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

TL;DR

This work develops an analytically tractable, first-principles model for stall-cell formation by coupling finite-length counter-rotating vortex filaments (separation and trailing-edge vortices) with an attached vortex sheet. Linear analysis yields the Crow-type growth and the most amplified wavelength, while a weakly nonlinear, multiple-scales treatment produces a Stuart–Landau amplitude equation that predicts finite-amplitude saturation. The coupling to the vortex sheet via the Birkhoff–Rott equation generates a vertical vorticity field $\Omega_y$ and an alternating spanwise velocity $w^*$ that together reproduce the observed stall-cell topology. Validation against high-fidelity DDES data shows good qualitative and quantitative agreement in sheet deformation, spanwise velocity, and $\Omega_y$ distribution, supporting the theory’s relevance for predicting stall-cell spacing and saturation dynamics with potential implications for design and flow control.

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 $Ω_y$ 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.

Figures (6)

• Figure 1: Time-averaged $Q = 7$ iso-surface from DDES showing the separation vortex tube and trailing-edge vortex tube forming a counter-rotating pair. The wave-like bending of both structures is evident.
• Figure 2: Contour plots from DDES at $x/c = 0.6$: (a) normalised vertical vorticity $\varOmega_y$ showing alternating positive and negative regions; (b) normalised spanwise velocity $w^*$ exhibiting the characteristic stall cell pattern. The black curves are showing in-plane streamlines.
• Figure 3: Schematic of the experimental model setup. Key features include the Separation Vortex Tube (SVT), the Trailing edge Vortex Tube (TVT), and the Separated Shear Layer (SSL).
• Figure 4: Comparison of vortex sheet deformation between the weakly nonlinear model (solid blue) and DDES (dashed red) at three downstream positions. The dimensional deformation $h_{\mathrm{sat}}/c$ shows the characteristic cosine pattern with exponential streamwise decay.
• Figure 5: Normalised spanwise velocity $w^*/w^*_{\max}$ along the deformed vortex sheet comparing the model prediction (solid blue) and DDES results (dashed red) at $x/c = 0.5$, $0.7$, and $0.9$.