The Reynolds-Averaged Vortex Force Map Method

Vortex-force mapping (VFM) links vortical flow structures to aerodynamic forces through compact-domain integrals weighted by geometry-only Laplace potentials, but existing formulations are tied to simple geometries and laminar flows. In this study, we derive a Reynolds-averaged vortex force map (RA-VFM) directly from the incompressible Reynolds-averaged Navier-Stokes (RANS) equations, augmenting the classical vortex-pressure (VP) term with a Reynolds-stress (RS) contribution based on the Laplace-potential-weighted divergence of the modelled Reynolds stress (Boussinesq eddy-viscosity form). The resulting framework reconstructs mean lift and drag from RANS mean fields while retaining spatial attribution of force production to specific regions and coherent structures within a compact control volume. We apply RA-VFM to unsteady RANS ($k$-$\omega$ SST) simulations of a realistic gliding goshawk with strong three-dimensionality and a matched GOE803 aerofoil section. For the aerofoil, the VP term alone reproduces the CFD force curves over the pre- and near-stall range, with RS contributions becoming appreciable only in deep stall. For the bird, by contrast, the VP term underpredicts both $C_L$ and $C_D$, whereas including the RS term reduces the mean absolute error relative to CFD from $6\%$ to $2\%$ in lift and from $5\%$ to $1\%$ in drag over an angle of attack range of $0^\circ$-$20^\circ$. RA-VFM thus extends vortex-force mapping to turbulent, 3-D RANS flows and enables quantitative attribution of mean lift and drag to specific coherent structures within compact domains.