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

2019 ◽  
Vol 1348 ◽  
pp. 012099
Author(s):  
Kseniia Kuzmina ◽  
Ilia Marchevsky ◽  
Evgeniya Ryatina
Keyword(s):  
Integral Equation ◽  
Exact Solutions ◽  
Boundary Integral Equation ◽  
Incompressible Flow ◽  
Flow Simulation ◽  
Boundary Integral ◽  
Vortex Methods

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

2019 ◽  
pp. 22-51
Author(s):  
I. K. Marchevsky ◽  
K. S. Kuzmina ◽  
I. A. Soldatova
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Boundary Integral Equation ◽  
Flow Simulation ◽  
Vortex Method ◽  
Vortex Sheet ◽  
Boundary Integral ◽  
Airfoil Surface ◽  
Analytical Formulae ◽  
Numerical Complexity

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.

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