A Continuous, Two-Way Free Boundary in the Unsteady Transonic Small Disturbance Equations

Abstract

We formulate a problem for the unsteady transonic small disturbance equations which describes a situation analogous to the reflection of a weak shock off a wedge, with the incident shock replaced by an incident rarefaction. We linearize this problem and solve it exactly, and we compute a numerical solution of the full nonlinear problem. The solution of this problem has several features in common with the solution of the weak shock reflection problem, known as Guderley Mach reflection. In both cases, a rarefaction wave reflects off a sonic line and forms a transonic shock. There is transonic coupling between the supersonic and subsonic regions across the sonic line and shock. In both situations, this sonic line/shock can be considered a free boundary in the formulation of a new type of free boundary problem which has not previously been formulated or analyzed. The free boundary problem that arises in the context of the problem considered here is, however, simpler than the free boundary problem that arises in the weak shock reflection problem.

High Resolution Solutions for the Supersonic Formation of Shocks in Transonic Flow

“…Analogous to these figures, Fig. 12(b) in [13] depicts velocity contours and the sonic line near the formation point of the first shock in a Guderley Mach reflection. In this case as in the other two, it is impossible to determine from the depicted numerical solution whether or not the shock begins on the sonic line.…”

“…8 and the inset plot in Fig. 12, for example, for illustrations of this claim for the rarefaction wave problem of [13] and for the transonic airfoil problem described above, respectively. Analogous to these figures, Fig.…”

“…In this paper, we present high resolution numerical solutions of two of our example problems -the rarefaction wave problem of [13], and the problem of steady transonic flow over an airfoil -formulated for the unsteady transonic small disturbance equations (TSDE). We use extreme local grid refinement in the region of the shock formation point (our finest mesh contains approximately 16×10 6 mesh points devoted to the local refinement), together with a high resolution numerical scheme and global grid continuation.…”

“…Where this rarefaction intersects the sonic line (2.5), as shown in [13], a shock is produced, and it is the formation point of this shock that we are focused on in this paper.…”

“…This shock hits the Mach shock and creates a new triple point, generating a new expansion fan which again reflects off the sonic line, and a whole cascade of shocks formed by characteristics reflecting off a sonic line is thus formed. A further example was introduced in [13]. There, a problem was studied numerically which is analogous to a shock reflection problem in which the incident shock is replaced by an incident rarefaction wave.…”