2D Linearized Harmonic Euler Flow Analysis for Flutter and Forced Response

Abstract

A two dimensional, unsteady, linearized Euler solver has been developed, and applied to both flutter and forced response problems. Solutions are obtained at a single frequency, with the time derivatives ∂Q/∂t replaced by −iωQ. The unsteady solver is derived from an existing steady flow Euler solver that uses adaptive triangular grids. The solver is not restricted to blade row geometries. The solution technique is the false time marching of Ni, used in conjunction with a Runge-Kutta scheme. As with most steady flow Euler solvers, shocks are captured, rather than fitted. The inlet and outlet boundary conditions are the two dimensional single frequency non-reflective boundary conditions due to Giles. For flutter problems, the grid moves with the blade: grid points on the blade surface remain fixed to the blade, and the motion of interior grid points is computed using a Laplacian smoother. Results are shown for three flutter calculations and one forced response calculation.

Potential methods are clearly more efficient than linearized Euler methods, but are restricted to shock free flows, or to flows with weak shocks and shock geometries simple enough to permit shock fitting. This and other investigations have shown that Euler methods can produce results comparable to potential methods. For transonic flows, the unsteady force applied to a blade by an oscillating shock — the “shock foot force” — can depend significantly on the way the shock is modeled. Improvements in the way unsteady Euler solvers treat shocks are required, possibly even extending to making the grid oscillate with the shock. Once the issue of the correct treatment of shocks can be resolved, Euler methods will be able to produce useful solutions in regimes inaccessible to potential methods.