Stall cells over an airfoil. Part 2: A vortex-based analytical model for their formation and saturation
LHEEA lab. - CNRS - Nantes Université, Centrale Nantes（南特大学中央理工学院）；FLOW, Department of Engineering Mechanics, KTH Royal Institute of Technology（皇家理工学院）
Rishabh Mishra, Emmanuel Guilmineau, Ingrid Neunaber, Caroline Braud
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.
The present paper numerically investigates flow control over NACA0021 (National Advisory Committee for Aeronautics) airfoil by applying a leading-edge tubercle as a passive flow control device at transitional Reynolds number (Re) regime. The study includes simulations over the unmodified airfoil and modified airfoil operating at Re=120000 using kT-kL-ω based Delayed Detached Eddy Simulation (DDES) model. The calculations include both the pre-stall and post-stall regions, while the performance of the mean flow aerodynamic properties like pressure, lift, skin-friction, and drag coefficients are …
Airfoil stall plays a central role in the design of safe and efficient lifting surfaces. We typically distinguish between static and dynamic stall based on the unsteady rate of change of an airfoil's angle of attack. Despite the somewhat misleading denotation, the force and flow development of an airfoil undergoing static stall are highly unsteady and the boundary with dynamic stall is not clearly defined. We experimentally investigate the forces acting on a two-dimensional airfoil that is subjected to two manoeuvres leading to static stall: a slow continuous increase in angle of attack with a…
Dynamic stall on airfoils negatively impacts their aerodynamic performance and can lead to structural damage. Accurate prediction and modelling of the dynamic stall loads are crucial for a more robust design of wings and blades that operate under unsteady conditions susceptible to dynamic stall and for widening the range of operation of these lifting surfaces. Many dynamic stall models rely on empirical parameters that need to be obtained from experimental or numerical data which limits their generalisability. Here, we introduce physically derived times scales to replace the empirical paramete…
Stall cells over an airfoil. Part 1: Three-dimensional flow organisation and vorticity dynamics
LHEEA lab. - CNRS - Nantes Université, Centrale Nantes（南特大学、中央南特学院）；FLOW, Department of Engineering Mechanics, KTH Royal Institute of Technology（瑞典皇家理工学院）
University of Naples Federico II（那不勒斯费德里科二世大学）；Stanford University（斯坦福大学）
A convolutional autoencoder is trained using a database of airfoil aerodynamic simulations and assessed in terms of overall accuracy and interpretability. The goal is to predict the stall and to investigate the ability of the autoencoder to distinguish between the linear and non-linear response of the airfoil pressure distribution to changes in the angle of attack. After a sensitivity analysis on the learning infrastructure, we investigate the latent space identified by the autoencoder targeting extreme compression rates, i.e. very low-dimensional reconstructions. We also propose a strategy to…
Tubercles are modifications to the leading edge of an airfoil in the form of blunt wave-like serrations. Several studies on the effect of tubercles on isolated airfoils have shown a beneficial effect in the post-stall regime, as reduced drag and increased lift, leading to a delay of stall. The prospect of delaying stall is particularly attractive to designers of axial compressors in gas turbines, as this leads to designs with higher loading and therefore higher pressure rise with fewer number of stages. In the present study, experiments were performed on a cascade of airfoils with NACA 65209 p…
The present study reports on flow past airfoils (stationary and moving) using sharp interface immersed-boundary approach. Non-boundary conforming approach like immersed-boundary method offers a viable alternative over traditional boundary conforming approach by allowing us to model flow past arbitrarily complex shapes, by eliminating the need to re-grid the flow domain as the body exhibits motion. We present flow past a NACA 0012 airfoil at stationary conditions as well as exhibiting pitching motion. Evolution of vortex dynamics and wake structures are presented to show that the developed shar…
Airfoils pitching in the stalled regime have been of keen interest in recent years due to their desirable aerodynamic force characteristics. In this study, we are numerically investigating the unsteady flow past a NACA 0012 airfoil under sinusoidally pitching motion, using a finite volume based sharp interface immersed boundary solver. The flow is investigated in the low Reynolds number regime (Re=3000) for reduced frequencies of 0.1 and 0.5, at three different pivot locations (c/3, c/2 and 2c/3 from the leading edge). The airfoil is subjected to sinusoidal oscillations with its incidence angl…
We numerically examine the mechanisms that describe the shock-boundary layer interactions in transonic flow past an oscillating wing section. At moderate and high angles of incidence but low amplitudes of oscillation, shock induced flow separation or shock-stall is observed accompanied by shock reversal. Even though the power input to the airfoil by the viscous forces is three orders of magnitude lower than that due to the pressure forces on the airfoil, the boundary layer manipulates the shock location and shock motion and redistributes the power input to the airfoil by the pressure forces. T…
We perform resolvent analysis to examine the perturbation dynamics over the laminar separation bubble (LSB) that forms near the leading edge of a NACA 0012 airfoil at a chord-based Reynolds number of 500,000 and an angle of attack of 8 degrees. Randomized SVD is adopted in the present analysis to relieve the computational cost associated with the high-Re global base flow. To examine the local physics over the LSB, we consider the use of exponential discounting to limit the time horizon that allows for the instability to develop with respect to the base flow. With discounting, the gain distribu…
The ability of vortices to induce aerodynamic loads on proximal surfaces plays a significant role in a wide variety of flows. However, most studies of vortex-induced effects primarily focus on analyzing the influence of the rotation-dominated cores of vortices. In this work, we show that not only are vortices in viscous flows surrounded by strain-dominated regions, but that these regions are dynamically important and can sometimes even dictate the induced aerodynamic loads. We demonstrate this for a pitching airfoil, which exhibits dynamic stall and generates several force-inducing vortices. U…
Numerical simulations of transition and long-term response of a wind turbine airfoil
FLOW, Department of Engineering Mechanics, KTH Royal Institute of Technology（KTH皇家理工学院工程力学系FLOW）；Department of Wind Energy, Technical University of Denmark, Risø Campus（丹麦技术大学风能系里斯厄校区）
Birds have a remarkable ability to perform complex maneuvers at post-stall angles of attack. The passive deployment of self-actuating covert feathers in response to unsteady flow separation while performing such maneuvers provides a passive flow control paradigm for these aerodynamic capabilities. Most studies involving covert-feathers-inspired passive flow control have modeled the feathers as a rigidly attached or a freely moving flap on a wing. A flap mounted via a torsional spring enables a configuration more emblematic of the finite stiffness associated with the covert-feather dynamics. Th…
Numerical simulations of plasma flow induced by dielectric barrier discharge plasma actuators (DBD-PA) are conducted with two different body-force models: Suzen-Huang (S-H) model and drift-diffusion (D-D) model. The induced flow generated in quiescent air over a flat plate in continuous actuation and the PA-based flow control effect with burst actuation in separated flow over NACA0015 is studied. In the comparative study, the body-force field and the induced velocity field are firstly investigated in the quiescent field to see the spatial difference and the temporal difference in a single disc…
