J. Plasma Fusion Res. SERIES, Vol.2 (1999) 339-342

Automatic Ghase of Low-Pressure vortex Axes
ADACHI Takahiro*, MIURA Hideakir and KIDA Shigeol
Doshisha IJniv., Kyotanabe, Kyoto 610-0321, JapantNational Institute for Fusion Science Toki, Gifu 509-5292' Japan(Received: 18 December 1998 / Accepted: 25 Februaty 1999)

Abstract
A new method by which the motion of the axis of a low-pressure vortex is chased automatically isthedeveloped for the purpose of understanding of vortical dynamics. An equation which describes
flowtoaapplied
ismethodThisnumerically.
solvedandtemporal evolution of a vortex axis is derived
field composed of two interacting pairs of anti-parallel vortex tubes to visualize their reconnection
process.

Keywords:
automatic chase, low-pressure vortex, vortex axis, reconnectionof the three eigenvalues are real. Then, we assume'

1. lntroduction

without loss of generality, that 2.(r) > L(2) > ,l,(3). The axisof a low-pressure vortex is defined as such a line on
which the pressure p takes a local minimum in crosssections normal to the third eigenvector ef;t, of H.6. lf
the pressure-gradient is normalized as I Vl | = 1, thenwe have the identity,

It is commonly recognized that coherent structures
such as vortex tubes and layers exist in fully developed
turbulence and believed that they may play essential
roles in turbulence dynamics. In order to investigate
their dynamics, it is necessary to develop an efficientmethod to educe their structure and to trace theirtemporal evolution in complicated flows. In this paper,
we present an automatic method for chasing a lowpressure vortex [1,2]. This method is successfully
applied to a flow field composed of two pairs of anti--aDgt^'' ="=:dx".

a)on the vortex axis, where p denotes the normalizedpressure. The direction of el3) is chosen so that it makes

parallel vortices placed orthogonally to each other.

an acute angle with afllax,.

The time derivative DlDt taken as moving with avortex axis is expressed as

2. Formulation
We derive here an equation which describes thetemporal evolution of the axis of a low-pressure vortex.
Let us define the eigenvalues 2(') (i = l, 2, 3) of pressure
Hessian H"u = d2pldx"dx6 (a, b = 1,2, 3) and theD -a *.dx, a
Dt: at d, d*, 'where xo = xo?) represents the position of the axis andd,r,/dr its translational velocity. DifferentiatinC eg. Q)

associated eigenvectors el') bY

H*"l' =f(i) tQt, e! e! = 1,(3)(l )with respect to time and using eq. (3), we obtain thewhere the summation convention is assumed forequation for the translational velocity as

repeated subscripts. Since F1,6 is a symmetric tensor, all
C o rre spondin g autho r' s e -mail : adach @ me. s aga- u. ac.j p*Present address: OTEC I'ab. Saga Univ., Saga 840-8502, Japan.
339@1999 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchAdachi T. et al., Aulomatic Chase of Low-pressure Vortex Axesaro (-----ald,l'I af 6,,;,,\ --=Er .
-L-H* -1l=i dtdx"
dt \dx,Cil,+At=@r*Al(4)As will be shown in Appendix, the second-order tensordef)lax,is written as( | aH^, ,,, ,.,\ ,i\di ?-l 1,t, Jr, Af e'l,'e'i'1e';'' (5)\-larf' S=The temporal evolution of a vortex axis is obtained byCCsolving eg. @), in which the translational velocity isexpressed only in terms of information of the pressure

Fig. 1 Sets of points in each step in the method ofautomatic chase of vortex axes. I, Ct, a vortex

field.

axis at t;2, Pt'^t, predicted points at t+ Lt;O,C't + ^t, candidate points; O-O-O-...-O-O, C, * o,,union of C't+^r and extra points; Ct*^t, vertex axis
at f + Atwhich has evolved from C,.

3. Numerical Method
Here, we take two snapshots at / and t + At, say,from a time series of the pressure field, which is
obtained by direct numerical simulation of the Navier-points until there are no more candidates within two grid

sizes in each of the three coordinate axes. Here,Stokes equation. Then we describe how to predict a newposition at t + At of a vortex axis which was picked up
arbitiarily at / by the use of eq. (4).

candidate points are such points on which L@ > O and aswirl condition are satisfied [1,2] (see Fig. l).
[2] Predict the position at t + At of each candidate pointof the vortex axis chosen in step [l] by the use of eq. (4)for the translational velocity. A set of these new

3.1 Finite-difference approximation of time
derivative
The time derivative in eq. (4) is approximated bypositions is denoted by P'* o'.

the first-order finite difference in terms of field[3] Find first a grid point which is closesr to eachelement in P' * at, then a candidate point which is locatedquantities at successive times r and t + /t asa'f

df'*o'/ dx" -df'l dx,

W=Nae'j)- el)twithin one grid-size from it in each of the threecoordinate axes, and which satisfies those conditions
given in step []. A set of these new candidate points is(6)denoted by (' t + dt .

a)[4] Connect each element in C't* ^t with its nearestneighbors if they are located within two grid-sizes inwherel and eG)'denote the normalized pressure and theeach of the three coordinate axes, and finish it

il =e(:)t+^Ia,tthird eigenvector at /. The difference of eqs. (6) and (7)yields the righrhand side of eq. (4) asa,i drf;, ai,*o, ldxo - s{r,*o,- a, ='
Arar,
Ar

otherwise. In this process, a point which does not belong

to C't * o'sometime be connected. The union of^uy
C't + at and these extra candidate points is denoted byC'Dt + At. The length of resultant Cxt + at is in general(8)

different from that of C' because separation andwhere use has been made of the relation.

ai',1,'=-,€.-gXareconnection of vortex axes may occur.

[5] Pick up only such sequences of points of C"t + atthat include at least two elements belonging to C't + at.
This condition should be introduced so as to exclude afalse termination caused by unavoidable numericalerrors. This condition is satisfied for the left one of thetwo parts of Cil' + ^t shown in Fig. I but not for the right.
Therefore, the latter is discarded, while the former isretained and identified as a vortex axis Ct * d' which has
evolved from C'.

(e)which follows from eq. (2).

3.2 Automatic chase
An automatic chase of vortex axes from t to t + At
is carried out according to the following steps:[1] At time /, choose a vortex axis C'which isconstructed by connecting nearest-neighboring candidate340

Adachi T. et al., Aulomatic Chase of Low-Pressure Vortex Axes(")

4. Application to a Pair of Anti-Parallel
Vortices
In order to examine the vortex interaction as wellas to check our new method described in the preceding
section, the method of automatic chase of vortex axes isapplied to a flow field which is obtained by the direct
numerical simulation of the Navier-Stokes equation. The
Fourier spectral method with resolution of 643 is used ina periodic cube of period2tr.
As the initial condition we take two pairs of antiparallel vortex tubes which are placed orthogonally toeach other. The vorticity distribution across each tube isgiven by the Gaussian distribution aso, = a1.,expl_ a{(y - yJ2 + (z + z)z ll- aroexp[- a{ (y +yo )2 + (z + z)2 l],ay - - aoexp[- a {(x - xn)' + (z - z)' l)+ o6exp[- a {(x + x)' + (z - z)' ll,a,=o(b)(10)

where, a = 50, foo = I00/n, Jo = ),o = 3Ax, and L.x = 2ttl
64 is the grid size. The circulation around each tube is f
= aoftla = 2. The vortex Reynolds number is then given

by Re7= lly = Asrylav = 200, where y = 0.01 is thekinematic viscosity of fluid. The motion of low-pressurevortex axes is traced by the present method of automaticchase every At = 0.1 time unit.
The temporal evolution of the vortex axes at t =0,1,5 is represented in Figs. 2(a)-(c). In each figure, we

display the loci of the sectionally local minimum ofpressure together with the edge of the periodic cube.
Two pairs of vortices are placed orthogonally to eachother at the initial instant (Fig. 2(a)). Each pair ofvortices travel perpendicularly to the other by their own
induced velocity and the two pairs approach each other.
At their closest positions, each pair is bended by therepulsive velocity induced by the other as shown in Fig.
2(b). In the meanwhile, parts of two vortices which
belong to different pairs approach each other withaligning their vorticity anti-parallel. Cancellation of antiparallel vorticity by viscous diffusion then becomes
effective, which causes vortex reconnections to change
topology of the vortex axes (Fig. 2(c)). As a result, thetwo pairs of anti-parallel vortices pass through eachother and are cut into four separate vortices (cf. Figs.

(c)2(a) and (c)).

Fig.2 Interaction of two pairs of anti-parallel vortex
tubes placed orthogonally to each other. (a) f= 0,5. Conclusions(b) t = 1, (c) t = 5. Double arrows denote theWe have derived equations which describe thedi rection of vorticitv.

34r

Adachi T. et al., Automatic Chase of Low-Pressure Vortex Axes

temporal evolution of the axis of a low-pressure vortex

orthogonality between dep/&, and e[). A differentiationand developed a new method which chases its axisof the first of eq. (l) with respect to rc, on the otherautomatically. It was successfully applied to a system ofhand, leads totwo pairs of anti-parallel vortex tubes to visualize adH^^
de'j.' aL(i\ -.de'l'e::t + l"' TL .
*dx, e'i'" + H,,n"" =:dx, - Qxc "dx.
-sequence of the vortex reconnections clearly.

References

On substitution of eq. ( 12) into eq. ( I 3), we obtaintll H.Miura and S. Kida, J. Phys. Soc. Jpn. 66,1331

Aet.i),i, r,r(Hob- A'i\6"b);; =e'^',dH^n e','e'j'(1997).

jfI2l S. Kida and H. Miura, J. Phys. Soc. Jpn. 67,2166(l998).

Y",,

gives

follows from definition (l) thatde',j'aH-^
17'j'-7'i'1at;'a?=- r':';; e'it, G+j), (15)(l l)Lo) - elt H"o eyt,which, after differentiating with respect to x., leads towhere e!)e[i) = 6, has been used. Since 0ef;)10x, is
orthogonal to e!''), then def)10x" should be written interms of the other two components (eq. 115';'1 of ef) (j +i ). Thus, we finally find thatalti\ del,detl,. aH^,-t--"" e'i,t + e(:\ H-^sxt.= =Lox,H.^ eti' + e')' ox(.
"" --!
dx.
0e1'
aH-^
de"',.
.
.
^"

t_\,

-^,,,dHou i,=e'i'Af e'i',t(14)

An inner product of eq. (14) and e!) (j = l,2,3, j + i)

Appendix
We derive here expression (5) of de(j)ldx.. lte,.l\ + e,)t :"= 7t') dxrdxr""dx.

(13)a"!'_ $ I

et:t + 7(i\ et:t -::ordH^n ,,, ,,,'l-,,,(16)a;- = ?- | 1,, Jriff- ""1"'';'1";"

(12)

I

iyril\'-

as long as LG) + 1(i\ fot i +where use has been made of eq. (l) and the342

j.