数理解析研究所講究録922 巻 1995 年 99-123

99

Vortex blob method and its application tovortex sheet motion.
T. Sakajo

Research Institute for Mathematical Sciences,
Kyoto University, Kyoto 606-01, Japan

1

Introduction

Vortex sheet is a surface in an incompressible fluid across which the tangential
component of fluid velocity has a jump discontinuity. It is often used as amathematical model for Kelvin-Helmholtz instability, where a thin vorticity
layer becomes unstable and large vortex structures appear. Such a surface
can be observed in many fluid flow, and so the study of the motion of vortex
sheet plays an important role in the field of fluid dynamics, especially in theunderstanding of turbulent flow. Although various researches have been done
so far, many questions still remain. The evolution equation of vortex sheet
contains singular kernel, which make it difficult to solve the equation both
numerically and analytically. Chorin’s vortex blob method is one of the mostsuccessful methods for numerical study of vortex sheet motions. This method
can be done by using desingularized kernel instead of singular kernel in theequation of vortex motion. The convergence of this method is showed by
Caflisch and Lowengrub[5]. This convergence theorem, however, is obtained
up to the critical time when the curvature of vortex sheet becomes infinity.
showed clearly how vortex sheets roll up after the criticaltime using this method and concluded that when reducing the regularization parameter, most of the numerical solutions have the roll-up structure
which is independent of the regularization parameter. Also, Nitsche andapplied the method to an axisymmetric vortex-sheet model inorder to simulate an real experiment in which vortex ring formation occurs.
They compared the numerical results with experimental ones, and noted that
their numerical solution is a good approximation of real fluid flow.
$\mathrm{K}\mathrm{r}\mathrm{a}\mathrm{s}\mathrm{n}\mathrm{y}[8,9]$$\mathrm{K}\mathrm{r}\mathrm{a}\mathrm{s}\mathrm{n}\mathrm{y}[15]$100

The purpose of the present paper is to add new examples which demonstrateuse-fullness of vortex blob method. We will show the results of two examples;(a) a vortex sheet motion in the presence of a uniform shear flow,(b)interactions of two vortex sheets.
In section 2, we state how to apply vortex blob method to the evolution
equation of vortex sheet motion with singular kernel and how to discritize it.
And we also explain numerical technique,
Fourier Filter”, in orderto observe the long time evolution of vortex sheets accurately. Numericalcomputations of one vortex sheet in the presence of a uniform shear flow
appears in Section 3. We consider the interactions of two vortex sheets in
Section 4.
$‘(\mathrm{K}\mathrm{r}\mathrm{a}\mathrm{s}\mathrm{n}\mathrm{y}’ \mathrm{S}$2

Chorins’s vortex blob method and Fourier
FilterA vortex sheet can be represented by the curve $\vec{R}\ni\Gamma-*z(\Gamma, t)\in\vec{C}$ withbeing taken along the curve. is called the circulation parameter and istime. We impose periodic boundary condition on vortex sheet,$\Gamma$$\Gamma$$\mathrm{t}$$z(\Gamma+1, t)=z(\Gamma, t)+1$.

The motion of a vortex sheet with periodic boundary condition is known
to be governed by the following Birkhoff-Rott equation,$(\mathrm{S}\mathrm{a}\mathrm{f}\mathrm{f}\mathrm{m}\mathrm{a}\mathrm{n}[17])$$\frac{\partial z(t,\Gamma)*}{\partial t}=\frac{1}{2i}\mathrm{p}.\mathrm{v}$(1). $\int_{0}^{1}\cot_{T}(z(t, \Gamma)-z(t, \Gamma’))d\Gamma’$ ,, andwhere the integral is Cauchy’s principal value,thecomplex conjugate. This equation is ill-posed (see) and integration of the solutions breaks down unless certain devices are used. In vortex
blob method, we desinguralize the equation (1) in such a way to replace thesingular kernel to a regular kernel as below.
$\dot{i}=\sqrt{-1}$$*\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{e}\mathrm{S}$$\mathrm{S}\mathrm{a}\mathrm{f}\mathrm{f}\mathrm{m}\mathrm{a}\mathrm{n}[17]$$\frac{\partial z(t,\Gamma)*}{\partial t}=\int_{0}^{1}K_{\delta(}Z(t, \Gamma)-Z(t, \mathrm{r}’))d\mathrm{r}’$where the kernel function$K_{\delta}$(2),

is defined as follows:$K \delta(_{X+i}y)=-\frac{1}{2}\frac{\sinh(2\pi y)+\dot{i}\sin(2\pi x)}{\cosh(2\pi y)-\cos(2\pi x)+\delta 2}$.

Note that the equation (2) reduces to (1) when $\delta=0$ . The equation iswell-posed for any time interval if $\delta>0$ . The high frequency modes arestabilized.

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We discretize (2) by the point vortices. Choosing a positive integer $N$ ,we consider the following system of ordinary differential equations:$\frac{\partial z_{n}(t)^{*}}{\partial t}=\sum_{n1\leq m\leq 1,m\neq}K_{\delta}(Z_{n}(t)-Z_{m}(t))$,(3), which approximate the vortex sheet at time
When we get. A fourth order Runge-Kutta, we integrate (9) to obtainmethod is used to integrate the ODEs. Then we let it go through the Fourier
filter as Krasny did. Repeating this process, we can simulate the dynamics
of the vortex sheet.
The desingularized equation (2) is well-posed, but in practical computation the round-off errors occur and deteriorate the computation eventually.
In order to get a long time evolution of vortex sheet accurately, “Fourier
is effective. Assume that theFiltering” scheme introduced bysolution of the equation (9) is given by the following form,$\{z_{n}(t)\}_{1}\leq n\leq N$$\{Z_{n}(t+\triangle t)\}_{1}\leq n\leq N$$t$$\mathrm{R}.\mathrm{K}\mathrm{r}\mathrm{a}\mathrm{S}\mathrm{n}\mathrm{y}[9]$$z( \Gamma, t)=\Gamma+n=-\sum_{\frac{N}{2}}^{\frac{N}{2}-1}a_{n}\exp 2\pi in\Gamma$

The second term is the disturbance from the steady solution $(z=\Gamma)$ of theequation (9).
“Fourier Filtering” scheme can be done by the following steps.

1. Calculate$z_{i}(t)arrow z_{i}(t+\triangle t)$using forth order Runge-Kutta method.

2. Apply Fast Fourier Transformation algorithm toobtain coefficients of the disturbance term .

$z_{i}(t+ \triangle t)-\frac{i}{N}$, and$a_{n}$

3. For each coefficients
it to O.(Filtering)
4. Get new
tered .

$z_{i}(t+\triangle t)$$a_{n}$, if$a_{n}$

doesn’t exceed some threshold, we reset

using Inverse Fourier Transformation with the fil-$a_{n}$

Vortex blob method with Fourier Filter make it possible to get a longtime vortex sheet calculation. For more detail, see [9].

3A vortex sheet in a uniform shear

We now consider a vortex sheet in a uniform shear.

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Equation and its linear analysis

3.1

Specifically, we consider the following equation:$\frac{\partial z(t,\Gamma)*}{\partial t}=\frac{1}{2_{\dot{i}}}\mathrm{p}.\mathrm{v}$. $\int_{0}^{1}\cot\pi(z(t, \Gamma)-z(t, \Gamma’))d\Gamma’+G{\rm Im} z(t, \Gamma)$,(4)

A uniform shear term, $GIm[Z(\Gamma, t)]$ , is added to the Birkhoff-Rott equation(1).
is the constant representing the strength of the shear. When $G=0$ ,the equation is the same as Krasny used in $[8, 9]$ . Note that we have neglected
the influence of the vortex sheet on the background flow.
Because of the simplicity of (4), we can use vortex blob method statedin Section 2 without any change. But by a certain reason which we explain
later, our equation with nonzero requires a larger number of point vortices
than are needed in the case of $G=0$ . This makes our problem technicallymore difficult.
However, the phenomenological difference between the case of $G=0$ andthat of $G\neq 0$ is far beyond the technical difference: we will show that thissimple equation yields some interesting phenomena if we vary . In fact,
when $G>0$ , we obtained some unexpected motions which the authors had
not imagined from the case of $G=0$ .
It well-known that the integral equation (4) with $G=0$ is ill-posed in
the sense that a disturbance of wave number is amplified by a factor pro(see Chapter 8 of Saffman [17], for instance). We study
portional to
the stability of the stationary solution
in the case of nonzero . This
analysis was done by Kiya and Arie [7] based on the linearization of the Euler equations. We study the linearized equation of (4) and derive the sameconclusion as theirs.
yieldsThe linearization of (4) around
$G$

$G$

$G$

$n$

$|n|$

$G$

$z\equiv\Gamma$$z\equiv\Gamma$$\frac{\partial\zeta(t,\Gamma)^{*}}{\partial t}=-\frac{\pi}{2_{i}}\mathrm{p}.\mathrm{v}$.

$\int^{1}0\frac{\zeta(t,\Gamma)-\zeta(t,\Gamma’)}{\sin^{2}\pi(\Gamma-\Gamma)},d\Gamma’+G{\rm Im}\zeta(t, \Gamma)$,(5)

where use has been made of$\mathrm{p}.\mathrm{v}$. $\int_{0}^{1}\cot(\pi x)dx=0$ .

We would like to analyze how unstable the zero solution
substitute$\zeta=\sum_{n=-\infty}^{+\infty}A(nt)e^{2}n\pi\Gamma i$

into (5) and take the$e^{2n\pi\Gamma i}$$\zeta\equiv 0$

is. We now,

mode only. We then obtain$\frac{dA_{-n}(b)^{*}}{dt}=\frac{\pi i}{2}\int_{0}^{1}\frac{1-\cos(2n\pi X)}{\sin^{2}(\pi X)}d_{X}A_{n}(t)+\frac{G}{2_{i}}(A_{n}-A_{-}^{*})n$

103

for each $n=\pm 1,$$\pm 2,$$\cdots$. On the other hand, we have$\int_{0}^{1}\frac{1-\cos(2n\pi x)}{\sin^{2}(\pi x)}d_{X}=2|n|$by the residue calculus. Therefore we obtain$\frac{dA_{-n}(t)^{*}}{dt}=Ti|n|A_{n}(t)+\frac{G}{2_{i}}(A_{n}-A_{-n}^{*})$

Replacing$n$with $-n$ and.

, we have$\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{n}\mathrm{g}*\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}$$\frac{dA_{n}(t)}{dt}=-\pi\dot{i}|n|A_{-}(nt)*+\frac{G}{2_{i}}(A_{n}-A_{-n}^{*})$.

These two equations are written in the following form:$\frac{d}{dt}(A_{-n}(A_{n}(t)t)^{*})=(\frac{G}{2i}-i|n|-\frac{G}{T2i}$$\frac{G}{2i}+T\dot{i}|n|\frac{G}{2i})$.

(6)

The characteristic polynomial of the matrix in the right hand side is(7)$X^{2}+ \frac{G^{2}}{4}-|-i\frac{G}{2}+\pi\dot{i}|n||^{2}$

The eigenvalues are:$\pm\sqrt{\pi^{2}n^{2}-c\pi|n|}$.

(8)

So, disturbances of mode are neutrally stable if $G>\pi|n|$ . This is exactly
what was obtained by Kiya and Arie [7].
The result of this analysis may be summarized as follows:$n$$\bullet$$\bullet$

negative$G$

enhances the instability of the flat vortex sheet;positive$G$

neutralizes perturbation of low frequency.

The second observation led the authors to imagine that the instability occurs
at a later time stage when $G>0$ . Also we thought that, if $G>\pi$ , then the
instability is realized as roll-ups with two or morein the fundamental
length $0<\Gamma<1$ . We began our numerical experiments to examine thisheuristic argument.
”(($\mathrm{s}$$\mathrm{e}\mathrm{y}\mathrm{e}$3.2Discritization and numerical results

We discretize (2) by the point vortices. Choosing a positive integer $N$ , weconsider the following system of ordinary differential equations:$\frac{\partial z_{n}(t)^{*}}{\partial t}=\sum_{n1\leq m\leq 1,m\neq}K_{\delta}(Z_{n}(t)-z(m\theta))+G{\rm Im} Z_{n}(t)$Parameters of discritization and desingularization are

,

(9)

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$N$

$\bullet$, spatial discretization parameter( number of vortices),$\triangle t$$\bullet$$\delta$$\bullet$, temporal step size,, desingularization parameter.

We computed vortex sheets when $G=-1.5\pi,$ $0,0.5\pi,$ $1.5\pi,$ $2.5\pi$ . In eachrun, the initial condition is$z(\mathrm{O}, \Gamma)=\Gamma+0.01(1-i)\sin 2\pi\Gamma$,

namely$z_{n}(0)= \frac{n}{N}+0.01(1-i)\sin(\frac{2\pi n}{N})$,$(n=1,2, \cdots , N)$

which is the same as what Krasny chose. Parameters are chosen as follows:$\delta=0.075,$ $N=512,$ $\triangle t=0.001$ . As for the choice ofand $N$ , we tested other$\delta=0.0375$
combinations; combinations of, 0.075, 1.0 and $N=256,512$ , 1024
were tested. We found that the above choice is most suited in the sense thatnumerical instability appears latest.
Figure 1 shows the time evolution when $G=0$ . Little visible difference isfound between ours and Krasny’s, showing our program’s validity. However,the scales of the -and -axes are slightly different, which make our rolls
look more oblate than Krasny’s. But this is simply a matter of scaling, only.
The same remark applies to the following figures, too. Figure 2 shows thecase of $G=-1.5\pi$ . This is similar to Figure 1, except that the roll is moreoblate, which is intuitively obvious, and that the roll appears earlier. Figure
3 shows the evolution when $G=0.5\pi$ . The figure displays no qualitative
difference from the previous cases except that the roll appears later, which
agrees with the linear analysis.
We next consider the case where $G=1.5\pi$ . In this case, the magnitudeofand
do not increase in the valid range of the linearized equation.
This suggests that the “one-eyed” roll-up observed by Krasny is unlikely to
be seen in this case. In fact our computation shows that more than one rollup appears in the fundamental wave length $0<\Gamma<1$ . The coefficients
andactually show no significant growth even in the nonlinear evolution,
which is clearly seen in Figure 6. Figure 4 shows the time evolution of thevortex sheet. In the early stage $(1 <t<1.6)$ , three vortex roll-ups are seen:
Two of them are larger than the one near the center. Since the vortex sheet
is symmetric about the origin, this suggests that two singularities occur firstand another singularity occurs at the center soon after that. As time goes on,
the rolls become larger and we see a-eyed” vortex sheet. At $t\approx 1.8$ ,another pair of singularities is created and we have a “five-eyed” vortex sheet
afterwards. We computed up to $t\leq 3.0$ but the numerical instability was
unavoidable when $t>2.6$ and we do not know the monotone increase of$\delta$$x$$A_{1}$$y$$A_{-1}$$A_{1}$$A_{-1}$(($\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{e}$105rolls manifests in the present situation. Anyway, this sounds curious since
implies stabilization of the vortex sheet as far as the linear
the positive
analysis can be applied. The naive stabilization is far from the truth. The
temporal increase of the number of singularities makes a contrast to the caseof $G=0$ . Krasny [10] shows the vortex roll-up for rather long time interval
when $G=0$ . It shows quite a big deformation of the roll but shows no signof the birth of new rolls.
The linear analysis suggests that the number of singularities may bigger
when $G>2\pi$ . We computed when $G=2.5\pi$ to obtain Figure 5. Seven eyes
are observed at time $t=2.6$ the first singularities appear near both ends ofthe interval. Figure 6 shows how the Fourier coefficients evolve with time.
$G$

3.3

Conclusion

The structure of singularities is qualitatively the same as that in the caseof zero shear, when the shear works in the direction which strengthens theinstability of the flat sheet. It is, however, quite different, when the shear
works in the stable direction. Our computation seems to indicate that the
number of singularities increases with time. Since the disturbance of low
frequency is stabilized by the presence of a uniform shear, the time when
first singularity emerges is delayed. As a result, however, the disturbance ofhigh frequency have a effect on vortex sheet motion and many roll-up are
observed.

Interactions of two vortex sheets4

Two vortex sheet model has been studied in very many researches in order
to simulate von K\’arm\’an-type flow which is a fluid flow through a bluff body.
observed that the wake in a streaming flow using vortex blob
method for vortex-dipole sheet model numerically and compared with realexperimental flow. B\"ogers [4] uses this method for two vortex sheet model,
too. Other numerical method was applied to this type of flow; Vortex in) and point vortex approximation (Lin andCell method (Aref andSirovich [12], Boldman, Brinch, and Goldstein [3] .
Our purpose is not to simulate von K\’arm\’an vortex streets, but to observe
the mutual interactions between two sheets with various parameters, which$\mathrm{K}_{\Gamma \mathrm{a}\mathrm{s}\mathrm{n}}\mathrm{y}[11]$$\mathrm{S}\mathrm{i}\mathrm{g}\mathrm{g}\mathrm{i}\mathrm{a}[1]$

$)$

are
$\bullet$$\bullet$$\bullet$distance between two sheets,
initial phase difference of two sheets,
strength of two sheets.

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4.1

Governing equations and its desingularization

The equation which describes the motion of two vortex sheet is a natural extension of the Birkhoff-Rott equation. We may add the contribution of othersheet to the Birkhoff-Rott equation itself. We define the complex-valued functions
as the representation of two vortex sheet’s positions.
is Lagrangian parameter, is time.) Periodic boundary condition is imposed
to both sheets,$z(\Gamma, t),$ $w(\Gamma, t)$$(\Gamma$$t$$z(\Gamma+1, t)$$=$$z(\Gamma, t)+1$$w(\Gamma+1, t)$$=$$w(\Gamma, t)+1$

Then, we obtain the governing equations of two vortex sheets,$\frac{\partial z^{*}}{\partial t}$$=$$\frac{\sigma_{1}}{2i}p.v$. $\int_{0}^{1}\cot\pi(Z(\Gamma, t)-z(\Gamma’, t))d\Gamma$$+ \frac{\sigma_{2}}{2_{\dot{i}}}\int_{0}^{1}\cot\pi(_{Z(\Gamma,)-w}t(\Gamma’, t))d\Gamma’$$=$$\frac{\partial w^{*}}{\partial t}$$\frac{\sigma_{2}}{2i}p.v$

’

,

(10)

. $\int_{0}^{1}\cot\pi(w(\mathrm{r}, t)-w(\Gamma’, t))d\Gamma’$$+ \frac{\sigma_{1}}{2_{\dot{i}}}\int_{0}^{1}\cot\pi(w(\Gamma, t)-z(\Gamma’, t))d\Gamma’$,complex conjugate operator,
are the strength of two vortex
sheets, respectively. $p.v$ . means the Cauchy principal value. Initial conditionsof two sheets are given by the same one as Krasny’s.
$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}*\mathrm{i}\mathrm{s}$$\sigma_{1},$$\sigma_{2}$$z(\mathrm{r}, \mathrm{o})$$=$$\Gamma+\epsilon\sin 2\pi\Gamma-i\epsilon\sin 2\pi\Gamma+\frac{iH}{2}$$w(\Gamma, 0)$$=$$\Gamma+\epsilon\sin 2\pi\Gamma-i\epsilon\sin 2T(\Gamma+\alpha)-\frac{iH}{2}$Here, $H$ is the average distance between two sheet and represents the phasedifference between two sheets, respectively. is the amplitude of disturbance.
We apply Chorin’s Vortex blob method as the numerical method. Usingsome non-negative parameter , we regularize the equation (10) as following.
$\alpha$$\epsilon$$\delta$$\frac{\partial x_{z}}{\partial t}$$=$$- \frac{\sigma_{1}}{2}\int_{0}^{1}\frac{\sinh(y_{z}-y’z)}{\cosh 2\pi(yz-y_{\mathcal{Z}}’)-\cos 2\pi(xz-x_{z}\prime)+\delta^{2}}$$- \frac{\sigma_{2}}{2}\int_{0}^{1}\frac{\sinh(y_{z}-y’w)}{\cosh 2_{T}(yz-y’w)-\cos 2T(Xz-x’)w+\delta 2}$$\frac{\partial y_{z}}{\partial t}$$=$,$\frac{\sigma_{1}}{2}\int_{0}^{1}\frac{\sin(x_{z}-X)z\prime}{\cosh 2\pi(yz-y_{z}’)-\cos 2T(x_{z}-X_{z}’)+\delta^{2}}$$+ \frac{\sigma_{2}}{2}\int_{0}^{1}\frac{\sin(x_{z}-X)w\prime}{\cosh 2_{T}(y_{z}-y’w)-\cos 2\pi(x_{z}-x_{w}\prime)+\delta^{2}}$

,

(11)

107

$\frac{\partial x_{w}}{\partial t}$$- \frac{\sigma_{2}}{2}\int_{0}^{1}\frac{\sinh(y_{w}-y_{w});}{\cosh 2\pi(y_{w}-y’w)-\cos 2\pi(_{X}w-x’)w+\delta^{2}}$$=$,$- \frac{\sigma_{1}}{2}\int_{0}^{1}\frac{\sinh(y_{w}-y_{z}’)}{\cosh 2_{T}(y_{w}-y’z)-\cos 2\pi(X_{w}-x’)z+\delta 2}$$\frac{\partial y_{w}}{\partial t}$$\frac{\sigma_{2}}{2}\int_{0}^{1}\frac{\sin(X_{w}-x)w\prime}{\cosh 2_{T}(y_{w}-y’w)-\cos 2\pi(_{X_{w}}-x’)w+\delta^{2}}$$=$$+ \frac{\sigma_{1}}{2}\int_{0}^{1}\frac{\sin(\begin{array}{l}\prime x_{w}-x_{z}\end{array})}{\cosh 2_{T}(y_{w}-y_{z});-\cos 2\pi(Xw-X_{z}’)+\delta^{2}}$

where

4.2

$z(\Gamma, t)=x_{z}(\Gamma, t)+\dot{i}y_{z}(\Gamma, t)$,

and $w(\Gamma, t)=x_{w}(\Gamma, t)+\dot{i}y_{w}(\Gamma, t)$ .

Numerical results

In what follows, only two types of the strength of two sheets are studied;$\sigma_{2}=-1$ . We pay attention to their mutual(2)(1)
interactions between two sheets. We investigate these characteristics with, and are chosen to bevarying parameters. Numerical parameters $N,$
2048, 0.01, and 0.1, respectively.
$\sigma_{1}=1,$ $\sigma_{2}=1,$$\sigma_{1}=1,$$\triangle t$

4.2.1

The case$\delta$$(\sigma_{1}, \sigma_{2})=(1,1)$

Figure 7 shows the time evolution of two sheets with initial conditions $H=$
$0.1$ and $\alpha=0$ . At nearly $t=0.6$ , one roll-up emerges on each sheet simultaneously. These roll-ups are getting closer as time goes. At $t=0.8$ newpair of roll-up appears. Consequently, at $t=1.0$ two pairs of roll-ups construct a core region. On the other hand, the distance between trailing arms
which connect with the core becomes thinner. Compared with the evolution, we can find that the structure of core regions are quiteof vortex
different. The core region constructed through two sheets motion has more
complicated structure than vortex layer’s. Does this core region structurechanges if the desingularized parameter of vortex blob method? In$\delta=0.05$, 0.0625, 0.075, 0.0875, 0.1, 0.1125,
Figure 8, the core regions with
0.125, 0.1375 at $t=1.0$ are given. Except for winding number of roll-ups,
the qualitative structure of core regions looks the same.
We consider the effect of initial distance $H$ between two vortex sheets.
Figure 9 is the time evolution of two vortex sheets with initial distance $H=$
$0.4,\alpha=0$ in the right column and $H=0.08,\alpha=0$ in the left column. When
$H=0.4$ , only one pair of roll-up appears and no core structure like $H=0.1$
can be seen up to $t=1.4$ . Upper vortex sheet flows toward left and lower
one goes to right, which seem to move independently of each other. When
$H=0.08$ , the same vortical core structure appears as $H=0.1$ . However,the size of core region is smaller and the distance of trailing arm is thinner.
We can conclude that the vortical core structure appears because of closer
initial distance between sheets.
$\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r}\mathrm{s}[2]$$\deltaarrow 0$

108

Our numerical results obtained above are executed with no initial phasedifference, i.e. $\alpha=0$ . In this case, complicated vortical core structure canbe found. It is natural to ask how does this core structure changes if we vary
the initial phase difference. Figure 10 includes the vortical core structures
formed through the motion of two vortex sheets with various initial phasedifferences; $\alpha=0$ , 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875. In all cases, initial
distance between two vortex sheets is fixed to $H=0.1$ . The positions andnumbers of roll-ups are different. These vortical structures for various initial
phase differences seem to change continuously with respect to .
$\alpha$4.2.2

The case$(\sigma_{1}, \sigma_{2})=(1, -1)$

This flow model corresponds to a flow through a bluff body. Many researches
has been studied this flow [1, 3, 4, 10, 12]. In the same way as used insubsection 3.1, we consider the center position of spiral and the critical timeof two vortex sheets with various combinations of initial conditions.
Although our purpose is not to observe von K\’armt vortex streets, it isnatural to begin our computation with the case that two vortex sheet arelocated to be symmetrical with respect to -axis. Figure 11 shows the time
evolution of two vortex sheets with initial condition $H=0.1$ and $\alpha=0.5$ .
This symmetry between two vortex sheets is conserved through the evolution,
which is obvious from the governing equation. Both upper and lower sheets
move towards right and at $t=0.5$ one roll-up on each vortex sheet emerge.
We can see a clockwise spiral on upper sheet and counterclockwise spiral on
lower sheet. This motion of two vortex sheets reminds us of a jet flow.
Figure 12 shows the evolution of two vortex sheets with initial difference$\alpha=0.5$ . The right column in the Figure shows the numerical solution with
$H=0.4$ and the left column is the solution with $H=0.1$ . We can find thatthe evolution doesn’t change qualitatively despite the change of $H$ , which is.
different from the case
Figure 13 are the numerical solutions at time $t=1.0$ with various initial
phase difference . The initial distance of each computation is fixed to $H=$
$0.2$ . When $\alpha=0$ , the position of upper vortex sheet spiral is located at theleft-hand side of the position of lower spiral. As a increase, the position of
upper spiral chase the positions of lower spiral. At $\alpha=0.5$ the upper one
catch up with the lower one and after pass it. However, at
0.875 theupper one is at the left-side of the lower one again.
$\mathrm{x}$$(\sigma_{1}, \sigma_{2})=(1,1)$$\alpha$$\alpha=$5

discussion

Chorin’s vortex blob method with Krasny’s Fourier Filter is quite useful toare proved
simulate vortex sheet motions. Vortex methods for vortex
However,
appearsany([5]).
converge
range
singularity
in the Time
beforeto$\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{e}\{_{J}\mathrm{s}$109

the behavior of the vortex sheet after any singularity is not completely understood. It might be that no legitimate solution is available after singularity,
let alone the validity of the Birkhoff equation (1). The authors learned thisview by ([6]), in which Caflisch suggested the necessity to add the vanishingviscosity, in order to get a unique solution. He argues that if the singularity
is a branch point, then only viscosity enables us to choose a unique solution.
We, however, obtained no numerical indication of convergence to more than
one solution after the singularity. Since we tested only one regularization,
it will be too bold to say, but we think that it might be possible to obtaina unique solution after the singularity even in the category of inviscid fluid
flow. Also, Tryggvason et al. [18] shows that a certain inviscid limit of viscous
flow simulation converges to a roll which is very close to Krasny’s roll. This
seems to be an indirect evidence to support the uniqueness of convergenceeven after the emergence of singularity. We also comment that the resolution of branch point by a viscosity might not always be possible. Masuda’s
surprising theorem ([14]) on a certain semi-linear diffusion equation can beinterpreted as such. His theorem says that a certain one-dimensional semilinear diffusion equation has a unique solution past a blow-up time only if
the solution is independent of the space variable, namely, only if the solution
is trivial.

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two dimensions, J. Fluid Mech. vol. 109 (1981), pp. 435-463.
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layers and vortex sheets, J. Fluid Mech. vol. 215 (1990), pp.161-194.

[3] D.R. Boldman, P.F. Brinich and M.E. Goldstein, Vortex shedding froma blunt trailing edge with equal and unequal external mean velocities,
J. Fluid Mech. vol. 109 (1981) pp. 435-463.

[4] C. B\"ogers, On the numerical solution of the regularized Birkhoff equation, Math. Comp. vol. 53 (1989), pp. 141-156.
[5] R.E. Caflisch and J.S. Lowengrub, Convergence of the vortex method
for vortex sheets, SIAM J. Numer. Anal., vol. 26 (1989), pp. 1060-1080.
[6] R.E. Caflisch, Analysis for the evolution of vortex sheets, Vortex Dynamics and Vortex Method, C.R. Anderson and C. Greengard eds. Amer.
Math. Soc., (1991), pp. 67-83.

[7] M. Kiya and M. Arie, Helmholtz instability of a vortex sheet in uniform
shear flow, Phys. Fluid, vol. 22 (1979), pp. 378-379.

110[8] R. Krasny, A study of singularity formation in a vortex sheet by the
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[9] R. Krasny, Desingularization of periodic vortex sheet roll-up, J. Comp.
Phys., vol. 65 (1986), pp. 292-313.
[10] R. Krasny, Computation of vortex sheet roll-up, Springer Lecture Notes
in Math., 1360 (1988), Eds., C. Anderson and C. Greengard, pp. 9-22.
$\#$[11] R. Krasny, Vortex Dynamics and Vortex Methods , Lectures in Applied
Mathematics (1991), vol. 28 , pp. 385-402.
[12] C. Lin and L. Sirovich, Nonlinear vortex trail dynamics , Phys. Fluids
vol. 31 (1988).
[13] D.W. Moore, The spontaneous appearance of a singularity in the shapeof an evolving vortex sheet, Proc. R. Soc. Lond. A, vol. 365 (1979), pp.

105-119.
[14] K. Masuda, Blow-up of solutions of some nonlinear diffusion equations,in “ Nonlinear PDE in Appl. Sci.”, H. Fujita, P.D. Lax, and G. Strang, eds., North-Holland, (1983), pp. 119-131.
[15] M. Nitsche and R. Krasny, A numerical study of vortex ring formation at
the edge of a cicular tube, J.Fluid Mech. (1994), vol. 276, pp. 139-161.
[16] C. Pozrikidis and J.J.L. Hidgon, Instability of compound vortex layers
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[17] P.G. Saffman, Vortex Dynamics, Cambridge Univ. Press, (1992).
[18] G. Tryggvason, W.J.A. Dahm, and K. Sbeih, Fine structure of vortex
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vol. 113 (1991), pp. 31-36.

111

0. Krasny’s result
Figure 1: Time evolution of the vortex sheet. G
the-and -axes are slightly
is reproduced accurately. The scales ofdifferent, which make our rolls look more oblate than Krasny s. But this issimply a matter of scaling, only.
$=$

$x$

$y$

112

Figure 2: Time evolution of the vortex sheet. $G=-1.5\pi$ . Qualitatively
the same as Figure 1 but the roll-up appears earlier and the roll size is moreoblate.

113

Figure 3: Time evolution of the vortex sheet. $G=0.5\pi$ .

114

0.5

$\mathrm{t}=1 000000$$\mathrm{t}=18(\mathrm{K}\mathrm{K})00$$arrow$$- 0.5$$\mathrm{t}=1 200000$$\mathrm{t}=2.000000$$\approx$$\mathrm{t}=] 400000$$\mathrm{t}=2.200000$$\infty$$\mathrm{t}=1 600000$$\mathrm{t}=2.400000$$\infty$

Figure 4: Time evolution of the vortex sheet. $G=1.5\pi$ . Singularities growin number from one to five.

115

0.5

$\mathrm{t}=1 200000$$\mathrm{t}=2.\propto)0000$$\mathrm{t}=14000\alpha)$$\mathrm{t}=2.200000$$\mathrm{t}=16\alpha)000$$\mathrm{t}=2.400(\mathrm{K})\mathrm{o}$$\mathrm{t}=1 800000$$\mathrm{t}=2.600000$$- 0.5$Figure 5: Time evolution of the vortex sheet. $G=2.5\pi$ .

116$arrowarrow\vee 0\circ\subset\omega\underline{\frac{\Phi}{\mathrm{o}}}$$\underline{=\mathrm{Q})}$$arrow\circ$’$\tau_{\supset}^{\Phi}$$\vdash=\vee\overline{\mathrm{o}_{-}\omega\varpi \mathrm{E}}-$$\veearrow\sim \mathrm{o}\mathrm{o}a)\underline{\omega\frac{\mathrm{Q})\subset}{\mathrm{o}}}$$=rightarrow \mathrm{t}\mathrm{D}$$arrow\circ$$0\supset\omega$,$\vdash=\vee\overline{\varpi \mathrm{E}0\mathrm{a}}-$Figure 6: Time evolution of the Fourier coefficients $a_{n}=1\cdots$ .
.
$(\mathrm{U}_{\mathrm{P}\mathrm{P}^{\mathrm{e}}}\mathrm{r}),G=2.5\pi(\mathrm{L}\mathrm{o}\mathrm{W}\mathrm{e}\mathrm{r})$$G=1.5\pi$117Condltlon[d]\tdnce$\cdot$o.

$100,$$\mathrm{V}_{0\Gamma}\mathrm{t}\mathrm{e}\mathrm{X}$$\mathrm{N}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{S}:2048,$)]$\mathrm{d}\mathrm{e}\mathrm{l}\uparrow \mathrm{a}:0.\mathrm{l}\mathrm{t}\mathrm{K}$Figure 7: The time evolution of two vortex sheets with the same vorticity.
The initial distance between sheets $H=0.1$ , initial phase difference $\alpha=0$ .

118$\mathrm{d}\mathrm{e}\mathrm{l}\{\mathrm{a}=0.1000$

deltd=0.1125$\mathrm{d}\mathrm{e}\mathrm{l}\uparrow \mathrm{a}=0.1250$

delta=o.1375

Figure 8: The core region of vortex sheet with various . The core structure
looks universal except for rotating number of roll-ups.
$\delta$

119$\mathrm{t}=0.600(\mathrm{K}\mathrm{X})$$\mathrm{t}=0.800000$$\mathrm{t}=1$000000$\mathrm{t}=1$200000Figure 9: The time evolution of two vortex sheets with the same vorticity.
The initial distance between sheets $H=0.4$ in the right column, and $H=$
$0.08$ in the left column. Their initial phase difference are $\alpha=0$ .

120

Figure 10: The numerical evolution of two vortex sheet in condition withvarious at time $t=1.0$ . The initial phase difference is fixed to $H=0.1$ .
$\alpha$121$\mathrm{C}_{0}\mathrm{n}\mathrm{d}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$[, Vortex$\mathrm{d}1\backslash \mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}:0.200$$\mathrm{N}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{r}.\backslash ’:5$

] $2,$ delta:o.]oo]

Figure 11: The evolution of two vortex sheets. Initial conditions of two vortex
sheets are $\sigma_{1}=1,\sigma_{2}=-1,H=0.2$ , andwith respect tox-axis).
$\alpha=0.5(\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}$122Figure 12: The evolutions of two sheets. The right column is the numericalsolution $H=0.4$ , the left is with $H=0.1$ .

123alpha $=0500$

alpha $=0.625$

alpha $=0750$

alpha $=0.875$

Figure 13: The numerical solutions of two sheet with the various initial
phase differences $\alpha=0.0$ , 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875. The initialdifference between them are fixed to $H=0.2$ .