NUMERICAL SIMULATION OF THE FLOW AROUND A C-CLASS CATAMARAN Filipe Santos Carvalho

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1 NUMERICAL SIMULATION OF THE FLOW AROUND A C-CLASS CATAMARAN Filipe Santos Carvalho filipe.s.carvalho@tecnico.ulisboa.pt Instituto Superior Técnico, Universidade de Lisboa, Portugal December 2016 ABSTRACT The main goal of this work was to study the hydrodynamic performance of a C-Class catamaran, determining forces acting on the vessel at different speeds, using RANS equations. In the first part of the work it was studied the flow around the catamaran modelling the free surface as a symmetry plane. A verification procedure was conducted leading to high numerical uncertainties. It was evaluated the suitability of the mesh and physical models setup. It was also evaluated the effectiveness of laminar-transition prediction, showing an overall small impact on the total resistance. In the second part of the work the free surface was modelled using VOF and VOF Waves, following STAR-CCM+ best practices for the setup. The results show that viscous resistance is dominant, reducing the overall importance of wave effects, leading to acceptable errors when neglecting free surface effects. Finally, it was evaluated the performance of the catamaran at different speeds allowing to compute and plot the resistance force with the Froude number. It was not possible to identify the take-off speed of the catamaran. Conclusions of this work present a physical model setup without free surface effects as a reliable basis for future works, allowing a more efficient process and weightless computational requirements. Keywords: catamaran; free surface; Volume of Fluid; RANS 1 INTRODUCTION A C-Class catamaran is a wind-sailing twohulled race vessel composed by a single rigid wingsail, a rudder and a daggerboard mounted in each hull. The C-Class catamaran in scope was designed and manufactured by Optimal Structural Solutions (Optimal), a Portuguese based company, aiming to compete in the 2013 International C-Class Catamaran Championship (ICCCC), in the UK. It was the first project for Optimal in the area. Nowadays, Optimal aims to develop a second prototype based on acquired experience, thus it is fundamental to understand the hydrodynamic performance of the catamaran. The main goal of this work is to study the hydrodynamic performance of the catamaran, determining forces acting on the vessel at different speeds. The work developed is divided in two main parts: hydrodynamic analysis of the catamaran neglecting free surface effects; hydrodynamics analysis considering free surface effect at different speeds. The first aims to get to know the flow around the catamaran and support decisions for the second part of the work, using a symmetry plane condition to model the free surface. At this stage, a verification exercise will be done to estimate numerical uncertainties associated with spatial discretization. The second part aims to find forces acting on the catamaran at different speeds considering free surface effects. It is assumed that the catamaran is sailing supported by one hull, with its appendices (daggerboard, rudder and winglets) and all the remaining components are above water level. It is assumed an equilibrium of moments in all directions, thus only force will be evaluated. Computational hardware is composed by an Intel Xeon CPU E GHz (16 cores), with 132 Gb of RAM and 1 Tb of HDD running on Ubuntu LTS x64. STAR-CCM+ v commercial code was used. The origin of the reference coordinate system is located at the most downward point of the 1

2 transom stern plane, with X oriented to forward (parallel to the waterplane), Y to port side (inwards) and Z oriented vertically (normal to the water plane). and its walls were named as: Inlet, Outlet, SlipWalls and SymmPlane. The problem is characterized by a Reynolds number of Re = and a Froude number of Fr = 0.62, computed with U = 10 knots = m/s and L = 7 m (waterline length) as reference velocity and length. The work uses the incompressible Navier- Stokes equations with constant properties to solve fluid dynamics through Menter s two-equation SST k ω turbulence model [1], with all y + wall treatment scheme, as recommended and widely used in external flow Naval applications [2]. Free surface was modelled using Volume of Fluid (VOF) model and VOF Waves, as also recommended for naval applications [3]. 2 ANALYSIS NEGLECTING FREE SURFACE EFFECTS 2.1 SETUP Waterline calculation The waterline defines the interface between floating and displaced volumes of the vessel. Moments in lateral and vertical directions are neglected, thus only vertical and longitudinal stability must be determined. Vertical stability requires solving Archimedes buoyancy equation I + P = ρ water displaced + P = 0, where the buoyancy must cancel the vessel weight. Longitudinal stability (heeling) requires a vertical alignment between the vessel s centre of gravity and the centroid of the waterline plane. Based on geometrical and components mass data, both stability requirements can be easily computed. The calculated displaced volume is = 0.29 m 3 and the waterline is at z = m. Figure 1 - Catamaran geometry for simulations neglecting free surface effects It was used an unstructured grid mesh, using triangular elements for the surface mesh, hexahedral for the volume mesh and prism layers near the wall envisioning boundary layer. The surface mesh size and curve size were defined aiming a proper spatial discretization of the geometry and of the zones with intense gradients. As for the prism layer, layer total thickness, stretching factor of geometric progression and number of layers were defined aiming y (1) + < 1 in the catamaran surface 1. These properties were estimated using y + definition and Schlichting friction factor correlation C f = (2 log 10 Re x=l 0.65) 2.3 [4], based on components Reynolds number. Domain size were set with 10 L length ( 1.5 L forward the bow and 7.5 L aft the stern), 10 L width and 5 L depth. Two conical volumetric refinement regions were defined envisioning a finer grid in the wake of the daggerboard and hull, with L and 2 L length, respectively. Velocity inlet and pressure outlet boundary conditions were specified for the inlet and outlet boundaries, respectively with inlet velocity U = 10 knots = 5.14 m/s and 0 Pa gauge pressure at the outlet. Turbulence intensity I and turbulence viscosity ratio β were set to I = 0.01 and β = Geometry, grid and physical setup The geometry in scope is presented in figure 1. The geometry is surrounded by a parallelepiped domain 2.2 VERIFICATION Verification procedure was followed using four grids, one coarser (h 4 ) and two finer (h 1 and h 2 ) relatively to a reference grid ( h 3 ), considering a 1 y + values at the walls were checked in the firsts simulations. 2

3 3 refinement rate of r = h coarser /h finer = and a constant mesh typology. Table 1 presents the results of the verification procedure for each variable of interest. Numerical errors for the estimated exact solution are below ~3.5% for the finest grid, which is acceptable. Due to the noisy, a data factor of safety F s = 3 was used leading to large numerical uncertainties. Smaller numerical uncertainties are only possible by successive grid refinements, say half the finest grid size used, leading to an increase of ~2 3 the number of mesh element, up to 253 million cells, which is not economically viable for the scope of this work. Numerical errors for the reference grid are below 4.5% of the estimated exact solution, thus it was decided to use this grid in the current work, saving up to 67% in the number of cells required. 12% of total resistance due to the magnitude of viscous drag. Fx [N] Fy [N] Fz [N] Daggerboard -7.18E E E+02 Hull -1.55E E E+02 Rudder -1.16E E E+00 Winglet -3.52E E E+01 Total -2.42E E E+02 Table 2 - Force results for the simulation without free surface Fx [N] Fy [N] Fz [N] Pressure -9.05E E E+02 Viscous -1.52E E E-01 Total -2.42E E E+02 Table 3 - Pressure and viscous contributions for the force results Fx Fy Fz h E+02 N 1.62E+03 N 4.37E+02 N h E+02 N 1.61E+03 N 4.38E+02 N Relative error 3.5% 0.5% 0.3% Uncertainty 11% 3% 1% Table 1 - Verification procedure results 2.3 RESULTS AND DISCUSSION Reference results Table 2 presents the reference results of forces acting on the catamaran. As expected, the hull represents the major contribution for the vessel s total resistance whereas the daggerboard for the lateral and vertical forces and heeling moment. Moreover, in table 3, lateral and vertical forces are dominated by pressure forces, while viscous forces represent the major contribution for the resistance component, due to the significant hull length (~7 m) and low impact on the undisturbed flow. Figure 2 presents the pressure field on the catamaran, showing a favourable pressure gradient up to the daggerboard s position. A detailed analysis of the axial velocity field on section planes over the wake of the catamaran show negligible impact of the vessel on the undisturbed flow. A similar analysis can be done for the axial vorticity showing insignificant vortices generated by the rudder and winglets, while the daggerboard s vortex has a higher size and intensity leading to an increase in the total resistance by Drag Lift 2. In spite of 48% of the total pressure resistance being generated by the daggerboard, it only represents Figure 2 Pressure field on the catamaran when free surface effects are neglected Mesh and physical model setup analysis Several analysis were performed to the mesh and physical model setups: wall treatment scheme ( low y + wall treatment instead of all y + wall treatment ); turbulence parameters (changing by a ratio of ½ upwards and downwards turbulence intensity and turbulence viscosity ratio); computational domain size (reducing individually each dimension successively by a ratio of ½); and mesh refinement regions (doubling regions length and not using refinement regions). Each analysis concluded negligible impact on the force results acting on the vessel. Due to significant saving on the number of cell, a typology with no wake related refinement regions was used in the simulations considering free surface effects (chapter 3). 3

4 2.3.3 Laminar-turbulent transition The favourable pressure gradient extension in the hull and the low Reynolds number (10 5 ) of the daggerboard, rudder and winglets suggest high impacts on resistance force by wrongly anticipating laminar-turbulent transition (a common characteristic of typical RANS turbulence models where transition Reynolds at zero pressure gradient is Re , [5]). The γ Re θ model was used to test this accuracy and influence. The free stream edge was defined based on wall distance, as recommended in [6], using δ/x = 0.37Re 1 5 x to compute maximum boundary layer thickness δ L = m. The free edge was defined when the wall distance equals 0.09 m, ensuring that the boundary layer is nowhere thicker than 0.09 m. Table 4 presents a decrease of 11.7% on total resistance when using the γ Re θ model, mostly due to the transition prediction on the daggerboard, rudder and winglet as shown in table 5. The typical transition Reynolds number Re tr = of a flat plate confirms the predicted location for the transition ~2% of the hull s length, ~50% of the daggerboard chord length, ~65% of the rudder chord length and lengthier than winglets chord length showing a significant difference to the case when γ Re θ model is not used ~0.01% of the hull s length, ~4% of the daggerboard chord length, ~5% of the rudder chord length and ~12% of the winglets chord length. In spite of the differences on the vessel s resistance, it was decided not to use the γ Re θ model due to the complexity increment of the overall physical model. Reference case γ Re θ Fx -2.42E+02 N 11.70% Fy 1.63E+03 N 4.41% Fz 4.37E+02 N 4.84% Table 4 - Variation in the force results when using γ Re θ model Reference case γ Re θ Daggerboard -7.18E+01 N 23.95% Casco -1.55E+02 N 2.57% Rudder -1.16E+01 N 52.39% Winglet -3.52E+00 N 30.51% Total -2.42E+02 N 11.70% Table 5 Variation in the resistance force of each component when using γ Re θ model 3 ANALYSIS CONSIDERING FREE SURFACE EFFECTS 3.1 SETUP Geometry An identical geometry of the one described on the previous chapter was used, with the addition of the floating part volume. Likewise, a parallelepiped domain was defined with the following named walls: Inlet, Outlet, SlipWalls, Top and Bottom Geometry, grid and physical setup It was used an unstructured grid mesh, using triangular elements for the surface mesh, hexahedral for the volume mesh and prism layers near the wall envisioning boundary layer. The surface mesh parametrization of the previous chapter remained unchanged. The prism layer parameters were defined using the approach of the previous chapter, aiming y (1) + < 1 in the daggearboard, winglets and the submerged part of the rudder, while the hull and the remaining part of the rudder were targeting 30 < y (1) + < 300 in order to avoid convergence issues and following common practices on VOF model, [7] [8] [9]. The domain used was similar to the one described in the previous chapter, with the addition of 1 L ref above the reference waterplane. VOF best practices [10] suggest the use of finer grid over the free surface. Therefore, three free surface refinement regions were defined: a coarser region covering the entire domain size near the free surface; a finer region covering the entire Kelvin wedge; and also a finer region around the transom wave. The regions heights were defined given a 10% margin of the estimated wave amplitude (the coarser regions considered a 15% margin). VOF waves best practices suggest that the length and width of a cell within the refinement regions should be d X = d Y = λ 80 and the height of d Z = ξ 40, where λ and ξ are the wave length and amplitude, and can be estimated using equations (1) and (2) (where S stands for cross sectional area). The coarser region was defined with an element size twice the size of the finer region. 4

5 ξ(x, 0)L S λ T L ref = 2πFr 2 (1) = 1 Fr 2 π [ 1 1 x cos ( Fr 2 (x L + 0.5) + π 4 ) L 1 1 x cos ( 0.5 L Fr 2 (x L 0.5) + π 4 )] (2) Physical models used were the same of the ones used in the previous chapter. VOF model was used with an implicit unsteady solver, using a time step Δt suggested by VOF best practices ( Δt = P/(2.4 80)) and lower and upper CFL of CFL l = and CFL u = , ensuring that HRIC scheme is always used. VOF Phase Replacement was used in order to avoid numerical ventilation of air under de hull, a common issue of VOF applications [7]. Finally, VOF waves model with a flat wave setup was used (current and wind speed of 10 knots) and VOF wave damping option activated with a damping length of twice the wave length [11]. Five inner iterations were defined as a stopping criteria. Simulation was stopped when force and moments monitors were within a ±1% range. Fx [N] Fy [N] Fz [N] Daggerboard -7.65E E E+02 Hull -1.70E E E+03 Rudder -1.71E E E+01 Winglet -3.61E E E+01 Total -2.67E E E+03 Table 6 - Force results for the simulation with free surface Fx [N] Fy [N] Fz [N] Pressure -9.96E E E+03 Viscous -1.68E E E+00 Total -2.67E E E+03 Table 7 - Pressure and viscous contributions for the results 3.2 RESULTS Reference results Table 6 and table 7 present the results for the catamaran sailing at a reference speed of 10 knots (Fr = 0.62). The total vertical force is 549 N higher than vessel s weight mainly due to the hull buoyancy, whereas the daggerboard s lift is ~1/ 3 of the hull s buoyancy. The lateral force is dominated by the daggerboard force. The hull is also the main contributor for the total resistance force, which is mainly due to viscous resistance showing minor influence of wave resistance (63% to 37%, respectively). Figure 3 shows the relative pressure distribution on the catamaran surface. Figure 4 and figure 5 show the calculated free surface, where dominant divergent wave pattern and a dry transom effect can be seen. Figure 6 shows the volume fraction of water on the catamaran surface, where a sharp air-water transition and the dry transom can also be seen. The dry transom effect is the result of low pressures at the stern avoiding water flow separating at the stern, thus pressure resistance savings comparing to an equivalent situation without dry transom effect. Figure 3 Relative pressure distribution on the catamaran, considering free surface effects Figure 4 Top view of the wave pattern 5

6 Figure 5 Free surface detail: the dry transom and the transom wave Table 9 present the resistance force result detailed per component of the catamaran, showing no variation on the hull s resistance in both CSL and SSL cases leading to the conclusion that, when comparing both cases, the wave resistance increase is compensated by the resistance savings due to dry transom (the dry transom effect does not occur in the Case SSL due to the absence of the free surface).the huge variations in the rudder results are due to its floating part, located at the beginning of the transom wave. The daggerboard and winglet results variations are acceptable within the uncertainty range of the problem. SSL CSL Δ Daggerboard -7.20E+01 N -7.65E+01 N 6% Hull -1.69E+02 N -1.70E+02 N 0% Rudder -1.15E+01 N -1.71E+01 N 33% Winglet -3.51E+00 N -3.61E+00 N 3% Total -2.56E+02 N -2.67E+02 N 4% Table 9 Resistance force result in both SSL and CSL cases Figure 6 Volume fraction of water on the catamaran surface Analysis on the impact of neglecting free surface effects It is relevant for future case studies on the topic to estimate the errors associated when neglecting free surface effect simulations described in the previous chapter. Table 8 presents the comparison between the results with and without free surface effects (Case CSL and Case SSL, respectively), indicating small variations in the results. Note that Case SSL results are based on the reference simulation of the previous chapter considering a log law setup to compute the boundary layer and summing the hydrostatic pressure to the relative pressure, ensuring equivalent setups in both cases (the use of log laws lead to an increase of 6% in the total resistance). Fx Fy Fz Case SSL -2.56E+02 N 1.63E+03 N 3.41E+03 N Case CSL -2.67E+02 N 1.52E+03 N 3.49E+03 N Δ 4% 7% 2% Table 8 Comparison between results with (CSL) and without (SSL) free surface effects. Δ represents the variation in the results Analysis at different Froude numbers The performance of the catamaran was evaluated for different Froude numbers. Table 10 present the results at Fr = {0.62 ; 0.5 ; 0.4 ; 0.3}, showing that the vertical force is always higher than the weight of the vessel, thus take-off speed determination is inconclusive. By computing the adimensional resistance force coefficient C X = F X / 1 2 ρau2, using A = /L ref, it was found that the resistance force coefficient varies with the increase of the Froude number, table 11. This conclusion is a common characteristic of free surface problems due to the increasingly importance of the wave resistance with the increasing of speed. Figure 7 shows the wave pattern at each Froude number, where transverse waves are dominant at low speed and divergent waves are dominant at higher speeds, as expected [12]. Fr U [m/s] Fx [N] Fy [N] Fz [N] E E E E E E E E E E E E+03 Table 10 Force results at different Froude numbers 6

7 C X U [m/s] Table 11 Resistance force coefficient at different Froude numbers 5.14 m/s 2.49 m/s Figure 7 - Wave pattern on the free surface at different speeds 4 CONCLUSIONS 3.31 m/s 4.14 m/s The main goal of this work was to study the hydrodynamic performance of the catamaran, determining forces acting on the vessel at different speeds. As a starting point, it was studied the flow around the catamaran neglecting free surface effects. A verification procedure to the spatial discretization was conducted leading to high numerical uncertainties, due to noisy data. A reduction of the uncertainty is only possible considering additional grid refinements that would have significant impact on the computational power required. It was evaluated the suitability of the wall treatment scheme, turbulent inlet parameters, computational domain dimensions and grid typology. It was also evaluated the effectiveness of laminar-transition predication using γ Re θ transition model, showing minor overall impact on the total resistance, despite the significant impact on the resistance of the catamaran appendices (daggerboard, rudder and winglets). In the second part of the work, free surface effects were considered using VOF and VOF Waves models. Both mesh and physical setup follow STAR- CCM+ best practices. The results show minor influence of pressure resistance (including wave resistance), as viscous resistance is the main contributor for the total resistance (37% and 63%, respectively). Therefore, small variations in the results when comparing to simulations without free surface effects (< 8%) were found the impact of 7

8 not considering wave resistance is attenuated by the pressure drag on the transom stern (in simulations without free surface). This is an important conclusion for future works on the area, presenting the free surface modelling as a symmetry plane as a reliable basis for preliminary designs. Finally, it was evaluated the performance of the catamaran at different speeds allowing to compute and plot the resistance force with the Froude number, envisioning a future comparison with wingsail data. It was not possible to identify the take-off speed of the catamaran. 5 REFERENCES [1] F. R. Menter, Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications, AIAA, vol. 38, no. 8, pp , Attitude, Chalmers University of Technology, Gothenburg, [9] M. Haase, J. R. Binns, G. Thomas and N. Bose, Resistance prediction of mediumspeed catamarans using free-surface viscous flow simulations, Australian Maritime College, Launceston, Australia, [10] STAR-CCM+ User Manual v [11] S. Gillis, How do I setup my mesh to best capture VOF Waves, [Online]. Available: ArticleDetail?id=kA PRbCAM&t ype=faq kav&searchterm=vof&product =&Type=&Faq=&_strSearchType=Basic&kn owledgebase=true. [12] O. M. Faltinsen, Hydrodynamics of highspeed marine vehicles, New York: Cambrige University Press, [2] L. Larsson and H. Raven, Ship Resistance and Flow, Jersey City, New Jersey: The Society of Naval Architects and Marine Engineers, [3] L. Larsson, F. Stern and M. Visonneau, Numerical Ship Hydrodynamics - An Assessment of the Gothenburg 2010 Workshop, Dordrecht: Springer, [4] H. T. Schlichting, Boundary Layer Theory, McGraw-Hill Book Company, [5] E. L. and H. M., The numerical friction line, Journal of Marine Science and Technology, vol. 13, no. 4, pp , November [6] P. Malan, K. Suluksna and E. Juntasaro, Calibrating the Gamma-ReTheta Transition Model for Commercial CFD, in 47th AIAA Aerospace Sciences Meeting, [7] I. M. Viola, R. Flay and R. Ponzini, CFD Analysis of the Hydrodynamic Performance of Two Candidate America s Cup, The International Journal of Small Craft Technology AC33 Hulls, Janeiro [8] D. Frisk and L. Tegehall, Prediction of High- Speed Planning Hull Resistance and Running 8

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