Viscous-flow calculations for bare hull DARPA SUBOFF submarine at incidence

Transcription

1 International Shipbuilding Progress 55 (2008) DOI /ISP IOS Press Viscous-flow calculations for bare hull DARPA SUBOFF submarine at incidence Serge Toxopeus Maritime Research Institute Netherlands (MARIN)/Delft University of Technology, Wageningen, The Netherlands As part of a multi-nation study regarding CFD validation, conducted by the Submarine Hydrodynamics Working Group, viscous-flow calculations on the unappended hull-form of the DARPA SUBOFF sailing straight ahead and at oblique motion were conducted in order to verify the accuracy of the predictions. The study shows good general agreement with the experiments for local field values as well as for global quantities. Keywords: DARPA SUBOFF, submarine, CFD, RANS, manoeuvring 1. Introduction By order of the Defence Materiel Organisation (DMO) of the Royal Netherlands Navy (RNLN) the Maritime Research Institute Netherlands (MARIN) conducts studies regarding the hydrodynamics and manoeuvring capabilities of submarines. Together, they are actively involved in the Submarine Hydrodynamics Working Group (SHWG). The SHWG is a group of international navies from Australia, Canada, Germany, France, UK, USA and The Netherlands. In general, the navies team up with local research institutes. Informal meetings are held every eighteen months to discuss non-classified subjects related to hydrodynamics of submarines. Within the SHWG, a study was initiated to assess the state-of-the-art in computations of viscous flows around a submarine and to accelerate and improve research and development of numerical submarine hydrodynamics. In the study, a common testcase was defined for which each navy was asked to generate predictions based on their best approach in viscous-flow calculations. The central question in the study is to demonstrate the capability of viscous-flow solvers to reproduce the flow field around and the forces acting on a submarine hull form. This article presents the work conducted by the Dutch team. * Address for correspondence: S. Toxopeus, Maritime Research Institute Netherlands (MARIN)/Delft University of Technology, P.O. Box 28, 6700 AA, Wageningen, The Netherlands. Tel.: ; Fax: ; S.L.Toxopeus@MARIN.NL X/08/$ IOS Press and the authors. All rights reserved

2 228 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF The submarine form considered is the DARPA SUBOFF hull form, as described in Groves et al. [7]. For this hull, ample comparison material is available. For the current work, only the bare hull, i.e. without sail, planes and propeller, is used. This condition corresponds to Configuration 3 as defined in Roddy [15] and to configuration AFF-1-* as defined in Liu and Huang [11]. In this paper viscous-flow calculations for the DARPA SUBOFF for straight flight and oblique motion are presented. Special attention is paid to the uncertainty in the predicted results and the correlation between the results and available experiments. 2. DARPA SUBOFF The main particulars of the DARPA SUBOFF as used in this work are specified in Table Grid generation A structured grid has been generated around the hull form. Use was made of a body-fitted non-orthogonal HO-type grid, which was strongly stretched towards the hull to capture the strong gradients in the boundary layer. Grid-refinement was also applied at the bow and at the stern for this reason. The grid was generated with MARIN in-house grid generation tools, see Hoekstra [8] or Eça et al. [6]. The size (length, depth and width) of this domain is based on experience with simulations of the viscous flow around conventional ships and on previous calculations done by MARIN for the DARPA hull form. During grid generation, first a base-grid is generated using a 3D elliptic grid generator, with a reasonably stretched grid node distribution in the normal direction, j, and the desired node distributions in stream-wise, i, and girth-wise, k, directions. By varying the control parameters in the grid generation process, the deviation from orthogonality is reduced as much as possible. Then grid stretching along the surfacenormal grid lines is applied to arrive at the desired grid spacing at the hull surface. Table 1 Main particulars of bare hull DARPA SUBOFF submarine Description Symbol Magnitude Unit Length overall L oa m Length between perpendiculars L pp m Maximum hull radius R max m Centre of buoyancy (aft of nose) FB L oa Volume of displacement m Wetted surface Swa m

3 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF 229 Only the starboard side lower half was modelled for zero drift since for this condition the flow may be assumed to be symmetrical with respect to the longitudinal horizontal and vertical planes. For non-zero drift angles, the port-side quarter of the hull was also modelled. Each grid block (port-side and starboard quarters) had nodes in stream-wise, wall-normal and girth-wise direction respectively, resulting in a total of over 1.8 million nodes for the straight-ahead condition and 3.7 million nodes for non-zero drift angles. For oblique motion, the domain is increased as a function of the drift angle in order to geometrically incorporate the drift angle and the larger perturbed flow area around the hull. On each side of the domain the grid consists of an inner block and an outer block. The inner block is the same for all calculations and the outer block can deform to allow for the drift angle and/or rotational motion of the ship. The outer blocks can be generated automatically by scripts and therefore grids for various manoeuvring motions can be made efficiently. The six boundaries of the computational domain are the following: the inlet boundary is a transverse plane located upstream of the forward perpendicular; the outlet boundary is a transverse plane downstream of the aft perpendicular; the external boundary is a circular or elliptical cylinder; the remaining boundaries are the ship surface, the vertical symmetry plane of the ship or coinciding block boundaries and the horizontal symmetry plane. The inflow plane was located at least 0.67L oa forward of midship (depending on the drift angle, a larger domain was used) and the outflow plane at least 0.78L oa aft of midship. The width and depth of the domain were both at least 0.23L oa away from the symmetry axis. In Fig. 1, a view of the inner block of the grid (coarsened for presentation purposes) used for this study is presented. The bow is directed to the left of the figure. Fig. 1. DARPA SUBOFF, computational grid (coarsened by factor 4), β = 0.

4 230 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF 4. Numerical procedures 4.1. Coordinate systems The origin of the right-handed system of axes used in this study is located at the intersection of the longitudinal axis of symmetry of the hull, midship and centre-plane, with x directed aft, y to starboard and z vertically upward. Note that all coordinates given in this paper are made non-dimensional with the overall length L oa of the submarine (L oa = m) unless otherwise specified. Furthermore, in the presentation of the results, the longitudinal positions along the hull are given with x = 0 defined at the bow. This facilitates easy comparison with other results published in literature. All velocities are made non-dimensional with the undisturbed velocity V 0. All integral forces and moments on the hull are based on a right-handed system of axes corresponding to positive directions normally used in manoeuvring studies. This definition was also used during the DARPA SUBOFF model tests. This means that the x-axis is directed forward, y to starboard and z vertically down. Similar to the results presented by Roddy [15], all moments are given with respect to the centre of gravity, which is located at L oa aft of the nose of the model. Nondimensionalisation is done with the length between perpendiculars (L pp = m) using X, Y, Z/ 1 2 ρv 0 2L2 pp and K, M, N/ 1 2 ρv 0 2L3 pp. The drift angle is defined by β = arctan u v which means that β is positive for flow coming from port side Boundary conditions At the hull surface, no-slip and impermeability boundary conditions are used. The velocities are set to zero (u = 0). Symmetry conditions are used on the longitudinal plane(s) of symmetry (for the definition of the curvilinear coordinates ξ, η and ζ,see Fig. 2): u ζ = 0, u ξ ζ = 0, u η ζ = 0, u ζ ζ = 0. Because the velocity and pressure behind the ship are unknown, a Neumann boundary condition is applied on the outflow plane: u i ξ = 0, p ξ = 0. The velocity components in the inflow plane and on the external boundary are taken from a potential flow calculation. For the inflow boundary, the three velocity components are taken from the potential flow solution, while in the external boundary, the tangential velocities and the pressure are set. During the viscous-flow cal-

5 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF 231 Fig. 2. Definition of curvilinear coordinate system (bow to the right). culation, the velocity normal to the external boundary is updated to allow for the displacement effect of the boundary layer Solver set-up Use was made of the in-house solver PARNASSOS, see Hoekstra [8]. PARNAS- SOS solves the Reynolds-averaged Navier Stokes equations for a three-dimensional, steady, incompressible flow around a ship hull, supplemented by a turbulence model, using finite-difference discretisation. For all calculations, use was made of Menter s one-equation turbulence model [13] (designated MNT) or two versions of the twoequation k ω turbulence model, i.e. the shear stress transport version [12] (designated SST) and the turbulent/non-turbulent version [10] (designated TNT). The MNT and TNT turbulence models were extended with the correction for longitudinal vorticity of Dacles-Mariani et al. [1]. The governing equations are integrated down to the wall, i.e. no wall-functions are used. For the present study, all calculations for straight-ahead condition were run until the maximum change of the pressure coefficient Δp max between successive iterations (the so-called L norm of Δp) had dropped well below For non-zero drift, the convergence criterion was set to Δp max < Based on these convergence criteria, iterative convergence errors in the calculations are assumed to be negligible compared to discretisation or modelling errors, see Eça and Hoekstra [5] Uncertainty analysis In order to determine and demonstrate the accuracy and reliability of solutions of viscous flow calculations, grid dependency studies are very important. Several methods for uncertainty analysis are available in literature. In the present work, the

6 232 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF method proposed by Eça and Hoekstra [4] is followed. The full details and background for the followed procedure for the uncertainty study can be found in their paper. For completeness however, a summary is given below. The procedure for uncertainty estimation as used in this project is based on a least squares version of the Grid Convergence Index (GCI) method proposed by Roache [14]. It uses two error estimators: δ RE and Δ M. δ RE is the discretisation error estimation obtained by Richardson extrapolation: φ i φ 0 = δ RE = αh p i, and Δ M is the data range: Δ M = max( φ j φ i ), 1 i, j n g, where φ i is the numerical solution of any local or integral scalar quantity on a given grid (designated by the subscript i), φ 0 the estimated exact solution, α is a constant, h i is a parameter which identifies the representative grid cell size, p is the observed order of accuracy and n g is the number of grids available. φ 0, α and p are obtained with a least squares fit of the data, see Eça and Hoekstra [2], minimizing the function: n g S(φ 0, α, p) = ( φi (φ 0 + αh p i )) 2. (1) i=1 When more than three grids are available and the least squares root approach is applied, it is not easy to classify the apparent convergence condition because the data may exhibit scatter [2]. First, the apparent order of convergence p from the least squares solution of Eq. (1) is established. Next, to identify the cases of oscillatory convergence or divergence, p is also determined using φ i = φ i+1 φ i in Eq. (1). This fit includes only n g 1 differences. The apparent convergence condition is then decided as follows: p>0forφ: monotonic convergence; p<0forφ: monotonic divergence; p < 0forφ : oscillatory divergence; Otherwise: oscillatory convergence. In the last few years, several possibilities for the uncertainty estimation based on the GCI using δ RE and Δ M have been tested. The present version of the procedure incorporates the experience obtained in several test cases and the suggestions and comments of the Uncertainty Workshop held in Lisbon in October 2004 [3]. The procedure for the estimation of the discretisation uncertainty U φ, valid for a nominally second-order accurate method, is summarised as follows:

7 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF 233 The observed order of accuracy is estimated with the least squares root technique to identify the apparent convergence condition according to the definition given above. For monotonic convergence: For0.95 p 2.05: U φ = 1.25δ RE + U s ; For0<p<0.95: U φ = min(1.25δ RE + U s,1.25δ M ); Forp>2.05: U φ = max(1.25δre + U s,1.25δ M ). If monotonic convergence is not observed: U φ = 3Δ M. U s stands for the standard deviation of the fit and δre is obtained with Richardson extrapolation using p equal to the theoretical value, which is 2 for PARNASSOS. The factors 1.25 and 3 used in these formulae are safety factors. It should be remarked that Richardson extrapolation is only allowed when the solutions fall in the asymptotic range where higher order terms can be neglected. In complex flows, this is at present never the case, which means that the apparent order of convergence can deviate considerably from the theoretical order of convergence. In the present method of determination of the discretisation uncertainty this has been incorporated by using the data range Δ M as an additional factor Grids for uncertainty analysis Three additional grids were derived from the original (fine) grid. The second grid was created from the first grid by removing every second point in all three dimensions. This grid is designated the medium-fine grid. The third and fourth grids are created by removing every second and third point and every second through fourth point from the finest grid respectively. These grids were designated the mediumcoarse and coarse grids. In Table 2 the particulars of the grids used during this study are presented. 5. Review of the calculations All calculations were conducted for deeply submerged condition, so that free surface effects are absent. Based on the Reynolds numbers Re used during the calculations (either or , depending on the test condition), the reference Table 2 Particulars of grids used during this study Identification n ξ n η nζ Points y + 2 (β = 10 ) Fine Medium-fine Medium-coarse Coarse

8 234 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF velocity V 0 corresponded to a model scale value of around 6.5 knots or 3.3 m/s.in this paper, the Reynolds number is based on the overall length of the model. 6. Sensitivity study 6.1. Iterative convergence For the straight-ahead sailing condition using the MNT turbulence model the socalled L norms, i.e. the maximum change of a given variable in the whole domain, of the pressure and velocity components converge smoothly to around The convergence history of the integral variables shows convergence to well below Similar convergence is found for the other straight-ahead sailing cases. For the non-zero drift angles, the desired (rather strict) convergence level was not reached when using the fine grid. In each case, the convergence stagnates due to local changes in the flow around the transition of the vertical flap into the bow. For the coarser grids for 10 drift angle, the desired convergence was obtained. In Fig. 3, the convergence history for the calculation with the medium-fine grid for β = 10 is presented as well as the convergence of the integral variable ΔY = Y it Y it 1 for the fine and medium-fine grids. The L norm of the local quantities reduces for the medium-fine grid smoothly to below and the difference ΔY between the integral variable Y between successive iterations to well below The other integral quantities show even better convergence. For the fine grid, it is seen that after sweep 800 the convergence of the integral variable ΔY stagnates at a level around Looking at intermediate results of the integral values during the iterations for β = 10, it is found that after iteration 800 the total force and moment components do not change more than 0.3%. Using a factor of safety of 3, it is, therefore, assumed that for oblique motion an iterative uncertainty U i of about 1% is appropriate. Fig. 3. Convergence history of local (left, medium-fine grid) and integral (right, fine and medium-fine grids) quantity ΔY, β = 10,MNT.

9 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF 235 Fig. 4. Pressure (left) and friction (right) coefficients along the hull, β = 10,MNT Grid sensitivity of local quantities In Fig. 4, the pressure coefficient C p = (p p )/( 1 2 ρv 0 2 ) and friction coefficient C f = τ /( 1 2 ρv 0 2) along the hull are presented for 10 drift angle with the MNT turbulence model. For each grid, two lines are plotted. One line (indicated by φ = 270 ) corresponds to the pressure or friction coefficient distribution along the keel line, incorporating the vertical flap at the bow and the vertical part of the flap at the stern. The second line (indicated by φ = 0 ) corresponds to the values along the starboard horizontal symmetry plane, i.e. the leeward side, incorporating the symmetry plane at the bow (in the grid used, only a vertical flap was modelled at the bow) and the horizontal part of the flap at the stern. Although the pressure coefficient distributions are practically identical, the difference in the distribution of the friction coefficient along the length of the leeward side of the hull between the fine grid and the coarser grid is judged to be relatively large, while further analysis showed that the difference between the two coarser grids was smaller. This means that the solution still differs largely depending on the grids used. This may be caused by the fact that the solution on the finest grid did not reach the desired convergence level, see Eça and Hoekstra [5]. The solution of the friction coefficient on the top or bottom side of the hull is however practically grid independent. For straight ahead sailing, a similar analysis shows that the differences between the medium-fine grid and fine grid are reasonably small for the calculations with the MNT turbulence model. For the TNT turbulence model however, somewhat larger differences were observed for the friction coefficient between the fine and the medium-fine grids. Apparently, the solution on the finest grid is more grid-dependent when the TNT turbulence model is used than when using the MNT turbulence model. Examining the fine-grid solutions for φ = 0 and φ = 270 for the straight-ahead condition, see also Figs 8 and 9, practically identical results are found. Only in the friction component, a slight difference is found. This difference is caused by the asymmetry introduced by the vertical flap at the bow.

10 236 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF Fig. 5. Axial and radial velocities at x = 0.978L oa, β = 0,TNT. Another comparison is made between the axial and radial velocity components at a transverse plane located at x = 0.978L oa, coinciding with the propulsor plane, see Fig. 5 for the straight-ahead condition. The radial velocity is defined to be positive for velocities away from the hull. Also from this figure, it is seen that the solutions for the fine grid and medium-fine grid are close together. Similar conclusions can be drawn regarding the MNT results. A calculation that was conducted using the TNT turbulence model, but without the correction for longitudinal vorticity, does not predict significantly other results than the calculations with correction. Therefore, these results are not further used in this study Discretisation error for integral quantities The results of the uncertainty analysis are presented in Table 3 and Fig. 6 for zero drift and in Table 4 and Fig. 7 for oblique motion condition. The relative step-size h i indicates the coarseness of the grid with respect to the finest grid. Theoretically, the discretisation error disappears if the solution is extrapolated to zero relative step-size. The first observation that can be made is the occurrence of high values and large variation of the apparent order of convergence p in Table 3. For this type of calculations it is common that the solutions on the different grids do not lie in the asymptotic range, see e.g. Eça et al. [3]. In the uncertainty estimation procedure, this has been incorporated by using the data range as additional uncertainty estimator, resulting in relatively conservative uncertainty values. For all forces and moments except the longitudinal force, the friction component is negligible compared to the pressure component. Comparing the convergence behaviour in Fig. 6 of the solutions using the MNT and TNT turbulence models, it is seen that the TNT turbulence model results in general are slightly more grid-dependent, as was already concluded earlier. This indicates that when using the TNT turbulence model, slightly finer grids should be used than when using the MNT model.

11 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF 237 Table 3 Uncertainty analysis, β = 0 Item MNT φ 0 φ 1 U φ (%) p X X f X p Swa TNT X X f X p Swa Fig. 6. Uncertainty analysis X components, β = 0 (squares: MNT, diamonds: TNT). The differences between the MNT and TNT results are mainly caused by the difference in the friction components X f, since the pressure components X p for the finest grid (and for the extrapolated values X p0 ) are practically identical. Based on the convergence behaviour found in the total force components and the relatively small differences in the solutions between the fine and medium-fine grids,

12 238 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF Table 4 Uncertainty analysis, β = 10,MNT Item φ 0 φ 1 U φ (%) p X X f (3) X p Y (2) Y f (3) Y p (2) K (3) K f (3) K p N N f (3) N p N/Y (2) Swa Note: Monotonic convergence (0); oscillatory convergence (1); monotonic divergence (2); oscillatory divergence (3). Fig. 7. Uncertainty analysis N components, β = 10 (squares: MNT). it is concluded that for zero drift the fine grid as used during the calculations is sufficiently accurate for the purpose of this study. For oblique motion, the iterative uncertainty for the finest grid was estimated to be about 1% which is not negligible compared to the discretisation error and therefore it is difficult to draw definitive conclusions regarding the required grid density. More study is required to determine the reason for the difficult convergence and to investigate whether alternative grid topologies, e.g. so-called CO-type grids, will improve the convergence as well as the accuracy of the solutions. Concluding, however, it is found that the results of the integral quantities for the fine grid for β = 10 are in line with the results for the coarser grids and therefore

13 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF 239 it is assumed that the fine grid solution is sufficiently accurate for comparison to the experiments. 7. Comparison with experiments, straight-ahead 7.1. Integral values Force measurement results were published by Roddy [15]. During these tests, the model was supported by two struts. The speed used during the experiments resulted in a Reynolds number of For this condition the averaged experimental value of the longitudinal force was found to be X = A detailed uncertainty analysis of the force measurements was not conducted. However, to obtain an estimate of the uncertainty in the experimental results, the uncertainty U D in the experimental data is estimated using the maximum difference between all measurements for β = 0 and a factor of safety of 1.25, resulting in U D = 1.2% X. Table 5 presents the longitudinal force components obtained from the calculations for the finest grids. As expected, the largest part (about 91%) of the total resistance is caused by friction resistance. Comparing the friction component with the ITTC friction coefficient for a flat plate, a slightly higher ( %) friction component is found in the calculations for the axi-symmetric body. In Fig. 6 and Table 3, it is Table 5 Comparison of integral values with experiments, β = 0 Item MNT TNT MNT TNT L oa m Swa m 2 X exp Re X f,ittc X f,schoenherr X X f X p X f /X (%) X f /X f,ittc (%) (1 + k) ITTC (1 + k) Schoenherr X/X exp 1(%) X X 0 /X exp 1(%)

14 240 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF seen that according to the grid dependency study the longitudinal force converges upon grid refinement to about X 0 = for MNT and about X 0 = for TNT, which are in magnitude slightly lower than the results for the fine grid. The difference of these predictions with the experiments is respectively 0.7% and 2.0%, which is judged to be very good for practical applications when also the uncertainty in the experimental data and the difference in Reynolds number are taken into consideration. The influence of the Reynolds number can be observed in the calculations. As expected, a slightly higher Reynolds number results in a slightly lower resistance coefficient, due to the decrease in frictional resistance coefficient. The error in the prediction of X changes from 1.3% resp. 3.7% for the lower Re value to 0.9% resp. 1.4% for the higher value. Taking into account the grid dependency and assuming a similar trend for the higher Reynolds number as for the lower number, the difference between the extrapolated value and the experimental value is estimated to be approximately 1.5% when using the MNT turbulence model and 0.3% when using the TNT turbulence model Local quantities Huang et al. [9] measured the flow field and pressure around the SUBOFF model. Figure 8 shows that the differences in pressure coefficient between the MNT and TNT turbulence models are negligible, which corresponds to the practically identical longitudinal pressure coefficients X p found in Table 5. For the friction coefficient, it is seen that for x<0.8l oa the MNT turbulence model predicts the distribution more accurately than the TNT model. However, for x>0.8l oa better agreement with the experiments is found for the TNT model. Especially the predicted distribution of the pressure coefficient is close to the experiments. The trends in the predicted distribution of the friction coefficient corre- Fig. 8. Pressure (left) and friction (right) coefficients along the hull, β = 0, MNT and TNT.

15 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF 241 Fig. 9. Axial and radial velocities (left) and Reynolds shear stress (right), β = 0, MNT and TNT, x = 0.904L oa. spond well to the trends found in the experiments. Although some discrepancies at the bow and stern area are found, it is concluded that the prediction of the pressure and friction coefficients is good. Figure 9 presents the streamwise and radial velocities at a specific longitudinal location in the aft of the hull. In these graphs, the difference between the MNT and TNT results is considered to be small. However, considering the axial velocity profile in more detail, it is seen that the slope of the curve near the wall is somewhat flatter (i.e. the velocity gradient is higher) for TNT than for MNT. Comparing the computed results with the experiments, it is observed that the trends in the development of the boundary layer are very well predicted by both turbulence models, but quantitative discrepancies are seen. Especially the magnitudes of the radial velocities appear to be over-predicted by both turbulence models. However, it is seen that in the experiments the radial velocity for the aftmost measurement planes changes sign between (r R 0 )/R max = 0.8 and (r R 0 )/R max = 2, suggesting outward radial flow in the far field. This may be caused by the use of an open-jet wind tunnel Reynolds shear stresses In Fig. 9, the Reynolds shear stresses U xu r/v 2 0 for x = 0.904L oa are presented. It is observed that in general, the MNT turbulence model provides a much better prediction of the Reynolds shear stresses compared to the TNT results. Also looking at other longitudinal locations along the stern, it is seen that the trends in the MNT results are quite close to the measurements, but the peaks are somewhat over-predicted in the two foremost measurement planes.

16 242 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF 8. Comparison with experiments, oblique motion 8.1. Integral values Table 6 and Fig. 10 present the force and moment components obtained from the calculations and the values from the experiments of Roddy [15] for various drift angles. The uncertainty in the experimental results is estimated in a way similar to the straight flight case. For drift angles larger than 10, the medium-fine grid results are given. This comparison shows that the error in the prediction for oblique motion is Table 6 Comparison of integral values with experiments, β > 0 Item β = 2 β = 4 β = 10 β = 18 L oa m Swa m 2 Re X exp X X f X p X f /X (%) X/X exp 1(%) X X 0 /X exp 1(%) 22.1 Y exp Y Y f Y p Y f /Y (%) Y/Y exp 1(%) Y Y 0 /Y exp 1(%) 25.3 N exp N N f N p N f /N (%) N/N exp 1(%) N N 0 /N exp 1(%) 8.6

17 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF 243 Fig. 10. Force and moment coefficients against drift angle. quite considerable for the longitudinal force X. The trends in the transverse force Y and yaw moment N and the de-stabilising arm N/Y on the other hand are predicted reasonably well Local quantities at leeward symmetry plane Figure 11 presents comparisons of the pressure coefficient along the hull, and the axial and radial velocities and Reynolds shear stresses, given at the leeward symmetry plane located at x = 0.978L oa, for a drift angle β of 2. These graphs show that the distribution of the pressure coefficient along the length of the ship and the velocity distribution at the stern is quite well represented. Also the magnitude of the Reynolds shear stress is predicted well Circumferential results In Fig. 12, the axial velocity components as a function of the circumferential angle for the aft-most longitudinal station (x = 0.978L oa ) are presented for a drift angle of β = 2. A circumferential angle φ = 0 corresponds to the leeward side and φ = 180 to the weather side.

18 244 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF Fig. 11. Pressure coefficients along the hull (top left), axial and radial velocities (top right) and Reynolds shear stress (bottom), β = 2. Fig. 12. Axial velocity, β = 2, x = 0.978L oa, (symbols: exp, lines: cfd). All calculations were conducted without modelling the struts at φ = 180 that were used to support the submarine model during the experiments. Therefore, discrepancies in the prediction of the flow around φ = 180 are inevitable. However, qualitative agreement for the flow for φ 180 is expected. For the flow at the leeward side of the hull (leeward meridian, φ = 0 ), the axial velocity for the different radii is well predicted. Noteworthy however is the asymme-

19 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF 245 try of the experimental results with respect to the leeward meridian (left-hand side of the graphs vs. right-hand side of the graphs). This may indicate that the struts used to support the model were positioned at an oblique angle with respect to the flow. Other deviations may be caused by the use of an open-jet wind tunnel. At the windward meridian (φ = 180 ), the wake of the struts is seen in the experiments and a comparison between the calculations and measurements cannot be made. 9. Influence of grid layout It was questioned whether the grid layout was the origin of the deviation from the measurements of the predicted longitudinal force X for oblique flow. Due to the H O grid topology, the hull form at the bow and stern is not exactly represented in the grid. For small angles of attack, the stagnation region coincides with the area in which the hull form is approximated by the H O shape of the grid and integration of the pressure may lead to inaccurate resolution of the forces generated in the stagnation region. To investigate whether an alternative grid layout would yield a better prediction of the pressure component of the longitudinal force, an additional study was conducted. Although it is expected that an O O grid will be the best option for this hullform at oblique flow, an H O topology was still used for the present study, but now the so-called flap was removed from the grid. The grid was first created in 2-D. To avoid a singularity in the grid at the longitudinal axis of symmetry, the points located in this axis were moved in radial direction by L oa. This distance was expected to be sufficiently small to not influence the flow around the hull. Subsequently, the grid was rotated around the x-axis to obtain the (axisymmetric) 3-D grid. The 2-D grid is presented in Fig. 13. Calculations were Fig. 13. Alternative grid layout, coarsened for presentation (bow to left).

20 246 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF Table 7 Comparison of integral values with experiments, modified grid layout Item β = 0 β = 2 β = 10 β = 18 β = 18 MNT MNT MNT MNT SST L oa m Swa m 2 Re X exp X X f X p X f /X (%) X/X exp 1(%) Y exp Y Y f Y p Y f /Y (%) Y/Y exp 1(%) N exp N N f N p N f /N (%) N/N exp 1(%) conducted with this new grid for several drift angles, using the MNT turbulence model. It is seen in Table 7 and Fig. 14 that the modified grid (designated cfd Axi) produces improved results for the longitudinal force compared to the original results (designated cfd). At β = 18, a discrepancy in the predicted longitudinal force X is still found. Also the prediction of the transverse force and yawing moment deteriorates slightly. However, in general better predictions are obtained. In Fig. 15, the calculated pressure distributions along the length are given for β = 10. Differences are seen, especially in the bow area. When the pressures are integrated over the hull area, the longitudinal force for the alternative grid improves considerably for the modified grid, see Table 7. For the new grid layout, it was possible to converge the results to the desired criterion, which was not achieved for the original results on the finest grid. Comparing the results obtained with the two different grids, it is found that the changes due to

21 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF 247 Fig. 14. Force and moment coefficients against drift angle. Fig. 15. Pressure coefficients along the hull, influence of new layout, β = 10. the different layout are mainly visible in the X-force results. The differences in the predictions for all other cases are not considered to be significant and therefore it is concluded that with the modified grid layout a more reliable solution is obtained. However, more study is required to further improve the overall correspondence of the CFD results with the measurements. To study the influence of the location of the external and outflow boundary planes on the solution, an additional calculation was conducted for β = 18 (using the MNT

22 248 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF Table 8 Comparison of results for different domain sizes, modified grid layout, MNT, β = 18 Item β = 18 β = 18 MNT L oa m Swa m 2 MNT External boundary 0.69L oa 1.49L oa Outflow boundary 0.94L oa 2.00L oa Re X exp X X f X p X f /X (%) X/X exp 1(%) Y exp Y Y f Y p Y f /Y (%) Y/Y exp 1(%) N exp N N f N p N f /N (%) N/N exp 1(%) turbulence model and the new grid layout) with an increased computational domain. The external boundary was located at 1.49L oa and the outflow boundary plane at x = 2.0L oa. A comparison of the forces and moments on the hull for the original domain and the increased domain are presented in Table 8. Based on the difference between the results, it is concluded that the influence of the domain size on the results is relatively small and that the original domain size was already appropriate for the calculations. 10. Influence of turbulence model, oblique motion Further calculations have been conducted using the modified grid layout in combination with k ω turbulence models. Several combinations have been tried, pro-

23 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF 249 Fig. 16. Pressure coefficients along the hull using new layout, β = 18, MNT vs. SST. viding similar results. The best results were obtained using Menter s SST version of the k ω turbulence model without using the Spalart correction for longitudinal vorticity, see Table 7. With the SST model, a considerable improvement of the prediction of the longitudinal force is obtained combined with a small improvement of the transverse force Y and yawing moment N. Figure 16 shows a comparison of the pressure distributions found for β = 18 when using the new grid layout in combination with respectively the MNT turbulence model and the SST turbulence model. The differences are seen aft of midship where the largest viscous effects can be expected. In Fig. 14, the forces and moments obtained using the SST model are included. These results clearly illustrate the importance of selecting an appropriate turbulence model. Compared to the straight-ahead case, the influence of the turbulence model is much more pronounced for oblique flow calculations. 11. Conclusions Based on this study, it is concluded that PARNASSOS is well capable of predicting the flow around a submarine at zero incidence. Additionally, the flow at oblique motion can be predicted well when selecting the proper grid layout and turbulence model. For zero incidence, the differences between the results for the two different turbulence models are judged to be relatively small, although the results with Menter s one-equation model tend to be slightly less grid dependent and provide a slightly better prediction of the Reynolds shear stress distribution in the aft ship. On the other hand, the predicted longitudinal force obtained with the TNT version of the two-equation k ω model is slightly closer to the measurements. The influence of the grid layout was studied to improve the prediction of the longitudinal force for oblique inflow. With a modified grid layout improved results were found.

24 250 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF Further improvements were obtained by varying the turbulence model. Using k ω models, the calculation results changed considerably, with smaller differences between the experimental results and the calculations. Best results were found using Menter s SST version of the k ω turbulence model. The results clearly demonstrate the importance of the selection of the turbulence model on the accuracy of the calculations for oblique flow conditions. Acknowledgements The work presented here was funded through TNO Defence, Security and Safety within the framework of Programma V705 carried out for DMO of RNLN. Their support is greatly acknowledged. References [1] J. Dacles-Mariani, G.G. Zilliac, J.S. Chow and P. Bradshaw, Numerical/experimental study of a wingtip vortex in the near field, AIAA Journal 33 (1995), [2] L. Eça and M. Hoekstra, An evaluation of verification procedures for CFD applications, in: 24th Symposium on Naval Hydrodynamics, Fukuoka, Japan, July [3] L. Eça and M. Hoekstra (eds), in: Proceedings of the Workshop on CFD Uncertainty Analysis, October [4] L. Eça and M. Hoekstra, On the influence of grid topology on the accuracy of ship viscous flow calculations, in: 5th Osaka Colloquium on Advanced CFD Applications to Ship Flow and Hull Form Design, Lisbon, Portugal, March 2005, pp [5] L. Eça and M. Hoekstra, On the influence of the iterative error in the numerical uncertainty of ship viscous flow calculations, in: 26th Symposium on Naval Hydrodynamics, Rome, Italy, September [6] L. Eça, M. Hoekstra and J. Windt, Practical grid generation tools with applications to ship hydrodynamics, in: 7th International Conference on Grid Generation and Computational Field Simulations, Hawaii, February [7] N.C. Groves, T.T. Huang and M.S. Chang, Geometric characteristics of DARPA SUBOFF models (DTRC Model Nos and 5471), Report No. DTRC/SHD , March [8] M. Hoekstra, Numerical simulation of ship stern flows with a space-marching Navier Stokes method, PhD Thesis, Faculty of Mechanical Engineering and Marine Technology, Delft University of Technology, October [9] T. Huang, H.-L. Liu, N. Groves, T. Forlini, J. Blanton and S. Gowing, Measurements of flows over an axisymmetric body with various appendages in a wind tunnel: the DARPA SUBOFF experimental program, in: 19th Symposium on Naval Hydrodynamics, Seoul, South Korea, August 1992, pp [10] J.C. Kok, Resolving the dependence on free-stream values for the k ω turbulence model, Report NLR-TP-99295, National Aerospace Laboratory, The Netherlands, July [11] H.-L. Liu and T.T. Huang, Summary of DARPA SUBOFF experimental program data, Report No. CRDKNSWC/HD , June 1998.

25 S. Toxopeus / Viscous-flow calculations for bare hull DARPA SUBOFF 251 [12] F.R. Menter, Two-equation eddy-viscosity turbulence models for engineering applications, AIAA Journal 32(8) (1994), [13] F.R. Menter, Eddy viscosity transport equations and their relation to the k ε model, Journal of Fluids Engineering 119 (1997), [14] P.J. Roache, Verification and Validation in Computational Science and Engineering, Hermosa Publishers, Albuquerque, NM, [15] R.F. Roddy, Investigation of the stability and control characteristics of several configurations of the DARPA SUBOFF model (DTRC Model 5470) from captive-model experiments, Report No. DTRC/SHD , September 1990.