Transonic shock in a nozzle I: 2D case

Abstract

In this paper we establish the existence and uniqueness of a transonic shock for the steady flow through a general two-dimensional nozzle with variable sections. The flow is governed by the inviscid potential equation and is supersonic upstream, has no-flow boundary conditions on the nozzle walls, and an appropriate boundary condition at the exit of the exhaust section. The transonic shock is a free boundary dividing two regions of C 1,1−δ 0 flow in the nozzle. The potential equation is hyperbolic upstream where the flow is supersonic, and elliptic in the downstream subsonic region. In particular, our results show that there exists a solution to the corresponding free boundary problem such that the equation is always subsonic in the downstream region of the nozzle when the pressure in the exit of the exhaustion section is appropriately larger than that in the entry. This problem is motivated by the conjecture of Courant and Friedrichs on the transonic phenomena in a nozzle [10]. Furthermore, the stability of the transonic shock is also proven when the upstream supersonic flow is a small steady perturbation for the uniform supersonic flow and the corresponding pressure at the exit has a small perturbation. The main ingredients of our analysis are a generalized hodograph transformation and multiplier methods for elliptic equation with mixed boundary conditions and corner singularities.

The existence and monotonicity of a three-dimensional transonic shock in a finite nozzle with axisymmetric exit pressure

“…The transonic shock problem in a de Laval nozzle is a fundamental one in fluid dynamics and has been extensively studied by many authors under the assumption that the transonic flow is quasi-one-dimensional or the transonic shock goes through some fixed point in advance [Chen et al 2006;Chen et al 2007;Chen and Feldman 2003;Chen 2008;Courant and Friedrichs 1948;Embid et al 1984;Glaz and Liu 1984;Kuz'min 2002;Liu 1982a;1982b;Xin and Yin 2005;2008a;2008b;Yuan 2006]. Courant and Friedrichs [1948, page 386] proposed a physically more interesting transonic shock wave pattern in a de Laval nozzle as follows: Given an appropriately large end pressure p e (x), if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle a shock front intervenes and the gas is compressed and slowed down to subsonic speed.…”

Two-Dimensional Regular Shock Reflection for the Pressure Gradient System of Conservation Laws

“…Let us explain. For each η(ξ) ∈ K, we use the equation η = σ − for ξ < 0 and η = σ + for ξ ≥ 0 to locate p on the boundary; i.e., formula (64). Along this curve as ξ increases, both p and ξ 2 + η 2 increase (see (71)) until p − ξ 2 − η 2 = 0 from which point we stop using this boundary, i.e., the point B.…”

“…Use p 2 to denote this value of p (at B) along the boundary where the ellipticity first vanishes. Use this p from (64) in the vector field l of the oblique derivative condition. We then have a linear tangential oblique derivative boundary value problem for p (1) :…”

“…The set is closed and convex in C 2,γ [−ξ 12δ , ξ 12δ ]. For any η ∈ B, we find p on it by (64). We estimate the minimum of p, called p δ m .…”

“…Simplified models that capture various isolated features of the Euler system may be proposed and studied to pave the road. Immediate models are the isentropic case, the irrotational (potential) flow equations, and the steady flows [53,[64][65][66], some of these assumptions are already made in some of the aforementioned papers. A remarkable and distinct model is the unsteady transonic small disturbance system (UTSD) which was proposed and studied ( [9][10][11]13,21,26,38,40,53,59,62]) for transonic solutions and the transition from Mach reflection to regular reflection.…”