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The von Neumann Triple Point Paradox Richard Sanders 1∗ and Allen M. Tesdall 2† 1 Department of Mathematics, University of Houston, Houston, TX 77204, USA sanders@math.uh.edu 2 Fields Institute, Toronto, ON M5T 3J1, Canada and Department of Mathematics, University of Houston, Houston, TX 77204, USA atesdall@fields.utoronto.ca Summary. We describe the problem of weak shock reflection off a wedge and discuss the triple point paradox that arises. When the shock is sufficiently weak and the wedge is thin, Mach reflection appears to be observed but is impossible according to what von Neumann originally showed in 1943. We summarize some recent numerical results for weak shock reflection problems for the unsteady transonic small disturbance equations, the nonlinear wave system, and the Euler equations. Rather than finding a standard but mathematically inadmissible Mach reflection with a shock triple point, the solutions contain a complex structure: there is a sequence of triple points and supersonic patches in a tiny region behind the leading triple point, with an expansion fan originating at each triple point. The sequence of patches may be infinite, and we refer to this structure as Guderley Mach reflection. The presence of the expansion fans at the triple points resolves the paradox. We describe some recent experimental evidence which is consistent with these numerical findings. Key words: self-similar solutions, two-dimensional Riemann problems, triple point paradox 1 Introduction Consider a planar normal shock in an inviscid compressible and calorically perfect gas which impinges on a fixed wedge with apex half angle θ w ,see Figure 1. Given an upstream state with density ρ = ρ r , velocity u = v =0 and pressure p = p r , one calculates that downstream of a fast (i.e., u + c) shock ∗ Research supported by the National Science Foundation, Grant DMS 03-06307. † Research supported by the National Science Foundation, Grant DMS 03-06307, NSERC grant 312587-05, and the Fields Institute.
114 R. Sanders and A.M. Tesdall I R θ w R I S M (a) (b) Fig. 1. A planar shock moving from left to right impinges on a wedge. After contact, I indicates the incident shock and R indicates the reflected shock. On the right, the dotted line S indicates a slip line and M is the Mach stem. Regular reflection is depicted on the left. Irregular reflection is depicted on the right. U l R I R U l I ? U r U U r S ? M (a) (b) Fig. 2. A blow-up of the incident and reflected shock intersection. Regular reflection is on the left and irregular on the right. The constant states upstream and downstream of the incident shock are denoted by U r and U l . Whether or not constant states indicated by the question marks exist depends on the strength of I. p l = 2γ p r γ +1 M 2 − γ − 1 γ +1 , ρ l ρ r = u l = 2 ( M − 1 ) , c r γ +1 M (1) 2 (γ +1)M 2+(γ − 1) M 2 , where γ denotes the ratio of specific heats, and M > 1 denotes the shock Mach number defined as the Rankine–Hugoniot shock speed divided by the upstream speed of sound c r = √ γp r /ρ r . Following interaction, a number of self-similar (with respect to the wedge apex) reflection patterns are possible, depending on the values of M and θ w . This wedge reflection problem has a rich history, experimentally, analytically, and numerically. Probably the earliest and most significant analytical result was found by von Neumann [Neu43]. In this work were first formulated the equations which describe two and three planar shocks meeting at a point separated by constant states, see Figure 2. The two shock theory leads to what is known as regular reflection. The three shock theory leads to Mach reflection. For supersonic regular reflection, state U immediately behind the reflected shock R is supersonic and becomes subsonic across a sonic line downstream (toward the wedge’s apex). When the incident shock angle is increased
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The von Neumann Triple Point Paradox<br />
Richard S<strong>and</strong>ers 1∗ <strong>and</strong> Allen M. Tesdall 2†<br />
1 Department of Mathematics, University of Houston, Houston, TX 77204, USA<br />
s<strong>and</strong>ers@math.uh.edu<br />
2 Fields Institute, Toronto, ON M5T 3J1, Canada <strong>and</strong> Department of<br />
Mathematics, University of Houston, Houston, TX 77204, USA<br />
atesdall@fields.utoronto.ca<br />
Summary. We describe the problem of weak shock reflection off a wedge <strong>and</strong> discuss<br />
the triple point paradox that arises. When the shock is sufficiently weak <strong>and</strong> the<br />
wedge is thin, Mach reflection appears to be observed but is impossible according to<br />
what von Neumann originally showed in 1943. We summarize some recent numerical<br />
results for weak shock reflection problems for the unsteady transonic small disturbance<br />
equations, the nonlinear wave system, <strong>and</strong> the Euler equations. Rather than<br />
finding a st<strong>and</strong>ard but mathematically inadmissible Mach reflection with a shock<br />
triple point, the solutions contain a complex structure: there is a sequence of triple<br />
points <strong>and</strong> supersonic patches in a tiny region behind the leading triple point, with<br />
an expansion fan originating at each triple point. The sequence of patches may be<br />
infinite, <strong>and</strong> we refer to this structure as Guderley Mach reflection. The presence<br />
of the expansion fans at the triple points resolves the paradox. We describe some<br />
recent experimental evidence which is consistent with these numerical findings.<br />
Key words: self-similar solutions, two-dimensional Riemann problems, triple<br />
point paradox<br />
1 Introduction<br />
Consider a planar normal shock in an inviscid compressible <strong>and</strong> calorically<br />
perfect gas which impinges on a fixed wedge with apex half angle θ w ,see<br />
Figure 1. Given an upstream state with density ρ = ρ r , velocity u = v =0<br />
<strong>and</strong> pressure p = p r , one calculates that downstream of a fast (i.e., u + c)<br />
shock<br />
∗ Research supported by the National Science Foundation, Grant DMS 03-06307.<br />
† Research supported by the National Science Foundation, Grant DMS 03-06307,<br />
NSERC grant 312587-05, <strong>and</strong> the Fields Institute.
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