Calculation Of Current Or Manoeuvring Forces Using A Viscous

Proceedings of ASME 28th International Conference on Ocean, Offshore and Arctic EngineeringOMAE2009

May 31-June 5, 2009, Honolulu, Hawaii

OMAE2009-79782

CALCULATION OF CURRENT OR MANOEUVRING FORCES USING AVISCOUS-FLOW SOLVER

Serge ToxopeusMARIN / Delft University of Technology

R&D Department, CFD ProjectsP.O. Box 28, 6700 AA Wageningen

The NetherlandsEmail: [email protected]

Guilherme VazMARIN

R&D Department, CFD ProjectsP.O. Box 28, 6700 AA Wageningen

The NetherlandsEmail: [email protected]

ABSTRACTTo investigate the capabilities of using a viscous-flow solver

to calculate current or manoeuvring forces on ship hulls, calcu-lations have been conducted for a test hull under several obliqueinflow conditions. All calculations were performed with a singlegrid topology to reduce the amount of grid generation time. Ver-ification and validation of the predicted loads and flow aroundthe hull has been performed through grid refinement studies andcomparison with experiments and results from previous calcula-tions. Furthermore, the influence of unsteady behaviour on theresults for large inflow angles is shown. Conclusions are drawnregarding the accuracy of the results and recommendations forimprovements and further work are given.

INTRODUCTIONThe application of viscous-flow solvers to calculate current

or manoeuvring forces on ship hulls has increased in the pastyears. Especially for submersibles, viscous-flow calculations arebecoming a useful alternative to conducting complex model tests.One of the main advantages is the absence of struts or stingersto support the model during the force measurements. However,much work is still to be done to demonstrate the capabilities andaccuracy of this kind of calculations.

In the present study, the solver FRESCO has been usedto calculate the forces on a test object. FRESCO is a newunstructured-grid hydrodynamic multi-purpose finite-volumeURANS solver for calculating viscous-flows around for exam-ple propellers with cavitation (Vaz et al. [1]), offshore structures(Waals et al. [2]), viscous free-surface flows (Vaz and Jaouen [3])and now ships in manoeuvring conditions. In this paper, the bare-hull DARPA SUBOFF submarine hull form has been selected astest case, because of the availability of extensive validation datafor field variables as well as for integral quantities. In litera-ture, several other studies concerning calculations on the bare-hull DARPA SUBOFF can be found, see e.g. Sung et al. [4,5,6],Bull [7], Jonnalagadda et al. [8], Bull and Watson [9] and Yang

and Lohner [10].Calculations have been conducted for straight flight as well

as for inflow angles ranging from 0◦ up to 90◦, representing purebeam inflow. The calculations for the different angles were con-ducted using the same grid. The change of inflow angle wasmodelled by adjusting the boundary conditions. This procedureminimises the time spent on grid generation.

Verification of the results has been conducted. By varyingthe grid density and solver parameters, the changes in the solu-tion as a result of these variations have been studied. Valida-tion of the flow field and the forces and moments has been per-formed. Comparison of the calculations with measurements andwith calculations conducted previously with the well-establishedsteady ship viscous-flow solver PARNASSOS (see Toxopeus [11])demonstrates the accuracy of the calculations. Furthermore, theinfluence of unsteady behaviour on the results for large inflowangles is shown. Conclusions will be drawn regarding the accu-racy of the results and recommendations for improvements andfurther work will be given.

NUMERICAL PROCEDURESCoordinate systems and definitions

The same definitions as used during the experiments wereadopted. The origin of the right-handed system of axes used inthis study is located at the intersection of the longitudinal axis ofsymmetry of the hull and the forward perpendicular plane, with xdirected aft, y to starboard and z vertically upward. Note that allcoordinates given in this paper are made non-dimensional withthe overall length Loa of the submarine (Loa = 4.356m). Thisfacilitates easy comparison with other results published in liter-ature. The velocity field V = (u,v,w) is made non-dimensionalwith the undisturbed velocity V0. The drift angle is defined byβ = arctan v

u , with u and v directed according to the x and y axesrespectively, which means that β is positive for flow coming fromport side.

All integral forces and moments on the hull are based on

1 Copyright c© 2009 by ASME

a right-handed system of axes corresponding to positive direc-tions normally applied in manoeuvring studies. This defini-tion was also used during the DARPA SUBOFF model tests.This means that the X force is directed forward, Y to star-board and Z vertically down. Similar to the results presentedby Roddy [12], all moments are given with respect to the cen-tre of gravity, which is located at 0.4621Loa aft of the noseof the model. Non-dimensionalisation is done with the lengthbetween perpendiculars (Lpp = 4.261m) using X ,Y,Z/ 1

2 ρV 20 L2

pp

and K,M,N/ 12 ρV 2

0 L3pp. The pressure coefficient Cp is defined as

Cp = (p− p0)/ 12 ρV 2

0 , p0 being the undisturbed pressure, and theskin friction coefficient C f as C f = |~τ|/ 1

2 ρV 20 .

In some of the results presented in this work, the circum-ferential angle α is used. This angle is defined according toFigure 1, with α = 0◦ corresponding to the leeward side andα = 270◦ to the keel for positive inflow angles.

y

z

er

eα

α

α = 270◦

α = 180◦

flow

PS SB

Figure 1. Definition of circumferential angle α

SolverIn this study, the flows were calculated with FRESCO, see

e.g. Vaz et al. [13] or Hoekstra et al. [14]. The code started as ajoint development by MARIN, HSVA and TUHH within the EUproject VIRTUE in 2005. FRESCO solves the multi-phase un-steady incompressible Reynolds-averaged Navier-Stokes equa-tions together with turbulence models and, if necessary, volume-fraction transport equations for each phase. The equations arediscretised using a flexible finite volume face-based discretisa-tion, which permits grids with elements with an arbitrary num-ber of faces (hexahedrals (structured grids), tetrahedrals, prisms,pyramides, etc.), and even h-refinement (hanging-nodes). Thecode is parallelised using MPI and sub-domain decomposition,and is targeted for Linux workstations, Linux Clusters and Super-Computers. It resembles in global lines the existing commercialCFD packages. However, contrary to the highly general commer-cial codes, FRESCO is intended to deal only with ship-building& offshore related flow problems, and is therefore optimised forand continuously applied to these types of flows. FRESCO hasalso some additional attractive points: 1) it is an in-house code,and therefore easy to trouble-shoot, extend, improve and cou-ple with other tools; 2) MARIN experience on CFD, free-surfacemodelling and cavitation modelling is incorporated in the code;3) no licence fees associated, which makes the computationscheaper. FRESCO is however a new code, which still has to ma-ture to the level of accuracy of PARNASSOS and therefore thecalculations need to be thoroughly validated.

Solver settingsMenter’s SST version of the two-equation k−ω turbulence

model [15] was selected for all calculations. The governing equa-tions were integrated down to the wall, i.e. no wall-functions areused. For the steady calculations, a 1st order time derivative to-gether with a time step size ∆t of 0.07s, was used. For a Reynoldsnumber of 1.2×107, this corresponds to about ∆t = tref/20, withtref = Loa/V0 = 1.575s. This time step size is too large to captureall time dependent fluctuations in the flow, but was adopted toaid the solution process to arrive to a steady solution. A higher-order (3rd order in structured grids) QUICK scheme was used todiscretise the momentum convection terms. For the turbulencequantities a first order upwind scheme was adopted for the con-vection terms. All diffusive terms were discretized by a 2nd orderscheme.

For the present study, all solutions were run until the max-imum change of the pressure coefficient ∆Cp (the so-called L∞

norm) between successive iterations had dropped below 1 ·10−5.

Boundary conditionsAt the hull surface a no-slip boundary condition is applied.

Symmetry conditions are used on the longitudinal plane(s) ofsymmetry. The exterior domain is formed by a half sphere (sym-metry is assumed between the bottom and top of the hull). Theboundary condition is set by calculating the angle between theinflow and the normal vector on a cell face on the boundary. Forangles larger than 90◦, the boundary condition of the cell face isset to inflow, otherwise to outflow. A Neumann boundary condi-tion stating that the normal gradient is equal to zero is applied onoutflow faces for all quantities.

The velocity components in the inflow faces are set to theundisturbed velocity components. The turbulent intensity I =u′/V0 is set to 0.05 and the eddy viscosity µt to 10µ . The valuesapplied at the inflow plane are also used as initial condition in thecomplete flow domain.

GRID GENERATIONHull form

The submarine form considered is the DARPA SUBOFFhull form, as described in Groves et al. [16]. For this hull, amplecomparison material is available. For the current work presentedin this paper, only the bare hull, i.e. without sail, planes and pro-peller, is used. This corresponds to Configuration 3 as defined inRoddy [12] and to configuration AFF-1-* as defined in Liu andHuang [17]. The main particulars of the DARPA SUBOFF arespecified in Table 1.

GridA structured grid has been generated around the hull form

using in-house tools. A body-fitted non-orthogonal O-O typegrid has been used. The bare hull of the submarine is axisym-metric and therefore the grid was made axisymmetric as well,by first generating the grid in 2D and then rotating the 2D gridaround the axis of symmetry. Hexahedral cells are used every-where except at the symmetry line where prisms are used insteadof degenerated hexahedral cells. The grid was strongly stretchedtowards the hull to capture the strong gradients in the boundary


Table 1. Main particulars of bare hull DARPA SUBOFF submarine

Description Symbol Magnitude Unit

Length overall Loa 4.356 m

Length between perpendiculars Lpp 4.261 m

Maximum hull radius Rmax 0.254 m

Centre of buoyancy (aft of nose) FB 0.4621 Loa -

Volume of displacement ∇ 0.708 m3

Wetted surface Swa 5.998 m2

layer. Grid-refinement was also applied at the bow and at thestern for this reason. In Figure 2 the 2D grid (coarsened for pre-sentation) is shown.

Apart from studying the feasibility of applying FRESCO tothe calculation of a submarine in oblique flow, one of the aimsof the present study was also to investigate whether the flow atdifferent drift angles could be calculated using a grid with thesame topology for all angles. The idea was to generate one gridand model the inflow angle by adjusting the boundary conditionson the exterior of the grid. To facilitate this, the exterior of thecomputational domain was formed by a half sphere. To avoidinteraction between the outer boundary, where undisturbed ve-locities are enforced, and the flow around the hull, the domainsize was large. The radius of the spherical exterior was 3.44 ·Loa.The maximum deviation from orthogonality in the grid was 51◦.

In Figure 3, a view of the grid used for this study is pre-sented. The bow is directed to the left of the figure. Table 2presents an overview of the number of nodes np used in theFRESCO calculations compared to the grid used previously inPARNASSOS, using a curvilinear coordinate system notation (seeFigure 4).

The number of points given are for the non-zero drift an-gle calculations. For zero drift, only a quarter of the hull wasmodelled and therefore only half the number of grid nodes wasused. For this study, most calculations were conducted usinggrid number 5. Additional calculations were done with othergrid densities, to investigate the grid dependency. Grid number5 is coarse compared to the PARNASSOS grids and also com-

Figure 2. 2D grid around the submarine (coarsened by factor 2)

Figure 3. 3D grid around the submarine (coarsened by factor 2), β 6= 0◦

Figure 4. Definition of curvilinear coordinate system (bow to the right)

pared to grids used in similar studies presented in literature whenwall-functions are not applied. For grid number 2, the number ofnodes on the hull surface is almost the same as for the PARNAS-SOS grid. However, the total number of nodes is less than halfthe number of nodes in the PARNASSOS calculations. This showsthat with O-O type grids, a large number of cells can be placednear the hull surface while keeping the total number of cells rel-atively low compared to H-O type grids. However, with an O-Otopology, the cell size in the wake of the hull also increases fastwhen moving away from the wall, and wake resolution is thennot optimal.

REVIEW OF THE CALCULATIONSCalculations for inflow angles ranging from 0◦ to 90◦ were

conducted for the DARPA SUBOFF bare hull form. In this paper,the following abbreviations are used to identify the results:

NL-TNT PARNASSOS with HO-grid and TNT tur-bulence model [18, 19, 11]

NL-Axi PARNASSOS with axisymmetric grid andMNT turbulence model [19, 11]

NL-Axi SST PARNASSOS with axisymmetric grid andSST turbulence model [19, 11]

NL-fresco-alt FRESCO with axisymmetric OO-grid andSST turbulence model

The parameter y+2 is the largest non-dimensional distance


Table 2. Comparison of grid densities between FRESCO and PAR-NASSOS (half ship)

Grid Id nξ nη nζ np×10−3

- NL-Axi 361 97 49×2 3432

1 NL-fresco-alt 241 105 113 2859

2 NL-fresco-alt 201 87 93 1626

3 NL-fresco-alt 171 75 81 1039

4 NL-fresco-alt 138 60 65 538

5 NL-fresco-alt † 121 53 57 366

6 NL-fresco-alt ] 101 31 47 147

7 NL-fresco-alt \ 86 38 41 134

8 NL-fresco-alt †† 61 27 29 48

NL-Axi grid used for PARNASSOS calculationsNL-fresco-alt grid used for FRESCO calculations† based on grid 1 by coarsening with factor 2] based on grid 2 by coarsening with factor 2\ based on grid 3 by coarsening with factor 2†† based on grid 1 by coarsening with factor 4

y+ of the first grid node to the wall. In order to capture the fullboundary layer and directly apply the no-slip condition at thewall, y+ should be equal or less than 1. Although most of thecalculations were conducted with a relatively coarse grid, the y+

2values in these calculations were between 0.7 and 0.9, dependingon the drift angle. For the finest grid, y+

2 was 0.33 for β = 0◦ and0.41 for β = 18◦.

All calculations were conducted for deeply submerged con-dition, so that free surface effects are absent. Based on referenceinflow velocities V0 on model scale of 2.7658 and 3.2268 m/s,the Reynolds numbers Re corresponded to respectively 12×106

(used for β = 0◦ and 2◦ only) and 14×106 (used for all drift an-gles). In this paper, the Reynolds number is based on the lengthoverall of the model.

NUMERICAL RESULTSIterative error

In Figure 5, the iterative convergence histories for thestraight-ahead sailing condition and for β = 18◦ are presented.All results are given in non-dimensional values. It is seen thatafter some initial transients, the L∞ norms of the pressure andvelocity components (top graph) converge smoothly to below1 ·10−7 for β = 0◦. The convergence history of the integral vari-ables shows convergence to well below 5 · 10−8. The conver-gence for the calculations with non-zero drift angles is similar,but for increasing drift angles, some stagnation in the conver-gence is found. Further examination of the different iterationsshows that the stagnation is caused by minor changes in the ve-locities, probably caused by instationary effects. The changes arewithin 0.01% of the final values.

Based on the aforementioned observations, iterative conver-gence errors in the calculations are assumed to be negligible withrespect to discretisation or modelling errors.

-10

-8

-6

-4

-2

0

0 1000 2000 3000 4000

log

L ∞

Re = 14×106,β = 0◦

∆u∆v∆w

∆cp

-10

-8

-6

-4

-2

0

0 1000 2000 3000 4000

time steps

log |∆X |log |∆Y |log |∆Z|log |∆K|log |∆M|log |∆N|

Figure 5. Convergence history of local (top) and integral (bottom) quan-tities, β = 0◦

Discretisation errorIn order to determine and demonstrate the accuracy and reli-

ability of solutions of viscous flow calculations, grid dependencystudies are very important. Several methods for uncertainty anal-ysis are available in literature. In the present work, the methodproposed by Eca and Hoekstra [20] is followed. The full detailsand background for the followed procedure for the uncertaintystudy can be found in their paper. A short summary of the pro-cedure can also be found in Toxopeus [11].

For Re = 14× 106, the discretisation error has been inves-tigated. In Table 3 and Figure 6, the results for β = 0◦ are pre-sented. The characteristic grid size hi is calculated for each gridi by hi = 1/(nξi − 1). The relative step-size hi/h1 indicates thecoarseness of the grid with respect to the finest grid h1. For in-stance h5/h1 = 2 indicates that grid 5 is two times coarser thangrid 1. The value of the solution obtained at the finest grid is indi-cated by φ1. The solution φ extrapolated to zero relative step-size(φ0) is an estimate of the solution for zero cell size 1. The graphsshow that scatter exists in the data: the data points are not ex-actly aligned according to the curve. Reasons for this might be

1Due to negligible but non-zero contributions of iterative and round-off errorsand the fact that the observed order of convergence may change during refine-ment, the extrapolated solution may however not be equal to the exact solutionobtained on an infinitely dense grid.


e.g. the non-evenly spaced cell nodes, the use of numerical lim-iters or lack of perfect geometrical similarity between the grids.For this high Reynolds number, i.e. when convection dominates,and when using an unstructured-grid QUICK scheme for con-vective fluxes, it is expected that FRESCO will be second orderaccurate. Based on the verification study, it is found that the ob-served order of convergence p is just below 2 indicating that theconvergence with grid refinement follows the expected order ofaccuracy of FRESCO. The uncertainty U in X is 2.2% which isjudged to be small.

Compared to the uncertainty study conducted using PAR-NASSOS calculations, see [11], it is found that uncertainty val-ues in the same order of magnitude are found: for PARNASSOSUX = 2.1% when using the TNT turbulence model. Interestingly,in FRESCO the absolute value of the friction component X f in-creases with increasing grid density, while in PARNASSOS theopposite trend is found. As expected, the pressure componentXp reduces in absolute value for both solvers when a finer grid isused.

Table 3. Uncertainty analysis, X, β = 0◦

Item φ0 φ1 Uφ p

X -1.05×10-3 -1.05×10-3 2.2% 1.75

Xp -8.78×10-5 -9.91×10-5 18.5% 1.92

X f -9.60×10-4 -9.52×10-4 2.2% 1.97

In Tables 4 through 6 and Figures 7 and 8, a selection of theresults for β = 18◦ are presented. In this case, the observed orderof convergence p ranges from 0.46 for X f to 3.91 for Xp. Thismay indicate that a finer grid needs to be used to obtain a reliablesolution.

For the transverse force Y and yaw moment N the observedorders of convergence are between 0.8 and 1.7, which may in-dicate that for the transverse force and yawing moment the griddensity is acceptable. The uncertainty in X is found to be rel-atively large. The large value is caused by the fact that for theoverall force X monotonic convergence was not obtained. How-ever, the value is acceptable from an engineering viewpoint. Theuncertainty in the overall transverse force or yawing moment isjudged to be small.

The verification study for FRESCO has been conducted forβ = 18◦ while for PARNASSOS the results for β = 10◦ were used.Therefore, the results cannot be compared directly. However,qualitatively, it is found that for FRESCO the uncertainty in Yis considerably smaller than for PARNASSOS due to monotonicconvergence of the values in FRESCO and monotonic divergencein PARNASSOS. The uncertainties in X and N are of similar mag-nitudes.

It is also concluded from this study, that using grid 5 (orrelative step size equal to 2) the numerical solutions for drift an-gles ranging up to 18◦ are within a plausible uncertainty region,where the discretization error is under control. For the rest ofthe results presented in the paper this grid will then be used withconfidence on the reliability of the numerical solution.

Figure 6. Discretisation uncertainty analysis, β = 0◦

Table 4. Uncertainty analysis, X, β = 18◦

Item φ0 φ1 Uφ p

X - -8.22×10-4 6.3% 1

Xp 2.62×10-4 2.41×10-4 24.7% 3.91

X f -1.18×10-3 -1.06×10-3 5.0% 0.46

1 Oscillatory convergence

Unsteady flowIn all calculations, the flow converged to a steady flow, al-

though time stepping was used. The main reasons for the ab-sence of unsteady effects are the poor grid resolution away fromthe hull surface and the coarse time step. Furthermore, unsteady


Table 5. Uncertainty analysis, Y, β = 18◦

Item φ0 φ1 Uφ p

Y 5.52×10-3 5.66×10-3 3.1% 1.17

Yp 5.20×10-3 5.36×10-3 3.7% 1.15

Y f 3.18×10-4 3.01×10-4 5.5% 0.85

Table 6. Uncertainty analysis, N, β = 18◦

Item φ0 φ1 Uφ p

N 3.43×10-3 3.41×10-3 0.7% 1.68

Np 3.42×10-3 3.40×10-3 0.7% 1.71

N f - 1.75×10-5 7.2% 2

2 Monotonic divergence

Figure 7. Discretisation uncertainty analysis, Y , β = 18◦

behaviour is suppressed by the assumption that the flow is sym-metric for the top and lower halves of the spherical domain. Thiseffectively prohibits vortex shedding in vertical direction aroundthe submarine hull form. However, for large inflow angles, it isexpected that vortex shedding will occur and that this might in-fluence the forces and moments acting on the hull. Although acomplete investigation into the influence of unsteady flow on theforces on the submarine is outside the scope of the present study,two additional calculations for β = 90◦ were conducted:

Figure 8. Discretisation uncertainty analysis, N, β = 18◦

• Full domain (also top half sphere modelled), same time iter-ation settings as before. This calculation was conducted toinvestigate the influence of the symmetry assumption.

• Full domain (also top half sphere modelled), improved timeiteration settings. This calculation was conducted to investi-gate whether with improved time resolution different resultswould be obtained.

The forces and moments for the first full domain calculationshow an oscillating behaviour, as presented in Figure 9. How-ever, the differences in the forces and moments predicted usingthe half domain and full domain, see Table 7, are judged to berelatively small, compared to the forces and moments found forother drift angles. When looking at the transverse velocity fieldaround the hull, pronounced vortex shedding is not clearly ob-served.

The calculation with improved time resolution was made us-ing a 2nd order implicit three-time-level integration scheme incombination with a time step ∆t of 0.015s or ∆t = tref/90. Ad-ditionally, 50 outer iterations were conducted for each time step,to sufficiently reduce the residuals between the time steps (about4 orders of magnitude, instead of 2).

In this case, as seen in Table 7 and Figure 10, the X force re-duces considerably, while the yawing moment N increases. In theout-of-plane forces, see the bottom graphs in Figures 9 and 10,a large change is seen compared to the calculation with less ac-curate time resolution: high-frequency oscillations of especiallythe vertical force are observed, caused by vortex shedding. Closeinspection of the solution shows that the pitch moment is in oppo-


site phase compared to the vertical force. This is caused by oscil-lating pressure fields in the bow area, due to the vortex shedding.This vortex shedding can also be seen in the transverse velocityfield. In the present calculation, however, the grid density in thewake is quite poor due to strong stretching of the cells towardsthe hull surface to capture the boundary layer adjacent to the hull.To obtain even more accurate vortex shedding, the grid should berefined in the wake region as well.

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0 1000 2000 3000 4000 5000

Re = 14×106,β = 90◦ (full domain)

XY/20

N

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0 1000 2000 3000 4000 5000

time steps

Z/2KM

Figure 9. Convergence history of forces and moments, β = 90◦, fulldomain, original settings

COMPARISON WITH EXPERIMENTS, β = 0◦

Integral valuesExperimental force measurement results are available for the

straight-ahead condition and were published by Roddy [12]. Theexperiments were conducted in the towing basin of the DavidTaylor Research Center. During the tests, the model was sup-ported by two struts. The speed used during the experimentsresulted in a Reynolds number of 14× 106. For this conditionthe experimental value of the longitudinal force was found to be:

X = average(Xtest1,Xtest2) = average(−1.061,−1.051)×10−3

= −1.056×10−3

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0 1000 2000 3000 4000 5000

Re = 14×106,β = 90◦ (full domain, time)

XY/20

N

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0 1000 2000 3000 4000 5000

time steps

Z/2KM

Figure 10. Convergence history of forces and moments, β = 90◦, fulldomain, time-accurate

Table 7. Forces and moments (× 1000), β = 90◦

Domain Half Full Full

Time Coarse Coarse Accurate

X 1.766 1.560 0.820

Y 44.653 44.029 44.489

Z - 0.954 0.729

K - 0.000 0.000

M - -0.024 0.080

N 0.531 0.345 1.579

A detailed uncertainty analysis of the force measurements wasnot conducted. However, to obtain an estimate of the uncertaintyin the experimental data, the uncertainty UD in the experimentaldata is estimated using a factor of safety of 1.25 by

UD = 1.25× abs(Xtest1−Xtest2) = 1.25×10−5 = 1.2%×X

Table 8 presents the longitudinal force components obtainedfrom the calculations. As can be expected for submarine hull


forms, the largest part (about 90%) of the total resistance iscaused by friction resistance. The difference εX between theFRESCO prediction of X and the measurement is about 0.5%,which is judged to be very good for practical applications whenalso the uncertainty in the experimental data is taken into con-sideration. Furthermore, it is found that the total resistance pre-dicted by FRESCO is close to the value predicted by PARNASSOS(NL-Axi): the skin friction coefficient is almost the same and thedifference is mainly caused by a slightly higher pressure coeffi-cient in FRESCO.

The influence of the Reynolds number can be observed inthe calculations. As expected, a slightly higher Reynolds num-ber results in a slightly lower resistance coefficient, due to thedecrease in frictional resistance coefficient.

Table 8. Longitudinal force X , Re = 1.4×107, β = 0◦

Solver Grid Integral values ×103

X X f Xp εX (%)Exp (DTRC) - -1.061 - - -Exp (DTRC) - -1.051 - - -

Mean µexp -1.056 - - -ITTC-57 - -0.936 - -

Schoenherr - -0.918 - -Katsui - -0.905 - -

Grigson - -0.932 - -NL-Axi - -1.027 -0.958 -0.069 -2.7NL-fresco-alt 1 -1.051 -0.952 -0.099 -0.5NL-fresco-alt 2 -1.040 -0.943 -0.097 -1.5NL-fresco-alt 3 -1.032 -0.935 -0.097 -2.3NL-fresco-alt 4 -1.022 -0.920 -0.102 -3.2NL-fresco-alt 5 -1.054 -0.926 -0.128 -0.2NL-fresco-alt 6 -1.017 -0.893 -0.124 -3.7NL-fresco-alt 7 -1.035 -0.883 -0.152 -2.0NL-fresco-alt 8 -1.144 -0.844 -0.300 8.3

Local quantitiesThe experimental values are obtained from flow field and

pressure measurements in a wind tunnel conducted by Huang etal. [21]. These experiments were conducted at a Reynolds num-ber of 12× 106. Figure 11 shows comparisons of the pressureand skin friction coefficients along the hull. It should be notedthat the results for NL-Axi were obtained for a Reynolds numberof 14×106.

These graphs show that the differences in pressure coeffi-cient between the FRESCO and PARNASSOS results are negligi-ble, which corresponds to a small difference in the longitudinalpressure coefficients Xp. For the skin friction coefficient, it isseen that the FRESCO results with the SST model are slightlycloser to the experimental data than the PARNASSOS results us-ing the TNT model, except for some points in the aft ship.

The predicted distribution of the pressure coefficient is closeto the experiments. The trends in the predicted distribution ofthe friction coefficient correspond well to the trends found in theexperiments. Although some discrepancies at the bow and sternarea are found, it is concluded that the prediction of the pres-sure and skin friction coefficients is good. It is noted that the

Figure 11. Pressure (top) and skin friction (bottom) coefficients alongthe hull, β = 0◦

discrepancies at the bow and stern were also present in calcu-lation results published by Bull [7] and Yang and Lohner [10]and in results submitted for a collaborative CFD study withinthe Submarine Hydrodynamics Working Group (SHWG, seewww.shwg.org) [22].

Figure 12 presents the streamwise and radial velocities atspecific longitudinal locations in the aft part of the hull.

In these graphs, the difference between the FRESCO andPARNASSOS results is considered to be small. Comparing thecomputed results with the experiments, it is observed that thetrends in the development of the boundary layer are very wellpredicted by both solvers, but quantitative discrepancies are seen.Especially the magnitudes of the radial velocities are different. Itis seen that in the experiments the radial velocity (positive cor-responds to a radial velocity away from the wall) changes signbetween (r−R0)/Rmax = 2 and (r−R0)/Rmax = 0.8, suggestingoutward radial flow in the far field. This may be caused by theuse of an open-jet wind tunnel.

Reynolds shear stressesIn this study, also the correlation between the measured and

the predicted Reynolds shear stresses is investigated by com-parison of −V ′xV ′r

V 20

. Following the eddy-viscosity assumption, the

Reynolds stresses are defined by

−V ′i V ′j = 2 ·νt ·Si j

νt being the eddy viscosity, and the strain rate tensor Si j beingcalculated by

Si j =12

(∂ui

∂x j+

∂u j

∂xi

).


http://www.shwg.org

x = 0.904Loa

x = 0.927Loa

x = 0.956Loa

x = 0.978Loa

Figure 12. Axial (lines in right-handed parts of each graph) and radial(lines in left-handed parts of each graph) velocities, β = 0◦

For α = 180◦ and α = 270◦ (see Figure 1), the V ′xV ′r componentsare given respectively by,

−V ′xV ′r =−νt ·(

∂u∂ z

+∂w∂x

)and −V ′xV ′r = νt ·

(∂u∂y

+∂v∂x

).

In Figure 13 and Figure 14, the Reynolds shear stresses forfour longitudinal stations are presented.

It is observed that curves representing the FRESCO resultscorrespond well with the measurements, except for the decay ofthe Reynolds stresses further away from the hull. The resultsare however better than for PARNASSOS with TNT, which under-predicts the peak of the distribution and shows even slower decay

x = 0.904Loa

x = 0.927Loa

Figure 13. Reynolds shear stress, β = 0◦, x/Loa = 0.904,0.927

of the stresses.

x = 0.956Loa

x = 0.978Loa

Figure 14. Reynolds shear stress, β = 0◦, x/Loa = 0.956,0.978

COMPARISON WITH EXPERIMENTS, β 6= 0◦

Integral valuesFigure 15 gives an impression of the results of the calcula-

tions for the bare hull DARPA SUBOFF for β = 18◦. In thispicture a pair of vortices on starboard is seen. These vorticesmodify the pressure distribution on the submarine and also thewake field of the submarine completely.


Figure 15. Impression of calculation results for bare hull DARPA SUB-OFF. Axial velocity contours, pressure contours on the hull, β = 18◦

Experimental results for oblique angles were published byRoddy [12]. The experiments were conducted at a Reynoldsnumber Re of 14× 106. Figure 16 presents the force and mo-ment components obtained from the calculations and the valuesfrom the experiments for oblique inflow.

In Tables 9 through 11 the results for β = 18◦ are shown.The comparison shows that FRESCO predicts a longitudinalforce X of the same order as magnitude as the value found withPARNASSOS using the SST turbulence model. Compared to thePARNASSOS MNT results, a considerable improvement is found(see Toxopeus [11] for a discussion on the differences betweenthe MNT and SST calculations with PARNASSOS). The deviationεX from the measurement is about 8%, which is within the un-certainty band of the measurements. The trends in the transverseforce Y and yaw moment N and the de-stabilising arm N/Y arepredicted reasonably well, but slightly less good than the PAR-NASSOS SST results. This may be caused by the coarse gridaway from the wall used in FRESCO.

Although experimental validation data are not available, cal-culations for large inflow angles were also conducted to inves-tigate the convergence behaviour and consistency of the forcesand moments calculated by FRESCO. It appeared that withoutany modification of the input files, reasonable results could beobtained. As seen in Figure 16, the trends in the forces and mo-ments are consistent and appear to be realistic. However, as dis-cussed previously, unsteady behaviour of the flow influences theflow and loads on the ship at the large drift angles and thereforethe results should only be interpreted qualitatively.

Local quantities at leeward symmetry planeFigure 17 presents comparisons of the pressure coefficient

along the hull, the axial Vx, tangential Vα and radial Vr veloci-ties and Reynolds shear stress respectively, given for the leewardsymmetry plane (α = 0◦, see Figure 1) located at x = 0.978Loa.The experimental values of the pressure distribution and the flowfield and Reynolds stresses were obtained using an inflow an-gle of β = 2◦ and for a Reynolds number of 12× 106. All cal-culations were therefore done for this Reynolds number, exceptfor the NL-Axi calculations which were obtained for a Reynolds

Figure 16. Force and moment coefficients against drift angle

number of 14×106.These graphs show that the distribution of the pressure co-

efficient along the length of the ship and the velocity distributionat the stern is quite well represented. The difference between theFRESCO and PARNASSOS results is considered to be small. Thedistribution of the Reynolds shear stress predicted by FRESCOshows good correspondence with the measurements.

CONCLUSIONSCalculations on the unappended hull-form of the DARPA

SUBOFF sailing straight ahead and at oblique motion were con-ducted in order to verify the accuracy of the predictions usingthe new viscous-flow solver FRESCO. Based on this study,it is concluded that FRESCO is well capable of predicting the


Table 9. Longitudinal force X , Re = 1.4×107, β = 18◦


X X f Xp εX (%)Exp (DTRC) - -0.670 - - -Exp (DTRC) - -0.852 - - -

Mean µexp -0.761 - - -NL-Axi - -1.071 -1.103 0.032 40.8NL-Axi SST - -0.813 -1.093 0.279 6.8NL-fresco-alt 1 -0.822 -1.063 0.241 8.0NL-fresco-alt 2 -0.814 -1.053 0.239 7.0NL-fresco-alt 3 -0.810 -1.044 0.234 6.4NL-fresco-alt 4 -0.817 -1.029 0.212 7.3NL-fresco-alt 5 -0.827 -1.020 0.193 8.7NL-fresco-alt 6 -0.959 -1.022 0.063 26.0NL-fresco-alt 7 -1.056 -1.017 -0.039 38.7NL-fresco-alt 8 -1.326 -0.999 -0.327 74.3

Table 10. Transverse force Y , Re = 1.4×107, β = 18◦


Y Y f Yp εY (%)Exp (DTRC) - 7.355 - - -Exp (DTRC) - 7.438 - - -

Mean µexp 7.397 - - -NL-Axi - 6.241 0.320 5.921 -15.6NL-Axi SST - 6.309 0.306 6.003 -14.7NL-fresco-alt 1 5.661 0.301 5.359 -23.5NL-fresco-alt 2 5.692 0.299 5.394 -23.0NL-fresco-alt 3 5.722 0.296 5.426 -22.6NL-fresco-alt 4 5.792 0.291 5.501 -21.7NL-fresco-alt 5 5.828 0.288 5.540 -21.2NL-fresco-alt 6 5.793 0.286 5.508 -21.7NL-fresco-alt 7 5.856 0.282 5.574 -20.8NL-fresco-alt 8 6.296 0.277 6.019 -14.9

Table 11. Yawing moment N, Re = 1.4×107, β = 18◦


N N f Np εN (%)Exp (DTRC) - 2.986 - - -Exp (DTRC) - 2.939 - - -

Mean µexp 2.962 - - -NL-Axi - 3.407 -0.041 3.448 15.0NL-Axi SST - 3.296 -0.041 3.337 11.3NL-fresco-alt 1 3.414 0.018 3.397 15.2NL-fresco-alt 2 3.406 0.017 3.389 15.0NL-fresco-alt 3 3.399 0.017 3.382 14.7NL-fresco-alt 4 3.384 0.017 3.366 14.2NL-fresco-alt 5 3.371 0.017 3.354 13.8NL-fresco-alt 6 3.417 0.017 3.399 15.3NL-fresco-alt 7 3.452 0.017 3.435 16.5NL-fresco-alt 8 3.517 0.017 3.500 18.7

flow around a bare-hull submarine. Comparison of the predictedresistance with experimental data gives very good agreement.The difference between the calculations and the measurementsis −0.5% for the straight-ahead sailing condition. This differ-ence is well within the estimated uncertainty band around theexperimental value and the numerical uncertainty.

VxVr Vα

Figure 17. Pressure coefficients along the hull (top), axial, tangentialand radial velocities (middle) and Reynolds shear stress (bottom), β = 2◦

For oblique motion, encouraging results are found for thepredicted trends in the forces and yawing moment as a func-tion of the oblique inflow angle. Comparisons between the re-sults from the new code FRESCO and the well established solverPARNASSOS show that only small differences exist between theresults of the two solvers.

The approach adopted to model the oblique flow by chang-ing the boundary conditions while using one common O-O typegrid has proven to be feasible for this hull form and thereforelarge amounts of grid generation time could be saved. However,for common applications, the wake is probably not sufficientlyresolved and therefore it is recommended to study the feasibilityof this approach for other ships as well.

FRESCO can be used to calculate the flow around the hullfor large inflow angles, both using a steady or unsteady computa-tion approach. For these large inflow angles, it was seen that un-steady vortex shedding influences the flow field around the hulland the forces and moments acting on the submarine. However,to obtain fully accurate unsteady results, adequately small timesteps and fine grid resolution on all possible shear-layers is re-quired. This has to be further investigated.


ACKNOWLEDGMENTPart of the work presented here was funded through TNO

Defence, Security and Safety within the framework of Pro-gramma V705 carried out for DMO of the Royal NetherlandsNavy. Their support is greatly acknowledged.

Another part of the work conducted for this article wasfunded by the Commission of the European Communitiesthrough the Integrated Project VIRTUE under grant 516201 inthe sixth Research and Technological Development FrameworkProgramme (Surface Transport Call).

REFERENCES[1] Vaz, G., Hoekstra, M., and Windt, J., 2007. Validation

Work on the Delft Twisted Wing. Wetted and CavitatingFlow. Tech. Rep. WP4-D4.2.3, VIRTUE, December.

[2] Waals, O., Vaz, G., Fathi, F., Ottens, H., Le Souef, T., andKwong, K., 2009. “Current Affairs - Model Tests, Semi-Empirical Predictions and CFD Computations for CurrentCoefficients of Semi-Submersibles”. In Proceedings ofOMAE2009, Honolulu, Hawaii, USA, June.

[3] Vaz, G., Jaouen, F., and Hoekstra, M., 2009. “Free-Surface Viscous Flow Computations. Validation of URANSCode FRESCO”. In Proceedings of OMAE2009, Honolulu,Hawaii, USA, June.

[4] Sung, C.-H., Griffin, M., Tsai, J., and Huang, T., 1993. “In-compressible flow computation of forces and moments onbodies of revolution at incidence”. 31st Aerospace SciencesMeeting and Exhibit, AIAA-1993-787, January.

[5] Sung, C.-H., Fu, T., Griffin, M., and Huang, T., 1995.“Validation of incompressible flow computation of forcesand moments on axisymmetric bodies at incidence”. 33rd

Aerospace Sciences Meeting and Exhibit, AIAA-1995-528,January.

[6] Sung, C.-H., Fu, T.-C., and Griffin, M., 1996. “Valida-tion of incompressible flow computation of forces and mo-ments on axisymmetric bodies undergoing constant radiusturning”. 21st Symposium on Naval Hydrodynamics, June,pp. 1048–1060.

[7] Bull, P., 1996. “The validation of CFD predictions ofnominal wake for the SUBOFF fully appended geometry”.21st Symposium on Naval Hydrodynamics, June, pp. 1061–1076.

[8] Jonnalagadda, R., Taylor, L., and Whitfield, D., 1997.“Multiblock multigrid incompressible RANS computationof forces and moments on appended SUBOFF configura-tions at incidence”. 35th Aerospace Sciences Meeting andExhibit, AIAA-1997-624, January.

[9] Bull, P., and Watson, S., 1998. “The scaling of highReynolds number viscous flow predictions for appendedsubmarine geometries”. 22nd Symposium on Naval Hydro-dynamics, August, pp. 1000–1014.

[10] Yang, C., and Lohner, R., 2003. “Prediction of flows overan axisymmetric body with appendages”. 8th InternationalConference on Numerical Ship Hydrodynamics, September.

[11] Toxopeus, S., 2008. “Viscous-flow calculations for barehull DARPA SUBOFF submarine at incidence”. Inter-national Shipbuilding Progress, DOI: 10.3233/ISP-2008-0048, 55(3), December, pp. 227–251.

[12] Roddy, R., 1990. “Investigation of the stability and con-trol characteristics of several configurations of the DARPASUBOFF model (DTRC Model 5470) from captive-modelexperiments”. Report No. DTRC/SHD-1298-08, Septem-ber.

[13] Vaz, G., Hoekstra, M., Hafermann, D., and Schmode, D.,2006. Definition Study of MARIN/HSVA Code on Mathe-matical and Numerical Models and Solution Strategy. Tech.Rep. WP4-D4.2.1, VIRTUE, February.

[14] Hoekstra, M., Vaz, G., Abeil, B., and Bunnik, T., 2007.“Free-surface flow modelling with interface capturing tech-niques”. MARINE International Conference on Computa-tional Methods in Marine Engineering, June, pp. 197–200.

[15] Menter, F., 1997. “Eddy viscosity transport equations andtheir relation to the k− ε model”. Journal of fluids engi-neering, Vol. 119, December, pp. 876–884.

[16] Groves, N., Huang, T., and Chang, M., 1998. “Geometriccharacteristics of DARPA SUBOFF models (DTRC ModelNos. 5470 and 5471)”. Report No. DTRC/SHD-1298-01,March.

[17] Liu, H.-L., and Huang, T., 1998. “Summary of DARPASUBOFF experimental program data”. Report No.CRDKNSWC/HD-1298-11, June.

[18] Toxopeus, S., 2006. “Viscous-flow calculations for DARPASUBOFF submarine”. MARIN Report 20294-1-CPM, Au-gust.

[19] Toxopeus, S., 2007. “Viscous-flow calculations for barehull DARPA SUBOFF submarine at incidence”. SHWGCollaborative CFD Excercise, Bare hull DARPA SUBOFFsubmarine at straight flight and drift angle, November,pp. 117–126.

[20] Eca, L., and Hoekstra, M., 2005. “On the influence of gridtopology on the accuracy of ship viscous flow calculations”.5th Osaka Colloquium on Advanced CFD Applications toShip Flow and Hull Form Design, March, pp. 1–10.

[21] Huang, T., Liu, H.-L., Groves, N., Forlini, T., Blanton, J.,and Gowing, S., 1992. “Measurements of flows over an ax-isymmetric body with various appendages in a wind tunnel:the DARPA SUBOFF experimental program”. 19th Sympo-sium on Naval Hydrodynamics, August, pp. 312–346.

[22] Toxopeus, S., ed., 2007. SHWG collaborative CFD ex-cercise - Bare hull DARPA SUBOFF submarine at straightflight and drift angle, MARIN Report 21668-1-CPM.


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