RS-DES of Unsteady Vortical Flow for KVLCC2 at Large ... - of Tao Xing

9th International Conference on Numerical Ship Hydrodynamics Ann Arbor, Michigan, 5-8 August 2007

BKW- RS-DES of Unsteady Vortical Flow for KVLCC2 at Large Drift Angles Tao Xing, Jun Shao, Frederick Stern IIHR-Hydroscience and Engineering The University of Iowa Iowa City, IA 52242-1585 ABSTRACT INTRODUCTION 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=0.28. No significant anisotropy effect is observed for DTMB 5415 due to the weaker anisotropy and insufficient grid resolution. KVLCC2 at drift angle 0, 12, 30, and 60 degrees are investigated neglecting the effect of free surface. Results of using BKW and RS RANS models for 0 and 12 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 30 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 NACA0024 and tip vortex instability for delta wings is extended to study flows at drift angle 30 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 KVLCC2 at drift angle 60 degrees. Further evaluation of relative merits for using BKW-DES and RS-DES for analysis of turbulent structures will be conducted in the future.

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 27 years. There have been five international workshops on the numerical prediction of ship viscous flow since 1980: the Gothenburg 1980 workshop, the SSPA-CTH-IIHR 1990 workshop, the Tokyo 1994 workshop, the Gothenburg 2000 workshop, and the Tokyo 2005 workshop. According to Larsson et al (2003), 16 out of 17 methods were based on the boundary layer approximation with only one RANS method in the first 1980 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 1990 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

hooklike shape pattern created by the bilge vortices. It was shown later (Deng et al. 1993) 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 2000 and 2005 workshops more modern hull forms were introduced including a KRISO tanker (KVLCC2M), 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 KVLCC2M, and diffraction condition for DTMB 5415. 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 2.5 to 7.5 times more iterations than a pure isotropic model (Bull, 2005). It was found that RS model shows significant improvements when separation is not severe (e.g. 0 degree yaw for ships) but show similar results as those predicted by twoequation isotropic model for oblique ship flow predictions (Pattenden et al., 2005; Gorski et al., 2005). 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 3D surface mounted hill and flow past an axisymmetric submarine hull (Bensow et al., 2006). 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. (2000) 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 -5/3 between 1 and 40 Hz, while the spectrum at the water surface follows a slope

of -2 between 0.1 and 5 Hz, which is related to the stretching of turbulent eddies from 3D into 2D 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 (1973). 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.2 (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. (2006) investigated the applicability of unsteady Reynolds averaged Navier-Stokes (URANS) with a surface-tracking method (URANStracking hereinafter, CFDSHIP-IOWA-V.3) to predict the organized vortical structures and instabilities for free-surface wave-induced separation around a surfacepiercing NACA0024 and compared predictions with experiments (Metcalf et al., 2006). 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.

These unresolved issues were resolved by Xing et al. (2007) 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., 2007). 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 0, 5, and 9.5 degrees was performed by Bridges et al. (2003). 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 (2002) investigated the effects of drift angles (β, up to 10 degrees) on a model ship (Series 60 CB=0.6 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 (2003) conducted rigorous verification and validation of a RANS code (CFDSHIP-IOWA-V.3) applied to a maneuvering problem covering the “static rudder” and “pure drift” conditions up to 12 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, 2005) 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 (2005) 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: 10, 30, and 60 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 60 degrees (Heredero, 2005) 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, 2005ab) 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 U0: St x = fx U 0 . 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, 2005a). 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., 2007) to study ship flows; (2) investigate anisotropy of turbulence for thick boundary layers, free surface, and especially unsteady separation; (3) 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

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; (2) extend the previous analysis for vortical structures, instabilities, and turbulent structures for NACA0024 and tip vortex instability for delta wings to study KVLCC2 flows at large drift angles. COMPUTATIONAL METHOD 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 3.03) is provided in Wilson et al. (2006). Version 3.03 is extended to version 4.0 (Carrica et al., 2007) 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 0 , 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: (1)

∂U i ∂U i ∂ ip 1 ∂ 2U i ∂ +U j =− + − ui u j ∂t ∂x j ∂xi Re ∂x j ∂x j ∂x j

(2)

In order to accommodate complex geometries, generalized curvilinear coordinates are used. The continuous governing Equations (1) and (2) are transformed from the physical domain in Cartesian

( x, y , z , t )

into

the computational

domain in non-orthogonal curvilinear coordinates

(ξ , η , ζ , τ )

1 ∂ ⎛ bi b ∂U i ⎜ J ∂ξ j ⎜⎝ J Re eff ∂ξ k j

+

k i

⎞ ∂U i 1 k ∂ ip ⎟ k = − bi J ∂ξ k ⎠ ∂ξ

(4)

⎞ b ∂ν t b ∂U j + Si ⎟⎟ + k l ⎠ J ∂ξ J ∂ξ k j

l i

The descretized momentum equations for any interior point can be written as:

Ui = −

∑

nb

anbU i ,nb − Si aijk

−

bik ∂p Jaijk ∂ξ k

(5)

Pressure Poisson equation The mass conservation Equation (3) 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., 2006):

∂ ⎛ bi j bik ∂p ⎞ ∂ bi j ⎛ ⎞ = anbU i ,nb − Si ⎟ ⎜ ⎟ ∑ ⎜ j ⎜ k ⎟ j ∂ξ ⎝ Jaijk ∂ξ ⎠ ∂ξ aijk ⎝ nb ⎠

(6)

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

∂U i =0 ∂xi

coordinates

∂x j ∂U i 1 k ⎛ + b j ⎜U j − ∂τ J ⎝ ∂τ

by applying the chain rule for partial

derivatives, which results in continuity and momentum equations given by (Carrica et al., 2006):

1 ∂ b jU i ) = 0 j ( i J ∂ξ

(3)

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 =0 ∂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:

∂U i nj ∂x j

=0

(8)

int

As a good approximation for air-water interfaces, the pressure on the air is equal to the atmospheric

pressure. The dimensionless piezometric pressure at the air-water interface is then given by:

pint =

zint Fr 2

(9)

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., 2006): n ⋅∇φ = sign (φ0 )

Where

φ0

(10)

is the non-reinitialized level set function and

n is the vector normal to the free surface defined by: ∇φ n=− ∇φ

(11)

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 j −ui u j = ν t ⎜ i + ⎜ ∂x ∂xi ⎝ j

⎞ 2 ⎟⎟ − δ ij k ⎠ 3

(12)

where δ ij is the Kronecker delta. The unknown eddy 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.

1 1/ Re+ σ k / ων t

ν t = k / ω , Pk / ω =

(14)

The sources for k and ω are:

sk = −G + β *ω k sω = −γ

ω

G + β *ω 2 − 2 (1 − F1 ) σ ω 2

k ∂U i G = τ ij ∂x j

1

ω

∇k ⋅∇ω (15)

The model constants, say φ , are calculated from the standard k-ω ( φ1 ), and k-ε ( φ2 ) values using a blending function (refer to Menter 1994 for the constant values):

φ = F1φ1 + (1 − F1 ) φ2

(16)

The blending function F1 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, 2000) 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 j ui u j = −ν T ⎜ i + ⎜ ∂x j ∂xi ⎝

⎞ 2 ( ex ) ⎟ + kδ ij + aij k ⎟ 3 ⎠

(17)

Details are given in Shao (2006). The momentum equations remain the same with one additional term ex ∂ aij( ) k ∂x j included in the source to account for the

(

)

( ex )

effect of the extra anisotropic tensor aij

:

1 ⎛ ⎞ = β 3 ⎜ Ωik Ω kj − II Ωδ ij ⎟ + β 4 Sik Ω kj − Ωik Skj 3 ⎝ ⎠ 2 ⎛ ⎞ + β 6 ⎜ Sik Ω kl Ωlj + Ωik Ω kl Slj − II Ω Sij − IV δ ij ⎟ 3 ⎝ ⎠

(

aij(

ex )

∂k 1 + ( v − σ k ∇ ν t ) ⋅ ∇ k − ∇ 2 k + sk = 0 ∂t Pk ∂ω 1 + ( v − σ ω ∇ν t ) ⋅∇ω − ∇ 2ω + sω = 0 ∂t Pω

The turbulent viscosity and the effective Peclet numbers are defined as:

(13)

(

+ β 9 Ωik S kl Ωlm Ω mj − Ωik Ω kl Slm Ω mj

)

) (18)

where the non-dimensional strain-rate and vorticity tensors are defined by: 1 ⎛ ∂U ∂U j Sij ≡ τ ⎜ i + ∂xi 2 ⎜⎝ ∂x j

⎞ 1 ⎛ ∂U ∂U j ⎟⎟ , Ωij ≡ τ ⎜⎜ i − ∂xi 2 ⎝ ∂x j ⎠

⎞ ⎟⎟ ⎠

(19)

⎛ 1 ν ⎞ The time scale is τ = max ⎜⎜ * ;Cτ ⎟ and the * β kω ⎟⎠ ⎝β ω invariants for the strain rate and vorticity tensors are:

II S = S kl Slk ,

III S = S kl Slm Smk

(20)

II Ω = Ω kl Ωlk , IV = Skl Ωlm Ω mk

The model coefficients are function of the invariants in Equation (18):

(

N 2 N 2 − 7 II Ω

β1 = −

Q

β4 = −

(

2 N 2 − 2 II Ω

),β

), β

6

3

=−

=−

12 IV NQ

)(

(

(

(

)

)

⎛ A′2 9 ⎞ ⎛ A′2 9 ⎞ 2 2 P1 = ⎜ 3 + II S − II Ω ⎟ A3′ , P2 = P12 − ⎜ 3 + II S + II Ω ⎟ 27 20 3 9 10 3 ⎝ ⎠ ⎝ ⎠ 9 9 ( eq ) A3′ = + CDiff max 1 + β1 II S , 0 5 4

(

( eq )

β1

N( ) 5 , =− 6 N ( eq ) 2 − 2 II Ω

(

)

N

( eq )

3

81 = A3 + A4 = 20

(21) The effect of the free surface on turbulence is considered by introducing a free-surface correction term to the Reynolds stress tensor:

τ ij = (1 + Δ ) Tij = Tij + ΔTij

(22)

Tij is the original Reynolds stress tensor without correction, and ΔTij is the correction term, which is: 3 3 ⎛ ⎞ ΔTij = CS ⎜ Tkm nk nmδ ij − Tki nk n j − Tkj nk ni ⎟ f ( y, z ) 2 2 ⎝ ⎠ CS = 0.5

With a damping function

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 (2006), including two-dimensional flat plate boundary layers and flow in square ducts with a quantitative verification and validation for the former.

k DRANS = ρβ *kω = ρ k 3 2 / lk −ω

)

eq

(24)

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., 2002):

)

)

C y = 1 and Cz = 4

DES models

6N 6 , β9 = Q Q

Q 5 2 Q = N − 2 II Ω 2 N 2 − II Ω 6 13 13 ⎧ A3′ + sign P1 − P2 P1 − P2 ( P2 ≥ 0 ) ⎪ 3 + P1 + P2 ⎪ N =⎨ ⎡ ⎛ ⎞⎤ 16 P1 1 ⎪ A3′ + 2 P 2 − P ⎟ ⎥ ( P2 < 0 ) cos ⎢ arccos ⎜ 1 2 ⎪3 ⎢3 ⎜ P 2 − P ⎟⎥ 1 2 ⎠⎦ ⎝ ⎣ ⎩

(

⎡ y ⎞⎤ ⎡ y ⎞⎤ ⎛ ⎛ f ( y, z ) = ⎢ min ⎜ 1, C y ⎟ ⎥ ⎢1 − min ⎜1, C z ⎟ ⎥ δ δ ⎝ ⎠ ⎥⎦ ⎣⎢ ⎝ ⎠ ⎥⎦ ⎣⎢

(23)

k DDES =ρk

(25) (26)

32

l

The length scales are:

lk −ω = k 1 2 ( β *ω )

(27)

l = min(lk −ω , CDES Δ )

(28)

where CDES is the DES constant, set at 0.65, 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, lk −ω , reflects the scale of the local energy-containing vortical structures. Inside the boundary layer of a wall or inside a region where no separation occurs, lk −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 lk −ω , and the LES is used ( l = CDES Δ ).

The

BKW-DES

model

has

been

validated by a simulation of massively separated flows around a NACA0012 aerofoil at 60 degrees angle of attack under the same conditions as the study by Shur et al. (1999).

APPLICATION OF RS MODEL TO FREE SURFACE FLOW

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 2nd order upwind (used for URANS) or 4th 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. 2007 for details). The pressure Poisson equation is solved using the PETSc toolkit (Balay et al., 2002). All the other systems are solved using an alternating direction implicit (ADI) method. The software SUGGAR (Noack, 2005) runs as a separate process from the flow solver to get the interpolation coefficients needed for overset grids.

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=1.0×106. The contour for axial vorticity and TKE are shown in Figures 1 and 2, 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 2 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. 2-5 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 10-3.

(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, Qcriterion 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, 1990):

Hn =

v ⋅ω | v || ω |

(29)

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

(a)

(b)

(c)

Figure 2: 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, 2006) 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 KVLCC2

As a modern ship hull form designed by the Korean Institute of Ships and Ocean Engineering (KRISO) in 1997, KVLCC2 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 1970. 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 3 shows the geometry. The focus of the flow around KVLCC2 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., 2003) and associated anisotropy of turbulence. The solution domain used in this study extends (-2L, 2L) in the streamwise direction (X), (-1.5L, 1.5L) in the transverse direction (Y), and (-1.2L, -0.1L) 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 =3×10−6 , i.e. y+ St x ( FSV ) > St x ( SV ) , with

the St x ( FSV ) is within the same range of that Figure 14: Karman-like vortex shedding at z=-0.0039.

reported for delta wing (Gursul 1994). All three vortices show similar trend as observed for delta wing

studies, i.e., fc U 0 and fL U 0 decrease as increase of x while St x are nearly constant for different x.

due to the larger Reynolds number and the much smaller distance to the no slip surface.

Figure 15: Variation of dimensionless frequency fL U 0 as a function of distance on the vortex core.

core

colored

by

vortex

Γ/U0x

Figure 17: Vortex strength/circulation.

Figure 18: Circulation of the vortex at breakdown locations.

KVLCC2 DRIFT ANGLE 60

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 U 0 x at 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 U 0 x in the current study is 3000 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 120 times that for the delta wing

The RS-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. The statistical convergence is assessed by examining the running mean on the time history of CT = C x2 + C y2 , as shown in Figure 19a, which establishes a statistically stationary unsteady solution. Compared with solutions for drift angle 12 and 30 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 20. 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.

EFFECT OF DRIFT ANGLES ON FORCES AND MOMENTS FOR KVLCC2

(a)

(b)

Figure 21 shows that the increases in force coefficients and moments with drift angle are quadratic, which is 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, RS agrees better with EFD (also evident in Table 3). For drift angle 12 degrees, BKW predicts CX and CY better than RS does and RS predicts CN better than BKW. For large drift angle 30 degrees, no significant differences are observed for forces and moments between BKW and RS models (note that CX and CN coincide in Figure 21a).

(a)

Figure 19: Time history and FFT of the total resistance coefficient C = C 2 + C 2 of KVLCC2 at drift angle 60 T x y degrees: (a) time history, (b) FFT.

(b) Figure 21: Force coefficients as a function of drift angles: (a) all drift angles, (b) drift angles up to 12 degrees.

CONCLUSIONS AND FUTURE WORK Figure 20: Vortex systems around KVLCC2 at drift angle 60 degrees.

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

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=0.28. No significant anisotropy effect is observed for DTMB 5415 due to the weaker anisotropy and insufficient grid resolution. KVLCC2 at drift angle 0, 12, 30, and 60 degrees are investigated neglecting the effect of free surface. Results of using BKW and RS RANS models for 0 and 12 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 12 degrees using BKW. It is likely attributed to the omission of the free surface effect, which needs further investigation. For drift angle 30 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 30 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 (CX, CY) and moments (CN) are performed on two sets of grids with refinement ratios 4 2 and 2 , respectively. CN monotonically converges on both sets of grids, CX monotonically converges on grids with the refinement ratio 2 , and CY oscillatorily converges on grids with the refinement ratio 4 2 . 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. Stθ is smaller and Sth 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 60 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 (2002) for the Series 60. 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. (2007); (2) evaluate RS and RS-DES models for free surface anisotropy, including wave-induced pressure gradient effect, for both simple geometries and practical ship flows; (3) 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 N00014-011-0073 and N00014-06-1-0420, 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 2.1.5, Argonne National Laboratory, 2002. Bensow, R.E., Fureby, C., Liefvendahl, M. and Persson, T., “A Comparative Study of RANS, DES and LES,” 26th Symposium on Naval Hydrodynamics, Rome, Italy, 17-22 September 2006. 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 2003, pp. 71-81. Bull, P.W., “Verification and Validation of KVLCC2M Tanker Flow,” Proceedings of CFD Workshop Tokyo 2005, National Maritime Research Institute, Tokyo, Japan, March 9-11, 2005, pp. 581-586. Carrica, P.M., Wilson, R.V., and Stern, F., “Unsteady RANS Simulations of the Ship Forward Speed Diffraction Problem,” Computers & Fluids, Vol. 35, No. 6, 2006, pp. 545-570.

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Table 1. Boundary conditions for all the variables.

ω

U

V

W

k fs = 10−7

ω fs = 9

U =1

V =0

W =0

∂p =0 ∂n

∂k =0 ∂n

∂ω =0 ∂n

∂ 2U =0 ∂n 2

∂ 2V =0 ∂n 2

∂ 2W =0 ∂n 2

∂φ =0 ∂n

0

∂k =0 ∂n

∂ω =0 ∂n

U =1

V =0

W =0

Far-field #2

∂φ =0 ∂n

∂p =0 ∂n

∂k =0 ∂n

∂ω =0 ∂n

U =1

V =0

W =0

Symmetry

∂φ =0 ∂n

∂p =0 ∂n

∂k =0 ∂n

∂ω =0 ∂n

∂U =0 ∂n

V =0

∂W =0 ∂n

No slip (ship hull)

∂φ =0 ∂n

Equation (6)

k =0

∂U =0 ∂n

V =0

W =0

φ

p

Inlet

φ = −z

∂p =0 ∂n

Exit

∂φ =0 ∂n

Far-field #1

k

ω=

60 β Re Δy12