A geometry aware arbitrary order collocation Boundary Element Method solver for the potential flow past three dimensional lifting surfaces

Abstract

This work presents a numerical model for the simulation of potential flow past three dimensional lifting surfaces. The solver is based on the collocation Boundary Element Method, combined with Galerkin variational formulation of the nonlinear Kutta condition imposed at the trailing edge. A similar Galerkin variational formulation is also used for the computation of the fluid velocity at the wake collocation points, required by the relaxation algorithm which aligns the wake with the local flow. The use of such a technique, typically associated with the Finite Element Method, allows in fact for the evaluation of the solution derivatives in a way that is independent of the local grid topology. As a result of this choice, combined with the direct interface with CAD surfaces, the solver is able to use arbitrary order Lagrangian elements on automatically refined grids. Numerical results on a rectangular wing with NACA 0012 airfoil sections are presented to compare the accuracy improvements obtained by grid spatial refinement or by discretization degree increase. Finally, numerical results on rectangular and swept wings with NACA 0012 airfoil section confirm that the model is able to reproduce experimental data with good accuracy.

Figures (13)

• Figure 1: The computational domain
• Figure 2: The computational domain including the presence of the wake sheet $\Gamma_W$ detaching from the trailing edge $\gamma_{TE}$
• Figure 3: A two dimensional sketch of the computational domain including details of the collocation nodes, coinciding with on the degrees of freedom (DOFs) of the discretization space. The image also indicates that in correspondence with the trailing edge $\gamma_{TE}$, a triple DOF is present to allow for the space $V_h$ functions to have different trailing edge values on the cells of the airfoil leeward side, windward side, and wake.
• Figure 4: A CAD rendering of the wing used in the first numerical tests, with relevant length measures.
• Figure 5: A top view of the automatically generated initial mesh on the wing and wake grid. On the left, the --- structured --- grid obtained setting maximum aspect ratio 2.5, and 3 uniform refinement cycles. On the right, the --- unstructured --- grid obtained setting maximum aspect ratio 2.5, 3 uniform refinement cycles, and 4 adaptive refinement cycles based on CAD curvature. Note that, regardless of the wing grid type, the wake mesh is structured. The interface between the two regions, occurring at the trailing edge, is conformal.