Stability of explicit navier-stokes procedures using k-ε and k - ε/algebraic reynolds stress turbulence models

Abstract A three-dimensional explicit Navier-Stokes procedure has been developed for application to compressible turbulent flows, including rotation effects. In the present work, a numerical stability analysis of the discrete, coupled system of seven governing equations is presented. Order of magnitude arguments are presented for flow and geometric properties typical of internal flows, including turbomachinery applications, to ascertain the relative importance of grid stretching, rotation and turbulence source terms, and effective diffusivity on the stability of the scheme. It is demonstrated through both analysis and corroborative numerical experiments that: (1 ) It is quite feasible to incorporate, efficiently, a two-equation k - ϵ turbulence model in an explicit time marching scheme, provided certain numerical stability constraints are enforced. (2) The role of source terms due to system rotation on the stability of the numerical scheme is not significant when appropriate grids are used and realistic rotor angular velocities are specified. (3) The direct role of source terms in the turbulence transport equations on the stability of the numerical scheme is not significant when appropriate grids are used and realistic freestream turbulence quantities are specified, except in the earliest stages of iteration (a result which is contrary to that generally perceived). (4) There is no advantage to numerically coupling the two-equation model system to the mean flow equation system, in regard to convergence or accuracy. (5) For some flow configurations, including turbomachinery blade rows, it is useful to incorporate the influence of artificial dissipation in the prescription of a local timestep. (6) Explicit implementation of an algebraic Reynolds stress model (ARSM) is intrinsically stable provided that the discrete two-equation transport model which provides the necessary values of k and ϵ is itself stable.

[1]  K. Chien,et al.  Predictions of Channel and Boundary-Layer Flows with a Low-Reynolds-Number Turbulence Model , 1982 .

[2]  A. Jameson,et al.  Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes , 1981 .

[3]  G. Stewart.  Introduction to matrix computations , 1973 .

[4]  Three-Dimensional Navier–Stokes Computation of Turbomachinery Flows Using an Explicit Numerical Procedure and a Coupled k–ε Turbulence Model , 1992 .

[5]  B. Launder,et al.  Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc , 1974 .

[6]  R. Maccormack,et al.  A numerical method for solving the Navier-Stokes equations with application to shock-boundary layer interactions , 1975 .

[7]  Budugur Lakshminarayana,et al.  Turbulence modeling for complex shear flows , 1985 .

[8]  G. Gerolymos.  Implicit multiple-grid solution of the compressible Navier-Stokes equations using k-epsilon turbulence closure , 1990 .

[9]  R. F. Warming,et al.  Diagonalization and simultaneous symmetrization of the gas-dynamic matrices , 1975 .

[10]  V. C. Patel,et al.  Turbulence models for near-wall and low Reynolds number flows - A review , 1985 .

[11]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[12]  T. J. Coakley,et al.  Turbulence modeling methods for the compressible Navier-Stokes equations , 1983 .

[13]  Dimitri J. Mavriplis,et al.  Multigrid solution of compressible turbulent flow on unstructured meshes using a two-equation model , 1991 .

[14]  David Gottlieb,et al.  Optimal time splitting for two- and three-dimensional navier-stokes equations with mixed derivatives , 1981 .

[15]  Hans Edelmann,et al.  Vier Woodbury-Formeln hergeleitet aus dem Variablentausch einer speziellen Matrix , 1976 .

[16]  Seungsoo Lee,et al.  Magnetohydrodynamic flow computations in three dimensions , 1991 .

[17]  Budugur Lakshminarayana,et al.  Explicit Navier-Stokes computation of cascade flows using the k-epsilon turbulence model , 1992 .

[18]  D. Choi,et al.  Computations of 3D viscous flows in rotating turbomachinery blades , 1989 .

[19]  D. Holmes,et al.  Solution of the 2D Navier-Stokes equations on unstructured adaptive grids , 1989 .

[20]  Klaus Bremhorst,et al.  A Modified Form of the k-ε Model for Predicting Wall Turbulence , 1981 .

[21]  Jubaraj Sahu,et al.  Navier-Stokes computations of transonic flows with a two-equation turbulence model , 1986 .

[22]  B. Lakshminarayana,et al.  Turbulence modeling for three-dimensional shear flows over curved rotating bodies , 1984 .