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24th Symposium on Naval Hydrodynamics Fukuoka, JAPAN, 8-13 July 2002 Efficient Methods to Compute Steady Ship Viscous Flow with Free Surface Hoyte C. Raven, Bram Starke, Maritime Research Institute Netheriands (MARIN), NetherIands. Abstract The computation of the steady viscous flow with free surface around a ship hull is addressed. To improve its practical applicability, two different alternative meth- ods are proposed: · a 'composite approach' in which the wave sur- face is computed beforehand using an inviscid- flow approximation; · a new formulation which permits to solve the vis- cous free-surface problem by iteration, without time integration. The former method is practical, provides accurate pre- dictions of many flow features, but neglects viscous ef- fects on the wave pattern. Applications are shown to the KCS test case, and to a case with wave-induced flow separation. The latter method solves the complete free- surface RANS problem by an unconventional approach which removes several difficulties and promises signif- icant reductions of computation time and complexity. Results are shown for a 2D bottom bump, a 3D pres- sure patch and a Series 60 case. 1 INTRODUCTION The practical application of Computational Fluid Dy- namics methods for predicting the steady flow around a ship hull has made much progress over the last decade. Today, several of the CFD tools play an important role in the ship hull form design. In particular nonlinear free-surface potential flow codes for predicting a ship's wave pattern are routinely used at institutes and ship- yards since several years. These codes are generally ro- bust, efficient and versatile, and give a good prediction of the quality of a design from the viewpoint of wave making (Raven, 1998~. Complementary to that, in recent years the use of RANS solvers for predicting the viscous flow around the hull has increased. This provides important new possibilities such as prediction of flow separation, wake fields, thrust deduction, local flow directions etc. Sub- stantial hull form improvements can thus be achieved at least for fuller hull forms. The complementary use of inviscid wave pat- tern predictions and RANS predictions for viscous flow generally implies a separate consideration of these as- pects. Often the viscous flow computations suppose symmetry boundary conditions at the undisturbed wa- ter surface, the so-called 'double-body' approximation. Thus neither an effect of the free water surface upon the viscous flow, nor an effect of the boundary layer and wake upon the wave pattern is incorporated. Ob- viously, the same approximation underlies most towing tank work, in particular resistance extrapolation meth- ods; but taking into account these interaction effects is a desired next step. This objective has given rise to perhaps the most active field of numerical ship hydrodynamics to- day, the development of solution methods for the prob- lem of the steady viscous flow around a ship hull with free-surface boundary conditions. A variety of methods has been proposed and appreciable progress has been made, as testified by the results presented at the recent Gothenburg Workshop (Larsson et al, 2000~. However, at present such RANS/FS (Reynolds-Averaged Navier-Stokes equations with Free Surface) methods are not widely applied in practical ship design yet. Besides still required further improvement in accuracy, the computation times and memory requirements of these methods are a draw- back. Within the short time frame of a usual merchant ship design project, one often resorts to the simpler double-body approximation to be able to complete the
study in time. Therefore, to have practical benefit of RANS/FS solvers, certain further improvements are desired. The present paper addresses two questions in this regard: · Can more efficient approaches be devised to solve the full RANS/FS problem? · Is a full RANS/FS method always necessary or are there suitable intermediate levels that include the relevant physics but require less computa- tional effort? In Section 2, we discuss the state of the art in RANS/FS methods, and indicate the aspects we believe should be improved. Section 3 describes a practical ap- proach that combines a free-surface potential flow code and a RANS solver. This appears to provide a solution that is equivalent to that of a full RANS/FS solver in the largest part of the domain, but at a fraction of the cost. In Section 4 we derive a new formulation of a full RANS/FS algorithm that solves the steady prob- lem by iteration, discarding any time-dependence; and we show the results obtained so far, which indicate fast convergence for the test cases tried. Prospects are dis- cussed and conclusions drawn in Section 5. 2 STATE OF THE ART IN RANS/FS METHODS Methods used While there is a variety of RANS/FS methods for ship hydrodynamic applications, they have several aspects in common, which we shall describe now to provide some background to further discussions. Almost all RANS/FS methods solve the RANS problem for the flow around the ship hull sub- ject to the usual boundary conditions on the hull (i.e. no-slip, sometimes wall functions); uniform-flow con- ditions at a remote outer boundary; and fully nonlinear free-surface boundary conditions (FSBC's) imposed at the actual water surface. The velocity components (in a coordinate system fixed to the ship, with x positive aft) are u,v,w, ((x,y) is the wave height, and all quantities are nondimensionalised using ship speed U. a reference length L, and gravity acceleration g. The general form of the FSBC's then is: · a kinematic condition that the free surface moves with the flow; ~t+U<x+v~y-w=o at z=; (1) · a normal component of the dynamic condition, requiring that at the surface the pressure is atmo- spheric in the water (p = 0); neglecting surface tension and viscous contributions this takes the form Fn2 Or~ = 0 at z = ~ (2) in which or = (p + pgz)/ (p U2) is the nondimen- sional hydrodynamic pressure. · two tangential components of the dynamic con- dition, requiring that no shear stress is exerted on the water surface: t.~.n=0 (3) Here, n is the unit vector normal to the wave sur- face, text are orthogonal unit vectors tangential to this surface and ~ denotes the viscous stress tensor. In 'moving-grid methods', the computational grid is updated at every time step such that it matches the (instantaneous) water surface. Thus the free surface is a boundary of the domain and the FSBC's can be imposed in a straightforward manner. Alternatively, in fixed-grid methods the grid is not free-surface conform- ing, and the boundary conditions either have to be inter- polated/extrapolated towards the free surface location, or are inherently satisfied by including the flow of the air above the water surface into the computation. Due to the large density ratio between air and water, the dy- namics of the air have no significant effect, and again the normal and tangential components of the dynamic condition are satisfied. Also, whatever the method, a kinematic condition is always there; either as a direct boundary condition, in a moving-grid method; or in the convection equation for the level set function, or in the algorithm to update the free surface in a VOF method. It is important to note that all RANS/FS meth- ods we have seen in ship hydrodynamics solve the un- steady form of the problem. In moving-grid methods the kinematic condition (1) then prescribes the displace- ment of the free surface in each time step, while the dynamic condition can be imposed to the RANS solu- tion in each time step. In fixed-grid methods a similar mechanism is used. The steady solution is thus obtained by computing the evolution of the flow after a grad- ual acceleration of the ship to the desired speed, and the subsequent evolution of the flow towards a steady state. This has proven to be a reasonably straightfor- ward extension of many RANS codes, and to provide a fairly stable approach. However, in practical applica- tions it is found to require substantial computation time, as a result of a slow approach to steady state, persistent unsteadiness and small permitted time steps. We come back to this issue in Section 4. Quality of the predictions At the recent Gothenburg Workshop (Larsson et al, 2000), almost all RANS/FS methods accurately pre-
dieted the wave profile along the hull. However, the wave pattern away from the hull was much more vari- able, and generally subject to appreciable numerical damping. In particular for the 5415 test case with its more diverging waves, several predictions had already lost half of the bow wave amplitude in the wave cut at y/L= 0.172. Also the predicted stern wave heights in the wave cuts presented at the workshop were generally poor and most variable for RANS/FS methods: The el- evation of the first stern wave crest was underestimated by 20 to 85 %. The only inviscid prediction presented, from RAPID, had just the right stern wave elevation at the position of the wave cut (but a too large amplitude further aft and right aft of the transom). Since solving the full RANS/FS problem would be expected to pro- vide a more accurate prediction due to incorporation of viscous effects on the stern wave system, there is still some work to do. To improve the predicted wave amplitudes at a distance from the hull, denser meshes should probably be used; but it still is difficult today to reach sufficient resolution to accurately predict the wave pattern even at some distance from the hull, and at the same time have sufficient resolution in the boundary layer; in par- ticular at high (full-scale) Reynolds numbers and low Froude numbers. Computation times are very substan- tial already, e.g. often of the order of 100 hours CPU on a fast workstation, for a grid of 1.5 2 million cells. Nevertheless, clear progress is evident com- pared to results published earlier, and the methods clearly have good prospects to produce accurate solu- tions in a near future. Desired improvements For a sensible study of ship wave making and to deduce desired modifications to the hull shape in order to re- duce dominant wave components, it is not sufficient to have only an accurate hull wave profile, since that may lead to wrong conclusions; and it is important to have a prediction of the wave pattern up to e.g. 0.3L off the ship's centreplane. Also short, diverging waves should be represented. Therefore, to improve the accuracy of the predictions, minimising the numerical wave damp- ing is indispensable in the first place. Secondly, we believe there is room for re- ducing computation time and complexity of RANS/FS computations; and a possible approach is presented in Section 4. In the third place, virtually all RANS/FS meth- ods proposed aim at solving the complete problem of the RANS equations subject to the nonlinear free- surface boundary conditions imposed at the real wave surface over the entire domain. It may be useful to have available a hierarchy of simpler approaches, from which the best compromise between completeness and complexity can be selected dependent on the strength of physical phenomena in an application considered. A possible hierarchy of this kind will be proposed in Sec- tion 5. Finally, all RANS/FS solutions we have seen so far have been for model scale. To really exploit the additional capabilities of RANS/FS methods, they must be able to give accurate predictions of the full- scale wave pattern as well. The particular advantage of RANS/FS methods, the incorporation of viscous effects on the wave pattern, is of little practical interest if it only applies to model scale; for the model-scale resis- tance prediction thus obtained we do not even see a rea- sonable way of extrapolating it to full scale, unless by techniques used for model tests which again disregard those same effects. The present paper addresses the second and third item mentioned above, and provides some indirect information on the 4th item. COMPUTING VISCOUS FLOW UNDER A PRECOMPUTED WAVE SURFACE Full RANS/FS incorporates the interaction between wave generation and viscous flow; both wave erects on the viscous flow and viscous effects on the wave gener- ation and propagation. Wave effects on the viscous flow around a ship hull may be significant for all cases with substantial wavemaking, the wavy upper boundary of the viscous flow domain having an effect on the devel- opment of the boundary layer all along the hull. On the other hand, viscous effects on the wave pattern are gen- erally insignificant along most of the hull (except those connected with wave breaking) and only substantial in the stern area, as is confirmed by many validations of nonlinear free-surface potential-flow methods (Raven, 1998~. Therefore, along most of the hull there is no in- teraction between the viscous flow and the free surface, but just a one-way action of the wave pattern on the vis- cous flow. This can be incorporated completely by first computing the wave pattern using a panel code, and next to compute the viscous flow under that wave sur- face, imposing free-slip boundary conditions at that sur- face. Surprisingly, this possibility seems to have been largely disregarded so far, except for the similar work at NTUA (Garofallidis, 1996) in which viscous flow was computed under a measured wave surface. Compared with a full RANS/FS solution we have to disregard one of the boundary conditions, the normal component of the dynamic boundary condition
at the wave surface. The degree to which that condi- tion appears to be violated by the solution, i.e. the re- sulting deviation from atmospheric pressure at the wave surface, indicates the validity of the approximation, as discussed below. This 'composite approach' has been briefly proposed in (Winds & Raven, 2000) and (Hoekstra et al, 2000), and in the meantime has been refined and ap- plied. It uses two essentially standard tools: an inviscid wave pattern prediction, and a viscous flow prediction without free surface. The methods we use will be briefly discussed next. Inviscid wave pattern computation The calculation of the wave pattern is made using a non- linear free-surface potential flow method. We use the code RAPID (Raven, 1996,1998), an iterative 'raised- panel' method. The hull surface and a surface at a spec- ified distance above the wave surface are covered with source panels that, together with the incoming undis- turbed flow, determine the velocity field. The strengths of the source panels are solved from the boundary con- ditions. On the hull, zero normal velocity is imposed; on the wave surface, the kinematic condition (eq. 1, in steady form) and the dynamic condition (2) are im- posed. Therefore, the free-surface boundary conditions correspond with those imposed in a viscous how, except for the tangential dynamic conditions (31. The boundary conditions being nonlinear and to be imposed on an unknown wave surface, the solu- tion needs to be found by an iterative process in which the free surface and the coefficients in the FSBC's are updated and the trim and sinkage of the hull are ad- justed. In practical cases, ~10) iterations are needed. The method is known to predict the wave pat- tern accurately, with as principal exception an overesti- mation of the stern wave system due to the neglect of viscous effects. Computation times are modest, of the order of 15 minutes on a usual PC for a practical case. Computation of the viscous flow All viscous-flow computations are carried out with the computer code PARNASSOS (Hoekstra, 1999; Hoek- stra & E,ca, 1998), which solves the steady Reynolds- averaged Navier-Stokes equations on a boundary-fitted mesh. The momentum and mass-conservation equa- tions are fully coupled and solved in primitive-variable form, without resorting to pressure correction meth- ods or the artificial compressibility approach. Addi- tional transport equations associated with the turbu- lence model are treated as uncoupled from the mass and momentum equations. Menter's one-equation model (Menter, 1997) is used, extended with the correction by Dacles-Mariani et al. (19951. The governing equations are integrated down to the wall (no wall-functions are used, not even for the full-scale computations). Mesh points are strongly clus- tered towards the hull to capture the gradients in the boundary layer. The resulting very high aspect ratio of the cells near the hull puts high demands on the solver for the linear systems, which is one of the motivations to maintain the coupling between the equations in the iterative solution. All terms in the momentum and mass- conservation equations are discretised at least second- order accurately. To avoid negative turbulence quanti- ties, only in the transport equations in the turbulence models we use a first-order upwind scheme for convec- tion. The resulting set of non-linear algebraic equations is quasi-Newton linearised. In order to reduce the size of the discrete equa- tion system we use a marching solution scheme. The velocities and pressure of a number of transverse grid planes are solved simultaneously. The grid planes are visited in downstream order, while the elliptic character of the RANS-equations is numerically recovered by it- eration; contrary to earlier versions no 'parabolisation' is applied. Each step of this iteration scheme includes not only the downstream sweep through the computa- tional domain, in which the eddy viscosity, the veloci- ties and the pressure are updated, but also an additional upwind sweep in which only the pressure is updated. A computation is continued until the maximum variation of the static pressure coefficient between successive it- erations drops below 1 x 10-4. In usual ship computations, the computational domain extends from 25% of the ship's length in front of the bow to 25 to 50% of the ship's length beyond the stern. The width and the depth of the mesh are taken approximately equal to twice the breadth and twice the draught of the ship, respectively. At the outer bound- ary of the viscous-flow domain, boundary conditions are imposed that are derived from a potential-flow cal- culation. The PARNASSOS code is applied on a regu- lar basis in practical ship design projects at MARIN, now in some 70 computations per year; a number that is quickly increasing. The composite approach The composite approach consists of first computing the wave pattern using the free-surface potential flow code; next, to generate a 3D grid in the domain under the re- sulting wave surface; and then to solve the RANS prob- lem on that grid. Mainly the last step has some particu- lar aspects, to be discussed now. The boundary conditions imposed to the vis- cous flow are the usual no-slip condition on the hull;
tangential velocities and pressure at the outer boundary, taken from the inviscid wave pattern calculation; and an outflow condition for the pressure derivative also taken from the inviscid flow. For the free surface, there is now one degree of freedom less compared to the original free-surface problem, since the wave elevation is now fixed. Conse- quently, like in any fixed-domain RANS problem only three boundary conditions can be imposed for veloci- ties and pressure at this surface, and one has to be sacri- ficed. In our implementation the imposed set of bound- ary conditions models a free-slip surface: the kinematic FSBC (in steady form) and the tangential components of the dynamic condition are retained, giving the fol- lowing boundary conditions for the velocity compo- nents: AX + vinyw = 0, or V.n = 0 t.~.n = 0 (4) (5) Zero normal derivatives are imposed for all turbulence quantities. The pressure is evidently coupled to the en- tire velocity field, and is primarily evaluated from the discretised momentum balance normal to the wave sur- face. We thus ignore the normal component of the dy- namic FSBC, eq. (2). The error in this condition, i.e. the deviation from the atmospheric pressure at the wave surface, gives an indication of the validity of the given wave surface for the viscous flow. From eq. (2) this er- ror can be expressed in a local wave-height difference according to A; = Fn2/`yr where ibid = a =~/(PU21- (6) I r rig/ kI- - ~ If the pressure deviation turns out to be zero everywhere, our solution is identical to one of a full RANS/FS solution. However, in practice at least some local pressure deviations will be found. The meaning of a solution having some local pressure deviations may not be immediately apparent but can be understood as follows. Suppose that the viscous flow under the pre- scribed wave surface is found to have a pressure devia- tion /~r~x,y). Then this prescribed wave pattern would be correct for an imaginary case in which the same pres- sure field llyr would act as an external pressure on the wave surface. The required change of the prescribed wave surface for our actual case is then the change of the wave pattern that would result from a removal of that external pressure field. In the spirit of a linearisa- tion we may assume this to be qualitatively equal to minus the wave pattern of such an external pressure dis- tribution, i.e. a local increase of the wave elevation ap- proximately in agreement with eq. (6), plus a trailing ~ ,---~00005~ ( Figure 1: Sten, wave pattern for Series 60 case, model scale; bottom: wave height from inviscid flow computation; top: wave height difference from pressure residual in RANS com- putation; wave pattern within a 'Kelvin wedge' downstream of it. Upstream of the location of the pressure residual, its effect decays quickly with distance. In other words: if our result has only a local pressure deviation at an iso- lated spot, we will have very nearly the full RANS/FS solution everywhere upstream of it, but not in a Kelvin wedge area downstream of it. The fact that the change of the trailing wave system is not observed in the pressure residual, is un- derstandable by considering the fact that such a change is a consequence of the generation and propagation of wave energy. By the elimination of the dynamic bound- ary condition from the system we solve, this compu- tation does not display any wave physics by itself. A local pressure error thus will not cause a trailing wave system. Besides the pressure deviations resulting from the modelling, there may be some due to numerical er- rors as well. Both the inviscid-flow and the viscous- flow computations will involve slight discretisation and other numerical errors. Since the numerical methods are entirely different, the numerical errors will have a different pattern. This again may cause some pres- sure residuals at the prescribed wave surface, which have no physical meaning. This kind of pressure resid- uals should vanish upon grid refinement, and Windt & Raven (2000) show an example of this. Results and applications Series 60 Figure 1 shows the pressure residual, expressed as a wave-height difference, for the Series 60 Cb = 0.60 model at Froude number En = 0.316 and model-scale Reynolds number Rn = 3.4 x 106, as reported by Windt and Raven (2000). Compared with the wave height itself, the wave-height difference is only substantial around the stern, due to viscous effects on the stern wave system. Besides, very close to the bow and the bow wave crest some small wave height differences oc- cur due to numerical errors (not shown in the figure). Everywhere else the difference is negligible (less than 5 x 10-4L or 2.7 % of the maximum wave elevation).
KCS container ship The composite approach has also been applied to the 'KRISO Container Ship' (KCS) at Froude number Fn = 0.26, both for model and full-scale Reynolds numbers (Rn = 1.4 x 107 and 2.5 x 109, respectively). For the computation on model scale the mesh consists of 345 points in main stream direction, 101 points in wall-normal direction and 53 points in girthwise direc- tion. For the computation on full scale the number of points in wall-normal direction is increased to 151. The maximum distance of the grid points adjacent to the hull is below y+ = 0.6. The availability of extensive experimental wave-height data allows a detailed comparison of the inviscid wave height prediction with reality. Fig. 2 com- pares the wave pattern from RAPID with the data and shows good agreement, with differences that are essen- tially confined to the region aft of the transom and the stern wave system. Figure 2: Wave pattern of KCS; top: RAPID prediction; bot- tom: experiment -0.006 ~.0045 ~.003 ~.00150.0005 0.002 0.0035 0.005 0.0065 0.008 Figure 3: KCS, model scale; top: wave height difference from pressure residual in composite solution; bottom: wave height difference between experiment and inviscid solution. Fig. 3 attempts to correlate these differences with the pressure residuals in the composite solution. The bottom half shows the predicted wave elevation Figure 4: As Fig. 3, close-up of stern area subtracted from the experimental wave pattern. Some minor deviations occur along the crests of the bow wave system. These can be due to small phase errors or insuf- ficient resolution; of course, subtracting the two wave signals is a severe test of the accuracy. However, the principal deviations occur aft of the transom. It seems rational to attribute these to the neglect of viscous ef- fects on the stern wave system, and this is essentially confirmed by the top half of Fig. 3, which shows the wave-height difference from the pressure residual in the viscous-flow computation at model scale. Again the principal deviations occur directly aft of the transom. Fig. 4 is a close-up of the stern area of the same figure. It illustrates that the differences between the inviscid prediction and the experimental data have the form of a persistent error in the entire stern wave system, while the pressure differences in the viscous solution are just local. This is a clear demonstration of the statement made above on the meaning of a pressure residual. ~ c ~ c o~_~ ~ i'' '"\\. ~1 ~ ~ experimen ~1 prescribed wave surface from Rapid - - corrected wave surface from Parnassos, model scale ~ ---;--- corrected wave surface from Parnassos, full scale JO ° 0.5 0.6 0.7 0.8 ,== x/L 1 0.9 Figure 5: Stern wave of KCS, at the centreline 1.0
For a more detailed comparison between the experimental and predicted stern wave systems, wave cuts have been made near the centreline aft of the tran- som, see Fig. 5. Substantial differences between the ex- periments and the inviscid predictions are found, e.g. a 30 % overestimation of the first wave crest at the centre- line. However, the first experimental marker lies higher than the transom edge (which is at the start of the full line in the graph), indicating the presence of a thin dead- water zone; the transom was not entirely cleared in the experiment. This, and the fact that a boundary layer is shed from the hull, leads to the reduced wave amplitude and the forward shift of the stern wave peak. If we now add the computed wave-height dif- ferences (eq. (6~) to the imposed wave elevation, we get the dashed line shown in Fig. 5, for the model-scale Reynolds number. This of course neglects the effect on the trailing wave system but gives an impression of the magnitude of the correction. It indicates a reduction of the stern wave height, of approximately the right size, confirming again that the neglect of viscous effects on the stern wave system is the main cause of errors in the inviscid wave pattern computation. The same can be done for full scale. Under the same wave surface we have generated an appro- priately refined grid and have run the RANS computa- tion for the full-scale Reynolds number of 2.5 x 109. Of course viscous effects decrease for increasing Reynolds number, most notably resulting in a decrease of the boundary-layer thickness and the width of the wake. This is e.g. reflected in the predicted wake fraction, which decreases from 0.29 to 0.18 between model and full scale in the present computations. The full-scale composite solution shows significantly smaller pressure deviations at the stern; and applying these to the im- posed wave elevations we obtain the dotted line in the same figure. The conclusion that at full scale the vis- cous effects on the wave pattern will be one half to 2/3 of those at model scale here seems justified. The effect of the wavy surface on the wake field is depicted in Fig. 6. In this figure a comparison is made between the double-body solution, the present solution beneath the precomputed wavy surface and ex- perimental data at the line z/L = - 0.03 in the propeller plane of the KCS container vessel. A moderate im- provementis foundin the axial velocity for y/L > 0.01, while further towards the centerline, that is, in the inner part of the wake, the predicted axial velocity is hardly affected by the shape of the surface. A more substan- tial improvement is found for the vertical velocity com- ponent (w/U), which shows an average increase of ap- proximately dw/U = 0.02 throughout the entire cross section. Similar results were reported at the Gothenburg Workshop (Larsson et al, 2000) for full RANS/FS com- 1 0.8 0.7 0.6 ~0.5 0.4 0.3 0.2 0.1 o ; . 1 ' ' ' ' I ' ' ' ' I UIU 0.9 o / / o o A _ o -O.1 0 0.005 0.01 y/L V I U 0.015 0.02 Figure 6: Wake field of KCS. Dahed line: without wavy sur- face; solid line, win wavy surface incorporated; symbols: ex- penment. putations. In general, the results of the composite ap- proach for the KCS case were competitive with those of RANS/FS codes presented at the Gothenburg Work- shop; and for the wave pattern, were even superior to most, with as single exception the area just aft of the transom. A considerable advantage from the practical point of view, however, is the large reduction in calcula- tion time. In the present computations both the conver- gence behaviour and the calculation time are compara- ble to double-body computations. An indication of the required computational effort (on a single R12000 pro- cessor of an SGI Octane workstation) is given in Table 1. The inviscid computation of the wave pattern preced- ing it took just a few minutes of CPU time. Table 1: Computation data for the KCS case Scale model full number of grid nodes 1.85M 2.75M number of global iterations 189 151 tolerance ACp~r 10-4 10-4 CPU time [h] 10 15.75 Tanker stern A second example to indicate the practical applicabil- ity of the composite approach is given in Fig. 7. Here, the result is shown of a tuft test performed in one of MARIN's model basins for a full-block ship (Cb= 0.89) at a Froude number Fn = 0.159 and a model- scale Reynolds number Rn = 1.3 x 107. This was part of a research project by the CRS-Pods working group of MARIN's Cooperative Research Ships. In the photograph from the experiment a sub-
Figure 7: Flow directions along the afterbody of a full tanker form. Top: experiment; middle: RANS without wavy surface; bottom: RANS with wavy surface. stantial region with flow reversal can be seen close to the waterline near the stern. In the tests it was observed that the extent of this flow separation depended on ship speed, and that the onset of separation coincided with the onset of wave breaking at the same location. There- fore this was an evident case of viscous / wave interac- tion. The experimentally determined flow pattern near the hull is compared with the viscous flow pat- tern predicted by a double-body computation as well as by the composite approach. As can be seen in the middle figure, the flow separation is not present in the double-body computation. The computation under a precomputed wave surface, however, does indeed pre- dict the experimentally observed area of flow reversal, albeit slightly smaller in size. The correct representa- tion of the stronger adverse pressure gradient in the free-surface flow appears to be important. Of course, some differences can be expected near the free surface, since the present approach does not take into account the interaction between the flow reversal and the free surface and does not model wave breaking. Neverthe- less, it can be seen that it is capable of capturing some relevant physical phenomena in the flow, and in the present case indicates the wish for a hull form modi- fication to suppress this wave-induced flow separation. 4 A STEADY ITERATIVE SOLUTION METHOD FOR FREE- SURFACE VISCOUS FLOWS Motivation While the composite approach described in the previ- ous section is a step forward and works efficiently, the fact that the wave pattern is determined by an inviscid approximation is a restriction. The absence of viscous effects on the wave pattern causes deviations from the experimental data at the stern, as was observed for the Series 60 and KCS cases considered. While these de- viations are localised, they make the prediction of the stern wave amplitudes, and thereby of the wave resis- tance, inaccurate; and in principle also the viscous flow then will be locally less accurate. Therefore, the method is limited and a complete, fully interactive RANS/FS method is still desired; albeit possibly confined to just the stern area. In setting up such a method, we have cho- sen a quite unconventional approach, first published in (Raven & Van Brummelen, 1999) and (Van Brumme- len & Raven, 20001. The motivation was that we want to avoid some drawbacks of most RANS/FS methods: the long computation times, the need to pass through all temporary stages to find the steady state solution,
the persistent time dependence, etc. Therefore, we shall first consider more closely the basis of these problems. Time dependence and integration times As discussed in Section 2, virtually all methods pro- posed so far follow a time-dependent procedure until a steady result is obtained. The wave surface is updated using the kinematic free-surface boundary condition, or a similar algorithm is applied for the convection of the wave surface in fixed-grid methods. However, this procedure is the origin of vari- ous difficulties: · The ship has to be accelerated smoothly to its de- sired speed, an impulsive start is usually not per- mitted. This takes time, extending the total inte- gration time to be covered by the computation to come to a steady result. · Once the ship speed has become constant, it still takes substantial time before a steady wave sys- tem has established in an area around a ship. Since in deep water the group velocity is just half the wave phase speed, the wave energy spreads rather slowly, and it takes time before a wave ele- vation at a distance has taken its steady value. It is impossible to find steady state earlier than that if time accuracy is conserved. To give an example: for a point at a distance of 1 ship length from the ship's centreline, a wave component with diver- gence angle of 60 degrees can only begin to settle to its steady elevation once the ship has travelled over 4.6 ship lengths at constant speed. · The initial acceleration of the ship generates tran- sient waves. Before a steady result can be ob- tained, these have to leave the domain or be ab- sorbed by damping regions. If artificial bound- aries are imperfectly transparent, persistent time- dependence may occur in calculations due to sloshing between the boundaries. · As is analysed in (Van Brummelen et al, 2001), the asymptotic decay of the unsteady effects is determined by one particular wave component, the transient wave that propagates in the same di- rection as the vessel and has phase speed twice the ship speed. Its group velocity equals the ship speed, such that its wave energy stays with the ship and only decays by dispersion. In 3D cases, the asymptotic decay of this wave is only ~(1/~. The effect of this wave is evident in many pub- lished convergence histories for wave resistance, as persistent, slowly decaying oscillations with nondimensional period AT.V/L= 8~Fn2. This only further prolongs the required integration, sometimes causing users to just take an aver- age over the last few periods instead of a desired steady solution. · Since the algorithm essentially uncouples the dy- namic conditions (imposed to the RANS solu- tion) and the kinematic free-surface conditions (used to update the wave surface) in each time step, small time steps must be made and a CFL condition usually needs to be respected. · If we refine the discretisation, the time step needs to be reduced. However, in order to have bene- fit of the better accuracy, also the unsteadiness must be reduced. As derived in (Van Brummelen et al, 2001), for a 3D method with second-order discretisation, this means that the number of time steps must be of ~(h-3) halving the grid spac- ing could ask for an 8-fold increase of the number of time steps! · The no-slip boundary condition on the hull, to- gether with the time-dependent kinematic free- surface condition, produces the 'contact line problem': the waterline location cannot move during the time integration. Generally this prob- lem is avoided rather than solved. Of course, there are ways to alleviate some of the difficulties mentioned, and not all methods suf- fer equally. Nevertheless, the number of required time steps generally ranges from 3000 to 30000 for pub- lished 3D ship cases. At each step the grid needs to be adjusted in a surface-fitting method, or the free surface reconstructed in a VOF method; and the effort is ambi- tious to say the least. The obvious solution for these difficulties is to avoid all unsteadiness and to solve a strictly steady form of the problem directly by iteration. The problem then only admits wave solutions that satisfy the steady dispersion relation, excludes any transients at startup, reduces reflection problems at artificial boundaries, and alleviates contact line problems. Exactly this has been done in free-surface potential flow solution methods since a long time. For these it is evident that for solving steady problems, steady solution methods are far more efficient than unsteady ones. Solving this steady prob- lem for RANS equations seems just as desirable the only question is: how? Derivation of the method The free-surface boundary conditions to be satisfied, in a steady form, are: u~x+v~y-w=o atz=; Fn2~r(=0 at z=; t.~.n = 0 (7) (8) (9)
The dynamics of the waves are essentially governed by the normal component of the dynamic con- dition, eq. (8), and the kinematic condition (7); the tan- gential components of the dynamic condition cause the appearance of weak free-surface boundary layers with little effect on the wave pattern. Iterative formulations for the steady RANS/FS problem could be based on alternatingly imposing the normal dynamic and the kinematic boundary condi- tion; like in time stepping approaches. One might im- pose to the RANS equations the kinematic condition on a guessed wave surface, together with the tangen- tial dynamic conditions; and next update the wave sur- face from the normal dynamic condition, i.e. from the pressure difference at the wave surface. This would be a straightforward extension of the composite approach described in Section 3. However, such a scheme would suffer from a drawback it shares with the usual time-stepping meth- ods: it uncouples the kinematic and normal dynamic condition. None of these two conditions has any wave- like character by itself: wave solutions, a dispersion re- lation, group velocity and all other properties of free- surface waves only arise from the combination of both conditions. Uncoupling leaves the task of generating a wave pattern to the iterative algorithm; and we believe we can do better by already building it into our RANS solution at each iteration. Here again, we look at what is being done in inviscid methods. All steady nonlinear inviscid meth- ods apply an iterative procedure, imposing a combined kinematic / dynamic FSBC in each iteration. Since our method is inspired by this, we now will pay some atten- tion to the derivation of the FSBC's in those. Free-surface potential flows For inviscid flows, a velocity potential ~ is usually in- troduced, the velocity v= V¢, and Bernoulli's equa- tion is invoked for replacing the pressure yt. In potential flow the kinematic and dynamic FSBC thus become: ~x; + by - 0z = 0 at z = ~ iFn2~1¢2_~2_~2~_~=0 (11) The next step then made is to combine both equations, by eliminating the wave elevation ~ from the kinematic condition: 2 Fn2 Fax ,33 + (p), ,,jG] + at ~ ~ (~2 + dp2 + (p2) + ¢ = 0 (12) at z = if. This condition forms the basis of iterative schemes for the nonlinear free-surface potential flow problem. After linearization it is imposed in each itera- tion, and the resulting velocity field is then substituted in the dynamic condition to obtain an updated wave sur- face. Unlike the separate FSBC's, the combined condi- tion already incorporates the essence of gravity waves and ship wave making. Even after linearization relative to a uniform flow and an undisturbed water surface, the resulting condition, the familiar Kelvin condition Fn2~+¢z=0 at z=0, (13) provides a qualitatively correct wave pattern in a single step without any iteration. It is this degree of 'implicit- ness' that we aim for in our RANS/FS method. Free-surface viscous flows Returning now to the viscous problem, we may not use Bernoulli's equation or a velocity potential, but the combination of kinematic and normal dynamic condi- tion can be made in exactly the same way. Substituting the wave elevation from the dynamic condition into the kinematic FSBC we get: Fn2(u~\U+v,~lv+w<~~ w=0 at z=~ (14) The set of the combined FSBC and the dy- namic conditions corresponds exactly with the original conditions, and ensures that the pressure variation, the normal velocity and the shear stress vanish at the wave surface. The advantage of the combined FSBC comes once we use it in the context of an iterative procedure. Our iterative process consists of solving the RANS equations under a guessed wave surface, on which the free-surface boundary condition (14) is im- posed, together with the tangential dynamic conditions; followed by a free-surface and grid update based on the normal dynamic condition (8~. The combined FSBC in itself demands neither a zero normal velocity nor a con- stant pressure at the estimated wave surface; but im- poses a relation between the normal velocity and the (10) pressure field, such that the velocity at z= ~ is par- allel to an isobar surface. Once the solution has been obtained, the dynamic FSBC updates the wave surface to that isobar surface. Van Brummelen et al (2001) give a more pre- cise derivation of the free-surface boundary condition (14) and derive that the theoretical asymptotic conver- gence rate of this iterative process is independent of the grid spacing. Again demanding that the residual at the free surface is reduced to the level of the discretisation errors, the number of iterations now is Clog 1/h), as opposed to the ~(h-3) number of time steps for a time- dependent approach.
0.1 i 1 vo.6. 0.4 0.2 o Figure 8: Example of a grid used in the numerical exper~- ments. The grid is coarsened for illustration purposes. -o. 05 Implementation Compared with the free-slip conditions imposed in Sec- tion 3, for a precomputed wave surface, the normal ve- locity component is not set to zero as in (4) but the com- bined FSBC (14) is imposed. This is nonlinear in the unknown pressure and velocities, and is Newton lin- earised. The explicit contributions are taken from the previous inner iteration (in the RANS solution). Second-order differences are used for the FSBC. An upstream difference is required for the gyr/9x term, contrary to its downstream difference in the ~momentum equation. Results 2D Bottom bump An initial, and so far the most complete, study has been done for a 2D test case. Use has been made of the code PARNAX (Hoekstra, 1999), which is a 2D finite-volume code very similar to PARNASSOS . To the solution oro- cess described before, an outer iteration loop was added in which the free surface and grid are adjusted using the normal dynamic condition (8) after a sufficient degree of convergence of the inner loop that solves the RANS equations. The test case is a bottom bump in a shallow 2D flow, at a depth Froude number of 0.43 and Re = 1.5 x lO5. The geometry of the obstacle is y= - 1+ 4 `?3x~(x~t), O ~ x~ ~ ~ , (15) with hb and e the height and length of the obstacle, non-dimensionalized with the undisturbed water depth. Choosing e = 2 and hb = 0.2, the setup is in accordance with (Cahouet, 1984). In this case rather strongly non- linear waves are generated (up to 70 % of the stagnation height). At inflow a boundary-layer velocity profile was specified in agreement with the data. Near the outflow additional wave damping has been applied to prevent possible problems there. Grids of 400 * 70 and 800 * 70 have been used. Figure 8 illustrates the geometry of the bump and the grid obtained after convergence. The main interest of the test is the convergence properties of the iteration for the free surface. Fig. 9 n no \ \ -0.1- O i Figure 9: Example of convergence of Me free-surface shape in consecutive iterations; 400*70 grid. shows the wave profile in consecutive free-surface it- erations, for the hb = 0.20 case. The calculations were started from a flat water surface. Notably, already the first iteration (i.e. a RANS solution under a flat water surface, subject to the FSBC's (14) and (9) ~ produces a wave pattern (labelled 'b') which is qualitatively cor- rect but has reduced amplitude due to the linearization. This entire free-surface adjustment is then applied, the grid updated, and a next iteration performed. The im- plicitness of the combined FSBC permits large free- surface adjustments in a single step and produces a very fast convergence of the wave profile. Notwithstanding the substantial nonlinearity, 9 iterations suffice. In (Van Brummelen et al, 2001) the convergence for this test case is further analysed, and the theoretical result of a convergence rate independent of the grid density is es- sentially confirmed. The convergence ratio is estimated as ~ = 0.45 to 0.52. In an additional test for a reduced bump height of hb = 0.15, generating less steep waves, convergence of the wave profile was obtained in just 3 iterations and the convergence ratio was co = 0.15. Compared to calculations using other methods for the same case, which e.g. required time integration up to a nondimensional time T = 60 (Vogt, 1998) or ~1000) free surface iterations (Tzabiras, 1997), we believe that the behaviour of our method is most en- couraging. Admittedly, from the point of view of the computation time one iteration in our method cannot be compared with one time step in other methods; the many time steps in time-dependent methods also repre- sent the iterative solution of the RANS equations them- selves. Nevertheless, we believe that the strong reduc- tion of the number of free-surface adjustments in itself is an important improvement. For the present case, the solution of the nonlinear free-surface problem takes just 2 3 times the computation time of a case with sym- metry boundary conditions. Figure 10 compares the result with the ex-
of o.os art -o~os 1 2 3 4 ~ x Figure 10: Computed wave elevation (solid line) and mea- surements from (Cahouet, 1984) (symbols), for hb = 0.20. The obstacle is located in the interval x ~ to, 2~. 800 * 70 and. perimental data from Cahouet. Agreement is generally good for the wave length, but the wave amplitude is overestimated and exceeds the experimental uncertainty quoted. Comparable levels of agreement for this test case have been shown in other publications. 3D Pressure patch Next we considere the wave pattern generated by a 3D pressure patch travelling over the free surface. The pa- rameters of the pressure distribution were taken equal to those of Wyatt (2000~. It has a Gaussian distribution, petty) =A.e-(r/B) ; r= ~/(xxO)2+y2 (16) in which we took A = 0.05, B= 0.5, x0 = -6.5. The Froude number based on the unit length was 0.6. The flow is essentially inviscid since no other perturbation is present. The domain considered extends fromx = - 7.5 to x = - 0.5. A 160 * 40 * 40 grid was used, which was essentially uniform at the water surface. The computa- tion was started from an undisturbed water surface and uniform flow. For not too large pressure amplitudes, no free- surface update and grid adjustment have been done. The RANS solution then predicts a wave patteIn shown in Fig. 11. To check the result, the same case has been computed using the inviscid-flow code RAPID, using a panel size comparable to the free-surface cell size. The agreement is fairly good, as shown by the two longitu- dinal wave cuts in Fig. 12. There is evidence of a some- what larger numerical damping and a slightly larger wave length in the RANS solution. Since these results were obtained without any free-surface update, they confirm that also in 3D the x z Yet Figure 11: Wave pattern of pressure patch at Fn = 0.6, from RANS computation. --- Rapid, nonlinear Parnassos, linear FSBC :, 7 ~ ~\ ~ \ , ~ By. . . ~ j ,. ~ , I . 1 , I . 1 . I . I . I -8.5 -7.5 -6.5 -5.5 4.5 -3.5 -2.5 -1.5 -( ).5 x/L Figure 12: Wave cut at centreline (top) and at y/L = 1.05 (bottom) for pressure patch at Fn = 0.6, comparison of RAPID and PARNASSOS . physics of wave making are embodied in the RANS problem subject to the combined FSBC; and that for mild, essentially linear waves no free-surface update is needed at all. The CPU time required to get this result amounted to 12.5 minutes CPU on a single SG R14000 processor at 500 MHz. Series 60 case The last application we show is the viscous flow and wave pattern around the Series 60 CB = 0.60 model at Fn = 0.316, Rn = 3.4* 106. The domain extends from
Figure 13: Wave pattern for Series 60 CB = 0.60 at Fn = 0.316, found from RANS with lineansed FSBC. 0.02 _ nn1 _ -0.01 -0.02 0.02 nn -0.0 ~- -0.5 o 0.5 X/L o- ~ _ ·\ ·7 2 ~ - ~ 1 1 ~ 1 1 1 , , , ~ 1 , , ~ , 1 ~ ~ -0.5 o 0.5 1 X/L 7.~ . · . al, - - ~ 1 Figure 14: Senes 60 CB = 0.60 at En = 0.316, longitudinal wave cuts at y/L = 0.0755 and 0.2067, compared with exper- imental data 0.25L upstream of the bow to 0.5L downstream of the stern, and had a half width of 0.5L. A grid of 320 * 121 * 44 = 1.7 M cells was used, under an undisturbed wa- ter surface. The computation was started from scratch using the free-surface boundary conditions and contin- ued until pressure changes between iterations were be- low 5 x 10-4 everywhere. At present the convergence of the computation with free surface is still slower than usual, and this needs to be addressed. Fig. 13 shows the wave pattern obtained. One should note that no free-surface adjustment has been made in this computation, and the wave pattern thus is a partly linearised result. Fig. 14 compares longitudinal cuts at y/L = 0.0755 and 0.2067 with experimental data from Toda et al(1991) and shows quite reasonable agreement. Further away from the hull the computed wave amplitude is some- what too small, partly due to the linearisation of the FSBC, partly due to numerical damping. This development is underway and progress is still made. However, the purpose of this computation was not to show any final agreement with the data, but to illustrate · that the mechanism of free-surface flow predic- tion using the combined FSBC works as in the other test cases; that the result, although provisional, shows the right wave physics; · that, without any grid update and by just mod- ifying a boundary condition, one already can get fairly close to the experimental wave pattern here; · that no further precautions at all were necessary at the bow or along the waterline: since there is no contradiction at the waterline between the no- slip condition and steady FSBC's, no problems were experienced. Of course, much more needs to be done. Specifically, stability and convergence rate of the com- putation with free-surface boundary conditions needs to be addressed, iteration for the free surface and grid needs to be performed, and efficiency improved. Even so we believe that the cases done so far hold promise for an efficient solution of the full RANS/FS problem in a limited number of iterations. 5 DISCUSSION AND CONCLUSIONS In search for a further extension of the already impor- tant role of CFD in practical ship design, we have con-
sidered possibilities to make the combined computation of viscous flow and wave pattern more readily applica- ble in practical ship design. The present paper describes two different methods: · a composite approach, in which the viscous flow is computed under a wave surface determined be- forehand from a potential flow calculation; · a full RANS/FS method that solves the problem by iteration rather than by time stepping. The composite approach is a combination of tools that are already extensively used in design, i.e. a nonlinear free-surface panel code and a RANS code without free surface. This makes it directly applica- ble in practice. Inspection of the results for Series 60 and KCS has shown the composite solution to be es- sentially equivalent to a full RANS/FS solution, except in a Kelvin wedge starting at the stern. Compared to a double-body RANS solution the composite approach uses no more computation time, but adds the wave ef- fects upon the viscous flow. This was found to be a rel- evant step forward: e.g. it leads to a small improvement in the wake field for the KCS case, but, more impor- tantly, permitted to predict the wave-induced flow sep- aration observed experimentally at the stern of a tanker. On the other hand, viscous effects on the wave pattern are neglected; and there seems to be no obvious way to extend the method to a more complete solution. The other method, iterative solution of the RANS/FS problem, is meant to reduce the long com- putation times needed by most RANS/FS methods. We have shown that by a particular formulation of the free- surface boundary conditions in steady form, all time- dependence can be omitted. This eliminates the long integration times, large number of time steps, resecting transient waves and contact line problems encountered in several other methods. Results for a 2D bottom bump show convergence of the wave pattern in 9 iterations. For a 3D pressure patch, a single iteration appears al- ready to provide an accurate wave pattern. For a Series 60 case, our initial results show that again the first iter- ation, a RANS solution on a grid under an undisturbed wave surface, provides a qualitatively correct wave pat- tern. While there is still much development needed, our present results seem to offer good prospects for a quick convergence to a final solution. These two methods can be regarded as mem- bers of a hierarchy of computational methods consisting of the following levels: · an inviscid wave pattern calculation, and a double-body RANS solution; as two separate components without any mutual influence. This is the traditional level that is widely used now in ship design. · the composite approach proposed here. Basically . the same components are used, but we impose free-slip conditions in the RANS solution. This adds the wave effects upon the viscous flow but lacks interaction near the stern. · RANS with linearised FSBCs. The RANS equa- tions are now solved on a fixed grid under an esti- mated wave surface (either undisturbed, or from an inviscid flow code) subject to the combined FSBC proposed in this paper. This adds viscous erects on the wave pattern and viscous/wave in- teraction. If the resulting wave elevation update is small, the linearisation is accurate and no grid update is needed. Finally, the fully nonlinear RANS/FS, by the iter- ative formulation proposed. This does need grid updates and iteration. Like the previous level, it adds the full viscous/wave interaction, but now without any assumption of the magnitude of the adjustments. The methods we propose in this paper illustrate and anticipate the various levels. We believe that efficient methods to compute steady ship viscous flow with free surface can be designed by choosing the right level from this hierarchy in the right case and in the right part of the flow domain; and we hope this will bring a large- scale and fruitful application of such methods in practi- cal ship design closer. Acknowledgement We thank Mervyn Lewis, Centre for Mathematics and Computer Science (CWI) in Amsterdam, for providing the Series 60 result in Figs 13-14. He cames out parallel work on the method of Section 4, in a project supported by the Netherlands Technology Foundation STW. We acknowledge that support and appreciate the benefit we have from our cooperation. The permission from the CRS-Pods working group of MARIN's Cooperative Research Ships pro- gram to publish Fig. 7 is gratefully acknowledged. REFERENCES Cahouet, J., Etude rlumirique et expenmentale du probleme bidimensionnel de la resistance de Agues non-lindaire", Ph.D. Thesis, ENSTA, Paris, 1984. (In French). Dacles-Mariani, J., Zilliac, G.G. , Chow, J. S. and Bradshaw, P., "Numerical / experimental study of a wingtip vortex in the near field", AIAA Journal, Vol. 33, 1995, pp. 1561-1568.
Garofallidis, D.A., "Experimental and numerical investigation of the flow around a ship model at various Fro ude numbers," Doctor's Thesis, National Techn. Univ. Athens, 1996. Hoekstra, M., "Numerical simulation of ship stern flows with a space-marching Navier-Stokes method", Doctor's Thesis, Delft Univ. Technology, Delft, Nether- lands, 1999. Hoekstra, M., and E,ca, L., "PARNASSOS :An efficient method for ship stern flow calcula- tion" 3rd Osaka Colloquium on Advanced CFD Applications to Ship Flow and Hull Form Design, Osaka, 1998. Hoekstra, M., E,ca, L., Windt, J., and Raven, H.C., "Viscous-flow calculations for KVLCC2 and KCS models using the PARNASSOS code", Gothenburg 2000 workshop. Larsson. L.. Stern. F.. and Bertram, V. (2000), . , , "Gothenburg 2000 A workshop on numerical ship hydrodynamics", Chalmers Univ., Gothenburg, Sweden. Menter, F.R., "Eddy viscosity transport equations and their relation to the k£ model", J. of Fluids Eng., Vol. 1 19, 1997, pp. 876-884. Raven, H.C., "A solution method for the nonlinear strip wave resistance problem", Doctor's Thesis, Delft Univ. Techn., Delft, Netherlands, 1996. Raven, H.C., "Inviscid calculations of ship wave making capabilities, limitations and prospects", 22nd Symp. Naval Hydrodynamics, Washington DC, ~ = ~ 1998. Raven, H.C., and Van Brummelen, H., "A new approach to computing steady free-surface vis- cous flow problems", 1 st MARNET-CFD work- shop, Barcelona, 1999. To be downloaded from http://www.marin.nl/publications/pg_ resistance.html Toda, Y., Stern, F., and Longo, J., "Mean-flow mea- surements in the boundary layer and wake field of a Series 60 Cb = .6 ship model for Froude numbers .16 and .316 IIHR Report No. 352, Iowa Institute of Hydraulic Research, August, 1991. Tzabiras, G.D., lion of 2D steady "A numerical investiga- free surface flows", Int. Jnl. Num. Methods in Fluids, Vol.25, pp. 567- 598, 1997. Van Brummelen, H., and Raven, H.C., "Numerical solution of steady free-surface Navier-Stokes flow", 15th Int. Workshop on Water Waves and Floating Bodies, Caesarea, Israel, 2000. To be downloaded from http://www.marin.nl/publications/pg_ resistance.html Van Brummelen, E.H., Raven, H.C., and Ko- ren, B. (2001), "Efficient numerical soludon of steady free-surface Navier-Stokes flow", Jnl. Computational Physics, Vol. 174, pp. 120-137. Vogt, M., "A numerical investigation of the level set method for computing free surface waves", Chalmers Univ. Technology, Report -CHA/NAV/R-98/0054, Gothenburg, 1998. Windt, J., and Raven. H.C "A composite pro- . . . . cedure for ship viscous flow with free surface", 3rd Num. Towing Tank Symp., Tjarno, Sweden, 2000. Wyatt, D.C., "Development and Assessment of a Nonlinear Wave Prediction Methodology for Surface Vessels", Journal of Ship Research, Vol. 44, No. 2, June 2000, pp. 96-107.
DISCUSSION H. Soding Technical University of Hamburg, Germany My colleagues in Hamburg and I stopped similar developments because we were not successful in handling breaking waves by solving Euler or Rans equations with grids fitted to the free surface. Neglecting breaking waves often leads to non- convergence. Thus I think one must either learn to handle breaking waves with surface-fitted grids, or abandon such grids for steady ship flow computations. AUTHORS' REPLY Prof. Soeding addresses the question whether one should use surface-fitting or surface-capturing methods. There does not seem to be a single answer to that question at present, and both classes of methods are being proposed in the literature. In any case, the composite approach will not be affected by wave breaking, as long as we are able to compute the wave surface beforehand by a panel method; which is no problem for the great majority of cases (except those having severe wave breaking in reality). As a matter of fact, our second method, the steady RANS/FS solution, probably will have problems for cases having substantial wave breaking in the steady flow, like other surface-fitting methods; but, as opposed to the time-dependent approaches, will be insensitive to any wave breaking in an intermediate stage. We expect that ad-hoc local models will be sufficient to remove these problems. DISCUSSION Chi Yang George Mason University, USA Authors have presented a very interesting and smart approach for computing the steady viscous flow with free surface. This approach is based on the assumption that the viscous effects on wave elevation can be neglected. It therefore leads to a very efficient method and also removes the difficulty associated with moving RAN S grids in the viscosity of the hull during the simulation. I would like to know how authors reconstruct the deformed free surface from the inviscid solution to generate RAN S grids. What type of RANS grids is used? The mesh movement will be relative small if the RAN S simulation starts from the inviscid solution. Have authors tried to update the wave elevation during RAN S simulation and compare the results with these obtained without the viscous effect on wave elevation? In order to save computing time, the inviscid solution can also be used as initial solution for RAN S computation to speed up the convergence. I would like to have your comment on that. AUTHORS' REPLY Our "composite approach" is based on the observation that viscous effects on the wave elevation can be neglected in the majority of the domain, except, most notably, the stern region. The free-surface for the viscous-flow computation is obtained by fitting a B-spline surface through the collocation points on the free surface in the inviscid solution. An orthogonal grid generator is then used to define the mesh on the free surface, which forms one of the boundary planes of the single-block, HO- mesh we used in these computations. With respect to the update of the wave elevation in the RAN S computations the answer is twofold. In the case of the composite approach, further grid updates based on the pressure residual at the free surface would correspond to a decoupling of the kinematic and dynamic FSBC's. This is what we have avoided using the combined conditions in our steady iterative method for free-surface flows; as motivated in our paper. Hence we do not intend to perform grid updates in the composite approach. In the case of,the steady iterative method, grid updates have been performed for the 2D bottom bump and the 3D pressure patch, but not yet for the Series 60 hull. Nevertheless, we expect to perform these updates in the near future and undoubtedly will compare the results with inviscid-flow predictions. The last question addresses the possible speed-up of the convergence of the viscous-flow computations using the inviscid-flow solution as an initial estimation. It is recalled that in our viscous-flow solver (PARNASSOS) we impose the pressure and the tangential velocities obtained from an inviscid- flow computation at the outer boundary of the domain. Rather than evaluating the inviscid velocities at every node in the interior of the grid as well, including the ones in the boundary layer, we basically start with a uniform pressure and velocity field. The initial sweeps on the coarser meshes in . our gee -sequencing procedure then essentially generate the initial estimation of the pressure and velocity field for the finer meshes. This procedure has been found to be quite stable and robust and leads to the computation times listed in our paper.
