Two Problems On Hypersonic-limit Flow Passing Bodies And Related Numerical Calculations

We study two types of problems on hypersonic-limit flows for the compressible Euler equations: the flow over two-dimensional curved wedge and three-dimensional conical flow.Limiting-hypersonic flow is the fluid with Mach number tending to infinity.Shock layer will be extremely thin and there will be concentration of mass of the gas on the boundary of the obstacle.In order to better depict the flow field,we generalize the definition of integral weak solution for the Euler equations to the class of Radon measures.For the problem of curved wedge,we propose three boundary value problems with damping for hypersonic-limit flow,calculate the corresponding Radon measure solutions and obtain the generalized Newton Busemann laws.For the three-dimensional conical flow,we mainly study its hypersonic limit when the attack angle is zero.Firstly,we prove the existence of shock wave solutions by analyzing the inverse problem of conical flow? secondly,we derive a second order ordinary differential equation for the radial velocity of the flow behind the shock wave,and by using Rankine-Hugoniot condition,we present numerical result for the relationship between the cone and the shock? finally,we prove that when the Mach number of the upcoming flow tends to infinity,the position of the shock wave will tend to the cone.We also use this conclusion to demonstrate that the sequence of measure solutions containing a shock wave converges to the measure solution of the hypersonic-limit flow as the Mach number approaches infinity.In addition,we also consider the hypersonic limit of conical Chaplygin flow.We prove that the mass will concentrate on the cone when the upcoming Mach number approaches a critical value(not infinity),and provide the corresponding Radon measure solution as well as the generalized Newton’s sine-squared law.For conical flows with attack angle,we obtain a highly singular and nonlinear ordinary differential equation(ODE)while finding the measure solution.For this ODE,we use the Fourier spectral method combined with Newton’s method to give numerical solution.Through the analysis of the numerical results,we explain specific physical phenomenon,and demonstrate the high accuracy and convergence of this numerical method.