Evaluation of near-singular integrals with application to vortex sheet flow View Full Text


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Article Info

DATE

2021-07-13

AUTHORS

Monika Nitsche

ABSTRACT

This paper presents a method to evaluate the near-singular line integrals that solve elliptic boundary value problems in planar and axisymmetric geometries. The integrals are near-singular for target points not on, but near the boundary, and standard quadratures lose accuracy as the distance d to the boundary decreases. The method is based on Taylor series approximations of the integrands that capture the near-singular behaviour and can be integrated in closed form. It amounts to applying the trapezoid rule with meshsize h, and adding a correction for each of the basis functions in the Taylor series. The corrections are computed at a cost of O(nw)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n_w)$$\end{document} per target point, where typically, nw\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_w$$\end{document}=10–40. Any desired order of accuracy can be achieved using the appropriate number of terms in the Taylor series expansions. Two explicit versions of order O(h2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(h^2)$$\end{document} and O(h3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(h^3)$$\end{document} are listed, with errors that decrease as d→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\rightarrow 0$$\end{document}. The method is applied to compute planar potential flow past a plate and past two cylinders, as well as long-time vortex sheet separation in flow past an inclined plate. These flows illustrate the significant difficulties introduced by inaccurate evaluation of the near-singular integrals and their resolution by the proposed method. The corrected results converge at the analytically predicted rates. More... »

PAGES

581-608

References to SciGraph publications

  • 2020-04-10. An inviscid model of unsteady separated vortical flow for a moving plate in THEORETICAL AND COMPUTATIONAL FLUID DYNAMICS
  • 2017-11-06. Ubiquitous evaluation of layer potentials using Quadrature by Kernel-Independent Expansion in BIT NUMERICAL MATHEMATICS
  • 2012-07-10. Simulating Vortex Wakes of Flapping Plates in NATURAL LOCOMOTION IN FLUIDS AND ON SURFACES
  • 2007-07-21. An inviscid model for vortex shedding from a deforming body in THEORETICAL AND COMPUTATIONAL FLUID DYNAMICS
  • 1988-06. Quadrature methods for periodic singular and weakly singular Fredholm integral equations in JOURNAL OF SCIENTIFIC COMPUTING
  • 2016-10-13. Error estimation for quadrature by expansion in layer potential evaluation in ADVANCES IN COMPUTATIONAL MATHEMATICS
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    http://dx.doi.org/10.1007/s00162-021-00577-9

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