Using XFOIL for Analysis of Airfoils under low Reynolds Number Conditions

Thanks to our #lemfev project, we have this wonderful opportunity to dive deep into low Reynolds number aerodynamics. Numerically, it is difficult to simulate low Re flows because of the significant role viscosity plays under these conditions. Nevertheless, we have a great zero-cost tool for a reasonably accurate numerical analysis of airfoils even when laminar bubbles form and travel along their surfaces, which is XFOIL, created by Mark Drela at MIT.

I have studied the documentation for XFOIL, which provided me with a good insight into its theoretical basis and also with guidance as to how to use the code correctly.

XFOIL uses a linear-vorticity stream function panel method, supplemented by a Karman-Tsien compressibility correction for inviscid compressible flow modelling, and a two-equation integral BL formulation, as well as an envelope e^n transition?criterion for BL modelling.

Transition in an XFOIL solution is triggered by one of two ways:

• ?free?transition, when the?e^n criterion is met;
• ?forced transition, when?a trip or the trailing edge is encountered.

The e^n method is only appropriate for predicting transition when the growth of 2-D Tollmien-Schlichting waves via linear instability is the dominant transition-initiating mechanism.??The linear-instability mechanism usually dominates in adverse pressure gradients.

Other possible mechanisms are:

• crossflow instabilities, typical for swept wings with significant?favorable chordwise pressure gradients;
• attachment-line transition, which occurs?with a large sweep, a large leading-edge radius, and a large Reynolds number;
• bypass transition characteristic for significant wall roughness and/or large freestream turbulence or vibration levels.

If any of these alternative transition mechanisms are present, in XFOIL, the trips must be set to mimic their effect.?

In a high-turbulence, dusty Martian atmosphere, the third mechanism is likely to be as important as the linear instability due to the adverse pressure gradients. In XFOIL, the bypass transition mechanism can be mimicked to some extent by the e^n method by?setting Ncrit to a small value.?This will?cause a transition just after linear instability begins.?However, for very large freestream turbulence or roughness in favorable pressure gradients, bypass transition can occur before the?linear instability threshold, and in this case, trips will have?to be set as well.

If the laminar bubble formed on an airfoil under the given conditions is thick, a boundary layer trip can actually decrease airfoil drag and increase lift.

I have written a code in Matlab that helps me run XFOIL, extract, and plot simulation results, as well as use them in aircraft analysis automatically.

When working with XFOIL, I was impressed by the accuracy of the numerical results when compared to test data. The figure below shows a polar plotted for the Ishii airfoil, and we see that the XFOIL accuracy is at the level of the ANSYS Fluent, while the computational time and effort are not comparable.

When plotting skin friction coefficient, I encountered an interesting oscillation in the region of the stagnation point, shown below.

As I learned, this feature is due to the following.

In XFOIL, the boundary layer equations are discretized with two-point central differencing, which is second-order accurate, but only marginally stable.?In particular, it has problems if the shape parameter must change very rapidly and the grid cannot resolve this rapid change.?To avoid this behavior, upwinding must be introduced, resulting in the Backward Euler Scheme, which is very stable, but has only first-order accuracy.?XFOIL automatically introduces upwinding into the equations only in regions of rapid change in the?interest of numerical accuracy.

Unfortunately, I failed to extract and plot the dissipation coefficient CD' in my code. CD'(x) is proportional to the local energy dissipation rate due to viscous shear and turbulent mixing.

It is useful for the sake of analysis since it indicates where on the airfoil drag is being created, both friction and pressure drag.

In any case, now, I have an extremely powerful automated analysis and plotting tool. I can even visualize, e.g., how the laminar bubble length changes with angle of attack!