BiGlobal instability analysis of turbulent flow over an airfoil at an angle of attack

Stability analysis can provide insight to aerodynamic flow control studies by numerical means. It is advantageous to use spectral numerical methods for the stability analysis as they are, at the same level of numerical effort, more accurate than standard finite volume or finite element alternatives. The disadvantage with classic spectral collocation methods, however, is the difficulty in handling geometry. While spectrally accurate and geometrically flexible methods (TSB) exist, based on the spectral/hp−element concept, this paper will present a means of undertaking a fluid mechanical instability analysis using spectral collocation numerical methods on a rectangular grid and conformal mapping techniques in order to represent the geometry of the problem. The flow control configuration of interest in this study is the leading edge separation of a NACA 0015 airfoil, at an angle of attack α = 18. Water tunnel experiments of this configuration, were undertaken by for a Rec ≡ U∞c/ν = 3× 10, where c is the length of the airfoil chord, U∞ is the freestream velocity, and ν the kinematic viscosity. The flow was perturbed via a zero-net-mass-flux (ZNMF) jet, normal to the surface, and spanned the entire leading edge. Frequencies F ≡ fc/U∞ = 0.65 and from F = 1.1 → 1.4 were found to enhanced the lift by more than 45%. A Large Eddy Simulation (LES) of the uncontrolled case was undertaken, and the largest frequency component of the lift force history was F = 0.63, corresponding to one of the frequencies that were found to significantly enhance the lift in the experimental study. The work presented within will introduce the development work of the conformal mapping and numerical techniques required to enable the spectral analysis of this flow configuration. In order to separate the conformal mapping development procedure from that of unsteadiness in the flow, here a lower Rec = 200 is first adopted at which the flow is laminar and steady, then the analysis is repeated at a slightly higher Rec = 300 at which the flow is laminar and unsteady. Results at the target Reynolds number of the turbulent flow at Rec = 3× 10 will be presented elsewhere. The paper will be organised as follows. Firstly an overview of the experimental study will be presented, followed by a comparison of experimental results with those obtained by applying the finite-volume Stanford CDP LES solver to this problem. In addition, a second order finite-element solver (ADFC) has been used to obtain basic states. Next, the derivation of the stability linear operator will be outlined, including a discussion of numerical aspects on the general curvilinear coordinate system, in particular the means of transforming the geometry and velocities between coordinate systems. Following this, two specific geometries are chosen in order to highlight the proposed analysis methodology. An 8:1 ellipse placed at an angle of attack α = 18 to the oncoming flow is first analysed, as the analytical derivatives required for the conformal mapping