Contact Discontinuities for 2-Dimensional Inviscid Compressible Flows in Infinitely Long Nozzles

Abstract

We prove the existence of a subsonic weak solution (u, ρ, p) to steady Euler system in a two-dimensional infinitely long nozzle when prescribing the value of the entropy (= p ρ γ ) at the entrance by a piecewise C 2 function with a discontinuity at a point. Due to the variable entropy condition with a discontinuity at the entrance, the corresponding solution has a nonzero vorticity and contains a contact discontinuity x 2 = g D (x 1 ). 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 piecewise smooth subsonic flow with a contact discontinuity and nonzero vorticity. We also analyze the asymptotic behavior of the solution at far field.

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

“…If ν en = 0 on Γ en , by the definition of Λ given by (2.10), then it follows from (2.6), where we extend the definition of p onto N \ N − gD by p = p 0 in N \ N − gD . And, this coincides with the result obtained from [3]. From this perspective, the two dimensional subsonic weak solution with a contact discontinuity, constructed in [3], can be considered as a three-dimensional subsonic weak solution with the zero-swirl boundary condition for ν en at the entrance of the cylinder N .…”

“…And, this coincides with the result obtained from [3]. From this perspective, the two dimensional subsonic weak solution with a contact discontinuity, constructed in [3], can be considered as a three-dimensional subsonic weak solution with the zero-swirl boundary condition for ν en at the entrance of the cylinder N . With this representation, (1.3) is decomposed as a system of second order elliptic equations for (ϕ, ψ), and transport equations for (S, Λ).…”

“…To analyze the asymptotic behavior of the solution, we use the stream function formulation and energy estimates. We emphasize that the asymptotic behavior of three dimensional subsonic weak solution with a contact discontinuity is completely different from the two dimensional solution, which are studied in [3]. Due to the non-zero angular momentum generated by the boundary condition at the entrance, the asymptotic limit of pressure p of three dimensional subsonic weak solution with a contact discontinuity does not converge to a constant p 0 at x = ∞.…”

“…There are many studies of smooth subsonic solutions to Euler system, see [5,6,14,15,16,17,18,21,22,23] and references cited therein. As far as we know, there are few results on the existence of solutions to Euler system with contact discontinuities [1,3,7,8,9,10,11,12,20]. In [20], supersonic contact discontinuities in three-dimensional isentropic steady flows were studied.…”

“…The first work to construct subsonic weak solutions with contact discontinuities to steady Euler system via Helmholtz decomposition is given in [3], in which new formulations of steady Euler system and Rankine-Hugoniot conditions via Helmholtz decomposition are introduced, and the existence of subsonic weak solutions with contact discontinuities and nonzero vorticity is proved in a two-dimensional infinitely long nozzle. Furthermore, it is proved that a two dimensional weak solution converges to a constant pressure state at far-field(x = ∞), if one side of the contact discontinuity has uniform state with (p, u) = (p 0 , 0) for a constant p 0 > 0.…”