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Example Two-dimensional Airfoil Analysis

The simplest aerodynamic analysis that can be considered is flow past a two-dimensional symmetric airfoil. We begin with the NACA0012 airfoil, which has been extensively studied and for which good-quality experimental data is available. First we show what can be achieved for this airfoil using XFoil; the NASA four-digit sections are built in to this toolkit directly so that a drag polar can be produced very rapidly, see Figure 13.9. Notice that very good agreement to experimental data is given until the section is highly separated and stall begins, when XFoil overpredicts the lift that is possible.

Next, to show the impact of meshing, we compare three two-dimensional RANS meshes with experimental results and tabulated data reported on the NASA turbulence modeling Web site. Figures 13.10 and 13.11 illustrate two NASA meshes, one with y+ ranging up to 2 and containing around 4000 cells, the second with y+ held below 0.35 and 57 000 cells. The third mesh is a block-structured model produced with ICEM and with y+ averaging 0.75 and 60 000 cells, see Figure 13.12. Note that the ICEM mesh achieves a worse y+ for a similar cell count because it is much less highly optimized to the section. The NASA meshes are the result of very careful placement of the mesh control points. Figure 13.13 shows the results from using these meshes with the Fluent k - m SST turbulence model and 6000 solver iterations (this yields a very highly converged solution and allows the use of a relaxation setting on momentum set down to 0.2 to aid stability of solution). Also included are three results tabulated on the NASA Web site from their CFD studies as well as the experimental results provided there.

Notice that for low AoAs (up to 6°), all the RANS meshes give similar lift and drag results and these agree well with experimental ones. At higher angles, where the onset of separation begins, none of the meshes correctly resolves the drag, overestimating this by up to 100% and

Lift versus drag polar for NAC0012 airfoil from XFoil and experiments

Figure 13.9 Lift versus drag polar for NAC0012 airfoil from XFoil and experiments. Note that when plotted in this way, both lift and drag coefficients may be found at a given angle of attack or, for a given lift coefficient, drag coefficient and angle of attack may easily be read off.

Low-resolution NASA Langley 2D mesh around the NACA0012 foil

Figure 13.10 Low-resolution NASA Langley 2D mesh around the NACA0012 foil.

Middle-resolution NASA Langley 2D mesh around the NACA0012 foil

Figure 13.11 Middle-resolution NASA Langley 2D mesh around the NACA0012 foil.

being not better than the much less expensive XFoil panel analysis. Also, none of the meshes accurately resolves the correct peak of the lift curve though the two finer meshes are within two degrees. The NACA0012 section is difficult for RANS-based CFD to resolve because (a) it has a low thickness to chord ratio, meaning that the zero AoA drag is very small; (b) when very smooth, the section is capable of attached flows up to 17° after which the stall is rather sudden; and (c) the studies carried out by NACA were for speeds where the boundary layer is thin requiring large meshes to resolve the viscous sublayer. This level of (dis)agreement is typical for RANS-based CFD near the onset of separation, even using high-quality validation meshes, since a separation bubble on the low-pressure side of the airfoil has to be correctly predicted from first principles if accurate results are to be generated, see Figure 13.14.

ICEM 2D mesh around the NACA0012 foil (courtesy of Dr D.J.J. Toal)

Figure 13.12 ICEM 2D mesh around the NACA0012 foil (courtesy of Dr D.J.J. Toal).

Experimental and 2D computational lift and drag data for the NACA0012 airfoil (using the k — a SST turbulence model). Adapted from Abbott and von Doenhoff 1959

Figure 13.13 Experimental and 2D computational lift and drag data for the NACA0012 airfoil (using the k — a SST turbulence model). Adapted from Abbott and von Doenhoff 1959.

 
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