Stall cells over an airfoil. Part 2: A vortex-based analytical model for their formation and saturation

Stall cells are spanwise-periodic flow structures that spontaneously form on airfoils operating near stall, fundamentally altering the aerodynamic loading distribution. Despite decades of experimental observations, a complete theoretical framework connecting vortex dynamics to the characteristic flow patterns has remained elusive. In this work, we develop an analytical model for stall cell formation based on the interaction between finite-length, counter-rotating vortex tubes representing the separation vortex and trailing-edge vortex. Linear stability analysis of the coupled vortex system yields the growth rate and wavelength selection of the Crow-type instability responsible for the wave-like bending of the vortex structures. A weakly nonlinear analysis using the method of multiple scales is performed to derive the Stuart–Landau amplitude equation, providing an explicit expression for the saturation amplitude at which nonlinear effects arrest the instability growth and establish quasi-steady cellular structures. The vortex sheet representing the separated shear layer is coupled to the vortex tube dynamics through the Birkhoff–Rott equation, from which we derive the induced vertical vorticity $Ω_y$ that drives the alternating spanwise velocity characteristic of stall cells. The model predicts quantitatively the spanwise velocity magnitude, vertical vorticity distribution, and vortex sheet deformation. The resulting framework provides a unified, first-principles description connecting the Crow-type instability of counter-rotating vortex tubes to the observed flow topology of stall cells. The model is validated against the DDES simulation data presented in the companion paper, demonstrating strong agreement.

💡 Research Summary

This paper presents a first‑principles analytical model that explains the formation and saturation of stall cells—spanwise‑periodic cellular flow structures that appear on airfoils operating near stall. The authors build on the long‑standing hypothesis that a pair of counter‑rotating vortex tubes—one representing the separated shear‑layer vortex (the “separation vortex”) and the other the trailing‑edge vortex—interact through mutual induction. By treating each vortex as a finite‑length vortex tube with circulation Γ, core radius a, length L and inter‑tube spacing b, they derive both linear and weakly‑nonlinear dynamics that capture the essential physics observed in high‑fidelity delayed‑detached‑eddy simulations (DDES).

Linear stability analysis revisits the classic Crow instability for counter‑rotating vortex pairs. Using the exact Biot–Savart law for finite tubes, the authors solve the eigenvalue problem and obtain the most amplified spanwise wavelength Λₛ ≈ 8.6 b and a growth rate σ ∝ Γ/b², modified by the finite‑core correction f(L/a). When the parameters extracted from the DDES (Γ≈0.12 U∞c, b≈0.23 c, L≈1.5 c) are inserted, the predicted wavelength matches the observed stall‑cell spacing of roughly two chord lengths (Λₛ≈2 c) and the growth rate agrees with the simulation’s exponential growth phase.

Weakly nonlinear analysis employs the method of multiple scales to derive a Stuart–Landau amplitude equation for the perturbation amplitude A(t): \