Positivity Preservation and Adaptive Solution for the k-? Model of Turbulence

A simple change of dependent variables that guarantees positivity of turbulence variables in numerical simulation codes is presented. The approach consists of solving for the natural logarithm of the turbulence variables, which are known to be strictly positive. The approach is valid for any numerical scheme, be it finite difference, a finite volume, or a finite element method. The work focuses on the advantages of the proposed change of dependent variables within the framework of an adaptive finite element method. The turbulence equations in logarithmic variables are presented for the standard κ-e model. Error estimation and mesh adaptation procedures are described. The formulation is validated on a shear layer case for which an analytical solution is available. This provides a framework for rigorous comparison of the proposed approach with the standard solution technique, which makes use of k and e as dependent variables. The approach is then applied to solve turbulent flow over a NACA0012 airfoil for which experimental measurements are available. The proposed procedure results in a robust adaptive algorithm. Improved predictions of turbulence variables are obtained using the proposed formulation

[1]  D. Pelletier,et al.  A pressure based adaptive finite element algorithm for compressible viscous flows , 1996 .

[2]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity , 1992 .

[3]  Tayfun E. Tezduyar,et al.  Massively parallel finite element simulation Of compressible and incompressible flows , 1994 .

[4]  T. Hughes,et al.  The Galerkin/least-squares method for advective-diffusive equations , 1988 .

[5]  D. Choudhury,et al.  Computations of complex turbulent flows and heat transfer using two-layer based wall functions , 1995 .

[6]  Michael S. Shur,et al.  Comparative numerical testing of one- and two-equation turbulence models for flows with separation and reattachment , 1995 .

[7]  Dominique Pelletier,et al.  Adaptive Remeshing for the k-Epsilon Model of Turbulence , 1997 .

[8]  F. Menter.  ZONAL TWO EQUATION k-w TURBULENCE MODELS FOR AERODYNAMIC FLOWS , 1993 .

[9]  D. Knight,et al.  A Navier-Stokes algorithm for turbulent flows using an unstructured grid and flux difference splitting , 1994 .

[10]  P. Spalart.  A One-Equation Turbulence Model for Aerodynamic Flows , 1992 .

[11]  T. Barth,et al.  A one-equation turbulence transport model for high Reynolds number wall-bounded flows , 1990 .

[12]  Dominique Pelletier,et al.  ADAPTIVE COMPUTATIONS OF TURBULENT FORCED CONVECTION , 1998 .

[13]  Dominique Pelletier,et al.  Adaptive Finite Element Solution of Compressible Turbulent Flows , 1998 .

[14]  F. Menter.  Improved two-equation k-omega turbulence models for aerodynamic flows , 1992 .

[15]  Dominique Pelletier,et al.  AN ADAPTIVE FINITE ELEMENT METHOD FOR A TWO- EQUATION TURBULENCE MODEL IN WALL-BOUNDED FLOWS , 1997 .

[16]  O. C. Zienkiewicz,et al.  Adaptive remeshing for compressible flow computations , 1987 .

[17]  T. Hughes,et al.  Convergence analyses of Galerkin least-squares methods for symmetric advective-diffusive forms of the Stokes and incompressible Navier-Stokes equations , 1993 .

[18]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[19]  D. Pelletier,et al.  An Adaptive Finite Element Scheme for Turbulent Free Shear Flows , 1997 .

[20]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuscka-Brezzi condition: A stable Petrov-Galerkin formulation of , 1986 .

[21]  L. Franca,et al.  Stabilized finite element methods. II: The incompressible Navier-Stokes equations , 1992 .

[22]  D. Pelletier,et al.  An adaptive finite element method for turbulent heat transfer , 1996 .

[23]  B. Launder,et al.  Mathematical Models of turbulence , 1972 .

[24]  Dominique Pelletier.  Adaptive finite element computations of complex flows , 1999 .