The no-slip condition is specified for the velocity on the airfoil surface while free-stream values are assigned for the velocity at the upstream boundary. The upper and lower boundaries are placed at 5 chord lengths, each, from the leading edge. The NACA 0012 airfoil resides in a rectangular computational domain whose upstream and downstream boundaries are located at 5 and 11 chord lengths from the leading edge, respectively. σ (p h, u h ) d Ω e +∑ e=1 n el ∫ Ω e δ ∇.u h d Ω +∑ e=1 n el ∫ Ω e 1 ρ (τ SUPG ρ u h.The stabilized finite element formulation of, is written as follows: find u h ∈ S h u and p h ∈ S p h such that ∀ w h ∈ V u h, q h ∈ V h p ∫ Ω w h
Let S u h and S p h be the appropriate finite element trial function spaces, and V u h and V p h the weighting function spaces for velocity and pressure. The domain Ω is discretized into subdomains Ω e, e=1,2,…, n el, where n el is the number of elements.
Here I is the identity Finite element formulation The stress tensor is related to the velocity u and pressure p by σ =−p I + T where T =2μ ε ( u ). u =0 on Ω for (0,T).Here ρ, u, f and σ are the density, velocity, body force and the stress tensor, respectively.The Navier–Stokes equations governing incompressible fluid flow are ρ ∂ u ∂ t + u The spatial and temporal coordinates are denoted by x and t and Γ represents the boundary of Ω. The spatial and temporal domains are denoted as Ω⊂ R 2 and (0, T), respectively. More details on the formulation can be found in the articles by Mittal and Tezduyar et al. Since the stabilization terms are weighted residuals, the consistency of the formulation is maintained in the sense that the stabilized formulations still admits the exact solution. These terms stabilize the computations against spurious numerical oscillations in advection dominated flows and enable the use equal-order-interpolation velocity–pressure elements. Terms that are based on the element level integrals of the residuals, are added to the basic Galerkin formulation. Stabilized finite element formulations are utilized to solve the flow equations in the primitive variables. For the relative performance of different turbulence models applied to geometries of aerospace interest the reader is referred to the review paper by Tulapurkara. The interested reader is referred to the articles by Kallinderis, Mavriplis and Anderson and Bonhaus for details. Despite the simplicity of the Baldwin–Lomax model, its implementation with unstructured grids is quite complex. In addition, the structured grid around the body allows for efficient implementation of the turbulence model. This type of a grid has the ability of handling fairly complex geometries while still providing the desired resolution close to the body to effectively capture the boundary layer flow, especially, in the context of unsteady flows. The finite element mesh consists of a structured mesh close to the body and an unstructured part, generated via Delaunay's triangulation, away from the body. The incompressible, Reynolds averaged Navier–Stokes (RANS) equations in conjunction with the Baldwin–Lomax model, for turbulence closure, are solved using stabilized finite element formulations. Carefully conducted computations are utilized to track the hysteresis loop in the flow close to the stall angle. The present work is an effort to study the behavior of the flow past a NACA 0012 airfoil near stall by solving the governing flow equations numerically. The Reynolds number for their experiments is 2.2×10 6. They also found that the hysteresis is not restricted to low Reynolds number flows but can also be found at relatively high Reynolds number. Hysteresis has also been reported by Biber and Zumwalt for a two-element GA(W)-2 airfoil and its extent was found to be a function of the angle of flap deployment, gap geometry and the history of changes in air speed or angle of attack. Hysteresis loop in the data for the aerodynamic coefficients was observed.
Hoffmann has reported his results for an experimental study for flow past a NACA 0015 airfoil at Re=250,000. While the hysteresis associated with the pitching motion of airfoil (also known as dynamic stall) has been investigated quite extensively (see, for example, ), the one observed for static stall has received much less attention. In certain cases, the stall is accompanied by a hysteresis in the flow. The massive flow separation on the airfoil surface results in a sharp drop in lift and increase in the drag. As the angle of incidence of an airfoil to the oncoming flow increases beyond a certain point, stall is observed.