Acta Mech Sin (2007) 23:369–382 DOI 10.1007/s10409-007-0073-6

RESEARCH PAPER

Prediction of separation flows around a 6:1 prolate spheroid using RANS/LES hybrid approaches Zhixiang Xiao · Yufei Zhang · Jingbo Huang · Haixin Chen · Song Fu

Received: 7 February 2006 / Revised: 28 December 2006 / Accepted: 29 December 2006 / Published online: 19 July 2007 © Springer-Verlag 2007

Abstract This paper presents hybrid Reynolds-averaged Navier–Stokes (RANS) and large-eddy-simulation (LES) methods for the separated flows at high angles of attack around a 6:1 prolate spheroid. The RANS/LES hybrid methods studied in this work include the detached eddy simulation (DES) based on Spalart–Allmaras (S–A), Menter’s k–ω shear-stress-transport (SST) and k–ω with weakly nonlinear eddy viscosity formulation (Wilcox–Durbin+, WD+) models and the zonal-RANS/LES methods based on the SST and WD+ models. The switch from RANS near the wall to LES in the core flow region is smooth through the implementation of a flow-dependent blending function for the zonal hybrid method. All the hybrid methods are designed to have a RANS mode for the attached flows and have a LES behavior for the separated flows. The main objective of this paper is to apply the hybrid methods for the high Reynolds number separated flows around prolate spheroid at high-incidences. A fourth-order central scheme with fourth-order artificial viscosity is applied for spatial differencing. The fully implicit lower–upper symmetric-Gauss–Seidel with pseudo time sub-iteration is taken as the temporal differentiation. Comparisons with available measurements are carried out for pressure distribution, skin friction, and profiles of velocity, etc. Reasonable agreement with the experiments, accounting for the effect on grids and fundamental turbulence models, is obtained for the separation flows.

The project supported by the National Natural Science Foundation of China (10502030 and 90505005). Z. Xiao · Y. Zhang · J. Huang · H. Chen · S. Fu (B) School of Aerospace Engineering, Tsinghua University, Beijing 100084, China e-mail: [email protected]

Keywords RANS/LES hybrid methods · DES · Zonal-RANS/LES · Weakly nonlinear correction

1 Introduction Flows around ships, submarines, hot balloons, airships and aircrafts are very complicate and three dimensional (3-D). At high angle of attack (AOA), the 3-D flow separation has been an interesting and challenging problem in fluid mechanics. Undesired effects such as loss of lift, increase of drag, amplification of unsteady fluctuations in the pressure fields and uncontrolled yawing moment caused by the asymmetric forebody vortices always accompany when the 3-D separation takes place. The 2-D separation flows are mainly dominated by the adverse pressure gradient, flow reversal, etc. In the 3-D separation flows, the separation features can be sensitive to the body configuration, roughness of surface, AOA and Reynolds number, etc. In addition to the complex topology of the flow patterns, the 3-D separation flows strongly challenge experimental equipments, analytical and predictive tools. The prolate spheroid, with a 6:1 major-minor semi-axis ratio, has very simple configuration; however, the flows present almost all the fundamental transition and separation phenomena of a 3-D flows. The flow separating from the leeward side of the spheroid rolls up into a strong primary vortex on each side of the spheroid and reattaches on the plane of symmetry. The primary vortex is always accompanied by one small secondary vortex, which separates and reattaches adjacent to the wall. The flow transition includes not only the streamwise Tollmien–Schilichting (T–S) wave instability but also crossflow instability, which is lack of effective predictive tools with the transition models until now. The flows around

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prolate spheroid have been studied both experimentally [1–3] and computationally using Reynolds-averaged Navier– Stokes (RANS) with k–ε model of Tsi and Whiteney [4], detached eddy simulation (DES, [5]) based on S–A [6] model of Constantinescu et al. [7], large-eddy simulation (LES) of Wikström et al. [8] and so on. Deng and Zhuang [9] applied a novel Slightly Compressible Model (SCM) to predict the flows at a relative little AOA. It has been observed that an individual RANS model is difficult to be reliably used to accurately calculate the flows with massive separations and unsteady flows. The relatively poor performance for separation using RANS models has motivated the increasing application of large eddy simulation (LES). LES is known to resolve directly the large eddy and only model the small-scale turbulent fluctuations. LES has also been shown to provide accurate turbulent flow simulation at a fraction cost of the direct numerical simulation (DNS). It is thus a powerful tool, providing a description of large, energy-containing scales of motion that are typically dependent on geometry and boundary conditions. The small scale motion is nearly homogeneous and is easily to be modeled. However, when LES is applied to boundary layers, the computation cost of whole-domain LES does not differ significantly from that of DNS. The “large eddies” close to the solid wall are physically small in scale. LES requires additional empiricism in the treatment of the boundary layer when LES is applied for the flows at high Reynolds number. Furthermore, the subgrid scale models for the boundary layer and compressible turbulent flows are not mature and need further improvement. To overcome the deficiencies of RANS models for separation flows and of LES for boundary layer flows at high Reynolds number, an alternative modeling strategy of turbulence flows, often called as the RANS/LES hybrid methods, has been proposed recently for predicting the unsteady and geometry-dependent separated flows. Such hybrid methods combine a high-efficiency turbulence model near the wall where the main flow features are dominated by small scale turbulent fluctuations with a LES-type treatment for the large scale motion in the core flow region far away from the wall. Many kinds of RANS/LES hybrid methods have been proposed in the recent decade. The first hybrid method named as DES, which replaces d, the distance from the nearest wall in S–A model, with d˜ = min(d, CDES ), was developed by Spalart et al.; Strelets [10] proposed a DES-type hybrid method based on Menter’s k–ω shear stress transport model (SST) through introducing a length scale L t in the turbulence kinetic energy transport equation and replacing it with the L˜ t = min(L t , CDES ); Baurle et al. [11] developed a kind of RANS/LES hybrid method for supersonic cavity flows, which refers to use of flow-dependent blending functions to shift the closure from RANS in near-wall region to LES in outer part of boundary layer and in regions of

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local separation; Menter and Kuntz [12] developed a kind of delayed-DES method to reduce the grid influence of the DES-limiter on the RANS part of the boundary, Liu et al. [13] investigated some complex flows using DES based on compressible SST model. Recently, to overcome the drawback of the original DES, which pre-transfers from RANS to LES because of the local refined grids in all directions, Deck et al. [14] proposed a zonal-DES method based on S–A model and applied it to predict the supersonic base-flow and threeelement airfoil flows. So far, the DES based on S–A model is one of the most widely used RANS/LES hybrid methods. The core of the hybrid methods is combination of the RANS near the wall with LES in the separation region. Therefore, the fundamental turbulence models show significant effect on the flow features near the wall. To construct the RANS/LES hybrid methods reasonably, we generally hope that the fundamental turbulence models can be integrated to the wall and have good numerical properties. The series of two-equation k–ω model, because of their favorable numerical aspects, are naturally thought as the appropriate choice although the original Wilcox’s [15] k–ω model suffers from free-stream dependence. Menter therefore developed a hybrid two-equation model through coupling the k–ω model in the near-wall region with the k–ε model, which does not suffer from the free-stream dependence, outside of the boundary layer. This model considers the transport of principal turbulent shear stress and shows good capability for adverse pressure gradients flows. Recently, non-linearity in the eddy viscosity has been reconsidered in view of realizability [16], which is not satisfied by any linear eddy viscosity formulation based on Boussinesq’s assumption. It is found that the k–ω weakly nonlinear eddy viscosity formulations (WD+) effectively improve the performance of turbulence prediction for flows in the presence of adverse pressure gradients, especially for the flows with shock-wave/boundary-layer interaction [17,18]. The primary objective of this work is to use the RANS/LES hybrid methods including DES based on S–A, SST and WD+ models and zonal-RANS/LES based on SST and WD+ models to predict the flows around the prolate spheroid at high AOAs and high Reynolds numbers. The turbulence models are applied near the wall while the LES methods are employed for the regions of separation and away from the wall. When solving the mean flow equations with RANS/LES hybrid methods, several numerical schemes, such as the fourth-order central scheme with modified artificial viscosity and second order fully implicit lower–upper symmetricGauss–Seidel with pseudo time sub-iteration (LU-SGS-τ TS) time marching method are implemented in this study. Details of the RANS/LES hybrid methods and their switching criterion, as well as the numerical algorithms are discussed in the following sections.

Prediction of separation flows around a 6:1 prolate spheroid using RANS/LES hybrid approaches

2 Turbulence models In the present work, the series of two-equation k–ω model and one-equation S–A model are taken as the fundamental turbulence models, and Cµ is incorporated into the definition of ω. 2.1 Wilcox’s k–ω model The original k–ω model developed by Wilcox [15] is given as
 ∂ρk ∂ ∂k ρu j k − (µ + σk µt ) + ∂t ∂x j ∂x j ∗ = τi j Si j − β ρkω, (1)
 ∂ρω ∂ ∂ω ρu j ω − (µ + σω µt ) + ∂t ∂x j ∂x j

model are a zonal weighting of model coefficients and a limitation on the growth of the eddy viscosity in rapidly strained flows. The zonal model uses Wilcox’s k–ω model, which are well behaved near solid walls without low-Reynolds number corrections and the standard k–ε model (in a k–ω formulation), which are relatively insensitive to free-stream values, near boundary layer edges and in free-shear layers. The switching is achieved with a flow dependent blending function (F1 ). The SST model also modifies the eddy viscosity by forcing the turbulent shear stress to be bounded by constant times the turbulent kinetic energy inside boundary layers. This modification improves the prediction of flows with strong adverse pressure gradients and separation. The SST turbulence model equations are given as

where Pω ≡ 2γρ(Si j − ωSkk δi j /3)Si j ; Si j is the strain rate which is defined as (∂u i /∂ x j + ∂u j /∂ xi )/2. The constants σk = σω = 0.5, β ∗ = 0.09, and β = 0.075. The eddy viscosity is defined as


 ∂ ∂k ∂ρk ρu j k − (µ + σk µt ) + ∂t ∂x j ∂x j ∗ = τi j Si j − β ρkω,
 ∂ρω ∂ ∂ω ρu j ω − (µ + σω µt ) + ∂t ∂x j ∂x j ρσω2 ∂k ∂ω = Pω − βρω2 + 2(1 − F1 ) , ω ∂x j ∂x j

µt,Wilcox = ρk/ω.

where

= Pω − βρω2 ,

(2)

(3)

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(5)

(6)

The linear eddy viscosity formulation of Eq. (3) was originally designed to accurately predict the boundary-layer flows with a zero pressure gradients.

⎧

√ 4⎫  ⎨ ⎬ k 500µ 4ρσω2 k F1 = tanh min max , 2 ; ⎩ ⎭ 0.09ωd ρd ω C Dkω d 2

2.2 k–ω Wilcox–Durbin+ (WD+) model [17,18]

and the cross diffusion

The linear eddy viscosity is known to be inadequate in the prediction of adverse pressure gradients. The weakly nonlinear correction is to produce asymptotic behavior of Cµ when Si j tends to infinity. The WD+ model equations are the same as those of the original k–ω model with different eddy viscosity definition. Here, an expression of weakly nonlinear eddy viscosity formulation accounting for the 3-D effect is given as ⎡ ⎤ ρk ρa k 1 ⎦, (4) µt,WD+ = min ⎣ ;  ω 2 2 ˜ ( + S )/2 where a1 = 0.31 S˜ 2 = 2(Si j − δi j Sll /3)(S ji − δ ji Skk /3) and   is the magnitude of the vorticity defined as  = 2Wi j W ji with Wi j = (∂u i /∂ x j − ∂u j /∂ xi )/2 denoting the rate of rotation tensor. 2.3 k–ω shear-stress transport model [19] The SST model combines several desirable elements of existing two-equation models. The two major features of this

 2ρσω2 ∂k ∂ω −20 . = max ; 10 ω ∂x j ∂x j


C Dkω

The parameter d is the distance from the nearest wall. Some other constants are calculated from φ = F1 φ1 + (1 − F1 )φ2 , where the φ’s are the constants: σk1 = 0.85, σω1 = √ 0.5, β1 = 0.075, γ1 = β1 /β ∗ − σω1 κ 2 / β ∗ = 0.553; σk2 = 1.0, σω2 = 0.856, β2 = 0.0828, γ2 = β2 /β ∗ − √ σω2 κ 2 / β ∗ = 0.44 κ = 0.41. The eddy viscosity of the SST model is given as µt,SST = min

 ρk ρa k  1 ; , ω F2

(7)

 √k 500µ 2  is another max 2 ; 2 0.09ωd ρd ω blending function. From Eq. (7), the eddy viscosity of the SST model, which is written in the form of Eq. (4), also includes weakly nonlinear effect. Therefore, from some point of view, the SST model can also be thought as a weakly nonlinear turbulence model. where F2 = tanh



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2.4 S–A model In the S–A model, a transport equation is used to compute the turbulent eddy viscosity, ∂ ∂(ρ ν) ˆ + (ρu j νˆ ) = Cb1 (1 − f t2 )ρ Sˆ νˆ ∂t ∂x j  ∂ νˆ  Cb1 f t2   νˆ 2 1 ∂  ρ ρ(ν + νˆ ) − Cw1 f w − + 2 κ d σ ∂x j ∂x j   2 1 ∂ νˆ + Cb2 ρ + f t1 ρU 2 , (8) σ ∂x j where the last term of the right hand is the trip term, which is not used in this paper. All the variables and constants in Eq. (8) are given as following, µt = ρ νˆ f v1 ; χ ≡ νˆ /ν; fv2 = 1−χ /(1+χ f v1 ); g = r + Cw2 (r 6 − r ); Sˆ =  + f v2 νˆ /(κ 2 d 2 ); Cb1 = 0.1355; σ = 2/3; Cw2 = 0.3; Cv1 = 7.1; and Ct4 = 2.

3 ); f ν1 = χ 3 /(χ 3 + Cν1 f t2 = Ct3 exp(−Ct4 χ 2 ); 6 )/(g 6 +C 6 )]1/6 ; fw = g[(1+Cw3 w3 2 2 ˆ d ); r = νˆ /( Sκ Cw1 = Cb1 /κ 2 + (1 + Cb2 )/σ ; Cb2 = 0.622;

Cw3 = 2; Ct3 = 1.1;

Poor convergence of the turbulence residual is presented for some cases, especially for the reattachment. It is found that Sˆ will go negative which disturbs r and results in blinking. Following modification proposed by Spalart is introduced in this paper Sˆ = f˜v3 (χ ) + f˜v2 (χ )ˆν /(κ 2 d 2 ),

(9)

where f˜v2 (χ ) = (1 + χ /Cv2 )−3 ; f˜v3 (χ ) = (1 + χ f v1 ) × (1 − f˜v2 )/χ , χ = max(χ , 10−4 ); now Sˆ ≥ 0; and Cv2 = 5. Modified f v2 , i.e., f˜v2 , remains positive along the wall; and the f˜v3 ranges notably from 1 in the vicinity of walls. This results in a modification of natural laminar–turbulent transition of the model.

3 RANS/LES hybrid methods The motivation of RANS/LES hybrid method is to combine the best features of both LES and RANS methods. RANS models can predict attached flows very well with relatively less computation cost. LES has demonstrated its ability to compute the separated flows accurately but with high computation cost for boundary layers flows. The Reynolds-stress tensor term is defined as τi j = −u i u j ,

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(10)

where u i denotes the fluctuating velocity u i -component and the overbar represents the mean or larger-scale values. The stress tensor is modeled with the LES-type Smagorinsky subgrid model, that is, ˜ τi j = 2νSma Si j ; νSma = Cs 2 S,

(11)

where Cs is the model constant;  is the grid scale. 3.1 DES based on S–A model The DES formulation is obtained through replacing the dis˜ which is tance to the nearest wall in the S–A model, d, by d, defined as d˜ = min(d, CDES ),

(12)

where  is the largest distance between the cell center under consideration and the cell center of the neighbors. The model constant CDES is taken as 0.65 in homogeneous turbulence. In “natural” applications of DES, the streamwise and spanwise grid spacing are the similar order of the boundary layer thickness. So, the S–A model is retained throughout the boundary layer, i.e., d˜ = d. When the production and destruction terms of the model are balanced, the length scale d˜ = CDES  in the LES region yields a Smagorinsky-like eddy ˜ In the region far away from the wall, viscosity ν ∝ 2 S. where d˜ > CDES , the length scale of model becomes griddependent. The model performs as a one-equation subgrid scale model. 3.2 DES based on two-equation models To construct DES-type hybrid method based on two-equation models, some transformation is adopted for the dissipation term in the turbulence kinetic energy transport equation. After introducing length scale, this equation is written as
 ∂(ρk) ∂ ∂k ρk 3/2 ρu j k −(µ + σk µt ) = τi j Si j − + , ∂t ∂x j ∂x j Lt (13) where the length scale L t is defined as L t = k 1/2 /(β ∗ ω). The ω-equation and the eddy viscosity are the same as those of the SST or WD+ model. The DES modification replaces the length scale L t by L˜ t = min(L t , CDES ) in Eq. (13) and CDES = (1 − F1 ) k -ε + F C k –ω for DES-SST, where the constants C k –ε = CDES 1 DES DES k –ω = 0.78, for DES-WD+, C 0.61, CDES DES = 0.78;  is the grid scale defined as  = max(x, y, z). When L t < CDES , the hybrid method acts in the SST or WD+ mode; When L t > CDES , the method acts in a Smagorinsky-like LES mode. When the turbulence production is balanced with the dissipation, Pk = ρνt S˜ 2 = Dk = ρk 3/2 /L t , k = β ∗ L 2t S˜ 2 and L t = CDES . Then the eddy

Prediction of separation flows around a 6:1 prolate spheroid using RANS/LES hybrid approaches

viscosity is given as

4 Numerical methods

˜ νt = (β ∗ )3/2 (CDES )2 S˜ ∝ 2 S.

(14)

From Eq. (14), the eddy viscosity is similar as that of Smagorinsky model. When the grid is locally refined, the hybrid method will act as in a LES mode. 3.3 Zonal-RANS/LES on two-equation models The zonal hybrid of RANS/LES method was originally developed from the idea of the SST model. This method utilizes a blending function, , which is dependent on the flow and distance from the nearest solid wall, to shift the closure from RANS model near the wall to a one-equation subgird model away from walls. The particular form chosen for the turbulence kinetic energy equation is as follows: ∂  ∂k  ∂(ρk) + ρu j k − (µ + σk µt ) ∂t ∂x j ∂x j = τi j Si j − ρ[β ∗ kω + (1 − )Cd k 3/2 /],

The computations in this article are all based on a 3-D compressible solver with a modified Jameson-type cell-centered finite-volume formulation. A modified implicit LU-SGS method with sub-iteration in pseudo time is taken as the time marching method when solving the mean flow equations and the turbulence model equations. Global non-dimensional time stepping (t = 0.01) is implemented to capture the unsteady properties of the separation flows. Implicit residual smoothing is employed to accelerate the convergence. 4.1 Jameson-type central scheme Two modifications are applied in the original scheme, they are 1. Anisotropic artificial dissipation is applied to reduce the effect on the high cell aspect ratios of the viscous grids near the wall; 2. fourth-order central scheme (e.g. in ξ direction) is applied, i.e.,

(15) F¯i+1/2 = (− F¯i−1 + 7 F¯i + 7 F¯i+1 − F¯i+2 )/12,

and the eddy viscosity is given as µt = (µt,SST or WD+ ) + (1 − )(ρCs k

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(18)

where F¯i is the advective flux. 1/2

),

(16) 4.2 LU-SGS-τ TS time marching

where Cd and Cs are the model constants, and  is a filter width defined as the cube root of the local cell volume. In the limit of balancing subgrid production and dissipation, the present model returns a Smagorinsky-type subgrid eddy vis√ ˜ Here, both Cd and Cs are set cosity: νt = Cs (Cs /Cd )2 S. as 0.01. Several blending functions can be found in Ref. [20]. Such functions have the following common feature:  is equal to one near the solid wall while it becomes zero out of the boundary layer. In this paper the blending function is taken as  = tanh(ζ 4 ), ζ = max(τ1 , τ2 ),

(17)

where τ1 = 500ν/(d 2 ω), τ2 = k 1/2 /(0.09dω). The turbulence specific frequency ω is obtained from its transport equation as presented in its original form, which is formulated so that its production term does not depend on the eddy viscosity. When the blending function is unity in Eqs. (15) and (16), the RANS component dominates, whereas the LES component dominates when the blending function is zero. All constants appearing in the modified blending function and in the SST and WD+ models are the same as the original forms.

The original LU-SGS scheme, which was developed by Yoon and Jameson, is unconditionally stable and completely vectorizable. However, the explicit treatment of the viscous terms, the approximation of the flux Jacobian matrices and the linearization procedure decrease its temporal accuracy. To achieve higher temporal accuracy, a Newton-like subiteration in pseudo time [17] is implemented. The time marching methods used in the article can be written as (L + D)D −1 (D + U ) Q¯ m ¯m = −(3 Q¯ i,m j,k − 4 Q¯ i,n j,k + Q¯ i,n−1 j,k )/2 − t Ri, j,k , + + + L = −α(Ai−1, j,k + Bi, j−1,k + Ci, j,k−1 ),

U=

− α(Ai+1, j,k

+

Bi,−j+1,k

(19)

+ Ci,−j,k+1 ),

D = [1.5 + αχ (σ A + σ B + σC ) + 2αθ ] × I, where  Q¯ m = Q¯ m+1 − Q¯ m ; R¯ i,mj,k are the residuals of N–S equations; α = t/V ol i, j,k ; A± , B ± and C ± are the matrices of advective flux in three directions; σ A , σ B and σC are the spectral radius in ξ , η and ζ -directions; χ is a constant, which is set to 1.01; θ is a stable factor which can improve the robustness of the computation procedure and is defined 2(µ + µt ) , where Re denotes the Reynolds number. as ρ|∇ζ |2 · Re

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The initial values for the sub-iteration are taken as Q¯ i,0 j,k = Q¯ i,n j,k . Starting with m = 0, the sequence of iteration Q¯ i,m j,k , m = 1, 2, 3, . . . converges to Q¯ i,n+1 j,k until the right-hand unsteady residuals are approach to 0. When  Q¯ i,m j,k → 0, it means that the residuals are converged in the pseudotime. The accuracy of the solution at each physical time step is the accuracy of the discrete unsteady governing equations. That is to say, in the case of conver¯ n+1 ¯ m+1 ¯ n+1 gence, Q¯ i,m+1 j,k → Q i, j,k , and Ri, j,k → Ri, j,k , then the following equation is valid, ¯n ¯ n−1 ¯ n+1 (3 Q¯ i,n+1 j,k − 4 Q i, j,k + Q i, j,k )/2 + t Ri, j,k → 0,

(20)

¯ n+1 ¯ V,n+1 while the residuals R¯ i,n+1 j,k = (δ Fi, j,k −δ Fi, j,k /Re)/V ol i, j,k , then Eq. (20) yields a second-order fully implicit scheme in physical time, V ol i, j,k

¯n ¯ n−1 3 Q¯ i,n+1 j,k −4 Q i, j,k + Q i, j,k 2t

+δ F¯i,n+1 j,k =

1 ¯ V,n+1 . δF Re i, j,k (21)

The computing practices show that the rate of convergence with the pseudo time level is very fast, and only a few sub-iterations are needed. 4.3 Boundary conditions At far-field, the 1-D Riemann characteristic analysis is employed to construct a non-reflection boundary condition. For smooth surfaces, no-slip boundary condition is used. Symmetric boundary condition is applied for half-model. Two layers of “ghost cells” (see Fig. 1, only the 2-D case is plotted and it can be easily extended to the 3-D cases) are used to treat all kinds of boundary conditions. These “ghost” cells contain the quantities of the adjacent cells. With the residual computation, which will be performed for each block separately, all cells can be treated in the

same way, without conditional statements for them near the block boundaries. The implementation without conditional statements will largely improve the computational efficiency at computers with vectorizing capabilities. 4.4 Solution of turbulence model equations The turbulence model equations are solved, decoupled with the mean flow equations, by using the LU-SGS methods. The production terms are treated explicitly, lagged in time whereas the destruction and diffusion terms are treated implicitly (they are linearized and a term is brought to the lefthand-side of the equations). Treating the destruction terms implicitly helps increase the diagonal dominance of the lefthand-side matrices. The advective terms are discretized by using second-order upwind and the diffusive terms are discretized by using second-order central scheme. 5 Results and discussion The simplicity of the body geometry should not give rise to the assumption that the flow features are as simple as the body configuration. In contrast, the flow field is rather complex, exhibiting pressure induced-vortex sheet-separation, which is similar with the circular cylinder flows [21]. At high AOA, several separation and reattachment lines appear on the leeward surface. The flow with natural transition exhibits two main phenomena: 1. On the windward side, a large laminar region appears; the flow undergoes both streamwise and crossflow transition. 2. On the leeward side, the flow is very complex with cross flow reversals indicating multiple separation and reattachment lines. It must be mentioned that the long-axis of prolate spheroid (2a) is chosen as the characteristic length (L). Two cases based on the measurements of Wetzel and Kreplin are investigated, separately. They are Case I: M∞ = 0.135, AO A = 20◦ , Re∞ = 4.2×106 [3] and Case II: M∞ = 0.25, AO A = 30◦ , Re∞ = 6.5 × 106 [1]. The initial conditions for hybrid simulations starting from the flow fields computed by RANS methods on S–A, SST and WD+ models, separately. All the numerical results are time averaged except those by RANS models. 5.1 Case I: M∞ = 0.135, AO A = 20◦ , Re∞ = 4.2 × 106 5.1.1 Effect on grids

Fig. 1 Two-dimensional view of block structure

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The grids around the prolate spheroid are shown in Fig. 2. Two grids including 121×101×86 points (about 1.0 M cells,

Prediction of separation flows around a 6:1 prolate spheroid using RANS/LES hybrid approaches

Fig. 2 The grids around the prolate spheroid

named as Grid A) and 131×121 ×121 points (about 2.0 M cells, called as Grid B) around the spheroid in the streamwise, circumferential and normal direction are used to predict the separation flows. The far-field boundary is located at a distance of 15 and the distance of the first cell center from the wall is about 5×10−6 . At the same time, the grids of leeward side and afterbody are clustered to predict more flow characteristics. Surface pressure (C p ), skin friction (C f ) and velocity profiles using Grids A and B computed by the WD+ model at a streamwise section, where x/L = 0.77, are shown in Fig. 3. The denser Grid B does not demonstrate more advanced performance than those of Grid A. The computational C p coefficients agree with the measurements very well although they are a little weaker at the second minimum, where  ≈ 160◦ , than those of experiments [2,3]; the second minimum of C p on Grid B is even weaker than that of Grid A. It is very difficult to predict the C f , because the transition in the 3-D separation flows is very complex. In fact, the experimental results are different, which reflects the difference of flow conditions and transition. All the results by the WD+ model based on Grid A and B present the inflection point in those vicinities, although the C f coefficients are much smaller than the measurements. Comparisons on velocity profiles at four azimuthal positions ( = 90◦ , 120◦ , 150◦ and 180◦ ) at this streamwise section are also shown in Fig. 3. 1. At  = 90◦ , where it is corresponding to the onset from the windward to leeward side of the spheroid, all the computational results agree well with the measurements and little difference occurs. The calculations in u/U∞ agree with the measurements adequately and they also present a logarithmic region, although they are slightly larger than the experiments. The v/U∞ is very small and the calculations show a little larger than the measured. The numerical results predict the peak in w/U∞ and its position near y/L ≈ 3 × 10−3 very accurately.

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2. At  = 120◦ , where it is a slightly leeward of the spheroid, all the predictions agree well with the measurements. The profiles in u/U∞ and w/U∞ are much sharper than those of  = 90◦ , which means that the low-momentum fluid accumulates between the primary vortex and the spheroid surface. If the numerical methods can calculate the velocity profiles at this position accurately, it means that the numerical methods can accurately predict the boundary layer separation. 3. At  = 150◦ , where it is near the second vortex separation, all the computational results in profile of u/U∞ match measurements well. However, the prediction in v/U∞ and w/U∞ show an obvious difference from those of experiments. 4. At  = 180◦ , where it is just located on the leeward symmetric plane, all the computational results in profile of u/U∞ and w/U∞ agree with measurement well. The normal velocity is a slightly larger than that of measurement. After analyzing the flows of both computations and experiments, Grid B shows little advantage on predicting the flow structure. Therefore, Grid A is taken as the baseline grid for the following researches. 5.1.2 Effect on fundamental turbulence models The SST model is a widely used turbulence model for flows with adverse pressure gradients. As mentioned above, the SST model can be thought as a weakly nonlinear model with its eddy viscosity definition. It is applied to validate the codes and to compare the performance of the WD+ model. Figure 4 presents comparisons on the surface pressure, surface skin friction and velocity profiles at streamwise section of x/L = 0.77. From these figures, the performance of SST model looks very similar with that of WD+ model, which means that the weakly nonlinear correction can effectively predict the adverse pressure gradients and the complex 3-D separation. The distinct difference between them is the skin friction coefficients. The skin friction coefficients computed by SST model are relatively larger than those of WD+ model at the leeward of the spheroid. There is little difference on the velocity profiles between them. 5.1.3 Effect on turbulence prediction methods Figure 5 presents the numerical results at x/L = 0.77 by three turbulence prediction methods, including RANS, DES and zonal-RANS/LES based on the SST model. The results by DES show little difference from those of RANS. Zonal-RANS/LES presents more satisfactory results than those of both DES and RANS.

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Fig. 3 Comparison on pressure, skin friction and velocity profiles based on Grids A and B

The zonal-RANS/LES shows a little stronger surface C p at  ≈ 160◦ than DES and RANS and it matches the measurements very well. For the C f , the second minimum value by zonal-RANS/LES matches the measured more accurately, while the positions by RANS and DES are slightly upwind in the circumferential direction comparing with the measurements. Furthermore, the skin friction by zonal-RANS/LES looks closer to the measurements, especially at  ≈ 135◦ and 165◦ . The zonal-RANS/LES presents more satisfactory results in the velocity profiles, especially for the u/U∞ at  = 120◦ and 150◦ . The azimuthal angle of primary and second separation lines by the three turbulence prediction methods are presented in Fig. 6. From the measurements, the primary separation line, which can be clearly described, occurs very upwind in the streamwise direction (the onset occurs about

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from x/L ≈ 0.15 and the secondary separation line is from x/L = 0.6 to 0.9). The calculations of streamwise range for the primary and secondary separation lines are shorter than those of measurements, because the onset of these lines are not easily distinguished on the spheroid surface. Little difference exists among the numerical results for the primary separation line, although they predict the separation about 5◦ upwind in the circumferential direction. However, the secondary separation lines computed by RANS and DES are more upwind (about 5◦ ) in the circumferential direction than measurements. DES presents the visible secondary separation lines (from x/L = 0.66 to 0.96) much longer than that of RANS (from x/L = 0.78 to 0.96). It indicates that DES can present more developed flow structures than those by RANS. Zonal-RANS/LES presents a well matched secondary separation lines again.

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Fig. 4 Comparison on pressure, skin friction and velocity profiles by SST and WD+ models

From the comparisons among the RANS, DES and zonalRANS/LES based on SST, zonal-RANS/LES can present more satisfactory results on surface pressure, skin friction, velocity profiles and primary and secondary separation lines.

5.1.4 Flow patterns In this subsection, the flow structures are demonstrated by analyzing the results of zonal-RANS/LES methods based on SST model. The vortices, which are plotted through the spacelines and total-pressure slices, are demonstrated in Fig. 7. The totalpressure slices at five streamwise sections including the typical location at x/L = 0.77 are shown for the development and evolvement of vortices over the prolate spheroid. Surface flow patterns are shown in the Fig. 8. The flow patterns on the leeward surface are very complex. The primary

and secondary separation and reattachment lines on the afterbody are clearly visible. The primary reattachment line lies on the plane of symmetry at  = 180◦ . The crossflow streamline patterns are also presented in this figure. Two visible vortices are presented. One is the large scale, strong and anticlockwise primary vortex detached the body and the other is the little scale, weak and clockwise secondary vortex close to the wall. At the same time, the primary separation and reattachment lines (S1 and R1) and the secondary separation and reattachment lines (S2 and R2) are also plotted in the same figure. The differences of the corssflow structure characteristics computed by all the methods mentioned above are presented in Table 1. Because the primary reattachment line (R1) lies on the plane of symmetry, i.e.  = 180◦ . The values of R1 is not listed in the table. The values of S1 by all numerical methods are about from 4◦ to 5◦ upwind in the circumferential direction comparing those of experiments. The values of S2

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Fig. 5 Comparison on pressure, skin friction and velocity profiles by different turbulence modeling methods

Fig. 7 Space lines and vortices

Fig. 6 Comparison on primary and secondary separation lines by different prediction methods

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by RANS-WD+ and zonal-SST are similar with the measurements, however, the RANS-SST and DES-SST present more upwind results.

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derivation of friction line from the constant azimuthal angle on the surface, and skin friction at four streamwise sections (x/L = 0.48, 0.56, 0.64 and 0.73) are compared with the measurements [1]. The turbulence prediction methods used for this case include the DES based on the S–A, SST and WD+ models and zonal-RANS/LES based on the SST and WD+ models. From this subsection, the performances of the hybrid methods based on different fundamental turbulence models are investigated. 5.2.1 DES on different models

Fig. 8 Surface flow and crossflow (x/L = 0.77) patterns: primary reattachment at R1; primary separation at S1; secondary reattachment at R2 and secondary separation at S2 Table 1 The crossflow patterns at x/L = 0.77

Exp.

Primary vortex core

Surface flow pattern

y/L

S1/(◦ )

–

z/L

R2/(◦ )

S2/(◦ )

–

114.9

–

147.2

RANS-WD+ 0.0876

0.0290

110.0

133.8

146.7

RANS-SST

0.0865

0.0300

110.5

134.8

141.5

DES-SST

0.0865

0.0301

110.6

134.5

140.0

Zonal-SST

0.0873

0.0313

109.4

133.4

146.9

If zonal-RANS/LES based on SST are taken as the benchmark, the performance of RANS based on WD+ looks very similar with it. The RANS and DES based on SST model present similar crossflow patterns and show a relatively distinct difference from those of measurements and the benchmark. Because there are no corresponding measurements about the turbulent variables, the turbulence kinetic energy and shear stress in the circumferential direction computed by zonal-RANS/LES based on SST model are presented in Fig. 9. From these figure, the distribution of them can approximately reflect the crossflow characteristics. 5.2 Case II: M∞ = 0.25, AO A = 30◦ , Re∞ = 6.5 × 106 In this case, the AOA is larger than that of Case I. The computational pressure, wall shear stress angle, which is the

It is very difficult to accurately predict the second minimum of pressure on the leeward side. The numerical results present a very good agreement in C p with measurements (see Figs. 10, 11). DES based on S–A model presents relatively good agreement with the measurement at x/L = 0.48, however, it gives a slightly large secondary separation region at the more downstream sections. The DES-SST and DESWD+ matches the measurement very well and show small difference from each other. All the hybrid methods show good capabilities of predicting the circumferential wall shear stress angle distribution (γ ) although the computational results show some departure from those of the experiments. For the skin friction C f , almost no turbulence prediction method used in this article matches the measurements, especial on the windward side, where  is from 0◦ to 90◦ . When  is greater than 120◦ , all the hybrid methods can present satisfactory skin friction and the position of maximum and minimum of C f , although the values are a little large. The SA-DES can present a little more suited results, where  ranges from 0◦ to 120◦ , than those of DES based on SST and WD+ models. The main reason is that the modified S–A model in this paper includes the natural transition criterion, while no transition models are implemented in the SST and WD+ models. In a sense, the performance of RANS/LES hybrid methods depends on the fundamental turbulence models. Therefore, good performance models should be chosen as the fundamental turbulence model when constructing the RANS/LES hybrid methods. 5.2.2 Zonal-RANS/LES on different models In this case, the zonal-RANS/LES method based on the SST and WD+ models cannot present more satisfactory results than those of DES. From the surface pressure C p , shear stress angle and skin friction C f , the zonal-RANS/LES based on SST matches the measurements better than those of hybrid method based on WD+ model. Both of them cannot well predict the skin

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380 Fig. 9 The turbulence kinetic 2 ) and shear stress energy (k/U∞ 2 ) at x/L = 0.77 (v  w  /U∞

Fig. 10 Comparisons on surface flow by DES

Fig. 11 Comparisons on surface flow by zonal-RANS/LES

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Fig. 12 The history of lift with the hybrid methods

friction because no transition model is introduced in the fundamental turbulence models. To predict the complex separation accurately, more advanced fundamental turbulence models including the transition model, especially crossflow transition model and strong nonlinear effect on adverse pressure gradients should be implemented. 5.2.3 Unsteady features of hybrid methods The convergence histories of the lift by all the RANS/LES hybrid methods used for this case are shown in Fig. 12. The unsteady lift fluctuations are observed and the irregular turbulent properties are presented.

6 Conclusions The performances of two RANS/LES hybrid approaches, the DES-type based on S–A, SST and WD+ models and the zonal-RANS/LES based on SST and WD+ models, are presented for the high Reynolds number flows past a 6 : 1 prolate spheroid at high AOA. Unsteady, large-scale motions are predicted successfully. In this study, the grids (including 1.0 M and 2.0 M cells) and fundamental turbulence models (including SST and WD+ models) effects on the flow structures are firstly investigated. One-equation S–A model is also applied to construct the DES method. Both two-equation k–ω models with weakly nonlinear eddy viscosity definition can accurately predict the flow structure, but not the windward skin friction which requires the crossflow transition model. The S–A model can partly predict the natural transition on the windward side of the spheroid, while the transition model is too simple to match well with the measurements. Compared with the measurements, zonal-RANS/LES method presents more satisfactory results than RANS and DES for Case I. The RANS and DES show little difference for the surface flow structures.

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In Case II, the DES based on S–A model, which accounts for the effect of natural transition, shows slightly better skin friction from the windward to leeward of spheroid surface. The hybrid methods based on the weakly nonlinear twoequation SST and WD+ models demonstrate poorly matched results, especially on the windward surface. From the original intention of RANS/LES hybrid method, the model acts as the RANS near the wall. It means that the intrinsic characteristics of RANS are preserved in this region. Therefore, the performance of the hybrid methods for the flow features near the wall mainly depends on the turbulence models. More powerful turbulence models should be chosen as the fundamental models when constructing the hybrid RANS/LES methods.

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