A Linearized Euler Method for Unsteady Flow in Axial-Flow Turbomachines

Doctoral thesis, 1994

A linear Euler method for analysing unsteady inviscid compressible quasi three- dimensional flows in oscillating cascades is developed. It is applicable to both blade flutter and forced response problems. The method considers linear time harmonic perturbations on a steady solution to the nonlinear Euler equations and it has the potential of becoming a computationally very efficient method. The perturbation flow is governed by the linearized Euler equations, which are linear differential equations whose coefficients depend on the steady flow field. To calculate the steady flow field an explicit flow solver, VOLFAP, is used. A grid generator based on the advancing front technique is coupled to the flow solver to provide for solution-adaptive remeshing. The idea of remeshing is especially useful for problems involving shocks, since shocks of any shape can be resolved as accurately as wanted. To discretize and solve the linear Euler equations a pseudo-time time marching Lax- Wendroff scheme is used. The last mesh used in the steady flow analysis is used in the linear unsteady analysis. The mesh is harmonically deformed for problems involving vibrating blades, flutter. Forced response problems are treated by prescribing incoming waves in the nonreflecting boundary conditions. Results for different standard configurations are presented for both flutter and forced response problems. Comparisons with other numerical methods show good agreement. A comparison between 1D and 2D nonreflecting boundary conditions show that spurious reflections from 1D boundary conditions can corrupt a solution severely even if the condition is not close to acoustic resonance. It is known that unsteady nonlinear solvers cannot effectively avoid spurious reflections from the boundaries. With the present method however, one has complete control of such phenomena. In a transonic fan cascade study the present method is found to produce results, for two cases, which qualitatively agree well with results obtained by time-accurate Navier- Stokes simulations.