An improved treatment of external boundary for three-dimensional flow computations

Abstract

We present an innovative n umerical approach for setting highly accurate nonlocal boundary conditions at the external computational boundaries when calculating three-dimensional compressible viscous ows over nite bodies. The approach is based on application of the di erence potentials method by V . S . R y aben'kii and extends our previous technique developed for the two-dimensional case. The new boundary conditions methodology has been successfully combined with the NASA-developed code TLNS3D and used for the analysis of wing-shaped congurations in subsonic and transonic ow regimes. As demonstrated by the computational experiments, the improved external boundary conditions allow one to greatly reduce the size of the computational domain while still maintaining high accuracy of the numerical solution. Moreover, they may provide for a noticeable speedup of convergence of the multigrid iterations.1 I n troduction Preliminaries. External ows over nite bodies or con gurations of bodies represent a wide class of important practical applications in uid dynamics. To treat this type of problems numerically, one typically truncates the original in nite domain. The resulting truncated problem is obviously subde nite unless supplemented by the proper closing procedure at the external computational boundary. The latter procedure is called the arti cial boundary conditions ABC's. In the ideal case, the ABC's would be speci ed so that the solution on the truncated domain coincides with the corresponding fragment of the original in nite-domain solution. The issue of ABC's is signi cant in CFD and in other areas of scienti c computing; theoretical estimates and computational experiments by di erent authors show that the proper treatment of external boundaries has a profound impact on the overall performance of numerical algorithms and interpretation of the results.Di erent ABC's methodologies have been studied extensively over the recent t w o decades. However, the construction of the ideal i.e., exact ABC's that would provide no error associated with the domain truncation and at the same time be computationally inexpensive, easy to implement, and geometrically universal, still remains a fairly remote possibility. Among the variety of approaches proposed to date only a few can be regarded as the commonly used tools in CFD. As a rule, these approaches are based on the essential model simpli cations e.g., locally one-dimensional treatment near the external boundary and, therefore, often lack accuracy in computations. This, in turn, necessitates choosing excessively large computational domains. On the other hand, these simple methods usually provide for local ABC's, and, therefore, for cheap, geometrically universal, and algorithmically simple numerical procedures, which are attractive for practical use.There are, of course, methods of another kind, which t ypically provide for highly accurate and robust numerical algorithms. These methods, however, are not used routinely because the corresponding ABC's in most cases appea...

Method of Difference Potentials and Its Applications

“…In all cases (see [79,80]), the MOP-based ABCs allow one to greatly reduce the size of the computational domain (compared to the standard local boundary conditions), while still maintaining the high accuracy of the numerical solution. This actually means an overall increase in accuracy due to the improved treatment of the artificial boundary; it also implies a substantial economy of computer resources .…”

On the combined performance of nonlocal artificial boundary conditions with the new generation of advanced multigrid flow solvers

“…(The latter point refers to the case when global ABCs could provide for convergence while the local ones diverged.) A similar kind of behavior has been observed previously for the global DPM-based ABCs combined with the older suboptimal multigrid methodologies [21,[25][26][27][30][31][32][33]. As at the moment different flavors of local boundary conditions still dominate the area of production computations in CFD, the new combined technique developed and tested in this paper provides a potential for creating new flow solvers far superior to those currently in use.…”

“…Second, in many other situations, the right-hand side /, although not exactly zero, decays sufficiently fast in the far field so that by neglecting it outside the computational domain (i.e., outside the artificial boundary) one does not introduce large errors. Experimentally, it is always possible to see whether or not the assumption of homogeneity in the far field is acceptable [26,27,30,31,33].…”

“…In other words, the principal gain from using the DPM is that the method allows one to simultaneously meet the high accuracy standards of the ABCs and the requirements of geometric universality and easiness in implementation. In addition to the review [21], there are original publications on the DPM-based ABCs, we mention here only those closely related to the steady-state compressible viscous aerodynamics, see [25][26][27][28][29][30][31][32][33].…”

“…Namely, boundary conditions (2.21a) are exact when the far field is homogeneous, f(p,9) = 0 for p > R. If a far-field inhomogeneity is introduced, its effect can, in principle, be incorporated into these boundary conditions making them nonhomogeneous as well. However, when the far-field inhomogeneity is small it can be neglected, which will not give rise to any solvability concerns for k ^ 0, and the corresponding error in the numerical solution will then be estimated by aposteriori numerical checks, see, e.g., [26,27,30,31,33]. Note, the incorporation of inhomogeneity into the boundary conditions (2.21a), k ^ 0, is not going to be as easy as obtaining relation (2.26) instead of (2.21b).…”