The Reynolds-Averaged Vortex Force Map Method

Abstract

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 ($k$-$ω$ 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.

Figures (7)

• Figure 1: (a) 3-D schematic of the Accipiter gentilis (Northern Goshawk) in gliding configuration. The ground-fixed coordinate system is defined with $x$ along the freestream direction ($U_\infty$), $y$ along the spanwise axis, and $z$ perpendicular to both. (b) Decomposition of the aerodynamic force into lift ${L}$ and drag ${D}$, or equivalently into normal ${N}$ and axial ${A}$ components. The body angle of attack is $\alpha_b$, defined between $U_\infty$ and the body $x$-axis; $\theta$ is the wing–body setting angle and $\gamma$ the body–tail setting angle. (c) Cross-sectional view of the wing, represented by a GOE803 aerofoil profile.
• Figure 2: CFD validation against theoretical and experimental results Riegels1961Usherwood2020Cheney2021 (a) Aerofoil, (b) Bird with wake comparison: CFD (top left), lab measurements (bottom left), and Q-criterion comparison: CFD (bottom right), PTV (top right).
• Figure 3: Comparison of force coefficients from CFD and RA-VFM method ($F^{(\mathit{vp})}_k + F^{(\mathit{rs})}_k$) for both the aerofoil (a–c) and the bird (d–f). (a,c,d,f) $k=L$; (b,e) $k=D$. For the aerofoil, XFOIL predictions are included. (c, f) Lift-component breakdown in the $(x, y, z)$ directions, where dual vertical axes are used to represent $F^{(\mathit{vp})}_L$ and $F^{(\mathit{rs})}_L$ separately.
• Figure 4: Visualisation of flow fields and dominant lift components for the aerofoil (a,c) and bird (b,d) under incidence-matched conditions ($\alpha_{\mathrm{eff}} = \alpha = 13^{\circ}$ and $22^{\circ}$). $1^{st}$ row: contours of $\omega_y$ overlaid with VFM for lift, contour lines show $|\Lambda_{L,y}|$ and arrows show its direction. $2^{nd}$ row: distribution of vortex–pressure lift density $f^{(\mathrm{vp})}_L$, mainly contributed by $f^{(\mathrm{vp})}_{L~y}$. $3^{rd}$ row: contours of $\zeta_z$ overlaid with contour lines of the gradient of the potential field $\phi_{L,z}$. $4^{th}$ row: Reynolds–stress lift density $f^{(\mathrm{rs})}_{L~z} = \phi_{L,z}\,\zeta_z$.
• Figure 5: Isosurfaces of the VP (a,c) RS (b,d) lift contribution with streamlines around the gliding bird ($1^{st}$ row) . Spanwise distribution of lift (a,c) and drag (b,d) ($2^{nd}$ row). (a,b) $\alpha_b = 0^{\circ}$and(c,d) $9^{\circ}$.