Exact solutions of boundary integral equation arising in vortex methods for incompressible flow simulation around elliptical and Zhukovsky airfoils

Abstract

The problem of 2D incompressible flow simulation around airfoils using vortex methods is considered. An exact solution for the boundary integral equation with respect to a free vortex sheet intensity at the airfoil surface line that arises in such problems is obtained. The exact solution is constructed for flows around elliptical and Zhukovsky airfoils using the theory of complex potentials and conformal mappings technique. It is possible to take into account the influence of singularities in the flow domain — point vortices which simulate vortex wake. The obtained exact solutions can be used to verify and estimate the accuracy of numerical schemes for the boundary integral equation solution: such procedure is also described in details.

The High-Accuracy Numerical Scheme for the Boundary Integral Equation Solution in 2D Lagrangian Vortex Method With Semi-Analytical Vortex Elements Contribution Accounting

“…It is well-known, that the contribution of the vortex element, placed at the point r w and having circulation Γ w to the vortex sheet intensity which is formed on the surface line of circular airfoil, can be found as the superposition of the influences of a system of mirrored vortices [12].…”

Improved Algorithm of Boundary Integral Equation Approximation in 2D Vortex Method for Flow Simulation Around Curvilinear Airfoil

“…We consider the elliptic airfoil with 2:1 semiaxes ratio (a = 1, b = 0.5, its parametric equations The exact solution for vortex sheet intensity can be easily obtained by using the complex potential theory and conformal mappings technique [26]: It is seen that both numerical solutions are \good", they are close to the exact solution. So the developed algorithm in this case is quite suitable for practical purposes, especially taking into account that for all the integrals in (21) there are exact analytical expressions derived in [25].…”

“…All the coefficients A 11 ij , including diagonal, can be neglected. Recall that the formulae (26) and (28){(29) are correct not only for smooth but also for nonsmooth airfoils, i.e., they can be applied for the neighboring panels, when there is the angle point of the airfoil at the common point of these panels.…”

“…However, in case of smooth airfoil, in practice the following approximate formulae, which are much simpler in comparison with (26) and (28){(29), can be used for the matrix coefficients which calculation requires integration over the neighboring panels (|i − j| = 1).…”

“…If the vortex element is placed close to the panel, it is hardly possible to derive rather simple and at the same time rather accurate approximate formulae, taking into account the curvilinearity of the panel. However, in this case the panel can be replaced with the arc of the osculating circle of radius R i = Under this assumption, it is possible to use the exact solution [26] and write down the following formulae:…”