On the Self-similar Diffraction of a Weak Shock into an Expansion Wavefront

Abstract

We study an asymptotic problem that describes the diffraction of a weak, self-similar shock near a point where its shock strength approaches zero and the shock turns continuously into an expansion wavefront. An example arises in the reflection of a weak shock off a semi-infinite screen. The asymptotic problem consists of the unsteady transonic small disturbance equation with suitable matching conditions. We obtain numerical solutions of this problem, which show that the shock diffracts nonlinearly into the expansion region. We also solve numerically a related half-space problem with a "soft" boundary, which shows a complex reflection pattern similar to one that occurs in the Guderley Mach reflection of weak shocks.

On Shock Reflection–Diffraction in a van der Waals Gas

“…It may be noticed that the inequality (32) is always true; this, indeed, implies that (23) follows from (29). Further it follows from (28) that the plus branch for tan φ r (with β i = 1) yields tan φ r = 0, whereas the minus branch yields tan φ r | β i =1 = − tan φ i , implying thereby that the branch with plus sign needs to be discarded because it is irrelevant here; for a valid solution, we use the minus branch for tan φ r .…”

“…In view of , Equation 2 implies that where is an arbitrary scalar valued function satisfying the following relation: Now, using and in (75) 3 , we obtain It may be remarked that the partial differential equations (PDEs) and bear a close structural resemblance with the self‐similar unsteady transonic small disturbance (UTSD) equations analyzed in . It may be recalled that the system – is the first‐order approximation to the flow near the point B ; to obtain a uniform solution valid throughout the flow field, boundary conditions for the system – must be specified in conformity with , , and .…”

“…It may be remarked that the partial differential equations (PDEs) (78) and (79) bear a close structural resemblance with the self-similar unsteady transonic small disturbance (UTSD) equations analyzed in [14,29]. It may be recalled that the system (78)-(79) is the first-order approximation to the flow near the point B; to obtain a uniform solution valid throughout the flow field, boundary conditions for the system (78)-(79) must be specified in conformity with (39), (40), and (44).…”

The unsteady transonic small disturbance equation: Data on oblique curves

“…A brief derivation of this system can be found in [11]. Here (u, v) is the perturbation velocity in the x and y directions; these are scaled differently from each other so that a weak nonlinearity in the dominant flow (in the x direction) balances weak diffraction in the y direction.…”

“…These properties make (1) a useful model for studying some features of multidimensional compressible flow. In particular, it is used to model reflection of weak shocks (where it gives results that are quantitively as well as qualitatively correct [11,20,22]). Shock reflection problems have an additional feature of self-similarity: solutions are functions of (ξ, η) = (x/t, y/t) alone.…”