Simulation of dynamic stall for a NACA 0012 airfoil using a vortex method

https://doi.org/10.1016/S0889-9746(03)00018-5Get rights and content

Abstract

The unsteady, incompressible, viscous laminar flow over a NACA 0012 airfoil is simulated, and the effects of several parameters investigated. A vortex method is used to solve the two-dimensional Navier–Stokes equations in the vorticity/stream-function form. By applying an operator-splitting method, the “convection” and “diffusion” equations are solved sequentially at each time step. The convection equation is solved using the vortex-in-cell method, and the diffusion equation using a second-order ADI finite difference scheme. The airfoil profile is obtained by mapping a circle in the computational domain into the physical domain through a Joukowski transformation. The effects of several parameters are investigated, such as the reduced frequency, mean angle of attack, location of pitch axis, and the Reynolds number. It is observed that the reduced frequency has the most influence on the flow field.

Introduction

Stall of an airfoil is characterized by a sudden drop in its normal force following an increase in the angle of attack that causes flow separation. Dynamic stall, on the other hand, is an unsteady phenomenon, where the airfoil pitches through the static stall angle, while the normal force continues to increase beyond its maximum value for unstalled conditions. This phenomenon is often associated with the formation of a leading-edge vortex, called the dynamic stall vortex, that travels along the airfoil surface as it grows, and finally separates from the airfoil at its trailing edge. However, some researchers (e.g., Jumper and Huge, 1991) attribute this lift overshoot to a delay in separation at high incidence, rather than the formation and subsequent convection of the dynamic stall vortex.

The dynamic stall phenomena was first encountered in the helicopter industry, where large torsional oscillations of the blades was observed and attributed to the periodic stalling and unstalling of each blade on the retreating side of the rotor disk. This was considered to be a serious problem, limiting the forward speed and gross weight of the helicopter (Crimi, 1973). This phenomenon may also occur, for example, on axial-flow compressor blades, large wind-mill rotors, or the wings of modern fighters during fast-pitching maneuvers.

Examples of early analytical studies of unsteady airfoil stall can be found in Carta 1967a, Carta 1967b, Ericsson (1967), and Ericsson and Reding (1971). Ericsson and Reding (1972) analyzed the dynamic stall of a helicopter blade section using a quasi-steady theory. Static experimental data were used as input to predict the dynamic stall characteristics of the airfoil; hence, the applicability of this theory was limited by the availability of the required static experimental data. They found the technique to be successful so long as the reduced frequency, κπfoc/U, was not high (κ<0.5), where fo, c, and U are the oscillation frequency, chord length, and free-stream velocity, respectively. Further improvements and modifications to their method can be found, for example, in Ericsson and Reding 1980, Ericsson and Reding 1988.

McCroskey et al. (1976) experimentally studied dynamic stall, and the associated unsteady boundary layer separation, in incompressible flow at high Reynolds numbers on leading-edge-modified NACA 0012 airfoils. They found the Reynolds number to have a small effect on the normal force and pitching moment coefficients, Cn and Cm. On the other hand, the effects of reduced frequency and leading-edge profile of the airfoil were found to be major. An experimental investigation was performed by Lorber and Carta (1988) to study dynamic stall aerodynamics of an airfoil at constant pitch rate and high Reynolds number (up to 4×106). It was observed that the strength of the dynamic stall vortex increased with pitch rate, and decreased with increasing Mach number. The starting flow over a NACA 0012 airfoil oscillating at large incidences was investigated by Ohmi et al. (1991), using flow visualization experiments and numerical calculations. Reynolds numbers from 1500 to 104, and reduced frequencies (in this case, f=fc/2U) of 0.1–1.0 were considered. The reduced frequency was identified as a key parameter in determining the vortex wake pattern of the oscillating airfoil. Reynolds number effects were found to be small compared to other parameters. Raffel et al. (1995) investigated the flow field over a NACA 0012 airfoil oscillating in pitch in a low-speed wind tunnel by means of particle image velocimetry (PIV). During the upstroke a dynamic stall vortex formed at the leading edge, which at α=24°, for example, extended over 75% of the upper surface of the airfoil. They observed a strong aperiodicity of the flow structure during the downstroke motion from cycle to cycle.

In the simulations presented here, the dynamic stall characteristics of a pitching NACA 0012 airfoil at Reynolds numbers of 3000 and 104 are studied using a vortex method for laminar flows, and the effects of the reduced frequency, mean angle of attack, and location of the pitch axis are investigated.

Section snippets

Solution method

The governing equations for the incompressible flow of a Newtonian fluid in two dimensions are the Navier–Stokes and continuity equations, which can be expressed in terms of the vorticity and stream-function as the following∂ω∂t+(u·)ω=ν2ω,2ψ=−ω.In these equations u=(u,v) is the velocity vector, ν is the kinematic viscosity, and ω and ψ are the vorticity and stream-function, respectively. Eq. (2) is the Poisson equation for stream-function which replaces the continuity equation.

Following

Airfoil profile

In this study, conformal transformations are used to map a circle onto an airfoil, and vice versa. The general form of the transformation F:z(x,y)→w(ξ,η) that transforms a circle onto our desired profile is given byw=F(z)=z+(C−B)2z−Peiα+P,where z=x+iy is a point on or outside the circle in the z-plane, w=ξ+iη is the corresponding point on or outside the transformed profile in the w-plane, and B, C, α, and P are control parameters that are described below with the help of Fig. 1.

As can be seen

Results and discussion

The simulation results from our parameteric study of the dynamic stall flow of the NACA 0012 airfoil are presented here. A summary of the case-studies in this investigation is shown in Table 1. In this work emphasis is put on investigating the effects of reduced frequency of oscillation of the airfoil. In all case-studies the flow is started impulsively at a steady angle of attack αmin. After the flow achieves steady state at this steady angle, the pitching oscillation of the airfoil is started

Conclusions

In this paper the pitching oscillations of a NACA 0012 airfoil were studied, and the effects of some parameters on the dynamic stall characteristics of the airfoil were investigated.

It was observed that the pitching oscillations of the airfoil delayed the flow separation to higher incidences compared to the static stall case. The normal force was observed to increase well beyond that at the static stall angle. Separation was observed to start from the leading-edge, followed by the formation and

Acknowledgements

The authors gratefully acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) and Les Fonds FCAR du Québec.

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