A Nonlinear Sub-grid Scale Model for Compressible Turbulent Flow

Transcription

1 hinese Journal of Aeronautics 0(007) hinese Journal of Aeronautics A Nonlinear Sub-grid Scale Model for ompressible Turbulent Flow Li Bin, Wu Songping School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing , hina Received 0 March 007; accepted 9 October 007 Abstract The governing equations for large eddy simulation (LES) are obtained by filtering the Navier-Stokes (N-S) equations with standard (non-favre filtering) spatial filter function. The filtered scale stress due to the standard filtering is then reconstructed by using the Taylor series expansion. The loss of information due to truncating the expansion up to the first derivative term is modeled by a dynamic nonlinear model (DNM), which is free from any empirical constant and wall damping function. The DNM avoids the singularity of the model and shows good local stability. Unlike the conventional dynamic Smagorinsky model (DSM), the DNM does not require the plane averaging and reduces the computational cost. The turbulent flow over a double ellipsoid for Reynolds number of and Mach number of 8.0 is simulated numerically to validate the proposed approach. The results are compared with experiment data, as well as the data of Reynolds averaged numerical simulation (RANS). Keywords: large eddy simulation; subgrid scale; compressible flow; nonlinear 1 Introduction Turbulence occurs frequently in engineering, and it has been studied for a long time. Even after many decades of intensive research, turbulent modeling remains one of the most difficult problems in computational fluid dynamics (FD). The widely used Reynolds averaged numerical simulation (RANS) approaches in which all scales are modeled are very efficient for certain classes of turbulent flows. RANS approaches fail to predict instantaneous information due to their results represent the mean properties of flow field. The direct numerical simulation (DNS) is the most straightforward method. The spatial and temporal scales of motion are solved without any model, which lead to enormous computational cost. The number of grid points is 9 4 about Re, which is computationally expensive. orresponding author. Tel.: address: baal@mse.buaa.edu.cn DNS is restricted to flows with low Reynolds number, which prohibits from practical applications. Large eddy simulation (LES) becomes a very promising technique for the prediction of turbulent flows because of large improvements in computing hardware and advances in both computational methods and sub-grid scale (SGS) modeling. In a LES, the large scales of 3-D unsteady turbulent motion can be calculated directly, but the eddy smaller than the grid size has to be modeled by SGS model. Since the small eddy can be considered universal theoretically, it is easier to construct a general model, and LES is believed to be more accurate than RANS. In recent applications, the equations of LES for compressible turbulent flow are founded in the framework of the Favre filtering. This method avoids some complexity, but gives rise to more difficulties, and cannot be compared with experimental data or DNS data, which are not Favre filtered. On

2 496 Li Bin et al. / hinese Journal of Aeronautics 0(007) the other hand, by using Favre filtering of the governing equations, some information of density fluctuation is lost. Some methods are developed to close the SGS stress in the framework of standard filtering. Adams proposed a model based on approximate deconvolution of the filtered field [1]. The deconvolution procedure is supplemented by the relaxation regularization. The eddy viscosity hypothesis used to construct the SGS model is the same as RANS []. Nelson [3] and Boersma [4] supposed the dynamic model respectively to simulate the mixed layers and jet flow. A dynamic equation model is developed [5] to obtain a more accurate correlation of density fluctuation in aero-optics. The incompressible tensor-diffusive model (TDM) based on the Taylor series expansion is extended to compressible flow [6]. None of the SGS stresses obtained by the standard filtering is neglected in Ref.[7]. All SGS stresses are modeled using eddy viscosity hypothesis [7]. This paper describes a LES method which uses the standard compressible LES filtered variables. Like the compressible TDM, only the first derivative term of Taylor series expansion is reserved, and the rest terms are dropped. The loss of information due to this truncation is modeled by a dynamic nonlinear model (DNM) [8]. DNM exhibits instantaneous local stability without using the conventional plane averaging technique or any arbitrary bounds for restricting the modeling coefficients. To validate the proposed model for turbulent simulation, LES of fully developed turbulent flow over a double ellipsoid is performed for a Reynolds number of (based on the length of the double ellipsoid) and Mach number of 8.0. omputational Method.1 Governing equations Using over-bar to designate the resolved part of a variable, the filtering quantity is defined as f( x) G( xx; ) f( x)dx (1) where f is arbitrary parameter of a turbulent field, G a filter function and the filter width. The LES equations are obtained by applying a filter on the compressible and unsteady Navier- Stokes equations. The resulting equations can be expressed in conservative form using Einstein summation notation [6] as follows ui A () t xi xi u u i i uj p ij ij (3) t x x x x j i j j E Euj puj ij ui q B B t x x x x x x 1 j j j j j j (4) where the independent variables t and x i represent time and spatial coordinate, respectively. The component of the velocity is denoted by u i. p is the static pressure, and E is the total energy of the filtered variables, defined as 1 E vt uk uk (5) where T is the static temperature, v the constant volume specific heat. The subgrid stress terms are A u u u u uu (6) i i, ij i j i j B1 E uj Euj, B puj puj (7) The components of the stress tensor and the heat flux vector can be expressed respectively as u u i j uk u u i j uk ij ( ij ) ( ij ) x j xi 3 xk xj xi 3 xk (8) T qj k( T) (9) x The Sutherland s law for the molecular viscosity coefficient is also employed. Finally the LES equation of state is writen as p RT (10). Subgrid scale model The complete but infinite expansion is obtained using Taylor series expansions [8-9] 1 ui ui ui 1 xl xl 1 1 ui (11)! 1 x x x x l m l m j

3 Li Bin et al. / hinese Journal of Aeronautics 0(007) Only the first derivative term is reserved. Similarly [6], u u uu i j i j uj ui uj u i ui uj 1 xm xm xm xm xm x m B B E u Eu pu pu 1 1 puj puj uk uk ui ukukui 1 u u u j j j j 1 p ui pui p ui 1 x x 1 1 p ui! 1 x x x x k k i k k i uuu l l m l m l (1) (13) (14) uk ui uk ui uk u k uk uk ui 1 xm xm xm xm xm x m ui uk u k uk uk ui uk ui uk 1 xm xm xm xm xm x m (15) Let superscript stand for the traceless part of a tensor, it has def 1 ij ij kk ij (16) 3 uj ui uj u i Oij ui uj 1 xm xm xm xm xm x m (17) Due to the truncation of the reconstruction series, the additional dissipation corresponded to the loss of information is required. It can be provided by an added dissipative term. The DNM term is added, which is expressed as a quadratic tensorial polynomial of resolved strain tensor s ij : ij Oij 1 IIij sij s sij (18) 3 in which the strain tensor is defined as 1 ui u j sij (19) x j x i and II is the second invariants of s ij, namely sij (0) omparing with the conventional Smagorinsky model [10] ij s s sij (1) one can observed that the nonlinear term coincides with conventional Smagorinsky model when 1 =0 and = s. From Eq.(18), it has O P ij Q u u uu () 1 ij ij ij i j i j where Pij ij sij, Qij s sij. 3 Parameters 1 and are determined by dynamic method. At the test filter grid, the SGS stress is T O P Q u u uu 1 ij ij ij ij i j i j (3) With the Germano identity [11] Lij Tij ij ui uj ui uj (4) The Leonard term L ij is solved from resolved scale. Inserting Eqs.()-(3) into Eq.(4) gives Lij Oij O ij 1 Pij Pij Q ij Q ij Oij O ij 1Mij Nij (5) Let E ij and F ij denote the local error tensor and the error density function respectively Eij L' ij 1Mij Nij (6) F EE (7) ij ij ij with L' ij Lij Oij Oij. Following the least square method, it has Fij 0 L' ijmij 1M ijmij Mij Nij 0 1 (8) Fij 0 L' ij Nij 1M ij Nij Nij Nij 0 viz. MijMij Mij N ij 1 L' ijm ij (9) M ij Nij Nij N ij L' ij N ij Due to the fact that the constitutive relation of the nonlinear term for the model presented here is an extension to a quadratic form from a linear one for the conventional dynamic model, the conventional dynamic Smagorinsky model of Lilly [1] is the first-order approximation of the nonlinear term for present model. Eq.(9) gives

4 498 Li Bin et al. / hinese Journal of Aeronautics 0(007) when 1 L' ijmij Mij Nij 1 D L' ij Nij Nij N ij (30) 1 MijMij L' ijmij D Mij Nij L' ij Nij ij ij Mij Nij M M D 0 M N N N ij ij ij ij In our numerical practice, it has when L' ijmij L' ij Nij 1, MijMij Nij Nij 6 D 10 with a small positive value. Ref.[10] shows that the isotropic component kk of the SGS stress tensor can be neglected without introducing appreciable errors. This assumption is used in present numerical test. The truncated terms denoted q j in energy equation is modeled using a simple gradient diffusion model [13] T qj cp s Pr x Pr t =0.9 is the SGS turbulent Prantdl number..3 Numerical scheme t j (31) The results of LES is sensitive with the numerical scheme [14]. The scheme used in LES must be of low-dissipation and high accuracy. For LES of hypersonic turbulent flow, the lack of shock capturing capabilities may result that the numerical noise will be generated near the boundary layer edge [15]. As the simulation march in time, the numerical noise penetrates the boundary layer and the solution is no longer physical. Thus, a shock capturing technique is necessary for performing DNS and LES of hypersonic and supersonic boundary layers. AUSM family schemes developed by Liou [16] have been used successfully in LES [17-1]. Advection upstream splitting method (AUSMDV) scheme [] is used for numerical simulation because of its low dissipation and the capabilities of shok capturing. The viscous terms are discreted by second order center scheme. The temporal integration is performed by using a dual time lower-upper symmetric Gauss-Seidel (LUSGS) method. 3 Numerical Results To validate the nonlinear modeling method for compressible turbulent simulations, the numerical test of hypersonic turbulent flow over a double ellipsoid is performed. The geometrical and flow parameters chosen for the test case are the same as given in Ref.[3], i.e., the free stream Mach number Ma = 8, tempeature T = K, density = 0.04 kg/m 3, and Reynolds number Re = The total grid points are 6.4 millions and the computaion is parallelized using message passing interface (MPI). The no-slip and adiabatic conditions are employed at the wall. For the exit boundary and farfield boundary, the flow parameters can be evaluated following the Riemann invariants. Fig.1 and Fig. give the upper and lower surface pressure distribution coefficients respectively on the symmetric planes of the double ellipsoid. The results obtained from the present nonlinear model, are compared with the results of RANS model [4] and the experiment data [3]. The pressure distribution coefficients obtained by LES and RANS agree well except at the vicinity of the pressure peak. The Fig.1 Pressure distribution coefficient at upper surface. Fig. Pressure distribution coefficient at lower surface.

5 Li Bin et al. / hinese Journal of Aeronautics 0(007) discrepancy before the second peak shows that the shock/boundary layer interaction is captured by nonlinear model, but RANS failed. Both pressure distribution coefficients are almost identical at lower surface. The contours of mean Mach number and density on the symmetric plane are plotted respectively on Fig.3 and Fig.4. The bow shock-wave and second shock-wave are captured distinctly. This means that the average flow fields computed by the nonlinear model are reasonable. The turbulent energy distribution computed by LES and RANS given in Fig.5 and Fig.6 respectively is similar. The profiles of the mean velocities at location x=0 and x=50 are shown in Fig.7. u + is nondimensional ve- Fig.6 Turbulent energy distribution on the symmetric plane from Ref.[3]. Fig.7 profile of the mean Van Driest equivalent velocity. Fig.3 Mean Mach number distribution on the symmetric plane. locity. The van Driest equivalent velocity is used to compute u +. The mean velocity distribution agrees well with the theoretical value. The local contour lines of instantaneous pressure and stream line near the second shock-wave are given in Fig.8 and Fig.9 respectively. Fig.8 shows that the instantaneous turbulent flow field is very complex. The shock/ boundary layer interaction induces a series of waves. Fig.9 shows the shock/boundary layer interaction induces separation and produces a series of vortexes. Fig.8 and Fig.9 show that the complex instantaneous turbulent flow field can be resoluted by our nonlinear model. Fig.4 Mean density distribution on the symmetric plane. Fig.8 Instantaneous pressure Fig.9 Instantaneous streamdistribution. line distribution. 4 onclusions Fig.5 Turbulent energy distribution on the symmetric plane. A nonlinear SGS model in the framework of standard filtering is proposed. The nonlinear model

6 500 Li Bin et al. / hinese Journal of Aeronautics 0(007) appears to be more robust than the conventional dynamic Smagorinsky model in the numerical simulation. The nonlinear model can be applied locally and simulation remains stable at each time step without the need for the plane averaging technique. This avoids the potential instability or singularity. The numerical tests conducted for hypersonic turbulent flows over a double ellipsoid show that the average flow field computed by the present nonlinear model is almost identical to the results computed by the RANS equations. Furthermore the more accurate instantaneous flow field is solved by using nonlinear model. The shock/boundary layer interaction is captured by the nonlinear SGS model, but the RANS model failed. As a result, the more accurate pressure distribution is obtained by using the nonlinear SGS model. The comparisons with RANS and experiment show that the present SGS model is available to predict the hypersonic turbulent flow. Acknowledgments The authors wish to thank the National Laboratory of omputational Fluid Dynamic for supplying computational devices. References [1] Adams N AStolz S. An approximate deconvolution procedure for large eddy simulation. Phys Fluids 1999; II: [] Freund J B, Moin P, Lele S K. ompressibility effects in a turbulent annular mixing layer. Flow Physics & omputation Division, Dept of Mechanical Engineering, Stanford University, Tech Rep TF-7, [3] Nelson, Menon S. Unsteady simulation of compressible spatial mixing layers. AIAA , [4] Boersma B J, Lele S K. Large eddy simulation of compressible turbulent jets. Stanford University: enter for Turbulence Research, [5] Han Z P. The research for infrared images and aero-optical transmitting effects of hypersonic aircrafts by using numerical simulations. PhD thesis, hina Aerospace Science and Industry orporation, 003. [in hinese] [6] Li B, Wu S P. Large eddy simulation for compressible turbulent flow based on non-favre filtering. Journal of Beijing University of Aeronautics and Astronautics (in press). [in hinese] [7] Zhou Y L. The research for turbulent model of compressible large eddy simulation. Postdoctoral thesis, Beijing University of Aeronautics and Astronautics, 003. [in hinese] [8] Leonard A. Large eddy simulation of chaotic convection and beyonds. AIAA Paper , [9] Bedford, K W, Yeo W K. onjunctive filtering procedures in surface water flow and transport. In: Galperin B, Orszag S A, editors. Large Eddy Simulation of omplex Engineering and Geophysical Flows. London: ambridge University Press, 1993: [10] Erlebacher G, Hussaini Y M, Speziale G, et al. Toward the large eddy simulation of compressible turbulent flows. Journal of Fluid Mechanics 199; 38: [11] Germano M, Piomelli U, Moin P, et al. A dynamic subgrid scale eddy viscosity model. Phys Fluids 1991; 3(7): [1] Lilly D K. A proposed modification of the Germano subgrid scale closure method. Physics of Fluids A 199; 4: [13] Kosovic B. Subgrid scale modeling for the large eddy simulation of high Reynolds number boundary layer. J Fluid Mech 1997; 336: [14] Sagaut P. Large eddy simulation for incompressible flows: an introduction. nd ed. Berlin: Springer, 00. [15] Martin M P. Shock-capturing in LES of high-speed flows. Stanford University: enter for Turbulence Research, 000. [16] Liu M S. Ten years in the making AUSM-family. AIAA Paper , 001. [17] Meinke M, Rister T T, Rutten F, et al. Simulation of intermal and free turbulent flows. Lecture Notes in Physics 1998; 515: [18] Louedin O, Billet G. Study of turbulence modeling for transient high speed flow with D and 3D large eddy simulation based on temporal filter. In: Liu, Liu Z, editors. First AFOSR international conference on DNS/LES, Louisiana, [19] Dailey L D, Pletcher R H. Evaluation of a second-order accruate compressible finite volume formulation for the large eddy simulation of turbulent flows In: Liu, Liu Z, editors. First AFOSR International onference on DNS/LES, Louisiana, [0] Mary I, Sagaut P. Large eddy simulation of flow around an airfoil near stall. AIAA J 00; 40(6): [1] Mary I, Sagaut P. Large eddy simulation of flow around an airfoil. AIAA Paper , 001. [] Wada Y, Liou M S. An accurate and robust flux splitting scheme for shock and contact discontinuties. SIAM J Sci omp 1997; 18: [3] Li S X, hen Y K, Li Y L. Hypersonic flow over double ellipsoid: experimental investigation. AIAA-97-87,1997. [4] Liu J Y, Li H, A two-equation model for high speed compressible turbulence. East West High Speed Flow Field onference 005, 005: