On The Shape Of Shock Waves In Steady Planar And Axisymmetical Mach Reflections

For supersonic internal flow, such as flow in the inlet and nozzle of the engine of an supersonic aircraft, a complicated oblique shock reflection may occur. The study of shock reflection phenomenon is important to the design and optimization of engines. In general, shock reflection includes two types, namely the regular reflection and the Mach reflection. Previous researches were mainly concentrated on the transition between two types of shock reflection and the height of the Mach stem, but few works contributed to the shape of shock waves in steady Mach reflections which is therefore the main theme of the present thesis.In this thesis, the flow fields behind the Mach stem in steady Mach reflections are analyzed theoretically and numerically. When the angle between the slipline and the reflecting plane is sufficiently small, the subsonic flow just behind the Mach stem can be described by the isentropic small-disturbance equation. Using this analytical model, a very simple algebraic expression for the shape of the Mach stem is obtained. It is found that the Mach stem is well approximated by a circular arc with very small curvature. The predicted shape of the Mach stem and the orders of the flow variables for justifying the isentropic small-disturbance equation agree very well with numerical results. The shape of the generatrix of Mach disk of axisymmetrical Mach reflection is also found to be a circular arc through similar approach.Different from the planar shock reflection problem, where the shape of the incident shock is a straight line, the shape of the incident shock wave in the inward-facing axisymmetric shock reflection in steady flow is an unknown curve. In this paper, a simple theoretical approach is proposed to determine the shape of this incident shock wave. The present theory is base on the steady Euler equations. When the assumption that the stream-lines are straight lines at locations just behind the incident shock is adopted, an ordinary differential equation is derived, and the shape of the incident shock wave is given by the solution of this ordinary differential equation. An approximate expression is obtained by solving the equation with perturbation method. The predicted curves of the incident shock wave at several inlet conditions agree very well with the results of numerical simulations.The influence of the geometrical parameters on axisymmetrical shock reflections are further studied by numerical simulations.