Stability of transonic shock fronts in two-dimensional Euler systems

Chen, Shuxing

doi:10.1090/s0002-9947-04-03698-0

Cited by 82 publications

(38 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The reason is that the above works [2,3,4,5,6,8,25,26,27] depend on a special transonic shock solution (the first class of transonic shocks as called in [28]), for which the nozzle is a straight duct, the transonic shock front is flat, and both the states ahead and behind of the shock front are uniform. Thus the position of the shock front may be arbitrary, and the back pressure does not depend on the position of the shock front (see [28]).…”

Section: Why This Happens?mentioning

confidence: 99%

“…The profile is solved by parts of the Rankine-Hugoniot conditions, while the position depends on the integral-like solvability conditions of elliptic boundary value problems. In [2,3,4,5,6,25,26,27], one fixes the position of the transonic shock front artificially.…”

Section: Why This Happens?mentioning

confidence: 99%

“…By applying the characteristic decomposition method as in [6,7,27], we can write the first correspond to the same streamline passing the point (r 0 , 0) in polar coordinates. So, the boundary conditions on them should be periodical with the period η 0 .…”

Section: Lagrangian Formulation Of Euler System In Polar Coordinates Andmentioning

confidence: 99%

“…Further understanding of the transonic shocks requires progresses in the theory of elliptic boundary value problems in non-smooth domains and new ideas to treat free boundaries -the transonic shock fronts. In recent years, there appear many outstanding works in this direction, for example, [2,3,4,5,6,7,13,25]. Yuan [27] established the ill-posedness of a class of transonic shock problems in a two-dimensional slowly-varying duct for given pressure at the exit.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Stability of Cylindrical Transonic Shocks for the Two-Dimensional Steady Compressible Euler System

Liu

Yuan

2008

J. Hyper. Differential Equations

View full text Add to dashboard Cite

We establish the stability of a class of cylindrical symmetric transonic shocks for two-dimensional complete compressible steady Euler system. This result partly explains the effectiveness of the popular quasi-one-dimensional model of nozzle flows used in aerodynamics. Mathematically, we solve a nonlinear free boundary problem of an elliptic-hyperbolic composite system, with the circular transonic shock front as the free boundary. We accomplish this by finding a (locally) unique fixed point of an appropriately defined boundary profile updating mapping. To define this mapping, we encounter a series of nonclassical boundary value problems in an annulus, which involve a new type of nonlocal elliptic problem, and integral-like solvability conditions to determine the position of the free boundary. This reflects an interesting new feature of boundary value problems of elliptic-hyperbolic composite systems.

show abstract

Section: Why This Happens?mentioning

confidence: 99%

Section: Why This Happens?mentioning

confidence: 99%

Section: Lagrangian Formulation Of Euler System In Polar Coordinates Andmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stability of Cylindrical Transonic Shocks for the Two-Dimensional Steady Compressible Euler System

Liu

Yuan

2008

J. Hyper. Differential Equations

View full text Add to dashboard Cite

show abstract

“…Another advantage in our analysis here is in the context of the real full Euler equations so that the solutions do not necessarily obey Bernoulli's law with a uniform Bernoulli constant, i.e., the Bernoulli constant is allowed to change for different fluid trajectories (in comparison with the setup in [9,10,13,16]).…”

mentioning

confidence: 99%

Transonic flows with shocks past curved wedges for the full euler equations

Feldman¹

2016

DCDS-A

View full text Add to dashboard Cite

We establish the existence, stability, and asymptotic behavior of transonic flows with a transonic shock past a curved wedge for the steady full Euler equations in an important physical regime, which form a nonlinear system of mixed-composite hyperbolic-elliptic type. To achieve this, we first employ the transformation from Eulerian to Lagrangian coordinates and then exploit one of the new equations to identify a potential function in Lagrangian coordinates. By capturing the conservation properties of the system, we derive a single second-order nonlinear elliptic equation for the potential function in the subsonic region so that the transonic shock problem is reformulated as a one-phase free boundary problem for the nonlinear equation with the shockfront as a free boundary. One of the advantages of this approach is that, given the shock location or equivalently the entropy function along the shock-front downstream, all the physical variables can be expressed as functions of the gradient of the potential function, and the downstream asymptotic behavior of the potential function at infinity can be uniquely determined with a uniform decay rate. To solve the free boundary problem, we employ the hodograph transformation to transfer the free boundary to a fixed boundary, while keeping the ellipticity of the nonlinear equation, and then update the entropy function to prove that the updating map has a fixed point. Another advantage in our analysis is in the context of the full Euler equations so that the Bernoulli constant is allowed to change for different fluid trajectories.

show abstract

Stability of a Mach configuration

Chen

2005

Comm. Pure Appl. Math.

Self Cite

View full text Add to dashboard Cite

We prove the stability of a Mach configuration, which occurs in shock reflection off an obstacle or shock interaction in compressible flow. The compressible flow is described by a full, steady Euler system of gas dynamics. The unperturbed Mach configuration is composed of three straight shock lines and a slip line carrying contact discontinuity. Among four regions divided by these four lines in the neighborhood of the intersection, two are supersonic regions, and other two are subsonic regions. We prove that if the constant states in the supersonic regions are slightly perturbed, then the structure of the whole configuration holds, while the other two shock fronts and the slip line, as well as the flow field in the subsonic regions, are also slightly perturbed. Such a conclusion asserts the existence and stability of the general Mach configuration in shock theory.In order to prove the result, we reduce the problem to a free boundary value problem, where two unknown shock fronts are free boundaries, while the slip line is transformed to a fixed line by a Lagrange transformation. In the region where the solution is to be determined, we have to deal with an elliptic-hyperbolic composed system. By decoupling this system and combining the technique for both hyperbolic equations and elliptic equations, we establish the required estimates, which are crucial in the proof of the existence of a solution to the free boundary value problem.

show abstract

Stability of transonic shock fronts in two-dimensional Euler systems

Cited by 82 publications

References 31 publications

Stability of Cylindrical Transonic Shocks for the Two-Dimensional Steady Compressible Euler System

Stability of Cylindrical Transonic Shocks for the Two-Dimensional Steady Compressible Euler System

Transonic flows with shocks past curved wedges for the full euler equations

Stability of a Mach configuration

Contact Info

Product

Resources

About