Aerodynamics and Design for Ultra-Low Reynolds Number Flight

Chapter 2
Conventional airfoils have round leading edges and trailing edges that approximate a
cusp. The sections considered here are typically only 2% thick with the added
complexity of manufacturing constraints requiring the consideration of blunt leading and
trailing edges due to minimum gage constraints. As a result, a significant portion of the
drag is being calculated in regions of very high surface curvature. Discretizing these
airfoils creates the possibility of incurring discretization and subsequent force integration
errors in the leading and trailing edge regions.
A second issue is the possibility of excessive numerical and spatial dissipation in the
solution affecting the results. INS2d has not previously been used for airfoil analysis at
ultra-low Reynolds numbers and excessive dissipation could introduce errors, and in
extreme cases result in solutions that erroneously appear to be steady-state.
Both of these issues have been assessed, and the forces have been verified by utilizing an
off-body control-volume approach. The method applies the two-dimensional
conservation of momentum equation which in integral form may be expressed as:
This far-field calculation is analogous to the surface force integration with the viscous
body force term replaced by the momentum flux across the control volume. As an
additional check, the post-processor calculates the total mass flux for the control volume.
Instead of the forces exerted on the airfoil by the fluid, the effect on the fluid from the
airfoil is calculated. The large rounded control volume does not have the regions of high
local curvature found at the surface, rather the integration area has been expanded,
reducing local sensitivities due to any particular region. Several issues arise with this
approach. These include numerical dissipation within the flow field, which the method
attempts to assess, and application too far from the airfoil, where the cell sizes generally
are larger and spatial discretization error and increased dissipation can become issues.
10
∫°
F′ =
ρV( V ⋅ n)
dS
+ ∫° pndS (2.4)