Upwind-biased finite-volume technique solving Navier-Stokes equations on irregular meshes

Abstract

New node-centered finite-volume discretizations of advective and diffusive derivatives on structured meshes with quadrilateral cells are presented. They are applied to the solution of Euler and full Navier-Stokes equations using a pseudotime-dependent approach. The advective derivatives are split and upwind biased. The most interesting aspect of the scheme lies in its ability to provide good accuracy on meshes with severe skewness and stretching distortions. A very sensitive detector is presented which is capable of selectively identifying shock waves and insufficiently resolved shear layers. It is used to automatically switch to a first-order upwind scheme in some regions and provides a sharp and monotone shock capture. Results obtained for plane and axisymmetric steady supersonic laminar flows are discussed. They include blunt-body flows, a shock/boundary-layer interaction, and a flow over a compression corner. OST classical finite-volume schemes for the solution of the Euler and Navier-Stokes equations require very regular and smooth meshes. Severe mesh skewness and stretching distortions can strongly deteriorate the accuracy and lead to significant parasitic diffusion or antidiffusion effects. These effects can be responsible for relatively poor evaluations of important flow properties and eventually for numerical instabilities. The accuracy of several spacecentered schemes using structured meshes has been theoretically and practically evaluated by Renard and Essers.1 For that purpose, they considered a simple model governed by a linear advectiondiffusion equation. Starting from an unstretched rectangular mesh, more and more severe distortions were introduced in the mesh using a special random grid distortion procedure. Different families of structured grids, each with different amounts of distortions measured by a parameter ranging from 0 to 1 were thus created and used to solve the model problem. Errors obtained with finer and finer grids presenting the same amount of distortion were evaluated. All of the tested schemes, which are frequently thought of as being second-order accurate, were found zeroth-order accurate, i.e., inconsistent, for diffusive derivatives. For the advective derivatives, the best schemes were found first-order accurate only, whereas many others were inconsistent. The main aim of this paper is to present new upwind-biased finite-volume schemes that do not exhibit the parasitic numerical effects appearing in other schemes and can, therefore, lead to a significantly improved accuracy on irregular meshes. They have been used to solve the steady Euler and Navier-Stokes equations for plane and axisymmetric supersonic and hypersonic flows using structured, distorted, node-centered meshes. Solutions are obtained from a pseudounsteady approach using a multistep Runge-Kutta scheme with variable time stepping. Our code simulates laminar perfect gas flows with a constant Prandtl number. The variable viscosity is given by the Sutherland law modified for temperatures below 120 K (see, e.g., Ref. 2 for more details). As real gas effects are not accounted for, some results are physically unrealistic but can be useful for comparisons with other codes.