Free boundary problems in shock reflection/diffraction and related transonic flow problems

Abstract

Shock waves are steep wavefronts that are fundamental in nature, especially in high-speed fluid flows. When a shock hits an obstacle, or a flying body meets a shock, shock reflection/diffraction phenomena occur. In this paper, we show how several long-standing shock reflection/diffraction problems can be formulated as free boundary problems, discuss some recent progress in developing mathematical ideas, approaches and techniques for solving these problems, and present some further open problems in this direction. In particular, these shock problems include von Neumann's problem for shock reflection–diffraction by two-dimensional wedges with concave corner, Lighthill's problem for shock diffraction by two-dimensional wedges with convex corner, and Prandtl-Meyer's problem for supersonic flow impinging onto solid wedges, which are also fundamental in the mathematical theory of multidimensional conservation laws.

1. Introduction

Shock waves are steep wavefronts that propagate in compressible fluids in which convection dominates diffusion. They are fundamental in nature, especially in high-speed fluid flows. Examples include transonic and/or supersonic shocks formed by supersonic flows impinging onto solid wedges, transonic shocks around supersonic or near-sonic flying bodies, bow shocks created by solar winds in space, blast waves caused by explosions and other shocks due to various natural processes. When a shock hits an obstacle, or a flying body meets a shock, shock reflection/diffraction phenomena occur.

Many such shock reflection/diffraction problems can be formulated as free boundary problems involving nonlinear partial differential equations (PDEs) of mixed elliptic–hyperbolic type. Understanding these shock reflection/diffraction phenomena requires a complete mathematical solution of the corresponding free boundary problems for nonlinear mixed PDEs. In this paper, we show how several long-standing, fundamental multidimensional shock problems can be formulated as free boundary problems, discuss some recent progress in developing mathematical ideas, approaches and techniques for solving these problems, and present some further open problems in this direction. In particular, these shock problems include von Neumann's problem for shock reflection–diffraction by two-dimensional wedges with concave corner, Lighthill's problem for shock diffraction by two-dimensional wedges with convex corner, and Prandtl-Meyer's problem for supersonic flow impinging onto solid wedges. These problems are not only long-standing open problems in fluid mechanics, but also fundamental in the mathematical theory of multidimensional conservation laws. These shock reflection/diffraction configurations are the core configurations in the structure of global entropy solutions of the two-dimensional Riemann problem for hyperbolic conservation laws, whereas the Riemann solutions are the building blocks and local structure of general solutions, and determine global attractors and asymptotic states of entropy solutions, as time tends to infinity, for multidimensional hyperbolic systems of conservation laws (see [1–8] and references therein). In this sense, we have to understand the shock reflection/diffraction phenomena in order to understand fully global entropy solutions to multidimensional hyperbolic systems of conservation laws.

We first focus on these problems for the Euler equations for potential flow. The unsteady potential flow is governed by the conservation law of mass and Bernoulli's law,

1.1

and

1.2

for the density ρ and the velocity potential Φ, where the Bernoulli constant B is determined by the incoming flow and/or boundary conditions, and h(ρ) satisfies the relation

with c(ρ) being the sound speed, and p is the pressure, which is a function of the density ρ. For an ideal polytropic gas, the pressure p and the sound speed c are given by p(ρ)=κρ^γ and c²(ρ)=κγρ^γ−1 for constants γ>1 and κ>0. Without loss of generality, we may choose κ=1/γ to obtain

1.3

This can be achieved by the following scaling: (t,x,B)→(α²t,αx,α⁻²B) with α²=κγ. Taking the limit γ→1+, we can also consider the case of isothermal flow (γ=1), for which

In §§2–4, we first show how the shock problems can be formulated as free boundary problems for the Euler equations for potential flow, and discuss recently developed mathematical ideas, approaches and techniques for solving these free boundary problems. Then, in §5, we present mathematical formulations of these shock problems for the full Euler equations and discuss the role of the Euler equations for potential flow, (1.1) and (1.2), in these shock problems even in the realm of the full Euler equations. Some further open problems in this direction are also addressed.

2. Shock reflection–diffraction and free boundary problems

We are first concerned with von Neumann's problem for shock reflection–diffraction [9–11]. When a vertical planar shock perpendicular to the flow direction x₁ and separating two uniform states (0) and (1), with constant velocities (u₀,v₀)=(0,0) and (u₁,v₁)=(u₁,0), and constant densities ρ₁>ρ₀ (state (0) is ahead of or to the right of the shock, and state (1) is behind the shock), hits a symmetric wedge

head-on at time t=0, a reflection–diffraction process takes place when t>0. Then, a fundamental question is what type of wave patterns of reflection–diffraction configurations may be formed around the wedge. The complexity of reflection–diffraction configurations was first reported by Mach [12] in 1878, who first observed two patterns of reflection–diffraction configurations: regular reflection (two-shock configuration; figure 1a) and Mach reflection (three-shock/one-vortex-sheet configuration; figure 1b); also see [1,13–15]. The issues remained dormant until the 1940s when John von Neumann [9–11], as well as other mathematical/experimental scientists (cf. [1,2,13–16] and references therein), began extensive research on shock reflection–diffraction phenomena, owing to their importance in applications. It was found that the phenomena are much more complicated than what Mach originally observed: the Mach reflection can be further divided into more specific subpatterns, and various other patterns of shock reflection–diffraction configurations may occur, such as the double Mach reflection, von Neumann reflection and Guderley reflection (see [1,2,13–16] and references therein). Thus, the fundamental scientific issues include the following:

• (i) structure of the shock reflection–diffraction configurations;

• (ii) transition criteria between the different patterns of shock reflection–diffraction configurations; and

• (iii) dependence of the patterns upon physical parameters such as the wedge angle θ_w, the incident-shock-wave Mach number and the adiabatic exponent γ≥1.

Figure 1.

(a) Regular reflection; (b) simple Mach reflection; and (c) irregular Mach reflection. From Van Dyke [13, pp. 142–144].

In particular, several transition criteria between the different patterns of shock reflection–diffraction configurations have been proposed, including the sonic conjecture and the detachment conjecture by von Neumann [9–11].

Careful asymptotic analysis has been made for various reflection–diffraction configurations in Lighthill [17,18], Keller & Blank [19], Hunter & Keller [20], Harabetian [21], Morawetz [22] and the references therein; also see Glimm & Majda [2]. Large- or small-scale numerical simulations have also been made; cf. [2,14,23] and references therein. However, most of the fundamental issues for shock reflection–diffraction phenomena have not been understood, especially the global structure and transition between the different patterns of shock reflection–diffraction configurations. This is partially because physical and numerical experiments are hampered by various difficulties and have not yielded clear transition criteria between the different patterns. In particular, numerical dissipation or physical viscosity smear the shocks and cause boundary layers that interact with the reflection–diffraction patterns and can cause spurious Mach streams; cf. [23]. Furthermore, some different patterns occur when the wedge angles are only fractions of a degree apart, a resolution that even sophisticated modern experiments (cf. [24]) have not been able to reach. For this reason, it is almost impossible to distinguish experimentally between the sonic and detachment criteria, as pointed out in [14]. In this regard, the necessary approach to understand fully the shock reflection–diffraction phenomena, especially the transition criteria, is still via rigorous mathematical analysis. To achieve this, it is essential to formulate the shock reflection–diffraction problem as a free boundary problem and to establish the global existence, regularity and structural stability of its solution.

Mathematically, the shock reflection–diffraction problem is a multidimensional lateral Riemann problem in domain .

Problem 2.1 (Lateral Riemann problem) —

Piecewise constant initial data, consisting of state (0) on and state (1) on {x₁<0} connected by a shock at x₁=0, are prescribed at t=0. Seek a solution of the Euler system (1.1) and (1.2) for t≥0 subject to these initial data and the boundary condition ∇Φ⋅ν=0 on ∂W.

Note that problem 2.1 is invariant under the scaling

2.1

Thus, we seek self-similar solutions in the form of

2.2

Then the pseudo-potential function satisfies the following Euler equations for self-similar solutions:

2.3

and

2.4

where the divergence div and gradient D are with respect to (ξ,η). From this, we obtain the following second-order nonlinear PDE for φ(ξ,η):

2.5

with

2.6

where B₀:=(γ−1)B+1. Then we have

2.7

Equation (2.5) is a nonlinear PDE of mixed elliptic–hyperbolic type. It is elliptic if and only if

2.8

If ρ is a constant, then, by (2.5) and (2.6), the corresponding pseudo-potential φ is in the form of

for constants u,v and k.

Then problem 2.1 is reformulated as a boundary value problem in the unbounded domain

in the self-similar coordinates (ξ,η).

Problem 2.2 (Boundary value problem) —

Seek a solution φ of equation (2.5) in the self-similar domain Λ with the slip boundary condition Dφ⋅ν|_∂Λ=0 on the wedge boundary ∂Λ and the asymptotic boundary condition at infinity:

where

and is the location of the incident shock in the coordinates (ξ,η).

By symmetry, we can restrict to the upper half-plane {η>0}∩Λ, with condition ∂_νφ=0 on Γ_sym:={η=0}∩Λ.

A shock is a curve across which Dφ is discontinuous. If Ω⁺ and are two non-empty open subsets of , and S:=∂Ω⁺∩Ω is a C¹-curve where Dφ has a jump, then is a global weak solution of (2.5) in Ω if and only if φ is in and satisfies equation (2.5) and the Rankine–Hugoniot condition on S,

2.9

and the physical entropy condition: the density function ρ(|Dφ|²,φ) increases across S in the relative flow direction with respect to S, where [F]_S is defined by

and ν_s is a unit normal on S.

Note that the condition requires another Rankine–Hugoniot condition on S,

2.10

If a solution has one of the regular reflection–diffraction configurations as shown in figures 2 and 3, and if φ is smooth in the subregion between the wedge and the reflection–diffraction shock, then it should satisfy the boundary condition Dφ⋅ν=0 and the Rankine–Hugoniot conditions (2.9) and (2.10) at P₀ across the shock separating it from state (1). We define the uniform state (2) with pseudo-potential φ₂(ξ,η) such that

the entropy condition holds,

for

the constant density ρ₂ of state (2) is equal to ρ(|Dφ₂|²,φ₂)(P₀) defined by (2.5),

and Dφ₂⋅ν=0 on the wedge boundary with unit normal ν. Then the Rankine–Hugoniot conditions (2.9) and (2.10) hold on the flat shock S₁={φ₁=φ₂} between states (1) and (2), which passes through P₀.

Figure 2.

Supersonic regular reflection.

Figure 3.

Subsonic regular reflection.

State (2) can be either subsonic or supersonic at P₀. This determines the subsonic or supersonic type of regular reflection–diffraction configurations. The supersonic regular reflection–diffraction configuration as shown in figure 2 consists of three uniform states (0), (1), (2), and a non-uniform state in domain Ω, where the equation is elliptic. The reflection–diffraction shock P₀P₁P₂ has a straight part P₀P₁. The elliptic domain Ω is separated from the hyperbolic region P₀P₁P₄ of state (2) by a sonic arc P₁P₄. The subsonic regular reflection–diffraction configuration as shown in figure 3 consists of two uniform states (0) and (1), and a non-uniform state in domain Ω, where the equation is elliptic, and φ_|Ω(P₀)=φ₂(P₀) and D(φ_|Ω)(P₀)=Dφ₂(P₀).

Thus, a necessary condition for the existence of a regular reflection–diffraction solution is the existence of the uniform state (2) determined by the conditions described above. These conditions lead to an algebraic system for the constant velocity (u₂,v₂) and density ρ₂ of state (2), which has solutions for some but not all of the wedge angles. Specifically, for fixed densities ρ₀<ρ₁ of states (0) and (1), there exist a sonic angle θ^s_w and a detachment angle θ^d_w satisfying

such that state (2) exists for all and does not exist for , and the weak state (2) is supersonic at the reflection point P₀(θ_w) for , sonic for θ_w=θ^s_w, and subsonic for for some .

In fact, for each , there exists also a strong state (2) with . There had been a long debate to determine which one is physical for the local theory; see [14,15,25] and references therein. It is expected that the strong reflection–diffraction configuration is non-physical; indeed, it is shown in Chen & Feldman [26] that the weak reflection–diffraction configuration tends to the unique normal reflection, but the strong reflection–diffraction configuration does not, when the wedge angle θ_w tends to π/2. The corresponding reflection–diffraction shock in the weak reflection–diffraction configuration is called a weak shock, since its strength is relatively weak in comparison with the other shock given by the strong state (2). From now on, state (2) always refers to the weak state (2).

If state (2) is supersonic, the propagation speeds of the solution are finite, and state (2) is completely determined by the local information: state (1), state (0) and the location of point P₀. That is, any information from the region of reflection–diffraction, especially the disturbance at corner P₃, cannot travel towards the reflection point P₀. However, if it is subsonic, the information can reach P₀ and interact with it, potentially altering a different reflection–diffraction configuration. This argument motivated the following conjecture by von Neumann [9,10]:

The sonic conjecture. There exists a supersonic reflection–diffraction configuration when θ_w∈(θ^s_w,π/2) for . That is, the supersonicity of state (2) implies the existence of a supersonic regular reflection–diffraction solution, as shown in figure 2.

Another conjecture is that a global regular reflection–diffraction configuration is possible whenever the local regular reflection at the reflection point is possible:

The detachment conjecture. There exists a regular reflection–diffraction configuration for any wedge angle . That is, the existence of state (2) implies the existence of a regular reflection–diffraction solution, as shown in figures 2 and 3.

It is clear that the supersonic/subsonic regular reflection–diffraction configurations are not possible without a local two-shock configuration at the reflection point on the wedge, so this is the weakest possible criterion for the existence of supersonic/subsonic regular shock reflection–diffraction configurations.

Problem 2.3 (Free boundary problem) —

For , find a free boundary (curved reflection–diffraction shock) P₁P₂ on figure 2, and P₀P₂ on figure 3, and a function φ defined in region Ω as shown in figures 2 and 3, such that φ satisfies the following:

• (i) equation (2.5) in Ω;

• (ii) φ=φ₁ and ρDφ⋅ν_s=Dφ₁⋅ν_s on the free boundary;

• (iii) φ=φ₂ and Dφ=Dφ₂ on P₁P₄ in the supersonic case as shown in figure 2 and at P₀ in the subsonic case as shown in figure 3; and

• (iv) Dφ⋅ν=0 on Γ_wedge∩Γ_sym;

where ν_s and ν are the interior unit normals to Ω on Γ_shock and Γ_wedge∩Γ_sym, respectively.

We observe that the key obstacle to the existence of regular shock reflection–diffraction configurations as conjectured by von Neumann [9,10] is an additional possibility that, for some wedge-angle , shock P₀P₂ may attach to the wedge tip P₃, as observed by experimental results (cf. [13, fig. 238]; also see figure 1c for irregular Mach reflection). To describe the conditions of such an attachment, we note that

Then, for each ρ₀, there exists ρ^c>ρ₀ such that

If u₁≤c₁, we can rule out the solution with a shock attached to the wedge tip.

If u₁>c₁, there would be a possibility that the reflection–diffraction shock could be attached to the wedge tip as experiments show [13, fig. 238].

Thus, in references [1,26], we have obtained the following results:

• (i) if ρ₀ and ρ₁ are such that u₁≤c₁, then the supersonic/subsonic regular reflection–diffraction solution exists for each wedge angle ; and

• (ii) if ρ₀ and ρ₁ are such that u₁>c₁, then there exists such that the regular reflection solution exists for each wedge angle . Moreover, if , then, for the wedge angle , there exists an attached solution, i.e. a solution of problem 2.3 with P₂=P₃.

The type of regular reflection–diffraction configuration (supersonic as in figure 2, or subsonic as in figure 3) is determined by the type of state (2) at P₀. For the supersonic and sonic reflection–diffraction cases, the reflection–diffraction shock P₀P₂ is C^2,α-smooth, and the solution φ is C^1,1 across the sonic arc for the supersonic case, which is optimal. For the subsonic reflection–diffraction case (figure 3), the reflection–diffraction shock P₀P₂ and the solution in Ω are both C^1,α near P₀ and away from P₀. Furthermore, the regular reflection–diffraction solution tends to the unique normal reflection, when the wedge angle θ_w tends to π/2.

To solve this free boundary problem (problem 2.3), we define a class of admissible solutions, which are the solutions φ with weak regular reflection–diffraction configurations, such that, in the supersonic reflection case, equation (2.5) is strictly elliptic for φ in , φ₂≤φ≤φ₁ holds in Ω, and the following monotonicity properties hold:

for e=P₀P₁/|P₀P₁|. In the subsonic reflection case, admissible solutions are defined similarly, with changes corresponding to the structure of the subsonic reflection–diffraction solution.

We derive uniform a priori estimates for admissible solutions with any wedge angle for each ε>0, and then apply the degree theory to obtain the existence for each in the class of admissible solutions, starting from the unique normal reflection solution for θ_w=π/2. To derive the a priori bounds, we first obtain the estimates related to the geometry of the shock: show that the free boundary has a uniform positive distance from the sonic circle of state (1) and from the wedge boundary and the symmetric line Γ_sym from P₂ and P₀. This allows us to estimate the ellipticity of (2.5) for φ in Ω (depending on the distance to the sonic arc P₁P₄ for the supersonic reflection–diffraction configuration and to P₀ for the subsonic reflection–diffraction configuration). Then, we obtain the estimates near P₁P₄ (or P₀ for the subsonic reflection) in scaled and weighted C^2,α for φ and the free boundary, considering separately four cases depending on Dφ₂/c₂ at P₀:

• (i) supersonic, |Dφ₂|/c₂≥1+δ;

• (ii) supersonic (almost sonic), 1<|Dφ₂|/c₂<1+δ;

• (iii) subsonic (almost sonic), 1−δ≤|Dφ₂|/c₂≤1; and

• (iv) subsonic, |Dφ₂|/c₂≤1−δ.

In cases (i) and (ii), equation (2.5) is degenerate elliptic in Ω near P₁P₄ in figure 2. In case (iii), the equation is uniformly elliptic in , but the ellipticity constant is small near P₀ in figure 3. Thus, in cases (i)–(iii), we use the local elliptic degeneracy, which allows us to find a comparison function in each case, to show the appropriately fast decay of φ−φ₂ near P₁P₄ in cases (i) and (ii) and near P₀ in case (iii). Furthermore, combining with appropriate local non-isotropic rescaling to obtain the uniform ellipticity, we obtain the a priori estimates in the weighted and scaled C^2,α-norms, which are different in each of cases (i)–(iii), but imply the standard C^1,1-estimates in cases (i) and (ii), and the standard C^2,α-estimates in case (iii). This is an extension of the methods developed in our earlier work [26]. In the uniformly elliptic case (iv), the solution is of subsonic reflection–diffraction configuration as shown in figure 3, and the estimates are more technically challenging than in cases (i)–(iii), owing to the lower a priori regularity of the free boundary and because the uniform ellipticity does not allow a comparison function that shows the decay of φ−φ₂ near P₀. Thus, we prove the C^α-estimates of D(φ−φ₂) near P₀. With all of these, we provide a solution to von Neumann's conjectures.

More details can be found in Chen & Feldman [1], and also see [26–28].

3. Shock diffraction (Lighthill's problem) and free boundary problems

We are now concerned with shock diffraction by a two-dimensional wedge with convex corner (Lighthill's problem). Consider a plane shock in the coordinates (t,x), , with left state (ρ,u,v)=(ρ₁,u₁,0) and right state (ρ₀,0,0) satisfying u₁>0 and ρ₀<ρ₁, going from left to right along a wedge with convex corner

The incident shock interacts with the wedge as it passes the corner, and then shock diffraction occurs (cf. [17,18]). The mathematical study of the shock diffraction problem dates back to the 1950s with the work of Lighthill [17,18] via asymptotic analysis; also see [29–31] via experimental analysis, as well as Courant & Friedrichs [15] and Whitham [32].

Similarly, this problem can be formulated as the following lateral Riemann problem for potential flow.

Problem 3.1 (Lateral Riemann problem; figure 4) —

Seek a solution of system (1.1) and (1.2) with the initial condition at t=0:

3.1

and the slip boundary condition along the wedge boundary ∂W,

3.2

where ν is the exterior unit normal to ∂W.

Figure 4.

Lateral Riemann problem.

Problem 3.1 is also invariant under the self-similar scaling (2.1). Thus, we seek self-similar solutions with the form (2.2) in the self-similar domain outside the wedge:

Then the shock interacts with the pseudo-sonic circle of state (1) to become a transonic shock, and problem 3.1 can be formulated as the following boundary value problem in the self-similar coordinates (ξ,η).

Problem 3.2 (Boundary value problem; figure 5) —

Seek a solution φ of equation (2.5) in the self-similar domain Λ with the slip boundary condition on the wedge boundary ∂Λ:

and the asymptotic boundary condition at infinity,

when in the sense that where φ₀, φ₁ and ξ₀ are the same as defined in problem 2.2 in the coordinates (ξ,η).

Figure 5.

Boundary value problem; see Chen & Xiang [33].

Because φ does not satisfy the slip boundary condition for ξ≥0, the solution must differ from state (1) in {ξ<ξ₁}∩Λ near the wedge corner, which forces the shock to be diffracted by the wedge. Then problem 3.2 can be formulated as the following free boundary problem.

Problem 3.3 (Free boundary problem) —

For θ_w∈(0,π), find a free boundary (curved shock) Γ_shock and a function φ defined in region Ω, enclosed by Γ_shock,Γ_sonic and the wedge boundary , such that φ satisfies the following:

• (i) equation (2.5) in Ω;

• (ii) φ=φ₀, ρDφ⋅ν_s=ρ₀Dφ₀⋅ν_s on Γ_shock;

• (iii) φ=φ₁, Dφ=Dφ₁ on Γ_sonic;

• (iv) Dφ⋅ν=0 on Γ_wedge;

where ν_s and ν are the interior unit normals to Ω on Γ_shock and Γ_wedge, respectively.

In domain Ω, the solution is expected to be pseudo-subsonic and smooth, to satisfy the slip boundary condition along the wedge, and to be C^1,1-continuous across the pseudo-sonic circle to become pseudo-supersonic. Then the solution of problem 3.3 can be shown to be the solution of problem 3.1.

The free boundary problem has been solved in [34,33]. A crucial challenge of this problem is that the expected elliptic domain of the solution is concave, so that its boundary does not satisfy the exterior ball condition, because the angle 2π−θ_w exterior to the wedge at the origin is larger than π for the given wedge angle θ_w∈(0,π), besides other mathematical difficulties including free boundary problems without uniform oblique derivative conditions. There is no general theory of elliptic PDEs on such concave domains, whose coefficients involve the gradient of the solutions. In general, the expected regularity in this domain, even for Laplace's equation, is only C^α with α<1. However, the coefficients in (2.5) depend on the gradient of φ so that the ellipticity of this equation depends also on the boundedness of the derivatives, which is one of the essential difficulties of this problem. To overcome the difficulty, the physical boundary conditions must be exploited to force a finer regularity of solutions at the corner to let equation (2.5) make sense. More precisely, the strategy here is that, instead of analysing equation (2.5) directly, we study another system of equations for the physical quantities (ρ,u,v) for the existence of the velocity potential.

A tempting try would be to differentiate first equation (2.3) to obtain an equation for v, then use the irrotationality to solve u (once v is solved), and finally use (2.4) to solve the density ρ. In order to show the equivalence between these equations and the original potential flow equation (2.5), an additional one-point boundary condition is required for v. However, it is unclear how the boundary condition is to be deduced for v for the problem. Moreover, along the boundaries Γ_shock and Γ²_wedge which meet at the corner, the derivative boundary conditions of the deduced second-order elliptic equation to v are second-kind boundary conditions, i.e. without the viscosity, compared to [26]. This implies that the results from [35,36] could not be directly used. To overcome this, the following directional velocity (w,z) is introduced whose relation with (u,v) is

such that the one-point boundary condition for w is not required for solving w, and then treat z as u.

On the other hand, for these equations, some new technical difficulties arise, for which new mathematical ideas and techniques have to be developed. First, it is a coupled system, so that the coefficients of the nonlinear degenerate elliptic equation for w depend on z, which makes the uniform estimates for w near the sonic circle more challenging. Second, the obliqueness condition on the free boundary deduced from the Rankine–Hugoniot conditions depends on the smallness of z. To overcome this, a degenerate elliptic cut-off function near the pseudo-sonic circle is introduced, which is more precise in comparison with [26]. The reason why the more precise degenerate cut-off function needs to be introduced is that the uniform estimates of w are required to obtain a convergent sequence near P₁, which is crucial for the equivalence between the deduced system and the potential flow equation (2.5) with degenerate elliptic cut-off. Third, because of the new feature that may be 0 along the pseudo-sonic circle, and the fact that there is no C²-regularity at P₁ where the shock and pseudo-sonic circle meet, from the optimal regularity argument by Bae et al. [27], more effort is needed to remove the degenerate elliptic cut-off case by case carefully, near and away from P₁, respectively. The final main difficulty is to show the equivalence of the original potential flow equation (2.5) and the deduced second-order equation for w with irrotationality and Bernoulli's law, which requires gradient estimates for w near the pseudo-sonic circle, but the estimates by scaling only provide a bound divided by the distance to this circle. This is overcome, thanks to the estimates involving ϵ.

When the wedge angle becomes smaller, several other difficulties arise. Owing to the concave corner of the solution domain at the origin, more technical arguments are required to obtain the existence of solutions to the modified problem. Unlike in [1], because it requires one to take derivatives along the shock to obtain a boundary condition for w, a new way to modify the Rankine–Hugoniot conditions is designed delicately, based on the nonlinear structure of the shock. From this modified condition, the Dirichlet condition is assigned on the shock where the modified uniform oblique condition fails. Thus, the uniform boundedness of solutions is required to be controlled more carefully. Finally, the existence of shock diffraction configuration is established.

4. Prandtl–Meyer reflection configurations and free boundary problems

We now consider Prandtl-Meyer's problem for unsteady global solutions for supersonic flow onto a solid ramp, which can also be regarded as portraying the symmetric gas flow impinging onto a solid wedge (by symmetry). For a steady supersonic flow past a solid ramp whose angle is less than the critical angle (called the detachment angle) θ^d_w, Prandtl [37] employed the shock polar analysis to show that there are two possible configurations: the weak shock reflection with supersonic or subsonic downstream flow and the strong shock reflection with subsonic downstream flow, which both satisfy the physical entropy conditions, provided that we do not impose additional conditions downstream; also see [15,38–40].

The fundamental question of whether one or both of the strong and the weak shocks are physically admissible has been vigorously debated over the past seventy years, but has not yet been settled in a definite manner (cf. [15,41–43]). On the basis of experimental and numerical evidence, there are strong indications that it is the weak reflection that is physically admissible. One plausible approach is to single out the strong shock reflection by the consideration of stability: the stable ones are physical. It has been shown in the steady regime that the weak reflection is not only structurally stable (cf. [44]), but also L¹-stable with respect to steady small perturbation of both the ramp slope and the incoming steady upstream flow (cf. [45]), whereas the strong reflection is also structurally stable for a large spectrum of physical parameters (cf. [46,47]).

We are interested in the rigorous unsteady analysis of the steady supersonic weak shock solution as the long-time behaviour of an unsteady flow and establishing the stability of the steady supersonic weak shock solution as the long-time asymptotics of an unsteady flow with the Prandtl–Meyer configuration for all the admissible physical parameters for potential flow. Our goal is to find a solution (ρ,Φ) to system (1.1) and (1.2) when a uniform flow in with , is heading towards a solid ramp at t=0:

Problem 4.1 (Lateral Riemann problem) —

Seek a solution of system (1.1) and (1.2) with B= and the initial condition at t=0:

4.1

and with the slip boundary condition along the wedge boundary ∂W,

4.2

where ν is the exterior unit normal to ∂W.

Again, problem 4.1 is invariant under the self-similar scaling (2.1). Thus, we seek self-similar solutions in the form (2.2), so that the pseudo-potential function satisfies the nonlinear PDE (2.5) of mixed type.

As the incoming flow has constant velocity , the corresponding pseudo-potential can be expressed as

4.3

Then problem 4.1 can be reformulated as the following boundary value problem in the domain

in the self-similar coordinates (ξ,η), which corresponds to in the (t,x) coordinates:

Problem 4.2 (Boundary value problem) —

Seek a solution φ of equation (2.5) in the self-similar domain Λ with the slip boundary condition:

4.4

and the asymptotic boundary condition at infinity,

4.5

along each ray with θ∈(θ_w,π) as in the sense that

4.6

In particular, we seek a weak solution of problem 4.2 with two types of Prandtl–Meyer reflection configurations whose occurrence is determined by the wedge angle θ_w for the two different cases: one contains a straight weak oblique shock attached to the wedge tip O and the oblique shock is connected to a normal shock through a curved shock when θ_w<θ^s_w, as shown in figure 6; the other contains a curved shock attached to the wedge tip and connected to a normal shock when , as shown in figure 7, in which the curved shock Γ_shock is tangential to a straight weak oblique shock S₀ at the wedge tip.

Figure 6.

Admissible solutions for in the self-similar coordinates (ξ,η); see Bae et al. [48].

Figure 7.

Admissible solutions for in the self-similar coordinates (ξ,η); see Bae et al. [48].

To seek a global entropy solution of problem 4.2 with the structure of figure 6 or figure 7, one needs to compute the pseudo-potential φ₀ below S₀.

Given , we obtain (u₀,v₀) and ρ₀ by using the shock polar curve in figure 8 for steady potential flow. In figure 8, θ^s_w is the wedge angle such that line intersects with the shock polar curve at a point on the circle of radius , and θ^d_w is the wedge angle such that line is tangential to the shock polar curve and there is no intersection between line and the shock polar curve when .

Figure 8.

Shock polars in the plane (u,v).

For any wedge angle , line and the shock polar curve intersect at a point (u₀,v₀) with and ; whereas, for any wedge angle , they intersect at a point (u₀,v₀) with u₀>u_d and . The intersection state (u₀,v₀) is the velocity for steady potential flow behind an oblique shock S₀ attached to the wedge tip with angle θ_w. The strength of shock S₀ is relatively weak compared with the other shock given by the other intersection point on the shock polar curve. Thus, S₀ is called a weak shock and the corresponding state (u₀,v₀) is a weak state.

We also note that states (u₀,v₀) depend smoothly on and θ_w, and such states are supersonic when θ_w∈(0,θ^s_w) and subsonic when .

Once (u₀,v₀) is determined, by (2.10) and (4.3), the pseudo-potentials φ₀ and φ₁ below the weak oblique shock S₀ and the normal shock S₁ are respectively in the form of

4.7

for constants u₀,v₀,u₁,v₁ and k₁. Then it follows from (2.6) and (4.7) that the corresponding densities ρ₀ and ρ₁ below S₀ and S₁ are constants, respectively. In particular, we have

4.8

Then problem 4.2 can be formulated into the following free boundary problem.

Problem 4.3 (Free boundary problem) —

For , find a free boundary (curved shock) Γ_shock and a function φ defined in domain Ω, as shown in figures 6 and 7, such that φ satisfies:

• (i) equation (2.5) in Ω;

• (ii)  and on Γ_shock;

• (iii)  and on when θ_w∈(0,θ^s_w) and on when for ; and

• (iv) Dφ⋅ν=0 on Γ_wedge;

where ν_s and ν are the interior unit normals to Ω on Γ_shock and Γ_wedge, respectively.

Let φ be a solution of problem 4.3 with shock Γ_shock. Moreover, assume that , and Γ_shock is a C¹-curve up to its endpoints. To obtain a solution of problem 4.2 from φ, we have two cases:

For , we divide domain Λ∩{η≥0} in the self-similar coordinates (ξ,η) into five separate regions. Let Ω_S be the unbounded domain below curve and above Γ_wedge (figure 6). In Ω_S, let Ω₀ be the bounded open domain enclosed by and {η=0}. We set . Define a function φ_* in {η≥0} by

4.9

By (2.10) and (iii) of problem 4.3, φ_* is continuous in Λ∩{η≥0}∖Ω_S and is C¹ in . In particular, φ_* is C¹ across . Moreover, using (i)–(iii) of problem 4.3, we obtain that φ_* is a global entropy solution of equation (2.5) in Λ∩{η>0}, which is the Prandtl–Meyer supersonic reflection configuration.

For , region in φ_* reduces to one point O, and the corresponding φ_* is a global entropy solution of equation (2.5) in Λ∩{η>0}, which is the Prandtl–Meyer subsonic reflection configuration.

The free boundary problem (problem 4.3) has been solved in Bae et al. [49,48]. Also see Elling & Liu [40] for earlier results for an important regime of physical parameters.

To solve this free boundary problem, we follow the approach introduced in Chen & Feldman [1]. We first define a class of admissible solutions, which are the solutions φ with Prandtl–Meyer reflection configuration, such that, when , equation (2.5) is strictly elliptic for φ in , holds in Ω, and the following monotonicity properties hold:

where e_S0 and e_S1 are the unit tangential directions to lines S₀ and S₁, respectively, pointing to the positive direction in ξ. For the case , admissible solutions are defined similarly, with corresponding changes to the structure of subsonic reflection solutions.

We derive uniform a priori estimates for admissible solutions for any wedge angle for each ε>0, and then employ the Leray–Schauder degree argument to obtain the existence for each in the class of admissible solutions, starting from the unique normal solution for θ_w=0.

More details can be found in references [49,48]; also see §2 and Chen & Feldman [1].

In Chen et al. [50], we have also established the strict convexity of the curved (transonic) part of the free boundary in the shock reflection–diffraction problem in §2 (also see Chen & Feldman [1]), shock diffraction in §3 (also see Chen & Xiang [34,33]), and the Prandtl–Meyer reflection described in §4 (also see Bae et al. [49]). In order to prove the convexity, we employ global properties of admissible solutions, including the existence of the monotonicity cone discussed above.

5. The shock reflection/diffraction problems and free boundary problems for the full Euler equations

When the vortex sheets and the deviation of vorticity become significant, the full Euler equations are required. Here, we present a mathematical formulation of the shock reflection/diffraction problems for the full Euler equations and the role of the potential theory for the shock problems even in the realm of the full Euler equations. In particular, the Euler equations for potential flow, (2.3) and (2.4), are actually exact in an important region of the solutions to the full Euler equations.

The full Euler equations for compressible fluids in , are of the following form:

5.1

where ρ is the density, v=(u,v) the fluid velocity, p the pressure and e the internal energy. Two other important thermodynamic variables are the temperature θ and the energy S. The notation a⊗b denotes the tensor product of the vectors a and b.

Choosing (ρ,S) as the independent thermodynamic variables, then the constitutive relations can be written as (e,p,θ)=(e(ρ,S),p(ρ,S),θ(ρ,S)) governed by

For a polytropic gas,

5.2

or equivalently,

5.3

where R>0 may be taken to be the universal gas constant divided by the effective molecular weight of the particular gas, c_v>0 is the specific heat at constant volume, γ>1 is the adiabatic exponent, and κ>0 is any constant under scaling.

Note that the corresponding three lateral Riemann problems in §§2–4 for system (5.1) are all invariant under the self-similar scaling: (t,x)→(αt,αx) for any α≠0. Therefore, we seek self-similar solutions:

Then the self-similar solutions are governed by the following system:

5.4

where , and (U,V)=(u−ξ,v−η) is the pseudo-velocity.

The eigenvalues of system (5.4) are

where is the sonic speed.

When the flow is pseudo-subsonic, i.e. q<c, the eigenvalues λ_± become complex, and thus the system consists of two transport equations and two nonlinear equations of elliptic–hyperbolic mixed type. Therefore, system (5.4) is hyperbolic–elliptic composite-mixed in general.

The three lateral Riemann problems can be formulated as the corresponding boundary value problems in the unbounded domains. Then the boundary value problems can be further formulated as the three corresponding free boundary problems. The free boundary conditions are again the Rankine–Hugoniot conditions along the free boundary S:

5.5

where L and N are the tangential and normal components of velocity (U,V) along the free boundary, that is, |(U,V)|²=L²+N². The conditions along the sonic circles are the Dirichlet conditions for (U,V,p,ρ) to be continuous across the respective sonic circles.

We now discuss the role of the potential flow equation (2.5) in these free boundary problems whose boundaries also include the fixed degenerate sonic circles for the full Euler equations (5.4).

Under the Hodge–Helmholtz decomposition (U,V)=Dφ+W with div W=0, the Euler equations (5.4) become

5.6

5.7

5.8

5.9

where ω=curl W=curl(U,V) is the vorticity of the fluid, and S=c_v ln(pρ^−γ) is the entropy.

When ω=0, S=const. and W=0 on a curve Γ transverse to the fluid direction, we first conclude from (5.8) that, in domain Ω_E determined by the fluid trajectories past Γ:

we have

This implies that W=const. since div W=0. Then we conclude

since W|_Γ=0, which yields that the right-hand side of equation (5.7) vanishes. Furthermore, from (5.9),

which implies that

By scaling, we finally conclude that the solution of system (5.6)–(5.9) in domain Ω_E is determined by the Euler equations (2.3) and (2.4) for self-similar potential flow, or the potential flow equation (2.5) with (2.6) for self-similar solutions.

For our problems in §§2–4, for system (5.6)–(5.9), we note that, in the supersonic states joint with the sonic circles (e.g. state (2) for problem 2.3, state (1) for problem 3.3, states (0) and (1) for problem 4.3),

5.10

Then, if our solution (U,V,p,ρ) is C^0,1 and the gradient of the tangential component of the velocity is continuous across the sonic arc, we still have (5.10) along Γ_sonic on the side of Ω. Thus, we have the following result.

Theorem 5.1 —

Let (U,V,p,ρ) be a solution of one of our problems, problems 2.3, 3.3 and 4.3, for system (5.6)–(5.9), such that (U,V,p,ρ) is C^0,1 in the open region formed by the reflection–diffraction shock and the wedge boundary, and the gradient of the tangential component of (U,V) is continuous across any sonic arc. Let Ω_E be the subregion of Ω formed by the fluid trajectories past the sonic arc. Then, in Ω_E, the potential flow equation (2.5) with (2.6) coincides with the full Euler equations (5.6)–(5.9), that is, equation (2.5) with (2.6) is exact in domain Ω_E for problems 2.3, 3.3 and 4.3.

Remark 5.2 —

The regions such as Ω_E also exist in various Mach reflection–diffraction configurations. Theorem 5.1 applies to such regions whenever the solution (U,V,p,ρ) is C^0,1 and the gradient of the tangential component of (U,V) is continuous. In fact, theorem 5.1 indicates that, for the solution φ of (2.5) with (2.6), the C^1,1-regularity of φ and the continuity of the tangential component of the velocity field (U,V)=∇φ are optimal across the sonic arc Γ_sonic.

Remark 5.3 —

The importance of the potential flow equation (1.1) with (1.2) in the time-dependent Euler flows even through weak discontinuities was also observed by Hadamard [51] through a different argument. Moreover, for the solutions containing a weak shock, the potential flow equation (1.1) and (1.2) and the full Euler flow model (5.1) match each other well up to the third order of the shock strength. Also see Bers [52], Glimm & Majda [2] and Morawetz [22].

6. Conclusion

As we have discussed above, the three long-standing, fundamental transonic flow problems can be all formulated as free boundary problems. Understanding these transonic flow problems requires a complete mathematical solution of these free boundary problems. Similar free boundary problems also arise in many other transonic flow problems, such as steady transonic flow problems including transonic nozzle flow problems (cf. [53–56]), steady transonic flows past obstacles (cf. [44,46,47,54,57]), supersonic bubbles in subsonic flow (cf. [58,59]), local stability of Mach configurations (cf. [60]), as well as the higher-dimensional version of problem 2.3 (shock reflection–diffraction by a solid cone) and problem 4.3 (supersonic flow impinging onto a solid cone). In §§2–5, we have discussed recently developed mathematical ideas, approaches and techniques for solving these free boundary problems. On the other hand, many free boundary problems arising from transonic flow problems are still open and demand further developments of new mathematical ideas, approaches and techniques.

Competing interests

We declare we have no competing interests.

Funding

The work of G.-Q.C. was supported in part by NSF grant no. DMS-0807551, the UK EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E035027/1), the UK EPSRC Award to the EPSRC Centre for Doctoral Training in PDEs (EP/L015811/1), and the Royal Society–Wolfson Research Merit Award (UK). The work of Mikhail Feldman was supported in part by the National Science Foundation under grants DMS-1101260 and DMS-1401490 and by the Simons Foundation under the Simons Fellows Program.