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The finite element numerical simulation of the transonic flow with condensation is studied in this paper. The transformed weak form is numerically discretized by using the classical Galerkin finite element spatial discretization scheme and the backward Euler difference time discretization scheme. Because the coupled Euler equations are convection equations, the results obtained by the classical Galerkin method often have nonphysical numerical oscillations, so the stabilization method is introduced. Another important feature of compressible Euler equation is the existence of shock wave. Because the thickness of shock wave is small and difficult to capture, the shock wave can be better captured by adding artificial viscosity term. Here, the streamline upwind/Petrov–Galerkin (SUPG) finite element method with good stability and the isotropic diffusion method with shock capture ability are used to solve the problem of transonic flow with condensation. In the simulation of steady problems, in order to improve the calculation accuracy of the existing area of the shock wave, the adaptive mesh refinement technology is added, which refines the mesh in the region of the shock wave to improve the resolution of the shock wave. Numerical experiments verify the feasibility and stabilization of the numerical method. Finally, the simulation of transonic flow around NACA-0012 and parallel-jet nozzle A1 has obtained the better numerical results.

In this paper, we propose a two-level nonconforming rotated finite element (TNRFE) method for solving the Navier–Stokes equations. A new nonconforming rotated finite element (NRFE) method was proposed by Douglas added by conforming bubbles to velocity and discontinuous piecewise constant to the pressure on quadrilateral elements possessing favorable stability properties. The TNRFE method involves solving a small Navier–Stokes problem on a coarse mesh with mesh size H and a large linearized Navier–Stokes problem on a fine mesh with mesh size h by the NRFE method. If we choose h = O H 2 , the TNRFE method gives the convergence rate of the same order as that of the NRFE method. Compared with the NRFE method, the TNRFE method can save a large amount of CPU time. In this paper, the stability of the approximate solutions and the error estimates are proved. Finally, the numerical experiments are given, and results indicate that the method is practicable and effective.