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

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.

Positivity of \(c^2\)

This section shows that the quantity

$$\begin{aligned} c^2= |{\dot{{\mathbf{x}}}}_p|^2 + \ddot{{\mathbf{x}}}_p\cdot ({\mathbf{x}}_p-{\mathbf{x}_0}) \end{aligned}$$

(37)

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.

Fig. 23

Sketch showing different scenarious: a flat curve, and b, c uniform arclength parametrization, $s_{\alpha }=$constant, with ${\mathbf{x}}_0$ outside of the curve in b and inside of the curve in c

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

$$\begin{aligned} c^2= |{\dot{{\mathbf{x}}}}_p|^2 >0 \end{aligned}$$

(38)

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

$$\begin{aligned} {\mathbf{N}}=\frac{1}{\kappa }\frac{d{\mathbf{T}}}{ds}~,\quad {\mathbf{T}}=\frac{d{\mathbf{x}}}{ds}~, \end{aligned}$$

(39)

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,

$$\begin{aligned} {\mathbf{T}}=\frac{{\dot{{\mathbf{x}}}}_p}{s_{\alpha }}~, \quad {\mathbf{N}}= \frac{d{\mathbf{T}}}{ds}/ \left| \frac{d{\mathbf{T}}}{ds}\right| =\frac{1}{\kappa }\frac{\ddot{{\mathbf{x}}}_p}{s_{\alpha }^2} ~. \end{aligned}$$

(40)

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

$$\begin{aligned} c^2=s_{\alpha }^2(1+d\kappa )> s_{\alpha }^2=|{\dot{{\mathbf{x}}}}_p|^2 >0~, \end{aligned}$$

(41)

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

$$\begin{aligned} c^2=s_{\alpha }^2(1-d\kappa )> \frac{1}{2}s_{\alpha }^2=\frac{1}{2}|{\dot{{\mathbf{x}}}}_p|^2 >0 ~, \end{aligned}$$

(42)

provided $d<1/(2\kappa )=R_{osc}/2$, where $R_{osc}=1/\kappa$ is the radius of the osculating circle.