Multigrid calculations for cascades

Abstract

Development of numerical methods for internal flows such as the flow in gas turbines or compressors has generally met with less success than that for external flows, due to the complexity of the flow pattern. Potential flow methods had been the major approach in cascade flow calculations until fairly recently, when solution of the Euler equations became practically feasible. The most widely used Euler method has been Denton's finite volume method [1]. Like all explicit methods, Denton's method suffers from the limit of the CFL condition. Implicit schemes have been developed to yield convergence in a smaller number of time steps. This will only pay, however, if the decrease in the number of time steps outweighs the increase in the computational effort per time step consequent upon the need tosolve coupled equations. A review of various time stepping schemes is given by Jameson [2]. The finite volume method with a multiple stage time stepping scheme developed by Jameson [3] has been very successful in calculating external flows. This method has the advantage of separated spatial and time discretizations and is easy to implement. The finite-volume discretization applies directly in the physical domain and is in conservation form. Adaptive numerical dissipation of blended first and third differences in the same conservation from as the convection fluxes is used to provide the necessary higher order background dissipation, and also the dissipation for capturing shocks. Subramanian [4] and Holmes [5] have successfully extended the 4---stage Runge-Kutta scheme to cascade calculations. Instead of the 4---stage scheme the present work uses a more flexible multistage scheme together with locally varying time steps, enthalpy damping and implicit residual averaging to extend the stability limit and accelerate convergence. Finally very rapid convergence is obtained by applying an effective multigrid method [6]. Since the necessary dissipation is added separately from the discretization of the conservation laws, the amount of dissipation can be carefully optimized to improve the accuracy. Two--dimensional subsonic, transonic and supersonic cascade flows have been calculated using this method. The results show excellent accuracy and convergence.