Contact discontinuities for 3-D axisymmetric inviscid compressible flows in infinitely long cylinders

Abstract

We prove the existence of a subsonic axisymmetric weak solution (u, ρ, p) with u = uxex + urer + u θ e θ to steady Euler system in a threedimensional infinitely long cylinder N when prescribing the values of the entropy (= p ρ γ ) and angular momentum density (= ru θ ) at the entrance by piecewise C 2 functions with a discontinuity on a curve on the entrance of N . Due to the variable entropy and angular momentum density (=swirl) conditions with a discontinuity at the entrance, the corresponding solution has a nonzero vorticity, nonzero swirl, and contains a contact discontinuity r = g D (x). We construct such a solution via Helmholtz decomposition. The key step is to decompose the Rankine-Hugoniot conditions on the contact discontinuity via Helmholtz decomposition so that the compactness of approximated solutions can be achieved. Then we apply the method of iteration to obtain a solution and analyze the asymptotic behavior of the solution at far field.

Steady Euler flows with large vorticity and characteristic discontinuities in arbitrary infinitely long nozzles

“…Then Chen-Deng-Xiang [11] focused on the full Euler equations for the infinitely long nozzle problem with general varying cross-sections by developing some useful new techniques; also see [24] for the existence and uniqueness of smooth subsonic flows with nontrivial swirl in axisymmetric nozzles. For vortex sheets, the stability of a subsonic flat contact discontinuity in nozzles by the perturbation argument has been established in Bae [1] and Bae-Park [3,4]. Some further related results can be found in [2,6,18,19,20,22,28,29,30,32,34] and the references cited therein.…”

“…All of these results are based on the perturbation analysis around a piecewise constant background solution (cf. [1,3,4,21,29]). As far as we know, the result in Theorem 2.2 is the first on the global existence of piecewise smooth solutions of multidimensional steady compressible Euler equations, which are not necessarily a perturbation of piecewise constat background solutions.…”

Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle

“…The study of the vortex sheets in steady compressible fluids is also an interesting topic, which has drawn a lot of attention recently. For the subsonic flow, the stability of an almost flat contact discontinuity in two dimensional nozzles with infinity length was established in [1]; and see [2,3] for further related results. Recently, the uniqueness and existence of the contact discontinuity, which is large, i.e., not a perturbation of the straight one, was obtained in [13].…”