28th Symposium on Naval HydrodynamicsPasadena, California, 12-17 September 2010

Adaptive Grid RefinementApplied to RANS Ship Flow Computation

J. Wackers, K. Ait Said, G.B. Deng, P. Queutey, M. Visonneau(Laboratoire de Mécanique des Fluides, CNRS / Ecole Centrale de Nantes, France),

I. Mizine (CSC Advanced Marine Center, Washington, DC USA)

ABSTRACT

An adaptive grid refinement method has been developedfor the ISIS-CFD flow solver. The method is fully inte-grated in the flow solver and is meant to be used for all thedifferent fields of application of this RANS solver. There-fore, the method is made to be as general, flexible androbust as possible, to enable directional refinement, un-structured grids, and fully parallelised computation. Theevaluation of the refinement criterion is made such, thatrefinement criteria can be easily exchanged.

In this paper, the refinement method is applied to twohighly different test cases, to show its versatility. In thefirst test, the double-model flow around the KVLCC2tanker, pressure-based refinement criteria are used to cre-ate a locally refined grid from a very coarse initial grid.Results in the propeller plane indicate that good accuracyis obtained with significantly less cells than on fine gridsmade by hand. The second test case is the HALSS fasttrimaran concept. For this case, parameter studies are firstperformed on non-refined grids, in order to determine theinfluence of the outer hull positions and the centre hullskeg design. Grid refinement around the position of thewater surface is then applied to selective cases in orderto get a clearer view of the wave field generated; theseresults allow us to confirm the conclusions from the pa-rameter study.

INTRODUCTION

Automatic adaptive grid refinement is a technique for op-timising the grid in the simulation of fluid flow, by adapt-ing the grid to the flow as it develops during the simula-tion. This is done by locally dividing cells into smallercells, or if necessary, by merging small cells back intolarger cells in order to undo earlier refinement. Grid re-finement is effective for flows that have localised struc-tures; fine cells can be used to keep the error near thesestructures low and to get good resolution, while coarsecells are used in the rest of the domain to reduce the totalcosts for the simulation.

Grid refinement is ideal for ship flow simulation. Theflow around a ship contains several different flow phe-nomena that are highly local in nature. The first of theseis the water surface; if this surface is modelled with anytype of capturing technique, then good grid resolution atthe surface is essential for the accurate computation ofthe ship’s wave pattern. Also the orbital velocity fieldsof waves are local phenomena. And lastly, many typesof vorticity, like bilge or keel vortices interacting with thepropeller or the rudder, consist of local very strong gradi-ents.

Thus, adaptive refinement is an interesting techniquefor simulating these flows. It can however be seen fromthis example that the local flow phenomena associatedwith a ship are fundamentally different. To be success-ful for ship flow, the adaptive refinement technique mustwork for all these different types of flows.

An automatic mesh adaptation method has been devel-oped for ISIS-CFD, the flow solver entirely developed bythe CFD group of the Fluid Mechanics Laboratory. Thegoal of the development was to produce a method thatcan be used in daily practice for all the different applica-tions of this code. Also, it should be easily combined withthe existing capabilities of ISIS-CFD, notably complexgeometries, steady and unsteady flows, free-surface cap-turing, and grid deformation for the imposed or resolvedmotion of ship hulls. Furthermore, the method needs to beeasily maintainable as the code develops over the years.For all these reasons, the mesh adaptation method hasbeen made as general as possible: it allows unstructuredgrids, directional refinement to keep the size of 3D re-fined grids low, and derefinement of refined grids to en-able unsteady flow simulation. The method is flexible toallow the easy changing of refinement criteria (that indi-cate where the grid is to be refined) and, like the flowsolver, it is completely parallel. The refinement methodis integrated in the flow solver.

This paper describes the application of the grid adapta-tion method to two very different test cases, in order to in-dicate its versatility and to show various mesh refinement

strategies. First, the technique underlying the grid adap-tation method is outlined. Then, the double-body flowaround the KVLCC2 tanker is computed. In this test case,a locally adapted fine grid is created entirely using auto-matic refinement, in order to compute accurately the flowin the propeller plane. The goal of this test is to show, thataccurate solutions can be obtained on grid with less cellsthan fine grids created by hand, and that these grids canbe created without requiring expert user knowledge.

Finally, the HALSS fast trimaran model test is used tocompute the resistance in calm water at both model andfull-scale for various trimaran configurations. The objec-tive of this last case is to automatically optimise the gridsaround the water surface and to confirm that favourableinterference can offset the side hull drag as shown fromtowing tank tests and from computations based on non-refined grids.

THE ISIS-CFD FLOW SOLVER

ISIS-CFD, developed by the CFD group of the FluidMechanics Laboratory and available as a part of theFINETM/Marine computing suite, is an incompressibleunsteady Reynolds-averaged Navier-Stokes (URANS)method. The solver is based on the finite volume methodto build the spatial discretisation of the transport equa-tions. The unstructured discretisation is face-based, whichmeans that cells with an arbitrary number of arbitrarilyshaped faces are accepted. A detailed description of thesolver is given by (Duvigneau and Visonneau, 2003) and(Queutey and Visonneau, 2007).

The velocity field is obtained from the momentum con-servation equations and the pressure field is extractedfrom the mass conservation constraint, or continuity equa-tion, transformed into a pressure equation. In the caseof turbulent flows, transport equations for the variablesin the turbulence model are added to the discretisation.Free-surface flow is simulated with a multi-phase flow ap-proach: the water surface is captured with a conservationequation for the volume fraction of water, discretised withspecific compressive discretisation schemes, see (Queuteyand Visonneau, 2007).

The method features sophisticated turbulence models:apart from the classical two-equation k−ε and k−ω mod-els, the anisotropic two-equation Explicit Algebraic StressModel (EASM), as well as Reynolds Stress TransportModels, are available, see (Deng and Visonneau, 1999)and (Duvigneau and Visonneau, 2003). The techniqueincluded for the 6 degree of freedom simulation of shipmotion is described by (Leroyer and Visonneau, 2005).Time-integration of Newton’s laws for the ship motion iscombined with analytical weighted or elastic analogy griddeformation to adapt the fluid mesh to the moving ship.

Furthermore, the code has the possibility to model morethan two phases. For brevity, these options are not furtherdescribed here.

REFINEMENT TECHNIQUE

During a flow calculation with adaptive grid refinement,the refinement procedure is called repeatedly to keepthe grid permanently adapted to the developing solution.Globally, the method works as follows: the flow solver isrun on an initial grid for a limited number of time steps.Then the refinement procedure is called. If a refinementcriterion, based on the current flow solution, indicates thatparts of the grid are not fine enough, these cells are refinedand the solution is copied to the refined grid. On this newgrid, the flow solver is restarted. Then the refinement pro-cedure is called again, to further refine or to derefine themesh. This cycle is repeated several times. When com-puting steady flow, the procedure eventually converges:once the flow starts to approach a steady state and the gridis correctly adapted to this state according to the refine-ment criterion, then the refinement procedure keeps beingcalled, but it no longer changes the grids.

Data structure

The refinement is cell-based. At this moment, it is limitedto unstructured hexahedral cells, but the code is writtensuch, that other cell types can be easily included. Thegrid data structure of ISIS-CFD, that uses node coordi-nates and pointers between cells, faces, and nodes, is usedas the basis for the grid refinement. The number of extrapointers is limited, the main addition is a system of cellfamily ties. These store the history of the refinement, sorefined cells can be derefined again to recover the originalgrid. When a cell is refined, all the new small cells get a‘mother’ pointer to the old big cell and ‘sister’ pointers toeach other. Thus, the group can be found again later andderefined back into the single large cell. The large cell issaved as a ‘dead’ cell, that has no faces nor a state vec-tor, only family ties. Thus, it conserves its own sisters andmother (probably ‘dead’ as well), in case it has to be dere-fined itself, after being restored. Family ties for other gridelements (faces, nodes) are not used, as their refinementhistory can be reconstructed from that of the cells.

Refinement decision

For maximum flexibility and to reduce the chance of er-rors, the code is divided in three separate parts, that ex-change only minimal information between them: the cal-culation of the refinement criterion, the refinement deci-sion, and the actual (de)refinement. The refinement cri-

terion indicates where the grid is to be refined; it can bebased on any desired aspect of the flow solution. To per-mit the user choice of refinement criteria and the easy in-corporation of new refinement criteria in the code, the cri-terion is computed as a field variable (comparable to thevelocity or the pressure). It does not depend explicitly onthe type or the orientation of the cells. A more detailed de-scription of the refinement criterion computation is givenin a following section.

In the second step, this refinement criterion is trans-formed into the decision of which cells to refine or to dere-fine. While this decision may depend on the type of thecells, it does not depend on the specific way in which therefinement criterion is calculated. It remains the same forany refinement criterion.

During the refinement decision step, the decision ineach cell is adapted to its neighbour cells. To guaranteethe quality of the mesh, extra cells may need to be refined,or derefinement of cells may be prevented. The most im-portant quality criteria are given in Figure 1. A face of acell may not be divided two times, which would cause toogreat differences in the sizes of its neighbour cells. Andthe angle between face normals and lines cell centre - facecentre may not be too great: this reduces the quality of thestate reconstruction at the faces, as indicated by (Queuteyand Visonneau, 2007).

b)a)

Figure 1: Grid quality criteria. Forbidden are: (a) facesthat are divided twice, (b) too large angles between facenormals and lines cell centre - face centre. The imagesrepresent 2D examples.

Finally, specific measures are taken in order to pro-tect the wall-aligned boundary layer meshes, if these arepresent. In general, to keep a good grid quality for thesehigh-aspect ratio cells, refinement parallel to the wall isnot permitted. Also, the refinement in each column ofcells is made the same, so that the boundary layer gridconserves its column structure.

At the end of the decision step, before a single cell isrefined, the refinement of the whole mesh is known; thismakes the actual refinement much easier.

Refinement

The actual (de)refinement is done cell by cell. Care istaken, after the treatment of each individual cell, to leavea valid mesh with all the pointers between cells, faces, andnodes in place, even if one knows that certain pointerswill be changed again when a neighbour cell is refinedlater on. This guarantees that, when a cell is refined, itdoes not have to distinguish between neighbour cells thatare refined, that will be refined, or that remain unchanged.The added flexibility and robustness of the code are wellworth the extra work this represents.

Furthermore, in the code, the refinement of cells andfaces is completely decoupled. These parts exchange onlyminimal information; a face does not need to know allthe details of the refinement of its neighbour cells. Thisgreatly facilitates unstructured-grid and directional refine-ment. Another advantage comes from the parallelisationof the algorithm: as the grid is divided in several blocks,the faces between these blocks exist in two blocks at once.In these two blocks, the face must be refined in exactlythe same way. This becomes easier when the refinementof the face is based on the properties of the face itself andnot strongly related with the refinement of a neighbourcell (that is not present in the two blocks).

Finally, the derefinement and refinement procedures areseparated, although they are very similar. Derefinement isperformed first.

Parallel redistribution

For efficient parallel flow computation, automatic redis-tribution of the cells over the processes is included in thegrid adaptation procedure. This redistribution is com-pletely integrated, it is performed between the derefine-ment and refinement steps; at that moment, the grid isat its smallest. Due to the preceding calculation of therefinement decision, the final topology of the grid is al-ready known then. Using this information, the grid isfirst repartitioned in parallel with the ParMeTiS library,see (Karypis and Kumar, 1999).

The next step is the actual displacement of cells be-tween the processes. This step is complicated by thestrictly local numbering used in each block. To solvethis problem, each block is domain-decomposed itself intosub-blocks to be dealt with by specific processes. Thedata structure for each sub-block is the same as the struc-ture of the blocks and each individual sub-block gets itsown local numbering. These sub-blocks are exchangedbetween the processes by MPI (Message Passing Inter-face) and then concatenated to produce the new blocks.

REFINEMENT CRITERIA

The refinement criterion is an important part of the algo-rithm, as it controls the location of the refinement and theshape of the refined cells. The refinement criterion has tobe carefully chosen depending on the flow problem that issimulated. In order to offer a flexible framework for thespecification of anisotropic grid refinement, we developeda metric-based method of criterion evaluation, in whichthe refinement criteria are specified as tensors. In this sec-tion, this method is explained first, then three differentrefinement criteria for ship flow are introduced: a crite-rion that refines at the water surface, a criterion based onsolution gradients, and one based on Hessian matrices.

Metric-based criteria

Our refinement criterion computation is based on met-rics. This technique has been used often for the gener-ation, and the adaptive refinement, of unstructured tetra-hedral meshes (see e.g. (George and Borouchaki, 1998)and (Alauzet and Loseille, 2010)). For the unstructuredhexahedral meshes that we use, it is the most flexible wayof specifying any type of anisotropic refinement.

In the method, the refinement criterion is computedas a field of 3×3 SPD tensors. The metric tensor Ci(R3→ R3) in each cell is considered as a geometric oper-ator that transforms each cell Ωi in the physical space intoa deformed cell Ωi in a modified space. Then, in each cell,directional refinement is applied to produce a grid that isas nearly uniform as possible in the modified space (seeFigure 2). Using suitable tensors Ci, desired cell sizes inany given direction can be specified.

c) Refinement.b) Modified space.a) Physical space.

Figure 2: Tensor refinement criterion. Cell Ωi and unitcircle (reference) in normal space (a), deformed cell Ωi

and deformed circle after application of the transforma-tion Ci (b), and refinement to create a uniform grid in themodified space (c).

As the metric formulation is very general, it can be usedto implement refinement criteria based on different quan-tities. These criteria can also be easily combined, it isenough to take, in each cell, the approximate maximum

of two or more criterion tensors using an approximationlike the one given by (George and Borouchaki, 1998).

Interestingly, the combination of metric-based refine-ment with a fixed original grid gives a highly robust pro-cedure. The major problem in metric-based tetrahedralgrid generation is the prevention of singular tensors, thatproduce infinite-sized cells. Here, the maximum cell sizeis limited by the original grid, so singularity in the ten-sors is not a problem. On the contrary, we use singulartensors to our advantage, to specify refinement in one di-rection only (see the next section for an example). Thus,any symmetric positive-semidefinite tensor can be used asa refinement criterion. We shall see three different exam-ples in the following subsections.

Free-surface criterion

Our first criterion refines in the neighbourhood of the wa-ter surface. Directional refinement is employed to refinethe grid in the direction normal to the surface only. Wherethe free surface is diagonal to the grid directions, isotropicrefinement is used, but where the surface is horizontal, di-rectional refinement is chosen; the resulting zone of direc-tional refinement includes the undisturbed water surface,as well as smooth wave crests and troughs. This is es-sential to keep the number of grid cells low, as the watersurface is often nearly undisturbed in most of the domain.Figure 3 gives an illustration of this refinement principle.

Figure 3: Isotropic and directional refinement at the freesurface. The curves represent volume fraction isolines.

Directional refinement normal to the free surface im-plies, that the refinement criterion imposes a constraint onthe cell size in that normal direction only. The cell size inall other directions is kept as large as possible. Thus, cellsthat are oriented diagonally to the interface must neces-sarily be refined in all directions, but cells that are alignedwith the interface can be refined in one direction only.In the context of tensorial refinement, this is obtained byspecifying tensors with only one non-zero eigenvalue, as-sociated with an eigenvector normal to the surface. As

explained above, this poses no problems from a numeri-cal point of view.

Gradient criteria

A second group of refinement criteria is based on theabsolute values of the gradients of solution quantities ineach cell. These criteria detect the regions where the flowfield changes rapidly; they react to most features of a flowand are thus more general than the free-surface criterion.Initially, these criteria have been applied to single-fluidflows.

Three criteria are chosen, one based on the gradient ofthe pressure:

Critp =√

( ∂p∂x )2 + ( ∂p

∂y )2 + (∂p∂z )2, (1)

one on the gradients of the three velocity components:

Critu =

√(∂u

∂x )2 + (∂u∂y )2 + (∂u

∂z )2 + ( ∂v∂x )2 + (∂v

∂y )2+(∂v

∂z )2 + (∂w∂x )2 + (∂w

∂y )2 + (∂w∂z )2

,

(2)and one on the vorticity:

Critω =√

(∂v∂z −

∂w∂y )2 + (∂w

∂x −∂u∂z )2 + (∂u

∂y −∂v∂x )2.

(3)Effectively, the vorticity can also be seen as a norm ofthe velocity gradients. These three are all scalar criteria:in each cell, the criterion gives only one cell size. In thecontext of metric-based refinement, this means that therefinement target is, to have the same cell size in all direc-tions. Therefore, the refinement tensors Ci in each cell arethe identity matrix I , multiplied by the value of Crit.

The main difference between the pressure gradient cri-terion and the two velocity-gradient based criteria is theireffect on boundary layers. There, the pressure varies lit-tle, while the velocity gradients are very strong. Thus,the velocity-gradient based criteria refine everywhere inboundary layers, but the pressure gradient criterion onlythere where the external flow creates a pressure gradient.In those cases where the flow outside the boundary lay-ers is the main interest of the computation, costly and un-necessary refinement in all boundary layer cells can beprevented by using the pressure gradient criterion. An ex-ample will be shown in the next section.

Hessian-based refinement

The Hessian matrix of second spatial derivatives of the so-lution is a common choice as a refinement criterion. Usingthe same reasoning as above, we base this criterion on the

pressure field, so it reacts to wave fields and to pressure-induced flow features at the hull, but not directly to thepresence of boundary layers.

Ci =

pxx pxy pxz

pxy pyy pyz

pxz pyz pzz

. (4)

The major difficulty in using the Hessian tensor as arefinement criterion is the accurate evaluation of the sec-ond derivatives, independent of the mesh. If the criterioncomputation is perturbed by local grid refinement, it mayreact more to existing grid refinement than to the pressurefield. To prevent this undesired effect, numerical errorsin the computed second derivatives must be significantlysmaller than the derivatives themselves in all cells.

A particular problem associated with unstructured hex-ahedral meshes, is that the grid remains irregular whenit is refined. For structured grids, and even for most un-structured tetrahedral meshes, when the grid is refined thecells get more and more the same shape and size as theirneighbours. On unstructured hex meshes however, therewill always be cells that are two times smaller than theirdirect neighbours. This means, that numerical schemeswhich rely on mesh regularity to get good accuracy arenot suited for these meshes; a useful scheme must givesufficient accuracy for arbitrary cell configurations.

For the computation of second derivatives, we use aleast-squares method based on third-order polynomials.In each cell, the polynomial is computed that best fitsthe pressure in the cell, its neighbours and its neighbours’neighbours, in the least-squares sense. The approximatedHessian is constructed from the second derivatives of thispolynomial. The least-squares procedure guarantees thatthe difference between the approximating polynomial andthe real pressure is not in the space of third-order polyno-mials; therefore, it is at least fourth-order. Hence, the ap-proximated second derivatives are second-order accurate,independent of the mesh geometry. For simpler meth-ods like the well-known Gauss integration, this cannot beguaranteed.

KVLCC2 TANKER

The first test that is computed here, is the double-modelflow around the KVLCC2 (KRISO Very Large Crude Car-rier) tanker. The goal of the test using this well-known testcase, is to study the gradient and pressure Hessian crite-ria and their capacity to create efficient fine meshes whenstarting from a very coarse initial mesh.

The study is concentrated on the flow in the propellerplane. The flow in this plane is strongly influenced bythe vorticity that is generated on the ship hull and shed

into the fluid at the rear of the ship, where the hull ta-pers into the stern. This vorticity generates a non-uniformflow field, that is an essential input for good propeller de-sign and a key flow feature in hull shape optimisation forpropeller efficiency. It is known that, while the resistanceof the KVLCC2 tanker is accurately predicted by RANSmethods on coarse grids, a fine grid of at least 1M cellsis needed to resolve correctly the propeller plane flow.Moreover, this grid must contain local refinement near thestern; the result is sensitive to the correct placement of thisfine zone. Thus, if the grid is to be created by hand usingstandard grid generator software, expert knowledge is re-quired from the person generating the grid.

We shall compute the propeller plane flow with auto-matic grid refinement, by starting from a coarse grid with-out any local refinement. The ultimate target is to provethat similar accuracy can be obtained using less cells thangrids refined by hand, using in particular the possibilityof directional refinement to reduce the number of refinedcells there, where the flow is aligned with a grid direc-tion. Alternatively, we strive to demonstrate that similaraccuracy can be obtained with the same number of cellsas on grids refined by hand, but without requiring the ex-perience needed to generate these grids.

Test case, setup, and grids

The test case is the KVLCC2 hull at model scale, Re =4.6 · 106, in double-body configuration and without pro-peller; the computations are performed with a single fluid(water) and a symmetry boundary condition is imposed atthe position of the undisturbed water surface. This test isone of the cases from the Gothenburg 2000 workshop, see(Larsson et al., 2000). Only half of the hull is meshed,so a symmetry boundary condition is imposed also on thehull Y symmetry plane. Concerning the far field, the ve-locity is imposed on the inflow boundary, the pressure onthe sides and the outflow boundary.

The initial coarse grid is an unstructured hexahedralmesh, generated with the HEXPRESS(TM) grid genera-tor. The cells on the hull have sizes parallel to the wall ofLpp/125, which is at least twice as coarse as usual. In thedirection normal to the wall however, given our boundarylayer refinement strategy (see above), cells have the smallsizes required for accuracy. For the wall-law boundarycondition used in ISIS-CFD, the first cells are chosen togive y+ = 30. The original grid has 58k cells.

Turbulence is modelled both with the two-equation k−ω SST model of (Menter, 1993) and with the anisotropictwo-equation EASM model (Deng and Visonneau, 1999).We have shown in the past, that anisotropic turbulencemodelling is required in order to correctly model vortexroll-up in ship aft-body flows. Here, we shall compare the

two turbulence models in order to confirm this conclusionfor automatically refined grids.

Gradient criteria

Initially, the flow is computed using the gradient-based re-finement criteria introduced in the previous section. Thesecriteria can be easily computed and give smooth, gradualrefinement. However, due to their nature they can only beused with isotropic refinement.

A first comparison is made between the three criteria,using the k − ω SST turbulence model. To obtain a rea-sonable comparison, the computation for each criterion isstarted from the converged solution on the original 58kcells grid. Then, the grid after one refinement step is stud-ied. The refinement threshold, which indicates the low-est value of the criterion that gives refinement, is adjustedfor each criterion until the number of cells per grid afterone refinement step are equal. Once these three thresh-olds are found, the computations and refinement steps arecontinued until the solution and the grid converge. In or-der to prevent excessive refinement in small regions withvery high gradients, the number of times that an origi-nal cell can be refined is limited to two. The resultinggrid sizes are 630k cells for the velocity gradient crite-rion, 620k cells for the vorticity criterion and 803k cellsfor the pressure gradient criterion.

Results in the propeller plane are shown in Figure 4.The ship hull can be seen at the top of these figures, thepropeller would be placed below the hull. The most strik-ing feature of the flow, as seen in the experimental result(Figure 5), is the hook-shaped region of low axial velocitycaused by the roll-up of vorticity created on the aft hull.In the computation on the original coarse grid, this hookshape is completely absent. On the refined grids, a ten-dency towards a hook shape can be seen, which is mostpronounced for the pressure gradient criterion.

Looking at the grids in the cross-section, it can be seenthat all three criteria detect the gradients that appear inthe propeller disk area. However, the pressure criterionrefines much more than the other two. This is becausethe velocity and vorticity criterion react to the strong ve-locity gradients in the boundary layer and thus refine theboundary layer grid over the entire ship length, even inthe middle section where the flow is nearly uniform in X-direction and where refinement is thus not necessary. (It isinteresting to see that these two criteria, while computedin a different way from the velocity gradients, give nearlyidentical results.) The pressure gradient in the middle partof the boundary layer is close to zero, so for a given num-ber of cells the pressure gradient criterion refines morein the region around the bow and the stern of the ship.As a result, the grid in the propeller disk is finer and the

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Figure 4: KVLCC2 tanker, cuts in the propeller plane atx/L = 0.0175. Grid cross-sections and axial velocityu/U∞ isolines are shown on the original coarse grid (58kcells) and the refined grids for the velocity gradient crite-rion (630k cells), the vorticity criterion (620k cells) andthe pressure gradient criterion (803k cells).

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Figure 5: KVLCC2 tanker, measurements of the axial ve-locity u/U∞ in the propeller plane at x/L = 0.0175, from(Larsson et al., 2000).

flow there is resolved better: it shows the beginning ofa hook shape and also in the region away from the sym-metry plane, the solution is smoother and closer to theexperimental results.

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Figure 6: KVLCC2 tanker with the EASM turbulencemodel. Grid cross-sections and axial velocity u/U∞ iso-lines are shown on the original coarse grid (58k cells) andthe refined grid for the pressure gradient criterion (1.07Mcells).

The remaining differences between the computationand the experiment are mostly due to the turbulencemodel. To study the effect of this model, the k − ω SSTmodel is replaced by EASM. Computations with this tur-bulence model are performed for the original coarse gridand for the pressure gradient criterion, using the samethreshold as before. The result is given in Figure 6. Therefined grid is similar to the grid for the pressure gradi-ent criterion and the k − ω SST model. However, due tothe more pronounced roll-up, the pressure gradients areslightly stronger so the refined grid has more cells: 1.07Mas opposed to 803k. Concerning the flow solution, on theoriginal grid the result has changed very little. But on therefined grid, the hook shape is now clearly visible and ingood agreement with the experiment. Overall, the qualityof this solution is excellent.

Hessian criterion

A first computation has been made with the criterionbased on the Hessian of the pressure, retaining the EASMturbulence model. With respect to the pressure gradientcriterion, this criterion has two advantages. First, the sec-ond spatial derivatives of the solution are generally con-sidered to be a better indicator of the local error than thesolution gradients. And second, the Hessian is a ten-sor criterion so it allows directional refinement, whichmeans that elongated cells can be produced in those re-

gions where the flow is uniform in one direction. Thisreduces the overall number of cells.

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Figure 7: KVLCC2 tanker, cross-sections of the gridbased on the Hessian criterion at Y/L = 0.02 at thefront of the ship (top) and at the back (bottom), showingisotropic and directional refinement.

Two cuts through the refined grid can be seen in Fig-ure 7. As the Hessian criterion is based on the pressure,it does not react to the presence of the boundary layer onthe middle part of the hull. In the zones of strong pres-sure variation near the bow and stern the flow is diagonalto the grid directions and the pressure varies in all direc-tions, so isotropic refinement is predominant. Closer tothe middle of the ship, the flow no longer varies stronglyin X-direction, so the refinement is mainly directional.The same can be seen in the wake flow behind the ship.We can also see that the Hessian-based criterion is moresensitive to small perturbations than the pressure gradientcriterion (despite the evaluation using third-order polyno-mials): the grid is slightly less smooth. Overall however,the grid quality is good.

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0RefinedEASM

Figure 8: KVLCC2 tanker, grid cross-sections and axialvelocity u/U∞ isolines for the refined grid based on theHessian criterion.

In the propeller plane, Figure 8, the grid refinement isconcentrated in the zone of the propeller disk, where thepressure changes most strongly. To the exterior, less re-

finement is applied than for the pressure gradient, so thesolution is not as well resolved there. However, the hookshape is perfectly resolved.

The size of this grid is 600k cells, compared to the1,07M for the pressure gradient criterion, but the solutionquality in the propeller disk is equivalent. This indicatesthe effectiveness of the directional refinement procedure.

HALSS TRIMARAN

The second case studied concerns a large trimaran ship,the Heavy Air Lift Support Ship (HALSS) tested atNSWCCD, where interference between the hulls has amajor impact on the wave making resistance. The HALSSconcept design (Mizine and Amromin, 1999) is a 35-knot ship with the following main dimensions: flight decklength 335m, flight deck width / docking hull beam 83.5m/ 54.9m, draft 11.5 m. A general view of the HALSS isshown in Figure 9.

Figure 9: General view of the HALSS concept.

The hydrodynamic research and model testing showedlarge changes in resistance due to moderate shifts in thelongitudinal position of the side hulls. Also, a strong in-fluence of the skegs on the stern flow was observed. Inorder to understand the factors leading to the interferenceeffects, several earlier CFD studies have been performed(Mizine and Karafiath, 2008), (Mizine et al, 2009).

It is a common point of view (Begovic et al, 2005),(Doctors and Scrace, 2003) that successful design of thetrimaran hulls is accomplished when the interference dragbetween hulls is zero. This means that the resistance ofthe trimaran is equal to the resistance of the centre hullplus resistance of the side hulls if each is operating alone.The model tests of the HALSS design showed howeverthat favourable interference can offset the side hull drag,leading to a total resistance that is lower than the summa-tion of the individual hull resistances.

In this section, the experimental model test is presentedfirst, then results of computations on non-refined gridssuitable for the prediction of the forces and main char-acteristics of the flow field will be presented. Some se-lected configurations are used to detail the influence of

Number DescriptionCentre Hull 1 No Skegs

only 3 Hull & SkegsCentre Hull 8 Hull & Skegs, SH. Fwd.

and 5 Hull & Skegs, SH. Mid.Side Hulls 9 Hull & Skegs, SH. Aft

Table 1: Limited list of experiments.

the skegs supporting the propeller shaft and of the sidehull longitudinal location. In a second part, to validate thefree-surface results and conclusions obtained on the non-refined grids, the optimal configuration relative to the lo-cation of the side hulls is selected for an adaptative grid re-finement exercise starting from the non-refined grids andusing the free-surface criterion as described previously.

Experimental model test

Model 5651 representing the HALSS trimaran conceptship was made to a scale ratio of 54.0 and consisted ofthree separate hulls, one centre hull and two identical sidehulls, connected together with aluminium cross structurepieces into a trimaran. Tests were performed on the cen-tre hull only, with and without skegs, and on the trimaranwith three longitudinal and three transverse positions ofthe side hulls. For each experiment, the model was re-strained in surge, sway, and yaw, but was free to pitch,heave, and roll.

Table 1 lists the experimental conditions used in thisarticle; for the trimaran cases, the side hulls are in theinnermost transverse position. A complete description ofthe Model 5651 tests is detailed in (Mizine et al, 2009).

A photograph of the HALSS model experiment atspeed 35 kts is shown in Figure 10 for the case referencedas Experiment 9. In this photograph it is clearly seen thatthe free surface can have strong deformations with break-ing transom waves. This requires adequate RANS CFDmethods because wave breaking is a real physical phe-nomenon that influences the optimization of multihulls.

Simulations on non-refined grids

The computed cases studied have a ship waterline length(LWL) of 289.8m and a ship draft of 11.5 m for the cen-tre hull and a draft of 7.5 m for the side hulls. The cor-responding model scale has a ship waterline length of5.367m for the centre hull and 3.271m for the side hulls.The pitch and heave are prescribed from the values ob-tained during the experimental campaign. Due to sym-metry considerations, only the Y>0 part of the model ismeshed with the help of the HEXPRESS(TM) automatic

Figure 10: HALSS model test in the configuration of Ex-periment 9, speed 35 kts.

grid generator. Figure 11 shows the computational do-main around the half-trimaran with the free surface at restrepresented by and internal refinement surface on whichspecific target cell sizes are prescribed.

Figure 11: Computational domain in HEXPRESS (red)with internal surface defining the location of the free sur-face at rest (green).

In the following, FS and MS indicate full scale simula-tions and model scale simulations, respectively. For nor-malised quantities, lengths are normalised by LWL andspeeds are normalised by the speed of the considered case.

While the experimental speed range is from 10 knots to45 knots, only five speeds between 25 knots and 40 knotswere selected for the numerical study. The model scalespeed for the 25 knots case is 1.7485 m/s and 2.8006 m/sfor the 40 knots case. Considering the Reynolds num-bers, Table 2, the 35 knots design speed was selected tobuild a unique mesh for a given trimaran configuration atall speeds. The y+ = 30 constraint on wall functions re-quires meshes of about 2.8M cells (hull+skeg+side hulls)for model scale simulations with 8 viscous layers insertedin the Euler mesh. For the full scale simulation, the wallfunctions method requires a y+ = 300 corresponding to a

Speed Fr Re(MS) Re(FS)25 kts 0.241 9.78 106 11.03 109

30 kts 0.289 11.73 106 13.24 109

32 kts 0.309 12.54 106 14.12 109

35 kts 0.337 13.69 106 15.44 109

40 kts 0.386 15.66 106 17.65 109

Table 2: Characteristics of the computed ship speeds.

distance between the first layer of grid points in the fluidand the first layer on wall of about 0.0004m (0.0003mfor the model scale) and the mesh size increases to about4.7M cells with the insertion of 24 viscous layers. Thevertical cell size in the vicinity of the free surface at rest isprescribed as 0.001LWL (5 mm at MS and 30 cm at FS)with a maximum cell aspect ratio of 100. For the samereasons as for the KVLCC2 case, the EASM anisotropicturbulence model is used in all cases.

Having followed the usual recommendations with theautomatic grid generator for the considered simulations,this defines the non-refined original grids suitable for acorrect prediction of the forces and the main flow charac-teristics. A typical MS run for these original grids on a10 processor IBM Power6 cluster takes about 6 hours toreach a well established solution with a time step of 0.02sfor 20s of simulation. In case of FS simulations the CPUeffort is similar using 16 processors and a time step of0.15s.

Forces will be presented in terms of normalised coeffi-cients following the classical definitions:

C. =F.

0.5ρU2Sw, PE = FXU, (5)

Cr = CT − Cf0, with CT = CP + CF , (6)

in which C. represents the normalised force coefficientF.. The axis X is aligned with the ship advancing direc-tion and PE represents the power required to maintainthe prescribed advancing velocity U . Sw is the wettedsurface, ρ is the specific mass of the water. The frictionalresistance coefficient CF0 from the ITTC’57 correlationline will be used to compute and compare the residuaryresistance coefficient, Equation 6, as reported experimen-tally. It will be derived from the computed total resistancecoefficient CT, from the computed pressure resistance co-efficient CP, and from the computed frictional resistancecoefficient CF.

The major objective of the computations is to see if thepower efficiency due to resistance, PE in Equation 5, isin agreement with the value obtained from towing tankmeasurements on the model-scale ship and extrapolatedto the full-scale trimaran. When full-scale simulations are

performed, the power efficiency is directly derived fromthe computed total resistance force at the correspondingspeed without any other approximations than those con-tained in the physical modelling. When model-scale sim-ulations are considered, the extrapolation to full-scale ofthe power efficiency will be obtained by computing theresistance FX force given by the reverse Equation 5 usingthe full-scale wetted surface, velocity, and the computedresiduary resistance coefficient which is supposed to bethe same at full scale as at model scale.

Influence of the skegs Together with the side hulls, thesternward extension on each side of the centre hull thatsupports the propeller shaft is a crucial element to investi-gate, since it implies severe modifications and constraintson the bare hull in the aft region. The flow field in frontof the propeller area and the changes in resistance must becarefully evaluated for further possible improved designs.

The influence of the skegs has been investigated on thecentre hull only from Experiment 1 (without skegs) andExperiment 3 (with skegs). The presence of the skegs in-creases the wetted surface by a factor 1.11. Table 3 com-pares the computed frictional, pressure, and total resis-tance coefficients for the two configurations at some se-lected speeds. First of all, one can notice that the viscousresistance is much larger than the pressure resistance forboth configurations. However, for the design speed of 35knots for instance, the pressure resistance represents 28%of the total resistance without skeg and 40% for the cen-tre hull with skegs. Although the viscous resistance is notstrongly affected by the presence of skegs, the pressureresistance may be multiplied by a factor 2 when skegs areincluded. At the design speed of 35 knots, the pressure re-sistance is multiplied by a factor 1.7. One can also noticethe relatively good agreement between the predicted vis-cous resistance and the values provided by the ITTC-57correlation.

The corresponding residuary resistance Cr is presentedin Figure 12 where one can notice a very satisfactoryagreement between measurements and computations, ex-cept for the lowest speed of Experiment 3. The reliabilityof the computations is confirmed with the spectacular in-fluence of the skegs especially for speeds ranging from 30to 40 knots where the computed residuary resistance co-efficient behavior is in agreement with the measured one.

Therefore, the main origin of the increased resistanceis the modification of the pressure field. The compari-son of the computed free-surface elevations in normalisedform is given in Figure 13. The large difference in termsof resistance between Experiments 1 and 3 comes from acompletely different behaviour of the free surface on the

Exp: Speed CF0 CF CP CT1: 25 3.012 2.982 0.225 3.2063: 25 2.984 0.883 3.8671: 30 2.919 2.953 0.913 3.8663: 30 2.059 1.695 4.6541: 32 2.886 2.936 1.122 4.0583: 32 2.942 1.990 4.9331: 35 2.843 2.910 1.116 4.0273: 35 2.913 1.906 4.8201: 40 2.779 2.877 1.651 4.5283: 40 2.888 2.105 4.992

Table 3: Computed resistance coefficients (x 1000) atmodel scale for Experiments 1 and 3 at various speeds.

Ship Speed (knots)

Res

idua

ryR

esis

tanc

eC

oeffi

cien

t-C

r(x

100

0)

10 15 20 25 30 35 40 450.0

0.5

1.0

1.5

2.0

2.5

3.0

Exp. 3

Exp. 1

Figure 12: Evolution of the residuary resistance with shipspeed for Experiments 1 and 3. Measured - lines withsmall symbols; computations - large empty symbols.

last quarter of the hull as illustrated in Figure 13. Theaddition of a skeg and the related strong modification ofthe pressure field result in a local suction on the free sur-face leading to a local breaking wave, which dramaticallyincreases the resistance.

The drastic change in pressure distribution due to thepresence of the skegs can also be clearly detected from asection-by-section analysis of the forces. The forces bysection distribution is obtained from the integration of theforces (frictional, pressure, and total) on the trimaran sur-face, on a prescribed number of sections. For all the pre-sented forces by section in this paper, 100 equi-distributedsections of width 0.01LWL are used. The distributionsin Figure 14 correspond to Experiment 1 at 35 knots andcontain the plots of the resistance coefficients, togetherwith the intersection of the free surface with the trimaranconfiguration in the background. Figure 15 plots the sameinformation for Experiment 3.

Up to X/L=0.6, the pressure distribution is similar be-

Figure 13: Free-surface elevation: centre hull with skeg(top) and without skeg (bottom) - 35 kts.

Figure 14: Forces by section:Center hull only withoutskeg - 35 kts

tween the two configurations, starting with a unfavourablepressure gradient until x/L=0.2 and the same unfavourablegradient until x/L=0.6 approximately. Further back, untilx/L=0.85, the Experiment 3 configuration produces a lessefficient pressure gradient than Experiment 1 does. Thena small pressure plateau occurs between 0.85L and 0.95Lwhere the minimum of the water elevation appears on thehull (-0.009L), as can be seen from the narrow shape ofthe geometry on the same figure and from Figure 13. Pastx.L=0.95L, the wave breaking takes place. Thus, the ef-fect of the skegs is to considerably reduce the extent ofthe region of favorable pressure gradients, with a reduc-tion starting early at x/L=0.65 approximately. This couldhave been observed from the wave elevation distributionconsidering the iso-0.0 as a reference on Figure 13, but notin a quantitative way as can be done from the longitudinaldistribution of the pressure force.

While not illustrated, it was observed that the skegstrongly modifies the wall pressure distribution, by modi-fying the curvature of the hull, leading to the developmentof a longitudinal vortex at the extremity of this appendage,although no local flow reversal can be detected. Figure 16

Figure 15: Forces by section: centre hull only with skeg -35 kts.

shows the computed and measured velocity field at thestarboard shaft at 30 knots immediately downstream ofthe end of the shaft. In the isowake distribution, onecan clearly see the wake of the skeg and two contrarota-tive longitudinal vortices in the computations, which arehardly visible in the measurements due to the measure-ment accuracy. However, the good agreement of this lo-cal flow field between the measurements and the viscoussimulations is very reassuring. Secondly, the modificationof the pressure field has an impact on the free-surface el-evation by creating, just behind the skeg, a deep troughfollowed by the strong breaking wave already mentioned.

Figure 16: Computed (left) and measured (right) isowakedistribution and secondary velocities at the starboard shaft- Experiment 3, 30 kts.

Influence of longitudinal location of side hulls The in-fluence of the longitudinal location for the side hulls ispresented using Experiment 8 for the forward longitudi-nal location, Experiment 5 for the middle longitudinal lo-cation, and Experiment 9 for the aft longitudinal location.

Excellent agreement between the measurements andthe computations is observed in the trends and differenceson the residuary resistance, which are remarkably cap-tured by the computations, see Figure 17. When the speed

is higher than 25 knots, the configurations of Experiments8 and 9 which correspond to the forward and aft longitu-dinal locations, respectively, are characterized by an in-crease of about 300% of the residuary resistance. To anal-yse the origins of this dramatic increase, the two extremecomputed speeds of 25 and 40 knots have been chosen tocompare the characteristics of the waves. The free-surfaceelevations for the three longitudinal locations are shown inFigure 18 for a speed of 25 knots and in Figure 19 for aspeed of 40 knots.

Figure 17: Longitudinal location of side hulls - evolu-tion of the residuary resistance with ship speed for Ex-periments 8, 5, 9. Measured - lines with small symbols;computations - large empty symbols.

At 25 knots, for Experiments 5 and 9 corresponding tothe middle and aft side hull longitudinal locations, thereis no significant interaction between centre and side hullbow waves, contrary to the forward longitudinal location(Experiment 8) where this interaction is very strong evenfor this moderate speed. In this case, the strong interac-tion leads to more complex wave trains between the hulls,which contribute to increase the resistance. However, theforward longitudinal location has no impact on the sternbreaking wave developing along the centre hull contraryto Experiment 9, for which one notices a very strong inter-action between waves emanating from the side and centrehulls.

At 40 knots, the situation is completely different be-cause of the magnitude and spatial extension of the bowwave created by the centre hull. A strong interaction be-tween the bow waves of both hulls can be observed for themiddle and forward longitudinal locations (Experiments 5and 8), with a very spectacular internal wave train in thelast case. In Case 8, the reflected wave on the centre hullis so intense that it leads to a rooster tail breaking wavebehind the side hull, which may explain the higher resid-uary resistance observed both in the experiments and the

Figure 18: Influence of the longitudinal location of sidehulls at 25 knots - wave elevations for Case 5 (top of topand bottom figures), Case 8 (bottom of top figure) andCase 9 (bottom of bottom figure).

computations for this side hull position.

Figure 19: Influence of the longitudinal location of sidehulls at 40 knots - wave elevations for Case 5 (top of topand bottom figures), Case 8 (bottom of top figure) andCase 9 (bottom of bottom figure).

Table 4 shows frictional, pressure, and total resistancecoefficients at the design speed of 35 knots for the tri-maran with side hulls in the three different longitudinallocations and also for the case without side hulls (Exper-iment 3). As computed for the cases without side hulls,the viscous resistance dominates the pressure resistancefor all configurations. The only exception is Experiment5 where the pressure resistance is only 20% of the totalresistance and with a slightly lower frictional resistancethan computed with the other longitudinal configurations.

The corresponding free-surface elevations at the designspeed, compared with the result obtained for the centrehull only (Experiment 3) are given in Figures 20, 21, and22. A common contour range is used based on Exper-iment 8 for the lowest elevation (-0.023LWL) detected

Simulation CF[% of CT] CP[% of CT] CTEXP 3 2.913 [60%] 1.906 [40%] 4.820EXP 8 3.205 [56%] 2.540 [44%] 5.745EXP 5 2.982 [79%] 0.784 [21%] 3.765EXP 9 3.050 [53%] 2.682 [47%] 5.733

Table 4: Computed resistance coefficients (x 1000) atmodel scale for Experiments 1, 3, 8, 5, 9 - 35 kts.

between the side hull and the centre hull and the high-est computated elevation (0.02LWL) detected behind thetransom. The corresponding values for the computedExperiment 5 are approximately 1.5 times lower with -0.015LWL for the minimum and +0.0135LWL for themaximum. This gives a first explanation for the low pres-sure resistance of Experiment 5.

Figure 20: Free-surface elevation: centre hull with skeg(top) and side hulls in forward longitudinal position (bot-tom) - 35 kts.

Figure 21: Free-surface elevation: centre hull with skeg(top) and with side hulls in middle longitudinal position(bottom) - 35 kts.

Favorable interference effects are effectively producedwith the side hull in the middle location since less waveinterference between the side and the centre hulls is foundin that case. Moreover, the wave system outside the side

Figure 22: Free-surface elevation: centre hull with skeg(top) and with side hulls in aft longitudinal position (bot-tom) - 35 kts.

hull changes so that one wave length corresponds to aboutone ship length, compared to about 0.8 ship length forExperiment 3, Figure 21. Also, the amplitude of this di-vergent wave system is lower in Experiment 5 even whencompared to the amplitude from Experiment 3 or 1.

To continue the design speed discussion, the longitu-dinal distribution of the forces is investigated followingthe approach already outlined for the skeg influence. Fig-ures 23, 24, and 25 give the forces by section distributionfor the three configurations. Since the computed resis-tance coefficient results from the integration over the sur-face of the ship, a high positive region observed in such alongitudinal distribution is not enough to conclude aboutits influence on the integrated coefficients since it is the re-sult of the balance between positive and negative regions.This is not the case with the CF frictional coefficient whenreverse flow is marginal.

For Experiment 8, Figure 23, the most significant ob-servation is the considerable increase of the unfavourablepressure gradient between the fore region of the centrehull and part of the front region of the side hulls. Thisis related with the strong wave interference observed inthe same region in Figure 20. The large reduction ofthe favourable pressure gradient starting at x/L=0.4 cor-responds to the presence of a deep trough (-0.020L) nearthe centre hull, centered at x/L=0.45. Later, past the endof the side hull location, the favourable pressure gradi-ent is still reduced. Concerning Experiment 9, Figure 25,since the side hull is shifted aft, the favorable pressuregradient starting at x/L=0.2 from the centre hull becomesunfavourable at the beginning of the side hull so that theregion of positive CP is higher in this case. For Exper-iment 5, Figure 24, the main characteristic of the longi-tudinal pressure coefficient is a quasi-linear behaviour inmost of the region corresponding to the side hull location.This remarkable behaviour indicates that very little inter-action takes place in that region, as already observed from

the wave elevations.

Figure 23: Forces by section: side hulls in forward longi-tudinal position - 35 kts.

Figure 24: Forces by section: side hulls in middle longi-tudinal position - 35 kts.

Simulations on refined grids

An important point to underline for the trimaran study is,that the original grids used are not “coarse” original gridsas for the KVLCC2 case. Instead, the non-refined grids ofthe earlier study are used. The experiment number 5 wasselected at 35 knots to check the free-surface criterion forboth MS and FS simulations. The refinement target is agrid spacing of about 0.0005LWL.

The solution is restarted from the original-grid solu-tion. Figure 26 plots the resulting evolution of the numberof cells during to the refinement process, related to thenumber of cells in the original grid. The refinement algo-rithm is applied every 25 time steps. While not fully com-pleted due to resource limitations, the refinement processconverges for both MS and FS. The refined grid containsabout 4.5M cells for the MS simulation and about 7.5M

Figure 25: Forces by section: side hulls in aft longitudinalposition - 35 kts.

cells for the corresponding FS simulation, which corre-sponds to a grid size increase factor of about 1.6.

Figure 26: Number of cells ratio increase during the re-finement process

Free-surface elevations for Experiment number 5 at the35 knots design speed are presented in Figure 27 for themodel scale case with the original grid solution on top andthe refined grid solution at the bottom. The same kind ofcomparison is presented in Figure 28 but for the full scalesimulations. The most visible effect of grid refinementis in the improvement of the details of the free surfacenear the rising wave behind the transom. Grid refinementalso confirms the very slight increase of the transom wavebetween model and full scale.

The effect of automatic grid refinement is clearly seenin Figure 29 showing the grid density on the free surfacefor the model scale case. The most evident observationis the increase of the grid density at and behind the skegregion where breaking wave occurs. The grid resolutionis also improved near the waves emanating from the bowregion and from the sharp ends of the side hulls.

Moreover, as seen in the enlarged view of Figure 30,the grid density is now improved to confirm that the di-vergent wave system outside the side hulls has a shorterwave length with lower amplitude compared to the result

Figure 27: Free-surface elevation with side hulls in mid-dle longitudinal position: original grid (top) and refinedgrid (bottom) - 35 kts.

Figure 28: Free-surface elevation at full scale with sidehulls in middle longitudinal position: original grid (top)and refined grid (bottom) - 35 kts.

of Experiment 3, Figure 13 or Figure 21. As a conse-quence, less wave breaking induced by the skegs, longerwave lengths, and lower amplitudes mean that less energyis spent to deform the water surface with the side hulls inmiddle location.

Figure 32 shows the cut plane X/Lpp=1.03 where thewave breaking phenomenon close to the centre hull hasbeen captured on the refined grid at model scale. Alsoimproved is the capture of the wave generated by the sidehull. This cut plane result is compared with the result ob-tained on the original grid, Figure 31. The free-surfacecriterion used is perfectly validated with respect to the re-quired threshold. In some regions, for quality preserv-ing reasons, the threshold value retained (0.0005LWL)can even produce one step smaller isotropic cell sizes(0.00025LWL): this is typically the case in wave breakingregions, see Figure 32. If an explicit refinement box (byhand) with isotropic target cell size of 0.00025LWL wasused on the original grid near the free surface, restricted tothe region where wave breaking occurs close to the centrehull, the additional grid size is roughly estimated as 20Mcells. This has to be compared with the 1.6 multiplication

Figure 29: Grid on the free-surface with side hulls in mid-dle longitudinal position: original grid (top) and refinedgrid (bottom) - 35 kts.

Figure 30: Free-surface elevation for side hulls in middlelongitudinal position - refined grid - 35 kts.

factor produced by the grid refinement.The effect of automatic grid refinement on the forces

is given in Table 5 at both model and full scale. Thefull-scale coefficients are expressed as percent changes ofthe corresponding model-scale coefficients. The expecteddecrease of the frictional coefficient from model to full-scale is the same with the same values (less than 0.01%)on the refined grid. Only small adjustments are producedbetween the original and the refined grid for the pressurecoefficient. The slightly equivalent effect of grid refine-ment on pressure resistance between model and full-scale(-2%) is diminished at model scale on the total coefficientsince the pressure coefficient represents 35% of the totalcoefficient at model scale but only 21% at full scale.

Power efficiency of the trimaran

Although not used for all the possible trimaran configu-rations, it was shown that the automatic grid refinementtechnique for optimising the grids around the water sur-face produces very small differences with respect to theresistance results obtained on the original grids. Thus, theresults from the originals grids can be used to evaluate theefficiency of various trimaran configurations.

Figure 31: Original grid in cut plane X/Lpp=1.03 withvolume fraction contours at model scale.

Figure 32: Refined grid in cut plane X/Lpp=1.03 with vol-ume fraction contours at model scale.

Table 6 summarizes the predictions and direct compu-tations of the power efficiency (PE) for the design speedof 35 knots with the wetted surface that depends on theconsidered case. When full-scale simulation (FS) is con-sidered, the power efficiency is directly obtained from theintegrated forces without any extrapolation. When model-scale simulation is considered (MS), the predicted powerefficiency is extrapolated as for the towing tank test. Ex-periment 1 is used to quantify the effect of the skegs andExperiment 3 is used to quantify the influence of the loca-tion of the side hulls on the effective power.

Although Experiment 1 has not been computed at full

Simulation CF CP CTOriginal: MS 2.982 0.784 3.765Refined: MS 0% -2.2% -0.5%Original: FS 1.390 0.744 2.134Refined: FS 0% -2.0% -0.7%

Table 5: Computed resistance coefficients (x 1000) onoriginal and refined grids for Experiment 5 - 35 kts.

Case SW (m2) PE (kW )EXP 1:Extrap.(MS) 9448 67085EXP 1:Towing Tank 63805EXP 3:Extrap.(MS) 10490 98614EXP 3:Towing Tank 99521EXP 8:Extrap.(MS)

16038194165

EXP 8:Comput.(FS) 191599EXP 8:Towing Tank 204035EXP 5:Extrap.(MS)

16038101554

EXP 5:Comput.(FS) 99977EXP 5:Towing Tank 100681EXP 9:Extrap.(MS)

16038193567

EXP 9:Comput.(FS) 192229EXP 9:Towing Tank 209406

Table 6: Power efficiency - 35 kts.

scale, the agreement of the extrapolated power from themodel-scale computation is less than 5% with the extrap-olated value from the corresponding towing tank test. Thesame remark holds about Experiment 3 where the agree-ment is lower than 1%. Comparing Experiment 1 and Ex-periment 3, the negative influence of the skeg is to con-siderably increase the power efficiency by 56% from thetowing tank value, to be compared with the increase of thewetted surface of 11% only. With the extrapolated model-scale computation predictions, the 47% increase is lowestsince the computed effective power for Experiment 1 isslightly over-predicted but slightly under-predicted withExperiment 3.

Concerning Experiments with side hulls (8, 5, and 9)the agreement of the extrapolation of the power efficiencybetween the model-scale computations and the direct full-scale computations is 1.3% for Experiment 8, 1.6% forExperiment 5, and 0.7% for Experiment 9. Compared totowing tank predictions, the same computed full-scale val-ues differ by 6%, 0.7%, and 8% respectively. Simulationsconfirm the extremely favorable interference with the cen-tre hull with side hull in middle locations since the powerefficiency is equivalent (around 1%) to the power obtainedwith the centre hull only of Experiment 3 while the tri-maran configuration has 18% more displacement and awetted surface increased by more than 50%. The otherconclusion is that, at least for the trimaran test case andconsidering the full-scale simulations results, the extrapo-lation technique of the resistance from model to full scalefor both experiments and simulations produces satisfac-tory results.

CONCLUSION

In this paper, an automatic adaptive grid refinementmethod for ship flow computation has been applied to twovery different test cases.

From the results of the KVLCC2 tanker case, we haveseen that for the simulation of viscous double-model shipflow, pressure-based gradient or Hessian criteria are to bepreferred due to their insensitivity to boundary layers. Asin computations without grid refinement, the anisotropicEASM turbulence model gives much better results thanthe k − ω SST model, confirming that anisotropic turbu-lence modelling is required for this type of flow.

Simple gradient criteria are able to detect the rele-vant flow features. In general, the results are equiva-lent to those obtained on manually generated locally re-fined grids (good resolution requires about 1M cells). Yet,with the automatic grid adaptation procedure, the expertknowledge needed to manually create these grids is notneeded. Hessian-based criteria, while sensitive to smallperturbations, show a real potential of being able to gen-erate equivalent results with less cells, thus reducing boththe effort needed in grid generation and in computationtime.

Concerning the HALSS model test and simulation re-sults, is has been demonstrated that a trimaran can bedesigned such that favourable hydrodynamic interactionsoffset almost all of the side hull drag over a practical rangeof speeds. It was found that in case of unfavourable in-terference we are dealing in particular with stern/transomwave breaking. For the highest computed speeds, non-linear effects are extremely large, which justifies hav-ing recourse to an accurate free-surface capturing viscoussimulation. Due to the large deformation of the free-surface and the occurrence of local breaking waves, auto-matic grid refinement has proved to be an excellent tool toproduce reliable simulations with quantitative agreement,by optimising the grids around the water surface in orderto account for these important physical phenomena.

ACKNOWLEDGMENTS

The initial HALSS concept evaluation and test programwork have been sponsored by Center for Commercial De-ployment of Transportation Technologies (CCDOTT) andthe Office of Naval Research (ONR). The FINETM/Marinecalculations and analysis have been performed based onCooperative Research Agreement between ECN/CNRS,CSC Advanced Marine Center, CCDOTT and ONR.

The French authors gratefully acknowledge the scien-tific committee of IDRIS (Insititut du Développement etde Ressources en Informatique Scientifique du CNRS,project 000129) for the attribution of CPU time.

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