Computation of Time Linearized Transonic Flow in Oscillating Cascades

A method for calculating the quasi-3D unsteady transonic flowfield in oscillating cascades is presented. The unsteady flow is assumed to be a small, harmonic perturbation of the non-linear steady flow, so that the steady flow problem is decoupled from the unsteady problem. As long as the vibration amplitudes remain moderate, the higher order terms in the governing equations derived under this assumption can be neglected and the describing unsteady flow equations become linear. Thus every frequency component can be calculated separately and the results be obtained by superposition. For the calculation of the steady state flow, about which the unsteady part is linearized, a finite-volume time-stepping Euler solver is used. Due to the similarity of the derived time-linear unsteady flow equations and the basic equations for the steady solver, the discretization is almost identical for both solvers. Thus it is possible to use much of the steady code with little modification for the time-linearized unsteady code. The time-linear unsteady flow equations are solved on a moving grid. This leads to a considerable simplification of the flow tangency boundary condition on the surfaces of the airfoils. Results obtained for various test cases compare favourably to flat plate theory and time-linearized potential methods as well as to experimental results from the Lausanne standard configurations. The approach presented is computationally more efficient than nonlinear unsteady Euler time-stepping methods, thus permitting application in the standard design procedure.