A CONTINUOUS, TWO-WAY FREE BOUNDARY IN THE UNSTEADY TRANSONIC SMALL DISTURBANCE EQUATIONS

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.

Weak shock reflection from an open end of a tube with a baffle plate

The present paper describes experimental and numerical investigations of a new boundary condition at an open end of a tube, in which a weak shock wave is discharged towards the surroundings. Experimental and computational investigations were performed on a simple shock tube with and without baffle plates. A numerical calculation was carried out for an unsteady, axisymmetric, inviscid, compressible flow. The size of baffle plate was varied in order to understand its effect on the reflection of the weak shock wave from the open end of the tube. With and without a baffle plate, the results of the experiment were in good agreement with those of numerical calculations. The results showed that an open end correction is subject to the presence of a baffle plate at the open end. An improved empirical equation for the reflection of the weak shock wave from the open end of a duct with and without a baffle plate was developed.

The von Neumann paradox in weak shock reflection

We present a numerical solution of the Euler equations of gas dynamics for a weak-shock Mach reflection in a half-space. In our numerical solutions, the incident, reflected, and Mach shocks meet at a triple point, and there is a supersonic patch behind the triple point, as proposed by Guderley. A theoretical analysis supports the existence of an expansion fan at the triple point, in addition to the three shocks. This solution is in complete agreement with the numerical solution of the unsteady transonic small-disturbance equations obtained by Hunter & Brio (2000), which provides an asymptotic description of a weak-shock Mach reflection. The supersonic patch is extremely small, and this work is the first time it has been resolved in a numerical solution of the Euler equations. The numerical solution uses six levels of grid refinement around the triple point. A delicate combination of numerical techniques is required to minimize both the effects of numerical diffusion and the generation of numerical oscillations at grid interfaces and shocks.