FAR - field boundaries and their numerical treatment

Abstract

Many computational problems of theoretical and practical interest are not naturally bounded by physical boundaries. Aerodynamic examples include flow calculations past aerofoils or past wing-body configurations, semi-bounded channel flows etc. Other examples include simulations of Turbomachinery flows, problems in Underwater Acoustics etc. To obtain a numerical solution, the problem has first to be converted to a finite region, by introducing an artificial boundary at some finite distance. Boundary conditions must be specified at the artificial boundary for well-posedness of the truncated problem. They should simulate an open boundary across which the fluid flows and should ideally allow outgoing waves to pass through without generating reflections. Indeed, reflections at the boundary not only degrade the accuracy of transient solutions but also inhibit convergence to steady-state. In many problems of practical interest, perfect absorption cannot be achieved. Instead one aims at minimizing the amount of reflected energy using asymptotic expansions based on various asymptotic arguments. The more accurate the boundary statements, the closer the artificial boundaries can be located to the regions of aerodynamic interest, thereby reducing the computational domain and costs. We present a thorough numerical study of the efficiency of several widely used boundary conditions in absorbing outgoing waves. We identify the key parameters upon which the level of absorption at the boundaries depends and expose the limitations of some of the existing recipes. We show that substantial reflections may occur even under conditions which are considerably milder than those encountered in practical calculations. We then introduce an unconventional approach to the treatment of artificial boundaries. It is proposed that in the far field the governing equations are modified in a boundary-layer like manner. Two closely related far field modifications are derived and analysed: (a) Slowing down the outgoing waves and (b) Attenuating the outgoing waves. Under the first modification the outgoing waves are prevented from reaching the boundary hence from reflecting. Under the second, the outgoing waves are attenuated to practically zero strength before reaching the boundary. Both modifications do not alter the propagation of the incoming waves to allow the launching of correct information from the boundary into the interior. Analytic conditions are derived to ensure that no reflections are generated due to the change of coefficients in the governing equations. Reflection analysis is also performed on the discrete level. Well-posedness of the modified systems is established as well as stability of the resulting interface problem. The modifications are extended to two space dimensions and are applied to a variety of one and multidimensional test problems. Results indicate that the proposed far field modifications are attractive in genuinely time-dependent calculations. Preliminary steady state calculations with the unsteady 2D Euler equations show significantly improved convergence properties.