激波反射问题对超声速飞行器动力装置进气道、喷口等的设计和优化具有重要意义；在理论上，该问题也是受到长期关注的前沿性课题。激波反射主要有两种形态，即正规反射和马赫反射。自从1878年马赫发现马赫反射现象以来，人们对正规反射和马赫反射之间转捩、滞后效应、马赫反射的稳定性、马赫杆高度等进行了大量的研究；但是，对激波形状的理论和数值研究还很少。本文重点关注两个方面，即平面与轴对称定常马赫反射中的马赫杆形状问题和定常轴对称激波反射中的入射激波形状问题。根据von Neumann的三激波理论，本文发现，在很多工况下，平面马赫反射中滑移线的倾角很小。在此条件下，通过量级分析，本文证明了马赫杆后紧邻马赫杆的亚声速流动区域中，流动可以近似用小扰动速度势方程来描述。通过求解上述方程和边界条件构成的定解问题，得到了马赫杆形状的解析表达式为一段圆弧。本文通过数值实验对马赫杆形状和马赫杆后亚声速区流动参数的量级进行了验证。结果表明，数值模拟得到的马赫杆形状与理论预测符合得很好，马赫杆后流动参数的量级分析也与数值结果相符。本文还将此方法推广到轴对称马赫反射中，得到的马赫杆形状也是一段圆弧。对于在轴线上发生反射的轴对称定常激波反射来说，入射激波形状是一段未知曲线。根据数值模拟结果，本文假定，在入射激波后的超声速流场内，流线在紧邻入射激波位置处为直线，由此可将轴对称欧拉方程简化为一个关于入射激波形状的常微分方程。用数值方法求解该方程，即可得到入射激波的形状曲线。本文还用摄动法求解该方程，得到了入射激波形状的近似解析表达式。该式说明，轴对称入射激波的形状曲线，是在相应的平面入射激波直线上增加一个修正量得到的。本文所给出的入射激波形状曲线与数值计算得到的结果符合得很好。本文还通过数值实验考察了几何构型的改变对定常轴对称激波反射的影响。

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.