Core dynamics on a vortex column

Fluid Dynamics North-Holland

Research

FLUID DUI!AMKS RESEARCH

13 (1994) 1-37

Core dynamics on a vortex column Mogens V. Melander

Fazle Hussain qf Mechanical Engineering, University of Houston, Houston, TX, USA

Department Received Accepted

13 January 1993 7 June 1993

Abstract. Through a detailed study of vortex core dynamics we show that variations in core size play a crucial role in the dynamics in 3D-vortex flows and that they in general cannot be ignored as has previously been assumed. To arrive at this conclusion we examine the core dynamics of an isolated axisymmetric vortex column with a nonuniform core. In this simple and idealized configuration, the effects of the core dynamics are seen most clearly because they are decoupled from other effects, such as interactions with other vortices and selfinduced displacement of the axis. We show analytically that the core dynamics results from a distinct physical mechanism, namely differential rotation along axisymmetric vortex surfaces. Furthermore, we show that the core dynamics is neither pure wavemotion nor pure mass transport, but a combination of both. We show and explain why core dynamics is highly Re-sensitive. Moreover, we find that the core variationswhich are likely in practical flow situations-do not disappear by inviscid effects. By examining viscous dissipation we find that core dynamics results in a significantly higher dissipation rate and that dissipation is the only effect that reduces core size variations. In fact, we find that the frequency of the core size oscillations increases with Re, but to a finite limit as Re -+ cc A striking, newly observed feature resulting from core dynamics is the appearance and disappearance of low enstrophy pockets inside the vortex.

1. Intr~u~tion

Turbulent shear flows are known to contain large-scale organized vertical structures-popularly called coherent structures. These structures dominate turbulence transport, and they are crucial to the understanding and modeling of turbulence physics, as well as to controlling turbulence phenomena like mixing, heat transfer, combustion, drag and aerodynamic noise generation. Currently, two principal obstacles inhibit the application of vortex dynamics to the study of coherent structures. The first of these two is that the influence of a vortex’s internal vorticity distribution on the dynamics of the vortex is not well understood. This is the problem of what we call “vortex core dynamics”. The second is that the structure and dynamics of a large scale vortex or coherent structure in a turbulent environment are not known. This is the problem of large scale-fine scale interaction. Virtually all studies have failed to address, let alone tackle, these two critical aspects of vortex dynamics. Coherent structures are vertical and carry significant circulation; thus we are of the opinion that their dynamics is best examined in the framework of vortex/vorticity dynamics; see Bridges et al. (1989), and Melander et al. (1991). In this way the velocity field induced by a coherent C~rTes~onde~ce Road, Houston,

to: Prof. F. Hussain, Department TX 77204-4792, USA.

0169-5983/94/$10.25 SSDI

0169-5983(93)E0041-I

0

1994 - The Japan

of M~hanical

Engineering,

Society of Fluid Mechanics.

University

All rights reserved

of Houston,

4800 Calhoun

2

M. V. Melander, F. Hussain / Core dynamics

structure is readily expressed in terms of a Biot-Savart integral. For the far field this integral can be simplified tremendously by introducing a perturbation expansion in a small parameter, namely a characteristic length scale of the structure divided by the distance to the coherent structure. The coefficients in this expansion involve physical space moments of the vorticity distribution within the coherent structure. As for the far field, only the first few of the terms in this expansion are of any consequence. The details of the internal vorticity distribution within the coherent structure as well as any fine scale structures in its immediate vicinity are not reflected directly in the first few terms of the expansion. This observation is in fact the key idea behind vortex filament methods. However, the detailed nature of the local vorticity distribution has an indirect influence on the far field through the time evolution of the coherent structure. That is, the motion and deformation of the large scale coherent structure may depend sensitively on the internal and nearby vertical structures. There are two issues to address here. One is the effect of core or internal dynamics on the motion of a vortex. The other is the interaction between large and small scales (i.e., coherent structure and nearby fine scale vertical structures), in particular the effect on the large scales. To understand and model these effects constitute a difficult and fundamental issue in vortex/vorticity dynamics. In this paper we put the first of these issues into an idealized setting by examining the dynamics of an axisymmetric vertical structure in an incompressible, viscous, constant property fluid. This is a useful simplification because the core dynamics decouples from the motion of the vortex’s axis. Note that, interactions between two structures of comparable scales, such as pairing and reconnection, are beyond the scope of this paper and has been treated elsewhere; e.g., Melander and Hussain (1988, 1989, 1990, 1991) Kida et al. (1989), Hussain and Zaman (1980) Husain and Hussain (1991). The internal dynamics of a coherent structure is caused by vortex lines which have torsion and thus are locally helical. Such helical vortex lines are typical for vortices in real 3D flows, for the deformation of a vortex under the influence of other vortices is sure to bend any straight vortex line. Likewise, a vortex typically acquires a nonuniform core diameter when advected in the far field of other vortices. There is generally a connection between helical vortex lines and the nonuniform core diameter of a 3D-vortex. That is, a vortex which has helical vortex lines exhibits an axial flow resulting in a nonuniform core size-provided that the axial flow is not uniform along the axis of the vortex. Vice versa, a vortex with a nonuniform core develops helical vortex lines due to different rotation rates along the axis of the vortex. Moreover, helical vortex lines can arise when the vorticity peak is not centered in the vortex core-a very likely situation in a turbulent flow. The importance of internal core dynamics is easily seen from the local induction approximation-which provides an expression for the selfinduced velocity of a curved vortex segment-because here the core size enters as a parameter. Owing to the difficulty involved in modeling the evolution of the vortex core (itself a vaguely defined concept in a viscous flow), the problem is often assumed away. For example, many theoretical studies of organized structures in turbulent flows (such as hairpins in boundary layers and homogeneous shear flows) have modeled them with a uniform core diameter, and have presumed that a nonuniform core has negligible effect on the flow evolution; e.g., Moore and Saffman (1972). Our interest in internal core dynamics was first spurred by the phenomena that we observed in the late stages of vortex reconnection (numerically studied by Melander and Hussain (1988, 1989, 1990, 1991) hereinafter abbreviated as MH). During the process, the reconnected vortex lines accumulate in “bridges” (fig. 1) located ahead of and aligned orthogonal to the annihilating pair of initially anti-parallel vortices (i.e., vortex dipole). At the end of the reconnection process, the bridges are in fact the reconnected vortices, as sketched in fig. 1. As a result of nonuniform accumulation of reconnected vortex lines in the bridges and vortex stretching experienced by the bridges during the reconnection, they have significant axial variations in their vorticity distribution. The vorticity magnitude is significantly higher where the reconnection has taken place than elsewhere (fig. 1). Also, the shape of the vortex core varies from crescent-shaped in the

M. V. Melander, F. Hussain / Core dynamics

Cross

section

A

Cross section

B

Fig. 1. Sketch of two vortices after reconnection has occurred. The figure shows an iso-vorticity surface and contours of the vorticity magnitude in two cross sections. The reconnected vortices are labeled “bridges” and the non-reconnected remnants of the original vortices are called “threads” in accordance with the terminology introduced in MH (1988, 1989, 1990). The important point to note here is the axial variation in the vorticity distribution in the reconnected vortices (bridges), as exemplified by the two cross sections.

reconnection region (cross section A in fig. 1) to circular away from it (cross section B). This axial variation in vorticity magnitude and core shape results in helical vortex lines and hence induces an axial flow. Directed toward the reconnection region, this axial flow appears to serve as an inviscid vorticity “smoothing mechanism” that reduces the axial variation of the vorticity magnitude; see MH (1990), and Moore and Saffman (1972). In order to focus specifically on this mechanism we have idealized the problem in the present study to that of an axisymmetric vortex column with an axial variation in the core size. This way, we eliminated two effects which complicate the evolution of reconnected vortices that we studied: namely, the curvature of the vortices and the induction by the two ‘threads” shown in fig. 1. These threads are the remnants or non-reconnected parts of the original vortices, and they remain as a part of the flow for a long time (MH, 1988, 1989). The threads induce an axial flow in the reconnected vortices. This flow is directed away from the reconnection region. (As a consequence, we have both positive and negative enstrophy productions within the same cross section of the bridges (MH, 1990).) We expect to shed some light on the essential hydrodynamical mechanism through the idealized problem we treat in this paper. Vortex core dynamics is also related to wave motion along a vortex. There are basically two kinds of such waves: axisymmetric and bending waves. In the former case, the waves are axial vorticity variations within the vortex core; such variations inevitably lead to locally helical vortex lines and hence an induced axial flow. In the case of bending waves, the axis of the vortex itself is also locally helical. Waves on vortices have been investigated by numerous authors. Hasimoto (1972) studied bending waves by means of the local induction approximation. He showed that the evolution equations thereby reduce to the nonlinear Schrodinger equation, and hence he found soliton solutions. However, the vortex core size was constant and the effects of vortex stretching and core dynamics were ignored. Small amplitude long axisymmetric waves have been studied by Pritchard (1970) and Leibovich (1970). In this limit the latter author finds that the amplitude obeys the KdV-equation and hence finds solitary waves. Large amplitude solitary waves of the axisymmetric type have been found by Leibovich and Kribus (1990) through the method of numerical continuation.

4

M.V. Melander,

2. The governing equations for axisymmetric

F. Hussain

J Core dynamics

vortex dynamics

We use the cylindrical coordinates (r*, 8, z*); throughout the paper * denotes dimensional quantities. Let the z-axis be the axis of the axisymmetric vortex, and let 8 increase in the direction directions are of the swirl; see fig. 2. The velocity components of g* along these coordinate designated (u*, u*, w*). In order to make the equation dimensionless we introduce characteristic time and length scales which we derive from the circulation r* and angular impulse -M*. That is, p

zz

sg* ’

gZ

(2.1)

dx* dy*

R2

and -M*=+

L* x (r* x w*)dx*

dy* dz*,

(2.2)

s R3

where Q* = V x g*. Because we consider an infinitely long vortex, the magnitude of M* is infinite. Therefore, we instead consider the axial average value per unit length, defined as-

n;i*

E

-L

lim

_

.+,4L

s(S L

-L

(x*’ + y*‘)g* .ezdx* dy*

(2.3)

R=

V-profile d

Vortex line

mmetric

vortex

surface

meridional \, streamline

Fig. 2. Definition sketch and cartoon of the dynamics at an early time (t z 1). The shaded surface is an axisymmetric vortex surface (5 = vr = constant) with a 90” wedge cut away. On this surface seven vortex lines are shown, these lines all have exactly the same shape-that is, appropriate rotations about the z-axis will make them fall on top of one another. Also shown are radial profiles of the azimuthal velocity u and the axial velocity w at various stations along the axis. Streamlines (IJ = constant) of the meridional flow are shown in one B = constant plane. The instantaneous 3D streamlines of the total flow field can be obtained by combining the meridional flow with the swirl u. One would thereby obtain streamlines which lie on axisymmetric stream surfaces, that are surfaces of revolution of rj = constant.

M.V. Melander, F. Hussain / Core dynamics

5

r* and k* have dimensions (m’/s) and (m4/s), respectively, thereby providing the units t; L”for time and length 2 = P/t = - sh2*/P,

z z L*/L = ( - 8@*/r)“,‘,

(2.4)

where t, L are the dimensionless versions oft* and L*. In these units we have r = 1 and $i = l/8. Moreover, Re = %/2v = T*/v, which is the Reynolds number traditionally used in vortex dynamics. The constant factor eight in (2.4) is chosen such that 2 is the vortex diameter at one standard deviation in a Lamb vortex (i.e., Gaussian vorticity profile). Consequently, i is l/7? x 0.10 of the turnaround time for a particle at the center of the Lamb vortex. Following Batchelor (1967), the cylindrical symmetry allows us to write the governing equations in terms of q = q/r, 5 E UY and the streamfunction II/for the meridional flow (u, w). We have in nondimensional form: u=--

D5

1w

WC

---

r az'

i

1w

(2.5)

r ar’

a25 iag a25

(

i3r2 Dt = Re --;z+aZ”

(2.7)

’ >

)

(2.8)

and for a passive scalar c, (2.9) Here D/Dt z a/at + u a/ar + w a/az, which represents the derivative following a particle in the meridional flow. SC is the Schmidt number v/D, where v is the kinematic viscosity, and D is the scalar diffusivity. Equations (2.5)-(2.9) clearly reflect the delicate nature of axisymmetric vortex dynamics with swirl. Surfaces of constant 5 are axisymmetric vortex surfaces (i.e., w, = - l/ra&jaz, w, = l/r a
3. Kinematics of the coupling between swirl and meridional flow In order to understand the physical meaning of the coupling term, we consider two material particles A and B on the same axisymmetric vortex surface < = to (Fig. 3). The angular velocity around the z-axis of particle A may be expressed in terms of 5 = rv,

4z.4,r.4) fj*=--_=_ IA

to ri

.

(3.1)

M.V. Melander, F. Hussain f Core dynumics

Fig. 3. Schematic for the physical interpretation of the coupling term. Shown are two material particles A and B located on the same vortex surface (5 = 5s). Because of the axisymmetry all particles on the circle z = z,,, r = rO have the same angular velocity. For simplicity of the figure, we therefore consider two particles in the same meridional plane.

Likewise,

the angular

velocity

of particle

B is

to

4ZB,IB) &-AL

= &,I

C~A + (TB- rA)12

4

rB

+ 26 + d2)-‘,

ri

(3.2)

where 6 = (rg - r-*)/r*. Using fig. 3 we can relate rB - rA to the vorticity components employing Taylor expansion of r along the vortex surface 5 = to on which A and B lie

rB

-

rA

In the following,

%@A, rA)

=

o,(zA,

rA)

tzB

-

zA)

+

OkB - zAj21.

=-

au 250 g w

r3

A

- l] = - ?[I

(zA

rA)

z

AZ+ 0 C(W’l =

Since 5 = ur = fh2, we obtain

= &(zA,rA) A

25,

z

az

(zA,rA)

AZ + O[(Az)‘]

Wz(ZA3rA)

a(@) A

AZ + O[(AZ)~] = o ;‘:* ;A) Az + 0

dzATr,4)

--i ap

+ O(S)]

>

aZ

= 24,

from (3.1)-(3.3)

a(0) (&T*)

ad

r4 a2

(3.3)

we let AZ E zg - zA. Since o, = - au/az, we obtain

6, - 6, = $ [(I + 26 + a2)-’

by

*

the following

expression

C(W21.

(3.4)

2

for the coupling

term in (2.8)

& - 6,

lim -. B+A

ZB -

zA

(3.5)

M.V. Melander, F. Hussain / Core dynamics

7

Note that the above limit does not equal ad/&-; instead, the limit represents unit length along the axisymmetric vortex tube r = to. That is,

the change in d per

C(~B - ~A)/(zB - zA)l

lim

B-A

equals the difirential rotation along an axisymmetric vortex surface. It is this differential rotation that makes the vortex lines locally helical and thus generates azimuthal vorticity oe = rn. Our result (3.5) states that the material rate of n-generation (i.e., the coupling term of (2.8)) at a given point equals the axial vorticity times the differential rotation along the axisymmetric vortex surface passing through the point. From (2.8) we observe that where the axisymmetric vortex tubes converge in the z-direction (i.e., a[‘/az > 0) there is material generation of positive q. Likewise, there is material generation of negative v] when the vortex tubes diverge (i.e., ag2/az < 0) in the z-direction. Note that the instantaneous value of v] does not influence the instantaneous material generation of ‘I, a fact that will prove to be useful for understanding the q-transport.

4. Formulas for dissipation and energy exchange between swirl and meridional flow In order to define a finite energy per wavelength 4(r, z) for the swirling flow. We define C$uniquely

ad

z

fj N Zf;llnr + O(r-‘),

= 4

Note that r = 1 in non-dimensional 1 @=%lnr-

as

of the vortex we introduce a stream function by the following two conditions r+

co.

units. It follows that the explicit

expression

m 2x5 - 1 pdp. sI 27V

for 4 is

(4.2)

Since the volume integral of v2 diverges logarithmically as r+ CC it is necessary to use 4 to introduce the excess energy in a wavelength A of the vortex (this terminology is commonly used in 2D vortex dynamics)

ss 1

E=n.

co

(u2 -

c#m, + w')

r dr dz.

(4.3)

0 0

This integral has a finite value and is obtained from the usual energy integral through integration by parts. Moreover, E is conserved in the absence of viscosity (easily seen by considering a cylinder of a very large radius R). The total energy dissipation per wavelength equals the dissipation of the excess energy and can be expressed in terms of a dissipation function e(r, 0, z), dE -_3 dt

1 - 2vlc ss0

Since the flow is axisymmetric

00 w2rdrdz 0

1 = - 2vrc

we can express

ss0

00 erdrdz. 0

(4.4)

E as the sum of four contributions, (4.5)

M. V. Melander, F. Hussain J Core dynamics

8

where (4.6)

(4.7)

(43)

(4.9) The first two terms in (4.5) represent contributions from the swirl, while the last two are from the meridional flow. When (4.5) is integrated over space we express the result in the form dE/dt = & + -Ijz + .& + &. s1 is the only non-vanishing contribution for a rectilinear vortex without axial variations. We now consider a vortex column which is periodic with period A in the axial direction. When integrated over space we obtain a rate of viscous energy decay, which is - v times the axial enstrophy SF’, per wavelength, i.e.,

= - 2vlt

ozrdrdz

+ 4vrt

‘[i?]:dz s0

A 00 = - 2vrc ozrdrdz ss0 0

= - vZZ. (4.10)

~z represents the dissipation due to the instantaneous axial convergence and divergence of the vortex tubes, regardless of any meridional flow. Since au/az = - w,, we obtain by integration over space,

I?*= - vzzr,

(4.11)

or - v times the radial enstrophy per wavelength. .s3 is the dissipation caused by extensional strain in the meridional flow. It is the only one of the four terms in (4.5) that is non-vanishing on the axis of the vortex. This follows simply from the axisymmetric geometry, which implies that u = O(r), II = O(r), w = O(1) near the axis. I+ vanishes where the streamlines are parallel to the axis, but becomes significant where the meridional flow converges or diverges on the axis. sq is the dissipation by shear in the meridional flow. This term gives a contribution in a cylindrical shell around the axis. From eqs. (4.4), (4.10) and (4.11) it follows that sj and .sq together yield a rate of viscous energy change proportional to the azimuthal enstrophy per wavelength when integrated over space: _& + .$ = - vZ&.

(4.12)

In addition to the simple formulas above we find a revealing formula for the energy transfer between the azimuthal swirl and the meridional flow. For the sake of simplicity we first derive this

M.V. Melander, F. Hussain / Core dynamics

formula

for the inviscid

case:

dEe

dt==

--

“5D(5/r)r&.dz=

-27~

Dt

(4.13)

We recognize the second factor in the last integral viscous effects (4.13) becomes

as the coupling

term (3.5). When we also allow

(4.14)

which is the rate of change of excess energy per wavelength we have a similar formula for the meridional flow:

of the swirling flow. Correspondingly,

(4.15)

Note that the formulas (4.14) and (4.15) are valid for all axisymmetric central role of the coupling term (3.5).

vortices, and emphasize

the

5. Initial conditions and numerical method We use a 3D spectral (Galerkin) method with periodic boundary conditions and a fourth order predictor-corrector scheme for the time advancement to simulate all flows considered in this paper. Although this method is not efficient for axisymmetric flows, it has the advantage that the flow may be given non-axisymmetric 3D perturbations, such as fine-grain turbulence, without changing the numerical method. In a companion study (MH, 1993b) we examine precisely such a turbulent vortex. In all the cases that we simulate (see table 1) the initial vortex is the same, but the Reynolds number varies; for our main case (Ll) Re = 665.2. Simulations for other Reynolds numbers are used to show the Re-sensitivity of the dynamics. The turbulent vortex considered in the companion study (MH, 1993b) consists of a coherent and an incoherent part. The initial coherent part is precisely the vortex of our main laminar case Ll. Only the properties of the incoherent part of the flow, which is a homogeneous isotropic turbulent background flow, vary. Case Ll was selected so as to satisfy two criteria. The first required that the evolution of the laminar vortex could be well resolved at a low resolution of 643, so that additional resolution can be used for fine scales (in the turbulent case). The second criterion was that the vortex should be sufficiently viscous so that the simulation can be continued until the vortex relaxes to the final state- that is axially uniform without axial flow-thereby providing a simple basis for comparison with the turbulent cases. The vortex initially has a sinusoidal axial variation of the core size, and the axial vorticity profile (w,(r, oo, zo)) is the same everywhere along the axis, when scaled with the core size. The radial vorticity component is determined such that the vorticity field is divergence free (i.e., circulation is constant in z). No axial flow is present initially as the azimuthal vorticity component

M.V. Melander,

10

Table 1 Parameters

for the simulations

Case

N

Ll L2 L3 L4 L5 L6

64 64 64 64 64 64

(dimensional

1.563 15.126 30.252 60.505 121.01 a

F. Hussain / Core dynamics

units)

0.5 0.5 0.5 0.5 0.5 0.5

0.6 0.6 0.6 0.6 0.6 0.6

2x 2x 2x 2x 2x 2n

20 20 20 20 20 20

0.6875 0.6875 0.6875 0.6875 0.6875 0.6875

0.09395 0.09395 0.09395 0.09395 0.09395 0.09395

Here N is the grid resolution, Y is the kinematic viscosity, p is the relative amplitude of the sinusoidal core size perturbation (see eq. (5.2)), r$ is the unperturbed core radius, A.*is the wavlength of the sinusoidal core size perturbation, o0 is the peak vorticity of the unperturbed vortex column (see section 5 for further specification of the laminar vortex), 2 and 7 are characteristic units for length and time (see eq. (2.4)). ’ In this simulation an artificial viscosity of the form v4A2 is employed, with vq = 6.989 x 10-s. b i and 7 are calculated on the basis of r* = 0.6989~~(r~)’ and M’* = 0.1 147w~(r~)4; note that the factor in the formula for M’* differs from that in (5.6), this is because of averaging over axial variations.

Table 2 Parameters

for the simulations

(non-dimensional

units)

Case

N

Re

P

r0

i

%

r

IGi

Ll L2 L3 L4 L5 L6

64 64 64 64 64 64

665.2 332.6 166.3 83.15 41.58 a

0.5 0.5 0.5 0.5 0.5 0.5

0.8727 0.8727 0.8727 0.8727 0.8727 0.8727

9.139 9.139 9.139 9.139 9.139 9.139

1.8752 1.8752 1.8752 1.8752 1.8752 1.8752

1 1 1 1 1 1

-

See table

1

Table 3 Times for frames Frame

label

t* 0

t :

f” E i j k 1 m n 0

P q r S

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 10.0 12.0

t 0

5.322 10.644 15.966 21.288 26.610 31.932 37.254 42.576 47.898 53.220 58.542 63.864 69.186 74.508 79.829 85.152 106.440 127.728

0.125 0.125 0.125 0.125 0.125 0.125

M. V. Melander, F. Hussain / Core dynamics

is set to zero. Thus the initial

vorticity

distribution

11

is,

w:(r*,O,z*)=P

(5la)

f$(r,

8, z) = 0,

(5.lb)

w,*(r,

8, z) = P

(-L >’ r0 s(z)

w2;

(UC)

here the function P represents the axial vorticity, where r. is the unperturbed vortex’s support radius, and s(z) gives the shape of the axisymmetric vortex surfaces. We have, s(z) = [ 1 - p cos (2rIz/A)],

(5.2)

where 2 is the wavelength (equal to the computational box size), and p is the amplitude of the sinusoidal perturbation along the vortex. Among many reasonable vorticity profiles the Gaussian seems to be a natural choice as it remains self-similar under the influence of viscous diffusion and also is a C” function with moderately varying gradients. Thus it can be represented numerically with spectral accuracy and is well resolved even at rather coarse resolution. However, since a Gaussian vorticity distribution is physically established by viscous diffusion of vorticity, we can infer that such a distribution is only representative of “old” vortices. Vortices which are “young” with respect to a viscous timescale have a more concentrated vorticity distribution without necessarily being sharply peaked. For example, when a vortex forms through roll up of a shear layer, the diffusion is appreciable only in the core of the vortex. As a model for such “young” vortices we choose a vorticity distribution of compact support and essentially Gaussian in the core region. A radial profile P satisfying these two conditions is oo*exp[

P(i) =

- 4c2/(1 - [2)]exp(4[4

+ 4P + 4P),

o

0 I i < 1, 1 I i,

i.

where IX: is the peak vorticity.

P is nearly

a Gaussian

for [ 4 1 and also C” because

exp[ - 4c2/(1 - [“)I exp (4c4 + 4c6 + 45’) = exp [ - 41 2( 1 + ._, C i 2n)]. The circulation

of a vortex column

with vorticity

P([)[ d[ z 0.6989 o0 z 0.8899 - 2x0~ 50

For a rectilinear if*=

vortex

00 exp C - 4(021idi.

s0 with vorticity

(5.4)

profile (5.3) is:

m 1-* = 2lc

(5.3)

(5.5)

profile P

- 0.06469 u,;

the corresponding value for the Gaussian profile is (7c/32)w0 z 0.098 17 oO. The laminar now completely specified by the parameters given in table 1.

(5.6) vortex is

6. Decay of vortex core size variations The concept of a core size is useful for answering quantitative questions, such as how fast the vortex relaxes to a rectilinear shape (i.e., without any axial variations), at what rate the vortex core expands as a result of diffusion and coupling between swirling and meridional flows, and how it

M. V. Melander,

12

F. Hussain 1 Core dynamics

grows by entrainment in the presence of turbulence. To answer such questions we need precise definitions of both a vortex and its core size. Currently, there is no consensus on such definitions. In our present problem the axis of the vortex is fortunately obvious, namely the z-axis (note that in other problems it can be difficult if not impossible to even define a meaningful axis). We can therefore focus on how to define a core size. We will first discuss some of the problems associated with such definitions, and then we state our own definition. A definition of a vortex core must be sensitive to the nonlinear dynamics of the Navier-Stokes equations. Not all definitions have this property. To illustrate this point, consider a core size (T {of a straight vortex column) defined on the basis of circulation r and axial angular impulse 16 per unit length in the axial direction: 0(z, t) =fi(

- A(& t)/r)r’z.

(6.1)

Arguably, this appears to be the most obvious of all definitions. Herefis f= 2 yields the core radius of an equivalent Rankine vortex, andf‘= ~2 an equivalent Lamb (“Gaussian”) vortex (by equ~vulef~t vortices we mean angular impulse and circulation). The inadequacy of (6.1) becomes clear axial variations. By doing so we obtain an average core size 6: c2(t) = lim i L o’(z, t)dz = -.f2R;i(t)/II.-*= 2L s -i. here we have used 2 as given in dimensional

a constant form factor; gives the core radius of vortices with the same when we average over

= -.f2[fi(0)/T

form by (2.3) and the Poincare’

+ 2vt], identity

(6.2) (see Howard

(1957)) dti/dt

= - 2vT.

(6.3)

(Note that normally dM/dt = Q in 3D viscous flows; this conservation law, however, implicitly assumes that there is no vorticity at inanity, so that the total circulation in any plane vanishes identically. Our vortex violates that assumption.) Equation (6.2) shows that 5 depends only on a(O), r and v. Thus for all vortices with the same values of these parameters, we have o(t)/QO) = (1 + 2rru0t/Re)“2,

(6.4)

where w. is the initial peak vorticity. This result holds regardless of the nature of the dynamics-laminar or turbulent. Obviously, we need a more sensitive core definition. We base our definition, which exhibits the required sensitivity to the effects of both core dynamics and turbulence, on “the kinematic vorticity number” !I& = (W2/2SijSij)i’2 introduced by Truesdell (1954), where Sij is the symmetric strain rate tensor $(aui/axj + aUj/axi). Using the dimensionless number !I& we define a vortex as any connected spatial region where %B, > 1. A detailed motivation for this definition is presented in Appendix A. Let A@,, t) be the area of the local core size as vortex’s cross section in the plane z = zO, we then define the instantaneous 0(z, t) = [A(z, t)/n]“*,

(6.5)

provided that the region in the plane z = z0 where $%I&> 1 is simply connected (if this is not the case we let ~(z, t) be the radius of the outermost %I$ = 1 contour). Figure 4 shows the evolution of the vortex core size a(z, t) in terms of the contour ‘lf3, = 1 in a meridional plane. We make the following observations. The initially narrow part of the vortex quickly thickens and splits up into two wavelike packets (discussed in detail later), which move in opposite directions (fig. 4A). Simultaneously, the initially thick part of the vortex shrinks. When the two wave packets collide a “bubble” forms inside the vortex (fig. 4B). This bubble is characterized by deformation dominated motion (!I& < t), indicating that the vortex locally is hollow. The bubble is only a temporary phenomenon, but after its occurrence the core size

M. V. Melander, F. Hussain / Core dynamics

13

t

(4

(B)

Bubble Fig. 4. The evolution of the local vortex core size O(Z, t) in case Ll is shown in terms of the contour 2Bk = 1. For contour labelling refer to the times given in table 3. In panel (A) we have: (a) heavy solid line; (b) dashed line; (c)dotted line; (e) chain dashed line; (g) thin solid line. Continuing in panel (B): (i) dashed line; (1) dotted line; (q) chain dashed; in addition, (a) and (g) from panel (A) are overlaid in panel (B) for comparison.

variations happen to be less dramatic and are eventually damped out almost entirely (fig. 4B). The contour %Bmk = 1 is overlaid in most subsequent figures so as to identify the core size as well as to provide a common reference between figures. In order to distinguish between viscous and inviscid effects, we consider the evolution of the same laminar vortex for a range of different Reynolds numbers (i.e., cases Ll-L6). Particularly we wish to determine whether the vortex’s relaxation to a rectilinear shape is a viscous or an inviscid phenomenon. Our vortex definition (6.5) is a very useful tool for examining this question. Figure 5A shows the growth of the average (area weighted) core size 5 with time. For cases Ll-L5 the core size grows proportional to (~t)‘/~, which is typical for viscous vortices. Case L6 represents a closer approximation to an inviscid flow due to the use of the “super-viscosity” - v4A2, where vq is chosen as small as the resolution allows. 0 increases only little in this simulation, which we shall therefore think of as representative for higher Re. Figure 5B is more revealing as it presents a measure for the vortex’s instantaneous deviation from rectilinearity (i.e., no axial variations). The quantity displayed in this figure is the standard deviation of A(z, to) normalized by the instantaneous average area A(&,) z nc2(tO). Figure 5B clearly shows that the axial variations in the core size progressively disappear with increasing viscosity and thus strongly suggests that the damping is purely viscous; compare the curves for Ll-L6. Moreover, the decay of the axial variations has an interesting dependence on Re. Namely, the decay is not monotonic, but rather like a damped oscillation, where both amplitude and frequency depend on Re. The frequency increases significantly with Re and appears to have a finite high Re-limit. Case L6 is indicative of the frequency in this limit. Figure 5 leads us to conclude that the disappearance of the core size variations (and hence also the disappearance of core and axial dynamics) is largely a viscous phenomenon. The Re-dependence of the core dynamics highlights the difference between local (i.e., internal) vortex dynamics, where local self induction and vortex stretching are of utmost importance, and

M.V. Melander,

14

0.60

F. Hussain J Core dynamics

I (B)

0.40

0.20

0.00 0

50

100

t

Fig. 5. (A) The ‘average’ core size is shown as a function of time for cases Ll-L6 (tables 1 and 2), where the ‘average’ core size o(t) has been calculated from the z-average of the core area, i.e., TVJ(~)~= A(z, t) = na(z, t)‘, where u(z, t) is defined in (6.5); (B) this panel shows the corresponding evolution of the standard deviation of the core area normalized by the instantaneous average core area, i.e., ([A(z, t) - A(z, t)]2)0.5/A(z,t). Note that the curve markers (a, 0, etc.) are consistent between panels (A) and (B).

classical 3D vortex dynamics (filaments) where interaction takes place over distances which are large compared to the core size. In the latter case only the circulation of the vortex (Reindependent) and its position are of importance. For internal vortex dynamics, on the contrary, the vorticity magnitudes (Re-dependent) are of critical significance. Interestingly, this observation has relevance for vortex reconnection as well. Because, as we have pointed out on numerous , Fig. 6. The instantaneous vorticity magnitude for case Ll is shown at times given in table 3. In each frame the iso-vorticity contours are drawn at equal increments 60. In addition, the boundary of the vortex given by the contour ‘I& = 1 is overlaid in each frame. Note that the contour lines have not been subjected to artificial smoothing, but are drawn on a 256’-mesh through exact trigonometric interpolation based on the spectral coefficients of the vorticity field. Along the axis some local minima and maxima of cu are indicated so as to enable ease of dynamical interpretation. The highest contour level u,, and the contour increment Fw change from frame to frame as shown in the following list, where both nondimensional and dimensional [*] values are indicated: (a) w,, = 6.58 [70], 6w = 0.94 [lo]; (b) w,, = 2.44 [26], 60 = 0.188 [Z]; (c)-(f) o,, = 1.88 [20], 6w = 0.188 [2]; (g) CU,,= 1.50 [l6], Fw = 0.188 121; (h))(l) ws = 0.846 [9], SW = 0.094 [I]; (q) Qh = 0.517 15.51, 601 = 0.047 10.53; (s) w,, = 0.328 [3.5], 6w = 0.047 [0.5].

M. V. Melander,

F. Hussain / Core dynamics

15

!a > wave packet

0 (> C

max

I

min

(e1 (9 (!J)

max

(h)

-----Y/-O

max

max

0i min m

(1) (k)

1max

max

min

max

max

max

max

(1) Cs) max

0

L

S

-

max

max

M.V. Melander,

16

F. Hussain / Core dynamics

(b) (c) (e 1

(0 (9) (h) 0i

* (0 (k) (1)

m () 0

0 S

Fig. 7. The enstrophy production P, for case Ll at times given in table 3. P, is negative in the gray-shaded regions and positive elsewhere. Positive values of P, indicate vortex stretching, while negative values represent reduction of vorticity’ by compression. The boundary of the vortex as given by the contour E& = 1 is overlaid in each frame. The contour increment W, is constant within each frame, but varies between frames. The highest and lowest contour levels (Pohr Pwlf as well as the contour increment &PO9 are given in the following list as both non-dimensional and dimensional [*] values:

M. V. Melander,

occasions (e.g. Melander, period in fig. 5B).

F. Hussain

1989), the reconnection

/ Core dynamics

timescale

decreases

17

with increasing

Re (like the

7. Vorticity evolution The vigor of the core dynamics is clearly reflected in fig. 6, which shows the evolution of the vorticity magnitude for case Ll. Here we analyze the details of this figure and explain the role of the enstrophy production (P, E OiSijOj) shown in fig. 7. Later, in section 8, we focus on the details of fig. 7 and explain how P, is determined by the meridional flow. Figure 6 shows cross sections of iso-vorticity surfaces not to be confused with vortex surfaces (fig. 10). Let us explain the difference. On an iso-vorticity surface the vorticity magnitude 101 is constant. Hence the enstrophy density -w - cc)is also constant on such a surface. A vortex surface is a surface to which the vorticity vector is everywhere tangent, i.e., a surface of vortex lines, see fig. 2. Thus a vortex line may cross an iso-vorticity surface; generally this is the case in 3D flows. The initial enstrophy peak splits into two equal peaks of considerably smaller amplitude as seen by comparing figs. 6a and 6b and recalling the periodic boundary conditions. The inviscid mechanism that alters the vorticity magnitude of a material particle is the enstrophy production, which represents the production (annihilation) of vorticity by local stretching (compression) of infinitesimal vortex tubes. Both positive (stretching) and negative production (compression) is seen in fig. 7. The initial vorticity peak splits up because of a strong local axial compression (see the negative production contours in fig. 7). The vorticity transport seen in fig. 6 appears to take place in the form of “wave packets”, because mass transport (given by the streamfunction in fig. 9) and vorticity transport are in opposite directions (compare figs. 6 and 9). This is possible because of the enstrophy production Pm, which allows variations in vorticity magnitude to propagate along vortex lines. In this respect it is important to observe, using, for example, figs. 6b, c and 7b, c that P, changes sign near each wave packet; enstrophy is produced at the front of each packet and annihilated at the back of the packet, hence producing the packet’s motion. Note that for later frames, where the packets are in close proximity or they overlap, the situation is less clear. Here we face a dilemma, which is roughly similar to that occurring for the wave motions governed by the linear 1D wave equation (a”4/at’ = CI’~/&‘): namely, are the packets reflected off each other when they collide (fig. 6f) or do we have a superposition of packets moving in opposite directions? For the linear lD-wave equation, the method of characteristics or d’Alembert formula elegantly resolves this problem. However, our present problem is nonlinear and the mathematical extraction of wave packets moving in opposite directions is not simple. In a separate study (MH, 1993a) we show how these wavepackets are readily separated by the “complex helical wave decomposition”.

(a) P, = 0; (b) Pwh= 0.497 [600], P,, = - 1.161 [ - 14001, 6P, = 0.166 [200]; (c) Pwh= 0.248 [300], P,, = - 0.498 [ - 6001, U’,,, = 0.0829 [lOO]; (e) Poh = 0.248 [300], P,, = - 0.207 [ - 2501, 6P, = 0.0415 [SO]; (f) Pm,,= 0.124 [lSO], P,, = - 0.166 [ - 2003, 6P, = 0.0207 [25]; (g) Pwh= 0.0497 [60], F’,, = - 0.149 [ - 1801, FP, = 0.0166 [20]; (h) Pwh= 0.0332 [40], P,, = - 0.116 [ - 1401, 6P,, = 0.0166 [20]; (i) Pwh= 0.0248 [30], P,, = - 0.0248 [ - 301, 6P, = 0.00415 [5]; (j) P,,,,, = 0.0207 [25], Pm, = - 0.0207 [ - 251, 6P, = 0.00415 [S]; (k) Poh = 0.0186 [22.5], P,, = - 0.0166 [ - 201, FP, = 0.00207 [2.5]; (1) Pwh= 0.0116 [14], Poh = 0.0116 [ - 143, 6P, = 0.00166 [2], (m) Poh = 0.00663 [8]. P,, = - 0.00995 [ - 121, 6P, = 0.00166 [Z-j; (0) P”,, = 0.00663 [8], P,, = - 0.0116 [ - 141, 6P, = 0.00166 [2]; (q) Pa,,, = 0.00332 [4], P,, = - 0.00498 [ - 63, 6P, = 0.00083 [l]; (s) Pm,,= 0.00145 [1.75], P,, = - 0.00145 [ - 1.75],6P, = 0.00021 [0.25]. The conversion formula used here is P, = P:. f3 (see eq. (2.4)). Note the dramatic decrease in the magnitude of the enstrophy production between frames (b) and (s).

18

M.V. Melander,

F. Hussain 1 Core dynamics

M. V. Melander, F. Hussain 1 Core dynamics

19

The formation of the low enstrophy bubble, fig. 6i, can also be explained in terms of enstrophy production. First notice that in general the P,-contours (fig. 7) are concentrated closer to the axis than the lol-contours (fig. 6). Furthermore, there is a prolonged strong local compression at the place where the bubble forms; see figs. 7g, h. Since the compression is only significant near the axis it follows that the vorticity magnitude will decrease rapidly near the axis, but not near the edge of the vortex core. Hence the vortex becomes locally hollow, as is clearly seen in figs. 6i, j. The bubble is not a permanent feature of the vorticity field (see figs. 6i-1). Its disappearance is mainly due to a change in the enstrophy production (figs. 7i-l), but “fill in” by viscous diffusion also contributes. From the last frames of figs. 6 and 7 we observe that the internal core dynamics fades away as the P, variations and its magnitude diminish. It is particularly clear that the axial dynamics disappear first near the edge of the core and then gradually retreats to an increasingly smaller region around the axis; compare figs. 6i-s. Figure 8 shows the evolution of the dissipation function E and each of the four components in (4.5). There is a dramatic decrease in the magnitude of E which is a result of the relatively low Reynolds number. We notice that Ed has a very small magnitude compared to the three other components; this is why Ed is plotted using different contour levels than those in the other columns. While &I dominates most of the time, Ed becomes significant during the formation of the low enstrophy bubble; see fig. 8. However, the largest contribution to dE/dt always comes from Ed, as is easily seen by comparing the frames of fig. 8 and recalling the cylindrical geometry. Moreover, we also notice that dissipation’associated with the flow reversal occurs when the secondary cells (see later) expand. Knowing that Ed is insignificant, we realize that it is possible to visually identify the components cl, Ed and E~ in the plots of E, as each component dominates at different radial locations.

8. Deformation

of vortex surfaces by the meridional flow

In this section we derive and apply an approximate kinematic relation between the enstrophy production Pm and the meridional streamfunction $. While this relation is inviscid in nature it is very useful for analyzing the viscous simulation because the dynamics is predominantly inviscid over short times. Equation (2.7) shows that, the circulation 27c5 = 27cur,(t) = 27-t

_o-e _= Fdr”

along any circular material circuit of the form (r, z) = (r,,,(t), z,(t)) is constant in the absence of viscosity. The axisymmetry guarantees that such a material circuit remains circular as it is displaced by the meridional flow. This is of crucial importance for understanding the nature of both vorticity transport and enstrophy production. Most of the vorticity is in the component o, whose inviscid evolution is uniquely determined by the motion of circular material circuits (2.7). Moreover, w, also gives by far the largest contribution to the enstrophy production P,. This is because P, is only appreciable near the axis of the vortex (fig. 7), where w, is the only non-vanishing vorticity component. Therefore the enstrophy production is largely determined by the motion of circular material circuits. Their motion is determined by the meridional flow (2.7), hence the importance of meridional streamfunction $. By eq. (2.6), the meridional flow vanishes in the absence of azimuthal vorticity. There is therefore no meridional flow at t = 0, but it develops immediately (for c > 0) as the vortex lines acquire torsion. Figure 9 shows the $-streamlines and corresponding contours of the meridional

M.V. Melander, F’. Hussain I Core dynamics

20

0i

M. V. Melander, F. Hussain / Core dynamics

(P)

Fig. 9. The evolution of the meridional flow in case Ll is shown at times given in table 3. Each frame includes both streamlines (above the axis) and contours of the meridional velocity magnitude (below the axis). Also shown in each frame is the boundary of the vortex given by the contour 2& = 1. The stream function is normalized to be zero on the axis; thus the orientation of the meridional flow’s circulatory motion is determined by the sign of 1+5. The motion is counter clockwise

M.V. Melander,

22

F. Nussain / Core dynamics

(a 1 (b) (cl

(e 1 (9) . (0

(1)

Fig. 10. Axisymmetric vortex tubes as given by the surfaces of constant g = ru. The frames are at times given in table 3 and correspond to case Ll. Note that for a given axisymmetric vortex tube 2nt equals the circulation of that tube. Thus the maximum value 5 is constant between frames, namely 1/2x (r = 1 in nondimensional units). The contour levels are tj =j/(16x), j = 1,2,. . , 15.The boundary of the vortex as given by the contour !IBL1, = 1 is also shown (heavy line). in the shaded regions ((/I i 0) and clockwise in the unshaded regions ($ > 0). The stream function is calculated spectrally plane, the spectral solution is then by solving the Poisson equation a2e/W + a2$/a z2 - rwg - w in a meridional evaluated on a 256’-mesh. In each frame the meridional velocity magnitude s E (u’ + w ’ )’ ’ is shown in terms of contours with a constant increment 6s, which however changes between frames. The highest contour level sh and 6s are listed for each frame below as both non-dimensional and dimensional [*] vatues: (a) $ E 0, s = 0; (b) sh = 0.301 [2.23,&s = 0.0273 [0.2];(d) s,, = 0.178 [1.3-J, 6s = 0.0137 [O.l]; (e ) st, = 0.0957 [0.7], 6s = 0.0137 [O,l f; (f) s,, = 0.0888 [0.65], 6s = 0.00683 [O.OS]; (g) s,, = 0.123 [0.9], 6s = 0.0137 [0.1-J; ()I sh= 0.0888 [0.65], 6s = 0.00683 [O.OS]; (j) s,, = 0.0683 [0.5]. Ss = 0.00683 [O.OS]; (k) s,, = 0.0383 [0.28], 6s = 0.00273 [0.02]; (I) s,, = 0.0246 [0.18-j, 6s = 0.00273 [O:o.OZ];(p) st, = 0.0410 [0.3], 6s = 0.00342 [0.025].

M. V. Melander, F. Hussain / Core dynamics

23

velocity magnitude. Note that the meridional flow has significant velocities near the axis, but not elsewhere (a consequence of the continuity equation and the axisymmetric geometry). The axisymmet~c vortex tubes (fig. 10) are surfaces of constant 5, and 5 is convected with the meridional flow (evolution equation (2.7)) in the absence of viscous effects. Hence the local cross section of a vortex tube shrinks when the meridional flow converges on the axis, and vice versa; compare the frames of figs. 9 and 10. The axial contribution to the enstrophy changes accordingly, due to the conservation of circulation. Using fig. 9b we see that P, at early times has to be negative near z = 0 and z = 21, because here the meridional flow diverges away from the axis. Further towards z = I the meridional flow converges on the axis, consequently P, has to be positive there due to stretching (as fig. 7c in fact shows). A frame by frame comparison of figs. 7 and 9 reveals that P, can be inferred almost perfectly from I,+.The underlying justification is an inviscid asymptotic relation between II/ and P,: p = w

Wo-w) ~

203

Dt

(8.1) where we have used the fact that IQ/azI < I&$43rl, as is obvious from fig. 10. Since u is proportional to r near the axis (as seen by a Taylor expansion of u in r, noting that u vanishes at r = 0), we can simplify the expression further:

4u02

Pw= _.z=

r

_#&@!!!

2 p2 aZy

(8.2)

which is asymptotically correct near the axis. Since P, is significant only when I G cr (fig. 7), eq. (8.2) is indeed applicable for inferring P, directly from the geometry of the meridional streamlines. In figs. 9b, d we see two counter circulating cells per wavelength 1 = 21 of the vortex. We call these the primary cells. If they were to persist for all times the vortex would continue to expand near z = 0 (z = 21)and contract near z = t, which does not happen as shown in fig. 6. The reason is that the circulatory motions in the meridional flow gradually reverse, starting with the emergence of small secondary cells near the axis (fig. 9e). Note that the primary and secondary cells are counter circulating. Thus the secondary cells arrest the expansion of the vortex near z = 0 and z = 21(easily seen with the help of the kinematic arguments of the previous paragraph). The gradual progression of the flow reversal is exemplified by figs. 9d-i. We observe that the reversal progresses along the axis and is nearly complete in fig. 9i, in which only the secondary cells are present in the region of appreciable vorticity. A second flow reversal can be seen in figs. 9i-p. The nature of this second reversal differs from the first in two respects. First, the new cells are not created along the axis, but outside the vortex core. Second, the growth of the new cells progresses differently, namely radially inward rather than outward as in the first reversal. In spite of these differences the second flow reversal plays the same important role as the first, namely to arrest expansions and contractions of the vortex tubes.

M. V. Melander, F. Hussain / Core dynamics

24

9. Evolution of the vortex line geometry Figure 11 shows a few characteristic vortex lines at selected times. Due to the axisymmetry, it is sufficient to consider vortex lines starting from a single radial rake. In all frames of fig. 11, the vortex lines are drawn starting from a radial rake at the same axial position (z = 0). The rake consists of four points uniformly distributed across the vortex core. At t = 0, all vortex lines lie in meridional planes (0 = constant) and therefore momentarily have no torsion (fig. lla). However, the vortex lines gain torsion immediately, because of axial variations in the swirling motion around the axis; 4 G d6/dt = u/r is large where the core size is small, and vice versa, as indicated in fig. 2. Near z = 1 = A/2 the vortex lines rotate slower around the z-axis than near z = 0 and 21. Consequently, the vortex lines start twisting between z = 0 and z = 1 (fig. llb). Vortex lines close to the axis tend to twist more than those near the edge of the vortex, because 4 is in general largest on the axis.

(a 1

r ’ helical twist

slow swirl

(b)

(C 1 I

(e) (9) innermostvortex line

0i

(1) (4) Fig. 11. The evolution of case Ll (see tables 1 and 2) is illustrated by four typical vortex lines at selected instants. The frame labeling (a,b,c,e,g, ,) refer to the times given in table 3. Note that the vorticity magnitude is not constant along the vortex lines but varies significantly; see fig. 6.

M. V. Melander, F. Hussain / Core dynamics

25

The helical twists propagate towards z = 1 (compare figs. 1 lb, c). Meanwhile the vortex core expands near z = 0 (21) and contracts near z = 1, as will be shown later. A comparison of figs. 1 lb and c also reveals the onset of a coiling reversal. Namely, the innermost vortex lines are wound less around the axis in fig. 1 lc than they are in fig. 11 b. This uncoiling has progressed further in fig. lie, which shows a reversal in the helical twist of the innermost vortex line near z = 0 and z = 21. At this instant, the reversed coiling has not yet affected the outermost vortex line, but later (fig. 1 lg) the helical twist of all vortex lines has reversed. Figure 1 lg superficially resembles the mirror image of fig. 1 lb (replace 8 by - 0). One might therefore think that the evolution is a damped periodic motion, with the damping provided by viscous effects and the time elapsed between 11 b and 1 lg representing roughly half a period. This, however, is only partly true, because the vortex lines are never torsion free everywhere at the same time, except at t = 0. In fact, it appears that not even a single vortex line is completely torsion free at any time t > 0, with the exception of the axis. Careful examination reveals that in fig. 1 li the innermost vortex line bulges significantly away from the axis near z = 1. This is associated with the “low enstrophy bubble” forming inside the vortex (see also fig. 10). The coiling and uncoiling of the vortex lines continue, see figs. 1 li, 1, q, but dampen with time, and thus the axial flow and the core dynamics die out. Initially, the ratio between the fastest and slowest swirl (dQ/dt) along the axis is 9.0; by the end of the simulation this ratio has been reduced to 1.2. Figure llq shows essentially a rectilinear vortex, except that close to the axis some core dynamics, though weak, still persists.

10. Coupling between swirl and meridional

flow

The physics of the coupling between swirl and meridional flow is most easily understood in terms of vortex lines on an axisymmetric vortex tube. When the vortex lines are coiled on the tube as in fig. 2 then the vortex has azimuthal vorticity, and q is therefore not identically zero. Note that the instantaneous winding of the vortex lines on an axisymmetric vortex surface (4 = constant) is not directly related to the shape of the surface. For example, it is possible to have the vortex lines wind in entirely different fashions on identical axisymmetric vortex surfaces. In this sense the coiling of the vortex lines on an axisymmetric vortex surface is independent of the instantaneous shape of the surface. However, the differential rotation is given by the shape of the tube and will increase or decrease y1,as discussed earlier. Let us consider a vortex, fig. 12A, where the differential rotation is decreasing the magnitude of ‘I, that is producing an uncoiling of the vortex lines. It clearly takes a finite time for the differential rotation to uncoil the vortex lines and thus change the sign of q (fig. 12C). While the differential rotation is uncoiling the vortex lines, the meridional flow distorts the shape of the vortex tube even further away from rectilinearity, because q has not yet changed sign and hence the orientation of the circulatory motion is preserved (figs. 12A, B). Since the axisymmetric vortex tubes are distorted well away from rectilinearity in fig. 12B, the differential rotation is large even though q is small. This results in a new coiling of the vortex lines, this time in the opposite direction, so as to produce a reversed meridional flow as required to bring the shape of the vortex tube back towards rectilinearity (fig. 12C). When eventually that is accomplished, the vortex lines are again coiled (fig. 12D). The meridional flow induced by these coiled vortex lines again deforms the vortex tubes (fig. 12E). From the kinematic analysis of the coupling term, it is easy to understand how the two cells in fig. 9b emerge. Namely, the initial local convergence and divergence of the axisymmetric vortex tubes immediately result in material generation of n by the differential rotation. For example, the clockwise circulating cell (fig. 9b), characterized by negative o0 and ye(fig. 13b), emerges as a result of the initial axial divergence of the vortex tubes between z = 0 and z = 1 (fig. 14a). Moreover, it

M.V. Melander,

26

F. Hussain / Core dynamics

Vortex

(4

(Q

Fig. 12. Schematic

of the coupling

between

swirling

and meridional

flows.

follows from (3.5) that the extreme values of the material q-generation occur at the inflection points of c2 (fig. 14a) and not necessarily at the narrowest part of the vortex or at the point of maximum differential rotation. At any fixed point between z = 0 and z = 1, both q and (l/r”)(@‘/az) (i.e., the material y-generation) are oscillating functions of time. By comparing corresponding frames of figs. 13 and 14 we observe that these oscillations are out of phase. The q-oscillation lags behind the (l/r4)(a<2/az)-oscillation by a phase shift, which depends both on spatial location (r, z) and time. We note that at early times (frames b-d) (l/r4)(a<2/az) changes sign near the z-locations where q has a maximum or a minimum. It may therefore be helpful to think of a roughly 90” phase difference between the two oscillations. This phase difference plays a critical role in the qtransport. To understand why we must recall the following three kinematic results. First, the shape of the vortex tubes (5 = &,) determines the material generation/annihilation of q through the coupling term (l/r4)(at2/az). S econd, the meridional flow determines how the shape of the axisymmetric vortex tubes (5 = t,,) changes with time. Here the orientation (clockwise or

M. V. Melander, F. Hussain / Core dynamics

27

counterclockwise) of the circulatory motion in the meridional flow cells (fig. 9) is critical in determining where the vortex tubes expand and where they shrink. Third, the meridional flow is uniquely given in terms of q. The sign of 11largely determines the circulatory motion in the cells. Specifically, the circulatory motion cannot change orientation, say from counterclockwise to clockwise, anywhere, until q has locally become sufficiently negative, because $ is given in terms of an infinite space Green’s function G(r, z; re, z,,) corresponding to eq. (2.6) i.e.:

fin

tw-,4 = Jrn -cc

c&o, zo) W,z; ro>zo)drodzo.

(10.1)

0

When the phase difference is viewed in the light of these three facts we see that the circulatory meridional motion cannot immediately flip from counterclockwise to clockwise anywhere when the vortex change from converging to diverging in the z-direction (because locally q does not change sign at the same time as the material q-generation). On the contrary, starting from the time when q-generation begins to be negative, a finite time elapses before the directional change occurs in the meridional flow. A comparison of frames from figs. 9 and 14 gives numerous examples of this time delay; consider for example the region near z = 0 in frames b through e. The q-transport differs significantly from the meridional flow’s material transport due to such time delays between the reversals in q-generation and subsequent reversals of the meridional flow. We proceed with a discussion of the differences and their causes. To fully appreciate these differences, try to infer the propagation direction of the local helical twists in fig. 11 by means of the local (self) induction approximation. For fig. 1 lb, local induction approximation would have both helical twists propagating away from z = 1. In reality, however, they both move towards z = 1, as seen by comparing figs. 1 lb, c. There are two q-transport mechanisms: convection of q with the meridional flow and material q-generation/annihilation by differential rotation. At early times, these two effects compete in the primary cells. In the interval between z = 0 and z = 1, the material transport of q by the meridional flow is towards z = 0 (figs. 9b, d). However, the differential rotation annihilates q near z = 0, while at the same time generating q further to the right (figs. 14b-d). The competition between the two effects is such that g is effectively transported away from z = 0 (figs. 13b-d) as explained schematically in fig. 15. In the secondary cell which forms later near z = 0 (fig. se), the meridional flow and the differential rotation both transport q in the same direction, namely away from z = 0 as seen by comparing frames e, g in figs. 9,13 and 14. The actual transport of q is thus faster than the material transport. We observe similar flow reversals in figs. 9k,l. We therefore conclude that the q-distribution moves in the direction of the material transport when a cell is expanding, only faster; in a contracting cell q moves against the meridional flow, but slower than in an expanding cell. In neither case does the q-transport behave exactly like the material transport of the meridional flow. Instead, the q-transport consists of a combination of material transport and wave motion. The latter is a consequence of the phase difference between q-generation and q. The emergence of the secondary cells (fig. 9e), marking the beginning of the reversal in the meridional flow, serves as an example illustrating the role of the phase difference. At early times (e.g., fig. 9b), the meridional flow in the primary cells expands the vortex tubes on one side of each cell (e.g., near z = 0) and contracts them on the other side of the cell (e.g., near z = I). As a result, the sign of Q2/i3z eventually changes, producing an oppositely signed q-generation near z = 0 and z = I (fig. 14b). This in turn reduces the amplitude of q until eventually the sign of the azimuthal vorticity (wO = rq) also reverses (figs. 13c-f). When a sufficient oppositely signed v has been generated then the secondary cells emerge (compare frames d, e of figs. 9 and 13).

hf. K Melander,

28

(9)

F. fiussain

/ Core dgr~nmics

M.V. Melander, F. Hussain / Core dynamics

29

11. Scalar transport Scalar transport is of importance for flow-visualization, mixing and chemical reaction, as well as for computational methods based on Lagrangian vortex elements. In laboratory experiments, instantaneous concentrations of dye or smoke are often used to infer structures and their evolutionary dynamics which we think can only be understood in terms of the vorticity field. Unfortunately, the vorticity field is not visible to the naked eye, nor can we measure the vorticity (vector) field instantaneously (everywhere) with the state-of-the-art measurement technology. We have often registered our objection to the almost universal use of flow visualization for interpreting complex vortex interactions and asserted the risk undertaken when one relies too heavily on flow visualizations in studies of vortex dynamics and coherent structures and other turbulence phenomena; see Bridges et al. (1989), and Hussain (1986). This is not only a problem of non-unity Schmidt number of flow markers. We continue to assert that while flow visualization is of great importance in qualitative interpretation of many flow phenomena, we may have to revise our perception of many flow events because of wrong inference based on flow visualization. The present numerical simulations illustrate how misleading such interpretations can be. We make two highly idealized choices for the initial scalar distribution. One choice (fig. 16, column one) is such that initially the scalar concentration equals the vorticity magnitude everywhere. Moreover, the Schmidt number is unity, so that the evolution of 101 and c differ only through action of the non-linear terms. Column one of fig. 16 shows the evolution of the scalar concentration. We observe that, the distributions of 101 and c begin to deviate immediately. This occurs because mass and vorticity transports are in opposite directions in the primary cells. Hence, the vorticity magnitude increases where the scalar concentration dilutes, and vice versa, (compare panels a and c of fig. 6 and column one of fig. 16). While mass is transported with the flow, the vorticity is also displaced by enstrophy production in the form of waves. Therefore, the two transports never match, not even when they are in the same direction, as in the secondary cells. A notable consequence of the different transports is that the boundary of the scalar does not faithfully represent the vertical fluid (compare the lower contour levels in panels of figs. 6 and 16). This difference is permanent, at the end of the simulation the vortex has essentially become rectilinear, but the scalar concentration has not, and will not since the axial dynamics is gone. Thus the final scalar distribution is lumped with a spatial period of 2 in z; flow visualization would indicate a wavy vortex where none exists! The second choice of initial scalar concentration is that c = 5: = vr everywhere. The evolution of this scalar concentration is shown in column two of fig. 16. A remarkable resemblance is observed between column two and fig. 10. The reason for this is that eqs (2.9) and (2.7) show that c and 5 evolve almost identically except for one term in the diffusive contribution. That is, in the inviscid case, the axisymmetric vortex tubes given by 5 = constant are exactly marked by the scalar concentration. This is in sharp contrast to the first choice (column one of fig. 16) where the difference between the evolutions of c and 1~1 is inherently inviscid.

Fig. 13. The evolution of the q-distribution (‘1 = we/r) for case Ll is shown at times given in table 3. n is positive inside the shaded regions and negative in the unshaded regions. The boundary of the vortex as given by the contour 21J1,= 1 is also shown (heavy line). The contour levels are equally spaced in r) with an increment of 6~. The peak contour level Q, and 6~) are listed below for each frame, dimensional values are included in brackets: (a) q = 0; (b) qh = 5.17 [SO], 6~ = 1.29 [20]; (c) qh = 2.58 [40], ST = 0.646 [lo]; (d) qh = 1.29 [20], 6~ = 0.323 [S]; (e) nh = 0.969 [lS], 6~ = 0.161 [2.5]; (f) )I,, = 1.94 [30], 6~ = 0.323 [S]; (g) qr, = 2.58 [40], 6~ = 0.646 [lo]; (h) q,, = 2.26 [35], 6~ = 0.323 [S];(i) n,, = 1.29 [20], 61 = 0.323 [S]; (j) Q, = 0.452 [7], 6~) = 0.0646 Cl]; (k) qr, = 0.322 [SJ, 6~ = 0.0646 Cl]; (1) nr, = 0.258 [4]. 8~ = 0.0646 [l] (m) qh = 0.388 [6], 6~ = 0.0646 [l]; (0) tf,, = 0.388 [6], 6~ = 0.0646 [l]; (q) q,, = 0.194 [3]. 6~ = 0.0322 [0.5]. The conversion formula used here is ? = (SL),*, see (2.4).

0i

. (I)

(1) PJ)

(1 0

(9)

M. V. Melander, F. Hussain / Core dynamics

31

q - distribution (q < 0) meridional streamlines

(A)

material

transport

03 ,/---.\

63

,’ /----.,

I If

I

,--5

I

\

‘1

material generation \

actual transport

(D) Fig. 15. Schematic

of the n-transport.

D gives the actual transport n-generation (C).

as a combination

of material

transport

(B) and

The differences between vorticity and scalar transport are especially interesting in the light of the recent popularity of Lagrangian vortex methods. Vortex particles (vortons/vortex sticks) are transported with the flow as a passive scalar, but also generate the flow. Each vortex particle’s position changes according to the flow generated by all other particles. Meanwhile, the particle’s vorticity vector changes by the strain rate field generated by all other particles. We believe that the present viscous vortex evolution would constitute a useful test case for viscous vortex methods such as that of Winkelmans and Leonard (1989) for two reasons. First, the evolution does in a simple way illustrate the differences between mass and vorticity transports. Second, the axial dynamics dies out in such a way that the final scalar and vorticity distributions are different. MH (1990) and Kida et al. (1989) give other examples of important differences between scalar and vorticity transport. We have previously emphasized the difference between marker boundaries and coherent vorticity boundaries in all cases of 3D flows even at SC = 1. Dynamically significant events characterized by intense vortex stretching are associated with depletion of scalars from the stretched (strengthened) vortices thus obscuring these events from visualization and instead emphasizing less significant zones of lower vorticity where markers accumulate. The present simulations show that even in an axisymmetric flow and even for SC = 1, the marker boundaries can markedly differ from vorticity boundaries and may mislead one about the flow physics. < Fig. 14. This figure shows the material n-generation (rj E Dq/Dt G r m4 agr/az) for case Ll at times given in table 3. q is positive inside the shaded regions and negative in the unshaded regions. The boundary of the vortex as given by the contour ZBr = 1 is also shown (heavy line). The contour levels are equally spaced in rj with an increment of 8d. The peak contour level 4s and Stj are listed below for each frame, dimensional values are included in brackets: (a) rjs = 7.28 [1200], SQ = 1.21 [200]; (b) rj,, = 1.82 [300], Srj = 0.152 [25]; (c) rj,, = 0.850 [140-j, 84 = 0.121 [20]; (d) tj,, = 0.668 [llO], ar) = 0.0607 [lo]; (e), (f) tj,, = 0.607 [lOO], SV)= 0.0607 [lo]; (g) & = 0.485 [80], &j = 0.0607 [lo]; (h) ij,, = 0.121 [20], 84 = 0.0121 [2]; (i) rjh = 0.212 [35], Srj = 0.0303 [S]; (j) rjh = 0.152 [25], Srj = 0.0152 [2.5]; (k) rj,, = 0.0849 [14], 8tj = 0.0121 [2]; (1) rjh = 0.0728 [12], Srj = 0.00607 [I]; (m) $, = 0.0607 [IO], Srj = 0.00607 [l]; (0) rj,, = 0.0273 [4.5], Srj = 0.00303 CO.51; (q) rjh = 0.0303 [S], Srj = 0.00303 [O.S]. The conversion formula is d = ri* (7’e). where i and J? are given in (2.4).

M. V. Melander, F. Hussain 1 Core dynamics

32

(a) (cl ((3 (9) (0 (k)

(9) Fig. 16. Evolution of the scalar concentration for case Ll (columns one and two) and L5 (column three). In column we have initially 1~1 = c, while in columns two and three c = 5 initially. In column three a few 5 = constant contours also shown.

one are

However, column two of fig. 16 shows that the scalar concentration can faithfully mark the axisymmetric vortex tubes provided that the Reynolds number is not too small as in column three of fig. 16.

12. Discussion

and conclusion

The internal core dynamics is usually ignored, for vortex dynamics is often conceived as a collection of many vortex filaments moving according to the collectively induced velocity field and by self-induction. The main reason for ignoring the core dynamics is the intrinsic difficulty in modeling this physical effect. Nevertheless, core dynamics is of crucial importance, for it indirectly affects the collectively induced velocity field. This happens because the position of each vortex filament depends sensitively on the self-induction which in turn is directly related to the internal core dynamics, particularly the core size. One particular aspect of this sensitivity is the dependence on the Reynolds number. We have demonstrated (section 6) that for the simple case of a single axisymmetric vortex the speed with which waves (i.e., core size variations) move along the vortex varies significantly with the Reynolds number. Moreover, the waves persist longer for higher Reynolds numbers and do not disappear in the high Re-limit. Thus core size variations cannot be ignored in filament models as assumed by Moore and Saffman (1972). We find that the local peak vorticity is critical in determining the speed of the wave motion. This is, in fact, a point

M. V. Melander, F. Hussain

/ Core dynamics

33

where internal vortex dynamics differs from the classical vortex dynamics (i.e., filament dynamics) where the circulation is the essential, but the peak vorticity is irrelevant. We have discovered a new phenomenon that results from core dynamics, namely that bubbles characterized by low vorticity form inside the vortex when the vortex core expands due to the converging axial flow arising as waves moving along the vortex’s axis collide. We find that core dynamics results in significantly higher energy dissipation. In particular, the bubbles, which are not permanent features, produce spots of very high dissipation. The core dynamics decays as the result of viscous diffusion of vorticity. The decay starts at the outside of the vortex and gradually progresses towards the axis, so that the core dynamics at late times takes place only in a small region near the axis. These effects are difficult to mode1 and incorporate in filament models. We have identified the mechanism responsible for the dynamics of an axisymmetric vortex with a nonuniform core. This mechanism reveals that internal vortex dynamics (i.e., core dynamics) is very different from classical vortex dynamics (filament dynamics). While the latter involves only the Biot-Savart law and the local induction approximation, the former is a delicate interaction between the geometry of axisymmetric vortex surfaces and the coiling of the vortex lines located upon them. For the sake of simplicity we discuss this mechanism for an inviscid flow. Here, the vortex surfaces (5 = constant) are deformed only by the meridional flow (given by the streamfunction I,+).Vortex lines on such a surface remain there for all time. However, the geometry of these lines, i.e., how they coil around the z-axis, changes with time. The meridional flow ($) is determined uniquely in terms of the coiling of vortex lines (i.e., the variable q). The interesting thing is that the rotation speed 8 around the axis varies along the vortex surface (i.e., 8 = Qr2) and the instantaneous geometry of the 5 = constant surface determines how the coiling of the vortex lines changes. Our kinematic analysis (section 3) shows that the rate of change of the vortex line coiling (Dv]/Dt) equals the product of the axial vorticity times the differential rotation rate along the axisymmetric vortex surface. This effect constitutes the entire coupling between swirl and meridional flow. Interestingly, the rate of energy transfer between the two types of flow is presicely the integral of the product of I,+and the coupling term; see (4.14), (4.15). By way of example (sections 8-10) we have shown that the observed features resulting from the core dynamics on an axisymmetric vortex with nonuniform core follow readily the mechanism described above. For example, the emergence, growth, decay and disappearance of cells in the meridional flow are easily understood in terms of coiling and uncoiling of vortex lines on axisymmetric vortex surfaces. Moreover, the fact that the dynamics is a combination of material transport and wave motion also follows readily. Likewise does the appearance and disappearance of the low vorticity bubble. We find flow visualization to be successful for higher Re only if a positive tracer concentration is injected in the flow so as to be constant on the axisymmetric vortex surfaces. In a follow up study (MH, 1993a), we apply and discuss a number of new mathematical tools for analyzing the flow’s helical structures. Included are the helicity integral, the helicity densities and the so-called “complex helical wave decomposition” (Lesieur, 1990). This decomposition is based on the expansion of the velocity field into eigenfunctions of the curl operator (Moses, 1971), but has never before been used to analyze physical space vorticity fields. By applying it to the laminar axisymmetric flow and comparing with the results obtained in the present paper, we find that the helical wave decomposition is a powerful mathematical framework of great physical relevance, as it easily separates wave packets moving in opposite directions; furthermore, it provides a simple explanation for the evolution and dynamics of vertical waves. In contrast to the analysis presented in the present paper, the new analysis in terms of the helical wave decomposition is not limited to the axisymmetric geometry. The decomposition contains the same information as the usual vorticity-velocity formulation, but it allows a complete description of the flow in terms of right and left handed vorticity components, plus a potential flow. In fact, the decomposition appears to open up a new frontier of vortex dynamics, namely the interaction and evolution of right/left polarized vertical structures.

M. V. Melander, F. Hussain / Core dynamics

34

Core dynamics is also of great interest for computations and modeling, because discretization of a vorticity field into a large number of vortex filaments constitutes one of the popular and attractive vortex methods (Leonard, 1985). Usually the self-induced axial flow inside the filaments is ignored, either by using a fixed core size e.g., Moore and Saffman (1972), Crow (1970), Schwarz (1985). or by using only the core size to conserve the volume of vertical fluid (e.g., Ashurst and Meiburg, 1988). A noticeable exception is the filament method recently developed by Lundgren and Ashurst (1989), who employed self-consistent model equations for the evolution of the variable circular core. Also note that Callegari and Ting (1978) have extended Moore and Saffman’s analysis to viscous, but high Reynolds number vortices. Our results here show that core dynamics is persistent in the inviscid limit and thus strongly suggest that core dynamics cannot be ignored. Moreover, our simulations of turbulent shear flows (MH, 1993b) show strong helical polarization of the vertical structures and thus suggest that it is perhaps also ill-advised to ignore the self-induced axial flow. We examine the effect of background fine-scale turbulence on the dynamics of a vortex, as well as how the turbulence itself is changed as a result of its interaction with the vortex in another paper (MH, 1993b). Thus this study can be viewed as that of coupling between large and fine scales-perhaps the most interesting but least understood facet of turbulence. There are a few particular concerns in this regard. In spite of typical assumptions made by theorists-of course, forced by analytical expediency and conceptual simplicity-there is no distinct scale separation in most turbulent flows, which instead consist of a continuum of scales from the largest determined by the flow geometry to the smallest at the Kolmogorov scale. Let us concede scale separation as a hypothetical possibility; then the conventional argument holds that the coupling between the largely disparate scales is weak and the fine scales are statistically isotropic in any turbulent shear flow. In fact, this local isotrophy hypothesis is the cornerstone of Kolmogorov’s theory as well as most modern theories of turbulence. But we have long insisted that in spite of wide separation of scales, the large and small scales are intimately coupled, thus making local isotropy quite unlikely; see Hussain (1983, 1984). In MH (1993b), we directly focus on this coupling by studying the interaction between a simple, well defined, yet dynamically evolving, large-scale vertical structure with fine-scale 3D turbulence. The geometry chosen seems to be the simplest to address the phenomenon and can indeed be viewed as a segment of typical large-scale coherent structures in practical turbulent shear flows such as mixing layers, jets, wakes, etc.

Acknowledgements This research has been supported by the Office of Naval Research under Grant NOOO1489-J-1361 and the Air Force Office of Scientific Research under grant F496620-92-5-0200.

Appendix A-Vortex

definitions

Perry and Chong (1987) define a vortex as any spatial region where the eigenvalues of deformation tensor Eij = ani/axj are complex. We may express this criterion in terms of the principal invariants Ii, I,, and Z3 of E. These are the coefficients in the characteristic polynomial Det (E - AZ), we have, Ii =v.u=o,

(A.1)

64.2) Z3 = Det E.

(A.31

M. V. Melander, F. Hussain / Core dynamics

The characteristic polynomial is the cubic A3 + IJ discriminant,

35

+ Z3. It has complex roots if and only if the

(A.4) is positive. At a critical point of the velocity field (u = OJ the instantaneous streamlines spiral or circle a unique axis if and only if D > 0. For steady flow this is an appealing result, and the vortex definition conforms with our intuitive concepts. For flows having a preferred reference frame (e.g., flow over a missile shaped body (Perry and Chong, 1987)) it also makes good sense to consider streamline patterns around critical points. However, for unsteady flows without an obvious reference frame, the Galilean variance of the streamlines and critical points is a source of ambiguity. For such flows the criterion D > 0 may not resemble our intuitive concept of a vortex. For a specific example of the definition’s shortcomings see Hunt et al. (1988). To overcome this and other difficulties, Hunt et al. (1988) employ a vortex definition based on threshold values for both ZZand the pressure p. Unfortunately the thresholds are ad hoc values. We propose a vortex definition based on the kinematic vorticity number 2&E

0.5

( 1 ?!z!L

2 SijSij

(A.5)

’

where Sij is the symmetric strain rate tensor $(aui/axj + auj/axi). The kinematic vorticity number is a dimensionless invariant, which was introduced by Truesdell (1953, 1954) as a measure of the quality of the motion (rotational versus deformational). It is mainly rotational when %I&> 1, and deformation dominated when ‘SBk< 1. In the extreme cases of potential flow and rigid rotation ?D3Lz)k attains its extreme values, zero and infinity. For a homogeneous incompressible fluid without extraneous body forces, (A.5) is equivalent to l---

2Bk=

(

2Ap -“.5 po2 > ’

64.6)

for proof see Truesdell (1953). It follows immediately from (A.6) that !&>loAp>O, !B&= loAp=O, !R$<

(A.7)

loAp
At a point where Ap > 0 the pressure p cannot have a local maximum. Likewise, p cannot have a local minimum at a point where Ap < 0. Thus follows Hamel’s theorem of pressure extremes: In any spatial region 9 where YBk 2 1 (‘%J < 1) the maximum (minimum) value ofp occurs on the boundary C@. (Hamel, 1936; Truesdell, 1953).

This theorem has generalizations, see the above references; these are, however, not relevant to the present discussion. Clearly ‘B&= 1 is a very special value. We base our definition of a vortex on this value: Definition: In a homogeneous incompressiblefluid connected spatial region where 2Blk > 1.

without extraneousforces

a vortex is any maximal

M.V. Melander, F. Hussain 1 Core dynamics

36

It is well known that under mild restrictions the rate of change of the total energy E is dE -_= dt

-

SijSijd~ = -

2~

s

w.odz,

V

s

where dz = dx dydz. However, this result is only a consequence of the theorem of average intensity balance between vorticity and deformation (Truesdell, 1948, 1957). For incompressible fluids (A.8), or the above mentioned theorem, yields

jco2(&l)dr=O,

(A.9)

where the integration extends over the entire fluid. With our vortex definition, (A.9) becomes an existence theorem for vortices. For (A.9) implies that only flows without vortices are potential flows, and flows having 2& = 1 everywhere. The latter class of flows is characterized by having harmonic pressure distributions; included in this class are parallel flows. When 2Bk > 1 the deformation tensor has complex eigenvalues, because the discriminant (A.4) may be expressed as follows, (A. 10) clearly 2Bk > 1 implies D > 0.For steady flows the stream and streak lines therefore spiral or circle around the principal axis of the deformation tensor when ‘2B1,> 1. Thus our definition of a vortex is also a vortex by Perry and Chong’s definition. However, D > 0 does not imply 2Bmk > 1. Thus our definition is more restrictive. The theoretical results described above justify our sharper definition. Unlike the eddy definition of Hunt et al. (1988), our definition does not require ad hoc thresholds.

References Ashurst, W.T. and E. Meiburg (1988) Three dimensional shear layers via vortex dynamics, J. Fluid Mech. 189, 87. Batchelor, G.B. (1967) Introduction co Fluid Dynamics (Cambridge University Press). Bridges, J., H.S. Husain and F. Hussain (1989) Whither coherent structures in: J.L. Humley ed, Whither Turbulence? Turbulence at the Crossroads (Springer Verlag) 132. Callegari, A.J. and L. Ting (1978) Motion of a curved vortex filament with decaying vertical core and axial velocity, SIAM J. Appl. Math. 35, 148. Crow, SC. (1970) Stability theory for a pair of trailing vortices, AIAA J. 8, 2172. Hamel, G. (1936) Eine besondere Art von Luftbewegung, Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Kl.5. Hasimoto, H. (1972) A soliton on a vortex filament, J. Fluid Mech. 51, 477. Howard, L.N. (1957) Close interaction of 3D vortex tubes, Arch. Rat. Mech. Anal. I, 113. Hunt, J.C.R., A.A. Wray and P. Moin (1988) Eddies, stream, and convergence zones in turbulent flows (Report S88, 193, Stanford University). Hussain, A.K.M.F. (1983) Coherent structures-reality and myth, Phys. Fluid. 26, 2816. Hussain, F. (1984) Coherent structures and incoherent turbulence, in: T. Tatsumi ed, Turbulence and Chaotic Phenomena in Fluids (North-Holland) 245. Hussain, F. (1986) Coherent structures and turbulence, J. Fluid Mech 173, 303. Husain, H.S. and F. Hussain (1991) Elliptic jets. Part 2: Dynamics of coherent structure pairing, J. Fluid Mech. 233, 439. Hussain, A.K.M.F. and K.B.M.Q. Zaman (1980) Vortex pairing in a circular jet under controlled excitation. Part 2: Coherent structure dynamics, J. Fluid Mech. 101, 493. Kida, S., M. Takaoka and F. Hussain (1989) Reconnection of two vortex rings, Phys. Fluids A I, 630. Leibovich, S. (1970) Weakly non-linear waves in rotating fluids, J. Fluid Mech. 42, 803. Leibovich, S. and A. Kribus (1990) Large-amplitude wavetrains and solitary waves in vortices, J. Fluid Mech. 216, 459. Leonard, A. (1985) Computing three-dimensional incompressible flows with vortex elements, Ann. Rev. Fluid Mech. 17, 523.

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Lesieur, M. (1990) Turbulence in Fluids (2nd Ed, Kluwer). Lundgren, T.S. and W.T. Ashurst (1989) Area-varying waves on curved vortex tubes with application to vortex breakdown, J. Fluid Mech. 200, 283. Melander, M.V. (1989) Close interaction of 3D vortex tubes, CTR-annual Res. Brief; 39. Melander, M.V. and F. Hussain (1988) Cut-and-connect of two antiparallel vortex tubes (CTR Report-S88,257, Stanford University). Melander, M.V. and F. Hussain (1989) Cross-linking of two antiparallel vortex tubes, Phys. Fluids A I, 633. Melander, M.V. and F. Hussain (1990) Topological aspects of vortex reconnection in: H.K. Moffatt and A. Tsinober eds, Topological Fluid Mechanics (Cambridge University Press) 485. Melander, M.V. and F. Hussain (1991) Reconnection of two antiparallel vortex tubes: A new cascade mechanism, Turb. Shear Flows 7, 9. Melander, M.V. and F. Hussain (1993a) Polarized vorticity dynamics on a vortex column, Phys. Fluids A5, 1992. Melander M.V. and F. Hussain (1993b) Coupling between a coherent structure and fine-scale turbulence, Phys. Rev. E, (in press). Melander, M.V., H. Husain and F. Hussain (1991) Coherent structures, vortex reconnection and turbulent mixing, in: L. Sirovich ed, New Perspectives in Turbulence (Springer) 195. Moore, D. and P. Saffman (1972) The motion of a vortex filament with axial flow, Trans. R. Sot. 272 A. 1226, 403. Moses, H.E. (1971) Eigenfunctions of the curl operator, rotationally invariant Helmholtz theorem, and applications to electromagnetic theory and fluid mechanics, SIAM J. Appl. Math. 21, 114. Perry, A.E. and MS. Chong (1987) A description of eddying motions and flow patterns using critical-point concept, Ann. Rev. Fluid Mech. 19, 125. Pritchard, W.G. (1970) Solitary waves in rotating fluids, J. Fluid Mech. 42, 61. Schwarz, K. (1985) Three-dimensional vortex dynamics in superfluid: 4He Line-line and line-boundary interactions, Phys. Rev. B 31, 5782. Truesdell, C. (1948) On the total vorticity of motion of a continuous medium, Phys. Rev. (2) 73, 510. Truesdell, C. (1953) Two measures of vorticity, J. Rat. Mech. Anal. 2, 173. Truesdell, (1954) The kinematics of Vorticity (Indiana University Publ. Science series No. 19). Winkelmans, G. and A. Leonard (1989) Improved Vortex methods for three-dimensional flows, in: R.E. Caflisch ed, Mathematical Aspects of Vortex Dynamics (SIAM) 25.