On transonic shocks in a conic divergent nozzle with axi-symmetric exit pressures

In this paper, we establish the existence and stability of a 3-D transonic shock solution to the full steady compressible Euler system in a class of de Laval nozzles with a conic divergent part when a given variable axi-symmetric exit pressure lies in a suitable scope. Thus, for this class of nozzles, we have solved such a transonic shock problem in the axi-symmetric case described by Courant and Friedrichs (1948) in Section 147 of [8]: Given the appropriately large exit pressure pe(x), if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes pe(x).     

Examples of steady subsonic flows in a convergent-divergent approximate nozzle

We construct special solutions of the full Euler system for steady compressible flows in a convergent-divergent approximate nozzle and study the stability of the purely subsonic flows. For a given pressure p₀ prescribed at the entry of the nozzle, as the pressure p₁ at the exit decreases, the flow patterns in the nozzle change continuously: there appear subsonic flow, subsonic-sonic flow, transonic flow and transonic shocks. Our results indicate that, to determine a subsonic flow in a two-dimensional nozzle, if the Bernoulli constant is uniform in the flow field, then this constant should not be prescribed if the pressure, density at the entry and the pressure at the exit of the nozzle are given; if the Bernoulli constant and both the pressures at the entrance and the exit are given, the average of the density at the entrance is then totally determined.     

Global transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone

In this paper, we are concerned with the global existence and stability of a steady transonic conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Theoretically, as indicated in [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, 1948], it follows from the Rankine-Hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the vertex of the sharp cone in terms of the different pressure states at infinity behind the shock surface, which correspond to the supersonic shock and the transonic shock respectively. In the references [Shuxing Chen, Zhouping Xin, Huicheng Yin, Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys. 228 (2002) 47-84; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Polytropic case, preprint, 2006; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Isothermal case, Pacific J. Math. 233 (2) (2007) 257-289] and [Zhouping Xin, Huicheng Yin, Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. 4 (2) (2006) 101-132], the authors have established the global existence and stability of a supersonic shock for the perturbed hypersonic incoming flow past a sharp cone when the pressure at infinity is appropriately smaller than that of the incoming flow. At present, for the supersonic symmetric incoming flow, we will study the global transonic shock problem when the pressure at infinity is appropriately large.     

Transonic shock in a nozzle I: 2D case

In this paper we establish the existence and uniqueness of a transonic shock for the steady flow through a general two‐dimensional nozzle with variable sections. The flow is governed by the inviscid potential equation, and is supersonic upstream, has no‐flow boundary conditions on the nozzle walls, and a given pressure at the exit of the exhaust section. The transonic shock is a free boundary dividing two regions of C flow in the nozzle. The potential equation is hyperbolic upstream where the flow is supersonic, and elliptic in the downstream subsonic region. In particular, our results show that there exists a solution to the corresponding free boundary problem such that the equation is always subsonic in the downstream region of the nozzle when the pressure in the exit of the exhaustion section is appropriately larger than that in the entry. This confirms exactly the conjecture of Courant and Friedrichs on the transonic phenomena in a nozzle [10]. Furthermore, the stability of the transonic shock is also proved when the upstream supersonic flow is a small steady perturbation for the uniform supersonic flow or the pressure at the exit has a small perturbation. The main ingredients of our analysis are a generalized hodograph transformation and multiplier methods for elliptic equation with mixed boundary conditions and corner singularities. © 2004 Wiley Periodicals, Inc.     

Transonic shocks in 3-D compressible flow passing a duct with a general section for Euler systems

This paper is devoted to the study of a transonic shock in three-dimensional steady compressible flow passing a duct with a general section. The flow is described by the steady full Euler system, which is purely hyperbolic in the supersonic region and is of elliptic-hyperbolic type in the subsonic region. The upstream flow at the entrance of the duct is a uniform supersonic one adding a three-dimensional perturbation, while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference. The problem to determine the transonic shock and the flow behind the shock is reduced to a free boundary value problem of an elliptic-hyperbolic system. The new ingredients of our paper contain the decomposition of the elliptic-hyperbolic system, the determination of the shock front by a pair of partial differential equations coupled with the three-dimensional Euler system, and the regularity analysis of solutions to the boundary value problems introduced in our discussion.

     

Uniqueness of transonic shock solutions in a duct for steady potential flow

We study the uniqueness of solutions with a transonic shock in a duct in a class of transonic shock solutions, which are not necessarily small perturbations of the background solution, for steady potential flow. We prove that, for given uniform supersonic upstream flow in a straight duct, there exists a unique uniform pressure at the exit of the duct such that a transonic shock solution exists in the duct, which is unique modulo translation. For any other given uniform pressure at the exit, there exists no transonic shock solution in the duct. This is equivalent to establishing a uniqueness theorem for a free boundary problem of a partial differential equation of second order in a bounded or unbounded duct. The proof is based on the maximum/comparison principle and a judicious choice of special transonic shock solutions as a comparison solution.     

Transonic nozzle flows and free boundary problems for the full Euler equations

We establish the existence and uniqueness of transonic flows with a transonic shock through a two-dimensional nozzle of slowly varying cross-sections. The transonic flow is governed by the steady, full Euler equations. Given an incoming smooth flow that is close to a constant supersonic state (i.e., smooth Cauchy data) at the entrance and the subsonic condition with nearly horizontal velocity at the exit of the nozzle, we prove that there exists a transonic flow whose downstream smooth subsonic region is separated by a smooth transonic shock from the upstream supersonic flow. This problem is approached by a one-phase free boundary problem in which the transonic shock is formulated as a free boundary. The full Euler equations are decomposed into an elliptic equation and a system of transport equations for the free boundary problem. An iteration scheme is developed and its fixed point is shown to exist, which is a solution of the free boundary problem, by combining some delicate estimates for the elliptic equation and the system of transport equations with the Schauder fixed point argument. The uniqueness of transonic nozzle flows is also established by employing the coordinate transformation of Euler-Lagrange type and detailed estimates of the solutions.     

The transonic shock in a nozzle, 2-D and 3-D complete Euler systems

In this paper, we study a transonic shock problem for the Euler flows through a class of 2-D or 3-D nozzles. The nozzle is assumed to be symmetric in the diverging (or converging) part. If the supersonic incoming flow is symmetric near the divergent (or convergent) part of the nozzle, then, as indicated in Section 147 of [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publ., New York, 1948], there exist two constant pressures P₁ and P₂ with P₁<P₂ such that for given constant exit pressure Pe∈(P₁,P₂), a symmetric transonic shock exists uniquely in the nozzle, and the position and the strength of the shock are completely determined by Pe. Moreover, it is shown in this paper that such a transonic shock solution is unique under the restriction that the shock goes through the fixed point at the wall in the multidimensional setting. Furthermore, we establish the global existence, stability and the long time asymptotic behavior of an unsteady symmetric transonic shock under the exit pressure Pe when the initial unsteady shock lies in the symmetric diverging part of the 2-D or 3-D nozzle. On the other hand, it is shown that an unsteady symmetric transonic shock is structurally unstable in a global-in-time sense if it lies in the symmetric converging part of the nozzle.     

Stability of transonic shocks in supersonic flow past a wedge

In this paper we study the stability of transonic shocks in steady supersonic flow past a wedge. We take the potential flow equation as the mathematical model to describe the compressible flow. It is known that in generic case such a problem admits two possible location of shock, connecting the flow ahead it and behind it. They can be distinguished as supersonic-supersonic shock and supersonic-subsonic shock (or transonic shock). Both these possible shocks satisfy the Rankine-Hugoniot conditions and entropy condition. In this paper we prove that the transonic shock is also stable under perturbation of the coming flow provided the pressure at infinity is well controlled.     

On subsonic compressible flows in a two-dimensional duct

This paper concerns subsonic flows passing a two-dimensional duct for the steady compressible Euler system. If the Bernoulli constant is uniform in the flow field, the density at the entry and both the pressures at the entrance and the exit are given, we show that the problem is generally ill-posed; but if we give the pressure at the exit with a constant difference, then under the same other conditions as above we establish the existence of subsonic flows.     

Transonic shock wave in an infinite nozzle asymptotically converging to a cylinder

We construct a single transonic shock wave pattern in an infinite nozzle asymptotically converging to a cylinder, which is close to a uniform transonic shock wave. In other words, suppose there is a uniform transonic shock wave in an infinite cylinder nozzle which can be constructed easily, if we perturbed the supersonic incoming flow and the infinite nozzle a little bit, we can obtain a transonic wave near the uniform one. As a consequence, we can show that the uniform transonic wave is stable with respect to the perturbation of the incoming flow and nozzle wall. Based on the theory of [G.Q. Chen, M. Feldman, Existence and stability of multi-dimensional transonic flows through an infinite nozzle of arbitrary cross-sections, Arch. Ration. Mech. Anal. 184 (2007) 185-242], the crucial parts of this paper are to derive the uniform Schauder estimates of the linear elliptic equation for the infinite nozzle asymptotically converging to a cylinder.     

Stability of transonic shock for supersonic flow past a wedge

When steady supersonic flow hits a slim wedge, there may appear an oblique transonic shock attached to the vertex of the wedge, if the downstream pressure is rather large. This paper studies stability in certain weighted partial Hölder spaces of the oblique transonic shock attached to the vertex of a wedge, which is against steady supersonic flows, under perturbations of the upstream flow and the profile of the wedge. We show that under reasonable conditions on the upcoming supersonic flow and the slope of the wedge, such transonic shocks are structural stable. Mathematically, we solve an elliptic–hyperbolic mixed type in an unbounded domain, and the flow field is proved to be C¹. Copyright © 2015 John Wiley & Sons, Ltd.     

STABILIZATION EFFECT OF FRICTIONS FOR TRANSONIC SHOCKS IN STEADY COMPRESSIBLE EULER FLOWS PASSING THREE-DIMENSIONAL DUCTS

Transonic shocks play a pivotal role in designation of supersonic inlets and ramjets. For the three-dimensional steady non-isentropic compressible Euler system with frictions, we constructe a family of transonic shock solutions in rectilinear ducts with square cross-sections. In this article, we are devoted to proving rigorously that a large class of these transonic shock solutions are stable, under multidimensional small perturbations of the upcoming supersonic flows and back pressures at the exits of ducts in suitable function spaces.This manifests that frictions have a stabilization effect on transonic shocks in ducts, in consideration of previous works which shown that transonic shocks in purely steady Euler flows are not stable in such ducts. Except its implications to applications, because frictions lead to a stronger coupling between the elliptic and hyperbolic parts of the three-dimensional steady subsonic Euler system, we develop the framework established in previous works to study more complex and interesting Venttsel problems of nonlocal elliptic equations.     

Global supersonic conic shock wave for the steady supersonic flow past a cone: Polytropic gas

In this paper, we establish the global existence and stability of a steady conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body as long as the vertex angle is less than a critical value. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Based on the delicate asymptotic expansion of the background solution, one can verify that the boundary conditions on the shock and the conic surface satisfy the “dissipative” property. From this property, by use of the reflected characteristics method and the special form of the shock equation, we show that the conic shock attached at the vertex of the cone exists globally in the whole space when the speed of the supersonic coming flow is appropriately large. On the other hand, we remove the smallness restriction on the sharp vertex angle in order to establish the global existence of a shock or a global weak solution, moreover, our proof approach is different from that in [Shuxing Chen, Zhouping Xin, Huicheng Yin, Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys. 228 (2002) 47-84] and [Zhouping Xin, Huicheng Yin, Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. 4 (2) (2006) 101-132].     

Numerical solutions of transonic two-dimensional flows at a ninety-degree wedge

We present numerical results on self-similar two-dimensional Riemann problems governed by the compressible Euler system and the nonlinear wave system, which give rise to a transonic shock. We consider a configuration for a vertical incident shock moving to the right above a rectangular object. The incident shock then interacts with a sonic circle soon after it moves beyond the object, and creates a transonic region. We implement Lax–Liu positive schemes and Strang splitting, and obtain linear correlations of the incident shock strength and the shock strength at the vertical wall. We further implement Roe average methods and finite volume methods on quadrilateral grids to capture a contact discontinuity of the Euler system near the corner of the object. The contact discontinuity creates a new supersonic state and a transonic shock inside the transonic region.     

A class of quasilinear degenerate elliptic problems

We establish the existence of solutions for a class of quasilinear degenerate elliptic equations. The equations in this class satisfy a structure condition which provides ellipticity in the interior of the domain, and degeneracy only on the boundary. Equations of transonic gas dynamics, for example, satisfy this property in the region of subsonic flow and are degenerate across the sonic surface. We prove that the solution is smooth in the interior of the domain but may exhibit singular behavior at the degenerate boundary. The maximal rate of blow-up at the degenerate boundary is bounded by the “degree of degeneracy” in the principal coefficients of the quasilinear elliptic operator. Our methods and results apply to the problems recently studied by several authors which include the unsteady transonic small disturbance equation, the pressure-gradient equations of the compressible Euler equations, and the singular quasilinear anisotropic elliptic problems, and extend to the class of equations which satisfy the structure condition, such as the shallow water equation, compressible isentropic two-dimensional Euler equations, and general two-dimensional nonlinear wave equations. Our study provides a general framework to analyze degenerate elliptic problems arising in the self-similar reduction of a broad class of two-dimensional Cauchy problems.     

Subsonic Euler flows in a divergent nozzle

We characterize a class of physical boundary conditions that guarantee the existence and uniqueness of the subsonic Euler flow in a general finitely long nozzle.More precisely,by prescribing the incoming flow angle and the Bernoulli’s function at the inlet and the end pressure at the exit of the nozzle,we establish an existence and uniqueness theorem for subsonic Euler flows in a 2-D nozzle,which is also required to be adjacent to some special background solutions.Such a result can also be extended to the 3-D asymmetric case.     

Continuous Subsonic-Sonic Flows in a Convergent Nozzle

This paper concerns continuous subsonic-sonic potential flows in a two-dimensional convergent nozzle. It is shown that for a given nozzle which is a perturbation of a straight one, a given point on its wall where the curvature is zero, and a given inlet which is a perturbation of an arc centered at the vertex, there exists uniquely a continuous subsonic-sonic flow whose velocity vector is along the normal direction at the inlet and the sonic curve, which satisfies the slip conditions on the nozzle walls and whose sonic curve intersects the upper wall at the given point. Furthermore, the sonic curve of this flow is a free boundary, where the flow is singular in the sense that the speed is only C^1/2 Hölder continuous and the acceleration blows up. The perturbation problem is solved in the potential plane, where the flow is governed by a free boundary problem of a degenerate elliptic equation with two free boundaries and two nonlocal boundary conditions, and the equation is degenerate at one free boundary.     

On global transonic shocks for the steady supersonic Euler flows past sharp 2-D wedges

In this paper, under certain downstream pressure condition at infinity, we study the globally stable transonic shock problem for the perturbed steady supersonic Euler flow past an infinitely long 2-D wedge with a sharp angle. As described in the book of Courant and Friedrichs [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948] (pages 317-318): when a supersonic flow hits a sharp wedge, it follows from the Rankine-Hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the edge of the sharp wedge in terms of the different pressure states in the downstream region, which correspond to the supersonic shock and the transonic shock respectively. It has frequently been stated that the strong shock is unstable and that, therefore, only the weak shock could occur. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand this open problem. More concretely, we will establish the global existence and stability of a transonic shock solution for 2-D full Euler system when the downstream pressure at infinity is suitably given. Meanwhile, the asymptotic state of the downstream subsonic solution is determined.     

Stability of transonic shock fronts in two-dimensional Euler systems

We study the stability of stationary transonic shock fronts under two-dimensional perturbation in gas dynamics. The motion of the gas is described by the full Euler system. The system is hyperbolic ahead of the shock front, and is a hyperbolic-elliptic composed system behind the shock front. The stability of the shock front and the downstream flow under two-dimensional perturbation of the upstream flow can be reduced to a free boundary value problem of the hyperbolic-elliptic composed system. We develop a method to deal with boundary value problems for such systems. The crucial point is to decompose the system to a canonical form, in which the hyperbolic part and the elliptic part are only weakly coupled in their coefficients. By several sophisticated iterative processes we establish the existence and uniqueness of the solution to the described free boundary value problem. Our result indicates the stability of the transonic shock front and the flow field behind the shock.