A TIME-DEPENDENT LINEARIZED EULER SOLVER FOR UNSTEADY FLOW THROUGH NOZZLES AND TURBOMACHINE-CASCADES

摘要

In the design of modern aircraft and land-based steam and gas turbines, one of the problems which must be solved is the flutter and forced response of turbomachinery bladings. To predict this flutter, an efficient and accurate numerical tool is required.This paper presents a time-dependent non-harmdhic linearized Euler method for solving linearized Euler equations of inviscid, compressible transonic unsteady flow using a shock capturing scheme. The present approach adopts a small perturbation assumption, which splits the flow into large steady-flow components and small unsteady-flow components. The unsteady-flow components are determined by the linearized unsteady-flow variable coefficient field equations whose variable coefficients depend on the large steady-flow components. The resulting linearized unsteady flow equations are integrated by an explicit time marching proceeding in finite difference formulation. Both one-dimensional and two-dimensional linearized unsteady flow Euler codes with added artificial viscosity constructed from second and fourth differences in the dependent variables have been addressed. In the one-dimensional code, a linearized Euler solver for unsteady transonic flow through nozzles was performed. The MacCormack scheme and the Lax-Wendroff scheme were used. Both reflecting and non-reflecting boundary conditions for linearized one-dimensional Euler code had been implemented at inlet and outlet. In the two-dimensional code, the linearized Euler solvers for unsteady transonic flow through nozzles and turbomachine-cascades have been carried out. The MacCormack-Richtmyer scheme and the Lax-Wendroff-Richtmyer scheme are employed. A no-penetration solid wall boundary condition and a periodic boundary condition, as well as reflecting and non-reflecting boundary conditions at inlet and outlet, are implemented.These model programs have been validated against different numerical test cases. The numerical results show that the time-dependent non-harmonic linearized unsteady flow Euler solver is robust. With the present method the non-reflecting boundary conditions can totally prevent non-physical, spurious reflection from the boundaries. This method can also obtain a converged periodic solution earlier than a non-linear Euler method and therefore reduce the total CPU time.