Stall cells over an airfoil. Part 1: Three-dimensional flow organisation and vorticity dynamics

TL;DR

This paper addresses stall-cell formation over an airfoil by performing high-fidelity three-dimensional simulations with a hybrid DDES-SST approach to resolve the coupled vorticity dynamics in separated flow. It reveals a Crow-type instability of a separation vortex tube interacting with a counter-rotating trailing-edge vortex, which deforms the vortex sheet and induces vertical vorticity that drives spanwise flow. A key quantitative finding is the linear rotation of the spanwise-velocity maxima with downstream distance, described by $\zeta = 14.5\left(\frac{x}{c}\right) - 0.8$, illustrating a robust 3D organization of stall cells. The study shows non-uniform spanwise load distribution and the sensitivity of stall-cell behavior to angle of attack, confirming stall cells under moderate turbulence and linking these dynamics to practical aerodynamic loading and control considerations; Part II extends these results with a vortex-sheet analytical model.

Abstract

This study investigates the three-dimensional organisation and evolution of stall cells in the separated flow region over an airfoil. Using a hybrid RANS/LES approach based on the DDES-SST turbulence model, we characterise the formation and development of these structures, which remain challenging to capture experimentally. Initial validation confirms accurate reproduction of global loads when comparing with both experimental data and RANS simulations. The complex three-dimensional flow organisation is analysed through investigating the vorticity, revealing that spanwise variation of the separation location leads to non-uniform load distribution along the airfoil span. The mid-span experiences premature separation due to flow bifurcation, while flow attraction at $\pm1$ chord length successfully sustains attached flow further along the chord. The separated flow generates a shear layer culminating in a separation vortex tube, which exhibits a Crow-type instability when interacting with the counter-rotating trailing edge vortex tube. This instability induces a wave-like bending of the vortex tubes and shear layer, generating significant vertical vorticity (y-vorticity) that drives spanwise flow. We identify a previously unreported phenomenon where the maxima of spanwise velocity structures exhibit rotation around fixed spanwise axes, with the rotation angle evolving linearly with downstream distance according to $ζ= 14.5(x/c) - 0.8$. This study provides new insights into the mechanisms underlying stall cell formation and highlights the importance of three-dimensional effects in separated flows, which has implications for aerodynamic load prediction and control strategies.

Figures (34)

• Figure 1: visualisation of the CFD-derived stall cell (SC) structures. In these images, the separation line vortex is designated as SL vortex and the stall cell vortex as SC vortex. The flow direction is indicated by blue arrows. (a) Depicts the surface flow lines on the wing's suction surface, complemented by in-plane flow lines on planes $z= 0$ (left side—symmetry plane), $z = -0.25$, and $z = -0.5$ (right side—inviscid wall). The 3D separation line is marked by blue dots. (b) Features a Q = 1 iso-surface with highlighted vortex core lines indicating vorticity direction. The distinctive open “U” shape of the Trailing Edge Line Vortex (TELV) inboard of the SC vortices is also shown. (c) Provides a side view detailing in-plane flow lines on plane $z = 0$ (symmetry plane—red lines) and $z = -0.25$. Specifications include an aspect ratio of 2.0, Reynolds number $Re_c = 8.7 \times 10^5$, and angle of attack $\theta = 10^\circ$ (adapted from manolesos2014study with permission from AIP publising under licence number: 6202370215761).
• Figure 2: Diagram illustrating vortex instability and the formation of cellular patterns in the separated flow over rectangular wings (taken from weihs1983cellular).
• Figure 3: Blade section at 82% of the radius of a full-scale wind turbine in comparison with a NACA63-3-420 profile with a modified camber of 4% instead of 2% (from Neunaber2022; published under CC BY-NC-ND 4.0.)
• Figure 4: Views of the mesh: (a) 3D mesh for the simulation of the flow over the airfoil (13 million cells); (b) side view of the mesh; (c) mesh around the airfoil; (d) front view of the mesh; (e) mid span view of the mesh at the end of the simulation after adaptive grid refinement (130 million cells). The figure is adopted from mishra2024developing, published under CC BY 4.0.
• Figure 5: Comparison of (a) $C_l$ and (b) $C_d$ curves obtained using the DDES-SST turbulence model with experimental data and simulation result from the $k-\omega$ SST Menter 2003 model. The experimental and $k-\omega$ SST Menter 2003 model simulation data are taken from mishra2024developing.