Stability of transonic shocks in steady supersonic flow past multidimensional wedges

Fang, Beixiang

doi:10.1016/j.aim.2017.04.019

Cited by 17 publications

(15 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…More recently, by constructing new background solutions that allow the incoming flows to tend to the speed limit, the global existence of steady symmetrically conical shock solutions was established in Hu-Zhang [19] when a supersonic incoming potential flow hits a symmetrically perturbed cone with the opening angle less than the critical angle. We also remark that some important results have been obtained on the stability of M-D transonic shocks under symmetric perturbations of the straight-sided cones or the straight-sided wedges, as well as on Radon measure solutions for steady compressible Euler equations of hypersonic-limit conical flows; see [2,3,4,23,27] and the references cited therein.…”

Section: Incoming Flowmentioning

confidence: 97%

Stability of Conical Shocks in the Three-Dimensional Steady Supersonic Isothermal Flows Past Lipschitz Perturbed Cones

Kuang¹,

Zhang²

2021

SIAM J. Math. Anal.

View full text Add to dashboard Cite

We are concerned with the structural stability of conical shocks in the threedimensional steady supersonic flows past Lipschitz perturbed cones whose vertex angles are less than the critical angle. The flows under consideration are governed by the steady isothermal Euler equations for potential flow with axisymmetry so that the equations contain a singular geometric source term. We first formulate the shock stability problem as an initial-boundary value problem with the leading conical shock-front as a free boundary, and then establish the existence and structural/asymptotic stability of global entropy solutions of bounded variation (BV ) of the problem. To achieve this, we first develop a modified Glimm scheme to construct approximate solutions via selfsimilar solutions as building blocks in order to incorporate with the geometric source term. Then we introduce the Glimm-type functional, based on the local interaction estimates between weak waves, the strong leading conical shock, and self-similar solutions, as well as the estimates of the center changes of the self-similar solutions. To make sure of the decrease of the Glimm-type functional, we choose appropriate weights by careful asymptotic analysis of the reflection coefficients in the interaction estimates, when the Mach number of the incoming flow is sufficiently large. Finally, we establish the existence of global entropy solutions involving a strong leading conical shock-front, besides weak waves, under the conditions that the Mach number of the incoming flow is sufficiently large and the weighted total variation of the slopes of the generating curve of the Lipschitz perturbed cone is sufficiently small. Furthermore, the entropy solution is shown to approach asymptotically the self-similar solution that is determined by the incoming flow and the asymptotic tangent of the cone boundary at infinity.

show abstract

Section: Incoming Flowmentioning

confidence: 97%

Stability of Conical Shocks in the Three-Dimensional Steady Supersonic Isothermal Flows Past Lipschitz Perturbed Cones

Kuang¹,

Zhang²

2021

SIAM J. Math. Anal.

View full text Add to dashboard Cite

show abstract

“…Remark 3.1. For the global stability of weak transonic shocks for the 3-D wedge problem, see [26,28]; also see the instability phenomenon for strong transonic shocks for the 3-D wedge problem in [75]. For the global stability of conical shocks for the M-D conic problem, see [27] for the transonic shock case, and [38,49,80] for the supersonic shock case.…”

Section: Approach II For Problem 32 (St) and (Wt)mentioning

confidence: 99%

“…In Chen-Fang [9], we developed a nonlinear approach and employed it to establish the stability of weak shock solutions containing a transonic shock for potential flow with respect to the M-D perturbation of the wedge boundary in appropriate function spaces. To achieve this, we first formulated the stability problem as an M-D free boundary problem for nonlinear elliptic equations.…”

Section: Further Problems and Remarksmentioning

confidence: 99%

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

Chen

2017

Sci. China Math.

Self Cite

View full text Add to dashboard Cite

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 or larger 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 PrandtlMeyer 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.

show abstract

“…1.2). The strong shock case has been studied in Chen-Fang [16] for the potential flow (also see [8]).…”

mentioning

confidence: 99%

“…There are other related papers about transonic shocks, such as [8,23] for transonic flows past three-dimensional wedges and [7] about transonic flows past a perturbed cone; see also [9,15] for the approaches developed earlier for dealing with transonic shock flows and [20] for the uniqueness of transonic shocks.…”

mentioning

confidence: 99%

Stability and asymptotic behavior of transonic flows past wedges for the full Euler equations

Feldman

2018

Interfaces Free Bound.

Self Cite

View full text Add to dashboard Cite

The existence, uniqueness, and asymptotic behavior of steady transonic flows past a curved wedge, involving transonic shocks, governed by the two-dimensional full Euler equations are established. The stability of both weak and strong transonic shocks under the perturbation of the upstream supersonic flow and the wedge boundary is proved. The problem is formulated as a one-phase free boundary problem, in which the transonic shock is treated as a free boundary. The full Euler equations are decomposed into two algebraic equations and a first-order elliptic system of two equations in Lagrangian coordinates. With careful elliptic estimates by using appropriate weighted Hölder norms, the iteration map is defined and analyzed, and the existence of its fixed point is established by performing the Schauder fixed point argument. The careful analysis of the asymptotic behavior of the solutions reveals particular characters of the full Euler equations.

show abstract

Stability of transonic shocks in steady supersonic flow past multidimensional wedges

Cited by 17 publications

References 30 publications

Stability of Conical Shocks in the Three-Dimensional Steady Supersonic Isothermal Flows Past Lipschitz Perturbed Cones

Stability of Conical Shocks in the Three-Dimensional Steady Supersonic Isothermal Flows Past Lipschitz Perturbed Cones

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

Stability and asymptotic behavior of transonic flows past wedges for the full Euler equations

Contact Info

Product

Resources

About