Long Time Behavior of Unsteady Flow Computations

Abstract

This paper addresses a specific issue of time accuracy in the calculations of external aerodynamic problems. The class of problems that is discussed here consists of inviscid compressible subsonic flows. These problems are inherently governed by a convective equation. This is readily seen by linearizing the Euler equation which results in a convective wave equation for the pressure. A key mathematical issue that is not well understood in literature for these problems is the long time behavior of the solution. This is an important aspect if one desires transient calculations of problems governed by the Euler equations or its derivatives such as the small disturbance equations or the potential formulations of the gust problem. In particular, difficulties arise for two-dimensional problems. In two dimensions the time decay rate of solutions of the wave equation is known to be rather slow. This applies to the convective wave equation as well. The consequences are rather severe if one focuses on the time accuracy of solutions of problems governed by the Euler equations.In concert with the above mentioned problem, exterior flows require proper modeling of boundary conditions. Computations of these flow problems require truncation of infinite regions into finite regions with the aid of artificial boundaries. On these boundaries one must impose boundary conditions that are consistent with the physics as well as guarantee consistency with the original problem posed in the unbounded region. Moreover, these boundary conditions must have accuracy in time as well as space. Some of the well-known procedures do address the issues of spatial accuracy and have remedy for these conditions. Unfortunately, these procedures do not address the time accuracies, which are crucial for the transient problems. Our treatment is discussed in detail and examples are presented to verify the results.