Predictions of compressible viscous flows at all Mach number using pressure correction, collocated primitive variables and non-orthogonalmeshes

Abstract

This paper is concerned with the development and application of an algorithm to highly interactive compressible flows at all Mach number regimes. The algorithm is based on the finite-volume method using non-staggered, non-orthogonal grids. The accuracy of different convection discretization schemes is investigated. The employed schemes are: second order upwind, central differencing and the quadratic upwind scheme, as well as a second order upwind flux limiter constructed from the TVDtheory. Flow problems with shock and shock/boundary-layer interactions are solved aiming to assess the relative accuracy of the schemes. It is shown that the SIMPLES algorithm together with the second order upwind flux limiter scheme is a robust and accurate method for solving full Navier-Stokes equations for a wide range of Mach and Reynolds numbers. Introduction In recent years, considerable progress has been made towards the unification of numerical methods developed for incompressible and compressible flows. The main aim consists in the development of methods for viscous flows at all Mach numbers that are as good as the current compressible flows solvers that employ density as a primary variable1,*,3,4 and those for incompressible flows based in pressure-correction method, (SIMPLE)S, that use pressure as dependent variable. Calculation procedures for viscous flows using pressure as primary dependent variable have been proposed based in extensions of the pressure-correction method6.7.8. Comparisons of several variants of the SIMPLE method for the solution of compressible flows have been carried out by Van Doormaal et aZ.9 and their extension to all speeds in arbitrary configurations by Karki and Patankarlo. In the framework of primitive variable formulation most of the authors have employed staggered grid systems in Cartesian9 and arbitrary configurationslOlll. The use of a staggered grid arrangement have overcame the pressure oscillatory behavior encountered with non-staggered grid systems. The pressure-weighted interpolation method proposed by Rhie and Chow12 and their applications for incompressible flows see e.g. Pen6 et al.13, Kobayashi and Pereira14~~5 have proved that pressure-velocity coupling in non-staggered can be at least as accurate as in staggered grid systems. Marchi et al.16 have recently reported compressible flow predictions using nonstaggered Cartesian grids together with the unsteady form of the Navier-Stokes and continuity equations. In this paper an extension of SIMPLE procedure and pressure weighted interpolation method for non-staggered grids is presented for all Mach numbers and arbitrary two-dimensional fluid flow configurations. The paper aims also to investigate the accuracy of different convection discretization schemes. The studied schemes are second order upwind central differencing and quadratic upwind schemes17,18, as well as non upwind biased second order monotonic scheme named Minmod after the work of Chakravarthyl9. The paper begins with the summary of the discretization procedure, convection discretization schemes and the SIMPLES algorithm. This is followed by three applications of the method corresponding to the supersonic viscous flow through a planar nozzle and a double throat nozzle, and the inviscid flow over a bump. The paper ends with final remarks. Mathematical and Numerical Model Using the Cartesian components of vectors and tensors the equations that describes the steady state of a compressible flow can be written in a general non-orthogonal coordinate system XI as follows: --j a ( p u k p 7 = o a x where h is the enthalpy, ui, TL, qi are the Cartesian components of the velocity vector, stress tensor and heat flux, respectively P’J represents the cofactor of the i-th row and j-th column in the Jacobian matrix of the coordinate transformation, yi = yi (d), yi is the reference Cartesian frame an J denotes the determinant of the Jacobian matrix. In the case of a Newtonian compressible fluid the stress tensor can be written as: (4) where p is the static pressure, p is the dynamic viscosity, 6ij is the delta of Cronecker and Eij is the rate of deformation tensor which is equal to the symmemc part of the velocity gradient. The heat flux vector is defined by a Fourier-type law, as follows: where k is the thermal conduction cp the specific heat. * MSc, PhD Student ** Associated Professor, Member of AIAA