数理解析研究所講究録
1495 巻 2006 年 112-117

112

Physicality of Weak Prandtl-Meyer Reflection*
Volker$\mathrm{E}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}^{\mathrm{T}}$Tai-Ping$\mathrm{L}\mathrm{i}\iota 1\ddagger$

May 6, 2006

Abstract
Consider supersonic flow around a ramp. There are two possible shockreflections, the weak and strong Prandtl-Meyer reflection. Experimentally,
it is the weak reflection that is observed. The purpose of this article isto present recent analytical formulations and computational results on
this problem. Through numerical computations, we show that the usual
time-asymptotic stability criterion fails to rule out the strong reflection.
Instead, we consider the self-sirnilar flows, the time-asymptotic states ofgeneral flows.

1

Introduction

Consider the isentropic Euler equations of compressible gas dynamics in two
space dimensions:$\rho_{t}+\nabla\cdot(\rho\vec{v})=0$$(\rho\tau))_{t}arrow+\nabla\cdot(\rho\vec{v}\otimes\vec{v})+\nabla(p(\rho))=0$

,

(1)
(2)

here is the density, $\vec{v}=(u, v)$ the velocity, $\vec{x}=(x^{1}, x^{2})=(x, y)$ the spacevariables, and, the spatial gradients. We consider the polytropic
is given byso that, after some rescaling, the pressure function$\rho$$\nabla=\partial_{\vec{x}}’$$\mathrm{g}\mathrm{a}‘ \mathrm{s}\mathrm{c}\mathrm{s}$$p(\rho)$$p(\rho)=\rho^{\gamma},$$\gamma\in(1, \infty)$.

The Euler equations possess two types of discontinuity, the shock waves as wellas the highly unstable vortex sheets. To focus on the shock waves, we will’This material is based upon work supported by the Institute of Mathematics, AcademiaSinica, and by the National Science Foundation under Grant no. DMS 0104019.
\dagger Division of Applied, Brown University, 182 Gcorge Street, Providence, RI$\mathrm{M}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}8$02912$(\mathrm{V}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}^{\mathrm{Q}\mathrm{s}\mathrm{t}\mathrm{a}}\mathrm{n}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{n}\mathrm{i}.\circ \mathrm{r}\mathrm{g})$\ddagger Department of Mathematics, Building 380, Stanford University, Stanford, CA 94305)(liuhath. stanford.
$\mathrm{e}\mathrm{d}\mathrm{u}$113consider the potential flow equations with the additional hypothesis that theflow is irrotational. That is, there exists a scalar potential function q5 such that$\vec{v}=\nabla\phi$.

FYom this and the Euler equations, (1), (2), the following Bernoulli law holds
for smooth flows:$0=\phi_{xt}:+\nabla\phi_{x}$.

$\cdot\nabla\phi+\pi(\rho)_{x_{i}}$$=( \phi_{t}+\frac{|\nabla\phi|^{2}}{2}+\pi(\rho))_{x_{\mathrm{i}}}$where$\pi(\rho)$

is related to the sound speed$\pi_{\rho}=c^{2}(\rho)/\rho,$

Thus, for some constant$A$$c(\rho)$,by$\mathrm{c}^{2}(\rho)=p_{\rho}(\rho)$.

,$\rho=\pi^{-1}(A-\phi_{t}-\frac{|\nabla\phi|^{2}}{2})$.

Substituting this (1) yields the potential flow equation:.

$\phi_{tt}+2\phi_{T},\phi_{tx}+2\phi_{y}\phi_{ty}+[(\phi_{x})^{2}-c^{2}]\phi_{xx}+2\phi_{x},\varphi_{y}’\phi_{xy}+[(\phi_{\tau},)^{2}-c^{2}]\phi_{yy}=0$(3)

The stationary potential flow equation$[(\phi_{x})^{2}-c^{2})]\phi_{xx,}+2\phi_{x}\phi_{y}\phi_{x,y}+[(\phi_{x})^{2}-\mathrm{c}^{2})]\phi_{yy}=0$(4)

is elliptic when the flow is subsonic, $|\nabla\phi|<c$ , and hyperbolic when it is supcrsonic $|\nabla\phi|>c$ .

The problem we are interested in is thea wedge of solid. The gas flow
is in the region outside of the wedge with anglc , in the polar coordinates$(x, y)=(r\cos\theta, r\sin\theta),$:$\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{w}\prime \mathrm{p}\mathrm{a}‘ \mathrm{s}\mathrm{t}$$\Omega$$2\alpha$$|\theta|\leq\pi,$

St $=\{(x, y) : |\theta|>\alpha\}$ .
For constant supersonic upstream flow$u>c$ , the compression that the wedge induces gives raise to sho . Using the shock polar analysis,
Figure 1,, Prandtl shows that there are two possible configurations, the weak shock reflection with supersonic downstream flow, Figure
2, and the strong shock reflection with subsonic downstream flow, Figure 3.
The weak reflection is the one observed experimentally. Our main purpose is toaddress this analytically.
$\nabla\phi(x, y)=(\overline{u}, 0),$$c,\mathrm{k}\mathrm{s}$$[\mathrm{C}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{t}rightarrow \mathrm{R}\mathrm{l}\mathrm{e}\mathrm{d}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}\mathrm{s}]$114

One plausible approach is to rule out the strong shock reflection by thc consideration of stability, the stable ones are physical. Both weak and strong reflections
are structural stable in that they exist upon srnall perturbation of the wedge.
The time-asymptotic stability is the relevant one here. In Section 2 wc present
computational results on the tirne-asymptotic stability on both refelctions. A
surprising finding is that both strong and weak reflections are time-asymptotic
consideration of timestable upon compactly supported perturbations. Thus
asyrnptotic stability fails to rule out the strong reflection. The main part of oureffort is to come up with an analytical setup of showing that the weak reflectionis the one observed experimentally in that it rcpresents the tirne-asymptotic
state of an accelerating flow. This is explained in Section 3.
$\mathrm{t}\mathrm{I}\iota \mathrm{e}$2

Computational Results on Time-Asymptotic
StabilityWe study the perturbation of the weak reflection, Figure 1, and of the strong
reflection, Figure 2. The perturbation is compactly supported. We present results for the familiar Godunov method. More sophisticated numcrical methods,
as wcll as changes to upstream Mach number and other pararneters, produce
qualitatively similar results. The computational results are presentcd in Figure
3 for weak reflection and in Figure 4 for strong rcflection. For weak reflection,
the perturbation moves away from the tip of the wedgc. This is because thedownstream flow is supersonic. Duc- to dispersion and the absorbing nature ofdownstrcam flow for
the leading shock, the perturbation decays in tiie.
strong reflection is subsonic and so the perturbation propagatev both away fromthe tip of the wedge as well as toward the tip, and it also decays in time. Insummary, both weak and strong reflections are nonlinear, time-asymptoticallystable upon compactly supported perturbation. Thus this analysis fails to cx$\mathrm{T}\}_{1\mathrm{e}}$115

Figure 3: Weak shock pcrturbation (dcnsity shown)

Figure 4: Strong shock perturbation (density shown)plain thc fact that cxpcrimcntally it is the weak, not strong, reflection that isproduccd. Wc do, howcver, observe the instability of the strong rcflection when
the downstrcarn state at $x=\infty$ is also perturbed; see Figurc 5.

3

Self-Similar FlowsTo produce the shock reflection., it is to accelerato the wedge to supersonic spccd;
or, equivalently, keep the wedge fixcd and accelerate thc upstream flow till thegivcn constant supersonic velocity. The acccleration yields conlplex fiow
patterns and rnany shock waves are generatcd in the proccss. What intcrcsted
us is what typc of shock rcflcctions will eventually be produccd. By thc tirne$sarrow 0_{+}$ , the proccss becomes the initialasyrnptotic scaling
value problem with initial values at timc $t=0$ :$(\overline{u}, \mathit{0})$$(x, y, t_{\text{ノ}})arrow s(x, y, t_{\text{ノ}}),$$\nabla\phi(x, y, 0)=(\overline{u}, 0)$.

In other words, the wedge is instantaneously accclerated to the supersonic statc. This simplifies greatly the thinking ,ause both the geometry of thedomain , the initial values, and the potential flow equation are invariant under$(\overline{u}, 0)$$\mathrm{b}\mathrm{e}\mathrm{c}$$\Omega$

116

Figlj
the$r\mathrm{e}\backslash \mathrm{r}_{):}$

Strong shock vanishes under downstream state perturbation; right:
ck appcars.

$\mathrm{v}\mathrm{v}^{\mathrm{Y}}\mathrm{a}\mathrm{k}\mathrm{s}\mathrm{h}o$the scaling$(x, y, t_{J})arrow s(x, y, t_{\text{ノ}})$. Thussolution is self similar:$\mathrm{t}_{J}\mathrm{h}\mathrm{e}$$\phi(\prime x, y, t_{\text{ノ}})=t\psi(\xi, \eta),$$\xi=x/t,$ $\eta=y/t$.

The sclf-similar potential flow cquation is.

$[c^{2}-(\psi_{\xi}-\xi)^{2}]’\psi_{J}\xi\xi-2(’\psi_{\xi}-\xi)(\psi_{\eta}-\eta)’\psi_{J}\xi\eta+[c^{2}-(\prime l^{y_{\eta}}-\eta)^{2}]\psi_{\eta\eta}=0$$(_{\mathit{0}}^{\ulcorner})$

The velocity takes similar form as before:$\nabla\psi=\vec{v}$.

The self-similar potential flow equation is of mixed type. It is hyperbolic if theflow is pseudo-supersonic:$|\vec{v}-(\xi, \eta)|>c$ ;and elliptic if it is pseudo-subsonic:$|\vec{v}-(\xi, \eta)|>C$.

The initial condition for the potential flow equation turns to the boundary. The self-similar solution
condition for the self-similar cquation at
represents the solution of the time-dependent
at time $t=1$ . Since thc
flow is clearly pseudo-supersonic around $|(\xi, \eta)|=\infty$ , the boundary condition
consists of one-dimensional shock waves parallel to the ramp. We have carricd
out numerical computations. Our result shows that the self-similar solution
consists of the weak shocks from the tip of the wedge, the one-dimensional
shocks paralell to the ramp, curved shocks connecting the thesc two set ofshocks, and a region of pseudo-subsonic flow bounded by the curved shock andtwo pseudo-sonic circles determined by the two constant states between the weak
shocks, curved shocks and the ramp, Figure 6. We are flnishing the analyticaljustification of this flow pattern. This would show thatweak shock reflectionis the physical one.
$|(\xi, \eta)|=\infty$$\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\dot{\mathrm{t}}\mathrm{i}\mathrm{o}\mathrm{n}$$\mathrm{t}\mathrm{h}\mathrm{c}$117

Figure 6: The weak shock appears spontaneously at the corncr. It is connectedto the straight reflected shock by a curved shock, with a nontrivial clliptic region
below.

References[CF48] R. Courant and K.O. Friedrichs, Supersonic fiow and shock waves, Interscience Publishers, 1948.
[EL05] V. Elling and Tai-Ping Liu, The ellipticity
for selfsimilar potential flow, J. Hyper. Diff. Eqns. 2 (2005), no. 4, 909-917.
$pr\dot{\mathrm{v}}ncipl,e$Acknowledgement: The second author would like to thank Professor Shinya
Nishibata for inviting him to participate in this Workshop at RIMS.