AIAA 2002-0120 Application of a Non-Linear Frequency Domain Solver to the Euler and Navier-Stokes Equations

This paper presents a technique used to accelerate the convergence of unsteady flows to a periodic steady state. The basis of this procedure is to assume the time period of the solution's oscillation and to transform both the solution and residual using a discrete Fourier transform. However, this paper also presents a method which iteratively solves for the time period during the process of calculating a solution. These methods are amenable to parallel processing and convergence acceleration techniques such as multigrid and implicit residual averaging. The accuracy and efficiency of the technique is verified by Euler and Navier-Stokes calculations for a pitching airfoil whose period of oscillation is forced. The capability to identify the natural frequency of oscillation is verified by Navier-Stokes calculations for laminar vortex shedding behind a cylinder where the time period of oscillation is unknown a priori. Results show that a limited number of modes can accurately capture the major flow physics of these model cases. Introduction Unsteady flows still present a severe challenge to Computational Fluid Dynamics (CFD). In general these flows can be subdivided into two general classes. The first are unsteady flows where the boundary conditions are forcing the unsteadiness at predetermined frequencies. Examples of this include the internal flows of turbomachinery, the external flow fields of helicopter blades or propellers, and certain aero-elastic computations. The second general class of unsteady flows are where instabilities in the fluid mechanical equations induces unsteadiness in the flow field. Examples in this class include (but are obviously not limited to) vortex shedding behind a cylinder, and other fluid dynamic cases involving separated flows and free shear layers. Without experimental or simplified analytic models, the temporal frequencies are difficult to determine a priori for this second class. This paper will present a reduced order scheme capable of solving unsteady flows for both classes of problem. The motivation for developing this scheme is the need to reduce the cost of unsteady CFD simulations for complex flows. In general, time accurate solvers are designed to capture any arbitrary time history in the evolution of the solution. There are many applications, however, such as helicopter rotors or turbomachinery where users are typically only concerned with the data once the solution has reached a periodic steady state. Nevertheless , the majority of the computational effort is expended in resolving the decay of the initial tran-sients. Algorithmic efficiency is a function of the …