A numerical method for the transonic cascade flow problem

An implicit time-marching finite-difference method for solving the three-dimensional compressible Navier-Stokes equations for the relative flow of a turbomachine impeller in general curvilinear coordinates is presented. The fundamental equations of the method have the distinctive feature that the momentums of the contravariant velocities are employed as the dependent variables. The use of the momentum equations of the contravariant velocities makes possible correct and simple treatments of some boundary conditions. In order to obtain the stable solution for high Reynolds number turbulent flow, the Navier-Stokes equations and the k-ε{lunate} turbulence model equations are solved simultaneously, and a high-resolution TVD upwind scheme is introduced. The calculated results of some two-dimensional turbulent flows agreed well with the experimental data. The calculated results of an axial-flow transonic compressor rotor flow showed that the leakage vortex from the tip clearance as well as the shock waves can be captured vividly, in spite of the relatively coarse grid.