Adaptive Solution of Transonic Flows

Abstract

Transonic flows imply that the flow is subsonic in certain parts of the domain of interest and is supersonic in others. In most cases the flow decelerates from the supersonic speed to the subsonic one through a fast compression layer. This compression layer is often very thin (of the order of several mean distance among the air molecules). The thin layer is called a shock-wave. Naturally, within the shock layers the continuum approximation of the fluid is not very good. However, if one uses the weak formulation (i.e. one seeks a weak solution) the continuum formulation, which accomodates discontinuities becomes a very good approximation to real flows. Transonic flows are non-linear in their character, and hence the location of shocks is unknown a-priori. Therefore, computing flows with shocks implies that one has to compute also the position of the shock. Computing a weak solution (in a strict sense) implies that the governing equations are valid only when the flow is smooth while across shocks one has to use the shock-jump (Rankine-Hugoniot, R-H) relations. As a consequence, the basic approach until the beginning of the seventies, was to use a so called “shock-fitting” technique. Shock fitting implies that the shape of the shock is adjusted so that the R-H relations aresatisfied. The approach is less complex for 2-D cases but becomes impractical for 3-D shocks. Thus, the competitive approach of “shock capturing” became the method of choice when computing general transonic flows. Shock capturing is possible by relaxing the assumption of computing the weak solution in the strict sense.