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 \(O(n_w)\) per target point, where typically, \(n_w\)=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 \(O(h^2)\) and \(O(h^3)\) are listed, with errors that decrease as \(d\rightarrow 0\). 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.
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The author thanks Robert Krasny for many discussions and helpful suggestions, and the anonymous reviewers for their supportive and constructive comments.
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Communicated by Jeff D. Eldredge.
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Positivity of \(c^2\)
Positivity of \(c^2\)
This section shows that the quantity
is positive in the cases considered in this paper, namely a flat boundary with nonuniform point distribution and curved boundaries with equally spaced points. For curved boundaries with target points on the “inside”, defined below, d must be sufficiently small.
In the flat case, see Fig. 23a, \(\ddot{{\mathbf{x}}}_p\) is tangent to the plate while \( {\mathbf{x}}_p-{\mathbf{x}_0}\) is normal to it, thus \( \ddot{{\mathbf{x}}}_p\cdot ({\mathbf{x}}_p-{\mathbf{x}_0})=0\) and
for all d, and \({\mathbf{x}}_p\) in the plate interior.
In the equally spaced case, \(|{\dot{{\mathbf{x}}}}_p|=s_{\alpha }=constant\), where s is arclength and \(s_{\alpha }=ds/d\alpha \). We consider two possibilities: either \({\mathbf{x}_0}\) lies on the “outside” of the curve, as in Fig. 23b, or \({\mathbf{x}_0}\) lies on the “inside” of the curve, as in Fig. 23c. Here, the “outside” is defined as the side opposite to that pointed into by the normal vector \({\mathbf{N}}\) at \({\mathbf{x}}_p\), where
while the “inside” is the side containing the osculating circle of the curve at \({\mathbf{x}}_p\). Here, \(\kappa =|d{\mathbf{T}}/ds|\) is the curvature at \({\mathbf{x}}_p\). Note that in the equally spaced case,
Thus if \({\mathbf{x}_0}\) is on the outside, \({\mathbf{x}}_p-{\mathbf{x}_0}=d{\mathbf{N}}\) and \(\ddot{{\mathbf{x}}}_p\cdot ({\mathbf{x}}_p-{\mathbf{x}_0})=d\kappa s_{\alpha }^2>0\) so
for all d. If \({\mathbf{x}_0}\) is on the inside, \({\mathbf{x}}-{\mathbf{x}_0}=-d{\mathbf{N}}\) and \(\ddot{{\mathbf{x}}}_p\cdot ({\mathbf{x}}_p-{\mathbf{x}_0})=-d\kappa s_{\alpha }^2\). Therefore
provided \(d<1/(2\kappa )=R_{osc}/2\), where \(R_{osc}=1/\kappa \) is the radius of the osculating circle.
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Nitsche, M. Evaluation of near-singular integrals with application to vortex sheet flow. Theor. Comput. Fluid Dyn. 35, 581–608 (2021). https://doi.org/10.1007/s00162-021-00577-9
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DOI: https://doi.org/10.1007/s00162-021-00577-9