeffect of the air cushion between the hulls of the SESs.The results were compared with the potential-flow-based method and it was found that the linearizedmethod was found to be attractive in terms of computa-tional requirements. The results of RANSEs and thelinearized method were in accordance with each other,correctly predicting the free-surface elevation and thetotal resistance. However, there was no comparisonmade in terms of wave resistance because RANSE-based CFD programs can only predict the pressure andfrictional resistances.

The other focus of this article is to remove the calcu-lation of the form factor which is dependent on theReynolds number. The general procedure in navalarchitecture to calculate the wave resistance is to com-pute the form factor first to determine the viscous resis-tance with the help of the ITTC correlation line andthen to subtract this value from the total resistance.The experimental determination of wave resistanceassumes the form factor to be constant for all Froude

2 Proc IMechE Part M: J Engineering for the Maritime Environment



and Reynolds numbers. However, a notable study onthis topic28 revealed that the effect of the Reynoldsnumber on the form factor cannot be neglected.Another study29 investigated the scale effect on theform factor and obtained the form factor of differentships with respect to the Reynolds number. Min andKang30 questioned the basic assumptions of the calcu-lation of form factor by the ITTC78 method and pro-posed a new extrapolation method for the prediction offull-scale ship resistance. The method used in this studyis the volume of fluid (VOF) method for tracking thefree surface, which is widely used in ship resistance cal-culations. However, the procedure followed in calculat-ing the wave resistance is novel. This study proposes animplicit methodology that automatically eliminates theflaws that may arise from the form factor—Reynoldsnumber dependency.

Mathematical modeling

RANSE method

In this study, RANSEs, implementing the k� e turbu-lence model, are used with commercial CFD software,ANSYS Fluent.31 ANSYS Fluent solves continuummechanics problems by discretizing the fluid domain bythe finite-volume method. RANSEs are time-averagedequations for motions of fluid flow. The basic RANSEswhich consist of conservation of mass and momentumare given as

∂Ui

∂xi=0 ð1Þ

∂Ui

∂t+

∂

∂xjUiUj

� �= � 1

r

∂p

∂xi+

∂

∂xj2nSij � u0iu

0j

� �

ð2Þ

where Ui is the mean flow velocity vector, n is the mole-cular kinematic viscosity and Sij is the mean strain-ratetensor. The strain-rate tensor is also defined as

Sij =1

2

∂Ui

∂xj+

∂Uj

∂xi

� �ð3Þ

The last term on the right-hand side of equation (2) isdenoted as the Reynolds stress tensor which is given by

tij = � u0iu0j ð4Þ

The Boussinesq eddy-viscosity approximation whichis used to connect the turbulence eddy viscosity mt toRANSEs is given as

mt = � 1

2

rtij

Sijð5Þ

k� e turbulence model dictates that turbulence eddyviscosity is computed by

mt = cmrk2

eð6Þ

k and e are parameters of the turbulence model andthey obey the transport equation. They are calculatedfrom equations (7) and (8). The obtained values forthese parameters are substituted into equation (6) todetermine the turbulence eddy viscosity, mt. With thehelp of equation (5), Reynolds stress tensor is solved ateach time step, the term that forms the core ofRANSEs. Equations (7) and (8) are given below

∂rk

∂t+

∂rUjk

∂xj=

∂

∂xj+ m+

mt

st

� �∂k

∂xj

� +Pk � re

ð7Þ∂re∂t

+∂rUje∂xj

=∂

∂xj

+ m+mt

se

� �∂e∂xj

� +

ek

ce1Pk � ce2reð Þð8Þ

Additional definitions of variables and detaileddescription may be found in Johansson et al.32

VOF method

The free surface is tracked by the VOF method and it isexplained in detail in Hirt and Nichols.33 Here, a shortexplanation of the method is given. The VOF methodis based on a concept fractional VOF and also a free-surface modeling technique that is tracking and locat-ing the free surface in multiphase flows. This methodprovides a simple way to track free boundaries in 2D orthree-dimensional grids.

The fraction of fluid for each cell can be shown by afunction F, whose value is unity in a cell full of fluid,while a zero value indicates that the cell contains nofluid. Cell with F values must be between zero and one.The evolution of the F field can be written as follows

∂F

∂t+

∂uF

∂x+

∂vF

∂y+

∂wF

∂z=0 ð9Þ

where t is the physical time, (x, y, z) denote theCartesian system and (u, v,w) are the components ofthe velocity.

This equation states that F moves with the fluid andis the partial differential equation analog of markerparticles.

CFD methodology and setup

Methodology

A traditional RANSE solving software divides the totalresistance into two parts which can be formulated as

CT =CP +CF ð10Þ

where CT denotes total resistance coefficient, while CP

is residual resistance coefficient and CF is frictionalresistance coefficient. In this study, coefficients are non-dimensionalized by

Kinaci et al. 3



Cx =2 � Px

r � V2ð11Þ

where x in the subscript of Cx may refer to any resis-tance component in that equation. The pressure resis-tance coefficient can be broken down into

CP =CW +CVP ð12Þ

where CW is the wave resistance coefficient and CVP isthe viscous pressure resistance coefficient. Note that afully submerged body inside a fluid of infinite depthhas zero wave resistance (no free-surface effect) and thepressure resistance coefficient in this case becomesequal to the viscous pressure resistance coefficient

CP =CVP ! at infinite water depth ð13Þ

Within this context, the viscous pressure resistancecoefficient of a body inside the fluid may be derivedfrom a single phase analysis (that only models theunderwater hull) which totally ignores the resistancecaused by the free surface. There are some studies in theliterature involving the calculation of viscous pressureresistance by single phase analysis. One of them is thework by Kinaci et al.34 for the resistance characteristicsof a benchmark Duisburg Test Case post-panamax con-tainer ship with CFD and another one is the experimen-tal work of the same ship by El Moctar et al.35

The methodology proposed in this article first com-putes the viscous pressure resistance of the body insidethe fluid with a single phase analysis. After determiningCVP in this way, a multiphase analysis (covering thefree-surface interface between air and water) for thesame geometry including the free-surface effects is car-ried out. The viscous pressure resistance coefficientfrom the single phase analysis can then be subtractedfrom the pressure resistance coefficient determined bythe multiphase analysis

CWof the body =CP(mp) � CVP(sp) ð14Þ

where mp and sp subscripts (symbols in the parentheses)refer to the multiphase and single phase, respectively.Multiphase analyses involve both air and water, whilesingle phase only covers water.

The multiphase analyses may be time-consumingbecause the acquisition of a meaningful free-surfacesolution is computationally costly. It will take 15times larger time for the computer to solve for themultiphase analysis. Inaccurate results may easily beobtained if only the residuals are followed to get sta-bilized. The residuals usually can be made stabilizedalthough the wave contours and wave resistancevalues are not acceptable, misguiding the user that acorrect physical incident is modeled by CFD. Thereare some parameters other than residuals that shouldbe checked closely before deciding that the analysisis finalized. The list below suggests a checklist to befollowed if a healthy and converged case is to befound.

Checklist

1. Follow all the residuals to get stabilized solution.All the stabilized residuals in this study were under1024.

2. Determine whether the total resistance isconverged.

3. Check whether the total flux is close to 0, ensuringthat the conservation of mass is achieved.

4. Observe if there is a Kelvin wake pattern on thefree surface.

5. Pay attention to the pressure resistance obtainedfrom the multiphase analysis. This value must begreater than the pressure resistance obtained fromthe single phase analysis. Wave resistance cannotbe less than 0.

6. Check out whether frictional resistance from themultiphase analysis is compatible with the fric-tional resistance obtained from the single phaseanalysis.

Points from 1 to 3 cover the most general computa-tional regulations and are the normal procedure of con-ducting RANSE-based calculations. Point 4 justinvolves a basic glance at the free surface and is notformulated. Capturing incorrect free surface deforma-tions in multiphase flows is a matter which is usuallystumbled upon and this is explained in detail in Jonesand Clarke.36 The flow chart of the CFD methodologyis given in Figure 1.

CFD setup

The CFD setup for single and multiphase analyses areall pressure-based steady solutions. The bodies insidethe fluid are fixed, so there are no sinkage and trimeffects. Considering the computer capabilities (memoryand CPU time) and mesh structure, the turbulencemodel is selected to be realizable k� e with standardwall function. k� e model is a two-equation turbulencemodel that is used extensively in industrial applicationsdue to its robustness and relatively easy implementa-tion. The general references and the model’s mathemat-ical background are given in Launder and Sharma37

and called in the literature as standard k� e model.38

The method is proved to work well with slender bodies.The ellipsoid is a streamlined body with low-pressuregradients and k� e turbulence model is known toreturn good results for these cases in Bardina et al.38

Realizable k� e model is an improved version of thestandard model and gives enhanced results.39 Due tothe relatively low number of mesh that this methodrequires (y+ is expected to lie in the range between 30and 300 as stated in ANSYS31), the applicability of thisturbulence method is feasible. For other applications ofthis turbulence model, refer Larsson et al.25 and Kinaciet al.34 Wall functions may be used in places whereadverse pressure gradients are not high. Flows around




bluff bodies will generally have boundary layer separa-tions of which the boundary layer should be very finelymeshed all the way through the wall. Bodies inside thefluid are generally well streamlined which allows wallfunctions to be used.

The SIMPLE algorithm is selected for pressure–velocity coupling and in spatial discretization pressure

is set to be body force weighted. SIMPLE is the moststraightforward CFD algorithm for predicting pressureand velocities in a fluid domain iteratively. Due to itssimplicity, computational efforts are relatively less com-pared to its peers. It has been reported in the book byVersteeg and Malalasekera40 that although SIMPLECis a more stable algorithm, calculations involved are

Figure 1. Wave resistance computation methodology with CFD.

Kinaci et al. 5



30% greater. Its disadvantage is that the algorithmneeds good initial and under-relaxation values or theanalysis may diverge.40 All other parameters under spa-tial discretization and under-relaxation factors areselected as default. Momentum is second-order upwind,while volume fraction, turbulent kinetic energy and tur-bulent dissipation rate are all first-order upwind.

For multiphase analyses, an open channel flow modeof the solver is activated. Open channel mode allowssimulations to be made that are similar to the experi-ments being made at the towing tanks. The free surfaceis quiescent in the beginning, both experimentally andnumerically. To satisfy this condition numerically in thesolver, open channel initialization method is selected asflat. Open channel uses VOF method to track compli-cated free-surface deformations. Waves generated dueto the existence of a hull with a submerged transom orbulb may be complex and VOF is a widely used methodfor even spilling breaking waves and bubbly free sur-face.41 Several numerical schemes that are tested involv-ing the multiphase flow problem of a surface-piercinghydrofoil can be found in Rhee et al.41 The volumefraction in spatial discretization is set to QUICK. Onedifference between these analyses in the inlet is velocityinlet in single phase and pressure inlet in multiphaseanalysis. Open channel flow of multiphase analysis onlyworks with boundary conditions of mass flow inlet andpressure inlet in the used software, and out of the two,pressure inlet is selected. The flow specification methodat the inlet is free-surface level and velocity where theuser must specify the vertical coordinates of free-surfacelevel and the bottom of the domain. The velocity mag-nitude specified in pressure inlet of open channel in mul-tiphase flow is also used as an input for the velocity inletin single phase flow. The boundary conditions used inthis work are given in Table 1. The explanations of all

the selections made here are explained in the ANSYSFluent Theory Guide.42

The domain extents are 4c in the flow direction (1cfor upstream, 1c for the geometry and 2c for down-stream), 2c in the depth direction and 1.25c in the direc-tion of breadth for multiphase analyses, where c is thecharacteristic length of the object inside the flow. Itmay be about the same for single phase analysesbecause air fraction of the volume only consists of asmall portion of the domain. If the analysis is multi-phase, the free-surface level changes according to thedraft/depth ratio of the body.

The multiphase analyses usually require finer mesh-ing due to the complexity of tracking the free surfacecorrectly and obtaining satisfactory results. A densergrid around the hull and the free surface will give muchdecreased errors and stable residuals in the calculation.The hexahedral elements are recommended especiallyfor multiphase solutions of ship flows. The tetrahedralelements are easier to use and manage to get satisfac-tory results for single phase analyses; however, they suf-fer to capture the free surface correctly in multiphaseanalyses. Due to the small Mach number for this typeof problem, air is assumed to be incompressible in allcalculations.

Grid independence

The study of grid independency of the CFD solutionsis given for the KRISO Containership (KCS) hull atFr=0:26. Due to the good balance between the com-puted results and used CPU time, medium grid isselected which has around 3.7 million elements. InTable 2, the different grids with their y+ values andcomputed resistance values are shown.

Table 1. Boundary conditions used in CFD setup.

Boundary conditions

Single phase analysis Multiphase analysis

Inlet Velocity inlet Pressure inletOutlet Pressure outlet Pressure outletWall Symmetry SymmetryTop No slip/stationary wall No slip/stationary wallBottom No slip/stationary wall No slip/stationary wallSymmetry Symmetry Symmetry

Table 2. Grid dependency results for different grids.

No. of elements Rt (N) 1000*Ct yav+

Coarser 1.2M 24.6993 6.0912 319Coarse 1.8M 18.6727 4.6038 222Medium 3.7M 15.9848 3.9327 187Fine 6M 14.712 3.6196 181Experiment 13.9191 3.4347




Uncertainty analysis of experiments

The uncertainty analysis is made for the experimentthat is carried out at the towing tank for KCS hull. Thebias and precision limits and total uncertainties for mul-tiple runs have been estimated for the total resistancecoefficient CT in model scale at Fr=0:26. The tests areperformed giving in total 12 test points in design speed.The uncertainty for the total resistance coefficient isgiven by

UCT=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBCTð Þ2 + PCT

ð Þ2q

ð15Þ

where UCTstands for the total uncertainty, BCT

for thebias limit uncertainty and PCT

for the precision limituncertainty. ITTC’s recommended procedure was fol-lowed when doing the uncertainty analysis and it isgiven in ITTC 7.5-02-01-03:1999.43 An example resis-tance test calculation can be found in ITTC 7.5-02-02-02:2002.44 The obtained results are given in Table 3 interms of total CT. A broader explanation about theuncertainty analysis of KCS can be found in Delen.45

Numerical and experimental results

Validation of the method by a fully submerged body:case 1

The proposed method is first validated by a fully sub-merged body moving under a free surface. The results

of this study are compared with the results of Farrell46

who has solved the flow around an ellipsoid at differentaspect ratios. The potential-flow theory was used togenerate the results and the comparison has been madefor an ellipsoid with an aspect ratio of 6.46 The distanceof the free surface to the ellipsoid is 0.126c. The waveresistance is calculated with the method explained inthe ‘‘CFD methodology and setup’’ section and it isnon-dimensionalized by the formulation as given inFarrell’s study

CW =RW

prgc3ð16Þ

The hybrid mesh system is used to discretize thedomain. Quadrilateral elements are preferred to repre-sent the ellipsoid and the region that contains the freesurface, while triangular elements are used in the rest ofthe domain. Figure 2 indicates a view of this type ofmeshing used in this study.

The wave resistance coefficients obtained byFarrell’s46 method as compared with those obtained bythe present methodology are shown in Figure 3. Bothresults seem in accordance with each other.

The compatibility of the results seems better forlower Froude numbers and the discrepancy betweenboth studies seems to get greater as the Froude numberincreases. There seems to be only one point where bothresults overlap in higher Froude numbers at aroundFr=0:78, but other than that the results seem to havesome discrepancy. This difference may be due to moreviolent wave patterns that may be observed at higherspeeds where the linearized potential-flow theory mayfail to grab the correct wave contours. RANSE is fullynonlinear and is more pertinent to catch wave patternsat higher speeds. The effect of the free surface on thepressure distribution of ellipsoid is given in Figure 4.To clearly examine the free-surface effect, hydrostaticpressure is not included in that figure.

Figure 2. Mesh system around the totally submerged ellipsoid: (a) a broad perspective, (b) a close-up view of the ellipsoid and (c) aclose-up view of the bow.

Table 3. Uncertainty analysis of KCS for total resistancecoefficient.

Term Value Percentage values of CT

BCT5.94 3 1025 1.38

PCT4.93 3 1025 1.15

UCT7.72 3 1025 1.79

Kinaci et al. 7



The wave pattern for another totally submergedellipsoid at different submergence depths for theaspect ratio of 8.33 is obtained by RANSE for thewhole domain. The wave pattern and the positions oflongitudinal wave cut are shown in Figure 5. The

results of the wave patterns at Fr=0:639 are shownin Figure 6.

It is clearly seen from Figure 6 that, as the distancefrom the free surface increases, the wave elevation peakdecreases. When the ellipsoid is closer to the free sur-face, the wave elevations at different wave cuts changegreatly (see Figure 6(a)). When the ellipsoid is far awayfrom the free surface, almost all wave cuts get on topof each other (see Figure 6(d)). Due to the limited fluiddomain at the computations, the radiation conditioncannot be seen in Figure 6. However, it is possible tovisualize the radiation of the waves at both ends of thedomain. This is a matter of time and computer capabil-ity as larger domains require more CPU time andmemory.

Validation of the method by a fully submerged body:case 2

A second validation for an ellipsoid was made with anaspect ratio of 8 to validate the robustness of CFD forfully submerged bodies. Total resistances obtained fromexperimental tests and CFD for Froude numbers rang-ing between 0.16 and 0.48 and h/c=0.3, 0.5 and 0.75were compared. The experiments were carried out at

Fr

1000*Cw

0.3 0.45 0.6 0.75 0.90

0.5

1

1.5

2

2.5

Farrell [46]Present study

Figure 3. A comparison of wave resistance coefficients.

Figure 4. Free-surface effect on pressure coefficient distribution over the ellipsoid. The figure on the left is at infinite depth (nofree-surface effect), while the one on the right has submergence depth (h = 0:126c) from the free surface at Fr = 0:5.

Figure 5. Wave contour of a submerged ellipsoid at Fr = 1:2771 for h/c = 0.1 and positions of wave cut.




the Ata Nutku Ship Model Testing Laboratory of theIstanbul Technical University. The towing tank testingof fully submerged ellipsoid is carried out using a strutarrangement which is between the towing tank carriageand the model. After each test, the strut is run indepen-dently at the same Froude numbers and depths to findits contribution to the total resistance. Total resistanceof the model ellipsoid is found by subtracting the strutresistance from the total combined (ellipsoid + strut)resistance.47 For an experiment carried out in the tow-ing tank, see Figure 7.

Towing tests were repeated two times to check outthat the results are consistent with each other and theresults were averaged before comparing with RANSE-based CFD results. Wave resistance obtained via CFDcannot be compared with the experiments becausethe experimental method implements the ITTCcorrelation line that is available for conventional

ships. Therefore, only total resistances were com-pared and good accordance was found between CFDpredictions and experiments. The results are shown inFigure 8.

Here, two types of CFD were implemented. The firstCFD method, which is named as CFD(Exp.Mth) inFigure 8, is the simulation of the experiment. The totalresistance with the strut was found first, and then theproblem was solved only with the presence of the strutitself in the flow. The resistance of the strut was thensubtracted from the strut-ellipsoid system to find theresistance of the bare ellipsoid. The second CFDmethod, which is named as solely CFD in Figure 8, cal-culates the resistance of the bare ellipsoid without tak-ing into account the strut. The results are in accordancewith each other and it can be said that the present CFDmethodology gives satisfactory resistance values forsubmerged bodies.

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x

waveheight

-1 -0.5 0 0.5 1 1.5 2 2.5 30.05

0.075

0.1

0.125

y=0y=0.03y=0.06y=0.09y=0.15y=0.3

a

b

c

d

e

f

h / c = 0.1, Fr = 0.64

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x

waveheight

-1 -0.5 0 0.5 1 1.5 2 2.5 30.225

0.25

0.275

y=0y=0.03y=0.06y=0.09y=0.15y=0.30

a

b

c

d

e

f

h / c = 0.25, Fr = 0.64

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x

waveheight

-1 -0.5 0 0.5 1 1.5 2 2.5 30.49

0.5

0.51

y=0y=0.03y=0.06y=0.09y=0.15y=0.30

a

b

c

d

e

f

h / c = 0.5, Fr = 0.64

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x

waveheight

-1 -0.5 0 0.5 1 1.5 2 2.5 30.997

0.998

0.999

1

1.001

1.002

1.003

y=0y=0.03y=0.06y=0.09y=0.15y=0.30

a

b

c

d

e

f

h / c = 1, Fr = 0.64

(a) (b)

(c) (d)

Figure 6. Wave pattern of a submerged ellipsoid at different depths at Fr = 0.64: (a) h/c = 0.1, (b) h/c = 0.25, (c) h/c = 0.5, and(d) h/c = 1.

Kinaci et al. 9



Validation of the method by a surface-piercing body:case 3

A third validation has been made for a Wigley hull onwhich many studies in the literature exist. In order toobtain a satisfactory result for wave resistance, thewave profiles in the downstream region of the hull andalong the hull must be correctly predicted. There arevarious studies (most of them relying on potential the-ory) to predict the wave profiles along the Wigley hulland all these results are tried to be validated withexperiments. One of these studies is the work ofTarafder and Suzuki.15 They have used modified

Rankine source panel method to obtain the wave pro-file along the hull and compared their results with anexperiment for Fr=0:316. The comparison of com-puted wave profile with that of RANSE is given inFigure 9.

The wave pattern computed by present methodologyis more compatible with the experiments, especially atthe bow region where there is a high free-surface eleva-tion. Another experiment is performed by He18 atFr=0:350 to validate the newly developed code basedon an iterative BEM. The RANSE result is comparedwith those calculations and experiments for the wave

Figure 7. An ellipsoid with the strut connected during experiments and its dimensions.47

Fr

TotalResistance(N)

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

CFD(Exp.Mth.)CFDExperiment

Figure 8. A comparison of total resistance for h/c = 0.5.

2x/L

2z/L

-1 -0.5 0 0.5 1-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

Present studyCalculation [15]Experiment [15]

Figure 9. Wave profile along the Wigley hull for Fr = 0:316.




pattern along the Wigley hull as shown in Figure 10.The wave pattern obtained by RANSE seems in betteraccordance with the experiments. The potential theorypredicts lesser wave elevations especially in placeswhere the perturbation of the free surface is greater asin the previous case.

The wave resistance values of the Wigley hull arenot published in He.18 Tarafder and Suzuki15 haveshared their results up to Fr=0:6. As mentionedbefore, RANSE is fully nonlinear and can give satisfac-tory wave profiles and wave resistance values even athigher Froude numbers. For wave resistance coefficientvalues computed with the proposed method, compari-sons are made with the calculations of Moraes et al.48

who computed the wave resistance up to Fr=0:9.However, experimental data are not included in theirstudy. Therefore, the experiments at different Froudenumbers are taken from the work of Millward andBevan.49 The compared results are shown in Figure 11.The wave resistance coefficient here is calculated as

CW =2RW

rSV2ð17Þ

The wave resistance results seem to be compatiblewith both the calculations of Moraes et al.48 and theexperiments of Millward and Bevan.49 It could also besaid that the proposed methodology gives satisfactoryresults.

The discrepancy between the results of present meth-odology and the experiments increases with the increasein Froude numbers as seen in Figure 11. The same pointapplies for Moraes et al.’s48 results as well. One reasonfor this may be due to the calculation of residual resis-tance in experiments. The resistance of the static wettedarea is subtracted from the residual resistance to obtainwave resistance in a towing tank. The wetted area ofthe hull does not deviate much due to smaller free-surface elevations at lower Froude numbers; however,

this is not the case at higher Froude numbers. Thedeviation of the wetted area of the hull is greater athigher speeds but during calculations of wave resistanceat the towing tank, only the static wetted area is calcu-lated and subtracted. Rigby et al.10 have suffered fromthe same problem and they have also made such a state-ment in their study. Another reason might be that dur-ing experiments the ship models are free to trim andsinkage effects may be present. At higher Froude num-bers, models are more prone to making high levels ofmotion, while in CFD analysis the model is fixed. Thesepoints must be kept in mind while comparing the calcu-lated and experimented results at high Froude numbers.The total resistance coefficients are compared with theexperimental data of Noblesse and McCarthy50 inFigure 12.

2x/L

2z/L

-1 -0.5 0 0.5 1-0.04

-0.02

0

0.02

0.04


Figure 10. Wave profile along the Wigley hull for Fr = 0:350.

Fr

1000*Cw

0 0.25 0.5 0.75 10

1

2

3

4

5


Figure 11. Comparison of wave resistance coefficients atdifferent Froude numbers.

Fr

Ct*1000

0.15 0.2 0.25 0.3 0.35 0.4 0.450

2

4

6

8

10

Experimental - Total resistance coefficient [50]

CFD - Total Resistance Coefficient

Figure 12. Comparison of total resistance coefficients forWigley hull.

Kinaci et al. 11



Comparison of free-surface deformation with Usluand Bal51—who made calculations with potentialtheory—for Fr=0:4 is given in Figure 13. It may besaid that there is a good agreement between the presentstudy and their work. The effect of free surface andviscosity on pressure distribution on the Wigley hull isgiven in Figure 14. It must be noted that the hydrostaticpressure is not included in this figure to correspond toUslu and Bal.51 There seems a good compatibility forcases of unbounded flow domain (which is a singlephase solution) and free-surface effect included domain(which is a multiphase solution). The free-surface eleva-tions change the pressure distribution along the hull, ascan be seen in Figure 14. The pressure decreases atpoints where wave troughs are found, while it increasesin places of wave crests. In Figure 14, there is some dif-ference at the bow in RANSE calculations and that isdue to RANSE being a fully nonlinear method. It maybe said that the pressure distribution obtained is betterthan the potential methods. Wigley hull has a slenderbody and the stern part of the hull is not complex. Thisis the place where the effects of viscosity must show upthe most, but due to the thin geometry of the hull, theresults are nearly the same.

The results may be improved by implementing anunsteady approach to the problem. However, unsteadysolutions are time-consuming when compared tosteady-state approach. Due to this reason, this studyonly covers the steady aspects of the fluid flow aroundthe Wigley hull. For further research on the unsteadyflow analysis of the Wigley hull, refer to the work byRhee and Stern.52

Validation for KRISO container ship: case 4

The proposed method has been validated for sub-merged and surface-piercing benchmark problems upto this point. The robustness of the method has alsobeen tested for the KCS. The experiments were carriedout at the towing tank of Ata Nutku Ship ModelTesting Laboratory at the Istanbul TechnicalUniversity.47 The main dimensions for the model aregiven in Table 4. A general view is given in Figure 15.The hybrid meshing method used for this vessel can beseen in Figure 16. The tetrahedral elements are used inthe vicinity of the vessel which has a complex geometryand is harder to mesh. The hexahedral elements areused to represent the rest of the domain. The fluiddomain around the ship is 8 � Lbp long, 8 � B wide and10 � T deep.

Figure 17 shows the wave resistance coefficient val-ues of KCS. There is good compatibility between theexperimental and calculated values of the wave resis-tance coefficient. The discrepancy is somewhat greater

Figure 14. Effect of free surface and viscosity on pressure coefficient distribution over the Wigley hull as compared to the inviscidwork of Uslu and Bal.51

Figure 13. Free-surface deformations for a Wigley hull atFr = 0:4 compared with the results of Uslu and Bal.51

Table 4. Hydrostatic properties of KCS hull in model scale.

Lbp (m) 4.014Lwl (m) 3.826B (m) 0.530T (m) 0.178Service speed 1.5839 m2/s




at lower Froude numbers. This difference may be dueto unsteady effects occurring at this region of the flow.The wave elevation on the hull may still be dependenton time. The effects of sinkage and trim also play a role;however, in this article, the ship is fixed. The bias ofwave resistance results with the experiments is generallyaccounted to the 2-degree-of-freedom movement of thevessel in the experiments. In numerical calculations, thevessel is bound to be at its fixed position in the flow,

while in the experiments the vessel is free to heave andpitch. This enforced stationary position of the vesselincreases the wave resistance of the ship. The same phe-nomenon was observed by Ogiwara and Kajatani,53

where they found out that there is an 8% increase intotal resistance of the Series 60 hull when the ship isforced to stay stationary in the experiments. This extraresistance is removed by Toda et al.54 and they havecalibrated their numerical results, reducing their

Figure 15. A general perspective view of the KRISO Containership: (a) bow side of the model, (b) stern of the model, (c) thewhole model45 and (d) the CAD model.

Figure 16. A perspective view of the hybrid mesh system around the vessel: (a) a broad perspective, (b) a close-up view of the hull,(c) a close-up view of the bow and (d) a close-up view of the stern.

Kinaci et al. 13



computational results by 8%. The flow around KCSwith 2 degrees of freedom is given in Simonsen et al.55

RANSE is very efficient in calculating the frictionalresistance coefficient as it can be seen from Figure 18.The calculated total resistance characteristics of the ves-sel seem compatible with the experimental results. Theform factor can be found via RANSE-based CFD andk is calculated in this study as

k=Cvp

Cfð18Þ

Viscous pressure and frictional resistances are com-puted by single phase analyses. k (the form factor)changes with the Reynolds number and this change canbe examined in Figure 19. The parameters in the formfactor equation are derived from calculated values inthis study. Therefore, the Reynolds number dependenceof the form factor is only dependent on the turbulence

model used. Different turbulence models may result indifferent form factors but the decision on which modelto be selected basically relies on the grid and the gridrelies on the computational capability. The range ofthis study was between Froude numbers 0.2 and 0.3(which corresponds to about Re’33106 � 73106),respectively. The form factor calculated in the experi-ments, implementing the method of Prohaska, was1.186. The calculated form factor at the ship’s servicespeed (at 24 knot) is 1.2083. The relative error betweenthe experimental form factor and the calculated formfactor is 1.8%.

The free-surface contours at Fr=0:26 is given inFigure 20. The Kelvin wave pattern can be clearlyobserved in the same figure. The computed and experi-mental wave elevation along the hull is given inFigure 21. A camera with a wide-angle lens is used inthis figure to closely show the elevations on the shiphull, therefore the ship seems bent. Experimental waveelevation could be understood following the ticks onthe hull. The bow wave formation obtained experimen-tally at the towing tank is given in Figure 22. The com-puted nominal wakes at the propeller disk are given inFigure 23 as compared with those measured byFujisawa et al.56 A satisfactory agreement is alsoobtained. The average nominal wake coefficient is, onthe other hand, found to be 0.75 with RANSE, whilethe experimental method returns a value of 0.77. Therelative error between experimental wake fraction andcomputational wake fraction is 2.6%.

Discussion

This work focuses on the prediction of wave resistanceby RANSE-based CFD. The generated results, usingthe opportunities of RANSE, seem in accordance withboth the results in the literature and the experimentsthat are carried out at Ata Nutku Ship Model Testing

Fr

Cw*1000

0.22 0.24 0.26 0.28 0.3 0.320

0.5

1

1.5

2

2.5

CFDEFD

Figure 17. Wave resistance coefficient of the hull, numericaland experimental.

Re

Cf*1000

3E+06 4E+06 5E+06 6E+06 7E+06 8E+063

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

CFDITTC 57

Figure 18. Frictional resistance coefficient of KCS.

Re

1+k

2.5E+06 5E+06 7.5E+061.15

1.175

1.2

1.225

1.25

Experimental Form Factor = 1.165

Figure 19. A linear form factor analysis via single phasesolutions by RANSE-based CFD.




Laboratory of Istanbul Technical University. This workhas two major contributions:

� The results of the methodology proposed in thisarticle are satisfactory for practical applications tocompute the wave resistance of ship-like bodies andthe free-surface deformations accordingly.

� The proposed method removes any flaws that mayarise from the calculation of the form factor by theITTC procedure.2 The methodology does not coverthe calculation of the form factor; instead, this isimplicitly done in this work. It is shown in this arti-cle that the implementation of this method removes

the Reynolds number dependence on the formfactor.

The study covers four different cases. These casesare selected carefully so that the prediction of waveresistance methodology by RANSE-based CFD coversunderwater bodies as well as surface-piercing bodies.The methodology has proven to generate good resultsfor slender bodies like a Wigley hull and for a well-rounded body like KCS. The method proposed in thisarticle may prove useful when there is a need to opti-mize a ship in terms of wave resistance which has sig-nificant share in total resistance at high speeds.

Figure 20. Free-surface contours of KCS at Fr = 0:26 by RANSE-based CFD.

x/Lwl

z/Lwl

0 0.2 0.4 0.6 0.8 1-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Free Surface Level

Figure 21. Wave elevation along KCS at Fr = 0:26 by RANSE-based CFD (top), experiment (bottom).45

Kinaci et al. 15



Conclusion

The flexibility of RANSE-based CFD paves the pathfor solving the flow around ship-like bodies either usingdouble-body assumptions or (along with VOF method)modeling the free surface. There are many parametersthat may be observed by single phase analysis. One ofthem is the viscous pressure resistance which helps todetermine the wave resistance of a ship. The proposedCFD methodology is an alternative way to the classicalmethod where the form factor is found with Prohaskamethod by subtracting the viscous resistance from totalresistance to calculate the wave resistance.

The classical method omits the dependency of formfactor on Reynolds number; however, in this methodol-ogy, the form factor does not need to be calculated.Single phase analysis automatically resolves the issue of‘‘zero wave resistance inclusion’’ inside the total resis-tance during the form factor calculation. In single phaseanalysis, the ratio of viscous pressure resistance to fric-tional resistance directly gives the form factor (1 + k)value of the subjected ship and it depends on theReynolds number. A quick linear calculation done by

single phase analyses leads to a compatible result withthe experiments. However, the form factor is not con-stant (as it is accepted in the experimental approach)and it is shown by CFD in this work that it is in fact afunction of Reynolds number.

Despite the conventional method that drives fromthe experiments that there is only one result for theform factor, this method includes the deviations thatmay happen in the form factor with increasingReynolds number. The methodology gives good resultsfor both submerged and surface-piercing benchmarkcases. It is also tested on KCS and the results onceagain promising. RANSE tends to return higher valuesfor wave resistance in lower Froude numbers and thisis considered to be due to Reynolds number gettingdigressed from the range that the used turbulencemodel supports. The turbulence models give goodresults in fully turbulent flows and if the flow is in tran-sition, the model may not work properly. As theFroude number is lowered, the Reynolds number is aswell lowered and the flow around the ship model maynot be fully turbulent. The discrepancy in lower

Figure 22. Experimental bow wave formation. Fr = 0:22 (left), Fr = 0:28 (right).45

Figure 23. Experimental56 and calculated nominal wakes at the propeller disk at Fr = 0:26.




Froude numbers will be investigated in a future workwith the effect of resolution for the free-surfacedeformation.

Acknowledgement

The authors would like to thank Cihad Delen for hishelp with the experiments.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research, authorship, and/or publi-cation of this article.

Funding

The author(s) received no financial support for theresearch, authorship, and/or publication of this article.

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Appendix 1

Notation

B breadthc length of the geometryFr Froude numberg gravitational accelerationh distance to the free surfaceLbp length between perpendicularsLwl length waterlineRe Reynolds numberS wetted areaSij mean strain-rate tensorT draftV velocity1+ k form factor

e rate of dissipationk turbulent kinetic energymt turbulence eddy viscosityr densitytij Reynolds stress tensor




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