A priori and a posteriori evaluations of sub-grid scale models for the Burgers’ equation

Abstract In large eddy simulations (LES) of turbulent flows, large scale motions are resolved by the numerical simulation while the effect of the small scale motions is represented as sub-grid scale (SGS) stresses computed with SGS models. In the present study, we perform a priori and a posteriori evaluations of five SGS models for the one dimensional Burgers’ equation with the high-order flux reconstruction (FR) or correction procedure via reconstruction (CPR) approach. It is shown that all models, except the scale similarity model (SSM) and the mixed model (MM), demonstrate very little correlation with the DNS results. The stability of the SSM is investigated. The effects of numerical dissipation and the models’ influence on LES are discussed. Based on the present study, we advocate the use of implicit LES (ILES) in LES in the context of discontinuous high-order methods, or any numerical methods with built-in numerical dissipation.

[1]  C. Meneveau,et al.  On the properties of similarity subgrid-scale models as deduced from measurements in a turbulent jet , 1994, Journal of Fluid Mechanics.

[2]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[3]  P. Durbin,et al.  Statistical Theory and Modeling for Turbulent Flows , 2001 .

[4]  P. Moin,et al.  A dynamic subgrid‐scale eddy viscosity model , 1990 .

[5]  H. T. Huynh,et al.  A Flux Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin Methods , 2007 .

[6]  Christer Fureby,et al.  Large-Eddy Simulation: Current Capabilities, Recommended Practices, and Future Research , 2009 .

[7]  M. Rogers,et al.  A Numerical Study of Self-Similarity in a Turbulent Plane Wake Using Large-Eddy Simulation , 1997 .

[8]  J. Ferziger,et al.  Improved subgrid-scale models for large-eddy simulation , 1980 .

[9]  P. Moin,et al.  A dynamic localization model for large-eddy simulation of turbulent flows , 1995, Journal of Fluid Mechanics.

[10]  Caskey,et al.  GENERAL CIRCULATION EXPERIMENTS WITH THE PRIMITIVE EQUATIONS I . THE BASIC EXPERIMENT , 1962 .

[11]  Harish Gopalan,et al.  A unified RANS-LES model: Computational development, accuracy and cost , 2013, J. Comput. Phys..

[12]  Haiyang Gao,et al.  A Unifying Lifting Collocation Penalty Formulation for the Euler Equations on Mixed Grids , 2009 .

[13]  Ugo Piomelli,et al.  Large-eddy simulation of rotating channel flows using a localized dynamic model , 1995 .

[14]  C. Meneveau,et al.  Experimental Study of Similarity Subgrid-Scale Models of Turbulence in the Far-Field of A Jet , 1995 .

[15]  J. Ferziger,et al.  Evaluation of subgrid-scale models using an accurately simulated turbulent flow , 1979, Journal of Fluid Mechanics.

[16]  N. Mangiavacchi,et al.  Subgrid-scale interactions in a numerically simulated planar turbulent jet and implications for modelling , 2000, Journal of Fluid Mechanics.

[17]  Zhi Jian Wang,et al.  A conservative correction procedure via reconstruction formulation with the Chain-Rule divergence evaluation , 2013, J. Comput. Phys..

[18]  K. Squires,et al.  Numerical investigation of the turbulent boundary layer over a bump , 1998, Journal of Fluid Mechanics.

[19]  S. Rebay,et al.  GMRES Discontinuous Galerkin Solution of the Compressible Navier-Stokes Equations , 2000 .

[20]  Victor Lee,et al.  New Trends in Large-Eddy Simulations of Turbulence , 2011 .

[21]  J. Smagorinsky,et al.  GENERAL CIRCULATION EXPERIMENTS WITH THE PRIMITIVE EQUATIONS , 1963 .

[22]  J. Koseff,et al.  A dynamic mixed subgrid‐scale model and its application to turbulent recirculating flows , 1993 .

[23]  P. Moin,et al.  Numerical investigation of turbulent channel flow , 1981, Journal of Fluid Mechanics.

[24]  Z. Wang,et al.  Adjoint Based Error Estimation and hp-Adaptation for the High-Order CPR Method , 2013 .

[25]  Meilin Yu,et al.  On the accuracy and efficiency of several discontinuous high-order formulations , 2013 .

[26]  梶島 岳夫.  乱流の数値シミュレーション = Numerical simulation of turbulent flows , 2003 .

[27]  H. T. Huynh,et al.  Differential Formulation of Discontinuous Galerkin and Related Methods for the Navier-Stokes Equations , 2013 .

[28]  K. Squires,et al.  Large Eddy Simulation of an Equilibrium Three-Dimensional Turbulent Boundary Layer , 1997 .

[29]  Zhi Jian Wang,et al.  On the Connection Between the Correction and Weighting Functions in the Correction Procedure via Reconstruction Method , 2013, J. Sci. Comput..

[30]  L. Margolin,et al.  Implicit Large Eddy Simulation: Computing Turbulent Fluid Dynamics , 2011 .

[31]  P. Moin,et al.  Large-eddy simulation of turbulent confined coannular jets , 1996, Journal of Fluid Mechanics.

[32]  Z. Wang,et al.  A priori and a posteriori evaluations of subgrid stress models with the Burgers' equation , 2015 .

[33]  Haiyang Gao,et al.  A High-Order Unifying Discontinuous Formulation for the Navier-Stokes Equations on 3D Mixed Grids , 2011 .

[34]  Yanan Li,et al.  Evaluation of Optimized CPR Schemes for Computational Aeroacoustics Benchmark Problems , 2013 .

[35]  C. Meneveau,et al.  Scale-Invariance and Turbulence Models for Large-Eddy Simulation , 2000 .

[36]  B. Geurts,et al.  On the formulation of the dynamic mixed subgrid-scale model , 1994 .

[37]  D. Lilly,et al.  A proposed modification of the Germano subgrid‐scale closure method , 1992 .