Artificial Boundary Conditions for Computation of Oscillating External Flows

Abstract

In this paper, we propose a new technique for the numerical treatment of external flow problems with oscillatory behavior of the solution in time. Specifically, we consider the case of unbounded compressible viscous plane flow past a finite body (airfoil). Oscillations of the flow in time may be caused, for example, by the time-periodic injection of fluid into the boundary layer, which in accordance with experimental data, may essentially increase the performance of the airfoil. To conduct the actual computations, we have to somehow restrict the original unbounded domain, that is, to introduce an artificial (external) boundary and to further consider only a finite computational domain. Consequently, we will need to formulate some artificial boundary conditions (ABCs) at the introduced external boundary. The ABCs we are aiming to obtain must meet the following fundamental requirement. One should be able to uniquely complement the solution calculated inside the finite computational domain to its infinite exterior so that the original problem is solved within the desired accuracy. Our construction of such ABCs for oscillating flows is based on an essential assumption: the Navier--Stokes equations can be linearized in the far field against the free-stream background. To actually compute the ABCs, we represent the far-field solution as a Fourier series in time and then apply the difference potentials method (DPM) of V. S. Ryaben'kii. This paper contains a general theoretical description of the algorithm for setting the DPM-based ABCs for time-periodic external flows. Based on our experience in implementing analogous ABCs for steady-state problems (a simpler case), we expect that these boundary conditions will become an effective tool for constructing robust numerical methods to calculate oscillatory flows.