Sv. Tsynkov, ARTIFICIAL BOUNDARY-CONDITIONS FOR COMPUTATION OF OSCILLATING EXTERNAL FLOWS, SIAM journal on scientific computing, 18(6), 1997, pp. 1612-1656
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 vi
scous plane flow past a finite body (airfoil). Oscillations of the flo
w in time may be caused, for example, by the time-periodic injection o
f fluid into the boundary layer, which in accordance with experimental
data, may essentially increase the performance of the airfoil. To con
duct the actual computations, we have to somehow restrict the original
unbounded domain, that is, to introduce an artificial (external) boun
dary and to further consider only a finite computational domain. Conse
quently, 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 exterior so that
the original problem is solved within the desired accuracy. Our constr
uction of such ABCs for oscillating flows is based on an essential ass
umption: the Navier-Stokes equations can be linearized in the far fiel
d 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. Thi
s paper contains a general theoretical description of the algorithm fo
r setting the DPM-based ABCs for time-periodic external flows. Based o
n our experience in implementing analogous ABCs for steady-state probl
ems (a simpler case), we expect that these boundary conditions will be
come an effective tool for constructing robust numerical methods to ca
lculate oscillatory flows.