LOCAL AND GLOBAL PERSPECTIVES IN FLUID DYNAMICS H . Keith Moffatt Isaac Newton Institute f o r Mathematical Sciences University of Cambridge, United Kingdom [email protected]1. INTRODUCTION It is a great honor to present this Closing Lecture at what has been an outstanding Congress, ICTAM 2000. These Congresse s have a proud t ra - dition, going back t o Del ft 1924, when Pr an dtl, vo n KBrm&n, G . I . Taylor and Burgers initiated the series and laid the ground rules. Sin ce then, every four years, with a brief interlude during World War 11, the world mechanics community has gathered together to share its knowledge and t o grapple with new emerging problems. This is truly a global enterprise, and we can all take pride and pleasure in our involve men t in it. Th e first ICTAM that I was privileged to attend was the tenth in the series, held at Stresa in 1960. And here we are now at the 20th ICTAM and actually the 10th that I have been personally involved in. The global participation at these Congresses is achieved in each case only through an effect ive local organization. Each Congress, whether at Stresa or Stanford, Lyngby or Haifa, Moscow or Toronto, has had its own distinctive local flavor that remains in the heart of each participant long after the details of individual lectures have faded from the memory. Chicago will be no different in this respect, and I would like to record my personal thanks to Hassan Aref and his great team on the Local Organizing Committee here, who have succeeded in making this 20th an d mil lennia1 Congress so uniquely memorable. Locality and globality: th e t wo complementary perspectives whose interplay is essential to the succ essful running of a World Congress. I t is a dif fere nt kind o f locality and globality th a t I propose t o discuss in this lecture; but certain parallels may emerge, and I shall return to the Congress scenario in my closing remarks . www.moffatt.tc
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8/3/2019 H.K. Moffatt- Local and Global Perspectives in Fluid Dynamics
In Fig. l( b ) by contrast, both boundaries are at rest and fixed at
angle a , and a two-dimensional flow is driven by some agency (e.g. a
rotating cylinder) far from the corner. In the Stokes approximation,
the streamfunction sufficiently near the corner satisfies the biharmonic
equation
with no-slip boundary conditions
v*+= 0: (2)
1+!1=+0=0 on O=O:a. ( 3 )
Here similarity solutions of the form II,= ~ ' f ( 0 ) ay be found in which
the parameter X must be determined from the eigenvalue problem result-
ing from ( a ) , ( 3 ) . This is evidently a similarity solution of the secondkind. The most interesting feature of the solution is that, for all acute
angles a (and actually over the wider range cy < 147"); all relevant eigen-
values X i are complex, and the corresponding flows are 'eddying' flows
as T + 0 (Moffatt 1964a;b). The fundamental structure is precisely as
indicated in Fig. l(b).
Figure 2
plane boundary (Taneda 1979).
Corner eddies in a Stokes flow over a cylinder at a small distance from a
Although this was all well understood in 1964, it was not until 1979
that the first photographs of such corner eddies were published (Taneda
1979). Taneda's beautiful photographs, many of which are reproduced
in Van Dyke's Album of Fluid Motion (1982), show a great variety of
situations in which corner eddies can arise. For example, shear flow over
a cylinder resting on a plane boundary shows the eddy sequence in the
8/3/2019 H.K. Moffatt- Local and Global Perspectives in Fluid Dynamics
530 UJ-CLOSIAGL E C T U R EH K E I T H f O F T A T T
Figuie 8 The rise of a T $ ~ \ T . s k i r te d bubb le t h r ough t he c om m on r oom of D A L I T PC a m br i dge T he bubb l e t ube o f d i a m e t e r 17 c m c on ta i n s g l \ ce r i ne T he po r t r a i t 1s
of X S E dd i ng t on
Figuie 9 Th e Japanese ba t h prob lem a l amina r s t ream of w a t e r flows i n to a deep
ba th , above a c r it ica l flow ra t e Q c , bubbles a re en t ra ined th rough the c i rcu la r cusp
formed at the free surface
and Kelvin, for which we know t ha t t he c i rcu la t ion round eve ry ma te -
r ia l ( i .e . Lag rang ian) c i rcui t is conserved. A closely related result ( J . - J .
8/3/2019 H.K. Moffatt- Local and Global Perspectives in Fluid Dynamics
Local and g lobal perspect ives in fluid dynamics 531
M oreau 1961, M offatt 1969) is th a t t h e helicity
Fl= 1 , u . w d V
for any m ater ial volume V . on whose surface w . n = 0. is also conserve d:here. w = c u r l u is th e vorticity field. T hi s conserv ation of helicity
ad m its topological in terpreta t ion : i t i s. in a sense that can be ref ined
(X rno l’d 19 74 ). th e degree of l inkage of co nsti tu ent vortex tub es in th e
flow. a linkage that. as recognized by Kelvin, is indeed conserved.
An ana logous resu lt ho lds in the mag ne tohydrodyn amics (M HD ) of
perfectly co nd uc tin g fluids, in which lines of m agne tic field B (or ‘B-
l ines’) are f rozen in th e f lu id . T h e analogous conserved qu ant i ty is the
magnetic helici ty
X.11= l7 . B d V
where B = c u r l A . an d i t ad mi t s s imi la r topolog ical in te rp re ta t ion . I t
is in this MHD context that the concept of helici ty has proved to have
profound s ignif icance. in two quite complementary s i tuations:
4.1. The turbulent dynamo
I be l ieve tha t the mos t impor tan t fundamenta l b reak th rough o f the
last half-century in our understanding of turbulent processes relates to
the dynamo problem: under what c i rcumstances wi l l a magnet ic f ie ld
B ( z . ) systematically increase in intensity un der th e dis tort in g an d dif-
fusive action of a field of s ta t i o n a ry h om og en eo us t u rb u le n c e u ( ~ , t ) ?
T h e ques tion was posed first in th is form by Batchelor (195 0) . T h e sim-
ple answer is tha t a suff icient condit ion for such dy na m o acti on is tha t
the turbule nce should . lack reflectional sy m m etry ’ : th e s imples t mea-
s u r e of such lack of reflectional sy m m et ry is th e m ean helicity of th e
tu rbu lence ( U w ) . an d if this mean helici ty is non-zero. the n dy na m o
action will in general occur (Steenbeck, Krause, and Radler 1966. Mof-
fa t t 1970) . Th is result h as huge impl icat ions for curren t un ders tan ding
of the process by which magnetic f ields are generated in planets , s tars
an d galaxies: i t is in more senses th a n one, a t ru ly g lobal phen omen on .
4.2. R,elaxation to equilibrium states
The relaxation process is the converse of the dynamo process and is
bes t i l lus t r a ted by s imple example . S uppose th a t a t some in i t i a l ins tan t
t = 0 , a magnetic field Bo(z )having some nontr ivial knotted or l inked
s t ruc tu re i s imbedded in a perfectly con du ctin g. bu t viscous. incom-
pressible fluid a t res t . T h e pr ot oty pe consis t ing of two l inked, un kn ot-
8/3/2019 H.K. Moffatt- Local and Global Perspectives in Fluid Dynamics
Local and global perspect ives in f l u i d d y n a m i c s 533
where h = p / p + i u 2 a n d p is the f luid densi ty (assumed co ns tan t) . To
every so lution of (8) , the re corresponds a solution of (9) via th e analogy
B + u , j + w , p + h o - h ,
where ho is an arb i t rary con s tant . Th us , magnet ic relaxation provides anindirect means of proving the existence of steady Euler flows (solutions
of (9)) of arb itra rily complex streamline topology; consideration of their
s tabi l i ty is quite another matter (Moffat t 1986; see also Vladimirov,
M offatt , and I lin 1999).
5. EXISTENCE OF SMOOTH SOLUTIONS
OF THE NAVIER-STOKES EQUATIONS
FOR ALL T > 0
T h e problem of existence an d smoothn ess for th e Navier-Stokes (N S)
equa tions was addressed by Leray (1934) an d has att ra ct ed intense effort
in the mathematical community s ince then. Despite these effor ts , the
problem remains open; it has recently achieved millennia1 status as one
of th e seven problems identified by th e Clay In sti tu te for which a prize
of one million d ollars is offered. Cha rles Fefferman has given d etails for
the NS problem on the websi te of the Clay Inst i tute.I do n’t propose t o solve thi s problem toda y! I do wish however to
make some observat ions about i t , because i t seems to me that both
local and global considerations are likely to play a part in its solution.
Leray himself was unable to prove the existence of smooth solutions
of the N S equations in 3 dimensions for all t > 0, nd recognized the
al ter nat iv e possibi li ty th a t a s ingular i ty may develop a t f inite time, t*
say. If such a singularity develops at a point that we may take to be
z =0, then i t seems plausible that the approach to this s ingular i ty (ast + *) hou ld be a t least locally self-similar. Leray n oted a remarkable
similar ity t ransfo rm ation of th e NS equations, or equivalently of the
vorticity equation
dW
at- v x ( U x w ) + v v 2 w
Th is t r ans format ion is
where I? is a constant with the dimensions of circulation (velocity x
length ) . Th e vor tici ty the n t ransforms as
w = c u r l u = -Cl((X) . (11)1
t* -0 = cur1,U
8/3/2019 H.K. Moffatt- Local and Global Perspectives in Fluid Dynamics
whereE= u/r, dimensionless parameter that we may assume to besmall. The (x,) problem of (10) is thus converted to a 'steady' problem
( 1 2 ) in the scaled space variable X . The term dw/d2 in (10) transforms
to the term i V x ( X x f-2) in (12). which represents outward transport of
the vorticity field f-2 by the steady spherically symmetric 'compressible
velocity field' i X . The question now is whether there exists a distribu-
tion of vorticity such that the corresponding velocity U in conjunction
with diffusion can compensate this outward transport.
For reasons developed in a recent paper (Moffatt 2000a), it is relevantto seek solutions of (12) in an inner region (in the sense of matched
asymptotic expansions) for which O ( X )matches with w ( x , ) (the outer
solution) in an overlap region X + m, x + 0. At leading order, this
seems to require that
1 11
. R ( X ) - L a s X + x ,IN2
( X )-1x1
with corresponding outer behavior
(This, incidentally, is suggestive of Stokeslet behavior, which might be
thought to require a source of momentum in the inner region. and thus
to be incompatible with unforced evolution!)
It has been shown by NeEas, R6iiEka and SverAk (1996) that no
smooth nontrivial solution of (12) exists (with E > 0) for which IUI E
L 3 ( X 3 ) . he behavior (13) is jus t at the boundary of this function space,
and is not excluded by the theorem.
A vorticity singularity as represented by (11) can occur only through
stretching of vortex lines at a rate that also becomes infinite as t + t * .Such a process can conceivably occur through the nonlinear interaction
of vortex tubes each one of which provides the strain field acting on the
other. Some mechanism must be present to cause all length scales in the
inner region t o decrease to zero (like (t*- ) ' I 2 ) as t + t * . Scenariosinvolving the collision of non-parallel vortex pairs seem plausible candi-
dates (Pelz 1997, Moffatt 2000a), the collision zone being then the inner
region in which the Leray transformation is relevant. An alternative
scenario is sketched in Fig. 11, which shows a vortex pair knotted in
the internal zone and arranged in the external zone in such a way as to
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Locul u n d global perspec t i ves in fluid dy numic s 535
.pul l th e knot t ig ht ‘ . Th is is cer ta in ly suggest ive of how a finite-time
s ingula r i ty may ar ise if th e ini tia l condit,ions are sufficiently ingeniously
contr ived.
Figuie 11 Conceptual sketch of a vortex pair configuration fo r which propagation of
the pair in the external region leads to tightening of any knot that may be contained
within t he inner region (here represented simply by a ’black box‘).
Topology evidently plays a pa r t in th i s . Note tha t th e con t r ibu t ion to
helicity from any region V ( t ) :1x1 < R ( w i th R co ns ta nt) of the inner
zone is given by
?i/ c . . u d 3 x = r l R U , 2 d 3 X ,
an d th a t th is is ind epe nd ent of t im e. Th is suggests conservation of
topology of vortex tubes in the inner region d e s p i t e the znfluence of
viscoszty. I t i s pe rha ps here th at one may f ind a clue for t h e relevance of
global ( i .e . topological) considerations for t h e NS s ingularity p roblem.
6. A TOY MODEL OF A FINITE-TIMESINGULARITY
I t may be app ropr ia te to conc lude th i s lec tu re wi th a real finite-times ingu la r i ty ! Th is is exh ibite d by t h e toy ’Euler’s disk’ , a heavy disk that
ro lls on i t s edge (F ig . 1 2 ) ’ and exh ib i t s a paradoxical increase of rolling
speed as i ts energy decreases (Bendik 2000) . T he energy E is equal to
:Mgact. (M = m as s , a = r ad ius ) when th e ang le Q between th e disk and
th e table is sma l l ( i . e . du r ing th e f inal ‘ shuddering’ phase of m ot ion) .
8/3/2019 H.K. Moffatt- Local and Global Perspectives in Fluid Dynamics
tha t the Narr io t t Hote l has ne i ther imploded nor exploded under the
extrenic pressure of activity th a t i t has experienced this week. Now
we are at the stage where the singularity must be resolved: we shall
returri to our homes aro und th e world like acoust ic pulses radia t ing from
this source. and th e local ly acquired imp act of th is C ongress will surelyinform and inspire future research in theoretical and applied mechanics
on a global scale . O n th at note , an d l ike a finite-t ime singularity, I mu s t
bring this ta lk t o a sudden end .
Acknowledgments
I t han k S arah Ki rkup an d Jon a than Chin fo r he lp in process ing th e t ex t a nd
f iguies , and SIu stap ha Xm rani for help in prepar in g the pow erpoint version of t h e
lec tu re
References
.- lrnol~d,1.. I . 1974 . The asympto t i c Hopf invariant and i ts applicat ions. [English
Ba r e u b l a t t . G. . 1979. S i mz l a r tt y , S e l f - s i mz l a r it y a nd In t e r m ed i a t e As ymp t o t z cs . New
Bat rhe lo r . G . K . 1950. On the spontaneous magnet ic f ie ld in a conduct ing l iquid in
Be n d i k . . J . 2000. IYebsitc: h t t p : / / w w w . e u l e r d i s k . c om
Burgers . , J , 11. 1948. -4 m a t h e m a t i c a l model i l l u s t r a t ing the theory o f tu rbu lence .
Adva nces i n Appl ied Mechanzcs 1 : 171-199.
Hancock. C . . H . K . hlof fa t t . and E . Leivis . 1981. Effects of inertia in forced corner
flows. J o u r n a l of Flu id Mechanics 112, 315-327.
Hills, C . P.: a n d H . K . hlof fa t t . 2000. Ro tary honing: a var iant of t he Taylor pa int -
sc raper p rob lem. J o u r n a l of Flu id Mechanics 418, 119-135.
Jeffrey. D. J . , a n d J . D . S h e r w o o d . 1980. S t reamline pat terns and eddies in lo iv-
Re y n o l d s - n u m b e r f l o w J o u r n a l of Fluzd Mechanics 96, 315-334.
J eo rig . J . - T . . a n d H . K . 1Io ffat t . 1992. Free-surface cusps associated w ith floxv a t lowReynolds number . J o u r n a l of Flu id Mechanics 241. 1-22.
J o s e p h , D . D . ; J . S e l so n , 11 .R e n a r d y , a n d Y. enard y. 1991. Two-dimensional cus ped
interfaces. ,Journal of Fluid Mech a n i cs 223. 383-409.
K e r r . R. 11. 1997. Euler s ingulari t ies a nd turbule nce. I n Proceedings of t h e 1 9 t h I n t e r -
n u t z o n a l Co n g r es s of Theoret ical a n d Appl ied Mechanics (T. T a t s u m i , E. IVatan-
ab e. arid T. Karnbe , ed s . ) . Am ste rdam : E l sev ie r Sc ience P ub l i sher s , 57-70.
Leray. .J. 1934. Siir le r r iouvement d 'un l iquide v isqueux empl issant l 'espace. A c t a
Muthernutrca 63 193-248.
Sloffatt, H . K . 1 96 4a . i ' i scous and resis t ive eddies near a sharp corner . J o u r n a l of
Fluzct M e c h a n z c s 18. 1-18.
SIoffat t . H . K . 19641). I ' iscous eddies near a sh ar p corn er . A r c h i w u m M e c h a n i k i
S t o s o w a n e j 2. 365-372.
S lo f f a t t . H . K . 1969. T he degree of kn otte dne ss of tang led vortex l ines. J o u r n a l of
Fl u i d Mech a n i cs 36. 117-129.
t ransla t io i i : Se l e c t u Mathematzca Souet ica 5 , 327-345 (1986)l
York : Plenull l .
t u r b u l e n t m o t i o n. Proctedrnys of the Royal Socie ty A 201. 405-416.
8/3/2019 H.K. Moffatt- Local and Global Perspectives in Fluid Dynamics
Local a'nd global perspec t i ves in f l u i d d y ~ n a m i c s 539
Lloffat t , H . K . 1 9 70 . T u r b u l e n t d y n a m o a c ti on a t low m a g n e t i c Re yn o ld s n u m b e r .
Journal of Fluid Mechanics 41, 35-452.
S lo f f a t t ; H . K . 1985. M agn etosta t ic equi l ibr ia and analogous Eu ler f lows of a rbi t rar i ly
complex topo logy , Par t 1) F u n d a m e n t a l s . Journal of Fluid Mechanics 159. 5%
378.
Mo f f a tt , H . K . 1986. M agn etosta t ic equi l ibria an d analogous Euler flows of arb i t rar i ly
complex topo logy , Par t 2 , Stabi l i ty considerat ions. Journal of Fluid Mechanzcs
Moffat t . H . K . 2000a. T he in terac t ion of skewed vor tex pai rs : a model for bloxv-up of
h lo f f a t t , H . K . 2000b. Eu ler ' s d isk an d i t s f in ite- time s ingular ity . Nature 404, 33-834.
N o r e a u , J.-J. 1961. Co nsta ntes d ' un Plot tourbi llonnai re en f lu ide parfa i t ba rot r op e.
C ompt es R endus d. 1'Acade'mie d e s Sciences, P ar is 252, 810-2813.
KeEas, J.. M . RCiiEka, and V . Serfik. 1996. O n Leray's self-similar solutions of t h e
Navier -Stokes equat ions. Acta Mathemat ika 176. 83-294.P e l z, R . B . 1997. Locally self-similar f ini te- t ime collapse in a high-symmetry vortex
f i lament model . Physical Review E 55, 617-1626.
Steenbeck. 11,.F. r a u se , a n d K . - H . R ad ler . 1966. Berechriurig d er rni t t lcren Lorentz-
Feldstarke U x B fur ein elektr isch lei tencles hlediurn in tu rb ul en te r . du rch
Coriolis---Kraftebeeinnflusster Beivegung. Zeztschrzft Naturforschung Tezl A 21.
T a n e d a . S. 979. Visua lizat ion of se pa rat in g Stoke s flows. ,Journal of the Physzcal
Soczety of J a p a n 46, 935-1942.
Taylor , G. . 1960. Similar i ty solutions of h y d r o d y n a m i c p r o b l e m s . I n Aeronautzcs and
As t ronaut ics (D ura nd Anniversa ry Volum e) . S e w York : Pergam on , 21-28.
Van Dy ke, 51. 1982. A n A lb um of Fluid Motion. Stan ford , Ca l if .: Pa rabo l i c P ress .
Vladimirov. V . A . , H . K . h l o f f a tt , a n d K . I. I lin . 1999. On genera l t r ans fo rm at ions and
var ia tional pr inciples for th e ma gne tohy drod yna mic s of ideal f lu ids . Par t 4, cner -
alized isoirorticity principle for three-dimensional flows. Journal of Fluzd Mechanzc.5
166: 59-378.
th e Navier-Stokes equ at ion s. Journal of Fluid Mechanics 409. 1-68.