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

Abstract

The problem of the accuracy improving is considered for vortex method. The general Galerkin-type approach is considered for the numerical solution of the boundary integral equation. It is shown that the airfoil surface line representation as a polygon consisting of rectilinear panels can lead to incorrect behavior of the numerical solution of the boundary integral equation with respect to unknown vortex sheet intensity, especially in case of considerably different lengths of the neighboring panels. However, in this case, the exact analytical formulae are derived for the coefficients of the linear algebraic system, which approximate the integral equation. The approach is developed which makes it possible to take into account explicitly the curvature of the airfoil surface line. The approximate analytical formulae are derived for the coefficients of the linear system, which are expressed through the integrals of the fractions, which denominators can be equal to zero. Numerical example is considered for the elliptic airfoil in the incident flow at its essentially non-uniform discretization. The suggested algorithm has acceptable numerical complexity, however it permits to obtain rather good numerical results.

Singularity resolving in solution of the boundary integral equation in two-dimensional vortex methods

“…If it is replaced by a polygon consisting of straight panels, the order of accuracy (in 1 L norm) cannot be higher than the second: the first order of accuracy is achieved for piecewise-constant solution representation, the second order of accuracy -for piecewise-linear distribution of the vortex sheet intensity [10]. In much more complicated schemes where the curvilinearity of the panels is taken into account explicitly [11], the third order of accuracy can be achieved at piecewise-quadratic solution representation. So, one can conclude that the problem of higher-order numerical schemes development is solved, more or less, for smooth airfoils.…”

Hierarchy of Numerical Schemes for the Bie Solution in 2d Flow Simulation Using Vortex Methods

“…If it is replaced by a polygon consisting of straight panels, the order of accuracy (in 𝐿 norm) cannot be higher than the second: the first order of accuracy is achieved for piecewise-constant solution representation, the second order of accuracy -for piecewise-linear distribution of the vortex sheet intensity [8]. In much more complicated schemes, where the curvilinearity of the panels is taken into account explicitly [9], the third order of accuracy can be achieved at piecewisequadratic numerical solution representation.…”

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

“…The last issue can be overcome by introducing curvilinear panels; however in this case A pq ij coefficients can not be calculated analytically. In [1,3,11] the original technique is developed for their calculation for curvilinear panels through Taylor expansions with respect to the panel length L i ; the resulting formulae are given in [3].…”

“…Finally, for the system (11), which correspond to the result of implementing the correction procedure, taking into account the above introduced notations for matrix coefficients A pq ij and D pp ii and the right-hand side coefficients (b w ) p i we obtain…”