Physical selection, uniqueness, and dynamical stability of detached shock solutions past a blunt body in supersonic potential flow

Bae and Xiang (Acta Applicandae Mathematicae 188 (2023); arXiv:1909.13281) gave the first rigorous construction of a detached shock for the two-dimensional steady Euler system of irrotational (potential) compressible flow past a symmetric blunt body: for every prescribed detached distance $d_0$ ahead of the body's nose and every sufficiently large incoming Mach number $M_\infty$, there exists an entropy solution whose transonic shock stands off the body at distance $d_0$ with far-field asymptotics given by the strong attached oblique shock state of the underlying wedge, and the shock is a convex curve when the body is convex. Because the Mach-number threshold can be made uniform over all $d_0\ge\underline d$, a single incoming flow admits a one-parameter family of such steady solutions, and the paper explicitly leaves open which member of the family — if any — is the physically realized one. This problem asks to close that gap: determine whether the family contains a distinguished physically valid solution, for instance one attained as the long-time limit of the unsteady irrotational Euler flow past the same body, determine the detached distance of that solution, settle uniqueness or non-uniqueness of the steady detached-shock solution for fixed $d_0$, and decide whether convexity of the blunt body is necessary for convexity of the detached shock.

Background

When a uniform supersonic stream of an inviscid gas with polytropic equation of state impinges on an obstacle and the flow is steady, the governing equations are the two-dimensional steady Euler equations. If the flow is moreover irrotational and isentropic, they reduce to the steady Euler system for potential flow, $\partial_{x_1}(\rho u_1)+\partial_{x_2}(\rho u_2)=0,\qquad \partial_{x_1}u_2-\partial_{x_2}u_1=0,$ closed by Bernoulli's law relating $\tfrac12|\mathbf u|^2$ and $\rho^{\gamma-1}$ ($\gamma>1$ is the adiabatic exponent, $\mathbf u=(u_1,u_2)$ the velocity, $\rho$ the density). Writing $c(\rho)$ for the sound speed, the Mach number $M=|\mathbf u|/c(\rho)$ governs the type of the system: it is elliptic in the subsonic region $M<1$ and hyperbolic in the supersonic region $M>1$. Across a shock, that is, a curve across which the solution jumps, the Rankine–Hugoniot conditions hold (conservation of mass and of normal momentum, continuity of tangential velocity), supplemented by the physical entropy condition; a weak solution satisfying these admissibility conditions is called an entropy solution.

For a straight symmetric wedge $W_0=\{x_1\ge|x_2|\cot\theta_w\}$, the classical shock polar analysis — the curve of downstream states in the $(u_1,u_2)$-plane compatible with the Rankine–Hugoniot conditions — yields, when the half-wedge angle $\theta_w$ is below the detachment angle and the incoming Mach number $M_\infty$ is large enough, two entropy solutions with straight oblique shocks attached to the wedge vertex: a weak shock solution, supersonic behind the shock, and a strong shock solution, subsonic behind it. If the wedge tip is rounded into a blunt body $W_b=\{x_1\ge b(x_2)\}$, where the profile $b$ is $C^3$, even in $x_2$, satisfies $b'(0)=0$ (a smooth nose perpendicular to the oncoming flow), $b''\ge0$, and coincides with the wedge profile away from the tip, a shock polar argument combined with the slip condition $\mathbf u\cdot\mathbf n=0$ on $\partial W_b$ shows that an attached shock is impossible: any shock must detach, standing a positive distance — the detached distance $d_0$, classically the standoff distance of the bow shock — ahead of the nose. The existence, shape, and standoff distance of detached shocks past blunt bodies is a classical problem of gas dynamics (Courant and Friedrichs, 1948), long treated only by numerical and asymptotic methods.

Bae and Xiang gave the first rigorous existence theory. Their main theorem states that for each fixed $d_0>0$ there exists $\bar\varepsilon>0$, depending on $\gamma$, the Bernoulli constant of the incoming flow, and $d_0$, such that whenever $M_\infty=1/\varepsilon$ with $\varepsilon\in(0,\bar\varepsilon]$, the potential-flow system admits an entropy solution in the exterior of $W_b$ whose shock $\Gamma_{\rm sh}=\{(f_{\rm sh}(x_2),x_2):x_2\ge0\}$: (i) detaches at $f_{\rm sh}(0)=b_0-d_0$, where $b_0=b(0)$ is the nose position; (ii) stays a uniform positive distance from the body; (iii) converges at far field to the strong attached oblique shock state of the wedge $W_0$, the shock slope converging to the slope of that straight shock; (iv) satisfies quantitative Hölder estimates; and (v) is transonic — the flow behind it is uniformly subsonic, $M\le1-\sigma$. Moreover, the threshold can be made uniform over $d_0\ge\underline d$ for any fixed $\underline d>0$, so one incoming flow possesses at least one entropy solution for every $d_0\in[\underline d,\infty)$: a one-parameter family of steady detached shocks (Remark 2.14 of the paper). When $b''\ge0$, the shock is moreover a convex curve ($f_{\rm sh}''\ge0$, Theorem 2.13(b)), proved via maximum principles in which the convexity of the body enters essentially.

The mathematics before and around this construction brackets the problem without touching its core. Chen, Christoforou, and Jegdić had constructed a family of detached-shock solutions of the potential-flow system past a wedge (not a blunt body) in a bounded domain, comparing them with the attached strong shock via the entropy functional $\int_\Omega\rho^\gamma\,dx$; Moiseev and Shifrin proved unique solvability for a linearized (Lavrent'ev–Bitsadze) model of weakly supersonic flow with a detached shock past a wedge. On the dynamical side — the natural route to any selection principle — the known rigorous results concern flat or straight shocks only: instability of transonic shocks attached to a cone or wedge in three space dimensions (Xu–Yin; Liang–Xu–Yin) and dynamical stability of a planar normal shock in a duct for unsteady potential flow (Fang–Huang–Xiang–Xiao). No dynamical result exists for curved detached shocks, and no known result singles out a detached distance $d_0$.

Problem Statement

Work in the setting of Bae–Xiang's construction: the two-dimensional steady Euler system for irrotational (potential) flow with adiabatic exponent $\gamma>1$ past the symmetric blunt body $W_b=\{x_1\ge b(x_2)\}$ of Definition 1.1 of the cited paper (the profile $b$ is $C^3$, even, $b'(x_2)>0$ for $x_2>0$ with $b'(0)=0$, $b''\ge0$ on $x_2\ge0$, and $b(x_2)=x_2\cot\theta_w$ for $x_2\ge h_0$), with the uniform supersonic incoming state $(\rho_\infty,u_\infty,0)$ of Mach number $M_\infty=1/\varepsilon$ sufficiently large. The construction yields, for any fixed $\underline d>0$, at least one entropy solution with a transonic detached shock $\Gamma_{\rm sh}$ at prescribed detached distance $d_0\in[\underline d,\infty)$ from the nose, each satisfying the far-field, separation, and subsonicity properties (i)–(v) of Theorem 2.13(a), and a convex shock when $b''\ge0$.

Determine the physical and geometric status of this family:

(1) Physical selection (Remark 2.14(i) of the cited paper). Does the family contain a physically valid detached-shock solution — that is, a solution satisfying properties (i)–(v) that is a long-time asymptotic limit of weak solutions of the unsteady Euler system for irrotational flow with the same incoming supersonic state past the same body $W_b$, the limit being taken from initial data other than the steady solution itself (equivalently, a dynamically stable steady state selected by the unsteady evolution)?

(2) Detached distance (Remark 2.14(ii)). If a physically valid solution exists, determine the value of the detached distance $d_0$ for it.

(3) Uniqueness and dynamical stability (Remark 2.14, preceding discussion). Is the entropy detached-shock solution unique for fixed $d_0$ within the class of solutions satisfying properties (i)–(v) — noting that the construction's estimate (2.27) controls only the far-field slope of the shock, so uniqueness is genuinely open — and is one of the constructed steady solutions an attractor of the unsteady irrotational evolution, i.e. dynamically stable, or unstable?

(4) Convexity necessity (Introduction of the cited paper; Remark 2.4 of the authors' 2020 survey). Is the convexity assumption $b''\ge0$ on the blunt body necessary for convexity of the detached shock, or does $f_{\rm sh}''\ge0$ (equivalently, convexity of $\Gamma_{\rm sh}$) persist under strictly weaker geometric hypotheses on $b$?

Items (1)–(3) concern one and the same object — the physically realized steady state among the constructed family — and item (4) asks which part of that state's convex geometry is genuinely forced by the body's convexity; together they supply the physical content that the existence construction deliberately leaves open. Item (2) is void if (1) is answered negatively (no physically valid member exists). The dynamical-stability half of item (3) asks the same selection question as item (1) — whether the unsteady evolution singles out a steady member of the family.

References

1. Myoungjean Bae and Wei Xiang. Detached shock past a blunt body. Acta Applicandae Mathematicae 188 (2023), Article 7. DOI: 10.1007/s10440-023-00617-y. https://doi.org/10.1007/s10440-023-00617-y (author preprint arXiv:1909.13281, https://arxiv.org/abs/1909.13281)
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