Upwind discretization of the steady Navier–Stokes equations

Abstract

A discretization method is presented for the full, steady, compressible Navier–Stokes equations. The method makes use of quadrilateral finite volumes and consists of an upwind discretization of the convective part and a central discretization of the diffusive part. In the present paper the emphasis lies on the discretization of the convective part. The solution method applied solves the steady equations directly by means of a non‐linear relaxation method accelerated by multigrid. The solution method requires the discretization to be continuously differentiable. For two upwind schemes which satisfy this requirement (Osher's and van Leer's scheme), results of a quantitative error analysis are presented. Osher's scheme appears to be increasingly more accurate than van Leer's scheme with increasing Reynolds number. A suitable higher‐order accurate discretization of the convection terms is derived. On the basis of this higher‐order scheme, to preserve monotonicity, a new limiter is constructed. Numerical results are presented for a subsonic flat plate flow and a supersonic flat plate flow with oblique shock wave–boundary layer interaction. The results obtained agree with the predictions made. Useful properties of the discretization method are that it allows an easy check of false diffusion and that it needs no tuning of parameters.

Design principles for bounded higher-order convection schemes – a unified approach

“…Hence the optimum member of the class is j = 0 which is in fact the van Albada limiter. Two other smooth, continuous PR limiters, of Spekreijse [43] and Koren [23] respectively, are included in Table A6, both of which share the problem of having W 0 ð0Þ ¼ 1 2 and therefore going outside the j-scheme region. The Gamma scheme of Jasak et al [21], proposed in NV form as a modified form of CDS respecting the CBC, blends CDS with an ad hoc quadratic function.…”

Multi-D Upwinding and Multigridding for Steady Euler Flow Computations

“…Further, they can be carried over to 3-D and extended to non-Cartesian grids. c. Limited ic= Vi-scheme [12]. Fig.…”

“…7c we still give the distributions obtained by the limited "' = ~-scheme from [12]. The monotone solution as obtained for 8 = i by the first-order accurate zero-crosswind diffusion scheme appears to be less diffused than that of the higher-order accurate "' = l-scheme.…”

Upwind Schemes for the Navier-Stokes Equations

“…(The choice of 20 equations allows us to consider a boundary layer flow in the analysis.) In [8], for both upwind schemes, the system of modified equations is derived, considering (i) a first-order accurate, square finite volume d.iscretization, and (ii) a subsonic flow with u and v positive, and pr:::. constant.…”