This paper concerns transonic shocks in compressible inviscid flow passing a two- dimensional variable-area duct for the complete steady Euler system. The flow is supersonic at the entrance of the duct, whose boundaries are slightly curved. The condition of impenetrability is posed on the boundaries. After crossing a nearly flat shock front, which passes through a fixed point on the boundary of the duct, the flow becomes subsonic. We show that to ensure the stability of such shocks, pressure should not be completely given at the exit: it only should be given with freedom one, that is, containing an unknown constant to be determined by the upstream flow and the profile of the duct. Careful analysis shows that this is due to the requirement of conservation of mass in the duct. We used Lagrangian transformation and characteristic decomposition to write the Euler system as a 2 × 2 system, which is valid for general smooth flows. Due to such a simplification, we can employ the theory of boundary value problems for elliptic equations to discuss well-posedness or ill-posedness of transonic shock problems in variable-area duct for various conditions giving at the exit.

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On transonic shocks in two-dimensional variable-area ducts for steady euler system ∗

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.

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Stability of cylindrical transonic shocks for the two-dimensional steady compressible euler system

We study the stability of transonic shocks in steady supersonic flow past a wedge. It is known that in generic case such a problem admits two possible locations of the shock front, connecting the flow ahead of 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 the entropy condition. We prove that the transonic shock is conditionally stable under perturbation of the upstream flow or perturbation of wedge boundary. Copyright © 2005 John Wiley & Sons, Ltd.

Stability of transonic shocks for the full Euler system in supersonic flow past a wedge

We prove that for the two-dimensional steady complete compressible Euler system, with given uniform upcoming supersonic flows, the following three fundamental flow patterns (special solutions) in gas dynamics involving transonic shocks are all unique in the class of piecewise C
 1 smooth functions, under appropriate conditions on the downstream subsonic flows: (i) the normal transonic shocks in a straight duct with finite or infinite length, after fixing a point the shock-front passing through; (ii) the oblique transonic shocks attached to an infinite wedge; (iii) a flat Mach configuration containing one supersonic shock, two transonic shocks, and a contact discontinuity, after fixing a point where the four discontinuities intersect. These special solutions are constructed traditionally under the assumption that they are piecewise constant, and they have played important roles in the studies of mathematical gas dynamics. Our results show that the assumption of a piecewise constant can be replaced by some weaker assumptions on the downstream subsonic flows, which are sufficient to uniquely determine these special solutions. Mathematically, these are uniqueness results on solutions of free boundary problems of a quasi-linear system of elliptic-hyperbolic composite-mixed type in bounded or unbounded planar domains, without any assumptions on smallness. The proof relies on an elliptic system of pressure p and the tangent of the flow angle w = v/u obtained by decomposition of the Euler system in Lagrangian coordinates, and a newly developed method for the L
 ∞ estimate that is independent of the free boundaries, by combining the maximum principles of elliptic equations, and careful analysis of the shock polar applied on the (maybe curved) shock-fronts.

Beixiang Fang

Papers

Stability of transonic shocks in supersonic flow past a wedge