Three-Dimensional Time-Marching Inviscid and Viscous Solutions for Unsteady Flows Around Vibrating Blades

Abstract

A three-dimensional nonlinear time-marching method of solving the thin-layer Navier–Stokes equations in a simplified form has been developed for blade flutter calculations. The discretization of the equations is made using the cell-vertex finite volume scheme in space and the four-stage Runge–Kutta scheme in time. Calculations are carried out in a single-blade-passage domain and the phase-shifted periodic condition is implemented by using the shape correction method. The three-dimensional unsteady Euler solution is obtained at conditions of zero viscosity, and is validated against a well-established three-dimensional semi-analytical method. For viscous solutions, the time-step limitation on the explicit temporal discretization scheme is effectively relaxed by using a time-consistent two-grid time-marching technique. A transonic rotor blade passage flow (with tip-leakage) is calculated using the present three-dimensional unsteady viscous solution method. Calculated steady flow results agree well with the corresponding experiment and with other calculations. Calculated unsteady loadings due to oscillations of the rotor blades reveal some notable three-dimensional viscous flow features. The feasibility of solving the simplified thin-layer Navier–Stokes solver for oscillating blade flows at practical conditions is demonstrated.