Application of a Flux Vector Splitting Methodology Towards the Solution of Unsteady Transonic Flows, With Future Emphasis On the Blade Flutter Problem

Abstract

The study presents a method, based on the flux vector splitting approach, to the problem of unsteady two-dimensional inviscid transonic flows, with emphasis on the numerical determination of the shock position through nozzles with varying backpressure. The methodology, governed by the Euler equations, is first explained for one and two dimensional steady state applications, and the accuracy of the results is validated by comparison with exact (one dimension) and numerical (two dimensions) solutions. The model is then applied to the problem of fluctuating backpressure in quasi one-dimensional and two dimensiona1 flows. The one-dimensional results are validated by comparison with a small perturbation analytical unsteady solution, where after sample cases are performed with the objective to understand fundamental aspects of unsteady flows. It is concluded that both the amplitude and frequency of the imposed fluctuating exit pressure are important parameters for the location or the unsteady shock. It is also shown that the average unsteady shock position is not identical with the steady state position, and that the unsteady shock may, under certain circumstances; propagate upstream into the subsonic flow domain. The pressure jump over the shock, as well as the unsteady post-shock pressure, is different for identical shock positions during the cycle of fluctuation, which implies that an unsteady shock movement, imposed by oscillating backpressure, may Introduce- a significant unsteady lift and moment. This may be or importance for flutter predictions. It is also noted that, although the sonic velocity is obtained in the throat of steady state quasi one-dimensional flow, this is not necessarily true for the unsteady solution. During part of the period with fluctuating backpressure, the flow velocity may be subsonic at the throat, and still reach a supersonic value later in the nozzle. This phenomenon depends on the frequency and amplitude of the imposed pressure fluctuation, as well as on the nozzle geometry.