Computation of Transonic Cascade Flow Using the Euler and Navier-Stokes Equations of Contravariant Velocities

Implicit time-marching finite-difference methods for solving the two- and three-dimensional compressible Euler and Navier-Stokes equations are presented. The distinctive feature of these methods is to use the momentum equations of contravariant velocity components. By using such equations, accurate and easy treatments of the solid wall boundary condition are realized for the Euler equations, and simple treatments of he periodic boundary condition become possible for the impeller flows. The numerical method are based on the Beam-Warming delta-form approximate-factorization scheme, and take into consideration the diagonalization and upstreaming by the theory of characteristics. The computations of turbulent flows are implemented using the two-equation k-ε turbulence model with the law of the wall. Finally, some numerical results of transonic cascade flow are shown.