Asymptotic analysis of transonic shocks in divergent nozzles with respect to the expanding angle

Abstract

In this paper we study the asymptotic behavior of the transonic shock solutions in divergent nozzles as the expanding angle goes to zero. It is well-known that there exist infinite shock solutions for steady 1-D flows in a flat nozzle with the position of the shock front being arbitrary, while there exists a unique shock solution in an divergent nozzle as the pressure at the exit is given within an appropriate interval. By analyzing the asymptotic behavior of the shock solutions as the expanding goes to zero, we are also trying to figure out a criterion which may be used to select the physical one among all shock solutions in the flat nozzle. It finally turns out that the limit shock solution as the expanding angle goes to zero strongly depends on the asymptotic behavior of the receiver pressure, which is assumed to be a function of the expanding angle, imposed at the exit. In particular, as the Mach number of the flow at the entrance is larger than γ+32, the function of the receiver pressure can be set identical to the value of the pressure behind the shock front for the flat nozzle. Then the limit shock solution may be considered as the physical one for the flat nozzle, and the limit position of the shock front being the admissible position of the shock front. To show these results, one of the key steps is to establish quantitative estimates for the position of the shock front for the given pressure at the exit. To this end, a free boundary problem for the linearized Euler system will be proposed which gives an approximation of the shock solution, then a nonlinear iteration scheme could be carried out to approach the shock solution. Moreover, the error estimates between the exact shock solution and the approximation are also established, which give quantitative information on the position of the shock front.