\,fAL 7om- n of y of lling 'Uter 59. The , on :ale s " "all VOL. 22, NO. 11, NOVEMBER 1984 AIAA JOURNAL /I. /,S 1609 Application of the Godunov Method and Its Second-Order Extension to Cascade Flow Modeling Shmuel Eidelman * Naval Postgraduate School, Monterey, California Phillip Colellat Lawrence Berkeley Laboratory, Berkeley, California and Raymond P. Shreeve:!: Naval Postgraduate School, Monterey, California The Godunov method and a new second-order accurate extension of the method are used for the solution of two-dimensional Euler equations. Both numerical schemes are described in detail. Their performances in the subsonic, transonic, and supersonic flow regimes are first tested on the problem of flow in a channel with a circular arc bump. The niethods are then applied to calculate the transonic flow through a supercritical com- pressor cascade designed by J. Sanzo For this case, the solution with the second-order extension of the Godunov method gives verygood agreement with the design distribution of parameters given by Sanzo Introduction I N recent years considerable attention has been given to numerical methods that use the analytical solution of the Riemann problem to calculate numerical fluxes at cell edges. 1,2 The first method to employ the solution of the Riemann problem in its formulation was introduced in 1957 by Godunov 3 and has been widely used in the Soviet Union since then. The numerical simulation of a wide variety of gasdynamic, magnetogasdynamic, and two-phase flow problems using the Gbdunov method in one, two, and three dimensions has been reported in Russian publications. 3 Unfortunately, very limited information is usually given concerning the accuracy and convergence of the method in multidimensional cases. Until recently, experience with tne Godunov method outside the USSR has been limited to one- dimertsiomll shock wave problems. 4 It has been shown that the method, which is first-order accurate, solves nonlinear one-dimensional problems with the same or even better ac- curacy than many second-order accurate methods. 4 Van Leer l was the first to extend the accuracy of the Godunov method to second order for solving gasdynamic problems in one space variable and in Lagrangian coor- dinates. These have since been reported in a number of in- vestigations and further developments of higher-order ex- tensions of Godunov's method. 2 In a comparative study by Woodward and Colella,2 it was shown that the second-order extensions of the Godunov method gave superior results for supersonic flows with multiple-shock reflections. In view of these results it was decided to examine the performance of these methods in computing internal steady flo",fields in comparison with the first-order accurate Godunov method. Godunov-type methods offer a distinct advantage in that they do not require the addition of artificial viscosity or- smoothing, which is very appealing for aerodynamic engineering applications. The second-order extension of the Godunov method reported in this study is more accurate than Presented as Paper 83-1941 at the AlAA Sixth Computational Fluid Dynamics Conference, Danvers, Mass., July 13-15,.1983; submitted July 25, 1983; revision received Dec. 20,1983. This paper is declared a work of the U.S. Government and therefore is in the public domain . • Adjunct Professor, Department of Aeronautics. Member AIAA. tStaff Scientist, Computer Science and Mathematics Department. :j:Director, TurbopropulsionLaboratory. Member AIAA. the first-order one, but also does not require artificial viscosity or smoothing. In the present work, results are reported for rotational subsonic, transonic, and supersonic inviscid internal flows obtained using a new code developed to implement the basic Godunov method and a second-order accurate extension of it. The overall purpose was to obtain accurate numerical simulations of transonic flows through cascades of turbo- machine blading. The present paper provides descriptions of the implementation of the Godunov method, the second-order extension of the method, and results of applying the code to specific problems. Although the test cases presented are for steady-state flows, the goal eventually is to use the developed code for nonsteady problems. For this reason, an artificial means to accelerate convergence was not attempted. Governing Equations and Boundary Conditions Equations The unsteady two-dimensional Euler equations can be written in conservation law form as where p pu pv pu p+pu 2 puv u= F= G= (1) pv puv p+pv 2 e (e+p)u (e+p)v where p is the density, u and v the velocity components in the X and Y coordinate directions, p the pressure, and y the ratio of specific heats. The energy per unit of volume e is defined by where E=p/(y-l)p is the internal energy. We look for the solution of the system of equations represented by Eq. (1) in