BKW- RS-DES of Unsteady Vortical Flow for KVLCC2 at Large Drift Angles

Transcription

1 9 th International Conference on Numerical Ship Hydrodynamics Ann Arbor, Michigan, 5-8 August 7 BKW- RS-DES of Unsteady Vortical Flow for KVLCC at Large Drift Angles Tao Xing, Jun Shao, Frederick Stern IIHR-Hydroscience and Engineering The University of Iowa Iowa City, IA ABSTRACT This study identifies the role of isotropic blended k-ε/kω (BKW) versus anisotropic Reynolds stress (RS) RANS models for steady ship flows and BKW-DES versus RS-DES for unsteady ship flows. Capability of RS model to predict anisotropy is shown for a solid/free-surface juncture and demonstrated for ship flow (DTMB 5415) at Fr=.8. No significant anisotropy effect is observed for DTMB 5415 due to the weaker anisotropy and insufficient grid resolution. KVLCC at drift angle, 1,, and 6 degrees are investigated neglecting the effect of free surface. Results of using BKW and RS RANS models for and 1 degrees show that RS model significantly improves the predictions of the resistance coefficients, axial velocity, and turbulent kinetic energy distributions at the propeller plane. For drift angle degrees, BKW and RS RANS models show steady solutions whereas BKW-DES and RS-DES models show unsteady solutions. In the latter case, limited differences on forces, moments and instabilities are observed. The previous analysis for vortical structures and instabilities for NACA4 and tip vortex instability for delta wings is extended to study flows at drift angle degrees using RS-DES, including quantitative verification. Two shear layer instability modes, a Karman-like vortex shedding, and three helical mode instabilities are identified. Compared to previous experimental and computational results, the Strouhal number (St) for Karman-like instability is in the same range whereas St for shear layer instability is smaller. Similarities and differences between the helical mode instabilities of the current study and those of delta wing flows are also discussed. RS-DES model is also applied to study KVLCC at drift angle 6 degrees. Further evaluation of relative merits for using BKW-DES and RS-DES for analysis of turbulent structures will be conducted in the future. INTRODUCTION Ship flows are challenging to computational fluid dynamics (CFD) due to unique physics and application conditions, ranging from resistance and propulsion to general 6 degree of freedom ship motions and maneuvering. Of interest herein is turbulence modeling that is of central importance for ship flow simulations. 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 (large drift angle ship flows), which require turbulence models accurately resolve the instabilities of the organized oscillations and turbulent structures. Wave breaking and bubbly air water mixture layers introduce additional challenges on turbulence modeling such as bubble entrainment, surface tension, and large density ratios. The importance of incorporating turbulence models to ship hydrodynamics has been shown as per CFD workshops in the past 7 years. There have been five international workshops on the numerical prediction of ship viscous flow since 198: the Gothenburg 198 workshop, the SSPA-CTH-IIHR 199 workshop, the Tokyo 1994 workshop, the Gothenburg workshop, and the Tokyo 5 workshop. According to Larsson et al (), 16 out of 17 methods were based on the boundary layer approximation with only one RANS method in the first 198 workshop. Since the boundary layer-based methods failed completely for the prediction of the flow near the stern, which is important for propeller design, most people turned to RANS method in the 199 workshop. Although the RANS method was able to predict the stern flow, it was not possible to predict the detailed shape of the velocity contour, i.e. the

2 hooklike shape pattern created by the bilge vortices. It was shown later (Deng et al. 199) that inadequate turbulence modeling, i.e. the overestimate of eddy viscosity in the bilge vortex, was responsible for the poor prediction of hooklike shape of velocity contour. This conclusion was confirmed again by Sotiropoulos and Patel (1994) who showed very good predictions of the hooklike shape velocity contour for the HSVA tanker by using a full Reynolds stress (RS) model. Some codes started to possess the capability of free surface computation in the 1994 workshop. In the latest and 5 workshops more modern hull forms were introduced including a KRISO tanker (KVLCCM), a KRISO container ship (KCS), and a US navy combatant (DTMB 5415), more advanced operating conditions were added as test cases, including self-propelled condition for KCS, obliquely towed condition for KVLCCM, and diffraction condition for DTMB Isotropic turbulence models are still dominant in these two workshops (notably blended k-ε/k-ω) and only few applications used anisotropic models because of the complexity, stability, and computational cost. For instance, a full RS model would require.5 to 7.5 times more iterations than a pure isotropic model (Bull, 5). It was found that RS model shows significant improvements when separation is not severe (e.g. degree yaw for ships) but show similar results as those predicted by twoequation isotropic model for oblique ship flow predictions (Pattenden et al., 5; Gorski et al., 5). Although natural unsteady separated flows would require high-resolution turbulence models, such as large eddy simulation (LES) or detached eddy simulation (DES), limited studies have applied these models for ship hydrodynamics, partly due to the computational cost as a result of the complex geometries, high Reynolds numbers, and the free surface. A comparative study of RANS, DES and LES was conducted for flows over a D surface mounted hill and flow past an axisymmetric submarine hull (Bensow et al., 6). In this study both LES and DES are more accurate than RANS. The interaction between a free surface and a turbulent boundary layer significantly modifies the separation mechanisms, unsteady flow field, and statistical dynamics of turbulence. One well known effect of a free surface on turbulence is the damping effect. Swean et al. (1991) measured the anisotropy in a turbulent jet near a free surface. They found that there is a thin region near the free surface in which the vertical velocity fluctuations are suppressed while the horizontal components are enhanced. Orlins et al. () reported further evidence of free surface dissipation of the normal component of velocity fluctuations. Measurements show that the bulk flow spectrum follows a slope of -5/ between 1 and 4 Hz, while the spectrum at the water surface follows a slope of - between.1 and 5 Hz, which is related to the stretching of turbulent eddies from D into D forms. Longo et al. (1998) conducted a LDV measurement of a solid/free-surface juncture flow and found that inner (near the plate and wake center plane and below the free surface) and outer (near free surface) regions of high streamwise vorticity of opposite sign. These two vortices are turbulence-driven secondary flows, which closely correlates with the cross-plane normal Reynolds stress anisotropy. The inner region vortex transports high mean velocity and low turbulence from the outer to the inner portion of the boundary layer and wake; while the out region vortex transports low mean velocity and high turbulence from the inner to the outer portion of the boundary layer and wake. A commonly used method to account for the effect of free surface damping on turbulence was proposed by Shir (197). In this method, a free surface correction term is added to the turbulence model, which damps the surfacenormal component of the Reynolds stress tensor and redistributes the energy among the other two components. This method was first used by Sreedhar and Stern (1998b) in a prediction of Solid/Free-Surface juncture boundary layer and wake of a surface-piercing flat plate using CFDSHIP-IOWA-V. (Tahara and Stern, 1996). Simulations showed the generation of secondary flows in the corner and the thickening of the boundary layer near the free surface, which were consistent with the experimental observations. Unsteady organized vortical structures, instabilities, and turbulent structures have been studied for wave-induced separation on idealized geometries. Kandasamy et al. (6) investigated the applicability of unsteady Reynolds averaged Navier-Stokes (URANS) with a surface-tracking method (URANStracking hereinafter, CFDSHIP-IOWA-V.) to predict the organized vortical structures and instabilities for free-surface wave-induced separation around a surfacepiercing NACA4 and compared predictions with experiments (Metcalf et al., 6). This study identified the organized vortical structures and associated instability mechanisms of unsteady freesurface wave-induced separation, with quantitative verification and validation. It predicted the dominant frequency in the separation region and traced its origin to the shear layer instability. Two additional dominant frequencies were identified in the separation region, i.e., Karman-like shedding and flapping. Detailed flow physics behind these three frequencies were explained. However, this study does leave some issues unresolved. URANS-tracking over-predicts wave elevation at the Kelvin wave crest, and also predicts a quicker pressure recovery after the separation. Root mean square (RMS) of the wave elevation and the foil surface pressure and the amplitudes of the three dominant frequencies were significantly underpredicted due to the dissipation of the RANS model.

3 These unresolved issues were resolved by Xing et al. (7) using DES with a single-phase level set method (CFDSHIP-IOWA-V.4), with further investigation of the turbulent structures. These two studies showed that URANS resolves most of the unsteady organized oscillations due to large-scale vortical structures and instabilities when there is a spectral-gap between the organized oscillations and random fluctuations. This facilitates URANS to capture the gross features of the unsteady separation and identify the important instabilities, but with significant deficiency in the amplitudes of the oscillation frequencies. DES has smaller modeling errors and thus likely resolves more turbulent structures than URANS. Anisotropy invariant maps show that turbulence is anisotropic in the middle of the separation region and is at a two-component state near the foil surface. The turbulent kinetic energy (TKE) and its budgets show similar feature to previous canonical flows but with large three-dimensional and free surface effects (Xing et al., 7). 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. These two studies also assessed the advantages and disadvantages of using surface-tracking and single-phase level set methods for free surface flows. Of interest herein are flows around ships at large drift angles. Most of the previous studies for ship flows are restricted steady and small drift angles. Experimental investigation of the flow past a submarine at angles of drift, 5, and 9.5 degrees was performed by Bridges et al. (). The tip vortex shed from the sail at angle of drift during a high-speed turn causes an pitching moment as a result of the vortex circulation that creates an equal and opposite circulation about the hull that results in a shift in the pressure distribution, increasing the pressure on the deck of the hull and decreasing the pressure on the keel. Longo and Stern () investigated the effects of drift angles (β, up to 1 degrees) on a model ship (Series 6 C B =.6 cargo/container) flow through towing tank tests with a free surface. The resistance increases linearly with β with same slope for all Fr, whereas the increases in the side force, drift moment, sinkage, trim, and heel with β are quadratic. The boundary layer and wake are dominated by the hull vortex system consisting of fore body keel, bilge, and wave-breaking vortices and after body bilge and counter-rotating vortices. Simonsen and Stern () conducted rigorous verification and validation of a RANS code (CFDSHIP-IOWA-V.) applied to a maneuvering problem covering the static rudder and pure drift conditions up to 1 degrees. Fair results of forces and moments were obtained for the bare hull quantities, but larger deviations between experiment and computations were observed for the forces and moments for the appended hull. The authors attributed this to the omission of the free surface and the limitations of the blended k-ε/k-ω model. A later study by the same authors (Simonsen and Stern, 5) further characterized flow pattern around the appended hull using the same code and correlated behavior of the integral quantities with the flow field. The flow pattern was characterized by fore and aft body bilge and side vortices, which are similar for straight-ahead and static rudder conditions, except in close vicinity of the rudder. The pure drift condition showed strong asymmetry on windward vs. leeward sides and a more complex vortex system with additional bilge vortices. For the considered drift and rudder angles, the friction was not particularly sensitive to the changed conditions, while more significant changes were observed for the pressure. Noting the limitation of URANS to resolve the unsteady separated flows, Heredero (5) used DES with a single-phase level set method (CFDSHIP-IOWA-V.4) for the free surface to study a Wigley hull at three drift angles: 1,, and 6 degrees. Only the flow at the largest drift angle is unsteady. Detailed information on the forces and vortical structures were reported. Karman instabilities were found to be present on the flow and its scaling showed good agreement with the universal Strouhal number. The present results (as will be shown later) and reevaluation of the flow around Wigley hull at drift angle 6 degrees (Heredero, 5) show that both flow patterns exhibit tip vortices similar to those formed in the flow over a delta wing at high angles of attack. Gursul (1994, 5ab) found that the tip vortices for delta wings have a helical mode instability that can be scaled using the distance to the leading edge of the delta wing (x) and the free stream velocity U : Stx = fx U. St x is nearly constant if the sweep angle and the angle of attack are fixed (Gursul, 1994), which suggests that f decreases as x increases. The breakdown location of the tip vortices is unsteady and follows a quasi-periodic antisymmetric pattern (Gursul, 5a). It was also reported by Gursul (1994) that the dimensionless circulation (vortex strength) Γ linearly increases as a function of x because of continuous feeding of vorticity from leading edge. The objectives of this study are to (1) implement a RS model with free surface damping effect and DES options to CFDShip-IOWA-V.4 (Carrica et al., 7) to study ship flows; () investigate anisotropy of turbulence for thick boundary layers, free surface, and especially unsteady separation; () identify the role of isotropic blended k-ω/k-ε (BKW) and anisotropic (RS) URANS models and BKW-DES and RS-DES for ship flows; and (4) investigate vortical structures, instabilities, and turbulent structures for ship flows. The approach is to use CFDSHIP-IOWA to: (1) use RS model with free surface damping effect to study

4 idealized geometries to assess the anisotropy of turbulence generated by the free surface, including a solid/free-surface juncture flow and flow around DTMB 5415; () extend the previous analysis for vortical structures, instabilities, and turbulent structures for NACA4 and tip vortex instability for delta wings to study KVLCC flows at large drift angles. COMPUTATIONAL METHOD U 1 1 k x i j Ui k p + bj U j = b k i k τ J τ ξ J ξ j k k l 1 bb b i i U i j ν U t b i j + S j k + + k l i J ξ J Reeff ξ J ξ J ξ The descretized momentum equations for any interior point can be written as: (4) The general-purpose RANS solver, CFDSHIP-IOWA, has been developed at Iowa Institute of Hydraulics Research (IIHR) over the past 15 years for support of students theses and research projects at IIHR, as well as transition to Navy laboratories, industry, and other universities. Documentation of the surface-tracking method (version.) is provided in Wilson et al. (6). Version. is extended to version 4. (Carrica et al., 7) with the use of a single-phase level set method, advanced iterative solvers, conservative formulations, and extension of the dynamic overset grid approach for free surface flows. Equations of motion All governing equations are non-dimensionalized using the free stream velocity U, the ship length L, and the water density ρ and viscosity μ. For Cartesian coordinates, the incompressible continuity and momentum equations in nondimensional tensor form are: U i = xi Ui Ui p 1 Ui U + = + uu t x x Re x x x j i j j i j j j (1) () In order to accommodate complex geometries, generalized curvilinear coordinates are used. The continuous governing Equations (1) and () are transformed from the physical domain in Cartesian coordinates ( x, y, z, t ) into the computational domain in non-orthogonal curvilinear coordinates ( ξ, η, ζ, τ ) by applying the chain rule for partial derivatives, which results in continuity and momentum equations given by (Carrica et al., 6): 1 j J ξ j ( bu i i) = () U i k anbu nb i, nb Si bi p (5) k aijk Jaijk ξ = Pressure Poisson equation The mass conservation Equation () can be enforced using the discretized form of the momentum Equation (5) resulting in a Poisson equation for the pressure of the form (Carrica et al., 6): j k j bb i i p bi j = a k j nbui, nb Si ξ Jaijk ξ ξ a (6) ijk nb Single-phase level set method only models water, leading to a Poisson equation with constant fluid properties (density and viscosity of air do not appear in the equations). Mass conservation in the air is not satisfied. Free surface modeling A single-phase level set method is used. The location of the free surface is given by the zero level set of the function φ, known as the level set function that is positive in water and negative in air. Since the free surface is a material surface, the equation for the level set function is: φ φ + U j = t x j (7) In the single-phase level set method, the jump condition at the free surface must be explicitly enforced since we solve the equations of motion only in water. Neglecting shear stress in the air, the jump condition at the free surface is: Ui n j x j int = (8) As a good approximation for air-water interfaces, the pressure on the air is equal to the atmospheric

5 pressure. The dimensionless piezometric pressure at the air-water interface is then given by: p z Fr = (9) int int The velocity is extended from the air-water interface to air by solving equation (8) over the whole air domain. The same extension procedure is performed for the turbulence quantities k and ω. The level set function is required to remain a distance function throughout the whole computation. Transport of the level set function with Equation (7) does not guarantee that φ remains a distance function as the computation evolves. To resolve this difficulty, an implicit extension is performed every time φ is transported. The first neighbors to the free surface are reinitialized geometrically. The rest of the domain is reinitialized using an implicit transport of the level set function with the normals (Carrica et al., 6): ( φ ) n φ = sign (1) Where φ is the non-reinitialized level set function and n is the vector normal to the free surface defined by: φ n = (11) φ RANS modeling Two-equation closure is used for the Reynolds stresses, where they are modeled as a linear function of the mean rate-of-strain tensor through an isotropic eddy viscosity ( ν t ), U U i j uu i j = ν t + δijk xj x i (1) where δ is the Kronecker delta. The unknown eddy ij viscosity is evaluated from the TKE (k) and the specific dissipation rate ω. Additional transport equations, presented below, are solved following Menter s (1994) blended k-ω/k-ε approach. k 1 + ( v σ k νt) k k + s = k t Pk ω 1 + ( v σω νt ) ω ω+ sω = t P ω (1) The turbulent viscosity and the effective Peclet numbers are defined as: ν = /, = t k ω Pk/ ω 1/Re+ σk/ ωνt The sources for k and ω are: * sk = G+ β ωk 1 (14) ω * 1 sω = γ G+ β ω 1 ( F1) σω k ω k ω Ui G = τ ij (15) x j The model constants, sayφ, are calculated from the standard k-ω ( φ 1 ), and k-ε ( φ ) values using a blending function (refer to Menter 1994 for the constant values): ( 1 F ) φ = Fφ + φ (16) The blending function F 1 is designed to be unity in the near-wall regions of boundary layers and gradually switches to zero in the wake region to take advantage of the strengths of the k-ω and k-ε models respectively. A RS model (Wallin and Johansson, ) is also available, which is based on a modified version of Menter s k-ε/k-ω turbulence model as the scale determining model, and an explicit algebraic Reynolds stress model as the constitutive relation in place of the Boussinesq hypothesis: U U i j ( ex) uu i j = νt + + kδij + aij k xj x i (17) Details are given in Shao (6). The momentum equations remain the same with one additional term ( ex) ( a k) x included in the source to account for the ij j ( ex) effect of the extra anisotropic tensor a : 1 a II S S + β6 SikΩklΩ lj +ΩikΩklSlj IIΩSij IVδij + β Ω Ω Ω Ω Ω Ω ( ex) ij = β ΩikΩkj Ωδij + β4( ikωkj Ωik kj ) 9 ( ik Skl lm mj ik kl Slm mj ) ij (18)

6 where the non-dimensional strain-rate and vorticity tensors are defined by: S ij 1 U U i j 1 U U i j τ +, Ωij τ x j x i xj x i (19) 1 ν The time scale is τ = max ;C * τ * β ω β kω and the invariants for the strain rate and vorticity tensors are: II = S S, III = S S S II =Ω Ω, IV = S Ω Ω S kl lk S kl lm mk Ω kl lk kl lm mk () The model coefficients are function of the invariants in Equation (18): ( Ω ) N N 7II 1IV β1 =, β = Q NQ ( N IIΩ ) 6N 6 β4 =, β6 =, β9 = Q Q Q 5 Q = ( N IIΩ)( N IIΩ) 6 A ( P1+ P ) + sign( P1 P ) P1 P ( P ) N = A 16 1 P 1 + ( P1 P ) cos arccos ( P < ) P1 P A 9 A 9 P1 = + IIS IIΩ A, P = P1 + IIS + IIΩ ( eq) A = + CDiff max ( 1 + β1 IIS,) 5 4 ( eq) ( eq) 5 N ( eq) 81 β1 =, N = A + A4 = 6 ( eq) ( N ) IIΩ (1) The effect of the free surface on turbulence is considered by introducing a free-surface correction term to the Reynolds stress tensor: ( 1 ) τ = +Δ T = T +Δ T () ij ij ij ij T is the original Reynolds stress tensor without ij correction, and Δ T is the correction term, which is: ij Δ Tij = CS Tkmnk nmδ ij Tkink nj Tkj nkni f ( y, z) C =.5 () S With a damping function y y f ( y, z) = min 1, Cy 1 min 1, Cz δ δ C = 1 and C = 4 (4) y z Here y and z are the distances from the wall and the free surface, respectively. The damping function controls the overall distribution of the normal Reynolds stress anisotropy. Actually it also accommodates the effect of the solid-wall so that the free surface correction part slowly decreases toward the solid wall and vanishes at the wall. For the current application, the boundary layer thickness δ at the deep was selected as an appropriate length scale. For surface ships, the boundary layer thickness at the mid-girth could be a good candidate for the length scale. More details of the development and validation of the RS model is presented by Shao (6), including two-dimensional flat plate boundary layers and flow in square ducts with a quantitative verification and validation for the former. DES models The BKW and RS models are extended to DES models in CFDSHIP-IOWA-V.4. The approach is to modify the dissipative term of the k-transport equation (Travin et al., ): k * D RANS ρβ kω ρk / lk ω k D k DES ρ = = (5) = l The length scales are: (6) 1 * lk ω = k ( β ω) (7) l = min( l, C Δ) (8) k ω DES where C DES is the DES constant, set at.65, the typical value for homogeneous turbulence, and Δ is the local grid spacing. Through this formulation, it is theoretically determined where the LES or the URANS will be applied. The length scale, l k, reflects the scale ω of the local energy-containing vortical structures. Inside the boundary layer of a wall or inside a region where no separation occurs, l k ω is small since TKE is small. Hence, l = l and the URANS is used. When k ω the flow separates, vortices generate a significant increase of the TKE and l k, and the LES is used ω ( l = CDESΔ ). The BKW-DES model has been validated by a simulation of massively separated flows around a NACA1 aerofoil at 6 degrees angle of attack under the same conditions as the study by Shur et al. (1999).

7 Numerical methods The resulting algebraic systems for the variables, u, v, w, p,φ, k, and ω are solved in a sequential form and iterated to achieve convergence within each time step. The equations are discretized using finite difference approach with body-fitted curvilinear grids. The convection terms uses nd order upwind (used for URANS) or 4 th order upwind biased schemes (for DES simulation), and the diffusion terms are solved using second-order central differences. A PISO (Issa, 1985) algorithm is used to obtain a pressure equation and satisfy continuity (cf. Carrica et al. 7 for details). The pressure Poisson equation is solved using the PETSc toolkit (Balay et al., ). All the other systems are solved using an alternating direction implicit (ADI) method. The software SUGGAR (Noack, 5) runs as a separate process from the flow solver to get the interpolation coefficients needed for overset grids. APPLICATION OF RS MODEL TO FREE SURFACE FLOW The capability of the RS model to predict the anisotropy caused by the free surface is tested using a flat-plate/free-surface juncture flow. Without waves, the free surface can be reasonably approximated as a rigid lid. The Reynolds number based on the length of the flat plate and the free stream velocity is Re= The contour for axial vorticity and TKE are shown in Figures 1 and, respectively. BKW, as expected, fails to predict all the important secondary flow effects associated with the normal stress anisotropy. The RS model result in Figure shows an increase in turbulent kinetic energy as the rigid lid boundary is approached. Similar trend was observed in the experiment as well as relevant DNS (Walker et al, 1996) and LES (Sreedhar and Stern, 1998a) studies. High performance computing A MPI-based domain decomposition approach is used, where each decomposed block is mapped to one processor. -5 nonlinear iterations are required for convergence of the flow field equations within each time step. Convergence of the pressure equation is reached when the residual imbalance of the Poisson equation drops six orders of magnitude. All other variables are assumed converged when the residuals drop to 1 -. (a) (b) (c) Figure 1: Axial vorticity distribution of the solid-rigid-lid juncture flow: (a) BKW, (b) RS, (c) EFD. Analysis method The Q-criterion (Hunt et al. 1988) is used to identify the vortex. The Q-criterion is based on the second invariant of velocity gradient tensor u. However, Q- criterion cannot be used to visualize the orientation of the vortex detected. To overcome this disadvantage, a unique quantity, i.e. the normalized helicity density, is proposed as follows and will be applied to color-code the Q-isosurfaces in the current study (Levy et al, 199): H n v ω = (9) v ω Where v and ω are the velocity and vorticity vector at the same point. The normalized helicity density H n represents the directional cosine between the vorticity vector and the velocity vector, 1 H n 1. The sign of H n indicates the direction of swirl of the vortex relative to the streamwise velocity component. (a) (b) (c) Figure : TKE distribution of the solid/rigid-lid juncture flow: (a) BKW, (b) RS, (c) EFD. The RS model was also applied to study DTMB 5415 with the free surface (Shao, 6) and compared with BKW predictions and the experimental data. RS model does not show significant improvements than BKW for axial velocity, vorticity, and Reynolds stresses. It is likely attributed to several factors. DTMB 5415 has a sonar dome vortex, but has thinner boundary layer without a bilge vortex, which causes weak anisotropy. The grid near the free surface used in the simulation was not fine enough to resolve the free surface damping effect. APPLICATION TO KVLCC

8 As a modern ship hull form designed by the Korean Institute of Ships and Ocean Engineering (KRISO) in 1997, KVLCC was used in the latest two workshops for ship hydrodynamics to replace the HSVA tanker, which had been used for previous workshops and was a representative of designs around 197. Extensive experimental investigations, including towing tank measurements (Van et al., 1997 and 1998) and wind tunnel experiments (Lee et al. 1998), have been done and data sets acquired. Figure shows the geometry. The focus of the flow around KVLCC is on the stern flow prediction. The velocity contour at the propeller plane is the hooklike shape pattern, which is mainly due to the strong bilge vortices (Larsson et al., ) and associated anisotropy of turbulence. The solution domain used in this study extends (-L, L) in the streamwise direction (X), (-1.5L, 1.5L) in the transverse direction (Y), and (-1.L, -.1L) in the vertical direction (Z). The negative Z ensures that the entire ship hull is submerged in the water without solving the level set transport equation. Instead, the top boundary is specified as symmetry boundary to mimic the double-tanker model in the experiments. Figure 4 shows the grid topology used for this geometry. Body-fitted O type grids are generated for ship hull and rectangular background grids are used for specifying boundary conditions away from the ship hull, with clustered grid near the free surface to resolve the wave flow pattern. Figure 4 also shows the specification of boundary conditions with details presented in Table 1. In order to resolve the boundary layer using the current turbulent model, the first grid spacing normal l to the ship hull is around Δy = 1 6, i.e. y + <1. The grid stretch function is the TANH. The grid dimensions are presented in Table. Grid is used for all simulations. The other three grids are only used for drift angle degrees for a verification study. The design of the grids enables two sets of grids with different grid refinement ratios, i.e., 4 for (1,, ) and for (1,, 5). The flow conditions are Re= , Fr=.. The coordinate systems for hydrodynamic forces and moments and velocity fields in the experiment and the current simulation are shown in Figure 5. The wake plane is located in the y-z plane located at x=.48l cos β, where β is the drift angle. pp Figure : KVLCC geometry Figure 4: Grids, solution domain, and boundary conditions Table. Grid dimensions for KVLCC simulations No. Block name (i mx, j mx, k mx ) Grid points Boundary layer ,76 1 (port/starboard) Background ,45 Total 589,97 Boundary layer ,561 (port/starboard) Background ,55 Total 957,67 Boundary layer ,767 (port/starboard) Background ,596 Total 1,6,1 Boundary layer ,7,861 5 (port/starboard) Background ,159,15 Total 4,64,874 Figure 5: Coordinate systems for hydrodynamic forces and moment and velocity fields KVLCC DRIFT ANGLE BKW and RS RANS models are applied to study KVLCC at drift angle degree. Table compares the predicted frictional resistance coefficient C XF with ITTC 1957 and the total resistance C X with experimental data. RS model agrees with EFD better than BKW does. Figure 6 shows comparison of the axial velocity contours at the propeller plane between CFD (BKW and RS) and EFD. Since the flow around KVLCC

9 features strong bilge vortices, the hooklike shape pattern of axial velocity distribution can be clearly seen in the EFD results. The results of the BKW model under-predict the size of the low speed region (~%U region) close to the center-plane. Results of RS model shows significant improvements on prediction of the hooklike shape pattern of axial velocity distribution. This suggests that the hooklike shape pattern is related to the anisotropy of the turbulence caused by the bilge vortices near the ship hull. The TKE distributions in Figure 7 show the similar trend as that for axial velocity distributions, i.e., the RS model gives better result than the BKW does. This suggests that there is a strong correlation between the axial velocity pattern and anisotropy of the turbulence near the ship hull. The BKW shows that the peak value of TKE (~.1%U ) is over-predicted by.5%, while the RS model predicts the same peak value (~1.7% U ) as the experimental data. Comparisons for cross-plane velocity and axial vorticity distributions show similar improvements by using RS model. The results using the current RS model are as good as the best solution at the CFD Workshop Tokyo 5. Table : Validation of resistance coefficients for KVLCC at drift angle zero. Turbulence model C XF 1 - (ε) (ITTC: ) C X 1 - (ε) (EFD: ) BKW -1.6 (-7.7%) (-6.%) RS -1.7 (-7.%) (-.1%) Figure 6: Axial velocity at propeller plane at drift angle. Figure 7: TKE at propeller plane at drift angle. KVLCC DRIFT ANGLE 1 The integral coefficients of forces in X and Y directions (C X and C Y ) and the moment around Z (C N ) axis using BKW and RS are compared with EFD data in Table 4 for KVLCC at drift angle 1 degrees. RS model improves the moment prediction but significantly over-predict the C X by 19%, higher than 11% for BKW. Similar trend was observed for full Reynolds stress model s prediction in CFD Workshop Tokyo 5 (e.g. NSWC-UNCLE-RS model, 6% higher than EFD for C X ). The EFD data in Table 4 are the same as those reported in the experimental study by Kume et al. (6). In their experiments, the Froude number and Reynolds number were.144 and , respectively. This suggests that the free surface effect may not be negligible and may cause the large difference for C X. Figure 8 shows the bottom view of the flow field. Three main vortices form around the ship hull: the fore-body bilge vortex (FBV), the fore-body side vortex (FSV), and the aft-body bilge vortex (ABV). These vortices are steady. Figure 9 shows the axial velocity contours at the propeller plane with the EFD data and the NSWC-UNCLE solution (one of the best solutions in CFD Workshop Tokyo 5). The solution using the current RS model shows similar trend as EFD and NSWC-UNCLE solution, with improvements on the distribution and magnitude of axial velocity near the location of the ABV. Unlike the zero drift angle case where there are two aft-body inboard rotating bilge vortices, the bilge vortex on the windward side disappears, the bilge vortex on the leeward side (ABV) become larger and stronger. There is a new fore-body side vortex (FSV) on the leeward side. Both the ABV and FSV carry relatively low turbulence. The current RS model results are as good as the best solution using a full Reynolds stress model in the CFD Workshop Tokyo 5.

10 Table 4: Force coefficients for KVLCC at drift angle 1 degrees. Turbulence model C X 1 - (ε) (EFD:-1.75) C Y 1 - (ε) (EFD: 7.8) C N 1 - (ε) (EFD:.59) BKW (11%) 6.85 (.%).689 (5.9%) RS -.86 (19%) 7.46 (.7%).578 (1.6%) Figure 8: Vortex systems around the ship hull at drift angle 1 degrees (Iso-surface of Q=1 colored by helicity). Figure 9: Axial velocity contour at propeller plane for drift angle 1 degrees: EFD (top), RS (middle), and NSWC- UNCLE solution (bottom). KVLCC DRIFT ANGLE For flows around KVLCC at drift angle degrees, BKW and RS RANS models show steady solutions whereas RS-DES model shows unsteady solutions. Thus, RS-DES is used for all the simulations at this drift angle. An additional simulation using BKW-DES model is also included to compare with RS-DES. three different filter widths are used. The minimum filter width equals the local grid spacing (1.,.,., and 5) and the other two filter widths are 4 (1.,.,.) and (1.1,.1,.1) times the local grid spacing, respectively. C X monotonically converges on grids (1.,., 5). C Y oscillatorily converges on grids (1.,.,.). C N monotonically converges on both (1.,.,.) and (1.,., 5). Table 6 summarizes the grid uncertainties with order of accuracy and correction factors for these three converged grid studies. It is obviously that all three grid studies are far from the asymptotic range (C G =1). The huge grid uncertainty for C X suggests that a much finer grid is needed for C X to reach the asymptotic range. This is also proved by Figure 1, where C Y and C N tends to converge after grid 5 but C X still requires finer grids to converge. For all grids, the increase of the filter width results in increase of forces and decreases of moments. The relative changes between the largest and smallest filter width for each grid are: Grid 1 (C X :.5%; C Y :.8%; C N :.8%), Grid (C X : 6.%; C Y :.5%; C N : 1.9%), and Grids (C X : 5.%; C Y :.44%; C N : 1.6%). It is the medium grid that is most sensitive to the change of the filter width. Solutions (1.,., and.1) have the same filter width and seem to be appropriate to be used to estimate the numerical errors only. Unfortunately, no solutions monotonically or oscillatorily converge. It is known that grid refinement for DES will change both the numerical errors and the sub-grid scaling modeling errors due to the dependence of the filter width on the local grid spacing. Even for the same filter width, different grids have different interface locations between the URANS and LES regions for DES, which change the DES modeling errors on different grids. How to quantitatively estimate the coupled numerical and modeling errors for DES needs further investigation. For drift angle degrees, the relative changes of forces and moments between BKW-DES and RS-DES are less than 6.5%. This suggests the anisotropy of turbulence near the ship hull has little effects on the forces and moments for oblique ship flow simulations, which is consistent with the conclusions by related previous studies (Pattenden et al., 5; Gorski et al., 5). Results for RS-DES model will only be presented in the following sections. Verification of integral forces Verification and validation are conducted for KVLCC at drift angle degrees using RS-DES. The methodology and procedure follow the quantitative procedures proposed by Stern et al. (6). Table 5 shows the converged running means for C X, C Y, and C N on grids 1,,, and 5. For each grid,

11 Table 5: Grid study for C X, C Y, and C * N. Grids/Model Case Filter width C X ( 1 - ) C Y ( 1 - ) C N ( 1 - ) 1.1 Δ x Δ x (RS-DES) 1. Δ x (RS-DES) (RS-DES) (BKW-DES) 5 (RS-DES).1 4 8Δ x Δ x Δ x Δ x Δ x Δ x Δ x Δx Vortex system, limiting streamlines, and frequencies Figure 11 shows the vortex system for KVLCC at drift angle degrees. The shear layer after the fore perpendicular becomes unstable due to the Kelvin- Helmholtz (KH) instability, followed by Karman-like vortex shedding. Open separation is detected on the leeward side, which causes five main unsteady vortices: fore-body bilge vortex (FBV), fore-body side vortex (FSV), aft-body bilge vortex (ABV), aft-body side vortex (ASV), and stern vortex (SV). Table 6: Uncertainties for C X, C Y, and C N with filter width equals the grid spacing * Grids Ratio P G C G U G U GC S C C X 1.,., ( 1 - ) C Y ( 1 - ) 1.,., C N 1.,., ( 1 - ) 1.,., *U G and U GC are based on the finest grid. (a) (a) (b) (b) Figure 11: Vortex system of KVLCC (coherent vortex colored by normalized helicity) for drift angle degrees: (a) bow view, (b) bottom view. (c) Figure 1 shows the limiting streamlines for KVLCC. There exists the primary separation line, primary reattachment line, secondary separation line, secondary reattachment line, and the third separation line. All three separations are open separation. Figure 1: Force coefficients on different grids (filter width equals grid spacing): (a) C X, (b) C Y, (c) C N.

12 Frequency amplitude Figure 1: Time history and FFT of the total resistance coefficient CT = Cx + C of KVLCC at drift angle y degrees (grid ): (a) time history, (b) FFT. Time history and FFT of the total resistance coefficient C T is shown in Figure 1. There are four dominant frequency modes, i.e.,, 17., 11, and 4. These frequencies are associated closely with the unsteady organized vortical structures, which will be analyzed in details in the following sections. Shear layer instability (b) Figure 1: limiting streamlines on KVLCC (port side, starboard side, and bottom views) for drift angle degrees. The two highest frequency contents, f 1 = (Figure 11b) and f =17. (Figure 11a), are caused by the KH instability. The frequency can be scaled using the momentum thickness (θ) and the velocity (U s ) at the separation point: Stθ = f θ U S () The scaled Strouhal numbers are presented in Table 8. Compared to St θ =.86 for a surface-piercing NACA4 hydrofoil (Kandasamy et al., 6). The present St θ are smaller, especially for the shear layer near the stern. This is due to a much smaller momentum thickness in the current study. (a) Table 8: Frequencies and associated scaling for shear layer and Karman-like vortex shedding. frequency Mechanism Scaled by St f 1 f f Shear layer instability Shear layer instability Karman-like shedding Momentum thickness θ=4.7e-5 Momentum thickness θ=1.7e-4 Half-width of wake h=6.68e Karman-like shedding instability

13 Karman-Like vortex shedding instability (f ) is due to interaction between opposite signed vortices (Sigurdson, 1995). The relationship between the shear layer instability and the Karman-like shedding instability is shown in Figure 14. Vortex V1 and V are generated by the shear-layer instabilities near the ship bow. V1 and V are merged together and form a new bigger vortex (V1+V), which later become V after new V1 and V are formed. The merged vortex V reattaches the ship hull and then shed following the Karman-like frequency, which is lower than the KH frequency due to the larger vortex size and smaller convection velocity. As suggested by Sigurdson (1995), the Karman-like frequency can be scaled using the half-wake width h and the velocity at the separation point: St h = fh U (1) St equals.75, which is within the range of the h universal St (.7~.9). Since there is no free surface in the current simulation (Fr=), the reduction of St h as reported for Karman-like shedding for the surfacepiercing NACA4 is not observed. Figure 14: Karman-like vortex shedding at z=-.9. S Helical instability The FFT of the total drag coefficient (Figure 1b) shows a dominant frequency mode at f 4 =4. Examining the vortex system with iso-surface of Q and the volume streamlines inside the Q-tube, we found that the fluid particle inside the Q-tube is very unsteady. As shown by the volume streamlines in Figure 11, a fluid particle originally moves on a straight line upstream (steady) and then follows a helical-shape path downstream. The transition point is regarded as a vortex break-down location. The wavelength of this helical streamline increases with the fluid particle convects further downstream. The helical path is unsteady and oscillating along the vortex core direction following different frequencies for different locations. It is likely that the f 4 =4 seen in Figure 11 is the dominant mode of the helical mode near the ship hull. The five vortices (FBV, FSV, ASV, ABV, and SV) are similar to the tip vortices found in the unsteady vortex flows over slender delta wings at high angle of attack (Gursul, 1994, 5) such that similar analysis can be used here. For each vortex, three points are selected at different streamwise locations on the vortex core and the time histories of the pressure at those points are reported and analyzed using FFT. Coordinate x is defined as the distance along the vortex core, whose origin is located at the intersection of the vortex core and the ship hull. Vortex breakdown location is defined as where the volume streamline changes from straight to helical shape. FFT shows that frequencies for FBV and ASV are at the order of magnitude of the shear-layer instability. This is because FBV is very close to the region dominated by shear layer instability and ASV is very close to the ship hull surface. Thus they are both inappropriate to be regarded as helical vortices. The non-dimensional frequency fl U (L is the ship length) and St for FSV, ABV, and SV are plot in x Figures 15 and 16, respectively. Examples of variations of dimensionless frequency fc U (c is the delta wing chord length) and fx U as a function of the streamwise distance for different delta wings are also shown to facilitate comparisons. The two curves for the delta wing in Figures 15 and 16 are the maximum and minimum values between which all other delta wing conditions fall between. Figure 15 shows the current dimensionless frequency is higher than those reported for different delta wings. This is likely due to the fact that the Reynolds number in the current study is 4 times that of the delta wing. St show different range x of constants Stx( ABV ) > Stx( FSV ) > Stx( SV ), with the Stx ( FSV ) is within the same range of that reported for delta wing (Gursul 1994). All three vortices show similar trend as observed for delta wing

14 studies, i.e., fc U and fl U decrease as increase of x while St are nearly constant for different x. x due to the larger Reynolds number and the much smaller distance to the no slip surface. Figure 17: Vortex core colored by vortex strength/circulation. Figure 15: Variation of dimensionless frequency function of distance on the vortex core. fl U as a Γ/Ux Figure 18: Circulation of the vortex at breakdown locations. KVLCC DRIFT ANGLE 6 Figure 16: Variation of dimensionless frequency as a function of streamwise distance along the vortex core. Although the delta wing theory on helical instability scaling seems to be applicable to the current study, it is worthy to note the differences. Unlike the periodic antisymmetric oscillation of the vortex breakdown location in the delta wings, the location of the vortex breakdown is steady. In contrast to the linear increase of vortex circulation along the delta wing chord, circulations of the vortices decrease with the increase of the distance along the vortex core (Figure 17), which is attributed to the strong viscous dissipation away from the ship hull surface. Figure 18 shows the dimensionless vortex circulation Γ Uxat the vortex breakdown location x for FSV, ABV, and SV. Compared with the same plot in the previous study for delta wing (Gursul et al., 1995), the maximum Γ Ux in the current study is times that of delta wing. The reason for the large differences is due to the breakdown locations in the current study are very close to the ship hull and the circulation itself is 1 times that for the delta wing The RS-DES model is used to study KVLCC at drift angle 6 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. The statistical convergence is assessed by examining the running mean on the time history of CT = Cx + C, as shown in Figure 19a, which y establishes a statistically stationary unsteady solution. Compared with solutions for drift angle 1 and degrees, FFT of the total drag coefficient (Figure 19b) shows a much broader range of frequency contents, which is consistent with a much broader range of scales of vortices visualized in Figure. Most of the energy are possessed 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.

15 EFFECT OF DRIFT ANGLES ON FORCES AND MOMENTS FOR KVLCC (a) Figure 1 shows that the increases in force coefficients and moments with drift angle are quadratic, which is consistent with the findings by Longo and Stern () for the Series 6 with the maximum drift angle 1 degrees. For drift angle degree, RS agrees better with EFD (also evident in Table ). For drift angle 1 degrees, BKW predicts C X and C Y better than RS does and RS predicts C N better than BKW. For large drift angle degrees, no significant differences are observed for forces and moments between BKW and RS models (note that C X and C N coincide in Figure 1a). (b) Figure 19: Time history and FFT of the total resistance coefficient C = C + C of KVLCC at drift angle 6 T x y degrees: (a) time history, (b) FFT. (a) Figure 1: Force coefficients as a function of drift angles: (a) all drift angles, (b) drift angles up to 1 degrees. (b) Figure : Vortex systems around KVLCC at drift angle 6 degrees. CONCLUSIONS AND FUTURE WORK The overall objective of this study is to identify the role of isotropic blended k-ε/k-ω (BKW) versus anisotropic Reynolds stress (RS) RANS models for steady ship flows and BKW-DES versus RS-DES for unsteady

16 ship flows. Capability of RS model to predict anisotropy is shown for a solid/free-surface juncture and demonstrated for ship flow (DTMB 5415) at Fr=.8. No significant anisotropy effect is observed for DTMB 5415 due to the weaker anisotropy and insufficient grid resolution. KVLCC at drift angle, 1,, and 6 degrees are investigated neglecting the effect of free surface. Results of using BKW and RS RANS models for and 1 degrees show that RS model significantly improves the predictions of the resistance coefficients, axial velocity, and TKE distributions at the propeller plane. One exception is the better prediction of forces for drift angle 1 degrees using BKW. It is likely attributed to the omission of the free surface effect, which needs further investigation. For drift angle degrees, BKW and RS RANS models show steady solutions whereas BKW-DES and RS-DES models show unsteady solutions. In the latter case, limited differences on forces, moments and instabilities are observed. The flows at degrees using RS-DES are investigated in details, including quantitative verification (no validation due to the lack of EFD data), limiting streamlines, vortical structures and associated instabilities. Verification for forces (C X, C Y ) and moments (C N ) are performed on two sets of grids with refinement ratios 4 and, respectively. C N monotonically converges on both sets of grids, C X monotonically converges on grids with the refinement ratio, and C Y oscillatorily converges on grids with the refinement ratio 4. The effect of the filter width on different grids is also qualitatively evaluated. Limiting streamlines show primary separation and reattachment lines, secondary separation and reattachment lines, and a third separation line. Two shear layer instability modes, a Karman-like vortex shedding, and three helical mode instabilities are identified. The Strouhal numbers (St) for shear layer instability and Karman-like shedding are scaled using the momentum thickness (θ) and half wake width (h), respectively. St θ is smaller and St h is larger than previous EFD/CFD results. The scaled St for the helical instabilities using the ship length decreases with the increase of the distance along the vortex core (x) while the scaled St for the helical instabilities using x remains constant, which is consistent with the previous scaling for tip vortices over delta wings. Unlike the periodic antisymmetric oscillation of the vortex breakdown location in the delta wings, the location of the vortex breakdown in the current study is steady. In contrast to the linear increase of vortex circulation along the delta wing chord, circulations of the vortices decrease with the increase of the distance along the vortex core, which is attributed to the strong viscous dissipation away from the ship hull surface. RS-DES is also applied to study flows at drift angle 6 degrees. Compared to small drift angle flows, the flow is featured by a broad range of vortical structures and frequency content. Overall comparison of forces and moment for different drift angles show that the increases in force coefficients and moments with drift angle are quadratic, which is consistent with the findings by Longo and Stern () for the Series 6. Future work is listed as follows: (1) evaluate relative merits for using BKW-DES and RS-DES for analysis of turbulent structures of flows around ships at large drift angles, including TKE and Reynolds stress budgets following Xing et al. (7); () evaluate RS and RS-DES models for free surface anisotropy, including wave-induced pressure gradient effect, for both simple geometries and practical ship flows; () apply RS and RS-DES models to more complex applications such as seakeeping and maneuvering; (4) develop new advanced turbulence models (e.g. hybrid URANS and LES), new methodologies and procedures for quantitative verification and validation of LES and DES, and advanced analysis methods for triple decomposition. ACKNOWLEDGEMENTS The Office of Naval Research under Grant N and N , administered by Dr. Patrick Purtell, sponsored this research. REFERENCES Balay, S., Buschelman, K., Gropp, W., Kaushik, D., Knepley, M., Curfman, L., Smith, B., and Zhang, H., PETSc User Manual, ANL-95/11-Revision.1.5, Argonne National Laboratory,. Bensow, R.E., Fureby, C., Liefvendahl, M. and Persson, T., A Comparative Study of RANS, DES and LES, 6th Symposium on Naval Hydrodynamics, Rome, Italy, 17- September 6. Bridges, D.H., Blanton, J.N., Brewer, W.H., and Park, J.T., Experimental Investigation of the Flow Past a Submarine at Angle of Drift, AIAA Journal, Vol. 41, No. 1, January, pp Bull, P.W., Verification and Validation of KVLCCM Tanker Flow, Proceedings of CFD Workshop Tokyo 5, National Maritime Research Institute, Tokyo, Japan, March 9-11, 5, pp Carrica, P.M., Wilson, R.V., and Stern, F., Unsteady RANS Simulations of the Ship Forward Speed Diffraction Problem, Computers & Fluids, Vol. 5, No. 6, 6, pp