Asymptotic expansion for 1D steady nonuniform enthalpy compressible supersonic flow at high Reynolds number

The exact solution of one-dimensional (1D) steady compressible Navier–Stokes (N-S) equations at high Reynolds number has not been given yet under nonuniform total enthalpy when the Prandtl number ( ) is not equal to 0.75 since Becker's work in 1922. In this paper, we give an asymptotic expansion of the solution of the above equations at high Reynolds number by using the method of matched asymptotic expansions and prove convergence of the asymptotic solution under some assumptions. By analyzing the dimensionless form of one-dimensional steady compressible Navier–Stokes equations, we find that if there are extreme points inside the boundary layer, the number of extreme points of velocity is at most one more than that of total enthalpy and the extreme points of each flow variables are different from each other. Based on the second-order asymptotic expansion solution, we show that there are extreme points inside the thin boundary layer under some special conditions. Examples are given to verify theoretical analysis. The present asymptotic expansion solution is valuable for verifying the efficiency of high-order numerical methods in flow simulation of high Reynolds number.