数理解析研究所講究録922 巻 1995 年 1-16

1

Acoustic signals associated with vortexline reconnection

in oblique collision of two vortex rings

Shizuko Adachi (安達静子)

Institute of Computational Fluid Dynamics

Katsuya Ishii

Department of Applied Physics, Nagoya University(石井克哉)

Tsutomu Kambe (神部勉)

Department of Physics, University of Tokyo

Abstract: A theory of vortex sound formulated in the form of multipole expansions isapplied to the oblique collision of two vortex rings at right angles. Using the theoreticalformula for the far-field acoustic pressure excited by a time-dependent localized vorticity
distribution, the coefficients of the multipole modes of the wave pressure are estimatedby the numerical data of computer simulation. Time evolution of the vorticity field isobtained by solving a viscous incompressible vorticity equation with a vorticity-potential
method developed for a three-dimensional vorticity field in a bounded domain. Numericalsimulations are carried out for two kinds of vortex rings with the same core parameters

but different ring rad\"u, and the details of the vortex motion during the oblique collision
are studied numerically. Computed main-mode amplitudes of the wave pressure excitedby the vortex motion are found to be consistent with those of the experimentally observed

acoustic wave not only qualitatively but also quantitatively unlike the previous study. Fromcomparison of the quadrupole and octapole modes of the far-field acoustic pressure between
the computation and the experiment, it is possible to estimate the slenderness ratio of acore radius to a ring radius of the experimental vortex ring. It is remarkable that thepresent simulation gives data identifying the source region for the wave emission, and thata characteristic evolutionary pattern of the vortex motion is responsible for the observed

up-down asymmetry of the radiation. Strong acoustic source is identified at the location

where vortexline reconnection occurs.

2

I. INTRODUCTION

The theory of vortex sound is developed successfully to describe the acoustic waves
driven by unsteady fluid motion with localized vorticity field at low Mach numbers and highReynolds numbers. Based on mathematical formulation in the form of multipole expansions,
explicit formulae of the wave profile are given in terms of the time-dependent vorticity field.

Under the condition that a typical Mach number $M$ is much less than unity, the whole field

is separated into two fields: inner flow and outer wave regions. Far-field expression of theacoustic pressure is obtained by matching of the two solutions of the inner flow region andthe outer wave region in an intermediate region. We present a review of the new generalmathematical formulation for the vortex sound [1-4] in Sec. II.
Kambe et $al.[4]$ compared the results from a laboratory experiment and a computer
simulation for an oblique collision of two vortex rings at right angles. While the normalized

mode coefficients from the computer simulation succeeded in reproducing the qualitative
feature of the experiment, they pointed out a quantitative discrepancy as follows. The

dimensionless quadrupole mode amplitudes from the simulation are by 25 times smaller
than the experiment and the octapole mode amplitudes by 125 times smaller. They argued
that this discrepancy resulted from the Reynolds number in the simulation which is muchsmaller than the experimental number. However there is another point to be examined
critically that they used vortex rings which are not thin even at the initial setup and thosevortex rings become too thick in the collision stage compared with the experimental vortex

rings. The point is that the large slenderness ratio may consequently be the reason of thediscrepancy. In the present paper we therefore examine the effect of the slenderness ratioon the mode coefficients. Two kinds of vortex rings with the slenderness ratio$\lambda_{0}=0.2$

and

0.1 are chosen for the investigation.

The vorticity-potential method $[5,6]$ for solving a viscous incompressible vorticity equation is reviewed in Sec. III. With a time-dependent vorticity distribution and a vector

potential obtained by this method the quadrupole and octapole mode amplitudes are eval-

3

uated.
Computational results of the vortex motion and acoustic mode amplitudes are presented
in Sec. IV. From comparison with the experiment we estimate a slenderness ratio of theexperimental vortex rings. In order to clarify which parts of the region in space contributeto the dominant mode amplitudes we present the spatial distribution of the third and thefourth order time derivatives of the impulse density of vorticity at the instance when those
amplitudes reach the maximum or the minimum. We focus on the vortex motion which ischaracteristic to the oblique collision and results in the excitation of asymmetric acoustic

waves. The location of the vortexline reconnection is shown to coincide with the strong
source region for the acoustic emission. Concluding remarks are given in Sec. V.

II. VORTEX SOUND FORMULATION

Neglecting the terms of $O(M^{2})$ , the basic equation in the inner region reduces to theincompressible Navier-Stokes equation. In the outer region the adiabatic relation $p-p_{0}=$$c^{2}(\rho-\rho 0)$

is satisfied approximately. Neglecting the term of $O(Re^{-1}M^{2})$ with the Reynolds

number of the inner source flow $Re$ which is assumed to be much larger than unity, we obtain

the wave equation for the pressure . Matching the outer solution to the inner solution is$p$carried out in an intermediate region. The details of the procedure is presented in Ref.[4].
In the case of unbounded space where there is neither a solid body nor an external force,the far-field expression retaining only the terms of $O(r^{-1})$ for sufficiently large$p^{(f)}( \mathrm{x},t)=-\rho_{0}P_{0}^{(1}())t_{r}\frac{1}{r}-\frac{\rho_{0}}{c^{2}}Q_{ij}(\mathrm{s})(t_{r})\frac{x_{i}x_{j}}{r^{3}}+\frac{\rho_{0}}{c^{3}}Q^{(}ij4)k(t)r\frac{X_{i}X_{J}xk}{r^{4}}+\cdots$$r=|\mathrm{x}|$is(1)where superscript $(n)$ denotes the n-th time derivative and $t_{r}=t-r/c$ the retarded time.

The coefficients$Q’ \mathrm{s}$are$Q_{ij}(t)=- \frac{1}{12\pi}\int(\mathrm{y}\cross\omega)_{i}y_{jy}d^{3}$

where$\omega$$Q_{ijk}(t)= \frac{1}{32\pi}\int(\mathrm{y}\cross\omega)_{iy_{j}}ykd^{3}y$,(2)

is obtained by solving the vorticity equation in the inner region. The first isotropic

term in (1) is related to the rate of energy dissipation as4$P_{0}(t)=- \frac{5-3\gamma}{12\pi}\frac{1}{c^{2}}I\mathrm{f}^{(1)}(t)$where $K$ is the total kinetic energy and$\gamma$,$K(t)= \frac{1}{2}\int v^{2}(\mathrm{y}, t)d3y$(3),

is the ratio of specific heats ( $\gamma=7/5$ for the air).

In this paper we study the oblique collision of two vortex rings at right angles. The

geometry of the initial set up of the experiment by Kambe et[4]. The centers of vortex rings are located on the$(X_{2}, x_{3})$$af$. is reproduced in Fig.1 fromplane with $\theta=\pi/2$ . Both vortex

rings move in the direction towards the origin. The expression of the pressure observed ata far point$(r_{obs}, \theta, \phi)$, satisfying the symmetry, is expressed as$p(\theta, \phi, t)=A_{0}(t)+A_{1}(t)P_{2}0(cos\theta)+A_{2}(t)P_{2}2(cos\theta)cos(2\phi)$$+B_{1}(t)P_{3}0(cos\theta)+B_{2}(t)P_{3}2(cos\theta)cos(2\phi)+\cdots$

where$P_{n}^{k}(cos\theta)$,(4)is the Legendre polynomial of the n-th order $(k=1, \ldots, n)$ . The coefficientsin the above expression are given by$A_{0=-} \frac{\rho_{0}}{r_{obs}}P0(1)$,$A_{1}=- \frac{\rho_{0}}{c^{2}r_{obs}}Q3(3)3$$B_{1}= \frac{\rho_{0}}{c^{3}r_{obs}}[Q_{3}33^{-}((4))Q_{3}(1/54)kk]$$\dot{\mathrm{w}}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$,’$A_{2}=- \frac{\rho_{0}}{6c^{2}r_{\circ}b_{S}}[Q^{(3}11)-Q_{22}(3)]$$B_{2}= \frac{\rho_{0}}{c^{3}r_{\mathrm{O}}b_{S}}\frac{1}{30}[\tilde{Q}_{11}^{(4)(}3-\tilde{Q}2234)]$

,

,

(5)
(6)

the tilde symbol denotes $\tilde{Q}_{113}=Q_{113}+Q_{131}+Q_{311}$ . With using the tensors$F_{ij}= \int y_{i}N_{j}d\mathrm{s}_{y}=\int(v_{jj}v_{i}-\frac{1}{2}v\delta 2)id^{3}y$$G_{jk}.= \int y_{i}y_{j}N_{k}d^{3}y$,$\mathrm{N}=\mathrm{v}\cross\omega$

,

(7)

,

(8)

the order of the time derivative in the expression of the mode amplitudes is reduced by one,and we have$A_{0=-} \frac{\rho_{0}}{15\pi c^{2}r_{ob}s}F_{i}^{(2}.\cdot)$

,

(9a)

$A_{1}=- \frac{\rho_{0}}{12\pi c^{2}r_{\circ}b_{S}}[F_{1}1+F_{22}-2F_{33}](2)$$A_{2}= \frac{\rho_{0}}{24\pi c^{2}r_{o}b_{S}}[F_{11}-F22]^{(2)}$

,

(9b)

,

(9c)

,

$B_{1}= \frac{\rho_{0}}{40\pi C^{3}r_{o}bS}[-2G_{31}1^{-2c_{3}}22+2G_{3\mathrm{s}3^{-G_{1}}}13^{-}G_{2}23](3)$$B_{2}= \frac{\rho_{0}}{240\pi c3r_{o}bS}[2G_{311^{-2G_{322}}}+G_{113}-G223]^{(3})$

We use these expressions in the present calculation.

.

(9d)
(9e)

5

III. VORTEX COLLISION SIMULATION

In order to obtain the mode coefficientsthe vorticity
vorticity$\omega$$\omega$$A$

and $B$ of the the far-field pressure in (4),in the inner region should be evaluated. The governing equations for theand the induced velocity$\mathrm{v}$

are given by$\partial_{t}\omega-\nabla\cross(\mathrm{v}\cross\omega)=\mathcal{U}\nabla^{2}\omega$

(10)

,

(11)

$\nabla\cdot \mathrm{v}=0$

Introducing a vector potential A by$\nabla\cdot \mathrm{A}=0$$\mathrm{v}=\nabla\cross \mathrm{A}$together with the solenoidal condition, the vector potential A satisfies the Poisson equation$\nabla^{2}\mathrm{A}=-\omega$. The solution

is given exactly by the Poisson integral$\mathrm{A}(_{\mathrm{X},t})=\frac{1}{4\pi}\int\frac{\omega(\mathrm{y},t)}{|\mathrm{x}-\mathrm{y}|}d^{\mathrm{s}_{\mathrm{y}}}$

.

(12)

The vorticity-potential method $[5,6]$ developed for a three-dimensional bounded vorticity

field is used in the present numerical simulation. When the vorticity distribution is given at
one time step, the vector potential (12) is evaluated at grid points on the boundary surfaceof the computational domain. Since$\omega$

is localized, the contribution of integrand to the

Poisson integral is confined in the localized region and$\omega$can be assumed to be zero on

the computational boundary during the simulation. Then the Poisson equation with theseboundary data is solved using the SOR method in order to obtain the values of the vector

potential on grid points in the interior domain. We use the second order central difference

scheme in space. Time marching to the next step following (10) is carried out with the fourthorder Runge-Kutta-Gill method using the vorticity and the vector potential obtained above.
In the present simulation, the number of operations is reduced as follows. The evaluation of
the Poisson integral is carried out for coarse grid points on the boundary surface, then they

are interpolated with a surface cubic Spline method to give the boundary values on fine grid
points. The whole integration domain in the r.h.s. of (12) is divided into subdomains. Since

the distance between boundary points and the vorticity field is large, the factor $1/|$ x–y

$|$

6

can be expressed by a power series inboundary point$\mathrm{x}$$r:=|\mathrm{x}-\mathrm{y}_{c}^{i}|$which is the distance between theand the center of the i-th subdomain of the integration

the first order term of the expansion. Then the integration only on$\omega$$\mathrm{y}_{\mathrm{c}}$, and we takeis evaluated for each

subdomain.
Initially two vortex rings are set to collide at right angles. The vortex ring radius is$R$

and the core radius is$a$. The vortex ring has a Gaussian vorticity distribution in thecore. In order to examine the effect of slenderness ratio$\lambda_{0}=a_{0}/R_{0}$we choose two sets ofparameters. Both sets have the same vortex core parameters $\Gamma=20\pi$ (circulation), $a_{0}=1.0$
and kinetic viscosity $\nu=0.1$ , but different vortex ring radii $R_{0}=5.0,10.0$ .

During the numerical simulation we checked the conservation property of the vorticity
distribution which is to be satisfied by motion of the incompressible fluid in unbounded
reglon.

IV. RESULTS AND DISCUSSION

Side view(from the$x_{1}$axis) and top view($\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}$theaxis) of the collision of two vortex$x_{3}$

rings at right angles for the two sets of parameters are presented in Fig.2. The isovorticity
surface of magnitude with 40% of the maximum at each instance is illustrated. Two vortex

rings app.ear to be translated with mostly constant speed except for the interacting parts.
The vortex cores come in contact with each other at about

elongated to the positive
stretched in the$x_{1}$$x_{3}$$t\simeq 1$. The contacting parts are-direction. As the contacting parts become large, the rings are-direction, which is due to the reconnection phenomena of vortex lines.

Finally the vorticity of the contact parts is diminished and two vortex rings become one
vortex ring.
If a vortex ring is very thin, the stable vorticity distribution in the core is considered to
be Gaussian. The translational velocity of the ring without the interaction of other rings isestimated by Saffman’s formula [7]$U= \frac{\Gamma}{4\pi R}\{\log\frac{8R}{a}-0.558\}$

.

(13)

7

Here, the core radius$a$is given by,$a=\sqrt{4\nu(t_{0}+t)}$$a_{0}=\sqrt{4\nu t_{0}}$

.

(14)

Using the above velocity of the ring, the Reynolds number of the ring motion is estimatedto be $Re=UR/\nu$ . On the other hand the Reynolds number of the circulation is $Re^{*}=\Gamma/\nu$ .
For the motion of one vortex ring several time scales are recognized: the first is the timescale of the translational motion as a whole $T_{1}=R/U$ , the second is that of the local
core deformation $T_{2}=a/U$ , the third is the period of the core rotation$\kappa=\Gamma/2\pi$$T_{3}=a^{2}/\kappa$, where, the fourth is the viscous life time of the core $T_{4}=a^{2}/(4\nu)$ and the fifth is thelife time of the vortex ring as a whole $T_{5}=R^{2}/(4\nu)$ . Velocities relating to the above timescales$T_{1},$$T_{3}$

and$T_{4}$

are $U_{1}^{*}=U,$$U_{3}^{*}=\kappa/a$

and $U_{4}^{*}=\nu/a$ respectively. Parameters of

the present simulation including the above time scales are listed in Table I. Those of thenumerical simulation and the experiment in Ref.[4] are also presented for comparison.
One of the necessary conditions for the numerical simulation is that a period of thesimulation must be much smaller than the life time of the vortex ring$T_{5}$. It is also desirable

that the interaction in which we are interested should finish before the core radius becomeslarge. Our calculation meets these conditions.
In order to compare the calculated mode amplitudes with the experiment, we normalize
them with the vortex ring radius$R_{0}$and the translational velocity$U_{0}$. The normalized

amplitudes are$\tilde{A}_{n}=A_{n}/[\rho_{0}R_{00}U^{4}/(c^{2}r_{O}b_{S})]$,$\tilde{B}_{n}=B_{n}/[\rho_{0}R_{0}U_{0}5/(c^{3}r_{O}b_{S})]$

.

(15)

The normalized time is $t^{\sim}=t/(R_{0}/U_{0})$ . The calculated mode amplitudes are plotted in Fig.3.
The origin of time$t^{\sim}$

is chosen at the time of the positive peak of$\tilde{A}_{1}$. Since the acoustic

waves prior to the interaction were removed in the experimental results, we subtract theamplitudes of non-interacting vortex rings from those of two interacting vortex rings andpresent the resultant mode amplitudes in Fig.3. The amplitudes are drawn with thick solidlines for $R_{0}=5$ and with thick dashed lines for $R_{0}=10$ . The observed amplitudes areshown with thin lines in the same figure for comparison.

8

In the case of the monopole emission, the mode amplitude is given by the rate of energy
dissipation which depends directly on the viscosity. It is noted that the Reynolds numbersin the simulation and in the experiment differ by about 50 times.
For both vortex ring radii $R_{0}=5$ and 10, the calculated mode amplitudes reproduce

qualitatively well the experimental amplitudes. While the main mode amplitudes with
$R_{0}=10$$\tilde{A}_{1}$andcoincide with the experimental ones quantitatively, the calculated peak value of$\tilde{B}_{1}$for $R_{0}=5$ is about a half of the experimental value. When we$\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{I}\mathrm{i}_{\mathrm{Z}}\mathrm{e}\mathrm{d}$the mode amplitudes presented in Fig.3, we used the translational velocity of the vortex

ring estimated with Saffman’s formula. The numerical simulation of a single vortex ringshows that the initial translational ring velocity is very close to Saffman’s value, but thering velocity decreases gradually because of the viscous effect. If we took account of thisslow-down of the vortex ring before the vortex collision and used a smaller value for thetranslational velocity in the normalization, the peak values of the mode amplitudes would

increase by about 50%. Then the experimental values of$\tilde{A}_{1}$and$\tilde{B}_{1}$fall in between thecalculated values for $R_{0}=5$ and those for $R_{0}=10$ . It might be concluded that theexperimental slenderness ratio would fall in between the two values used in the simulation.
In addition, considering the fact that the vortex cores grow in size during the interaction,

we estimate that the slenderness ratio of the experimental vortex ring would be between

0.15 and 0.25.
Since all parameters in the present computer simulation are different from those in Ref.[4],it is not easy to conclude what brings the large difference between these two simulations.
But one of the most striking difference seems to be the fact that the life time of the vortex

core$T_{4}$and the life time of the vortex ring$T_{5}$(See Table I) are much smaller in Ref.[4]than those in the present calculation. In the numerical simulation of Ref.[4] the vortex rings

are not very thin even at the initial state (the slenderness ratio is two or four times of ourcomputation) and they become thick during the collision stage. On the other hand thevortex rings in the present study keep thin even during the collision process. The differencein the time evolution of the vortex ring structure must be one of the most important reason

9

why these two simulations bring so different magnitude to mode amplitudes.
We next look into the details of the vorticity field related to the amplitudes$B_{1}$, because the amplitude

compared to$B_{1}$$A_{2}$

is much smaller than$A_{1}$and the amplitude$A_{1}(t)$and$B_{1}(t)$is negligible

depend on the spatial distribution and time evolutionof only the third component of the impulse density of the vortex system$A_{1}(t)$andin both the experiment and the simulation. As seen from Eqs.(2) $,(5)$ and(6) the amplitudes

amplitudes$B_{2}$$A_{1}$

and$B_{1}(t)$$\mathrm{p}=\frac{1}{2}\mathrm{y}\cross\omega$. The

are proportional to the space integral of

.

(16)

$b_{1}( \mathrm{y},t)=\frac{2}{32\pi}p^{(4)}3(\mathrm{y}, t)(y^{2}3-\frac{1}{5}y^{2})$$a_{1}( \mathrm{y}, t)=\frac{2}{12\pi}p_{3}^{(3)}(\mathrm{y}, t)y_{3}$In order to see such part of the vortex ring which gives significant contribution to the above

integrand, we firstof$p_{3}$$\mathrm{c}\mathrm{a}1_{\mathrm{C}\mathrm{u}}1.\mathrm{a}\mathrm{t}\mathrm{e}$the spatial distribution of the third and fourth time derivatives

at the time around the peaks of the amplitudes

positive peak of the amplitude$B_{1}$(time a), both$p_{3}^{(3)}$$A_{1}$

andand$p_{3}^{(4)}$$B_{1}$. At the time around thehave large magnitude in thelower part of the vortex ring. They have large magnitude in the upper-right and upper-leftpart of the ring at the time around the positive peak of the amplitude$A_{1}$(time ), and then$b$

the regions with large magnitude move further left or further right and downward direction

at the time around the negative peak of
distribution of$p_{3}^{(3)}$and$p_{3}^{(4)}$(time$B_{1}$$\beta$). This time evolution of the space

is well understood from that of the vorticity field during thecollision. The projective side view(from the$x_{2}$

with 40% of the maximum is presented in Fig.

axis) of the isovorticity surface of magnitude$4(\mathrm{a})$. The vorticity profile of Fig.

$4(\mathrm{a})$showsthat the first collision occurs at the top part of the two vortex rings. Subsequently, the top
part stretches upward around the time a. As the collision proceeds the interacting region
widens sideward along the cores. In the upper right and left parts, the vorticity field changes

very rapidly and those parts stretch in the outward direction of the vortex ring at the time$b$

. Then the total downward motion of the vortex ring pulls down the collision region in theupper part which expands outward around the time .
$\beta$We next calculate the integrandsof$a_{1}$

and$b_{1}$

from the$x_{2}$$a_{1}$

and$b_{1}$. The projective side views of the distribution

axis are presented in Figs.

$4(\mathrm{b})$and (c) for the vortex ring with

10

$R_{\mathrm{O}}=10$

at the timeand . The region with large magnitude of$b$$\beta$$\alpha,$$a_{1}$

andis drawn with$b_{1}$

black. The $x_{3}=0$ line is shown with a thick line on the grid. Since the lower part of thevortex ring locates near the$(X_{1}, x_{2})$plane, the weighting factor$y_{3}$

on$p_{3}^{(3)}$or$y_{3}^{2}- \frac{1}{5}y^{2}$on$p_{3}^{(4)}$reduces the contribution from the lower part but enhances the contribution from the upper
part. Therefore the regions with large magnitude of$a_{1}$

and$b_{1}$

at the time$\alpha$exist in both thelower and the upper parts. On the other hand the regions with large magnitude ofexist only in the upper part at the time

$b$

and

magnitude in the upper region. The signs of$\beta$$a_{1}$

becauseand$b_{1}$$p_{3}^{(3)}$and$p_{3}^{(4)}$$a_{1}$

and$b_{1}$

themselves have largechanges alternatively like a wave inthe region with large magnitude according to the spatial distribution of the time derivativesof$p_{3}$, and there occurs strong cancellation between contribution with opposite signs. But

even after the strong cancellation, the upper part integral has dominant contribution to both$a_{1}$

at the time

$b$

and

$b_{1}$

at the timeand$\alpha$$\beta$. These features are also seen in the smaller

vortex ring with $R_{0}=5$ .

Then let us consider the time dependence of the octapole mode amplitude
tions (16) show that the functionof$a_{1}(\mathrm{y}, t)$$b_{1}(\mathrm{y}, t)$$B_{1}(t)$. Equa-is approximately proportional to the time derivative

when the acoustic source is localized in a narrow region which does not move dur-ing the rapid change of the vorticity field. In the case of oblique collision, this condition issatisfied at around the time

as shown above. This explains the behaviour of$b$passes zero from positive to negative sign around the maximum of$A_{1}$$B_{1}$, which.

The time evolution of the vorticity distribution shows that the vorteAine reconnection

begins during the time
nitude ofFig.

$4(\mathrm{d})$$\omega_{2}$and . The contours $on$ the symmetry plane$b$. The large magnitude of$b$$(x_{1}, x_{3})$of the mag--component which is perpendicular to the symmetry plane are illustrated in

nection. At the time

the time$\alpha$and$\beta$$\alpha$$\omega_{2}$-component means that there occurs vortexline recon-

there exists almost no

large amplitude of$\omega_{2}$$\omega_{2}$-component on the symmetry plane, but atexists in the region where vortex rings are in contact

with each other. Comparing these contours with the spatial distribution of$a_{1}$

and$b_{1}$, wefind that strong acoustic source exists in the region where vorteAine reconnection occurs.

11

V. CONCLUDING REMARKS

A general formulation for the acoustic wave generated by a vortex motion is applied tothe oblique collision of two vortex rings at right angles. The vortex motion in the collision
process is simulated numerically by solving a viscous incompressible vorticity equation witha vorticity-potential method. The mode amplitudes of the far-field acoustic pressure are
obtained by calculating the moment integrals of the vorticity distribution following themathematical formula presented in Ref.[4]. Numerical simulations were carried out for two
parameter sets of vortex rings which have the same vortex core but different ring radii. The
slenderness ratio of a core radius to a ring radius is 0.2 for a smaller vortex ring and 0.1 fora lager vortex ring. It is shown that the slenderness ratio is one of the important factors
to determine the acoustic emission from the vortex ring collision. Comparing the numericalmode amplitudes with the experimental ones, it is estimated that the slenderness ratio of theexperimental vortex ring might be between 0.15 and 0.25. We have calculated the momentsof the third and the fourth time derivatives of$p_{3}= \frac{1}{2}(\mathrm{y}\cross\omega)_{3}$of the space contribute dominantly to the mode amplitudes
the$B_{1}$or$A_{1}$$A_{1}$and confirmed which regions
and$B_{1}$

at the instance whenmode amplitude has its peak value. Characteristic features of the local vortex

motion relating to the quadrupole and the octapole acoustic emission are illustrated. It has
been clearly shown that the region of strong acoustic source coincides with the region where

vortexline reconnection occurs in the oblique collision of vortex rings.

References
1. T. Kambe, J. Sound Vib. 95, 351 (1984).

2. T. Kambe, J. Fluid Mech. 173, 643 (1986).
3. T. Kambe, Theoretical and Applied Mechanics, 1992, edited by S.R. Bodner, J. Singer,
A. Solan and Z. Hashin (Elsevier Science, Amsterdam, 1993), pp. 239-255.

12

4. T. Kambe, T. Minota and M. Takaoka, Phys. Rev. E48, 1866 (1993).
5. L. Ting, J. Fluid Mech. 127, 497 (1983).
6. K. Ishii, K. Kuwahara and C.H. Liu, Computers Fluids 22, 589 (1993).

7. P.G. Saffman, Stud. Appl. Math. 49, 371 (1970).

13

TABLES
I. Parameters and Time Scales.
$-\underline{\mathrm{T}\mathrm{A}\mathrm{B}\mathrm{L}\mathrm{E}}$
Case I

Case II

Simulation

5.0

10.0

0.49

1.0

1.0

0.20

$a_{2}(t=2)$

1.34

1.34

0.35

$\Gamma$$62.8(20\pi)$

62.8

2.99

$U_{0}$

3.13

1.91

1.17

$27.3\cross 10^{3}\mathrm{m}\mathrm{m}/\mathrm{s}$

$\nu$

0.10

0.10

0.01

$15\mathrm{m}\mathrm{m}^{2}/s$

Re $=UR/\nu$

157

191

57.3

8550

$Re^{*}=\Gamma/\nu$

628

628

299

$\lambda_{0}=a\mathrm{o}/R_{0}[a_{2}/R_{0}]$$0.2[0.27]$

0.1[0.13]$0.4[0.71]$$T_{1}=R_{0}/U_{0}$

1.60

5.23

0.42

$T_{2}=a_{0}/U_{0}[a_{2}/U_{0}]$0.319[0.427]$0.524[0.702]$$0.17[0.3]$$T_{3}=a_{0}^{2}/\kappa[a_{2}^{2}/\kappa]$

0.10 [0.18]$0.10[0.18]$

0.0841 [0.251]$T_{4}=a_{0}^{2}/(4\nu)[a_{2}^{2}/(4\nu)]$$2.5[4.5]$$2.5[4.5]$

1 [3]$T_{5}=R_{0}^{2}/(4\nu)$

62.5

250.0

6.0

0.368s$U_{1}^{*}=U_{0}$

3.13

1.91

1.17

$27.3\cross 10^{3}\mathrm{m}\mathrm{m}/\mathrm{s}$$U_{3}^{*}=\kappa/a_{0}[\kappa/a_{2}]$$10.0[5.56]$$10.0[5.56]$$2.38[1.36]$$U_{4}^{*}=\nu/a_{0}$

0.1

0.1

0.05

$R_{0}$$a_{0}$$(\mathrm{R}\mathrm{e}\mathrm{f}.[4])$$\mathrm{E}\mathrm{x}\mathrm{p}.(\mathrm{R}\mathrm{e}\mathrm{f}.[4])$4.7mm

0.172ms

14

FIG. 1. Geometrical setup of the collisionof two vortex rings and$\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{I}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{e}$system.

(a) $R_{0}=8$(b) $R_{0}=10$

FIG. 2. Side view ($\{\mathrm{r}\mathrm{o}\mathrm{m}|$the$x_{1}$axis) and top view (from thevortex rings at right angles for (a) $R_{0}=5$ and (b) $R_{0}=10$ . The
with 40% of the maximum at each instance are illustrated.

$x_{3}$axis) of the collision of two$\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{v}\mathrm{o}\mathrm{I}\mathrm{t}\mathrm{i}_{\mathrm{C}}\mathrm{i}\mathrm{t}\mathrm{y}$surfaces of magnitude

15

FIG. 3. Main mode amplitudes for (a)$\tilde{A}_{0},$$(\mathrm{b})\overline{A}_{1},$$(\mathrm{c})\tilde{A}_{2},$amplitudes are drawn with thick solid lines for $R_{0}=5$ and with

ObseIved amplitudes are shown with thin lines.

$(\mathrm{d})\overline{B}_{1}$$\mathrm{t}1\iota \mathrm{i}_{\mathrm{C}}\mathrm{k}$and (e)$\tilde{B}_{2}$. Calculated

dashed lines for $R_{0}=10$ .

16$a_{1}$$b_{1}$$\omega_{2}$FIG. 4. Projective side views (bution of (b)$a_{1}$and (c)and (negative peak of$\beta$$b_{1}$$\overline{B}_{1}$$\mathrm{h}\mathrm{o}\mathrm{m}$the$x_{2}$axis) of (a) the isovorticity surface and the distri-for $R_{0}=10$ at the time). The distIibu$\iota \mathrm{i}\mathrm{o}\mathrm{n}$$\alpha$of (d)($\mathrm{p}\mathrm{o}\mathrm{S}\mathrm{i}\iota \mathrm{i}\mathrm{v}\mathrm{e}$$\omega_{2}$on thepeak of$(x_{1}, x_{3})$$\overline{B}_{1}$

),

$b$

(positive peak ofplane $(x_{2}=0)$ .

$\tilde{A}_{1}$)