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Calculating pressure
Doublet gradient
The APM Solver determines the surface pressure coefficient from the gradient of the doublet strength. The gradient of the doublet strength distribution is not readily available. The solver uses several algorithms to calculate the gradient. Depending on the problem, some algorithms might perform better than others.
If the doublet_gradient_scheme option in the .conf file is set to 0, the doublet gradient will be calculated with a Constrained Hermite Taylor Series Least Squares (CHTLS) algorithm [1]. This is the default APM Solver doublet gradient scheme.
If the doublet_gradient_scheme option in the .conf file is set to 1, the doublet gradient will be calculated with a Weighted Least Squares (WLS) algorithm.
If the doublet_gradient_scheme option in the .conf file is set to 2, the doublet gradient will be calculated with a Linear Shape Functions (LSF) algorithm.
If the doublet_gradient_scheme option in the .conf file is set to 3, the doublet gradient will be calculated with a Finite Differences (FD) algorithm.
If the doublet_gradient_scheme option in the .conf file is set to 4, the doublet gradient will be calculated with the Hess & Smith doublet - vortex ring equivalence (HS) algorithm.
The Finite Difference (FD) doublet gradient scheme can be used only on structured meshes!
Gradient smoothing
The doublet gradient estimated with either of the methods above can be further smoothed via a Gauss-Seidel smoother. The options doublet_gradient_smoothing_iterations and doublet_gradient_smoothing_factor control the strength of the smoothing. The default recommended settings are 3 iterations and a smoothing factor of 0.5. A smoothing factor near 0 results in weak smoothing, and a smoothing factor close to 1 results in strong smoothing.
The HS algorithm, coupled with gradient smoothing, is recommended on complex meshes. It will give superior results to the CHTLS algorithm.
The images below show a comparison of the surface pressure coefficient obtained with different doublet strength gradient algorithms.
CHTLS (left), WLS (middle), and LSF (right) results for a sphere geometry
CHTLS (left), WLS (middle), and LSF (right) results for the AGARD-AR-303 E6
The image below compares the CHTLS (left) with HS (right). The effect of doublet gradient smoothing can be clearly seen in the pressure coefficient distribution.
CHTLS (left), and HS + gradient smoothing (right) results for the Cruiser UAV.
Trefftz-plane loads
An alternative to the surface pressure integration is to use the far-field of Trefftz-plane approach. In general this is a more reliable approach, especially for the induced drag component. To enable the Trefftz-plane analysis set the trefftz_plane_analysis option in the .conf file to 1.
Trefftz-plane used to calculate the loads of a fixed-wing UAV
For best results when using the Trefftz-plane analysis perform an unsteady solution. The unsteady solution will allow the wake to roll-up. The Trefftz-plane analysis will be performed at the location of the third to last row of wake panels.
