h.k. moffatt and d.w. moore- the response of hill’s spherical vortex to a small axisymmetric...

8/3/2019 H.K. Moffatt and D.W. Moore- The response of Hills spherical vortex to a small axisymmetric disturbance

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J . Fluid Mech. (1978), vol . 87 , part 4, pp. 749-760Printed in Great Britain

749

The response of Hills spherical vortex to asmall axisymmetric disturbanceBy H. K. MOF FAT T

School of Mathematics, University of Bristol, EnglandA N D D. W. MOORE

Department of Mathematics, Imperial College, London(Received 16 January 1978)

The response of a Hills spherical vortex to an irrotational axisymmetric small per-turbation is examined on the assumption that viscous diffusion of vorticity is negligible.The problem of determining the response is first reduced to a system of differentialequations for the evolution of the Legendre coefficients of the disturbance streamfunction. This system is then solved approximately and it is shown that , if the initialdisturbance is such as to make the vortex prolate spheroidal, the vortex detrains afraction $c of its original volume, where e is the fractional extension of the axis ofsymmetry in the imposed distortion. The detrained fluid forms a thin spike growingfrom the rear stagnation point. If the vortex is initially oblate, irrotational fluid isentrained a t the rear stagnation point to the interior of the vortex.

1. IntroductionHills spherical vortex (Hill 1894) provides one of the best-known examples of a

steady rotational solution of the classical equations of inviscid incompressible fluidflow, and it is rather remarkable that its stability characteristics have not beeninvestigated in detai1.t The spherical vortex is known t o be an extreme member of aone-parameter family of steady vortex rings (Norbury 1973) whose global stabilityto axisymmetric disturbances has been inferred by Benjamin (1976) in an approachto the stability problem based on the methods of functional analysis. Benjamin showsessentially that, subject to certain subsidiary conditions, the kinetic energy associatedwith any steady vortex ring is greater than the kinetic energy associated with anyneighbouring axisymmetric unsteady solution of the governing equations; thisproperty guarantees that disturbance energy cannot spontaneously grow at theexpense of the parent vortex. The analysis below, based on linear perturbation theory,is consistent with this conclusion, bu t a t the same time reveals a type of behaviourthat lies outside the scope of the functional-analytic approach, viz. if the sphericalsurface of the Hills vortex is subjected to an axisymmetric perturbation, then thisperturbation decays exponentially a t all points except in an exponentially decreasingneighbourhood of the rear stagnation point of the vortex (in a frame of reference in

t During the course of this work, it was drawn to our attention tha t Bliss (1973) has studied theproblem by methods similar to those described here, and has obtained equations equivalent to(2.19) below. He did not attempt an analytical solution of these equations, but drew similarconclusions to ours, from a numerical study, concerning the failure of a truncation technique.

www.moffatt.tc


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750 H . K . Moffatt and D . W . Moorewhich the undisturbed flow is steady); in this neighbourhood, the perturbationincreases without limit !

Physically, it is quite easy to see what is happening: since the vortex lines arefrozen in the fluid, the surface of the vortex behaves like a material surface. Theirrotational flow outside the vortex tends to sweep the perturbation towards the rearstagnation point where it develops an increasingly spiky structure. This sweepingprocess is modified by the self-induced velocity associated with the vorticity per-turbation at the vortex surface; but the modification is not such as to prevent con-centration and amplification of the disturbance at the rear stagnation point.

This type of spatially non-uniform behaviour is evidently associated with thepresence in the undisturbed flow of a rear stagnation point. It seems most unlikely th atsimilar behaviour could occur in the case of other (non-degenerate) members of theone-parameter family of vortex rings mentioned above. The process does neverthelesshave some points of contact with the processes of entrainment and detrainment ofvorticity in real ring vortices, as described by Maxworthy (1972) on the basis of visualstudies. Maxworthy found that vorticity is in general spread throughout the wholevolume of fluid that is carried along with a conventional vortex ring (although it issignificantly weaker outside the vortex ring than inside). Viscous diffusion plays adual role, in part causing a spread of vorticity and so an increase in the volume of thetranslating mass of fluid, and in part permitting detrainment of rotational fluid intoa thin wake which emanates from the rear stagnation point. We shall find that, forHills vortex, both entrainment and detrainment processes can in fact be understoodwithin the context of a purely inviscid analysis.

The rapid growth of the spike at the rear stagnation point of Hills vortex is similarto the rapid growth of the viscous boundary layer a t the rear stagnation point of animpulsively started cylinder (Proudman & Johnson 1962). In both cases, vorticity isfound a t small times in a region of outwardly convecting flow (viscous diffusion beingthe agency responsible for the initial generation of vorticity in the problem studiedby Proudman & Johnson).

An axisymmetric perturbation of a Hills vortex may be conceived in terms of anexternally imposed irrotational distortion. For example, if a small Hills vortex isembedded in irrotational flow through a contracting duct, then, relative t o axesmoving with the vortex, it experiences an axisymmetric irrotational strain, tendingto make it prolate, with major axis along the axis of symmetry. Alternatively, if thesmall vortex is carried through a ring vortex with the same axis of symmetry i t willexperience first an extensive strain (tending to make i t prolate), then a compressivestrain (tending to restore the spherical shape); nonlinear evolution of the Hillsvortex during the process of interaction will presumably lead to a net residual axi-symmetric perturbation which will continue to evolve after the actual interactionbecomes negligible.

The particular property of the Hills vortex that makes a stability analysis tractableis the uniformity of o/s throughout the vortex, where w is the azimuthal componentof vorticity, and s represents distance from the axis of symmetry. This propertypersists under unsteady irrotational perturbations, since the vorticity equation in aninviscid incompressible fluid may be written in the simple form


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Response of a spherical vortex to a disturbance 751where D / D t is the Lagrangian (or material) derivative. We shall restrict attention toexternally imposed irrotational disturbances of the kind discussed above, so that, ifw = As in the undisturbed vortex, where h is a constant, then w = As in the disturbedvortex also. Attention may then be focused on the evolution of the bounding surface8(t) f the region of rotational flow.

The equations governing small axisymmetric perturbations are derived in $ 2 .They can be reduced to an infinite set of ordinary differential equations for the co-efficients in a Legendre expansion of the perturbation. This infinite system whentruncated can be integrated without difficulty, but the truncation is valid only forcomparatively small times. For longer times, the coefficients fail to decay as theirorder increases. An approximate treatment, valid for large times, is presented in 0 3.

2. The evolution equationsThe spherical vortex may be represented by the Stokes stream function

in spherical polar co-ordinates ( r ,8 , x ) relative to an origin 0 at the centre of thevortex, with 8 = 0 parallel to the uniform stream U a t infinity. Note that, with thisorientation of axes, the rear stagnation point is at r = a , 8 = 0. The distance from theaxis of symmetry is s = rsin 8. The vorticity distribution is w = V A U = (0,0, wo ) ,where

i.e.0 (r> a)As ( r < a)O = {with h = - 15U/2a2.

Suppose now th at a small axisymmetric perturbation of the surface r = a of thevortex is imposed a t some initial instant t = 0. Then it s surface 8(t)will subsequentlydistort according to a n equation of the form

r = a(1+eh(O,t)). (2.4)Provided the perturbation is irrotational in the sense indicated in $ 1, the streamfunction $ ( r , 8)of the disturbed flow must satisfy( - . \ r y e in V+

in V-D2$ = - sin 8w ( r ,8 ) =where V, denote the exterior and interior of s(t)espectively. Let

Then, by virtue of (2 .3 )and (2.5), we have evidentlyD2$-,- = D2$-,+= 0,

i.e. the disturbance stream function $-,epresents an irrotational flow.(2.7)


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752 H . K . Moflatt and D. . MooreThe general axisymmetric solutions of (2.7), with $1- regular at r = 0 and

giving at most a dipole velocity field as r -+CO, are

m$l+(r, 8 , ) =where p = cos0.

The velocity field is (U, ,O ) , where1 a$ U=--- 1 a$= --2sin08' rsine ar 'and is thus given by

Continuity of U (or of $) across S(t) o order E givesAn = B,. (2.12)

Continuity of U (or equivalently of the pressure fieId) acrosss(t)o order E then gives

Now the surfaceS(t) s a material surface in the fluid; henceDFt(r-a-aeh(O,t)) = 0,

or equivalentlyu = a e -+-- on S(t).( 2)

Hence, correct to order E, we have, using (2.4) and (2.10),

(2.13)

(2.14)

(2.15)or, using the expansion (2.13), and the Legendre equation satisfied by Pn(p),

(2.17)


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Response of a spherical vortex to a disturbance 753The fkst term on the right of (2 .15) [or ( 2 . 1 6 ) ] epresents convection of the surface ofthe vortex by the velocity field associated with the undisturbed flow, while the secondterm represents the effect of the induced velocity a t the unperturbed surface r = adue to the vorticity perturbation distributed over the whole surface. It is evidentfrom comparison of the two series on the right of (2 .16) hat the former effect dominateswhenn 1.From (2 .13) , (2 .17) ,and the identity

( 2 n+ 1)P,(p) P;+l(p)-P;-,cu)we now easily find the equations satisfied by the coefficientsA , ( t ) , viz.

(n = 192, . ) I ( 2 . 1 8 )( 2 n + 1)--=& A, 3 ( n - 1)[n(n- )An-1 - n+ 1)(n+ 2 )A,,,] (n = 1 , 2 , ...). (2 .19)2 n + 3dt

Note immediately that d A , / d t = 0, so thatA , = constant = C say. ( 2 . 2 0 )

This merely reflects the conservation of impulse for the disturbed vortex. The impulseI and kinetic energy T of the vortex (Batchelor 1967, chap. VII) are in fact given byI / p U a s = - n+4nA1+ O(sa),

T / p U a a 3= -+Qn- nsA, +O(s2),The coefficientsA , ( t ) are simply related to the coefficientsh,(t) defined byand constancy of both quantities (atorder E ) is guaranteed by ( 2 . 2 0 ) .

0Comparing ( 2 . 2 3 )and (2 .13) ,andusing (2 .18)again, we have-,-' -- - -- ( 2 n + 1 ) A , (n = 1 , 2 , 3 , ...).2n-1 2 n + 3 15Summing the odd and even members of this set of equations gives

2 *15 12 " 315 1

h , = -- C ( 4 n - 1)Aan-,,i h , = -- ( 4 n + 1 ) A 2 , .

(2 .21)( 2 . 2 2 )

(2 .23)

(2 .24)

(2 .25)(2 .26)

Now conservation of the volume of rotational fluid implies that h, = 0 for all t ; in fact,inspection of the system (2 .19) does confirm that

( 2 . 2 7 )"-dt 1 ( 4 n - 1)A z n - l ( t )= 0,consistent with (2 .25) .We also obtain

(2 .28)(2 .29)so that, from ( 2 . 2 6 ) , hl = - CUt /5a+D ,


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754 H . K . Mofa t t and D . W . Moorewhere D is a constant of integration; we may choose the time origin t = 0 so thatU = 0. It is evident that (2.29)describesa translation of the vortex without distortionrelative to the origin in which the unperturbed vortex is stationary.

We expect that, so long as the surface of the vortex does not exhibit any largegradients, the coefficientsA , will decrease rapidly with n. This suggests solving thesystem (2.19)by setting A N + 1 , N+ , , .. equal to zero for some value of N . (This wasthe approach adopted by Bliss 1973.)

The equations were integrated (with N = 50) using a fourth-order Runge-Kuttascheme, the choice of time step being dictated by the shortest time scale occurringin the solution; this decreases with N in a way we estimate theoretically from theresults of 5 3. The initial disturbance is a spheroidal distortion, i.e.

(2.30)

the values of A,(O) are obtained from (2.24) and the results are shown in table 1.Clearly when U t l a = 3.0, the coefficients are not declining rapidly enough with n tojustify the truncation.

The reason for this behaviour can be deduced from an approximate solution of thegoverning equations (2.19) valid for n $ 1 . As a first step, the equations (2.19)areexpressed in a simpler form in terms of new variables. Let

A n +, (n= 0 , 1,2, . )n(n+ 2) (n+3)2 n + 5, = (2.31)and 7 = 3UtIPa. (2.32)The equations (2.19) then become

da,/d.r = c,(a,-,-a,+,) (n = 1,2, ...) (2.33)where c, = n [ l - (2n+5)-,]. (2.34)When a,,a%, s, have been determined by solving this system of equations, thevalues of A , , A ,, A,, . are determined a t once. Then A , and A , are given by

A , = C , (2.35)The character of the solutions of (2.33) may be anticipated by the following argu-

ment. For n $ 1, c, - n, and (2.33) may be compared with the partial differentialequation atqa7 = - 2naa/& (2.36)for a function a(n,T),n being regarded as a continuous variable. The solution of(2.36)satisfying an initial condition a(n,0) = H(n) s

a(n,7) H(ne-2').For example, if

then 10 elsewhere.

(n, 2T Q n Q n2eZ7),a(n,7)=

(2.37)

(2.38)

(2.39)


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Response of a spherical vortex to a disturbance 755Ut lo AI0 A*, Am A 50

3.0 - .894 x 10-3 - .691x 10-3 - 1.063x lops - .752 x lO-' -4.437 x 10-41.0 - .698 x 10-3 - .144 x 10-6 - .526 x 10-* - .233 x 1O- l 0 - .332 x 1O - l a2.0 - .352 x 10-3 - 1.174x 10-3 - .910 x l OP 4 - .026 x lows - .345 x 10-64.0 - .605 x 10-3 - 1-729x 10-8 - .121 x 10-4 - .836 x 10-4 - -414x low4

TABLE1. The Legendre coefficients of the stream function obtainedfrom the governing equations (2.19) truncated at Ab0.

This ndicates the manner in which the 'spectrum' of the disturbance evolves towardshigher values of n as 7 increases.

3. The behaviour for large 7It is evident from (2.34) that replacement of c, by n is a good approximation for all

n 2 1; he error when n = 1 is only 2% and it decreases rapidly as n increases. Wetherefore study here the behaviour of the system

da, /d~= n(an-l-a,+l) (n= 1 , 2 , ...),where a0(7)= 0.The initial values of a, are given in terms of the initial distortion by

(n = 1 , 2 , ...), (3 .3)or, consistent with the approximation leading to (3 .1) ,

a,(())= -- (n= 1 , 2 , ...). (3 .4 )The system (3 .1 )admits a particular solution

4 4 7 ) = (th7)" (n = 0 , 1 , 2 , ...) (3 .5 )(where th 7 3 tanh 7); from this we can construct a family of particular solutions

each of which satisfies (3 .2 )and the initial conditions1 (n = r ) ,0 (n > r ) .ag'(0)= (3 .7 )

Thus if a,( 0) = 0 for all n > nowe can construct a solution of (3 .1 )and ( 3 . 2 )by forminga linear combination

it is easy t o see tha t the constantspr are uniquely determined.As in the numerical work described in 8 2, we select for detailed study the case of an


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756 H . K . Moj&tt and D . W . Mooreinitially spheroidal distortion, so that the constants h,( 0) have the values specifiedin (2.30). It follows from (3.4) that

al(0) = - 3 ,a,(O) = 0 (n 2 2).

This is the caseno= 1and thus(3.9)

(3.10)d8 d7,@) = --- (th7), (n = 0,1,2, ...).It is easy to verify that , for largen, la,(~)Ihas a maximum when

and the maximum value itself is(3.11)

(3.12)This confirms the predicted shift of the spectrum towards large values of n as timeincreases. Note that (3.11) is compatible with the approximation embodied in (2.36),whichimplies that a(n, ) s constant on the characteristic curves 27 = logn+ constant.

The initial conditions (2.30) and the relation (2.24) give A,(O) = $ and hence, from(2.35), = 4. Thus, from (2.13) and (2.31)

h = - +-2A2(t)p+&sech27 ~ (th7)m-3Pk(p). (3.13)2%+ 1)2n=3 n(n+ I )Now (2n+ l), 1

n(n+1) (3.14)so that i t is consistent with the approximations already made in obtaining (3.1) and(3.4) to replace the left-hand side of (3.14) by 4 in working out the sum in (3.13). Thuswe find

(3.15)The infinite series is easily summed and finally

h = - - A2(t)p+6 cosech2 [ (1- pth7+ h27)-*- 1- pt h ~ ] . (3.16)The asymptotic behaviour for 7 9 1 is now easily determined. With the approxi-

mations(3.16) gives cosech27N 4e-2T and th7 N 1- 2e-ZT,h N - + - A2(7)- e - 2 T ) p +2e-2T{[4e-4T+2(1 p)]-8 -l}, (3.17)an expression that is uniformly valid in p. The singular behaviour near the rearstagnation point ,U = 1 is now readily apparent. If p is fixed and not equal to 1 hen

h N - 6 - - ,",,a(. + constant)+O(e-2'), (3.18)using (2.35) to determine the asymptotic behaviour of A , . The term proportional top represents a steady drift without change of shape; the propagation speed is altered


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Response of a spherical vortex to a disturbance 757by a factor 1-Qe.Thus, apart from a small residual change of radius, the perturbationdies away at every point of the surface except in a decreasing neighbourhood of therear stagnation point. Right a t the rear stagnation point, where p = 1, (3.17) gives

h N &e4?, (3.19)so tha t the perturbation here increases exponentially. Thus we can form the followingpicture of the development of the perturbation. When E > 0 (the perturbed vortexbeing then slightly prolate) vorticity is swept off the surface r = a (leaving i t a t theslightly diminished radius a(1- 8)) and is shed into a spike in the neighbourhoodp z 1. When E < 0 (corresponding to an initially oblate perturbation) the radiusincreases to a(1 - e ) , this increase being brought about by entrainment of irrotationalfluid into the region r < a in the neighbourhood p E 1. The change of propagationspeed is just that required by this change of radius.

The detailed shape of the spike can be found from (3.17); restoring the physicalvariables r, 0 and t,

r = a + E exp (3~ t / a )I +$eZexp (3Ut/a)]-+. (3.20)According to (3.20) the volume of the spike is V ,where

V = +m3eexp (3Ut/a) 0[1 +$@exp (3Ut /a)]- )d~, (3.21)cwhere 8 is any angle such that

e x p ( - 3 ~ t / a a )< 0 4 I . (3.22)The integral is easily evaluated to give (for sufficiently large t),

v N 97Tea3, (3.23)which is just the volume lost by the Hills vortex when its radius shrinks from a toa(1-@). Thus, when E > 0, a definite volume of rotational fluid is detrained from theHills vortex into a spike growing from the rear stagnation point. Similarly, whenE < 0 , a definite volume of irrotational fluid is entrained in an indented spike at therear stagnation point. The form of (3.20)reveals tha t the growth of the spike in eithercase is controlled entirely by the convective effects of the undisturbed flow field, asanticipated in 02. The motion of a fluid particle near the rear stagnation point in theflow field given by (2 .1) is governed by the equations

adrldt = 3U(r-a), ad@/&= -gue, (3.24)which may be integrated t o give

I- a = (r0-a)exp[3Ua-~(t-to)],e = e,exp[-3~( t - t , ) /2~,] , (3.25)where ro and do are the co-ordinates of the particle a t t = to. I f we use the solution(3.25) to determine the motion of fluid particles which lie a t t = toon the surface

ro- = &ze exp (3Uto/U)[I + $6); exp (3U t , / a ) ] - ~ , (3.26)we see a t once that at time t ( > o ) hey lie on the surface (3.20).


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758 H . K . MouSfatt and D . W . MooreThis suggests a way of removing the restriction to small disturbances. Equation

h s e x p ( 3U t l a )< 1, (3.27)(3.20) s valid so long astha t is, so long as the spike is small. Pick a time to for which (3.27) holds and followthe subsequent evolution of the surface (3.26)using the full equations for convectionof particles near the axis. When E > 0 , so that the spike grows outwards, these are

dr/dt = U ( - s/@),rdO/dt = - U ( i +d/2@),

and when E < 0, so that the spike grows inwards,(3.28)

(3.29)

Results obtained in this way for E = + 10-4 are shown in figure 1. Evidently the spikegrows very rapidly even for a small initial disturbance. This conclusion is supportedby calculating the time needed for the spike to reach a point 2a downstream of therear stagnation point.

The calculation is in two stages. First, according to linear theory the fluid particleinitially at r = a(1 + E ) , 8 = 0 is at r = a(1+ 7 ) , 8 = 0 at t = to , where, according to(3.19), f to9 a / U , 7 = &cexp (3Uto/a), (3.30)and this is valid so long as 7 < 1. Secondly, the time T a t which a fluid particle startinga t r = a(1+7 ) at t = to arrives at r = 2a can be found from the first of equations (3.28)and satisfies U ( T- o)/a= 1.824.. ++log (7-l) +O ( 7 ) . (3.31)Thus on substituting for 7 from (3.30)we find th at to cancels and

U T / a = 2.592.. +6 log ( E - ~ )+O(E) . f3.32)When E = 10-4, U T / a = 5.66. ..,so th at the spike is one diameter downstream by thetime the vortex has travelled about 24 diameters.

When E < 0 , he indented spike will approach the oruurd stagnation point, although,under the effect of convection alone, i t will never quite reach i t ; its surface will insteadpresumably be increasingly distorted by the internal convective action of the vortex,its meridian section ultimately developing a tight double spiral structure (see, forexample, figure 12 of Maxworthy 1972). Viscosity would of course in time tend toeliminate variations in o /s within the new vortex.A surprising feature of the results presented in this section is tha t the response ofHill's vortex contains no oscillatory components. This conclusion is, however, basedon the approximate treatment of the differential equations embodied in (3.1) and(3.4).This approximation can be shown not to violate conservation of volume ormomentum and to lead to a set of coefficientsA , which differ from the values obtainedby truncation by only 5-10%; of course, th e time a t which the comparison is mademust be small enough for the truncation method to be valid.


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Response of a spherical vortex to a disturbance 759

I.1FIGURE. The shape of the spike in the case B = iO - 4 for Ut/a= 3 .2(dashedcurve) and Ut /a= 3 .7 (continuouscurve).

However, it is worthwhile t o examine the exact system ( 3 . 1 9 ) o see whether or notit permits free oscillations. If we writeA& ) = 2,exp ( i o U t / a ) , ( 3 . 3 3 )and substitute in the differential equations ( 2 . 1 9 ) ,we find th at

n ( n - 3 ) - ( n + l ) ( n + 2 ) -A,,1] (n = 1 , 2 , 3 , ...). ( 3 . 3 4 )2 n + 32n+ l)iwA, = 3 ( n - 1 )- n-1 -[ 2 n - 1The asymptotic form of the solution of this system of difference equations for large nis easily determined :where C and D are constants. An acceptable solution must satisfy B,+O at anexponential rate as n+ 0 and this behaviour cannot be secured by any real choiceofo. hus free oscillations are not possible.

3, Cn-@u+D( )" n&, ( 3 . 3 5 )

We are grateful to Prof. P. G. Saffman for drawing our attention to the work ofBliss in the course of a valuable discussion of the problem with one of us (DWM).The help of Dr J. Toomre in some early computational studies of the system ( 2 . 1 9 )is also gratefully acknowledged.


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760 H . K . Moflatt and D . W . MooreREFERENCES

BATCHELOB,. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.BENJAMIN,. B. 1976 The alliance of practical and analytical insights into the nonlinearprobloms of fluid mechanics. Lecture Notes in Mathematics, no, 503 (ed. P. Germain & B.Nayroles), pp. 8-29. Springer.BLISS,D. B. 1973 The dynamics of flows with high concentrations of vorticity. Ph.D. thesis,Dept. of Aeronautics and Astronautics, MIT.B m , M. J. M . 1894 On a spherical vortex. Ph2. Tram.Roy. Soc. A 185,213-245.MAXWORTBY,T. 1972 The structure and stability of vortex rings. J . Fluid Mech . 51, 15-32.NORBURY,. 1973 A family of steady vortex rings. J . Fluid Mech. 57, 417-431.PROUDMAN,. & JOHNSON,. 1962 Boundary layer growth near a rear stagnation point.

J . Flzcid Mech. 12, 161-168.

h.k. moffatt and d.w. moore- the response of hill’s spherical vortex to a small axisymmetric...

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