Transonic Shocks in Compressible Flow Passing a Duct for Three-Dimensional Euler Systems
- First Online:
- 131 Downloads
In this paper we study the transonic shock in steady compressible flow passing a duct. The flow is a given supersonic one at the entrance of the duct and becomes subsonic across a shock front, which passes through a given point on the wall of the duct. The flow is governed by the three-dimensional steady full Euler system, which is purely hyperbolic ahead of the shock and is of elliptic–hyperbolic composed type behind the shock. The upstream flow is a uniform supersonic one with the addition of a three-dimensional perturbation, while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference. The problem of determining the transonic shock and the flow behind the shock is reduced to a free-boundary value problem. In order to solve the free-boundary problem of the elliptic–hyperbolic system one crucial point is to decompose the whole system to a canonical form, in which the elliptic part and the hyperbolic part are separated at the level of the principal part. Due to the complexity of the characteristic varieties for the three-dimensional Euler system the calculus of symbols is employed to complete the decomposition. The new ingredient of our analysis also contains the process of determining the shock front governed by a pair of partial differential equations, which are coupled with the three-dimensional Euler system.
Unable to display preview. Download preview PDF.
- 1.Cole J.D., Cook L.P. Transonic Aerodynamics, vol. 30. North-Holland, Amsterdam, 1986Google Scholar
- 13.Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equation of Second Order, 2nd edn. Grundlehren der Mathematischen Wissenschaften, 224. Springer, Berlin, 1983Google Scholar
- 14.Hörmander L. (1985) The Analysis of Linear Partial Differential Operators, 3. Springer, BerlinGoogle Scholar
- 15.Kuz’min A.G. (2002) Boundary Value Problems for Transonic Flow. John Wiley, LondonGoogle Scholar
- 17.Mcowen R.C. (2003) Partial Differential Equations, Methods and Aplications, 2nd edn. Pearson Education, Upper Saddle River, NJGoogle Scholar
- 25.Zeidlerm E. (1986) Nonlinear Functional Analysis and its Applications. Fixed-Point Theorems, vol. 1. Springer, New YorkGoogle Scholar