We have used the NASA developed production Navier-Stokes code TLNS3D by V. Vatsa to calculate the transonic viscous flow over the ONERA M6 wing for the Mach number M=0.84, Reynolds number Re=11.7 million, and the angle of attack of 3.06 degrees. The code has been supplemented by the global DPM-based artificial boundary conditions (ABCs) for treatment of the external boundary. The calculations were done on a one-block curvilinear C-O grid. The results obtained using the DPM-based ABCs were compared with the ones based on the standard external boundary conditions (locally one-dimensional characteristics-type treatment).
First, we we have been able to achieve a major accuracy improvement: the external boundary was brought from the distance of 10 root chords of the wing to as close as 3 root chords away of the wing's surface. (Note, for the subcritical case M=0.5 we could bring the external boundary even closer, up to 1.25 root chords of the wing.)
Second, we have been able to significantly speed up the convergence of the multigrid iterations.
On the figures below, we present the convergence histories for two different computational domains of the approximate radii of 3 and 10 root chords of the wing, respectively.
Convergence history for the residual of the continuity equation. Average "radius" of the computational domain is 3 root chords of the wing; grid dimension is 193x49x33.
Convergence history for the number of supersonic points in the domain. Average "radius" of the computational domain is 3 root chords of the wing; grid dimension is 193x49x33.
Convergence history for the residual of the continuity equation. Average "radius" of the computational domain is 10 root chords of the wing; grid dimension is 209x57x33.
Convergence history for the number of supersonic points in the domain. Average "radius" of the computational domain is 10 root chords of the wing; grid dimension is 209x57x33.
The next series of computations has been conducted for the same ONERA M6 wing, Mach number M=0.84, Reynolds number Re=11.7 million, and the angle of attack higher than before, 5.06 degrees. Unlike in the previous case, for which the flow is attached, there is now a stronger adverse pressure gradient, which causes boundary layer separation downstream of the shock wave on the upper surface of the wing. Most of the separation occurs in the area close to the wing tip. We calculated this flow for the two different computational domains of the radii 10 and 3 root chords of the wing, respectively; in both cases we used 193x57x33 grids, the grid in the smaller domain was finer in the near field. The two-equations Menter's model has been used for simulating the turbulence inside the computational domain. In the Figure below, we compare the results for the different domains and different types of ABCs. Note, the procedure with the standard ABCs fails to converge for the small computational domain.
Surface pressure distribution at the 90% of semi-span for the ONERA M6 wing, M=0.84, Re=11.7 million, angle of attack 5.06 degrees.
Convergence history for the residual of the continuity equation. Average "radius" of the computational domain is 3 root chords of the wing; grid dimension is 193x49x33.
Convergence history for the residual of the continuity equation. Average "radius" of the computational domain is 10 root chords of the wing; grid dimension is 193x49x33.