Singularity Based Method for Small Perturbation Unsteady Aerodynamics Using Higher Fidelity Steady State Pressure Profiles

A low-cost, singularity-based method that builds upon the classic potential flow solution procedures is presented in this paper. The classic potential flow solution method distributes elementary solutions to the Laplace’s equation such as sources, vortices or doublets, on the aerodynamic surface. The strengths of those singularities are calculated to satisfy the non-penetration boundary condition and the flow smoothness at the trailing edge. The potential flow predictions are improved to include some compressibility and viscous effects by superposing additional singularities, which strengths are computed to match pressure profiles obtained from a higher resolution conmputational fluid dynamics (CFD) solver. Four variants of the procedure are presented and discussed. Classic potential solution methods can be based on the camber-line (e.g. Lumped Vortex Method in 2D for airfoil) or on the actual airfoil (respectively wing) surface (e.g. source-double panel method). Likewise, the four variants differ by the types of singularities used and/or the aerodynamic surface where the boundary condition is applied: Method 1 optimizes the camber-line shape so that when distributing discrete vortices along that curve, the correct pressure jump across it is obtained. Method 2 duplicates and offsets the geometric camber-line into an “upper” and a “lower” camber curves. Singularities are then distributed on those curves to match the pressure distribution across the airfoil upper and lower surfaces. Method 3 overloads the airfoil surface panels with multiple singularities to meet the pressure requirements. Finally, Method 4 combines continous and discrete singularities to achieve that goal. The methods are surveyed upon three airfoils: NACA006 (thin, symmetric airfoil), NACA4412 (thin, cambered airfoil) and a two-element airfoil, S207. The methods are compared in terms of accuracy in predicting pressure coefficient, aerodynamic loads and applicability for future unsteady simulations. Based on this steady-state survey, Method 4 is the most accurate and stable, with the appropriate choice of singularities. One key element for small perturbation analysis is to capture accurately the reference position, particularly the mean wake on which the vortices are shed, which method 4 is found to be capable of predicting.