Numerical accuracy
Table of contents
Panel density requirements
If strong separation bubbles are present in a viscous solution, then it is very important to have good panel resolution in the region of the bubble(s). The large gradients at a bubble tend to cause significant numerical errors even if a large number of panels is used. If a separation bubble appears to be poorly resolved, it is a good idea to re-panel the airfoil with more points, and/or with points bunched around the bubble region. The paneling is controlled from the PPAR menu. A good rule of thumb is that the shape parameter Hk just after transition in the bubble should not decrease by more than 1.0 per point. Likewise, the surface velocity Ue/Vinf should not change by more than 0.05 per point past transition, otherwise there may be significant numerical errors in the drag. The point values can be observed by issuing SYMB from the VPLO menu. Moderate chord Reynolds numbers (1-3 million, say) usually require the finest paneling, since the bubbles are still important, but very small. On many airfoils, especially those with small leading edge radii, the development of the small bubble which forms just behind the leading edge can have a significant effect on CLmax. For such cases, the default paneling density at the bubble may not be adequate. In all cases, inadequate bubble resolution results in a “ragged” or “scalloped” CL vs CD drag polar curve, so fortunately this is easy to spot.
Differencing order of accuracy
The BL equations are normally discretized with two-point central differencing (i.e. the Trapezoidal Scheme), which is second-order accurate, but only marginally stable. In particular, it has problems with the relatively stiff shape parameter and lag equations at transition, where at high Reynolds number the shape parameter must change very rapidly. Oscillations and overshoots in the shape parameter will occur with the Trapezoidal Scheme if the grid cannot resolve this rapid change. To avoid this nasty behavior, upwinding must be introduced, resulting in the Backward Euler Scheme, which is very stable, but has only first-order accuracy. Previous versions of XFOIL allowed a specific constant amount of upwinding to be user-specified. Currently, XFOIL automatically introduces upwinding into the equations only in regions of rapid change (typically transition). This ensures that the overall scheme is stable and as accurate as possible.
Since only a minimal amount of upwinding is introduced in the interest of numerical accuracy, small oscillations in the shape parameter H will sometimes appear near the stagnation point if relatively coarse paneling is used there. These oscillations are primarily a cosmetic defect, and do not significantly affect the downstream development of the boundary layer. Eliminating them by increasing upwiding would in fact produce much greater errors in the overall viscous solution.