数理解析研究所講究録908 巻 1995 年 103-110

103

Wave generation and interaction with vortex motions:sound and shock waves$\mathrm{T}$Kambe

Department of Physics, University of Tokyo
Hongo, Bunkyo-ku, Tokyo 113, Japan

ABSTRACT Theory of vortex sound in the aeroacoustics can describe characteristicfeatures of the waves radiated by localized vortex motions at low Mach numbers. Someexperimental evidences and computational examples are presented from our studies toshow remarkable agreement between observed and computed wave profiles emitted by
two kinds of vortex motions in free space: head-on collision and oblique collision of two
vortex rings. In the oblique collision of two vortex rings, reconnection of vortex-lines
causes rapid change of geometrical structure of vortex-lines, resulting. in excitation of acharacteristic type of acoustic wave. As a last exmple, a supersomc interaction of avortex ring with an impinging plane shock wave is presented. A computer simulation
illustrates a strong local intensification of the wave caused by the presence of the vortex.
1. Introduction
Wave generation by the vorticity dynanics governed by the vorticity equation is
now recognized to be formulated in the frame of the theory of vortez sound. Recent
studies of the vortex sound are illustrated here by presenting two cases of the head-on.collision and obliqu.e collision of two vortex rings. In addition, strong interaction of ahigh-speed vortex nng with an impinging shock wave is described, in which remarkable. Characteristic Mach number
local intensification of the wave is observed
while the one in the latter isunity,
than
less
be
much
to
in the former case is assumed
larger than unity.
In the theory of vortex sound at low Mach number and high Reynolds number [1vortex motion scaled on a length8], the source flow is characterized by a
by outer wave field scaled onsurrounded
is
field
flow
This
size.
vortexrepresenting the$\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{i}_{\mathrm{C}}\mathrm{a}\mathrm{U}\mathrm{y}$$l$$1_{\mathrm{o}\mathrm{C}\mathrm{a}}\mathrm{J}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{d}$.

, because
enables separate analysis of the two fields: inner flow and outer wave.
of the compactness of the source flow,
In our investigation of the vortex sound in the past decade or so, evidences showing
the validity of the theory are accumulating. Regarding the vort collisions, the firststage of the vortex interaction is almost inviscid. This is followed by the second viscousstage : cancellation of opposite signs of vorticity at the time of head-on collision [4],, and so on. Acoustic waves
reconnection of vortex-lines in the oblique
the waves can be detected . erimentally [4, 5, 7].
are.generated by these motions andinterest here is comparison of the wave proffies thus determned computationallywith the corresponding profiles deternined experimentally.
$\mathrm{r}\mathrm{e}\mathrm{g}_{\mathrm{l}\mathrm{o}\mathrm{n}\mathrm{S}}$$l\ll\lambda$$e\mathrm{x}$$\mathrm{c}\mathrm{o}\mathrm{U}\mathrm{i}_{\mathrm{S}}\mathrm{i}_{0}\mathrm{n}[7,15]$$\mathrm{e}\mathrm{x}\mathrm{p}$$\mathrm{M}\mathrm{a}\ln$.2. Mathematical Formulation of Vortex Sound
Basic equation of the aerodynamic sound is given by the Lighthill’s equation [8]:$\rho_{u}-C^{22}\nabla\rho=\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}\tau_{:}j$

(1)

104

where

$T_{ij}=\rho v_{i}v_{\mathrm{j}}+(p-c^{2}\rho)\delta:\mathrm{j}-\mathcal{T}_{i\mathrm{j}}$,$x=(x_{i})$ is the space coordinate vector,
the pressure, the density,
the i-th component of velocity andthe viscous stress tensor. Summation convention is assumed
ontime , the ratio of the two termson the left hand side is estimated as. In the first
approximation negl.eCting terms of $O(M^{2})$ and hence the termthequation (1),reduces to the Navier-Stokes equation for an incompressible fluid, governing solenoidal
vort motion treat as a source flow.
In the outer region, the rhs of (1) becomes vanishingly small and the pressure isgoverned by the wave equation: $p_{\ell t}-C^{2}\nabla 2p=0$ , because of the (assumed) adiabatic
relation. In the theory $[7, 10]$ , an asymptotic representation of the velocity
potential of the inner flow is given first, and then a solution of the outer wave equation isdetermined so as to be matched to the inner solution. The wave pressure
matching$p$$v_{i}$$\rho$$\tau_{i\mathrm{j}}$$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}(\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{S}\mathrm{C}\mathrm{r}\mathrm{h}\mathrm{s}\mathrm{J}_{\mathrm{e}\mathrm{d}\mathrm{b}}\mathrm{o}\mathrm{f}(1)\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{y}\mathrm{t}\mathrm{h}\mathrm{e}1\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{h}l\mathrm{a}\mathrm{t}\mathrm{C}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{h}\mathrm{n}\mathrm{d}$$\tau$$|\rho_{u}|/|c^{2}\nabla^{2}\rho|=O(l^{2}/\mathrm{c}^{2}\tau^{2})=O(M^{2})$$e$$\rho u,$$e\mathrm{d}$$e\mathrm{x}$$\mathrm{d}p=c^{2}\mathrm{d}\rho$$p^{(\mathrm{w})}$.

to

fluid. The pressure takes a simple form in the acoustic far-field $(|x|/\lambdaarrow\infty)$ . The
wave pressure observed at a point $x=(x_{1},x_{2},x_{s})$ in the acoustic far-field is given as(2)$p^{(\mathrm{w})}(X,t)=- \rho \mathrm{o}P0((1)\iota_{\mathrm{r}})\frac{1}{r}-\frac{\rho_{0}}{c^{2}}Q_{i}(3)(jtr)\frac{x_{i^{X}\mathrm{j}}}{r^{3}}+\frac{\rho_{0}}{c^{3}}Qi\mathrm{j}k((4)t_{r})\frac{x_{i}x_{j^{X_{k}}}}{r^{4}}+\cdots$

where superscript $(n)$ denotes the n-th time derivative, and$Q_{i}(t)= \frac{1}{8\pi}\int(y\cross\omega)i\mathrm{d}^{\epsilon}\mathrm{y}$,$Q: \mathrm{j}(t)=-\frac{1}{12\pi}\int(y\cross\omega)_{i}y\mathrm{j}\mathrm{d}^{3}\mathrm{y}$(3),$Q_{*\dot{g}k}.(t)= \frac{1}{32\pi}\int(y\cross\omega)_{i}yjyk\mathrm{d}\mathrm{s}_{\mathrm{y}}$

(4)

,

(5)

is the vorticity:, andvector $x=(x_{i})$ is used for the point ofobservation and $y=(y_{i})$ for the point of integration (the souce point). The resultant.impulse of the vort system is defined by, defined by (3), which is an invariantof motion in free space. Hence the dipole term disappears because the dipole mission
is related to the rate of change of the impulse.
’
The second
le terms (quadrupole [3]) can be shown to derivefrom the non-isotropic part of the Reynolds stress[11]. The first isotropic(monopole) term arises when the total kinetic energy $K$ of the system changes, where$\mathrm{t}\dot{\mathrm{h}}\mathrm{e}$$\omega=\Delta\cross v$$\omega=\omega(y,t)$$4\pi Q_{i}$$e\mathrm{x}$$e$$\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}\mathrm{r}\mathrm{u}\mathrm{p}\dot{\mathrm{o}}$$\mathrm{M}\tilde{\mathrm{o}}\mathrm{h}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}_{\mathrm{S}}$$\rho_{0:}vv_{\mathrm{j}}$$P_{0}(t)=- \frac{5-3\gamma}{12\pi}\frac{1}{c^{2}}K^{(1)}(t)$,$K(t)= \frac{1}{2}\int v^{2}(\mathrm{y},t)\mathrm{d}^{\epsilon}\mathrm{y}$,(6)$K$ is the total kinetic energy.nd 7 the ratio of specific heats ( $\gamma=7/5$ for the air). $K$
vanishes identically in an invlscid fluid. Here we have written the formula up to thethird order terms. An experimental observation (described in \S 4) shows existence of
this order.
Using the length scale , the vorticity scale $\omega=u/l$ and the time scale $\tau=l/u$ , thescaling law of the wave pressure of the quadrupole sound is deduced as follows. The
tensoris normalized byand henceby $l^{4}u/\tau^{3}=lu^{4}$ (the first monopole
term gives the same by (6) . Thus we find the scaling law for the quadrupole sound as$l$$l^{4}u$$Q_{:\mathrm{j}}$$Q_{:\mathrm{j}}^{(3)}$$)$$pQ \sim\frac{\rho_{\mathrm{O}}}{c^{2}}lu\frac{1}{r}4=\frac{\rho_{0}u^{4}}{c^{2}}\frac{l}{r}$105

The sound intensity
law $[5, 8]$ :$I_{Q}$

is given by$p_{Q}^{2}/\rho_{0}c$$I_{Q}\sim(\rho_{0}u^{8}/c)5(l/r)2\mathrm{u}\propto 8$.

. Hence we obtain the well-known intensity

3. Head-on Collision of-Two Vortex Rings
Axisymmetric collision of two vortex rings [4] is a particularly simple example ofof the formula (2) vanishes and thethe vortex sound. In an inviscid fluid, the first
reduces tosecond quadrupole$\mathrm{t}e\mathrm{m}$$\mathrm{t}e\mathrm{r}\mathrm{m}$$p=- \frac{\rho_{\mathrm{O}}}{12c^{2}}Q^{(}3)(t_{f})\frac{1}{r}(1-3\cos\theta 2)$,$Q(t)= \int\omega(\sigma,z)\sigma^{2}Z\mathrm{d}\sigma \mathrm{d}Z$(7)

where $t_{r}=t-r/c$ , and the vorticity is assumed to have only the azimuthal -component
and, with $z=x_{3}$ andin the cylindrical coordinate systembydescribed
theis$\theta=\arccos(z/r)$ . In this case the temporal behavior of the pressure-order derivative $Q^{(3)}(t)$ , and its spatial distribution is given
single scalar function of
by $P_{2}^{0}=(3\cos^{2}\theta-1)/2$ , an axisymmetric four-lob directivity (Figure 1).
$\phi$$\sigma=\sqrt{x_{1}^{2}+X^{2}2}$$(z, \sigma,\phi)$$\omega(\sigma,z)$$3\mathrm{r}\mathrm{d}$$e$Figure 1. Directivity of the acoustic.
pressure: 1–3$\cos^{2}\theta$ot two vortex
Figure 2. Head-on
rings (definition sketch).
$\mathrm{c}\mathrm{o}\mathbb{I}\mathrm{l}\mathrm{s}\mathrm{l}\mathrm{o}\mathrm{n}$Computation : Suppose that we have two vortex rings having a common symmetry
axis with one vortex being a mirror image of the other with respect to the plane$z=0$ (Figure 2), and that they approach each other according to the equation of, ring radius $R(t)$ (coremotion. When the vortex is charact rized by its strength
radius ) and the distance $Z(t)$ from the symmetry plane $z=0$ , the interacting motionof two vortex rings of very thin cores is described by a system of first-order ordinary.
differential equations [9]. Then the profile function $Q(t)$ is expressed
The wave pressure form is given by $Q^{(3)}(t)$ , which is calculated numerically by solving
the system of differential equations and shown by the curve I in Figure 3, together withthe observed one E. The first peak of the profile corresponds to the initial inviscid stageof the colliding motion. Effect of finite core-size and core deformation on the wave
profile is studied in detail by the inviscid contour dynamics [12]. This analysis suggests) $(t)$ is due to core deformation.
in thethat the-symmetric viscous vortex collision was carried out toA numerical simulation of$Q^{(3)}(t)$ , shown by the curve V in Figure 3, at the Reynolds
estimate the profile function
number $Re\approx 1.3\cross 10^{3}$ , based on the initial translation velocity $U=|\dot{Z}(0)|$ and ring.
diameter 2
$z$

$e$

$-\Gamma$$\delta$$\mathrm{b}\mathrm{y}-2\Gamma R^{2}(t)Z(t)$$\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{v}\mathrm{e}-Q^{(}3$$\mathrm{d}\mathrm{i}\mathrm{p}_{\mathrm{o}\mathrm{b}}.\mathrm{s}\mathrm{e}\mathrm{r}\mathrm{v}\mathrm{e}\mathrm{d}$$\mathrm{a}_{1}\dot{\mathrm{n}}$$R_{\mathrm{O}}=2R(\mathrm{o})[13]$Observation : Experimental observations of the corresponding acoustic waves due tothe vortex collision are reported in [4] and [14]. The Reynolds number $Re$ was of theor larger in the latter study.
order$10^{4}$

106

Figure 3.
Temporal profiles $-Q^{(3)}(t)$ ;I: inviscid,: ex.perimental,
V : viscous simulation.
$\mathrm{E}$The observed pressures at 35 stations are represent

in the truncated Fourier series:. The two terms
are found to be dominant and rewritten as$e\mathrm{d}$$p(\theta,t)=a_{0}(t)+a_{1}(t)\cos\theta+b_{1}(t)\sin\theta+a_{2}(t)\cos 2\theta+b_{2}(t)\sin 2\theta$and$a_{2}(t)\cos 2\theta$$p_{\mathrm{m}\mathrm{q}}(\theta,t)=p\mathrm{m}(t)+p\mathrm{q}(t)(1-3\cos\theta 2)$$a_{\mathrm{O}}(t)$,represent the monopolar and quadrupolar components respectively.
where
and
are detected, andandSignificant amplitudes ofis shown as the curvein Figur $e3$ .
$p_{\mathrm{m}}$$p_{\mathrm{q}}$$p_{\iota \mathrm{n}}(t)$$p_{\mathrm{q}}(t)$$p_{\mathrm{q}}(t)$$\mathrm{E}$

4. Oblique Collision of Two Vortex Rings.

)
that in the previous case of head-on collision. At the oblique collision, opposite sensesof vortex lines are forced to come into contact at the inner part. This event is followedby rapid change of vorticity structure and excitation of acoustic waves. Analysis of
the wave data provides some information of the complex vortex motion associated withvortexline reconnection.
Consider a problem that the initial state is g.iven in such a way t.ha. two vortex rings
are set to move along the paths intersecting at right angles at the ongin and collide with
one another. The bisecting straight line etween the two paths of the vortex center istaken as the polar nis $\theta=0$ (along theaxis) of the spherical coordinate system. The
plane perpendicular to theplane on which $\theta=.\pi/2$ . There areaxis is the.
two symmetry plan es including theaxis: one includes the trajectones of the vortex
plane and the plane
centers which is defined as
perpendicular to it is also
a symmetry plane which bisects the two trajectori . The plane $\phi=0$ is taken along
the positivea.ris. From the geometrical arrangement just mentioned, the acousticpressureis characterized by the symmetry:$\mathrm{C}\mathrm{o}\mathrm{m}_{\mathrm{P}^{\mathrm{u}}}\mathrm{o}\mathrm{b}1\mathrm{i}\mathrm{q}\mathrm{u}_{\mathrm{i}\circ \mathrm{n}\mathrm{a}1}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{i}_{\mathrm{S}}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{t}1\mathrm{y}[7]$$\mathrm{E}\mathrm{v}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{l}\mathrm{y}\mathrm{t}\mathrm{t}\mathrm{f}\mathrm{t}\mathrm{w}\mathrm{o}\mathrm{v}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{x}\mathrm{r}\mathrm{i}\mathrm{n}_{1}\mathrm{g}\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{h}\mathrm{i}8\mathrm{o}\mathrm{b}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{C}\mathrm{o}\mathrm{u}^{\mathrm{h}1}\mathrm{S}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{n}\xi_{\mathrm{F}\mathrm{i}\mathrm{u}}\mathrm{e}\mathrm{s}\mathrm{i}_{\mathrm{S}\mathrm{S}}\mathrm{t}$$\mathrm{g}\mathrm{r}\mathrm{e}4\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{n}\mathrm{o}\mathrm{a}\mathrm{X}\mathrm{i}_{\mathrm{S}}\mathrm{y}\mathrm{m}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}\mathrm{l}\mathrm{i}\mathrm{d}\mathrm{u}\mathrm{i}\mathrm{e}\mathrm{d}\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{u}\mathrm{y}\mathrm{a}\mathrm{n}\mathrm{k}\mathrm{e}\mathrm{d}$$\mathrm{t}$$\mathrm{b}$$x_{3}$$(X_{1}, x_{2})$$x_{3}$$x_{3}$$(X_{1}, x_{3})$$(X_{2}, X_{3})$$e\mathrm{s}$$x_{1}$$p(\theta, \emptyset,t)$$p(\theta, -\phi, t)=p(\theta,\phi,t)$,$p(\theta, \phi+\pi,t)=p(\theta, \phi,t)$.

(8)

The formula (2) is rewritten by using the spherical polar coordinates $(r, \theta, \phi):x_{1}=$$x_{2}=r\sin\theta\sin\emptyset,$ $x_{3}=r\cos\theta$ . In view of the symmetry (8), the pressure isrepresented as$r\sin\theta\cos\emptyset,$$p(\theta, \emptyset,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$,(9)where higher order terms are omitted since observed amplitudes are not significant.
Thus it is found that the far-field acoustic pressure (2) is presented in terms of the$\mathrm{r}e$107five normal modes with five coefficient functions of time $[A_{\mathrm{o}}(t), A_{1}(t), A2(t), B1(t), B2(t)]$ .

Here the Legendre functions are $P_{2}^{0}=(1/2)(3\cos\theta 2-1),$$P_{3}^{0}=(1/2)(5\cos^{3}\theta-3\cos\theta),$$P_{3}^{2}=15(\cos\theta-\cos^{\epsilon}\theta),$$\cdots$.

$P_{2}^{2}=3\sin^{2}\theta$,$\cdot$.., andcollision were detected at 102
The acousttic wavesemitted by theof radius $r=620$ mm on the threedifferent angular positions on the three great
orthogonal planes: (1) $\phi=\pi/2,3\pi/2;(2)\phi=0,\pi;(3)\theta=\pi/2$ . The trajectories ofplane observed by a photosensor are shown in Figure 4,
the vortex cores in the.
of the single (unperturbed) vortex’was 4.
where the ring radius. Observation:$90^{\mathrm{o}}$$\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{l}\mathrm{e}8$$(X_{2}, x_{3})$$7\mathrm{m}\mathrm{m}$$R_{\mathrm{O}}=R(0)$Figure 4. Observed core trajectori es of
two colliding vortices in theplane including the vortex center.
The marked positions andcorrespond to the times of Figure 5(broken lines).
$(X_{2}, X_{3})$

$a$

$b$

$t(ms)$

Figure 5. Main mode amplitudes ofobserved waves.

The symmetry relation (8) is found to be consistent with the observed data. In
fac.t, the acoustic pressure detected in the plane $\theta=\pi/2$ can be expanded into Fourier
senes with respect to the angle , and it is found that the Fourier coefficients of $\sin m\phi$$(m=1\sim 4)$ and $\cos m’\emptyset(m’=1,3,4,5)$ ar negligible. Three sets of profile functionsof $[A_{0}, A_{1,2}A, B_{1}, B_{2}]$ can be determined from the observed data. It is found that they
coincide almost with each other. These profiles are shown in Figure 5.
Computation: A computer simulation [15] of the vortex collision is newly carried out inorder to have more realistic estimation of the acoustic emission than the previous one
potential [16]. The viscous incompressible[7], using the method of vorticity andvorticity equation is solved numericaly, together with the continuity equation, by a new
grid points. Boundary condition of zero-vorticity is imposed on thealgorism on
cubic bounding surface.
The isotropic component is proportional to the second time derivative of the kinetic
energy from (2) and (6), and the quadrupole components and higher modes are related
to the change of moments of vorticity distribution. Using the data from the simulaof (4) and (5). Thus we readily
andtion, we can calculate the tensors. It is remarkable that the main
obtain the main mode coefficientsmode amplitudes obtained from the computation are in quantitative agreement withthe observed ones, unlike the previous computation for fatter cores of vortex.
$\phi$$e$$ve\mathrm{c}t_{op}$$141^{3}$$Q_{*\mathrm{j}}.(t)$$Q_{-}i\mathrm{j}k(_{-}t)$$[\overline{A}_{\mathrm{O}},\tilde{A}_{1,2}\tilde{A},B_{1}, B_{2}]$108

Signifi cance of dominant wave modes:

Observed amplitudes$A_{1}$

and$A_{2}$of the two, while significant amplitude of the modeofis negligibly
is observed.
To see the significance of the wave signals, let us look into the distributions of the wave
andsource. The amplitudes
are represented by the integrals,$P_{3}^{2}\cos 2\phi$$\mathrm{s}\mathrm{m}\mathrm{a}\mathrm{U}$$P_{3}^{0}$$A_{1}(t)$$B_{1}(t)$,$a_{1}( \mathrm{y},t)=-\frac{1}{12\pi}y_{3}p3((3)\mathrm{y},t)$$A_{1}(t)= \int a_{1}(\mathrm{y},t)\mathrm{d}^{\epsilon}\mathrm{y}$$B_{1}(t)= \int b_{1}(\mathrm{y},t)\mathrm{d}^{\mathrm{s}_{\mathrm{y}}}$

where

,

$b_{1}( \mathrm{y},t)=\frac{1}{32\pi}(y_{3}^{2}-\frac{1}{5}y^{2})p\mathrm{s}^{4)}((\mathrm{y},t)$$p_{3}=(y\cross\omega)_{3}$.

andare considered to represent the source distributions
The integrands
is projectedof each wave mode. In figure 6, distribution of the source functionaxis) at three times: the time (positive
to the plane $(x_{1,s}x)$ (viewed from thepeak), where projectedpeak) and the time (negativepeak), the time . (positive
contours of the lso-vorticity surface are also plotted at the corresponding times. It isremarkable that the wave sources at the times and are localized in the upper left
and right regions where connection of vortex lines are observed. This is considered to
be a first identification of the acoustic signal related to the vortex-line reconnection.
$b_{1}(\mathrm{y},t)$$a_{1}(\mathrm{y},t)$$b_{1}(\mathrm{y}, t)$$\alpha$$x_{2}$$b$$A_{1}$$b$$B_{1}$$B_{1}$$\beta$$\beta$$\mathrm{r}e$

(a)

(b)

-vorticity surfaces andof (a) the
Figure 6. Projection diagrams in the planeand (see Fig. 5).
at the three times:(b) the source fimction$(x_{1}, x_{3})$$b_{1}(\mathrm{y},t)$$\mathrm{i}\mathrm{s}\mathrm{o}$$\alpha,$$b$$\beta$5. Interaction of a Vortex Ring with a Shock Wave
Problems of strong interaction between a rotational motion and compressive waves
are suited for computational study. Recently, experimental investigation is made for an.axisymmetric problem of strong shock-vortex interaction [17], and computer simulations
show significant nonlinear effects in the interaction of a shock wave with a vortex. Thissection is a brief account of a computational study by Takayama et al. [18] to simulate
the experiment.

109

The computer simulation is carried out by discretizing.the axisymmetric forms ofthe coservation laws of mass, momentum and energy for a VISCOUS and heat-conducting
gas model supplemented by the species conservation equation. Time integration of
the hyperbolic set of partial differential equations is made by the explicit second-order
capturing FCT smoothing techniquemethod, com.plemented by the[19]. Initially a vortex nng is generated in the computational domain by driving a
shock wave of nitrogen gas (wiht Mach number 1.4) through a small circular orifice
into a cylindrical tube of much larger dimension (Figure 7). Another plane shock wave
(driven at the other end of the tub ) is advancing&om the opposite direction, whichis shot at a later tine and impinging on the vortex with a shock disk, after interactingwith the first wave. The gaseous fluid in the cylindrical tube is as sumed to consist ofandnitrogen and oxygen, being at rest initiaUy, with the pressure , temperature
and $\gamma=1.3982$ , respectively. The
specific heat ratio set as. The results arenomalization parameters are given such that $l_{0}=1$ cm andobtained for the mesh size of $\triangle x/l_{0}=\triangle r/l_{0}=0.01$ and the starting time of the thefrom the ffist shock.
second shock wave was at
A remarkable property found in the simulation is that, during the passage of theshock wave over the vortex ring, the part of the wave propaga.ting through the inside of(Figure 8). Maximum
the ring-vortex is intensified spontaneously at a localized
pressure occurs inside the vortex and attains a high value, about several times that of
the impinging shock of Mach number of around 1.2, the vortex translation Mach number
being 0.60 (Figure 9).
From detailed study. of the structure of the interaction, it is found that there exists
a gas dynamic mechamsm of intensification of shock waves in the flow field in which
two step processes are observed. The double-step mechanism of the self-intensificationconsists of the first slight intensification of the impinging shock wave by the interaction
with the opposing flow due to the vortex, and the second intensification by the convergence of outer diffract shock wave around the vortex on the central axis at the backof the vortex ring. A possible application of the present mechanism is the spontaneous
ignition of a gas mixture with a reactive component by a shock wave.
$\dot{\mathrm{s}}\mathrm{h}\overline{\mathrm{o}}\acute{\mathrm{C}}\mathrm{k}$$\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{c}_{\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{C}}$$e$$p_{0}$$p_{0}=101.3\mathrm{k}\mathrm{P}\mathrm{a},$$\gamma$$T_{0}$$T_{\mathrm{O}}=300\mathrm{K}$$t_{0}=1\mu \mathrm{s}\mathrm{e}\mathrm{c}$$t_{2}/t_{\mathrm{O}}=108$$\mathrm{r}\mathrm{e}\mathrm{g}_{\mathrm{l}\mathrm{o}\mathrm{n}}$$e\mathrm{d}$Acknowledgement: The author wishes to gratefully thank Dr. S. Adachi and Prof.
K. Ishii and also Profs. T. Minota and F. Takayama for collaborative work on various
aspects of the present subjects.

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11 T. Kambe, J. Sound Vib. 95 (1984) 351.
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110. Liu, Computers Fluids 22 (1993), 589.
16 K. Ishii, K. Kuwahara and17 T. Minota, Fluid $Dyn$ . Res. 12 (1993), 335.
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1993; Elsevier), to appear in 1995.
$\mathrm{C}.\mathrm{H}$$\mathcal{B}$$|_{\mathrm{D}/}\text{し}$Figure 7. Computational domain and setup.

Figure 9. Perspective pressure plot
at $t/_{t_{2}=}153.32$ .

$t/t_{0}=$137.31$t/t_{0}=$148.45$t/t_{0}=$153.78Figure 8. Pressure contours at three times for $M_{2}=1.225$ .