27th Symposium on Naval Hydrodynamics Seoul, Korea, 5-10 October 2008

Vortical and Turbulent Structures Using Various Convection Schemes with Algebraic Reynolds Stress-DES Model for the

KVLCC2 at Large Drift Angles

Frederick Stern, Farzad Ismail, Tao Xing, Pablo Carrica IIHR-Hydroscience and Engineering

The University of Iowa Iowa City, IA 52242-1585

ABSTRACT Vortical and turbulent structures for the KVLCC2 at a range of drift angles are studied using various convection schemes coupled with algebraic Reynolds stress detached eddy simulation (ARS-DES) turbulence models. The convection schemes and the turbulence models are evaluated quantitatively using rigorous verification and validation (V&V), including comparisons with available EFD data. For 0˚ drift, the integral forces are most accurately predicted by the fourth order (hybrid) interpolation scheme (FD4h) coupled with ARS but the local quantities (velocities and turbulent quantities) are best predicted by the second order TVD scheme with Superbee limiter coupled with ARS (TVD2S-ARS). For 12˚, the local quantities and the integral forces and moments are most accurately predicted by TVD2S-ARS. The vortical structures are also least dissipated when computed with TVD2S-ARS at 12˚ and 30˚ drift angles. Turbulent structures are analyzed using ARS-DES model with a pure FD4 scheme. The turbulent kinetic energy (TKE) and Reynolds stresses peak near the separation point at the bow and within a certain distance to the vortex core of all the vortices. Near the bow, turbulent structures are similar to those for a separated turbulent boundary layer and the recirculation region of a backward-facing step flows. Overall Reynolds normal stresses have similar distribution and magnitude compared to TKE. uw and vw are an order of magnitude smaller than uv , which has the same order of magnitude with the normal stresses. TKE and Reynolds stresses reach a local maximum right after the vortex breakdown points for helical vortex tubes and are intensified along the vortex core further downstream, likely due to the enhanced unsteady oscillation caused by the helical instability,

which is consistent with previous studies on vortex breakdown for flows over a delta wing. INTRODUCTION

Ship flows are challenging to computational fluid

dynamics (CFD) due to unique physics and application conditions, ranging from resistance and propulsion to general six degree of freedom ship motions and maneuvering. Of interest herein are vortical and turbulent structures for ship flows at large drift angles, which cast challenges to develop robust and accurate convection schemes and advanced turbulence models.

Most finite difference/volume convection schemes suffer from artificial diffusion and phase errors. In ship hydrodynamics, these errors cause undesirable effects such as smoothed and/or shifted free surface waves and also under prediction of vorticity, yielding inaccurate prediction of integral quantities such as forces and moments, and local quantities such as propeller plane velocities. Very high order accurate (3rd order and above) convection schemes can be used to reduce numerical diffusion and phase errors. Di Masscio et al. (2008) evaluated several convection schemes on a NACA0012 airfoil and DDG51 surface combatant and showed that high order schemes are more accurate than low order schemes but at the expense of losing computational robustness. To achieve stability without compromising the solution accuracy, most ship hydrodynamics computations rely on second order convection schemes as witnessed in Tokyo 2005 CFD Workshop (Hino, 2005). However, different second order convection schemes have different magnitudes of numerical diffusion and phase errors, thus each scheme may produce results that vary significantly (Jasak et al., 1999). More importantly, the accuracy of the convection schemes depends on the grids used,

particularly on the metrics of grid distortions which include grid aspect ratio (AR), grid stretching or expansion ratio (ER) and grid skewness (Q). Unfortunately, the evaluation of convection schemes is usually performed on uniform and orthogonal grids in one (Waterson and Deconinck, 2007) or two dimensions (Tammamidis and Assanis, 1993) without investigating the effects of the grid distortions.

Accurate turbulence modeling is also important in ship hydrodynamics computations. An ideal turbulence model should have the capability to address the anisotropy that is caused by the presence of a thick boundary layer or free surface. Unsteady vortical structures can be formed by the anisotropy of turbulence, forced separation (propeller, waves), or natural separation (transom sterns, control surfaces, large drift angle ship flows), which require turbulence models to accurately resolve the instabilities of the organized oscillations and turbulent structures. Bubble entrainment caused by wave breaking, bubbly air water mixture layers, and bubble/air-layer induced skin-friction drag reduction cause the needs to develop complex turbulence models including surface tension and large density ratios. In the CFD Tokyo 2005 workshop, computations using anisotropic turbulence model greatly improved the flow predictions over isotropic models when flow separation was not severe but showed similar results for oblique ship flows (Hino, 2005). Xing et al. (2007a) analyzed the role of isotropic blended k-ε/k-ω (BKW) versus anisotropic algebraic Reynolds stress (ARS) RANS models for steady ship flows and BKW based detached eddy simulation (BKW-DES) versus ARS-DES for unsteady ship flows. Results using BKW and ARS Reynolds averaged Navier-Stokes (ARS-RANS) models for 0 and 12 degrees showed that ARS model significantly improved the predictions of the resistance coefficients, axial velocity, and turbulent kinetic energy distributions at the propeller plane. For 30˚ drift, BKW-RANS and ARS-RANS show steady solutions while DES predicts unsteady flows but limited differences on forces, moments and instabilities were observed between BKW-DES and ARS-DES. KVLCC2 at 60˚ drift was also studied using ARS-DES.

The applicability of URANS and DES to predict organized vortical structures, instabilities, and turbulent structures is investigated in the complementary study by EFD (Metcalf et al., 2006), URANS with surface tracking method (Kandasamy et al., 2008), and DES with single-phase level set method (Xing et al., 2007b) for free-surface wave-induced separation around a surface-piercing NACA0024. These studies identified the organized vortical structures and associated instability mechanisms and turbulent structures of unsteady free-surface wave-

induced separation, with quantitative verification and validation. These studies also predicted three dominant frequencies in the separation region, i.e., shear layer instability, Karman-like shedding and flapping, and explained the detailed flow physics behind these three frequencies. URANS can capture the gross features of the unsteady separation and identify the important instabilities, but with deficiency in the amplitudes of the oscillation frequencies. Compared to URANS, DES predicts much broader frequency content that are similar to the EFD data. Without the aid of URANS, it is difficult to isolate those frequencies and relate them to instability theories. However, once those instabilities are identified with the aid of URANS, DES can resolve more flow physics and provide a much better analysis of the turbulent structures. Anisotropy invariant maps show that the turbulence is anisotropic in the middle of the separation region and that it is at a two-component state near the foil surface. The turbulent kinetic energy (TKE) and its budget show similar features to previous canonical flows but with large three-dimensional and free surface effects. The free surface damps velocity and pressure fluctuations and moves the peaks of turbulence quantities from the high-speed to the low-speed side of the free shear layer.

The analysis for vortical structures and instabilities for NACA0024 and tip vortex instability for delta wings was extended by (Xing et al., 2007a) to study KVLCC2 flows at drift angle 30 degrees using ARS-DES, including quantitative verification but without any turbulent structure analysis. Two shear layer instability modes, a Karman-like vortex shedding, and three helical mode instabilities were identified. Compared to previous experimental and computational results, the Strouhal number (St) for Karman-like instability was in the same range whereas St for shear layer instability was smaller. Similarities and differences between the helical mode instabilities of the current study and those of delta wing flows were also discussed. Tip vortices were strongly associated with spiral type breakdowns where the vortices underwent a sudden expansion. It was shown that the breakdown led to significant turbulence intensities at the breakdown position and increased turbulence levels further downstream (Breitsamter, 1997). Quasi-periodic velocity fluctuations occurred downstream of vortex bursting corresponding to a helical mode instability of the breakdown flow field. The dominant frequency can be scaled using the wing leading-edge normal velocity and a length scale for the lateral expansion of the burst vortex core, which resulted in a universal frequency parameter at around 0.28 (Breitsamter, 2008).

To the authors’ best knowledge, rigorous verification and validation (V&V) of the convection schemes coupled with isotropic or anisotropic

turbulence models on the KVLCC2 and the study of its turbulent structures have not been performed. Also, guidelines for constructing computational grids in terms of the grid aspect ratio (AR), stretching or expansion ratio (ER) and skewness (Q) for an effective use of the convection schemes in industrial CFD have not been reported. This study combines the complementary efforts of the evaluation of various convection schemes for 0˚ and 12˚ drift angles (Ismail et al., 2008a and 2008b) and the analysis of vortical and turbulent structures for 30˚ drift using ARS-DES (Xing et al., 2008a.)

The approach is to compare the two best schemes (TVD2S, FD4) obtained by Ismail et al. (2008a) (refer to appendix A for summary) and the baseline FD2 scheme for prediction of the vortical and turbulent structures, integral forces and moments, and the local quantities. An algebraic Reynolds stress turbulence model (ARS) is used (Wallin and Johnson, 2000) and a DES computation is performed for the unsteady case at 30˚ drift. The overall V&V methodology and procedures follow Stern et al. (2006) with an improved correction factor formula for grid/time-step uncertainty estimates (Xing and Stern, 2008b). Validation for comparison of multiple convection schemes/turbulence models follows Coleman and Stern (1997). Complete V&V analyses are performed for the integral forces and moments for 0˚ and 12˚ drifts. Analysis of turbulent structures for a surface-piercing NACA24 (Xing et al., 2007b) is extended to study unsteady flows around KVLCC2 at 30˚ drift with additional Reynolds stress budgets. To ensure minimal degradation of the convection solution, the KVLCC2 grids are constructed as closely as possible to the guidelines proposed by Ismail et al. (2008a). COMPUTATIONAL METHODS The general-purpose solver CFDShip-Iowa-V.4 (Carrica et al., 2007) solves the unsteady RANS or DES equations in the liquid phase of a free surface flow. The free surface is captured using a single-phase level set method. The blend k-ε/k-ω model (Menter, 1994) and an explicit algebraic Reynolds stress model (Wallin and Johnsson 2000) are used for isotropic and anisotropic turbulence modeling, with a DES option (Xing et al., 2007a) when flow is unsteady. Initial boundary value problems can be solved using either absolute inertial coordinates or relative inertial coordinates (Xing et al., 2008c). Numerical methods include advanced iterative solvers, conservative upwind-type finite difference convection schemes (Wilson et al., 2004a), parallelization based on a domain decomposition approach using the message passing interface (MPI), and dynamic overset grids for local grid refinement and large amplitude motions.

Herein, only details of the different convection schemes and analysis methods are presented. Discretization of Convection Terms Since all the convection terms in CFDShip-Iowa-V.4 are computed in the computational domain with each dimension using the same convection scheme, the convection discretizations are presented only in one dimension. The 1D convective term transporting any quantity at node is discretized as

(1)

where the convection velocity is linearized within the momentum and the k-ε/k-ω loop but are changed within the nonlinear iteration loop. There are several linear convection schemes available in the code, two of which are used here. At the interface , the corresponding fluxes can be written as:

(2)

(3)

The first method is the second order upwind (FD2) and the second is the fourth order cubic interpolation upwind-biased scheme (FD4). Strictly speaking, FD4 is only second order accurate (Ismail et al., 2008a) but has a fifth order diffusion term which means the scheme has very little numerical diffusion. The velocities in Equations (2-3) are written so that they provide directional upwinding. A similar approach is used for fluxes in other faces. The nonlinear convection schemes are discretized according to a flux-limiter method (Boris and Book 1973), appropriately blending the first order upwind ( ) with FD2 as the high order flux :

(4)

To compute the limiting coefficient , the upwind

and downwind slope ratios need to be

computed first. If the flow at is in an upwind direction, is determined via several choices of TVD limiters (Sweby 1984) available in the code: (5)

(6)

(7) Equations (5), (6) and (7) represent the Minmod (most diffusive), van Albada and Superbee (most compressive) limiters. In this study, only the Superbee limiter is used. Note that is used if the flow is in

the downwind direction. A similar approach is performed when computing . The resulting algebraic systems for the variables u , v , w , p , k , and ω are solved in a sequential form and iterated to achieve convergence within each time step. A PISO (Issa, 1985) algorithm is used to obtain a pressure equation and satisfy continuity (cf. Carrica et al. 2007 for details). The pressure Poisson equation is solved using the PETSc toolkit (Balay et al., 2002). All other systems are solved using an alternating direction implicit (ADI) method. The software SUGGAR (Noack, 2005) is used as a processing step to obtain the interpolation coefficients needed for overset grids. Analysis Methods The Q-criterion (Hunt et al. 1988) is used to identify the vortical structures. The Q-criterion is based on the second invariant of the velocity gradient tensor, ∇u. The Q-criterion cannot be used to visualize the orientation of the vortices detected. To overcome this disadvantage, the normalized helicity density is applied to color-code the Q-isosurfaces in the current study (Levy et al, 1990):

| || |v ωv ωnH ⋅=

(8)

where v and ω are the velocity and vorticity vectors at the same point. The normalized helicity density Hn represents the directional cosine between the vorticity vector and the velocity vector, 1 1nH− ≤ ≤ . The sign of Hn indicates the direction of swirl of the vortex relative to the streamwise velocity component.

The analysis method for turbulent structures follows Xing et al. (2007b) and Xing and Stern (2008a). TKE, or the total TKE, consists of resolved TKE and modeled TKE (Hedges et al. 2002). The resolved TKE ( )Rk x is evaluated using the resolved

unsteady velocity components ( ),iu tx :

( ) ( )21 ,

2R ik u t=x x

(9)

where

( ) ( ), , ( )x x xi i iu t U t U= − (10)

represents the unsteadiness caused by organized oscillations and random fluctuations and ( )iU x represents the time-averaged velocity components.

( )iU x is evaluated using:

( ) ( ) ( )0

0

1, ,x x xt T

tU U t U t dt

T+

= = ∫

(11)

where T0 is the averaging time. The resolved Reynolds

stresses i j Ru u are evaluated using:

( ) ( )0

0

1 , ,x xt T

i j i jtRu u u t u t dt

T+

= ∫

(12)

The resolved TKE budget is evaluated using the time-averaged exact TKE transport equation (Le et al. 1997):

1 1 10Re Re 2

i i j ji i iR Rj i j R

j j k k j j j j

C P T PTD

u u u p uU u uk kU u ux x x x x x x x

ε

⎛ ⎞ ′∂ ∂∂ ∂ ∂∂ ∂∂= − − − + − −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

(13) where the right hand side terms of the equation are, from the left to right, turbulent convection (C), turbulent production (P), viscous dissipation (ε), viscous diffusion (D), turbulent transport (T), and pressure transport (PT). These terms are evaluated similarly to ( )iU x (Equation (11)) and ( ),p t′ x is defined similarly as ( ),iu tx

(Equation (10)). The

resolved ε cannot be evaluated since viscous dissipation occurs at the Kolmogorov scale and requires a DNS grid, which is unaffordable for such a high Re flow. The time-averaged modeled TKE ( )Mk x is calculated by time-averaging the ( ),Mk tx calculated

by the two-equation turbulence model. The modeled Reynolds stresses

i j Mu u are evaluated by:

23

jii j T ij

j i

UUu u kx x

ν δ⎛ ⎞∂∂− = + −⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

(14)

where ijδ

is the delta Kronecker. The modeled TKE budget terms are evaluated using the averaged modeled equation:

* 10Re 2

iM M T Mj i j MM

j j j j j j

C P D T PT

Uk k kU u u kx x x x x x

ε

νβ ω

+

⎛ ⎞ ⎛ ⎞∂∂ ∂ ∂∂ ∂= − − − + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

(15) Reynolds stress budget is given by:

2

2

0

1 2Re Re

ij ij ij

ij ij ij

i j j i j kik i k j k

k k k k

C P T

i j jii j

k k k j i

D

u u U u u uUU u u u ux x x x

u u uu p pu ux x x x x

ε ∏

∂ ∂ ∂∂= − − − −∂ ∂ ∂ ∂

∂ ∂ ′ ′∂ ∂ ∂+ − − −∂ ∂ ∂ ∂ ∂

(16) Methods for computing time averaged, resolved, and modeled budget terms are similar to those used for TKE budget analysis and thus not presented herein. In the current study, T0 is chosen to be 5 flow times. The turbulent structures are insensitive to the averaging time by comparing with those computed using 3 or 4 flow times. 5 flow times correspond to 110/86, 55, and 20 times the typical periods for shear layer, Karman-like shedding, and helical instabilities, respectively. Both resolved and modeled viscous diffusion terms are negligible due to the very high Re. The modeled TKE is shown for the solution at one instantaneous time. For time-averaged turbulent statistics, DES resolves about 85% of the total TKE and Reynolds stresses. The modeled turbulent statistics show similar trends as the resolved turbulent statistics. For completeness, the resolved TKE, TKE budget, and Reynolds shear stresses budgets for KVLCC2 at drift angle 30 degrees are presented with the exception of the viscous dissipation (modeled). The total turbulent statistics are similar to the resolved and are not shown. 3D view of turbulent structures will be shown on the vortex surfaces and in eight y-z planes starting near the bow and ending at the trailing edge of the helical vortex ABV (X = -0.4, X = -0.27, X = -0.15, X =0, X = 0.2, X = 0.39, X = 0.6, X = 0.8).

To facilitate the discussion, streamlines dividing separation bubbles and outer regions are called ‘free-shear lines.’ It should be noted that the word ‘center’ of the shear layer used in the canonical 2D flows is the locus of the inflection points of the mean streamwise velocity profile, which corresponds to the high-speed side of the ‘free shear line’ (Xing et al., 2007b). The improved correction factor formula for grid/time-step uncertainty estimates (Xing and Stern, 2008b) is:

[ ]

( )

1

1

1

1

*

3 2 *

3 2 *

*

2(1 ) 1 0 0.875

25.6(1 ) 12.8(1 ) 1.1 0.875 1.0

135.8( 1) 49.4( 1) 1.1 1.0 1.125

2 1 1 1.125 22

k

k

k

k

k RE k

k k RE k

kk k RE k

kk RE k

k

C C

C C CU C C C

C C CC

δ

δ

δ

δ

⎧ − + < ≤⎪⎪ ⎡ ⎤− − + − + < ≤⎣ ⎦⎪⎪= ⎨ ⎡ ⎤− − + − + < <⎣ ⎦⎪⎪⎧ ⎫⎪ − + ≤ <⎡ ⎤⎨ ⎬⎣ ⎦⎪ −⎩ ⎭⎩

(17) As shown in Figure 1, the improved uncertainty estimates were shown to provide more reasonable intervals of uncertainty when the estimated order of accuracy is larger than the theoretical order of accuracy, i.e.,

thk kp p> .

Figure 1: Factors of safety for correction factor and GCI verification methods. Validation of a single turbulence model will be conducted using the comparison error E and validation uncertainty UV. When a validation effort involves multiple convection schemes/turbulence models, additionally the comparison will include the quantity |E|+UV. COMPUTATIONAL SET-UP & TEST MATRIX To mimic the double tanker experiments of (Lee et al., 2003), the model KVLCC2 is computed with the top

boundary specified as symmetry, simulating a flat free surface, and the Reynolds number is set to

. The KVLCC2 geometry is shown in Figure 2 and the computational domain, grids and boundary conditions are shown in Figure 3. The computational domain in this study extends (-2L, 2L) in the longitudinal (streamwise) direction X, (-1.5L, 1.5L) in the transverse direction Y and (-1.2L, -0.1L) in vertical direction Z.

Figure 2: KVLCC2 geometry.

Figure 3: Grids, solution domain and boundary conditions.

Body-fitted double-O type grids are used to discretize the points near the ship hull and Cartesian grids are used as background grids. To accurately resolve the boundary layer, the first grid spacing normal at the ship hull is at least 1.2y+ < . The ship hull grid was generated using a hyperbolic solver. The grid dimensions and the measurements of the grids in terms of the grid distortion metrics (AR, ER, Q) are presented in Table 1. The grid ER and Q are the two most sensitive grid metrics but the factor grid AR is less critical (refer to Table A.2). Most of the grid points (75 % or more) comply with quality requirements of Q and ER (grid stretching) proposed by (Ismail et al., 2008a). However, the grids near the wall within the boundary layer are highly stretched with high AR. Also, some parts of the double-O grids are highly skewed particularly around the poles. The design of the grids 1,2,3 enables a set of verification study with grid refinement ratio of . All results shown in this paper are based on Grid 2 (medium) except the V&V.

(a) (b)

Figure 4: Coordinate systems for (a) hydrodynamic forces and moments, and (b) velocity fields. The coordinate systems for the hydrodynamic forces and moments are shown in Figure 4a while for the velocity fields are shown in Figure 4b.

Table 2: Test Matrix.

The coupling of the convection schemes and the turbulence models is in short notation. For example the isotropic turbulence model coupled with the second order upwind is referred as FD2-BKW. Coupling between the anisotropic turbulence model and the second order upwind with Superbee limiter is TVD2S-ARS, and so forth. The complete simulation strategy for KVLCC2 is included in Table 2. RESULTS AND DISCUSSION The numerical results in this study are compared with the EFD results of (Lee et al., 2003) and (Kume et al., 2006) for 0° and 12° drift angles. Unfortunately, there are no EFD results available for 30° drift. Xing et al. (2007a) and the Tokyo 2005 workshop (Hino, 2005) showed that overall the ARS turbulence model is better than the BKW model. However, both over-predict the integral forces and moments for 12º drift with ARS predictions higher than BKW. Herein, the focus is to evaluate the various convection schemes with the ARS model. For 30º drift, FD4 with ARS-DES is used for analysis of turbulent structures.

Drift

°

BKW RS

Scheme Grid Study

Grids Used

Scheme Grid Study

Grids Used

0

FD2 Yes 1,2,3 FD2 Yes 1,2,3 TVD2S No 2 TVD2S Yes 1,2,3 FD4h No 2 FD4h Yes 1,2,3

12

FD2 Yes 2 FD2 Yes 1,2,3 TVD2S No 2 TVD2S Yes 1,2,3 FD4h No 2 FD4h Yes 1,2,3

30

DES

FD2 TVD2S No 2 TVD2S FD4h No 2 FD4h FD4 No 2

For a particular choice of convection scheme for the momentum equations, varying the convection scheme for the turbulent transport equations produces results that are almost identical but the opposite is not true. This is because the choice of convection scheme highly influences the results. Only results of identical convection schemes for momentum and turbulence transport are included. The discussion begins with the V&V for the integral forces and moments followed by the study of the evolution of the local quantities before presenting the results at the nominal wake. Note that the uncertainty for all of the integral forces and moments for the experimental data is UD = 0.7% (Hino, 2005). KVLCC2 Drift Angle 0 For this case, the global quantities studied include the integral force CTX and the surface pressure distribution. The local quantities studied include the axial velocity U, turbulent kinetic energy (TKE) and its components (i.e uu ) and the turbulent shear stresses uv and uw . V&V of integral force The average total force coefficient CTX of different convection schemes is within 1.8% of Xing et al. (2007a) and 5.0% to the average CTX obtained in Tokyo 2005 CFD workshop (Hino, 2005). The difference of results between this study and (Xing et al., 2007a) is due to a different convection scheme used by the latter. Xing et al. (2007a) used a pure FD4 scheme, which is very sensitive to numerical instability especially for fine grids. To make it more robust for computations on several grids, a hybrid fourth and second order interpolation scheme (FD4h) is used. The computations for CTX are monotonically converged (Table 3 and Figure 5). CFX (friction coefficient) and CPX (pressure coefficient) also monotonically converge (Ismail et al., 2008b). The computations are reasonably near to the asymptotic range (average 1-CG = -0.55). As the grids are refined, the average relative solution change (ε) is about 1.2%. The iterative errors are at least 1-2 orders of magnitude smaller than the relative changes of the force coefficients hence neglected in validating the numerical results. Since this is a steady-state computation, UT is also neglected hence the numerical simulation uncertainty is approximated as USN ≈ UG . FD4h-ARS has the highest convergence ratio (RG=0.43) and closest to the asymptotic range (1-CG=-0.33) and the theoretical order of accuracy (PG=2.4). This is followed by the FD2-BKW, TVD2S-ARS and FD2-ARS. The FD2-BKW has the smallest grid uncertainty (UG=0.97%), followed by FD4h-ARS (UG=2.2%), which implies that these two methods have the smallest numerical simulation uncertainties.

CTX predicted by FD4h-ARS has the smallest error (E=-1.5%) followed by FD2-ARS (E=-1.9%), TVD2S-ARS (E=-4.3%) and FD2-BKW (E=-6.15%) where negative values indicate under-prediction. These errors are based on the difference of CTX using the finest grid (grid 1) and EFD (Kume et al., 2006). In terms of UV FD2-BKW is the smallest (UV=1.21%) followed by FD4h-ARS (UV=2.28%), TVD2S-ARS (UV=5.93%) and FD2-ARS (UV=6.14%). The overall comparison between the three convection schemes and the two turbulence models results based on |E|+UV yields FD4h-ARS producing the smallest interval of 3.76% followed by FD2-BKW (7.36%), FD2-ARS (8.04%) and TVD2S-ARS (10.2%). However, the computation with FD2-BKW is not validated since |E|>UV. This indicates a deficiency of the BKW turbulence model predicting anisotropic boundary layer. In fact, FD2-BKW predicts CTX that converges to a different value compared with other methods (Figure 5a). Only computations with ARS turbulence model (FD2-ARS, TVD2S-ARS and FD4h-ARS) are validated. Table 3: Verification and validation study for CTX at 0° using grids 1,2,3 ( ).

(a) (b) Figure 5: Verification for resistance at 0˚ drift angle: (a) Total force coefficient CTX and (b) Relative change of CTX, εN = |SN-SN-1|/S1 × 100 and iterative errors. Surface pressure distribution Figure 6 shows the pressure distribution on the hull surface predicted by the FD2-ARS, TVD2-ARS schemes and EFD (Kume et al., 2006). The results of FD4h-RS are also very similar (Ismail et al., 2008). There are very little differences between the results produced by the three convection schemes with maximum between them. Overall, the CFD pressure contours are similar to EFD with notable differences at the stern.

Method RG PG 1-CG UG

% E %

UV

% |E|+UV

%FD2-BKW

0.40 2.64 -0.50 0.97 -6.15 1.21 7.36

FD2- ARS

0.28 2.83 -0.67 6.09 -1.90 6.14 8.04

TVD2S-ARS

0.37 2.89 -0.72 5.89 -4.27 5.93 10.2

FD4h-ARS

0.43 2.44 -0.33 2.20 -1.48 2.28 3.76

(a)

(b)

(c)

Figure 6: Surface pressure distribution predicted by (a) EFD (Kume et al., 2006), (b) FD2S-ARS and (c) TVD2S-ARS. Local flow quantities The cross-sections of the local flow quantities from the midships to the wake are of interest, particularly at the nominal wake plane ( ). From the midships to , the qualitative features of the velocities and turbulent quantities of the numerical solutions are very similar with each other and with EFD (Lee at al., 2003) hence not included. Differences between CFD/EFD results start to show beginning at and can be seen in the evolution of the velocities downstream of this point (Figure 7). The outer most contours (U=0.95) of the numerical solutions have a wider wake compared to EFD (Figures 7a-7b) due to excess numerical diffusion, although there are only small contour differences between three convection schemes. The differences between CFD/EFD results for U are more significant at the nominal wake plane (Figures 7c and 8) and downstream to the wake (Figures 7d-7e). At the nominal wake plane, the ‘hook’ is best captured using the TVD2S-ARS followed by FD4h-ARS and FD2-ARS. The results of FD4h-ARS and FD2-ARS are very similar at the nominal wake, so only the latter is included with TVD2S-ARS in Figure 8. However, the magnitude of U at the ‘hook’ is under-predicted by the all three schemes even with the ARS turbulence model compared to EFD. TVD2S-ARS produces the smallest magnitude differences of U at the hook (max E=-16.2%) followed by FD4h-ARS (max E=-21.4%) and FD2-ARS (max E=-25.1%). The ‘hook’ predicted by FD2-BKW is severely dissipated and in fact, using a less diffusive convection scheme like TVD2S with BKW does not improve the prediction of the ‘hook’ (Ismail et al., 2008). This may

imply that accurate ‘hook’ prediction at the nominal wake is mainly due to the anisotropic turbulence model and not the convection scheme. Beyond the nominal wake plane, the flow separates and massive vortical and turbulent structures propagate downstream to the wake. This is a matter of accurate prediction of the turbulent structures which requires LES/DES (Bensow et al., 2006) with much finer grids and also accurate transport prediction of the bilge vortex which requires a minimally dissipative or a very high order convection scheme. This explains why the TVD2S-ARS scheme produced results that are closest to EFD at the centerline regions (Figures 7d-7e) but not accurate enough compared to EFD since only RANS coupled with a moderately coarse grids (1.6 M points) are used.

(a)

(b)

(c)

(d)

(e)

Figure 7: Cross sections of velocities U (contours) and V and W (vectors) at 0˚ drift angle compared with EFD (Lee et al., 2003) at (a) X=0.4, (b) X=0.45, (c) X=0.48, (d) X=0.52 and (e) X=0.6.

(a)

(b)

Figure 8: U velocity at nominal wake (X=0.48) computed with (a) FD2-ARS and (b) TVD2S-ARS compared with EFD (Lee et al., 2003). The evolution of the flow patterns for the turbulent quantities is similar to the evolution of the axial velocity U, thus only the results at the nominal wake plane are included. In terms of TKE, uu and vv , the contour lines predicted by TVD2S-ARS are qualitatively closer to EFD (Figures 9-11). However, the TKE is over-predicted by TVD2S-ARS in the centerline region of the bilge vortex (max E=12.0%) as depicted in Figure 9b, due to perhaps the anti-diffusion mechanism embedded in the scheme. TKE is under-predicted in the centerline region by both FD2-ARS (max E=-5.0%) (Figure 9a) and FD4h-ARS (max E=-4.7%) compared to EFD. uu is also over-predicted by TVD2S-ARS (max E= 8.1%) as shown in Figure 10b but again under-predicted by FD2-ARS (max E=-3.8%) (Figure 10a). vv is slightly over-predicted by TVD2S-ARS (max E= 3.1%) but under-predicted by (max E= -4.1%) (Figures 11a-11b). In terms of the turbulent shear stresses (Figures 12-13), uv is very much under-predicted by FD2-ARS (max. E=-40.2%) unlike TVD2S-ARS (max. E=-5.3%). For uw , the maximum E=-43.4% is obtained with FD2-ARS, which is a very large error compared to TVD2S (max. E=-12.2%). Note that the turbulent normal stresses in this study are at least twice larger in magnitude compared to the shear stresses, reconfirming turbulence anisotropy.

(a)

(b)

Figure 9: TKE at nominal wake (X=0.48) computed with (a) FD2-ARS and (b) TVD2S-ARS compared with EFD (Lee et al., 2003).

(a)

(b)

Figure 10: uu at nominal wake (X=0.48) computed with (a) FD2-ARS and (b) TVD2S-ARS compared with EFD (Lee et al., 2003).

(a)

(b)

Figure 11: vv at nominal wake (X=0.48) computed with (a) FD2-ARS and (b) TVD2S-ARS compared with EFD (Lee et al., 2003).

(a)

(b)

Figure 12: uv at nominal wake (X=0.48) computed with (a) FD2-ARS and (b) TVD2S-ARS compared with EFD (Lee et al., 2003).

(a)

(b)

Figure 13: uw at nominal wake (X=0.48) computed with (a) FD2-ARS and (b) TVD2S-ARS compared with EFD (Lee et al., 2003). KVLCC2 Drift Angle 12 V&V of integral forces and moments The integral forces and moments predicted here are within 2% of the results produced by (Xing et al., 2007a). The differences are due to the same reason as explained in the previous case. Compared to Tokyo CFD Workshop 2005, the average forces and moments obtained in this study differ by 5.2% (CTX), 8.6% (CTY), 1.2% (CN) (Tables 4-6). Only TVD2S-ARS and FD4h-ARS have all forces and moments that monotonically converge (Figures 14-16). FD2-BKW and FD2-ARS does not produce converge solutions except for the total friction (CFX) (Ismail et al., 2008b). Between TVD2S-ARS and FD4h-ARS, the latter has a larger convergence ratio for forces and moments. However, TVD2S-ARS computations are closer to the theoretical order of accuracy (also to the asymptotic range) and smaller grid uncertainties compared to FD4h-ARS for all of the forces and moments. Since TVD2S-ARS has the smallest grid uncertainties, it also has the smallest numerical simulation uncertainties for all of the forces and moments since this is also a steady state problem (UT ≈ 0) and the iterative errors are at least two orders of magnitude smaller than the relative changes of the forces and moments. TVD2-ARS also produces smaller errors for all of the predicted forces (CTX:E=12.7%, CTY:E=2.2%) and moments (CN:E=1.2%) compared to FD4h-ARS (CTX:E=13.7%, CTY:E=6.9%, CN:E=4.4%). In terms of |E|+UV, TVD2S-ARS also has smaller values when

predicting the forces and moments compared to FD4h-ARS. For CTX, TVD2S-ARS yields |E|+UV=13.84% compared to FD4h-ARS (17.79%). For CTY TVD2S-ARS yields |E|+UV=2.00% compared to FD4h-ARS (8.59%). For CN, TVD2S-ARS yields |E|+UV=2.95% compared to FD4h-ARS (7.60%). Unfortunately for this drift angle, none of the results for forces and moments are validated. Table 4: Verification and validation study for CTX at 12° using grids 1,2,3 ( ).

Table 5: Verification and validation study for CTY at 12° using grids 1,2,3 ( ).

Table 6: Verification and validation study for CN at 12° using grids 1,2,3 ( ).

(a) (b)

Figure 14: Verification for x-resistance at 12˚ drift angle: (a) Total force coefficient CTX and (b) Relative change of CTX, εN = |SN-SN-1|/S1 × 100 and iterative errors.

(a) (b)

Figure 15: Verification for y-resistance at 12˚ drift angle: (a) Total force coefficient CTY and (b) Relative change of CTY, εN = |SN-SN-1|/S1 × 100 and iterative errors.

(a) (b)

Figure 16: Verification for z-moments at 12˚ drift angle: (a) Total force coefficient CN and (b) Relative change of CN, εN = |SN-SN-1|/S1 × 100 and iterative errors. Vortical structures and flow at nominal wake The transport of the vortices for FD2-ARS and TVD2S-ARS schemes are shown in Figures 17a-17b. Vortex results of FD4h-ARS are almost identical to results produced by FD2-ARS hence not included. The TVD2S-ARS is the least dissipative, preserving the fore-body bilge vortex (FBV), fore-body side vortex (FSV) and aft-body bilge vortex (ABV) at much longer distances compared to FD2 (and also FD4h). Figure 18 depicts the nominal wake plane FSV is best captured by TVD2S-ARS compared to EFD. As a result, the nominal wake U velocity is also best captured by TVD2S-ARS (Figure 19c). The vortex system and

Method RG PG 1-CG UG

% E% UV

% |E|+ UV

%FD2-BKW

— — — — — — —

FD2- ARS

— — — — — — —

TVD2S-ARS

0.46 2.22 -0.16 0.92 12.69 1.15 13.84

FD4h-ARS

0.67 1.41 0.50 4.02 13.71 4.08 17.79

Method RG PG 1-CG UG

% E% UV

% |E|+UV

%FD2-BKW

— — — — — — —

FD2- ARS

— — — — — — —

TVD2S-ARS

0.59 1.53 0.30 0.32 1.23 0.77 2.00

FD4h-ARS

0.74 0.87 0.64 4.17 4.37 4.22 8.59

Method RG PG 1-CG UG

% E% UV

% |E|+UV

%FD2-BKW

— — — — — — —

FD2- ARS

— — — — — — —

TVD2S-ARS

0.47 2.18 -0.13 0.16 2.23 0.72 2.95

FD4h-ARS

0.61 1.44 0.36 0.13 6.89 0.71 7.60

nominal wake velocity and vorticity of TVD2S-ARS are very similar to TVD2S-BKW (not included), implying that the TVD2S scheme improves the flow predictions for 12˚drift.

(a)

(b)

Figure 17: Vortical structures (iso-surface of Q=100 colored by helicity) at 12˚ drift angle computed with (a) FD2-ARS and (b) TVD2S-ARS.

(a)

(b)

(c) Figure 18: Nominal wake at 12˚ drift angle: (a) EFD (Kume et al., 2006), (b) FD2-ARS, and (c) TVD2S-ARS.

(a)

(b)

(c)

Figure 19: Nominal wake U at 12˚ drift angle: (a) EFD (Kume et al., 2006), (b) FD2-ARS, and (c) TVD2S-ARS. KVLCC2 Drift Angle 30 Vortical structures There are very little differences in predicting the fore-bilge-vortices (FBV) and stern-vortices between the FD4h-ARS and TVD2S-ARS (Figures 20a-20b). The main differences lie in predicting the aft-body-side vortices (ASV), FSV and ABV with the biggest differences on the last two vortices. There only perhaps about 5-10% magnitude differences for these vortices between the two schemes but the FSV and ABV of the TVD2S-ARS are preserved at much longer distances. This is observed before in the 12º drift, but for this case it is more obvious since stronger vortices are shed. This reaffirms the importance of not only DES or anisotropic turbulence model but also a minimally dissipative convection scheme to accurately predict propagation of massive vortical structures occurring in ships with large drift angles.

(a)

(b)

Figure 20: Vortical structures (iso-surface of Q=100 colored by helicity) at 30˚ drift angle computed with (a) FD2-ARS and (b) TVD2S-ARS. Turbulent structures near the bow Turbulent structures are first studied near the shear-layer and Karman-shedding dominant regions by examining a slice at z=-0.0039 (Figures not shown). The TKE peaks near the separation point at the bow and on the high-speed side of the free-shear line, which are similar to those of a separated turbulent boundary layer and the recirculation of a backward-facing step flow, respectively. P and ε are the main producing and consuming terms, respectively. P peaks where the TKE peaks. Maximum P occurs on the high-speed side of the free-shear line. The main contribution to the production term is P11, which is consistent with that in the backward-facing step flow. The largest magnitude of convection is near the separation point and on the free-shear line. The main contribution to C is from the longitudinal component, which indicates C transports energy from upstream to downstream. Convection shows negative and positive values on the high-speed and low-speed sides of the free-shear line, respectively. This is similar to the ‘TKE growing’ region observed in the free-surface wave-induced separations (Xing et al, 2007b). T is negative where P is large and positive where P is small, which suggests that T transports the energy from the largest TKE regions (high-speed side of the free-shear line) toward the outer region and the inside of the separation bubble, which more evenly distributes TKE. The same observation is valid in the recirculation region for a backward-facing step flow. The main contribution to the turbulent transport is T11. The ratio of T peak to P peak is (T/P)=0.83. PT is negative on free-shear line right after the separation

point and along the high-speed side of the free-shear line further downstream. It is at the same order of magnitude as the convection term. Large magnitudes of viscous dissipation are at the same locations as P peaks but with opposite sign and a more uniform distribution. The ratio of ε peak to P peak on the high-speed side of the free-shear line is (ε/P)=0.17. Overall the turbulent structures near the bow are similar to those of the free-surface wave-induced separations around NACA24 but without free-surface effect. 3D TKE distribution and budget A 3D view of the TKE and its budget is shown in Figure 21. Overall the TKE peaks near the separation point at the bow and within a certain distance to the vortex cores of all vortices. TKE reaches local maximum right after the vortex breakdown points, which are X=-0.15, X=0.36, and X=0.38 for FSV, ABV, and SV, respectively. TKE is intensified along the vortex cores further downstream likely due to the enhanced unsteady oscillation caused by the helical instability. Similar trend was observed for vortex breakdown for the leading-edge vortex over delta-wing at high-angle of attack (Breitsamter, 1997). P and ε are the main producing and consuming terms, respectively. Production peaks where the TKE peaks. Maximum production occurs near the bow with main contribution to the production term is P11. The largest magnitude of convection is near the separation point at the bow and within a certain distance around the vortex cores. As in the case of the bow turbulent structures, the main contribution to the convection term is from the longitudinal component. The ratio of C peak to P peak is: (C/P)=0.20. The largest magnitude of T is at the separation region near the bow and within a certain distance to the SV core. T is almost negligible for other regions. As in the case for turbulent structures near the bow, T more evenly distributes TKE. The ratio of T peak to P peak is (T/P)=0.20. The PT is negative at the vortex cores except FBV. It is at the same order of magnitude as the convection. Large magnitudes of viscous dissipation are at the same locations as P peaks but with opposite sign and a more uniform distribution. The ratio of ε peak to P peak is |ε/P|=5.

(a) (b)

(c) (d)

(e) (f) Figure 21: Resolved TKE budget: (a) TKE, (b) production (resolved), (c) convection (resolved), (d) transport (resolved), (e) pressure transport (resolved), (f) viscous dissipation (modeled). 3D Reynolds normal stress distribution and budgets Overall uu , vv , and ww show similar magnitude and distribution as TKE. Production peaks where the normal stresses peak. The main contributions to the production terms are uv U Y∂ ∂ , uv V X∂ ∂ and vw W Y∂ ∂ for uu , vv , and ww , respectively. C of uu and ww show opposite signs to C of vv . The main contributions to the C term are from U uu X∂ ∂ , V vv Y∂ ∂ , and V ww Y∂ ∂ for uu , vv , and ww , respectively . The ratio of C peak to P peak is the same for all three normal stresses: (C/P)=0.20. The largest magnitude and transport behavior of T are as in the case for TKE. Transport behavior of T is similar to that of TKE. The main contributions to the turbulent transport are uuu X∂ ∂ , vvv Y∂ ∂ , and vww Y∂ ∂ for uu , vv , and ww , respectively. It is negative where P is large and positive where P is small, which suggests that T more evenly distributes the normal stresses. The ratios of T peak to P peak are the same for all three normal stresses: (T/P)=0.20. PT is negative at the vortex cores except positive at the cores of ABV and FSV for ww . Large magnitudes of viscous dissipation are at the same locations as P peaks but with opposite sign and a more uniform distribution. The ratios of ε peak to P peak are around 1 for the three normal stresses. 3D Reynolds shear stress distribution and budgets 3D views of the uv , uw and vw with budgets are shown in

Figures 22, 23, and 24, respectively. Overall uw and vw are one order of magnitude smaller than uv , that has the same order of magnitude as the three normal stresses. uv peaks near the separation point at the bow and within a certain distance to the vortex core of all vortices inside the volume. uw and vw also peak near the separation point at the bow and within a certain distance to the vortex core of SV only inside the volume. The three shear stresses reach local maximum right after the vortex breakdown points and are intensified along the vortex core further downstream likely due to the enhanced unsteady oscillation caused by the helical instability. P and ε are the main producing and consuming terms, respectively. Production peaks where the TKE peaks. Maximum production occurs near the bow. The main contribution to the production terms are uu V X uv U X∂ ∂ + ∂ ∂ , uv W Y vw U Y∂ ∂ + ∂ ∂ , and uv W X uw V X∂ ∂ + ∂ ∂ for uv , uw and vw , respectively. The largest magnitudes of convection for the three shear stresses are near the separation point at the bow and within a certain distance around the vortex cores. The main contributions to C terms are from V uv Y∂ ∂ , V uw Y∂ ∂ , and V vw Y∂ ∂ for uv , uw and vw , respectively. The ratio of C peak to P peak is: (C/P)=0.20. The largest magnitude and transport behavior of T are as in the case for TKE. The main contributions to the turbulent transport are uuv Y∂ ∂ , uvw Y∂ ∂ , and vww Z∂ ∂ for uv , uw and vw , respectively. The ratios of T peaks to P peaks are (T/P)=0.20 for the three shear stresses. For uv , PT is negative at all the vortex cores. For uw and vw , PT is negative and positive for the windward and leeward side of ABV, respectively, which is opposite to SV and FBV. For all three shear stresses, PT are at the same order of magnitude as the convection but with opposite sign. Large magnitudes of viscous dissipation are at the same locations as P peaks but with opposite sign and a more uniform distribution. The ratios of ε peak to P peak are around 1 for the three Reynolds shear stresses.

(a) (b)

(c) (d)

(e) (f) Figure 22: Resolved uv budget: (a) uv , (b) production (resolved), (c) convection (resolved), (d) transport (resolved), (e) pressure transport (resolved), (f) viscous dissipation (modeled).

(a) (b)

(c) (d)

(e) (f) Figure 23: Resolved uw budget: (a) uw , (b) production (resolved), (c) convection (resolved), (d) transport (resolved), (e) pressure transport (resolved), (f) viscous dissipation (modeled).

(a) (b)

(c) (d)

(e) (f) Figure 24: Resolved vw budget: (a) vw , (b) production (resolved), (c) convection (resolved), (d) transport (resolved), (e) pressure transport (resolved), (f) viscous dissipation (modeled). KVLCC2 Drift Angle 60 As shown in Xing et al. (2007a), the ARS-DES model is used to study KVLCC2 at drift angle 60 degrees. As the drift angle gets large enough, the flow field changes to a deadwater-type flow. Deadwater flow has a large flow recirculation (deadwater) zone, i.e., a broad wake containing a broad range of scales of vorticies. Compared with solutions for drift angle 12 and 30 degrees, FFT of the total drag coefficient shows a much broader range of frequency contents, which is consistent with a much broader range of scales of vortices. Most of the energy is contained by the low-frequency modes, especially the dominant one at f=1.7. The complexity of the vortex system makes the isolation of the frequencies and instability analysis to be extremely difficult. EFFECT OF DRIFT ANGLES ON FORCES AND MOMENTS FOR KVLCC2 The forces and moments that agree best with the EFD data are presented in Figure 25. Overall it shows that the increase in force coefficients and moments with drift angle is quadratic, consistent with the findings by

Longo and Stern (2002) for the Series 60 with the maximum drift angle 10 degrees. For drift angle 0 degree, CX predicted with pure FD4 with ARS by Xing et al. (2007a) compared to TVD2S-ARS are very similar to each other and to EFD as shown in Figure 25b. For drift angle 12 degrees, TVD2S-ARS predicts CX, CY much closer to EFD compared to results of Xing et al (2007a) but results of CN are very similar. For large drift angle 30 degrees, no significant differences are observed for forces and moments between computations of Xing et al. (2007a) and TVD2S-ARS. In this study, the KVLCC2 is not computed at 60˚ but was computed by Xing et al. (2007a).

(a)

(b)

Figure 25: Force coefficients as a function of drift angles: (a) all drift angles, (b) drift angles up to 12 degrees. CONCLUSIONS AND FUTURE WORK Two important factors to accurately predict the KVLCC2 flows are analyzed in this paper. First is accurate prediction of the anisotropic boundary layer with/without flow separation which can be achieved through an algebraic Reynolds stress (ARS) turbulence model, and with DES when the flow separates. The second factor is accurate transport of the flow quantities (i.e. velocity, vorticity, TKE, etc.) which can be achieved by using a convection scheme with minimal numerical dissipation. A very high order convection scheme might also work but most likely at the expense of losing computational robustness for practical ship hydrodynamics.

For straight ahead (0˚ drift) condition, using an ARS turbulence model is crucial in order to accurately predict the integral forces and also local flow quantities, particularly for capturing the ‘hook’ shape. Using less dissipative convection schemes such as TVD2S and FD4h (compared to FD2) produce limited improvements on the forces. However, the least dissipative TVD2S scheme produces some improvements in the flow fields particularly at the nominal wake plane and better captures the ‘hook’ shape. For 12˚ drift, there are almost no improvements for the local flow quantities when anisotropic turbulence model is used. In fact, the anisotropic model over-predicts the integral forces and moments (relative to the isotropic model) compared to EFD, consistent with the conclusions in Tokyo 2005 CFD Workshop. For this 12˚ drift, predictions of the transported flow quantities and integral forces and moments are improved by using the least dissipative scheme TVD2S. For 30˚ drift, the vortical structures are preserved at much longer distances when computed with TVD2S-ARS compared to FD4h-ARS. Turbulent structures are analyzed using TKE and Reynolds stress budgets. The TKE and Reynolds stresses peak near the separation point at the bow and within a certain distance to the vortex core of all vortices. Near the bow, turbulent structures are similar to those for a separated turbulent boundary layer and the recirculation region of a backward-facing step flows. Overall Reynolds normal stresses have similar distribution and magnitude as TKE. uw and vw shows one order of magnitude smaller than uv that has the same order of magnitude as the normal stresses. TKE and Reynolds stresses reach local maximum right after the vortex breakdown points for helical vortex tubes and are intensified along the vortex core further downstream likely due to the enhanced unsteady oscillation caused by the helical instability. Turbulent production (P) and viscous dissipation are the main generator and consumer terms, respectively. Production peaks where the TKE peaks. For TKE and the Reynolds normal stress vv , the main contribution to the convection (C) term is from the longitudinal component. For other Reynolds stresses, C is dominant by the spanwise component. The turbulent transport (T) is negative where P is large and positive where P is small, which suggests that T distributes TKE and Reynolds stresses more evenly. Large magnitude of T is only observed near the bow and SV vortex and almost negligible for other regions. The pressure transport is at the same order of magnitude as the convection but with opposite sign, compared to its small magnitudes for 2D canonical flows. Large magnitudes of viscous dissipation are at the same

locations as P peaks but with opposite sign and a more uniform distribution. Future work includes the evaluation of the turbulent structures using the TVD2S-ARS scheme at 30 degrees and the evaluation of KVLCC2 with free surface and with propellers and appendages. Triple decomposition is also needed to isolate the organized oscillations and the random fluctuations. ACKNOWLEDGEMENTS

This research was sponsored by Office of Naval Research grant N00014-01-1-0073 and N00014-06-1-0420 under the administration of Dr. Patrick Purtell. Computations were performed on IBM Power 4 machines at DoD NAVO MSRC. REFERENCES Balay S., Buschelman K., Gropp W., Kaushik D., Knepley M. et. al. “PETSc User Manual”. ANL-95/11 Rev. 21.5, 2002, Argonne National Laboratory.

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Table 1: Grids used for verification study. AR, ER and Q together with % of grid points complying with the guidelines of best practices of using upwind-type finite difference schemes (Ismail et. al, 2008a).

Grid 1 2 3 Ship 287×174×69=3,445,722 203×122×49=1,213,534 144×88×35=443,520

Background 152×93×82=1,159,152 107×66×58=409,596 76×47×41=146,452 Total 4,604,874 1,623,130 589,972

y+ 0.56 0.80 1.13

ER Range 0.30< ER < 1.0 0.32 < ER < 1.0 0.38 < ER < 1.0

Average 0.95 0.93 0.91 %Comply 84 82 75

Q

Range 0.0 < Q < 0.88 0.0 < Q < 0.86 0.0 < Q < 0.85 Average 0.19 0.17 0.18

%Comply 76 75 76

AR Range 1.0 < AR < 3329 1.0 < AR < 3129 1.0 < AR < 3019

Average 115.21 111.33 111.21 %Comply N/A N/A N/A

APPENDIX A Table A.1: The magnitude of errors associated with the three upwind-type finite difference convection schemes (Ismail et al., 2008a) evaluated on uniform Cartesian grids . For the errors in numerical diffusion and dispersion, negative errors denote under-prediction or lagging phase errors while positive errors indicate over-prediction or leading phase errors. Clearly, FD4h and TVD2S are better than FD2.

Metric

FD2-baseline TVD2S FD4 Smooth Problem

Discontinuous Problem

Smooth Problem

Discontinuous Problem

Smooth Problem

Discontinuous Problem

Diffusion Error %

-11.53 -40.82 -2.30 -4.32 -1.72 14.52

Dispersion Error %

0.54 3.30 0.05 -0.39 -0.15 -1.57

% 0.72 6.10 0.39 1.41 0.19 3.23 Order of Accuracy

1.68 0.79 1.72 1.04 1.80 0.81

Table A.2: The table summary presenting effects of grid distortion metrics (AR, ER, Q) on the numerical diffusion, dispersion, L1 errors and the order of accuracy of upwind-type finite difference convection schemes. The impact is measured in terms of minimal (O(0.1%)), moderate (O(1%)) and severe (O(10%)). The restrictions for constructing the grids in terms of the grid distortion metrics, particularly on Q and ER are presented (Ismail et al., 2008a).

Metric AR ER Q Diffusion minimal to moderate moderate severe

Dispersion minimal minimal to moderate minimal to moderate minimal moderate moderate

Order of Accuracy

minimal to moderate moderate to severe severe

Restrictions None 0.9 < ER < 1.0 0.0 < Q < 0.2

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