Non-reflecting Inflow and Outflow in a Wind Tunnel for Transonic Time-Accurate Simulation

Abstract

We consider a time-dependent flow problem in an infinitely long wind tunnel of circular cross-section. It is assumed that beginning with a distance up-and downstream from the stream-lined body, the flow is approximately described by the time-dependent Euler equations linearized about a uniform free-stream flow. By using this linear model, we transfer the boundary conditions from infinity to the front and back boundaries of a finite computational domain. The obtained bound-Ž . ary conditions called transparent ones TBCs are represented by explicit formulae in the physical space. The TBCs contain operators of Fourier expansion with respect to eigenfunctions of the cross-section and convolutions with respect to time, i.e., they are non-local in both space and time. One possibility of their numerical implementation is described; the main attention is paid to the evaluation of the convolution operator. Due to the representation of the convolution kernel by an exponential polynomial, the evaluation formulae become even local in time.

Artificial boundary conditions for the numerical simulation of unsteady acoustic waves

“…The key consideration of interest is that if the RHSs of system (4) are compactly supported in space and time on some domain Q ⊂ R 3 × [0, +∞), then the RHSs of both equation (7) and equation (8) will also be compactly supported on the same domain Q. Consequently, solutions of equations (7) and (8) will have lacunae of the same shape as prescribed by formula (3). Thus, we have arrived at the following Proposition 2 Let the RHSs of system (4) be compactly supported in space and time: supp q vol (x , t) ⊆ Q and supp b vol (x , t) ⊆ Q, where Q ⊂ R 3 × [0, +∞).…”

“…Several other nonlocal ABCs' methodologies for unsteady waves have been recently advocated in the literature, most notably [4][5][6][7][8][9][10]; we also mention the survey paper [11] and the bibliography there. Compared to these techniques, our approach has a number of distinctive characteristics.…”

Well-posedness and Stability of Exact Non-reflecting Boundary Conditions

“…As in the continuous case we limit ourselves to zero growth, which means that η ≤ 0 in (26) is needed. Hence, to prove stability, we must show that the boundary terms in (30) and (31) …”

Exact Non-reflecting Boundary Conditions Revisited: Well-Posedness and Stability

“…When it comes to the implementation of exact NRBC's it is, for special geometries, possible to localize the boundary conditions in time while still keeping them exact. This is exemplified in [9] where computations are performed for the wave equation on a spherical domain, and in [29] where highly accurate boundary conditions are used for a flow in a cylinder. See [15] for more details on exact and approximate NRBC on special computational domains.…”