VALIDATION OF CALCULATIONS OF THE VISCOUS FLOW AROUND A SHIP IN OBLIQUE MOTION

Transcription

1 The First MARIN-NMRI Workshop in Toko (Oktober 25-26, 24) VALIDATION OF CALCULATIONS OF THE VISCOUS FLOW AROUND A SHIP IN OBLIQUE MOTION Serge TOXOPEUS Maritime Research Institute Netherlands (MARIN), Wageningen, The Netherlands. ABSTRACT The capabilit of the viscous-flow solver PARNASSOS to simulate the flow around a ship in stead oblique motion has been studied. To obtain insight into the reliabilit and accurac of the results, grid dependenc studies were conducted. Local flow quantities as well as integral variables were compared to measurement values. Two different hull forms were considered: the Series 6 of which extensive flow measurements are available and the KVLCC2M which is the subject of the coming CFD Workshop in Toko in 25. KEY WORDS: CFD, RANS, viscous-flow, manoeuvring, hdrodnamic forces, validation. INTRODUCTION For studies into the manoeuvrabilit of ships onl limited use of viscous-flow calculations is nowadas made. One of the major reasons for this is that these calculations involve substantial computing requirements. Another reason is that validation material is sparse and onl a few successful demonstrations of the practical application of these calculations are available. Furthermore, in order to stud the manoeuvrabilit of a ship, either time-domain viscous-flow calculations for the ship with active propulsion and steering appendages should be made, or a dedicated mathematical model of the forces on the ship should be derived from viscous flow calculations for stead aw and drift conditions. Although the first approach theoreticall models more accuratel the hdrodnamics around the appended ship, the second approach is better suited for practical application at least for the near future. At MARIN, viscous-flow computations are used frequentl for the optimisation of hull forms from a resistance and powering point of view. Currentl, studies are conducted to extend the capabilities of this solver to also simulate the flow around ships in oblique flow. The first results have alread been presented b Toxopeus (24). Parts of these results are reproduced in this present paper. Besides the possibilit to stud the detailed flow around the ship in various conditions, this is the first step towards using the viscous flow solver as a tool to predict the forces and moments on manoeuvring ships. Extensive flow field measurements on the well-known Series 6 hull form were conducted at Iowa Institute of Hdraulic Research (IIHR) and the results were made available to the public. Therefore, this case is ver suitable for validation studies of viscous-flow calculations. Various researchers have alread reported such validation studies, such as Alessandrini and Delhommeau (1998), Cura Hochbaum (1998), Di Mascio and Campana (1999) and Tahara et al. (22). Validation data is also available for the KVLCC2M hull form which will be used during the coming CFD Workshop in Toko in March 25. This paper presents viscous-flow calculations conducted on the Series 6 hull form sailing at a drift angle. In order to examine the reliabilit of the predicted flow fields and integrated quantities derived from the solution, the discretisation error in the result is studied based on grid refinement. To validate the results, comparisons are made between model measurements conducted at IIHR and simulations of the viscous flow around the model. More detailed comparisons can be found in Toxopeus (24). Although the full stud is not et finalised, the first results of flow simulations for the KVLCC2M hull form are presented. PARTICULARS OF THE SHIPS AND VALIDATED CONDITIONS Series 6 The first hull form under consideration is the Series 6 with a block coefficient C B of.6. The particulars of this hull form are presented below, taken from Longo (1996): Main particulars of Series 6 Designation Model scale (1:4) Full scale Length L pp 3.48 m m Beam B.46 m m Draught T.163 m 6.52 m Block coefficient C B.6.6 Wetted area S m m 2 Bilge keels, rudder and propeller were not present during the model tests and are therefore not modelled in the calculations. 91

2 As a first step, the calculations are conducted with an undisturbed water surface, i.e. neglecting the generation of waves. The measurements were conducted at two different Froude numbers, i.e. Fn=.16 and Fn=16. For the validation stud of the flow fields presented in this paper, the Fn=.16 results are used. The measurements of the mean velocities and pressure fields were conducted with the model restrained from moving in an direction relative to the carriage. For the measurements of the forces and moments, however, the model was free to heave, roll and pitch. Boundar conditions ξ η Z X Y ζ hull η Because of the definition of the coordinate sstem used in the calculations for the present stud, the data of the IIHR measurements have been mirrored to allow for easier comparison with the numerical results. This means that in this stud, the starboard side is the pressure side and the port-side the suction side. KVLCC2M The second hull form under consideration is the KVLCC2M. The particulars of this hull form are presented below, taken from the website of the coming CFD Workshop in Toko, 25 ( Main particulars of KVLCC2M Designation Model scale (1:64.4) Full scale Length L pp 4.97 m 32. m Beam B 8 m 58. m Draught T 231 m 2 m Block coefficient C B 1 1 Wetted area S m m 2 Bilge keels, rudder and propeller were not present during the model tests and are therefore not modelled in the calculations. Conform the specifications for the workshop, the calculations are conducted with an undisturbed water surface, i.e. neglecting the generation of waves. The measurements were conducted with the model restrained from moving in an direction relative to the carriage. NUMERICAL PROCEDURES Coordinate sstem The origin of the right-handed sstem of axes used in this stud is located at the intersection of the water-plane, midship and centre-plane, with x directed aft, to starboard and z verticall upward. Note that all coordinates given in this paper are made non-dimensional with L pp unless otherwise specified. All velocities are made non-dimensional with the undisturbed velocit U. Figure 1: Definition of curvilinear coordinate sstem At the hull surface, no-slip and impermeabilit boundar conditions are used. The velocities are set to zero ( u = ) while the pressure is olated from the interior. Smmetr conditions are used on the water surface: uξ uη p uζ =, =, =, = ζ ζ ζ Because the velocit and pressure behind the ship are unknown, Neumann boundar conditions are applied on the outflow plane: ui ξ p =, = ξ The velocit components in the inflow plane and off-bod plane are taken from a potential flow calculation while the pressures in both planes are calculated using the Bernoulli equation. During the viscous-flow calculation however, the velocit normal to the off-bod plane is updated to allow for the displacement effect of the viscous boundar laer. Solver set-up Use was made of the in-house solver PARNASSOS, see Hoekstra (1999). It solves the discretised Renolds-averaged Navier-Stokes equations for a stead, three-dimensional incompressible flow around a ship hull, supplemented b a turbulence model. For all calculations, Menter's oneequation turbulence model (1997), extended with the correction for longitudinal vorticit of Dacles-Mariani (1995), was used. The governing equations are integrated down to the wall, i.e. no wall-functions are used. The solver automaticall applies grid-sequencing in order to accelerate convergence on the final grid. This technique starts the calculation on a coarse grid (designated istep=8) obtained b taking ever 8 th stream-wise plane and converges until the maximum changes in the pressure are below a certain limit. Then the grid is refined (istep=4) and the calculation is continued until the required convergence. This procedure is conducted until the solution satisfies the specified convergence criterion on the finest grid (istep=1). 92

3 For the present stud, all solutions were converged until the maximum change of the pressure coefficient between successive iterations had dropped well below Iterative convergence errors are therefore assumed to be negligible with respect to discretisation errors. Uncertaint analsis In order to determine and demonstrate the accurac and reliabilit of solutions of viscous flow calculations, grid dependenc studies are ver important. Several methods for uncertaint analsis are available in literature. In the present paper, the method proposed b Eça and Hoekstra (22) is followed, which amounts to establishing Roache s Grid Convergence Index (GCI) (see Roach (1998)) b a leastsquares method. The full details and background for the followed procedure for the uncertaint stud can be found in their paper, but the main idea is as follows: block consists of an inner block and an outer block, see Figure 2. The inner block is the same for all aw 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 efficientl. Due to the division of the computational domain in an inner and outer block, control of the qualit of the grid near the hull is obtained. Use is made of bod-fitted, nonorthogonal HO-tpe grids, which are strongl stretched towards the hull to capture the strong gradients in the boundar laer. Richardson olation is used to estimate the value φ of a specific quantit for an infinitel dense grid. The numerical prediction error ε i is estimated b: ε i =φi φ =α hi p The olated value φ, the constant α and the apparent order of convergence p are found b appling a least-squares method using the grids with different representative grid-cell sizes h i. This method requires at least four geometricall similar grids of various densities. Using the values found from the least-squares method, the uncertaint in the prediction is in the case of grid convergence (p > ) specified to be as follows, with the factor of 1.25 a safet factor based on experience: U = 1.25 φ φ 1 If divergence or oscillator convergence is found (p <= ) then the uncertaint U is found b: ( ( i) ( i) ) U = 3 max φ min φ Once again, 3 is a safet factor based on experience. Based on the uncertaint analsis, it is assumed that the true solution will be bound with 95% confidence b the interval: φ=φ 1 ± U It must be stressed that in this paper, onl the number of nodes in stream-wise direction was varied. This means that the grids are not geometricall similar. COMPUTATIONAL DOMAIN AND GRID Grid generation For each calculation, separate blocks are used for the port side and starboard side of the computational space. Each Figure 2: Inner (top) and outer (bottom) block structure The inner block is generated with a number of cells corresponding to normal MARIN practise in resistance and powering optimisation calculations, see Hoekstra (1999) or Eça et al. (22). The size (length, depth and width) of this domain is based on experience with simulations of the viscous flow around a ship sailing with zero drift angle. During the grid generation, first a base-grid is generated using a 3D elliptic grid generator, with a reasonabl stretched grid node distribution in the normal direction, j, and the desired node distributions in stream-wise, i, and girth-wise, k, directions. B varing the control parameters in the grid generation process, the deviation from orthogonalit is reduced as much as practicall possible. Then grid stretching along the surface-normal grid lines is applied to arrive at the desired grid spacing at the hull surface. To incorporate the drift angle of the ship, the inner block is rotated around the vertical z-axis over the desired aw angle. Then the outer block is generated around the inner block. The interfaces between the blocks are matching to allow for subsequent merging of the inner and outer blocks. The cell stretching used in the inner block is automaticall applied to the outer block as well. The size of the outer blocks is chosen such that the rotated inner block can smoothl 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 off-bod planes. Before starting the calculations, the separate blocks are merged into one block on the port side of the ship and another block on the starboard side of the ship. 93

4 Series 6 The first grid used in this stud incorporates the hull form of the Series 6 for 1 degrees of drift angle. The port-side and starboard blocks together had for the finest grid nodes in stream-wise, wall-normal and girthwise direction respectivel, resulting in a total of over 2 million nodes. Compared to similar studies presented in literature this number of grid nodes is relativel large. The maximum deviation from orthogonalit in this grid was 45. The inflow plane was located at.65 L pp forward of midship and the outflow plane 2 L pp aft of midship. The width and depth of the domain were 8 L pp and.27 L pp respectivel. In the Figure 3, a top view of the grid (coarsened for presentation purposes) used for this stud is presented. The bow is directed to the left of the figure. Figure 4: KVLCC2M, computational grid, -3 drift REVIEW OF THE CALCULATIONS Series 6 The following calculations were conducted for the Series 6 at 1 drift angle: istep ni nj nk nodes p max Figure 3: Series 6, computational grid, 1 drift KVLCC2M The grid used in this stud incorporates the hull form of the KVLCC2M for and -3 degrees of drift angle (drift angle negative when bow rotated to starboard w.r.t. the incoming flow). For the zero drift case, onl the starboard side of the ship is modelled due to the smmetr with respect to the centre-plane. On the finest grid, the block consisted of nodes in stream-wise, wall-normal and girth-wise direction respectivel. This number of nodes amounts to almost 1.3 million nodes. For the drift calculation, the port-side and starboard blocks together had for the finest grid nodes, resulting in a total of over 2.9 million nodes. Compared to similar studies presented in literature this number of grid nodes is relativel large. The inflow plane was located at.76 L pp forward of midship and the outflow plane 3 L pp aft of midship. The width and depth of the domain were.42 L pp and 6 L pp respectivel. In the following figure, a top view of the grid (coarsened for presentation purposes) used for this stud is presented. The bow is directed to the left of the figure. The reference velocit U for the calculations corresponded to a model scale value of 75 m s -1. With a kinematic viscosit ν of m 2 s -1 this results in a Renolds number of All calculations were conducted with undisturbed water plane. KVLCC2M The following calculations were conducted for the KVLCC2M hull form: drift istep ni nj nk nodes p max The reference velocit U for the calculations corresponded to a model scale value of 4 m s -1. With a kinematic viscosit ν of m 2 s -1 this results in a Renolds number of All calculations were conducted with undisturbed water plane. 94

5 GRID DEPENDENCY STUDY Series 6 Based on the results of the calculations using different grid densities, the discretisation error in the solution can be examined. In Figure 5, the value of the longitudinal velocit u is presented against the transverse position with respect to the centreline. The figure contains the results for a transverse cut close to the bilge and is located at x= (i.e. aft of midship) and z=-.48. This figure shows that the results on the finest grid are not et grid-independent. But it is encouraging that the differences between successive numerical solutions decrease with increasing mesh densit (within 3.3%)..6 hull surface.4 exp istep=1 istep=2.2 istep=4 istep= Figure 5: Comparison grid densit and experimental data, x=, z=-.48, longitudinal velocit u 1 In Figure 6 the differences between the calculated longitudinal velocit and the experimental values are given for the four grids. Especiall for this transverse cut, which is placed directl through the vortex originating from the bilge, the solution still changes with increasing mesh densit. However, once again the differences between successive solutions decrease with increasing mesh densit. u() -u(exp) istep=1 istep=2 istep=4 istep=8 hull surface Figure 6: Difference in longitudinal velocit between calculations and experiments, x=, z=-.48 Also the dependenc of the integrated forces on the ship for the various grids has been studied. In Figure 7, the nondimensional transverse force Y and awing moment N for the different grid densities are given. Corresponding to the definitions used b Longo, Y is made non-dimensional with.5 ρ U S and N with.5 ρ U L pp 3. The relative step-size indicates the coarseness of the grid with respect to the finest grid. Theoreticall, the discretisation error (in stream-wise direction) disappears if the solution is olated to zero relative step-size. The olated value is the value that should be compared to the experiments. The horizontal dashed lines in the figure indicate the uncertaint interval. The figure illustrates that for the transverse force as well as the awing moment the difference in the results for progressivel finer grids reduces to less than 2.5%. The figure also shows that olation of the results from grid dependenc studies to zero for this tpe of computations is not trivial. Although the differences between the solutions for the aw moment are relativel small and large for the transverse force, the uncertaint interval for the aw moment is large due to the fact that oscillator convergence was found. This illustrates that for these cases, the uncertaint estimation method is not well suited. It should be noted however, that onl grid refinement in longitudinal direction was applied. Y' [-] N' [-] Figure 7: Grid dependenc of integrated variables Y' Y' exp N' N' exp The table below shows the quantitative results of the uncertaint analsis, with φ 1 the solution on the finest grid, φ the estimated value using olation to an infinitel fine mesh, U the uncertaint and p the apparent order of convergence: Quantit φ φ 1 U p X' %.43 Y' % 1.85 Z' % 1.52 K' % 1.2 M' % Divergence N' % Oscillator convergence KVLCC2M, zero drift case The following figures illustrate the longitudinal velocit u and transverse velocit v behind the stern of the KVLCC2M for the straight-ahead condition for the four different grids. 95

6 In these figures comparisons are made between the results of the different grids at a transverse cut located at x=.48 and z=-.5. It is seen that for >.1, the results are more or less grid-independent. However, closer to the centreline, the solution still changes considerabl between the second finest grid and the finest grid. Noteworth is the change of the transverse flow direction between <.1. u (,istep=1) u (,istep=2) u (,istep=4) u (,istep=8) z u v Figure 8: Grid dependenc of axial velocit behind stern, x=.48, z=-.5, drift v (,istep=1) v (,istep=2) v (,istep=4) v (,istep=8) Figure 9: Grid dependenc of transverse velocit behind stern, x=.48, z=-.5, drift The reason for this grid dependenc is the resolution of the grid between the aft part of the ship and the position at which the solution is resolved. If insufficient grid cells are present, the transition between the no-slip boundar condition and free-stream condition at the solution position is too abrupt and the development of the flow is not well modelled. In Figure 1, the position of the measurements is compared to the grid for the finest mesh. It is seen that onl two grid nodes are present between the no-slip hull-surface and the measurement position. For the coarser grids, one (for istep=2) or even no node (for istep=4 and istep=8) is present and subsequentl interpolation between discrete results at neighbouring grid nodes will result in inaccurate predictions of the flow at the measurement position. It is therefore proposed to use grid refinement at the stern to resolve the velocities at the measurement position more accuratel x Figure 1: Discretisation of stern and location of measurements (white dot) The dependenc of the integrated variables on the grid densit in stream-wise direction is also studied. In this case, not onl the total forces (subscript t) but also the separate components due to pressure (subscript p) as well as due to friction (subscript f) are considered. Figure 11 presents the results for the longitudinal force X and vertical force Z graphicall. In this case, the forces for the KVLCC2M are made non-dimensional using.5 ρ U L pp T, conform specifications for the CFD Workshop 25. X p ' X f ' X t ' exp Z p ' Z t ' Figure 11: Grid dependenc of integrated variables, drift Z f ' 96

7 Quantitativel, the results are presented in the table below (with the pitch moment M made non-dimensional using.5 ρ U L pp2 T). Quantit φ φ 1 U p X p ' % Oscillator convergence X f ' % 1.92 X t ' % Divergence Z p ' % 1.54 Z f ' % 1.2 Z t ' % 1.52 M p ' % 1.15 M f ' % 1.86 M t ' % 1.15 It is found that with the grid geometr used, the longitudinal force due to the pressure cannot be resolved accuratel. Since this component is in the same order of magnitude as the friction component, the total longitudinal force is also inaccuratel predicted. An increase of the grid densit at the bow and stern will result in a better description of the hull form and the local velocit gradients and subsequentl to a more accurate prediction of the longitudinal force. COMPARISON WITH EXPERIMENTS Series 6 Alread in the previous sections, the results of the calculations have been presented along with results of the experiments. As can be observed in Figure 5 and Figure 6 with the transverse cut, the results of the calculations for the finest grid are quite close to the measurements. The table below presents the maximum differences obtained from the comparison of the transverse cuts: z istep=1 istep=2 istep=4 istep=8 u max u max These results show that for istep=1, the maximum error in the longitudinal velocit at the transverse cut at x= is less than 5% of the undisturbed velocit. The results show that the vortex from the bilge in the calculations is slightl shifted awa from the hull, resulting in relativel large differences when subtracting the experimental results from the calculations. Qualitativel, the phsics of the flow are well represented. Quantitativel, the comparison of the integral forces in Figure 7 shows that for the istep=1 results a good correlation of the aw moment with the measurements is found. However, a deviation of the side force from the experimental values is also present. A reason for the discrepanc between the simulations and the experiments can be that the experiments were conducted with the model free to trim and sink while the calculations were done with the ship on even keel. Furthermore, differences ma be introduced because in the calculations an undisturbed water plane was assumed. The influence of this assumption has to be examined in further studies. In Figure 12 through Figure 14, the calculated longitudinal velocit fields based on the finest grid for several transverse planes are given together with the velocit field obtained from the experiments. Note that the calculated results have been projected on the measurement points and therefore some irregularities that are present in the measurement data are introduced in the simulation results. The dashed lines represent the measurements and the continuous lines represent the calculations. Figure 12: Axial velocit contours, x=, 1 drift Figure 13: Axial velocit contours, x=.5, 1 drift Figure 14: Axial velocit contours, x=.7, 1 drift The correspondence between the measurements and the simulations is ver promising and the majorit of the calculated contour lines follow the contours from the experiments closel. The calculated velocit profile at the keel is ver close to the measured profile while the position and magnitude of the vortex developing at the keel is well predicted. The strength of the vortex from the bow area is slightl less than found during the measurements and the location is somewhat shifted downward and further awa from the ship. KVLCC2M, zero drift case Comparisons between the longitudinal and transverse velocities obtained from the calculations and from the measurements at x=.48/z=-.5 are given in Figure 15 and Figure 16. It is seen that the phsics of the flow are captured but some discrepancies are seen close to the centre line, especiall in the transverse velocit. In further research, 97

8 .7 additional calculations will be done with a refined grid in order to investigate whether this is caused b the poor grid resolution at the stern. The prediction of the longitudinal force is rather poor. The resistance force from the simulation is 46% lower than the measured resistance. As alread suggested, grid refinement around the bow and stern will improve the prediction. u (,istep=1) u (exp,mirrored) u (exp) KVLCC2M, -3 drift case Onl integral variables are available from the measurements for the -3 drift case. The table below presents the comparison between the simulated and experimental results. Quantit Pressure Friction Total Exp Error X' % Y' % Z' K' M' N' % u During the experiments, also a 3 drift angle condition was tested. When assuming a smmetric flow around the ship model for equal positive or negative drift angles, these results ma also be used to compare to the simulation results. This leads to the following comparison: v Figure 15: Axial velocit at x=.48/z=-.5, drift v (,istep=1) v (exp,mirrored) v (exp) Quantit CFD Exp (-3 ) Error Exp (3 ) Error X' % % Y' % % N' % % Apparentl, the awing moment measured during the test with -3 drift angle differs significantl from the result for 3 drift angle. This comparison leads to more favourable error values, but still large discrepanc is found in the transverse force. For practical application the predictions should be improved. Figure 18 presents the axial velocit contours at x=.48. The asmmetr in the flow due to the drift angle is clearl present Figure 16: Transverse velocit at x=.48/z=-.5, drift Figure 18: Axial velocit contours, x=.48, -3 drift Figure 17: Axial velocit contours at x=.48, drift Figure 17 presents the axial velocit contours at x=.48. The horizontal line in the figure represents the transverse cut at z=-.5 at which the velocities were resolved. 98

9 CONCLUSIONS Simulations of the viscous-flow around two different hull forms have been conducted. In order to determine the grid dependenc several grid densities were used for some of the calculated conditions. Although still notable differences between the two finest meshes for the Series 6 were found, detailed comparisons with experimental data show that the phsics of the flow are well predicted even when looking at discrete positions in the flow field. However, for the second hull form, the KVLCC2M, strong grid dependence was found at the propeller plane. This is caused b poor grid resolution between the hull surface and the propeller plane. Qualitativel, promising results are obtained. For practical purposes however, the accurac of the results should be improved. For the current calculations, the predicted aw moment is close to the measurements but the side force is under-predicted. Reasons for these discrepancies might be the neglect of trim and sinkage for the Series 6 and of the water surface deformation. For the KVLCC2M, this is probabl also caused b insufficient grid resolution at the bow and stern. This should be studied in future research. Tahara, Y., Longo, J. and Stern, F. "Comparison of CFD and EFD for the Series 6 CB=.6 in stead drift motion". Journal of Marine Science and Technolog, pages 7:17 3, 22. Toxopeus, S.L. "Simulation and Validation of the Viscous Flow around the Series 6 Hull Form at 1 Drift Angle". 7th NuTTS Numerical Towing Tank Smposium, Hamburg, October 24. REFERENCES Alessandrini, B. and Delhommeau, G. "Viscous free surface flow past a ship in drift and in rotating motion". 22 nd Smposium on Naval Hdrodnamics, pages , August Cura Hochbaum, A. "Computation of the turbulent flow around a ship model in stead turn and in stead oblique motion". 22 nd Smposium on Naval Hdrodnamics, pages , August Dacles-Mariani, J., Zilliac, G.G., Chow, J.S. and Bradshaw, P. "Numerical / experimental stud of a wingtip vortex in the near field". AIAA Journal, Vol. 33, pp , September Di Mascio, A., and Campana, E.F. "The numerical simulation of the aw flow of a free surface ship". 7 th International Conference on Numerical Ship Hdrodnamics, pages , Jul Eça, L., Hoekstra, M., and Windt, J. "Practical grid generation tools with applications to ship hdrodnamics". 7 th International Conference on Grid Generation and Computational Field Simulations, Februar 22. Eça, L. and Hoekstra, M. "An evaluation of verification procedures for CFD applications". 24 th Smposium on Naval Hdrodnamics, Jul 22. Hoekstra, M. Numerical Simulation of Ship Stern Flows with a Space-Marching Navier-Stokes Method. PhD thesis, Delft Universit of Technolog, Facult of Mechanical Engineering and Marine Technolog, October Longo, J.F. Effects of Yaw on Model-Scale Ship Flows. PhD thesis, Universit of Iowa, Ma Menter, F.R. "Edd viscosit transport equations and their relation to the k-ε model". Journal of Fluid Engineering, Vol. 119, pp , Roach, P.J. Verification and Validation in Computational Science and Engineering. Hermosa Publishers,