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

TL;DR

This paper presents a vortex-based analytical model for stall cell formation on airfoils, linking vortex dynamics to flow patterns and validating predictions with simulation data.

Contribution

It introduces a first-principles analytical framework connecting vortex interactions and instabilities to stall cell formation, advancing theoretical understanding.

Findings

Quantitative predictions of spanwise velocity and vorticity.

Validation against high-fidelity simulation data.

Identification of the instability mechanism driving stall cells.

Abstract

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 $\Omega_y$ 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.