数理解析研究所講究録989 巻 1997 年 158-168

158

The application of PVM to the computation ofvortex sheet motion
Hisashi OKAMOTO and Takashi SAKAJO

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

1Vortex sheet and its governing equation

We consider a motion of incompressible, inviscid fluid. Attention is restricted to a vortex
sheet motion. A vortex sheet is a surface, along which the velocity changes discontinuously. The vorticity concentrates on the surface, outside which the flow is irrotational. We
assume a further simplification that the flow is two-dimensional. Mathematically, twodimensional vortex sheet is represented by a curve. When we identify the two-dimensional
space with complex plane, a vortex sheet in two-dimensional space is expressed by a complex valued function, whereis a Lagrangian parameter along the curve,
which represents the circulation of the flow. represents time.
We consider the dynamics of a two-dimensional vortex sheet with aboundary$z(\Gamma,t)$$\Gamma\in \mathrm{R}$$t$.eriodic$\mathrm{p}$condition;$z(\Gamma+1, t)=z(\mathrm{r}, t)+1$.

The equation which describes the motion of vortex sheet is known as the Birkhoff-Rottequation ([16]):$\frac{\partial z^{*}(\Gamma,t)}{\partial t}=\frac{1}{2\pi i}\mathrm{p}.\mathrm{v}$.

$\int\frac{d\Gamma’}{z(\Gamma,t)-z(\Gamma’,t)}$

The integral on the right hand side is Cauchy’s principal value. is the imaginary un\‘it.
complex conjugate. Taking the periodic boundary condition into account, werewrite the equation and obtain $([8])$ :$i$$*\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{S}$$\frac{\partial z^{*}(\Gamma,l)}{\partial t}$$=$$\frac{1}{2i}\mathrm{p}.\mathrm{v}$.

$\int_{0}^{1}\cot\pi(Z(\Gamma, t)-z(\Gamma’, t))d\mathrm{r}’$

.

(1)

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This is the equation which we are going to study numerically.
, is an equilibrium solution of theA flat vortex sheet of constant strength,equation (1). This equilibrium is known to be unstable: even a small perturbation cangrows very rapidly and the vortex sheet shows extreme complexity for large $t([8,9,17])$ .
The following properties are known:$z(\Gamma, t)\equiv\Gamma$$\bullet$Linearized stability analysis shows that perturbations of short wave length growexponentially. The shorter the wavelength is, the faster the perturbation grows $([16])$ .
(Kelvin-Helmholtz instability.)$\bullet$$\bullet$$\bullet$$\bullet$If the initial perturbation is an analytic function of , then the sheet remains analytic
for a positive time interval ([2, 20]).
$\Gamma$

The vortex sheet loses analyticity in finite time $([14])$ .

The initial value problem for (1) is ill-posed in the sense of Hadamard $([3])$ .

A vortex sheet evolves into a complex form having spirals (see, for. instance, [8, 9,17]).

Because of the ill-posedness of the equation, it is difficult to apply naive numericalmethods to the computation of a vortex sheet. We apply Chorin’s vortex blob method,which we are going to explain in the next section.

2

Numerical method

Instead of the original equation, we consider the following smoothed equation. (Thisequation is given by Krasny $([8])$ . ).
,(2)$\frac{\partial z^{*}(\Gamma,t)}{\partial t}=\int_{0}^{1}K_{\delta}(z(\mathrm{r}, t)-Z(\Gamma’, t))d\Gamma’$

where.

$K_{\delta}(X+ \dot{i}y)=-\frac{1}{2}\frac{\sinh(2\pi y)+i\sin(2\pi x)}{\cosh(2\pi y)-\cos(2\pi x)+\delta^{2}}$This equation is well-posed for any time interval if. is an artificial parameter thatmakes the equation well-posed. When, the equation reduces to the original equation.
The convergence of the solution of the smoothed equation to that of the Birkhoff-Rottequation is proven as far as the solution of (1) is smooth. However, after the appearanceof the singularity, only convergence in some weak sense is proven ([12, 13]).
$\delta>0$$\delta=0$$\delta$

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2.1Discretization

In order to compute (2), we approximate the vorte.x sheet by $N$ points.
$\Gamma_{i}=\frac{i}{N}$,$z(\Gamma_{i}, t)=z_{i}(t)$,$i=0,$$\cdots,$$N-1$ .

Then, we discretize the integral by trapezoidal rule and obtain the following system ofordinary differential equations:$\frac{\partial z_{n}^{*}(t)}{\partial t}=\frac{1}{N}\sum_{m=0}^{N}K_{\delta}(_{Z_{n}}(-1t)-zm(t))$,$i=0,$$\cdots,$$N-1$ .

(3)

In order to integrate the system of O.D.E, we use the fourth-order Runge-Kutta method.
The parameters we can change are as follows:$\bullet$the number of vortices$N\cdots$$\bullet$$\triangle t\cdots$$\bullet$$\delta\cdots$time step size for Runge-Kutta methodsmoothing parameter of vortex blob method

This is a rough description of the vortex method. For more details, see, e.g., Puckett [15].
Numerical computation of the right hand side of (3) requires $O(N)$ multiplications foreach vortex point. Since there are $N$ vortices, $O(N^{2})$ operations are necessary to computethe velocity fields at all the positions of particles. In order to obtain an accurate numericalsolution, we need a large number of vortices to discretize the vortex sheet. Thus thecomputation becomes too slow when $N$ is large. This is the most serious disadvantageof the vortex method. There are some algorithms which evaluate the velocity field in$O(N\log N)$ operations within some errors $([4,6,7,18])$ . Although they are promising,
they have some defects, too $([7,18])$ . In the present paper, we would like to study anothermethod, Parallel Virtual Machine software, to evaluate the velocity field at the vortex
points. In the following section, we will explain how to implement this tool for numericalcomputations and the efficiency will be examined.

3Parallel Virtual Machine

The Parallel Virtual Machine (shortly PVM) is a software framework for heterogeneous
parallel computing in networked environments ([5]). PVM supports complete message
passing model and it emulates a distributed memory model in heterogeneous network.
Our virtual parallel machine consists of four computers. They have the following
CPU’s, memory, and operating systems, respectively$\bullet$$\mathrm{A}\cdots$

PA-RISC 7000,$32\mathrm{M}\mathrm{B}$, HP-UX 9.0.5

161$\bullet$$\mathrm{B}\cdots$

Pentium$120\mathrm{M}\mathrm{H}\mathrm{Z},$$\bullet$$\mathrm{C}\cdots$

Pentium$100\mathrm{M}\mathrm{H}_{\mathrm{Z},6}4\mathrm{M}\mathrm{B}$$\bullet$$\mathrm{D}\cdots$

Pentium$64\mathrm{M}\mathrm{B}$, FREEBSD, FREEBSD$60\mathrm{M}\mathrm{H}\mathrm{Z},$$24\mathrm{M}\mathrm{B}$, FREEBSD

These computers are connected through 10Base-T Ethernet.
We use the following master-slave type algorithm to implement the .omputation ofvelocity field:$\mathrm{c}$1. Divide $N$ points by group ( is the number of computers).
is the numberof vortices, the velocities of which are computed in the j-th computer, whence
$k$

$k$

$n_{j}$$\Sigma_{j=1j}^{k}n=N$.

2. Send the position of$n_{j}$

vortices to each slave computers

3. Each slave computer evaluates the velocities at the positions of$n_{j}$

points4. S. end back the results to the master computer

5. loop to 2$n_{j}$

is determined by CPU speed to make the evaluation time of each computers as evenas possible.

3.1

Test problem : Two Vortex Sheets

Using PVM and vortex blob method for a vortex sheet, we compute the motion of twovortex sheets. We consider two, nearly parallel, vortex sheets. We denotes upper vortex
sheet and lower vortex sheet byand, respectively. Then the equation ofmotion of two vortex sheets is written as follows:$z(\Gamma, t)$$\frac{\partial Z^{*}(\Gamma,t)}{\partial t}$$\frac{\partial w^{*}(\Gamma,t)}{\partial t}$

where$\sigma_{1}$$w(\Gamma, t)$’$=$$\frac{\sigma_{1}}{2_{\dot{i}}}\mathrm{p}.\mathrm{v}.\int_{0}^{1}\cot\pi(Z(\Gamma, t)-z(\mathrm{r}’, t))d\mathrm{r}$$+$$\frac{\sigma_{2}}{2i}\int_{0}^{1}\cot\pi(Z(\Gamma, t)-w(\Gamma’, t))d\mathrm{r}’$$=$$\frac{\sigma_{2}}{2i}\mathrm{p}.\mathrm{v}.\int_{0}^{1}\cot\pi(w(\Gamma, t)-w(\Gamma’, t))d\Gamma’$$+$$\frac{\sigma_{1}}{2i}\int_{0}^{1}\cot\pi(w(\mathrm{r}, t)-Z(\Gamma’, t))d\Gamma’$is the vorticity density of upper vortex sheet and$\sigma_{2}$,,is that of lower vortex sheet.

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The initial value of$z$

and$w$$z(\Gamma, \mathrm{O})$$=$$w(\Gamma, 0)$$=$is taken as follows:$\Gamma+\epsilon\sin 2\pi \mathrm{r}-i\epsilon\sin 2\pi\Gamma+i\frac{H}{2}$,$\Gamma+\epsilon\sin 2\pi\Gamma-i\epsilon\sin 2\pi(\Gamma+\alpha)-\dot{i}\frac{H}{2}$$(0\leq\alpha<1, H\neq 0,0\leq\Gamma<1)$,

where $H$ is average distance between two vortex sheets and a is the phase difference of
two vortex sheets.
The numerical parameters of the computations are$\bullet$

the number of vortices$N\cdots$$\bullet$$\Delta t\cdots$$\bullet$$\delta\cdots$$\bullet$$\bullet$$\bullet$$\bullet$

3.2.

$\epsilon\cdot\cdot$$\cdot$smoothing parameter of vortex blob method

the amplitude of the disturbance$H\cdots$$\alpha\cdots$time step size for Runge-Kutta method

the average distance between two vortex sheets

initial phase difference of two vortex sheets$\sigma_{1},\sigma_{2}\cdots$

the vorticity of two vortex sheets.

Reduction of execution time

We show the execution time when we use PVM. The execution time (in second) is measured by one time step of Runge-Kutta method. (The time to evaluate velocity field fourtimes. ) Among four computers, machine is the fastest of the four computers. The. We divide $N$ vortex
ratio of CPU speeds is approximately equal topoints according as this ratio. The following are the list of of performances.
$\mathrm{B}$$\mathrm{A}:\mathrm{B}:\mathrm{c}:\mathrm{D}=9:10:8:5$

3.2.1

The result for $N=2048$

1. Single processor $k=1$$\bullet$$\mathrm{A}\cdots$$\bullet$$\mathrm{B}\cdots$$\bullet$$\mathrm{C}\cdots$$\bullet$$\mathrm{D}\cdots$

40.68 seconds
33.79 seconds
41.19 seconds
67.22 seconds

2. Two processors $k=2$

163$\bullet$$\mathrm{A}+\mathrm{B}\cdots$

20.87 seconds $(\cross 1.62)$

3. Three processors $k=3$$\bullet$$\mathrm{A}+\mathrm{B}+\mathrm{C}\cdots$

14.81 seconds $(\cross 2.28)$

4. Four processors $k=4$$\bullet$$\mathrm{A}+\mathrm{B}+\mathrm{C}+\mathrm{D}\cdots$

13.26 seconds $(\cross 2.54)$, for instance, implies that the computation is 1.77 times faster
Here and hereafter,
than the computation of single processor.
$\cross 1.77$$\mathrm{B}$3.2.2

The result for $N=4096$

1. Single processor$\bullet$$\bullet$$\mathrm{A}\cdots$$\mathrm{B}\cdot\cdot$‘$\bullet$$\mathrm{C}\cdots$$\bullet$$\mathrm{D}\cdots$

153.09 seconds
134.98 seconds
263.29 seconds
159.49 seconds

2. Two processors$\bullet$$\mathrm{A}+\mathrm{B}\cdots$

75.90 seconds $(\cross 1.77)$

3. Three processors$\bullet$$\mathrm{A}+\mathrm{B}+\mathrm{C}\cdots$

55.29 seconds $(\cross 2.44)$

4. Four processors$\bullet$$3.2.3$$\mathrm{A}+\mathrm{B}+\mathrm{c}+\mathrm{D}\cdots$

49.19 seconds $(\cross 2.74)$

The result for $N=8192$

1. Single processor$\bullet$$\mathrm{A}\cdots$$\bullet$$\mathrm{B}\cdots$$\bullet$$\mathrm{C}\cdots$$\bullet$$\mathrm{D}\cdots$

635.06 seconds
521.05 seconds
640.54 seconds
1050.41 seconds

2. Two processors

164$\bullet$$\mathrm{A}+\mathrm{B}\cdots$

308.12 seconds $(\cross 1.69)$

3. Three processors$\bullet$$\mathrm{A}+\mathrm{B}+\mathrm{C}\cdots$

219.82 seconds $(\cross 2.37)$

4. Four processors$\bullet$$\mathrm{A}+\mathrm{B}+\mathrm{C}+\mathrm{D}\cdots$

188.56 seconds $(\cross 2.76)$

The above results show that the usefulness of PVM at least when a small number ofcomputers are combined.

3.3

Numerical results

We choose $N=4096$ . Figure 1 is the long time evolution of two vortex sheets. Initial. Figure 2 isaverage distance $H$ is 0.2, initial phase difference a is O,and$H$
the long time evolution of two vortex sheets. Initial average distance is 0.2, initial phase. Both figures show complicated spiral structures.
difference is 0.
, such complexities seem
Since the vorticity is not of distinguished sign
to comply with what are predicted in [13].
$(\sigma_{1}, \sigma_{2})=(1, -1)$$\alpha$$5,\mathrm{a}\mathrm{n}\mathrm{d}(\sigma_{1}, \sigma_{2})=(1, -1)$$(\sigma_{1}, \sigma_{2})=(1, -1)$

4

Summary and acknowledgment

When the number of vortices exceeds a few thousands, the efficiency of PVM is satisfactory. PVM is a useful tool for particle simulations. However, if PVM environment
consists of a lot of node computers, the rapid increase of data transfer may well makeit impossible for us to execute fast computation. To make more efficient computation,
we must choose algorithms with more effective data transfer. There is a possibility tocombine the fast algorithms $[4, 6]$ and PVM effectively.
Professor O. Pironneau advised us to try computations with adaptive increase of thenumber of vortices. This attempt is in progress and will be reported elsewhere. We thankhim for the advice.

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Figure 1: Long time evolution of two vortex sheets. The initial parameters are $H=0.2$ ,.
$\alpha=0.0,\mathrm{a}\mathrm{n}\mathrm{d}(\sigma_{1}, \sigma_{2})=(1, -1)$

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Figure 2: Long time evolution of two vortex sheets. The initial parameters are $H=0.2$ ,..
.
$\alpha=1.0,\mathrm{a}\mathrm{n}\mathrm{d}(\sigma_{1}, \sigma_{2})=(1, -1)$

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