1 Introduction

Owing to the obvious importance in a wide range of engineering applications,massively separated flow and shock/turbulent-boundary-layer interaction in the flow-field over the bluff bodies have received a great deal of attention. Numerical techniques can give insight in the understanding of turbulent flow in the geometries of engineering interest ^[1] . Nowadays,there are four prevalent methodologies to simulate the turbulent flow,i.e.,the Reynolds averaged NavierStokes equations (RANS),the direct numerical simulation (DNS),the large eddy simulation (LES),and the RANS/LES hybrid methods.

Extensive approaches applied in engineering problems are mainly based on the RANS method. Unfortunately,since all the turbulent scales must be modeled via turbulence models, the RANS method usually fails to provide the reliable simulations in the large separated flow regions. The DNS can resolve all scales of turbulent motion accurately as long as the grid is sufficiently fine,that is,the grid number is proportional to 9/4 power of the Reynolds number ^[1] . Thus,the present computer resources have limited its application only to low Renolds number flows with simple configuration. The LES is an approach intermediate between the DNS and the RANS. In the LES method,the turbulent scales larger than the filter length are resolved directly,and the scales smaller than the filter length are modeled using the sub-grid scale model ^[2] . Comparing to the DNS method,the LES technique can save some computer resources,however,it also requires a number of grid points to simulate the turbulent boundary layer. For the complex engineering problems,e.g.,the airborne or ground vehicle applications, a pure LES will use over 10 11 grid points,which is estimated to be possible 30 years later ^[3] .

The RANS can predict well boundary layers and their separation,but not large separation regions. Recently,many researchers combine the RANS and the LES to acquire a RANS/LES hybrid method. For the RANS/LES hybrid method,observers are hopeful that the RANS operates in the boundary layer region and the LES acts in the massively separated flow region. Spalart et al. ^[4] proposed a RANS/LES hybrid method,i.e.,the detached-eddy simulation (DES) method,which is derived from the Spalart-Allmaras (SA) one-equation turbulence model ^[5] . In the DES approach,the RANS and LES computational zones are divided directly via a gridspacing limiter,which causes that computations are sensitized to the grid resolution. The space region between the two areas can result in a gray area problem ^[3] ,which has motivated many researches on the improvement of the original DES version,such as the delayed-detached eddy simulation (DDES) approach ^[6] .

Another popular RANS/LES hybrid method is the scale-adaptive simulation (SAS) proposed by Menter et al. ^[7] . In the SAS method,the RANS switches to the LES through the resolved small scale,but not the grid spacing. The SAS has a pure RANS nature but achieves the LES behavior unlike any traditional RANS model. For saving computer resources,Xu and Yan ^[8] proposed an SAS model based on the SA one-equation turbulence model,named as the XY-SAS model. For the high wave number damping,a limiter is also used in the original XY-SAS model. Gao et al. ^[9] has demonstrated that the XY-SAS model without the limiter can also give a reliable prediction of the separated flow such as the incompressible flow around a square cylinder. In this study,a modified SAS model based on the XY-SAS model is used. The purpose is to estimate the capabilities of this model on predicting the massively separated flow and shock/turbulent-boundary-layer interaction. 2 Mathematical formulation and numerical methods 2.1 Governing equations and boundary conditions

The governing equations used in our code are the three-dimensional Favre-averaged compressible Navier-Stokes equations,which can be written in the generalized coordinates (ξ,η,ζ) as

where Q = [ρ,ρu,ρv,ρw,E] is the conservative flow variables with the density ρ,the velocity component u_i,the pressure p,and the specific total energy E,J represents the transformation Jacobian,F,G,and H are the inviscid flux terms,and Fv ,Gv,and Hv are the viscous flux terms. To non-dimensionalize the governing equations,we use the free-stream variables including the density ρ_∞,the temperature T_∞,the velocity U_∞,and the diameter of the cylinder D as the characteristic scales. For neatness,the detailed formulations of these terms are not shown here,and readers can find in our previous paper^[10].

In this work,the initial condition is set as the free-stream quantities. No-slip and adiabatic conditions are applied to the body surface. Far-field boundary conditions are treated by the local one-dimensional Riemann invariants. For the circular cylinder,the periodic condition is used in the spanwise direction. 2.2 DES method

The DES methodology used in this study is the original DES version introduced by Spalart et al. ^[4] ,which is derived from the SA one-equation model ^[5] . The model is provided with a destruction term for the eddy viscosity that depends on the distance to the nearest solid wall d. This term adjusts the eddy viscosity to scale with the local deformation rate producing an eddy viscosity given by ~ d² . Spalart et al. ^[4] proposed to replace d to the closest wall with defined by

where ∆ represents a characteristic mesh length and is defined as the largest of grid spacing in all three directions,i.e.,∆ = max(∆x,∆y,∆z ),and the constant C_DES is taken as 0.65 from a calibration of the model for isotropic turbulence ^[11] . 2.3 DDES method

To reduce grid dependency of the RANS computation on the attached boundary layer, Spalart et al. ^[6] developed the DDES method. The DDES version in our code is also based on the SA one-equation model. In the DDES formulation,a new length scale can be implemented through a function ƒ_a as follows:

The function ƒ_a is defined as where µ represents the molecular viscosity,µ_t is the turbulent eddy viscosity,and κ is the von Karman constant. 2.4 SAS method

The SAS method allows the simulation of unsteady turbulent flows without limitations of most unsteady RANS models. The first SAS version developed by Menter et al. ^[7] is derived from KE1E one-equation model. Subsequently,Menter and Egorov ^[12] developed the SAS technique using the shear-stress transport (SST) two-equation model. To obtain the SAS,the von Karman length scale L_vk is introduced into the turbulence scale equation. A complete recipe for L_vk in the SAS model can be written as

where α,β,ζ₂,c_µ,and CS are the model constants, is a scalar invariant of the strain rate tensor represents the magnitude of the Laplacian of the velocity.

Xu and Yan ^[8] developed a new SAS method based on the SA one-equation model,named as the XY-SAS model. Lvk in the XY-SAS model represents the scale of the vortices statistically, which is calculated through

where is the magnitude of vorticity with For the high wave number damping,a DES limiter is also added into the original XY-SAS model ^[8,13] Gao et al. ^[9] suggested that the DES limiter C_DES∆ in the XY-SAS model could be removed.

Both S and |∇² u| denote the strains,and Ω represents the rotation. Thus,it can be seen that the mathematical formulas of L_vk in Eqs. (5) and (6) only involve alternative effect of strain or rotation. Considering these two effects,a modified von Karmann length Lvk is used in the present SAS model (called as the SAS-SA),

In particular,the DES limiter is removed in the SAS-SA model,i.e., = min

It is noted that Lvk in Eq. (8) also can represent the scale of the vortices statistically. In addition,a strain or compressible effect has been considered in this SAS-SA model due to the addition of |∇² u|. 2.5 Numerical method

We briefly summarize the numerical method here,and the reader can refer to the given references ^[2,10] for details. The governing equations are numerically solved by the finite-volume method. The convective terms are discretized with the second-order central/upwind hybrid scheme,and the diffusive terms are discretized with the second-order central scheme. The temporal integration is performed using an implicit approximate-factorization method with sub-iterations to ensure the second-order accuracy. This numerical strategy has been applied with success to a wide range of turbulent flows such as compressible flow past an aerofoil ^[14] and transonic flows over bluff cylinders ^[2,15,16] . We have carefully examined the numerical approach used in this study and verified that the calculated results are reliable. 3 Results and discussion

Massively separated flow and shock/turbulent-boundary-layer interaction often occur in the engineering field. A reliable and efficient numerical technique is needed in the practice applications. To estimate the reliability of the present numerical methodologies,two typical cases are calculated,i.e.,the subcritical flow past a circular cylinder and the transonic flow over a hemisphere cylinder. To improve the computational efficiency,the present code is equipped with a multi-block domain decomposition feature to facilitate parallel processing in a distributed computing environment. 3.1 Subcritical flow past circular cylinder

The subcritical flow past a circular cylinder at Reynolds number 3 900 is a typical case for computational validations due to its abundant experimental data ^[17,18,19] . The O-type grid is used with a far-field boundary at 31D away from the cylinder in the cross-section,and the spanwise length is 4D. After our test ^[20] ,the grid number 97 × 97 × 41 in the radial,azimuthal,and spanwise directions is enough for this physical problem. The time step is chosen as 0.01D/U_∞. As a check,the average value of ∆_z⁺ corresponding to the first grid point away from the body is less than 1 for all the simulations,which indicates that the grid size nearest the wall is sufficient.

To assess quantitatively predictions of the four simulations,Table 1 shows the time-averaged drag coefficient C_D,the recirculation length L_r/D,and the Strouhal number St as well as their comparisons with some typical experimental data ^[17] . We firstly compare with the values of C_D and note that all the computational results are in good agreement with the experimental data. Comparisons of the Strouhal number St of vortex shedding show that all simulations also agree well with the experiment. Apparently,the quantities C_D and St are not very sensitive to the simulation methods. Comparing with the values of the recirculation length L_r/D,it is identified that L_r/D computed by the DDES is larger than the experimental data. The L_r/D values obtained by the two SAS models are all in the reliable range,however,the result calculated by the XY-SAS model is lower than that computed by the present SAS method, which indicates that there is a slight difference between the flow fields predicted by the two SAS models. To clearly present the difference,some profiles are analyzed,including the pressure coefficient C_p ,the streamwise velocity u,and the Reynolds stress.

Distributions of the mean pressure coefficient _p on the cylinder surface obtained by computations and experiments ^[17] at Re = 3 000 and 8 000 are shown in Fig. 1. The _p curves obtained by the DES and the DDES agree well with the experimental data at Re = 3 000. However, the results predicted by the two SAS approaches show good agreement with the experimental data at Re = 8 000. Comparing to the two DES methods,the _p values are underestimated slightly by the two SAS models. Figure 2 shows the mean streamwise velocity along the wake centerline. The two DES methods overestimate the length of recirculation zone. The minimum velocity in the recirculation region computed by the XY-SAS model is slightly lower than that obtained by the present SAS model. Figure 3 shows the Reynolds stress profiles in the near wake at x/D = 1.54. All the Reynolds stresses are overpredicted by the two DES methods. The shear Reynolds stress and the streamwise Reynolds stress predicted by the XY-SAS model are a little better than those simulated by the present SAS model. However,for the transverse Reynolds stress the present SAS model can give a better prediction than the XY-SAS model.

3.2 Transonic flow over hemisphere cylinder

To test the proposed methodology applied on prediction of the shock/turbulent-boundarylayer interaction,the transonic flow over a hemisphere cylinder at zero incidence is investigated. Based on the experiment ^[21] ,the free-stream Mach number M_∞ is chosen as 0.85,and the Reynolds number based on the cylinder diameter is chosen as 4 × 10⁵ . Figure 4 shows the geometry and computational mesh of the hemisphere cylinder. The O-H-type grid is used with clustered distributions in the vicinity of the wall. The grid number is 149 × 149 × 65 in the streamwise,normal,and azimuthal directions,respectively,with the outer boundary 30D. The time step is 0.002D/U_∞. Corresponding to the aforementioned physical problem,the mean ∆_z⁺ of the nearest gird size away from the wall is also less than 1 for all the simulations. For the transonic flow around a hemisphere cylinder,shock and flow separation coexist between M_∞ = 0.7 and 0.85 ^[21] . The most pronounced separation region occurs at M_∞ = 0.85 due to the shock/turbulent-boundary-layer interaction. Notably,this shock-induced turbulent separation is of less scope. Thus,this problem can be considered as a formidable test-case for the turbulent simulation.

The streamwise distributions of the mean pressure coefficient _p on the body surface are shown in Fig. 5. For the predictions of the two DES methods,poor agreement with the experimental data is exhibited near the separated region (0.4 < x/D < 2) as expected,while good agreement downstream the separation region (x/D > 2). Both the two SAS models can predict well the separation point,however,compared with the XY-SAS result,better agreement with experiment is obtained by the present SAS computation. In the transonic flow over a hemisphere cylinder,the boundary-layer separation is induced by the shock wave ^[21] . Due to the existence of gray area in the two DES models,the flow field including the shock position cannot be predicted accurately,which may be why the two DES methods cannot give a reasonable prediction of the shock/turbulent-boundary-layer interaction. For the further estimation, Fig. 6 shows the streamwise velocity profiles at eight different positions. There are no discrepancies upstream the separation region x/D < 0.4 for all simulations. Corresponding to C_p ,the velocity profiles predicted by both the SAS methods show better agreement with experiment than those obtained by the DES and the DDES near the separation region 0.4 < x/D < 2. Moreover,a little better prediction is obtained by the present SAS model than the XY-SAS model at x/D = 1,where near the core region of shock/turbulent-boundary-layer interaction. In summary,we can see that the present SAS model can give an improvement for the prediction of shock/turbulent-boundary-layer interaction.

4 Conclusions

A modified SAS technique based on the SA one-equation model is proposed. Two typical cases,i.e.,the subcritical flow past a circular cylinder and the transonic flow over a hemisphere cylinder,are investigated to estimate the proposed SAS approach. For comparison,the same cases are computed by the DES,the DDES,and the XY-SAS methods. For the subcritical flow past a circular cylinder,the wall pressure coefficients are predicted well for all the simulations, while the recirculation length and the Reynolds stress are overpredicted and underpredicted for both the DES and DDES methods. The transverse Reynolds stress predicted by the present SAS model is better than that calculated by the XY-SAS model. For the transonic flow over a hemisphere cylinder,poor agreement with the experiment is obtained by the DES and the DDES approaches near the separation region. The wall pressure computed by the present SAS technique shows better agreement with the experiment than the XY-SAS method. For the velocity profiles,the present SAS method can give better prediction than the XY-SAS model near the core region of shock/turbulent-boundary-layer interaction. In conclusion,all comparisons indicate that the present SAS method can give better prediction of massively separated flow and shock/turbulent-boundary-layer interaction than the DES and the DDES approaches. Furthermore,comparing with the XY-SAS model,the present SAS model also can give some improvements.