Transonic small disturbance theory with strong shock waves

Abstract

Introduction T most common methods of predicting aerodynamic characteristics at transonic speeds are either the transonic small disturbance (TSD) theory or the full potential equation (FPE) theory. The more accurate Euler equation solutions are expensive to obtain, although for flows with strong shock waves such solutions are essential. The FPE theory requires that the flow is irrotational and treats the wing boundary conditions exactly (numerically). The TSD theory is an approximation to the FPE theory. One advantage of the TSD theory is the flexibility in deriving the approximate equation. This flexibility is generally utilized by a choice of a transonic scale parameter. The basic assumption of irrotationality in both these theories is only valid when the flow is shock free or contains only weak shocks. Both TSD and FPE solutions are in satisfactory agreement with realistic Euler equation solutions, provided that the basic restriction to weak shock waves is not violated. The thin wing boundary conditions can also introduce errors into the TSD solutions. If the flow has strong waves, however, then there is considerable disagreement among all three theories. Generally the predicted shock locations for the potential theories are much further aft than for the Euler equations. The problem addressed in this paper is to examine the error in the shock location in the TSD theory in two-dimensional flow and to derive a correction procedure within the confines of small disturbance theory. The basic hypothesis of the present theory is that the error in shock location is primarily due to the stronger shock strength predicted by TSD theory compared to that of the Euler equations. The technique uses two TSD solutions with different scaling parameters and an interpolation scheme derived for discontinuous transonic flows to give a corrected shock strength.