Transonic Shock Solution Research Articles

Structural stability of transonic shocks for the full compressible Euler system in a two-dimensional general nozzle with an external force

In this paper, we investigate the transonic shock solutions to the full compressible Euler system in a general two-dimensional nozzle with an external force. In their book, Courant and Friedrichs [Supersonic Flow and Shock Waves (Interscience Publishers, Inc., New York, 1948)] described the transonic shock phenomena in a de Laval nozzle: given the appropriate receiver pressure, if the upcoming flow is still supersonic after passing through the throat of the nozzle, then a shock front intervenes at some place in the diverging part of the nozzle, and the gas is compressed and slowed down to subsonic speed. We first establish the existence and uniqueness of one-dimensional transonic shock solutions to the steady full Euler system with an external force by prescribing suitable pressure at the exit of the nozzle when the upstream flow is a uniform supersonic flow. With the help of external forces, both existence and uniqueness are established by solving a nonlinear free boundary value problem in a weighted Hölder space.

Structural stability of transonic shock flows with an external force

This paper is devoted to the structural stability of a transonic shock passing through a flat nozzle for two-dimensional steady compressible flows with an external force. We first establish the existence and uniqueness of one-dimensional transonic shock solutions to the steady Euler system with an external force by prescribing suitable pressure at the exit of the nozzle when the upstream flow is a uniform supersonic flow. It is shown that the external force helps to stabilize the transonic shock in flat nozzles and the shock position is uniquely determined. Then we are concerned with the structural stability of these transonic shock solutions when the exit pressure is suitably perturbed. One of the new ingredients in our analysis is to use the deformation-curl decomposition to the steady Euler system developed by Weng and Xin [Sci. Sinica Math., 49 (2019), pp. 307–320] to deal with the transonic shock problem.

Existence of transonic shocks in a cylindrical nozzle with given exit normal momentums

In this paper, we study the existence of transonic shocks in a cylindrical nozzle. More precisely, given the compatible outer normal momentum at the exit of a cylindrical nozzle, there exists a transonic shock solution such that the incoming supersonic flow is compressed to some subsonic flow. The shock position is uniquely determined by the exit normal momentum when we fix one point of the shock. This problem can be reduced to solve a nonlinear free boundary value problem for a hyperbolic–elliptic coupled system and the key ingredient is to solve a second‐order quasilinear elliptic equation with a nonlocal term by the method of separation of variables.

Non-uniqueness of transonic shock solutions to Euler–Poisson system with varying background charges

Non-uniqueness of transonic shock solutions to Euler–Poisson system with varying background charges

Sonic–supersonic–shock–subsonic–sonic solution to Euler–Poisson system with varying background charges

In this paper, steady transonic shock solutions to the Euler–Poisson system are investigated with respect to varying background charge. More precisely, the density of the background charge is prescribed as a piecewise constant function whose discontinuity is determined only by the shock front. The Euler–Poisson system with varying background charge can describe the motion of the electrons in the special semiconductor devices. We establish the well-posedness of sonic–supersonic–shock–subsonic–sonic solutions, with sonic physical boundary conditions at the upstream and downstream respectively.

Asymptotic analysis of transonic shocks in divergent nozzles with respect to the expanding angle

In this paper we study the asymptotic behavior of the transonic shock solutions in divergent nozzles as the expanding angle goes to zero. It is well-known that there exist infinite shock solutions for steady 1-D flows in a flat nozzle with the position of the shock front being arbitrary, while there exists a unique shock solution in an divergent nozzle as the pressure at the exit is given within an appropriate interval. By analyzing the asymptotic behavior of the shock solutions as the expanding goes to zero, we are also trying to figure out a criterion which may be used to select the physical one among all shock solutions in the flat nozzle. It finally turns out that the limit shock solution as the expanding angle goes to zero strongly depends on the asymptotic behavior of the receiver pressure, which is assumed to be a function of the expanding angle, imposed at the exit. In particular, as the Mach number of the flow at the entrance is larger than γ+32, the function of the receiver pressure can be set identical to the value of the pressure behind the shock front for the flat nozzle. Then the limit shock solution may be considered as the physical one for the flat nozzle, and the limit position of the shock front being the admissible position of the shock front. To show these results, one of the key steps is to establish quantitative estimates for the position of the shock front for the given pressure at the exit. To this end, a free boundary problem for the linearized Euler system will be proposed which gives an approximation of the shock solution, then a nonlinear iteration scheme could be carried out to approach the shock solution. Moreover, the error estimates between the exact shock solution and the approximation are also established, which give quantitative information on the position of the shock front.

Transonic shock solutions of the steady Euler flow in quasi-one-dimensional convergent nozzles

In this paper, we investigate the well-posedness of transonic shock solutions of the steady Euler flow in quasi-one-dimensional convergent nozzles under the physical boundary conditions proposed by Courant-Friedrichs in [12]. Inspired by [20], for any supersonic incoming flow at the entrance, we derive the critical length of the nozzle and show that as long as the nozzle length is less than or equal to the critical length, the existence and uniqueness of transonic shock solutions with the exit pressure lying in a certain range can be obtained. Moreover, the transonic shock position is monotonically determined by the exit pressure.

Radial Transonic Shock Solutions to Euler-Poisson System with Varying Background Charge in an Annulus

Radial Transonic Shock Solutions to Euler-Poisson System with Varying Background Charge in an Annulus

Dynamic stability of transonic shock solutions for one-dimensional Euler equations with external forcing

Dynamic stability of transonic shock solutions for one-dimensional Euler equations with external forcing

Non-uniqueness of transonic shock solutions to non-isentropic Euler–Poisson system with varying background charges

Non-uniqueness of transonic shock solutions to non-isentropic Euler–Poisson system with varying background charges

Nonlinear structural stability and linear dynamic instability of transonic steady-states to hydrodynamic model for semiconductors

For unipolar hydrodynamic model of semiconductor device represented by Euler-Poisson equations, when the doping profile is supersonic, and the boundary data are in subsonic region and supersonic region separately, the system possesses the shock transonic steady-states and the smooth transonic steady-states. In this paper we study the nonlinear structural stability and the linear dynamic instability of these steady transonic solutions. For any relaxation time: 0<τ≤+∞, by means of elaborate singularity analysis, we first investigate the structural stability of the C1-smooth transonic steady-states, once the perturbations of the initial data and the doping profiles are small enough. We note that, when the C1-smooth transonic steady-states pass through the sonic line, they produce singularities for the system, and cause some essential difficulty in the proof of structural stability. Moreover, when the relaxation time is large enough τ≫1, under the condition that the electric field is positive at the shock location, we prove that the transonic shock steady-states are structurally stable with respect to small perturbations of the supersonic doping profile. Furthermore, we show the linearly dynamic instability for these transonic shock steady-states provided that the electric field is suitable negative. The proofs for the structural stability results are based on singularity analysis, a monotonicity argument on the shock position and the downstream density, and the stability analysis of supersonic and subsonic solutions. The linear dynamic instability of the steady transonic shock for Euler-Poisson equations can be transformed to the ill-posedness of a free boundary problem for the Klein-Gordon equation. By using a nontrivial transformation and the shooting method, we prove that the linearized problem has a transonic shock solution with exponential growths. These results enrich and develop the existing studies.

On Admissible Positions of Transonic Shocks for Steady Isothermal Euler Flows in a Horizontal Flat Nozzle under Vertical Gravity

In this paper we are concerned with the existence of transonic shocks for 2-D steady isothermal Euler flows in a horizontal flat nozzle under vertical gravity. In particular, we focus on the contribution of the vertical gravity in determining the position of the shock front. For steady horizontal flows, the existence of normal shocks with the position of the shock front being arbitrary in the nozzle can be easily established. This paper will try to determine the position of the shock front as the state of the flow at the entrance of the nozzle and the pressure at the exit are slightly perturbed. To this end, this paper proposes a free boundary problem of the linearized Euler system with vertical gravity, whose solution could be an initial approximation for the shock solution with the free boundary being the approximation for the shock front. Due to the existence of the vertical gravity, difficulties arise in solving the boundary value problem in the approximate subsonic domain behind the shock front. The linearized Euler system is elliptic-hyperbolic composite for subsonic flows, and the elliptic part and the hyperbolic part are coupled in the $0$-order terms depending on the acceleration of gravity $g$. Moreover, the coefficients are not constants since the unperturbed shock solution depends on the vertical variable. New ideas and techniques are developed to deal with these difficulties and, under certain sufficient conditions on the perturbation, the existence of the solution to the proposed free boundary problem is established as the acceleration of gravity $g>0$ and the perturbation are sufficiently small. Then, with the obtained initial approximation of the shock solution, a nonlinear iteration scheme can be constructed which leads to a transonic shock solution with the position of the shock front being close to the initial approximating position.

Structural stability of transonic shock flows with external force in a divergent 3-D axisymmetric perturbed nozzle

This paper concerns with the structural stability of the transonic shock for steady Euler flows with external force in a divergent 3-D axisymmetric perturbed nozzle. We first establish the existence and uniqueness of spherical symmetric transonic shock solution to steady isentropic Euler system with external force by prescribing suitable exit pressure at the exit of a straight divergent nozzle when the incoming flow at the entrance is spherical symmetric supersonic flow. Then we investigate the stability of this spherical symmetric transonic shock solution to the 3-D axisymmetric steady isentropic Euler system with external force when the incoming supersonic flow and the nozzle boundary and the exit pressure and the external force are suitable perturbed. One of new ideas in the analysis is to use the deformation-curl decomposition provided in [21] to deal with the transonic shock problem.

On admissible positions of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle

This paper concerns with the existence of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle, which are governed by the Euler equations with the slip boundary condition on the wall of the nozzle and a receiver pressure at the exit. Mathematically, it can be formulated as a free boundary problem with the shock front being the free boundary to be determined. In dealing with the free boundary problem, one of the key points is determining the position of the shock front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution gives an initial approximating position of the shock front. Compared with the 2-D case, new difficulties arise due to the additional 0-order terms and singularities along the symmetric axis. New observations and careful analysis will be done to overcome these difficulties. Once the initial approximation is obtained, a nonlinear iteration scheme can be carried out, which converges to a transonic shock solution to the problem.

3-D Axisymmetric Transonic Shock Solutions of the Full Euler System in Divergent Nozzles

We establish the stability of 3-D axisymmetric transonic shock solutions of the steady full Euler system in divergent nozzles under small perturbations of an incoming radial supersonic flow and a constant pressure at the exit of the nozzles. To study 3-D axisymmetric transonic shock solutions of the full Euler system, we use a stream function formulation of the full Euler system for a 3-D axisymmetric flow. We resolve the singularity issue arising in stream function formulations of the full Euler system for a 3-D axisymmetric flow. We develop a new scheme to determine a shock location of a transonic shock solution of the steady full Euler system based on the stream function formulation.

Non-uniqueness of transonic shock solutions to non-isentropic Euler–Poisson system

Non-uniqueness of transonic shock solutions to non-isentropic Euler–Poisson system

On Admissible Locations of Transonic Shock Fronts for Steady Euler Flows in an Almost Flat Finite Nozzle with Prescribed Receiver Pressure

AbstractThis paper concerns the existence of transonic shock solutions to the 2‐D steady compressible Euler system in an almost flat finite nozzle (in the sense that it is a generic small perturbation of a flat one), under physical boundary conditions proposed by Courant‐Friedrichs in [14], in which the receiver pressure is prescribed at the exit of the nozzle. In the resulting free boundary problem, the location of the shock front is some of the most desirable information one would like to determine. However, the location of the normal shock front in a flat nozzle can be anywhere in the nozzle so that it provides little information on the possible location of the shock front when the nozzle's boundary is perturbed. So one of the key difficulties in looking for transonic shock solutions is to determine the shock front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution will be taken as an initial approximation for the transonic shock solution. Once an initial approximation is obtained, a further nonlinear iteration can be constructed and proved to lead to a transonic shock solution. In this paper, a sufficient condition in terms of the geometry of the nozzle and the given exit pressure is derived that yields the existence of the solutions to the proposed free boundary problem. By this condition, it will be shown that, for the proposed linear free boundary problem, if the nozzle is either strictly expanding or contracting, there exists only one solution as long as the given pressure at the exit lies within a certain interval; while if the nozzle is generic with both expanding and contracting portions, there may exist more than one solution for the same given receiver pressure, which implies the existence of more than one initial approximating shock front. This nonuniqueness of the initial approximations will lead to the nonuniqueness of the transonic shock solutions for generic nozzles with the same given receiver pressure, which shows the instability of the unperturbed normal shock solution under general perturbations of the nozzle boundary and how this instability may behave. © 2020 Wiley Periodicals LLC

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.

Supersonic flow onto solid wedges, multidimensional shock waves and free boundary problems

When an upstream steady uniform supersonic flow impinges onto a symmetric straight-sided wedge, governed by the Euler equations, there are two possible steady oblique shock configurations if the wedge angle is less than the detachment angle—the steady weak shock with supersonic or subsonic downstream flow (determined by the wedge angle that is less than or greater than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which satisfy the entropy condition. The fundamental issue—whether one or both of the steady weak and strong shocks are physically admissible solutions—has been vigorously debated over the past eight decades. In this paper, we survey some recent developments on the stability analysis of the steady shock solutions in both the steady and dynamic regimes. For the static stability, we first show how the stability problem can be formulated as an initial-boundary value type problem and then reformulate it into a free boundary problem when the perturbation of both the upstream steady supersonic flow and the wedge boundary are suitably regular and small, and we finally present some recent results on the static stability of the steady supersonic and transonic shocks. For the dynamic stability for potential flow, we first show how the stability problem can be formulated as an initial-boundary value problem and then use the self-similarity of the problem to reduce it into a boundary value problem and further reformulate it into a free boundary problem, and we finally survey some recent developments in solving this free boundary problem for the existence of the Prandtl-Meyer configurations that tend to the steady weak supersonic or transonic oblique shock solutions as time goes to infinity. Some further developments and mathematical challenges in this direction are also discussed.

Global stability of steady transonic Euler shocks in quasi-one-dimensional nozzles

We prove global in time dynamical stability of steady transonic shock solutions in divergent quasi-one-dimensional nozzles. We assume neither the smallness of the relative slope of the nozzle nor the weakness of the shock. Key ingredients of the proof are an exponentially decaying energy estimate for a linearized problem together with methods from Luo et al. (2011) [12].