Existence of Global Solutions for Unsteady Isentropic Gas Flow in a Laval Nozzle

Abstract

In this paper, we study the motion of isentropic gas in the Laval nozzle. The Laval nozzle is the most important type of nozzle utilized in some turbines. In particular, we consider unsteady flows, including transonic gas flows, and prove the existence of global solutions for the Cauchy problem. In spite of its importance, this problem has received little attention until now. The most difficult point is to obtain bounded estimates for approximate solutions. To overcome this, we introduce a modified Godunov scheme. The corresponding approximate solutions consist of piecewise steady-state solutions of an auxiliary equation and yield a sharper bounded estimate. As a result, we find an invariant region for our solutions. Finally, in order to prove their convergence, we use the compensated compactness framework.

Existence and stability of solutions to the compressible Euler equations with an outer force

“…The following proposition and theorem can be proved in the same manner to [9][10][11]. We can similarly obtain (4.1) 2 .…”

Global Entropy Solutions to the Gas Flow in General Nozzle

“…Liu [19] first proved the existence of a global solution with initial data of small total variation and away from sonic state by a Glimm scheme. Tsuge [27][28][29] first studied the global existence of solutions for Laval nozzle flow and transonic flow for large initial data by introducing a modified Godunov scheme. Recently, Chen and Schrecker [4] proved the existence of globally defined entropy solutions in transonic nozzles in an L p compactness framework, whose uniform bound of approximate solutions may depend on time t. In our paper, we are focusing on the L ∞ compactness framework.…”

Global entropy solutions to the compressible Euler equations in the isentropic nozzle flow for large data: Application of the modified Godunov scheme and the generalized invariant regions

“…Lu [11], Gu and Lu [12] extended [19] to the nozzle flow with a monotone cross section area and the general pressure by using the vanishing viscosity method. In addition, the author [20] treated the Laval nozzle. In these papers, the monotonicity of the cross section area plays an important role.…”

“…(M2) The second is to consider the nozzle without the monotonicity of the cross section area. In fact, we cannot apply the method of [11], [12], [19] and [20] to such a nozzle as the supersonic wind tunnel. (M3) The third is to consider large data.…”