High-Order Numerical Scheme for Vortex Layer Intensity Computation in Two-Dimensional Aerohydrodynamics Problems Solved by Vortex Element Method

The study deals with the numerical simulation of two-dimensional viscous incompressible flow around airfoils by using vortex element method. The numerical scheme and the corresponding algorithm for this method usually presuppose the replacement of the airfoil with the polygon which consists of panels, and the unknown vortex layer intensity is assumed to be piecewise-constant on the panels. The accuracy of this scheme varies from O(h²) to O(h³) for different airfoils (h is the panels’ length). In the present research we developed a new high-order numerical scheme. The new approach assumes the vortex layer intensity to be not piecewise-constant, but piecewise-linear or piecewise-quadratic on each panel. It is also important that the solution is not assumed to be continuous along the airfoil; it is most needed for correct simulation of the flow around airfoil with the angular points and sharp edges. Moreover, we take into account the fact that the airfoil’s boundary is curvilinear: each part of the airfoil’s boundary is approximated by a cubic spline instead of a straight panel. In order to obtain linear algebraic equations, we used the least squares method instead of collocation-type conditions in separate control points or on average on the panels. We applied higher order of accuracy Gaussian quadrature formulas for approximate integrals calculation. The results of the research show that the developed scheme has higher accuracy order than the previously known schemes. For some particular model problems (flow around circular, elliptical and Zhukovsky airfoils) this approach allows us to obtain solution with accuracy O(h⁵).

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