## Flow over an Airfoil with a Protuberance

Here we show some very recent Navier-Stokes solutions carried out to study the aerodynamic effect of a small protuberance extending from the bottom surface of an airfoil. These calculations represent an example of the state-of-the-art of full Navier-Stokes solutions at the time of writing. The work was carried out by Beierle (Reference 89). The basic shape of the airfoil was an NACA 0015 section. The computational fluid dynamic solution of the Navier-Stokes equations was carried out using a time-marching finite volume code labeled OVERFLOW, developed by NASA (Reference 90). The flow was low speed, with a freestream Mach number of 0.15 and Reynolds number of 1.5 x 106. The fully turbulent flow field was simulated using a 1-equation turbulence model.

Using a proper grid is vital to the integrity of any Navier-Stokes CFD solution. For the present case, Figures 20.8-20.11 show the grid used, progressing from the

(a) |

x~xo So (b) |

Figure 20.7 Effects of shock-wave/boundary-layer interaction on (a) pressure distribution, and (b) shear stress for Mach 3 turbulent flow over a flat plate.

big picture of the whole grid (Figure 20.8) to the detail of the grid around the small protuberance on the bottom surface of the airfoil (Figure 20.11). The grid is an example of a chimera grid, a series of independent but overlapping grids that are generated about individual parts of the body and for specific flow regions.

Some results for the computed flow field are shown in Figures 20.12 and 20.13. In Figure 20.12, the local velocity vector field is shown; the flow separation and locally reversed flow can be seen downstream of the protuberance. In Figure 20.13, pressure contours are shown, illustrating how the small protuberance generates a substantially asymmetric flow over the otherwise symmetric airfoil.

Finally, results for a related flow are shown in Figure 20.14. Here, instead of a protuberance existing on the bottom surface, an array of small jets that are distributed

z Figure 20.8 Individual grid boundary outlines used in the chimera grid scheme for calculating the flow over an airfoil with a protuberance. |

over the bottom surface alternately blow and suck air into and out of the flow in such a manner that the net mass flow added is zero, so-called “zero-mass synthetic jets.” The resulting series of large-scale vortices is shown in Figure 20.14—another example of a flow field that can only be solved in detail by means of a full Navier-Stokes solution. (See Hassan and JanakiRam, Reference 91, for details.)