Calculation of the Flow around the KVLCC2M Tanker

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1 Calculation of the Flow around the KVLCC2M Tanker L. Eça 1, M. Hoekstra 2 and S.L. Toxopeus 2 1 Instituto Superior Técnico, Portugal 2 Maritime Research Institute, Netherlands SUMMARY The flow around the double-body KVLCC2M tanker is computed in grids with H-O topology. First the zerodrift angle case is considered. Two turbulence models are used: s one-equation model and Kok s version of the k ω model. A thorough uncertainty analysis based on solutions on 8 grids of various density is carried out. In the second part of the paper the flow predictions for drift angles 3, 6, 9 and 12 degrees are discussed. Encouraging results are obtained, but the relatively high level of uncertainty in the pressure drag evaluation requires further attention. INTRODUCTION The calculations for the KVLCC2M tanker hull form were performed with the flow solver PAR- NASSOS (Hoekstra & Eça, 1998), which is based on a finite-difference discretisation of the Reynoldsaveraged continuity and momentum equations with fully-collocated variables and discretisation. The equations are solved with a coupled procedure, retaining the continuity equation in its original form. In PARNASSOS several eddy-viscosity turbulence models are available. In a numerical calculation of a ship stern flow, the turbulence model selection is not only based on the quality of the predictions, but also on the numerical robustness and the ability to converge the solution, i.e. reduce the iterative error to the desired value. Most applied in PARNASSOS is the one-equation model proposed by (1997), which leads to a remarkably robust method and allows convergence of the solution to machine accuracy in almost any case. However, several validation studies have shown that the predictions of the flow field in the bilge vortex region are not as good as the ones obtained with the k ω model. Therefore, we have also performed calculations for the zero-drift case using the TNT version of the k ω model (Kok, 1999). The main advantage of the TNT version when compared with the popular BSL and SST versions (, 1994) is the absence of references to the distance to the wall. The calculations for the drift cases were conducted using s turbulence model only. The Spalart correction to account for the effects of streamwise vorticity, described in Dacles-Mariani et al. (1994), is adopted in both turbulence models. No attempts have been made to add special features for modelling transition. So the basis turbulence model acts as the transition model as well. The Reynolds number is equal to All computations were performed using dimensionless quantities with L PP and U as the reference length and velocity scales. COMPUTATIONAL DOMAIN AND GRID TOPOLOGY All calculations described in this paper were conducted for the unappended hull form. Several grid topologies have been tested for the calculation of the flow around the KVLCC2M double model (Eça & Hoekstra, 25). The results presented in this paper were all obtained on structured grids with H-O topology with some extra grid clustering close to the propeller plane. For the zero-drift case, a singleblock calculation was conducted while for the nonzero drift case the domain was decomposed into effectively two blocks. The six boundaries of the computational domain are the following: the inlet boundary is a x = constant plane located upstream of the forward perpendicular; the outlet boundary is also 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 symmetry plane of the ship and the undisturbed water surface. Zero-drift case For the zero-drift case, the inlet boundary was located at.25l PP upstream of the forward perpendicular. The outlet boundary was located at.25l PP downstream of the aft perpendicular. The radius of the cylinder defining the external boundary was.18l PP. Eight geometrically similar grids have been generated with in-house codes (Eça, Hoekstra & Windt, 22) for the estimation of the discretisation error. The variation in the number of grid nodes in the streamwise, n ξ, normal, n η and girthwise, n ζ directions is

2 Figure 1: Grid at boundaries of the computational domain. as well. It was decided to have matching interfaces between the blocks so that the inner and outer blocks can be merged. The size of the outer blocks is chosen such that the rotated inner block can smoothly be incorporated in the outer grids. This means that increasing drift angles will result in wider domains. The size of the domain is based on the assumption that a solver for potential flow is used to calculate the velocities in the inflow and external planes. Before starting the calculations, the separate blocks are merged into one block for the port side of the ship and another block for the starboard side of the ship. presented in Table 1, which includes also the maximum y + obtained at the first grid node away from the wall (y + 2 ) with s one-equation model. Figure 1 gives an impression of the (coarsened) point distribution on the boundaries of the computational domain. Grid n ξ n η n ζ y + 2 G G G G G G G G Table 1: Number of grid nodes and y + at the first grid node away from the wall. Non-zero drift cases The flow around the hull at non-zero drift angles has no port-starboard symmetry and the computational domain must be extended to cover the port side as well. Furthermore, a larger domain is required in order to incorporate the drift angle. On each side of the domain the grid consists of an inner block and an outer block, see Figure 2. The inner block is the same for all yaw angles and the outer block can deform to allow for the drift angle of the ship. Therefore grids for various drift angles can be made efficiently. The inner block is generated with a number of cells similar to the grids as used for the zero-drift case. Based on early calculations (Toxopeus, 24), grid clustering at the propeller plane and the bow of the ship was applied to resolve the gradients of the flow at these locations more accurately. To incorporate the drift angle of the ship, the inner block is rotated around the vertical z-axis over the desired yaw angle. Then the outer block is generated around the inner block. The cell stretching used in the inner block is automatically applied to the outer block Figure 2: Inner and outer blocks (coarsened) at 12 drift angle. The number of nodes in the grids used for the drift cases are presented in Table 2, which includes also the maximum y + value for the cells adjacent to the hull that was obtained during the calculations. A positive drift angle β corresponds to the flow coming from port side. Note that also a calculation with zero drift angle was conducted with a grid similar to the grids used for non-zero drift in order to be able to determine the relation between the drift angle and integral or local variables consistently. β n ξ n η n ζ nodes y Table 2: Number of grid nodes and y + 2 for drift cases. Table 3 presents the sizes of the computational domains for the drift case calculations. For increasing drift angles, the computational domain size is increased in order to be able to incorporate the inner block in the outer deforming mesh.

3 β inlet outlet width depth [L PP ] [L PP ] [L PP ] [L PP ] Table 3: Size of computational domain for drift cases. BOUNDARY CONDITIONS At the ship surface the no-slip condition is applied directly and the normal pressure derivative is assumed to be zero. The undamped eddy viscosity, the variable in s one-equation model, vanishes at a no-slip wall. With the present formulation of the k ω model (Kok and Spekreijse, 2), all the turbulent quantities are zero at a solid wall. Symmetry conditions are applied at the undisturbed water surface and on the ship symmetry plane (for the zero-drift condition). At the inlet boundary, the velocity profiles are obtained from a potential flow solution, which also determines the tangential velocity components and the pressure at the external boundary. At the outlet boundary, streamwise diffusion is neglected and the streamwise pressure derivative is set equal to zero. For the drift cases, the lift generated by the hull form is modelled in the potential flow solution by applying a vortex sheet on the symmetry plane of the ship. At the stern of the ship, the Kutta condition (the flow leaves the trailing edge smoothly) is applied, which allows the solution of the unknown vortex strengths on the sheet. Since the only purpose of the potential flow solution is to set the boundary conditions for the viscous flow solution at the inlet and external boundaries, vortex shedding from the bilge of the ship is omitted. UNCERTAINTY ESTIMATION We only deal with the discretisation error, assuming the iterative and round-off errors to be negligible. The uncertainty, U φ, of any integral or local flow quantity is estimated with a procedure based on a least squares root version (Eça and Hoekstra, 22) of the Grid Convergence Index (GCI), proposed by Roache (1998). Two basic error estimators are involved in the present procedure for uncertainty estimation: the extrapolation to grid cell size zero performed with Richardson extrapolation, δ RE ; and the maximum difference between the data points available, M. We have collected some experience with several variants of uncertainty estimation procedures (Eça and Hoekstra, 24). In the present calculations we have adopted the following options: Determine the observed order of accuracy, p, from the available data. For.95 p < 2.5, U φ is estimated with the GCI and the standard deviation U fit of the fit: U φ = 1.25δ RE +U fit. For < p <.95, the same error estimate is made but is then compared with the value of M multiplied by a factor of safety of 1.25, so that U φ is obtained from: U φ =min(1.25δ RE +U fit,1.25 M ). For p 2.5, U φ =max(1.25δre +U fit,1.25 M ), where δre is also calculated in the least squares root sense with p = 2. If monotonic convergence is not observed, U φ = 3 M. RESULTS FOR ZERO DRIFT Numerical Convergence In the present calculations we have adopted as convergence criterion the reduction of the maximum difference between consecutive iterations of the three velocity components and of the pressure to 1 12, which is equivalent to machine accuracy. ( φ) max φ=c p, φ=ν -2 t, φ=c p, -4 φ=ν t, Iteration Figure 3: Convergence history on grid. Unfortunately, we were not able to satisfy this criteria for the 8 grids with the TNT k ω model. In the grids G1, G3 and G5 the convergence stagnates at a level that does not allow to neglect the iterative error and so we have dropped the results obtained in these 3 grids. The convergence histories obtained for the G7 grid with the two turbulence models are illustrated in Figure 3.

4 .9 Resistance Coefficients The predicted values of total resistance, C T, friction resistance, C F, and pressure resistance, C P, are presented in Table 4 with the estimated uncertainties. These force components have been made non-dimensional using 1 2 ρu 2 S with S the wetted surface at rest. Velocity field at the propeller plane The selection of the turbulence model has a significant effect on the prediction of the velocity field at the propeller plane. The isolines of axial velocity obtained with the two turbulence models and the transverse velocity fields are plotted in figure 5..5 C T U CT C F U CF C P U CP a b Table 4: Predicted resistance coefficients and their estimated uncertainties (a=, b=). z/l PP z/l PP y/l PP U C F x U=.72 p=.8 U=.8 p= 4.7 p*=2 Figure 5: Velocity field at the propeller plane (top:axial velocity, bottom: transverse velocity). There is a more pronounced hook shape in the k ω solution than in the prediction with s model. There are also differences in the bilge vortex, specially in the lower part close to the symmetry plane. y/l PP C P x h i /h U=.344 p=.4 U=.111 p= h i /h 1 U u z/l PP y/l PP Figure 6: Uncertainty in the axial velocity field at the propeller plane. An interesting result is the uncertainty of the axial velocity field prediction at the propeller plane. The values of U u are below.1 for most of the field. However, at the bilge vortex region the maximum values of U u reach levels above.1, with the k ω predictions exhibiting the largest values of uncertainty. Figure 4: Convergence of the friction and pressure resistance with grid refinement. In both cases, the uncertainty of C P is clearly larger than the one of C F. The estimated uncertainties are much larger for the k ω model than for s one-equation model. As illustrated in Figure 4, the observed order of accuracy is below 1 for the solutions obtained with the k ω model, whereas the p obtained with s model 1 is 4.7 for C F and 2 for C P. 1 The fit to C F plotted in Figure 4 is made with p=2. RESULTS FOR DRIFT ANGLES Numerical Convergence In the calculations of the drift cases a reduction of the maximum difference in pressure between consecutive iterations to was adopted as the convergence criterion. In these cases there is no attempt to estimate the discretisation error. Therefore, there is no need to reduce the iterative error to machine accuracy.

5 Integral Coefficients In this section, the forces and moments presented are made non-dimensional using respectively 1 2 ρu 2 L PP T and 1 2 ρu 2 L 2 PPT, in accordance with specifications for the CFD Workshop 25. Table 5 presents the results of the calculations for each drift angle β as well as a comparison between the calculated variables and the measured ones. CX is the longitudinal force, CY the transverse force and CN the yawing moment with respect to the origin of the xyz coordinate system, which is located at station 1. The results for zero drift angle are comparable to the results as presented in Table 4 (C T = , C F = and C P = ). cfd exp β CX CY CN CX CY CN β ε CX ε CY ε CN -3-2% -27% 8% -1% % -28% 25% 6 1% -12% 5% 9 4% -7% 3% 12 2% -9% -1% Table 5: Integral variables. Except maybe for the results for 3 drift angle, the predictions obtained by the calculations are very promising. In almost all cases the prediction is within 1% from the measurements. Noteworthy is the consistent underprediction of the transverse force, while both the longitudinal force and yawing moment are predicted quite accurately. Figure 7 presents the yawing moment as a function of the drift angle. More results can be found in the proceedings of the CFD Workshop 25. Compared to the results presented for earlier calculations for the KVLCC2M at a drift angle, see Toxopeus (24), the grid refinement at the bow and stern has improved the prediction of the longitudinal force. The earlier calculations were conducted with an equidistant grid with 251 nodes in longitudinal direction along the hull surface. For the present calculations, the grid was non-equidistant with 342 nodes along the hull surface. The improvement in the prediction is mainly caused by the change in the prediction of the pressure component CX P, since the friction component CX F is practically equal for the two CN beta Figure 7: Yawing moment against drift angle. (exp: open circles, cfd: solid line) different grids, see Table 6 and the results for zero drift angle presented in Table 4. The pressure component however decreases considerably for the nonequidistant grid. This finally results in a better prediction of the longitudinal force for the non-equidistant grid. β grid CX P CX F CX equidistant non-equidistant equidistant non-equidistant Table 6: CX components for different grids. Also for the transverse force CY the difference in pressure component determines the difference in the total transverse force, see Table 7. Furthermore, the friction component is an order of magnitude smaller than the pressure component and therefore is almost negligible in the total force. However, when comparing the measurements with the results for the two different grids, it is seen that in this case the finer grid does not lead to a better prediction. A grid refinement study should be conducted to verify whether sufficient grid nodes in girth-wise direction are applied. β grid CY P CY F CY equidistant non-equidistant Table 7: CY components for different grids. Side force distribution To understand the manoeuvrability of ships and to be able to generate reliable generic mathematical manoeuvring models, the longitudinal distribution of the

6 side force is of interest. Therefore, the predicted longitudinal distribution of the lateral force has been compared to the experimental values to determine the accuracy of the predictions, see Figure 8. The comparison shows that although the side force according to Table 5 is systematically underpredicted, the predicted distribution is very close to the measurements and therefore the accuracy of this prediction is judged to be good. dy/dx dy (exp) 6 dy (cfd) 6 dy (exp) 12 dy (cfd) x/lpp CONCLUSIONS Figure 8: Side force distribution. At zero drift angle, we have performed calculations with two eddy-viscosity models: the one-equation model proposed by and the TNT version of the k ω model. Grid refinement studies have been performed with both models to estimate the numerical uncertainty of the predictions. The results show that the uncertainty of the pressure resistance coefficient is at least one order of magnitude larger than the uncertainty of the friction resistance coefficient. The level of uncertainty of the selected flow quantities depends on the turbulence model choice. The k ω model leads to higher levels of uncertainty than the model. However, the comparison of the predicted velocity fields at the propeller plane shows a better agreement of the k ω predictions with the experimental results. For the non-zero drift cases, which were all conducted using s one-equation model, the compliance of the predicted results with the measurements is good and within 1% from the measurements for most cases. Based on a variation of the grid density and grid node spacing it was found however that the uncertainty in the pressure component of all integral forces is large. A grid sensitivity study is therefore recommended in order to verify the calculated results. REFERENCES Dacles-Mariani J., Zilliac G.G., Chow J.S., Bradshaw P., Numerical/experimental study of a wing tip vortex in the near field, AIAA Journal, Vol. 33, September 1995, pp Eça L., Hoekstra M., Windt J., Practical Grid Generation Tools with Applications to Ship Hydrodynamics, 8 th International Conference in Grid Generation in Computational Field Simulations, June 22, Hawaii, USA. Eça L, Hoekstra M., An Evaluation of Verification Procedures for CFD Applications, 24 th Symposium on Naval Hydrodynamics, July 22, Fukuoka, Japan. Eça L., Hoekstra M., A Verification Exercise for Two 2-D Steady Incompressible Turbulent Flows, 4 th European Congress on Computational Methods In Applied Sciences And Engineering, ECCOMAS 24, July 24, Finland. Eça L., Hoekstra M., On the influence of grid topology on the accuracy of ship viscous flow calculations, 5 th Osaka Colloquium on Advanced CFD Applications to Ship Flow and Hull Form Design, 25, Osaka, Japan. Hoekstra M., Eça L., PARNASSOS : An Efficient Method for Ship Stern Flow Calculation, Third Osaka Colloquium on Advanced CFD Applications to Ship Flow and Hull Form Design, May 1998, pp , Osaka, Japan. Kok J.C., Resolving the Dependence on Free-stream values for the k ω Turbulence Model, NLR-TP , July 1999, National Aerospace Laboratory, NLR, The Netherlands. Kok J.C, Spekreijse S.P., Efficient and Accurate Implementation of the k ω Turbulence Model in the NLR multi-block Navier-Stokes system, NLR-TP , May 2, National Aerospace Laboratory, NLR, The Netherlands. F.R., Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications, AIAA Journal, Vol.32, August 1994, pp F.R., Eddy Viscosity Transport Equations and Their Relation to the k ε Model, Journal of Fluids Engineering, Vol. 119, December 1997, pp Roache P.J., Verification and Validation in Computational Science and Engineering, Hermosa Publishers, Toxopeus S.L. Validation of calculations of the viscous flow around a ship in oblique motion, The First MARIN-NMRI Workshop, October 24, pp

Calculation of hydrodynamic manoeuvring coefficients using viscous-flow calculations

Calculation of hydrodynamic manoeuvring coefficients using viscous-flow calculations Serge Toxopeus Maritime Research Institute Netherlands (MARIN), Wageningen, The Netherlands Delft University of Technology,