Saturation-Free Numerical Scheme for Computing the Flow Past a Lattice of Airfoils with a Sharp Edge
2019, Vol. 15, no. 2, pp. 135-143
Author(s): Petrov A. G.
The Joukowski – Chaplygin condition, which allows us to determine the circulation of the flow past contour with a sharp edge, is one of the most important achievements of S. A.Chaplygin, whose 150 birthday is celebrated this year. This research is devoted to this problem.
We consider the flow past of a lattice of airfoils by a potential fluid flow. The profile line is determined parametrically by an equation in the form of two dependences of the Cartesian coordinates on the parameter. For a smooth closed loop the Cartesian coordinates are periodic analytical functions. Their Fourier series coefficients decrease exponentially depending on the harmonic number. In the meantime, for a sharp edge loop they decrease much slower — inversely to the square of the harmonic number. Using the symmetric continuation of the profile with a sharp edge, a method of presenting it as a Fourier series with exponentially decreasing coefficients is proposed. Based on this idea, a quickly converging numerical scheme for the computation of the flow past airfoils lattice with a sharp edge by a potential fluid flow has been developed.
The problem is reduced to a linear integro-differential equation on the lattice contour, and then, using specially developed quadrature formulas, is approximated by a linear system of equations. The quadrature formulas converge exponentially with respect to the number of points on the profile and can be rather simply expressed analytically.
Thanks to its quick convergence and high accuracy, this method allows one to optimize profiles by using a direct method by any given integral characteristics. We can find the distribution of the shear stress and the breakaway point on the calculated velocity distribution on the profile from the solution of the boundary layer equation. In the method suggested we do not need to perform the difficult work of constructing the lattice. Also, the problem of scheme viscosity at high Reynolds numbers is omitted.
We consider the flow past of a lattice of airfoils by a potential fluid flow. The profile line is determined parametrically by an equation in the form of two dependences of the Cartesian coordinates on the parameter. For a smooth closed loop the Cartesian coordinates are periodic analytical functions. Their Fourier series coefficients decrease exponentially depending on the harmonic number. In the meantime, for a sharp edge loop they decrease much slower — inversely to the square of the harmonic number. Using the symmetric continuation of the profile with a sharp edge, a method of presenting it as a Fourier series with exponentially decreasing coefficients is proposed. Based on this idea, a quickly converging numerical scheme for the computation of the flow past airfoils lattice with a sharp edge by a potential fluid flow has been developed.
The problem is reduced to a linear integro-differential equation on the lattice contour, and then, using specially developed quadrature formulas, is approximated by a linear system of equations. The quadrature formulas converge exponentially with respect to the number of points on the profile and can be rather simply expressed analytically.
Thanks to its quick convergence and high accuracy, this method allows one to optimize profiles by using a direct method by any given integral characteristics. We can find the distribution of the shear stress and the breakaway point on the calculated velocity distribution on the profile from the solution of the boundary layer equation. In the method suggested we do not need to perform the difficult work of constructing the lattice. Also, the problem of scheme viscosity at high Reynolds numbers is omitted.
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