Development of meshless method boundary conditions for high speed flows

Abstract

SHOCK STRUCTURE MODELDuring numerical flow simulation, diffusion can be induced through artificial viscosity term to control development of flow. Early researchers included the dissipation term in the formulation of finite volume Euler equations to suppress odd-even point oscillations arising from pressure-velocity coupling. It was found that second difference diffusion terms with second and fourth difference pressure coefficients rendered wiggle-free computation. The highorder central differencing schemes produce accurate results in smooth flow regions but give rise to oscillations at shocks or discontinuities. This problem has been resolved through special upwinding methods and by Roe linearization technique [1], with or without characteristic decomposition. The development of Euler codes has paralleled the development of positivity or monotonicity preserving schemes that satisfy Jameson's local extremum diminishing principle (LED criterion) [2] in order that physically meaningful values of pressure and density could be computed. Customarily, the residual is evaluated as convective and diffusive fluxes separately in order to improve the accuracy of computation of flow properties. If meshless technique is employed then the diffusive flux can be obtained as the product of point fluxes and least square (LSQ) coefficients along coordinate directions prior to time integration to steady state solution. Irregular boundaries and fine broken cells at the wall, which were constraints to computational accuracy of Cartesian mesh solvers, are now implemented efficiently by meshless wall boundary techniques [3]. The accuracy with which shock waves reflecting at the wall boundaries can be captured depends both on the computational schemes used and the shock structure model adopted. Even after a century old research, a concrete shock wave development model does not exist. Among various models suggested by researchers, the three-shock theory [4, 5] (3ST) putforth by von Neumann stands out as the most realistic one to describe the commonly encountered shock wave phenomena. In situations where only a regular shock reflection is expected as in ducted flows, a Mach wave reflection too may occur. A consequence of this effect is the appearance of Mach stem at the wall accompanied by a regular shock wave. Now, according to 3ST, non-regular shock reflection called Mach reflection (MR) is produced by three intersecting shock waves: incident wave, reflected wave and Mach stem. Their point of intersection called triple point (T) may move toward, away or parallel to the reflecting surface, giving rise to direct, inverse or stationary MR as shown in the Figures 1. These occurrences are termed as Mach stem and it forms the subject matter of this paper.