Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections

Abstract

Abstract. We establish the existence and stability of multidimensional transonic flows with transonic shocks through an infinite nozzle of arbitrary cross-sections, including a slowly varying de Lavel nozzle. The transonic flow is governed by the inviscid steady potential flow equation with supersonic upstream flow at the entrance, uniform subsonic downstream flow at the infinite exit, and the slip boundary condition on the nozzle boundary. The multidimensional transonic nozzle problem is reformulated into a free boundary problem, for which the free boundary is a transonic shock dividing two phases of C 1,α flow in the infinite nozzle, and the equation is hyperbolic in the supersonic upstream phase and elliptic in the subsonic downstream phase. We further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain and to solve the multidimensional transonic nozzle problem in a direct fashion. Our results indicate that, for the transonic nozzle problem, there exists a transonic flow such that the flow is divided into a C 1,α subsonic flow up to the nozzle boundary in the unbounded downstream region from the supersonic upstream flow by a C 1,α multidimensional transonic shock that is orthogonal to the nozzle boundary at every intersection point, and the uniform velocity state at the infinite exit in the downstream direction is uniquely determined by the supersonic upstream flow at the entrance which is sufficiently close to a uniform flow. The uniform velocity state at the exit can not be apriori prescribed from the corresponding pressure for such a flow to exist. We further prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance.

Uniqueness of symmetric steady subsonic flows in infinitely long divergent nozzles

“…[10]). In recent years, there is an increasing interest in studying subsonic flows and transonic shocks in nozzles (see, for example, [4][5][6][7][8][13][14][15][16][17][19][20][21][22][23] and references therein). All of these works concentrate on flows in either finitely long nozzles or infinitely long nozzles with finite width (like a cylinder).…”

Transonic shock wave in an infinite nozzle asymptotically converging to a cylinder

“…While for transonic flows with shocks, most previous studies involve either experimental and numerically simulations or special wave patterns [1,7,11], except the rigorous results on the existence and stability of the quasi one-dimensional transonic shocks, see [8,17]. Recently, some important wave patterns involving truly multi-dimensional transonic have established for various models and geometries, especially for the transonic wave pattern in a nozzle, see [2][3][4][5]25,26].…”

“…To outstand the background and motivation of this paper, we would like to discuss some of the recent notable studies on multi-dimensional transonic shocks for the potential flow in a nozzle, see [3][4][5]25,26]. Roughly speaking, there are mainly two kinds of nozzle problems.…”

“…We should note that in [3,4] Chen and Feldman developed an iteration scheme which is an effective tool to deal with some kinds of transonic shock problems. The second one is the study for the curved nozzle [5,25,26]. Xin and Yin [25,26] established the existence and stability of a steady multi-dimensional transonic shock in a finite general curved nozzle which is a small perturbation of the cylinder nozzle.…”

“…It is governed by the potential flow equation for a velocity potential ϕ : Ω ⊂ R n → R, which can be deduced from the conservation of mass and the Bernoulli law for the velocity, and is a second order nonlinear equation of mixed elliptic-hyperbolic type for the transonic problems (see [3][4][5][6][7]), div ρ |Dϕ| 2 Dϕ = 0, x ∈ Ω, (1.1) where the density ρ(|Dϕ| 2 ) is ρ |Dϕ| 2 = 1 − θ |Dϕ| 2 1/(2θ) and θ = (γ − 1)/2 with the adiabatic exponent γ > 1. It is easy to verify that the nonlinear equation (1.1) is elliptic if ρ |Dϕ| 2 + 2|Dϕ| 2 ρ |Dϕ| 2 > 0, which corresponds to a subsonic flow, while is hyperbolic if ρ |Dϕ| 2 + 2|Dϕ| 2 ρ |Dϕ| 2 < 0, which corresponds to a supersonic flow.…”