Fundamentals of vortex dynamics

Andrew David Gilbert,

Mathematics Department, University of Exeter, UK

Leverhulme Trust Research Fellowship - 2019 Motivation: vortices and more vortices

• wing-tip vortices (NASA)

• draining Lake Texoma, USA Turbulence

• Leonardo da Vinci’s sketch

• vortices in turbulence simulations

• vortices in quantum turbulence Instabilities

• von Karman vortex street

• Crow instability

• Kelvin-Helmholtz instability

• Widnall vortex ring instability or or Dt⇢ + ⇢ u =0, D ⇢ + ⇢ u =0(2.25), (2.25) r · t r · which captures nicely the ideawhich that the captures density nicely of a the fluid idea element that the changes, density by of a fluid element changes, by D ⇢or, decreasing if the divergence of the fluid flow u > 0, and increasing if t Dt⇢, decreasing if ther divergence· of the fluid flow u > 0, and increasing if u < 0. u ⇢0, andu =0(2.25) increasing, if (2.25) 2.5 Navier–Stokeswhich captures equation nicely again2.5 the Navier–Stokest idea thatr · the equation density againr of· a fluidr element· changes, by uwhich< 0. captures nicely the ideawhich that captures the density nicely of the a fluid idea element that the changes, density of by a fluid element changes, by r · Dt⇢, decreasing if theD divergence⇢, decreasing of the if the fluid divergence flow ofu the> 0, fluid and flow increasingu > 0, if and increasing if Dt⇢, decreasing if the divergencet of the fluid flow u > r0,· and increasing ifr · u < 0. We nowu < have0. the Navier–Stokesr · equation (2.14) and the continuity equation We now haveru <· the0. Navier–Stokesr equation· (2.14) and the continuity equation (2.23),2.5 amountingr Navier–Stokes· to 4 equations equation(2.23), in 3-d again amounting space, for to 5 4 quantities, equations in the 3-d components space, for 5 quantities, the components u of u, p and ⇢. The system would need to be completed by an equation of ui of u,2.5p and2.5 Navier–Stokes⇢ Navier–Stokes. The system equation would equationi 2.5 again need Navier–Stokes again to be completed equation by again an equation of state, for example linking pressurestate, to for density. example linking pressure to density. We now have the Navier–Stokes equation (2.14) and the continuity equation We will however adopt the simplest fluid dynamical system in which to under- We willWe howeverWe now now have adopt have the the Navier–Stokes the simplest Navier–StokesWe fluid now equation havedynamical equation the (2.14) Navier–Stokes system and (2.14) the in and continuity which equation the to continuity under- equation (2.14) and equation the continuity equation (2.23), amounting to 4 equationsstand vortex in 3-d dynamics space, and for topological 5 quantities, aspects, the components taking the dynamic viscosity stand vortex(2.23),(2.23), dynamics amounting amounting and to 4 topological equations to 4(2.23), equations in amounting aspects, 3-d space, in 3-d taking to for space, 4 5 equations quantities, the for dynamic 5 quantities, in the 3-d components space, viscosity the for components 5 quantities, the components ui of u, p and ⇢. The systemµ and woulddensity need⇢ to be to constant be completed (and uniform by an in equation space). Then of the continuity µ and densityu of u, ⇢p toand be⇢. constant The system (andui of wouldu uniform, p needand ⇢ in to. The space). be completed system Then would by the an need continuity equation to be completed of by an equation of state,i forui of exampleu, p and linking⇢. Theequation pressure system expresses would to density. need that to the be fluid completed flow u is by divergenceless, an equation of and with this we equationstate, expresses for example that linkingthe fluid pressurestate, flow foru to exampleis density. divergenceless, linking pressure and with to density. this we state, for example linkingcan also pressure rewrite the to density. Navier–Stokes equation as: can also rewrite the Navier–Stokes equation as: WeWe will will however however adopt adopt the the simplestWe simplest will however fluid fluid dynamical adopt dynamical the system simplest system in which fluid in dynamical to which under- to system under- in which to under- We will however adopt the simplest fluid dynamical system in which2 to under- standstand vortex vortex dynamics dynamics and andstand topological topological vortexDtu aspects, dynamics=2 aspects,@tu taking+ andu taking topological theu = dynamic thep dynamic+ aspects, viscosity⌫ u, taking viscosity theu dynamic=0. viscosity(2.26) standDtu vortex= @tu + dynamicsu uNavier-Stokes= and topologicalp + ⌫ u, aspects,and· rEuleru taking=0r. equations the dynamic(2.26)r viscosityr · µ and density ⇢ to be· constantr µ andr (and density uniformr⇢ to in be space). constantr · Then (and the uniform continuity in space). Then the continuity µ and density ⇢ to be constantHere we have (and divided uniform through in space). by the Then constant the⇢ continuityand replaced p/⇢ by p (to equationµ and expresses density that⇢ to the beequation constant fluid flow expresses (andu is divergenceless, uniform that the in fluid space). and flow with Thenu is this divergenceless, the we continuity and with this we Hereequation we have expresses divided through thatavoid the by the fluid too constant many flow symbols!).u⇢isand divergenceless, replaced The kinematicp/⇢ by and viscosityp (to with⌫ this= µ/⇢ weis a constant, and canequation also rewrite expresses the Navier–Stokes that•canTake the also constant fluid equation rewrite flow density, the as:u Navier–Stokes isconstant divergenceless, viscosity, equation incompressible and as: with this flow we and write avoidcan too also many rewrite symbols!). the Navier–Stokes Thewe kinematic usually refer viscosity equation to this⌫ as: as= justµ/⇢ theis a viscosity constant, in and what follows. can also rewrite the Navier–Stokes equation as: 2 we usually refer to this as just the viscosityD intu what= @2tu follows.+ u u = p + ⌫ u, u =0. (2.26) Dtu = @tu + u u = p + ⌫ u, u =0. (2.26) · r r r 2 2r· r· r r r · Dtu =D@tutu=+@tuu + uu =u = p +p⌫+ ⌫u, u, u u=0=0. . (2.26)(2.26) Here we have divided through2.6Here· Eulerr by we· ther have equationr constant dividedr for⇢r ideal throughandr flow replaced byr ther·p/· constant⇢ by p (to⇢ and replaced p/⇢ by p (to 2.6 Euler equation for ideal flow Hereavoid weHere too have we many divided have symbols!). divided through• Theavoidwith through kinematic kinematic too by many the by the viscosityconstantviscosity symbols!). constant ⌫ =⇢ The µ/and ⇢ ⇢ kinematicand isand replaced a replaced constant,replacing viscosityp/ andp/ ⇢ ⇢ by ⌫byby= p pµ/ for(to(to⇢ convenienceis a constant, and we usually refer to this as just the viscosity in what follows. avoidwe too usuallyavoid many too refer many symbols!). to this symbols!). asThe just The Euler the Thekinematic viscosity equation kinematic in viscosity is what the viscosity case follows.⌫ of=⌫ (2.26)=µ/µ/⇢ is⇢ withoutis a a constant, constant, viscosity and and⌫ =0,andforms Thewe Euler usuallywe equation usually refer is to refer the this case to asthe this• of justEuler basis as (2.26) thejust equation of withoutmuch viscosity the viscosityfor of ideal our viscosity in discussion flow what in what ⌫ follows.=0,andforms follows.below: (highly singular limit) the basis2.6 of much Euler equation of our discussion for ideal2.6 flow below: Euler equation for ideal flow Dtu = @tu + u u = p, u =0. (2.27) 2.6 Euler equation for ideal flow · r r r · 2.6 Euler equationDtu = @tu for+ idealu u flow= p, u =0. (2.27) The Euler equation is theSetting·The caser Euler of⌫ =0in(2.26)isahighlysingularlimitas (2.26)r equation withoutr is· the viscosity case of⌫ (2.26)=0,andforms without viscosity⌫ multiplies⌫ =0,andforms the highest Settingthe⌫ =0in(2.26)isahighlysingularlimitas basis of much of ourderivatives discussionthe basis below: in of the much equation! of our⌫ multiplies discussion A fluid with below: the⌫ highest=0isalsodescribedasanideral The Euler equation is the case of (2.26) without viscosity ⌫ =0,andforms derivatives in the equation! Afluid fluid—thereisnofrictionandnodissipationofenergy.TheEulerequation with ⌫ =0isalsodescribedasanideral The Euler equation is the case of (2.26)D withouttu = @tu + viscosityu u =⌫ =0,andformsp, u =0. (2.27) the basis ofD muchtu = of@cantu our+ beu discussion writtenu = in below: thep, attractiveu =0 form·.r r (2.27)r · fluidthe—thereisnofrictionandnodissipationofenergy.TheEulerequation basis of much of our discussion· r below:r r · can be written in the attractive formSetting ⌫ =0in(2.26)isahighlysingularlimitas⌫ multiplies the highest Setting ⌫ =0in(2.26)isahighlysingularlimitasDtu = @tu + u u = p,⌫@ multipliesu = u u =0! the. highestP, (2.27) (2.28) derivatives in· r the equation!r t A fluidr · ⇥ withr⌫ =0isalsodescribedasanideral derivatives in theDtu equation!= @tu A+ fluidu withu =⌫ =0isalsodescribedasanp, u =0. ideral (2.27) fluidSetting—thereisnofrictionandnodissipationofenergy.TheEulerequation⌫ =0in(2.26)isahighlysingularlimitas@t(usingufluid= u vector—thereisnofrictionandnodissipationofenergy.TheEulerequation·!r calculusP,r identityr (2.5)),· ⌫ wheremultiplies(2.28) the Bernoulli the highest function is can be⇥ writtenr in the attractive form Settingcanderivatives be⌫ written=0in(2.26)isahighlysingularlimitas in in the the attractive equation! form A fluid with ⌫ =0isalsodescribedasan⌫ multiplies the highestideral (using vector calculus identity (2.5)), where the Bernoulli functionP = p +is1 u2 (2.29) fluid —thereisnofrictionandnodissipationofenergy.TheEulerequation2 derivatives in the equation!@ Au fluid= u with! ⌫P,=0isalsodescribedasan@tu = u ! (2.28)P, ideral (2.28) t 1⇥2 r ⇥ r fluid —thereisnofrictionandnodissipationofenergy.TheEulerequationcan be written in theandP attractive the= pvorticity+ 2 u formis given by (2.29) (using vector calculus identity (2.5)), where the Bernoulli function is can(using be written vector incalculus the attractive identity (2.5)), form where the Bernoulli function is and the vorticity is given by @tu = u ! P, ! = u1 , 2 (2.28) (2.30) 1 2⇥ r P =r⇥p + 2 u (2.29) P = p + 2 u (2.29) (using vector calculus! identity@=tu = (2.5)),uu, ! whereP, the Bernoulli function(2.30) is (2.28) and the vorticity⇥ isr given by and the vorticity is given by r⇥ 6 (using vector calculus identity (2.5)),P where= p + the1 u2 Bernoulli function is (2.29) ! = u, 2 ! = u,(2.30) (2.30) 6 r⇥ r⇥ and the vorticity is given by 1 2 P = p + 2 u (2.29) 6 6! = u, (2.30) and the vorticity is given by r⇥ ! = u, (2.30) r⇥6

6 which is another divergence-free vector field

! =0, (2.31) r · (by vector calculus identity (2.10)). We have u =0byincompressibility,but that is only an approximation (no fluid is absolutelyr· incompressible), whereas ! =0exactlyas! is defined as the curl of u. r · Note that by taking the divergence, the pressure is given by

2P = (u !). (2.32) r r · ⇥ Actually we have made a slightwhich lapse is another here: P divergence-freeis actually the vector Bernoulli field function, but we often slip into callingwhich the pressure. is another Whatever divergence-free it is called,vector field obtaining P is thus a non-local procedure:which the is another flow and divergence-free vorticity at allvector points field! in=0 the, domain (2.31) a↵ect all other points. In a numerical code, this Poisson equationr!· =0, has to be (2.31) r! ·=0, (2.31) solved each time-step. (by vector calculus identity (2.10)).r · We have u =0byincompressibility,but that(by vector is only calculus an approximation identity (2.10)). (no We fluid have is absolutelyur=0byincompressibility,but· incompressible), whereas that is only an approximation (no fluid is absolutelyr· incompressible), whereas (by vector! calculus=0exactlyas identity! (2.10)).is defined We have as the curlu =0byincompressibility,but of u. thatr is· only! =0exactlyas an approximation! is defined (no fluid as is the absolutely curlr· of u. incompressible), whereas 2.7 Vorticity equation, advectionr · and di↵usion of vorticity !Note=0exactlyas that by taking! is the defined divergence, as the curl the of pressureu. is given by r · Note that by taking the divergence, the pressure is given by Note that by taking the divergence,2 the pressure is given by Taking the curl in (2.28) eliminates the pressure term involving2P P= = (uP(u,! and).!) gives. (2.32)(2.32) rr r ·rr· ⇥ ⇥ the vorticity equation 2P = (u !). (2.32) ActuallyActually we we have have made made a ar slight slight lapser lapse· here: here:⇥P isP actuallyis actually the Bernoulli the Bernoulli function, function, @t! = (u !), (2.33) Actuallybutbut we we we often haver⇥ slip made slip into into⇥ a slight calling calling lapse the the pressure. here: pressure.P is Whatever actually Whatever it the is it Bernoullicalled, is called, obtaining function, obtainingP is P is or (using (2.7)), thus a non-local procedure: the flow and vorticity at all points in the domain but wethus often a non-local slip into calling procedure: the pressure. the flow Whatever and vorticity it is called, at all points obtaining in theP is domain a↵ect all other points. In a numerical code, this Poisson equation has to be thusDt!a a=↵ect non-local@t! all+ otheru procedure:! points.= ! the Inu a flow. numerical and vorticity code, at this(2.34) allPoisson points in equation the domainhas to be solved each time-step.· r · r a↵ectsolved all other each points. time-step. In a numerical code, this Poisson equation has to be This is forVorticity ideal flow, whereas solvedequation each for time-step.⌫ > 0 we have instead This can also be written in2.7 the Vorticity attractive equation, form advection2 and di↵usion of vorticity Dt! = @2.7t! + Vorticityu ! = equation,! u + advection⌫ !, and di↵usion(2.35) of vorticity 2.7 Vorticity· r equation,· advectionr r and di↵usion of vorticity • Take the curl of ⇥ u = u P or we can replace either theTaking nonlineart the curl terms⇥ in (2.28)⌅ or the eliminates viscous the term pressure giving term involving P , and gives Takingthe vorticity the curl equation in (2.28) eliminates the pressure term involvingr P , and gives where the Bernoulli functionTakingthe theisvorticity curl in equation (2.28) eliminates the pressure term involving P , andr gives • to obtain vorticity equation for ideal flow @t! = (u !), r (2.33) @t!the= vorticity(u equation!) ⌫ ( !). r⇥ ⇥ (2.36) r⇥ ⇥ r⇥ r⇥@t! = (u !), (2.33) or (using (2.7)), 1 2 r⇥ ⇥ P = p + u @t! = (u !), (2.33) The vorticity equation, howeveror (using it (2.7)), is written,2 D lookst! = much@tr⇥! + u simpler⇥! = than! u the. (2.34) or (using (2.7)), · r · r Navier–Stokes equation as the pressure has been eliminatedDt! = @t! from+ u the! equation.= ! u. (2.34) and the vorticity is given byThis is for ideal flow,D ! whereas= @ ! for+ u⌫ > 0! we=· r have! insteadu. · r (2.34) In many ways it is much easier to think of thet behaviourt of· r vorticity· r without • or for non-zero viscosity = u the non-local pressure fieldThis to deal is forwith, ideal and flow, this whereas is the great for ⌫ benefit> 0 we of have working instead2 This is for ideal flow,⌅⇥ whereasDt! = @ fort! +⌫ u> 0 we! = have! insteadu + ⌫ !, (2.35) whichwith is vorticity another instead solenoidal of velocity. vector But, field some of this simplicity· r is illusory· r as ther D ! = @ ! + u ! = ! u + ⌫ 2!, (2.35) flow velocity• eliminatesu has to pressure be foundor we but can from still replace have inverting either thet tricky the the nonlineart curllink ! = terms oru the. This viscous2 term giving Dt! = @t! + u ! =· r! u +· r⌫ !, r (2.35) can be written as a Biot–Savart integral (compare=0 finding· r the magnetic·r⇥r fieldr b or we can⌅ replace· either@ ! = the nonlinear(u !) terms⌫ or( the! viscous). term giving(2.36) arising from a given currentor we distribution can replace eitherj = thet b nonlinearinr⇥ magnetostatics).⇥ terms orr⇥ the Overall viscousr⇥ term giving Notesolving that this by equationtaking the! = divergence,u for u theamounts pressurer⇥ to obtaining is given by the pressure by r⇥The vorticity equation,@t! however= ( itu is written,!) ⌫ looks( much!) simpler. than the(2.36) solving the elliptic equation (2.32). @t! = (r⇥u !) ⇥ ⌫ ( r⇥!r⇥). (2.36) Navier–Stokes2P = equation(u r⇥ as) the pressure⇥ hasr⇥ beenr⇥ eliminated from the equation. In many ways it is much easier to think of the behaviour of vorticity without TheThe vorticity⌅ vorticity equation,⌅ equation,· ⇥ however however it is written, it is written, looks muchlooks simplermuch simpler than the than the the non-local pressure field to deal with, and this is the great benefit of working Actually we have lapsedNavier–Stokes here:Navier–StokesP is equation the equation Bernoulli as the as pressure the function, pressure has been hasbut eliminated been we eliminatedoften from slip the from equation. the equation. with vorticity7 instead of velocity. But, some of this simplicity is illusory as the In manyIn many ways ways it is much it is much easier easier to think to think of the of behaviour the behaviour of vorticity of vorticity without without into calling the pressure. Whateverflow velocity itu ishas called, to be found obtaining from invertingP is thus the curl a non-! = u. This the non-localthe non-local pressure pressure field to field deal to with, deal with,and this and is this the great is the benefit great benefitr⇥ of working of working local procedure: the flow andwithcan be vorticity vorticity written instead as at a Biot–Savart all of points velocity. integral in But, the (compare some domain of this finding a simplicityect the all magnetic is illusory field b as the witharising vorticity from instead a given of current velocity. distribution But, somej of= this simplicityb in magnetostatics). is illusory as Overall the other points. In a numericalflowflow velocity code, velocity thisu hasuPoisson tohas be to found beequation found from frominvertinghas to invertingr⇥ be the solved curl the! curl each= ! =u. Thisu. This solving this equation ! = u for u amounts to obtaining ther⇥ pressurer⇥ by time-step. can becansolving written be written the as elliptic a Biot–Savart as a equation Biot–Savartr⇥ integral (2.32). integral (compare (compare finding finding the magnetic the magnetic field b field b arisingarising from from a given a given current current distribution distributionj = j =b in magnetostatics).b in magnetostatics). Overall Overall r⇥ Taking the curl eliminatessolvingsolving the this pressure equationthis equation and! = gives! =u thefor uuvorticityforamountsur⇥amounts equation to obtaining to obtaining the pressure the pressure by by r⇥ solvingsolving the elliptic the elliptic equation equationr⇥ (2.32). (2.32).7 ⇥ = (u ) t ⌅⇥ ⇥ or 7 7 D = ⇥ + u = u t t · ⌅ · ⌅ This latter equation shows the is transported with the fluid flow (left-hand side) but also undergoes stretching and rotation (right-hand side). Technically it is Lie-dragged in the flow and for this reason vorticity (rather than velocity) plays a central role in discussion of topological fluid mechanics. In other words given a flow u, the above equation carries vectors as though they each join two infinitesimally close fluid particles. Here the Lie bracket of two vector fields is defined as [u, v]=u v v u · ⌅ · ⌅ It is antisymmetric [u, v]= [u, v] and satisfies the Jacobi identity [[u, v], w] + [[v, w], u] + [[w, u], v]=0

For more information, see books on dierential geometry, for example that of Schultz [16]. In the ideal case we also have boundary conditions: in a finite domain we impose a no normal flow condition D

n u =0 (r ) · ⇤ S

5 Vortex filament motion

• local approximations giving the motion of a thin tube of vorticity - a vortex filament

• by Helmholtz and Kelvin, the filament moves and stretches with the fluid motion

• we can also invert by the Biot-Savart law

• combines dynamics and differential geometry of curves Vortex filament: Biot-Savart integral

• integral links velocity to vorticity (suppress time-dependence)

• take vorticity confined to a thin tube along a curve and has circulation (integral of vorticity across a surface area) of

• orthonormal Serret-Frenet basis

• arclength , curvature , torsion Local velocity from a filament

• filament through origin O with axes aligned with (at O)

• as

• look at velocity at a point in plane perpendicular to vortex at O Integration to give local flow

• Biot-Savart along a filament

• from a local length is

• put point and

• we want to be close to the filament , leaving Local flow

• at position

• flow is

• including strong local circulation, which does not move the filament

• and a weaker flow in the binormal direction

• has a logarithmic dependence on cut-off and vortex filament width

• treat as a constant: velocity of vortex filament is now

• or by rescaling time, Local induction approximation (LIA)

• points on the curve with Serret-Frenet

• and velocity

• or

• beautiful but highly idealised : no vortex stretching, only local induction, vortex width and cut-off scale fudged Evolution of curvature and torsion - I

• dash for derivative

• general motion (for present)

• now and so with

• have and

• equate these gives with Evolution of curvature and torsion - II

• dash for derivative

• general motion (for present)

• have and

• equate these gives with

• have linked A, B, C, D, E, F, G, H, K to velocity components in Evolution of curvature and torsion - III

• to close the system we use the fact that is an orthonormal basis

• and so

• with

• we have Equations for curvature and torsion

• gives equation from arc-length parameterisation

• and give

• or for LIA Equations for curvature and torsion under LIA

• a lot of manipulation… gives

• prime denotes derivative with respect to arclength

• …link to nonlinear Schrodinger equation (integrable PDE)… Knot evolution under LIA

• Ricca, Samuels, Barenghi: evolve a torus knot under LIA Evolution of F(2,3) and F(3,2) under LIA Evolution of F(3,2) under LIA and Biot-Savart William Irvine and collaborators (Chicago)

• vortex rings created by dragging a knotted aerofoil through water:

• https://www.youtube.com/watch?v=YCA0VIExVhg (1:10)

• https://www.youtube.com/watch?v=9CnilX-oLrI

• https://www.youtube.com/watch?v=LdOX24KwSUU

• https://www.youtube.com/watch?v=CoUglS21w6c Vortex stretching

• this important phenomenon is not in the LIA though it appears in more sophisticated models

• intense fine-scale vortices seen in 3-d turbulence

• vortex stretching creates fine scales

• question of the regularity of the 3-d Euler equation: Jörg Schumacher

• starting with smooth initial conditions, does the solution remain smooth for all time?

• fundamental, unsolved problem: Clay Millenium prizes

• In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) has named seven Prize Problems. The Scientific Advisory Board of CMI selected these problems, focusing on important classic questions that have resisted solution over the years.

• Birch and Swinnerton-Dyer Conjecture • Hodge Conjecture • Navier-Stokes Equations • P vs NP • Poincaré Conjecture --- proven! • Riemann Hypothesis • Yang-Mills Theory Navier-Stokes equations Idealised vorticity stretching

• full equation

• idealised ODE

• solution

• singular blow-up at time

• but: vorticity tends to stretch perpendicular vorticity, not itself

• problem of geometrical complexity

• e.g. no stretching (no singularity) in two dimensions Beale-Kato-Majda theorem

• rigorous result

• Suppose we start with a smooth Euler flow at time and that at time it is no longer smooth. Then, necessarily

• clear numerical criterion to capture any loss of smoothness

• eliminates certain types of singularities, e.g. if the maximum then Exact solutions of blow-up

• Let A be any symmetric trace-free matrix, then

• satisfies the Euler equation. But infinite energy, blows up everywhere at once, even in 2-d

• flows of the form

• e.g., in 2-d channel

• can show blow-up, e.g., Colliding vortices

• in 2-d a vortex pair of opposite signs translates, and similarly in 3-d

• no vortex stretching though

• try two pairs at right angles Colliding vortex pairs: Moffatt

• idea: two vortex pairs propagate towards, and stretch, each other

• vorticity intensified, feedback to faster evolution

• singularity? not clear ; viscosity may not stop a singularity if it occurs Colliding vortex pairs: Pelz

• 8 pairs colliding; highly symmetrical flow

• using vortex filaments under Biot-Savart blow-up very clean

• but actual vortices tend to flatten, depleting nonlinearity in simulations Evolution of anti-parallel vortices: Kerr/Bustamante

• vorticity intensifies strongly

• and flattens to form tadpole structures

• singularity at t* = 18.7 ? Vortex ring collisions in three dimensions

• https://www.youtube.com/watch?v=XJk8ijAUCiI

• https://www.youtube.com/watch?v=USzOciNHeh0&t=182s

• vortex rings move with the fluid (Helmholtz)

• then stretch (vortex line stretching) and accelerate outwards - how quickly?

• geometry:

• approximate 2-d dipole travelling outwards Theoretical ideas

• Childress, G, Valiant 2016

• major and minor axes ,

• conserve volume:

• problem… energy diverges: Theoretical ideas

• Childress, G, Valiant 2016

• major and minor axes ,

• conserve energy:

• necessarily, volume goes down, vorticity shed Vorticity shedding

• original picture a bit naive

• conserve energy:

• necessarily, volume goes down

• must `lose’ volume: shedding of vorticity in a tail behind the propagating vortex ring pair (visible on movies)

• `tadpole’ or `snail’ structure emerges Simulations

• in axisymmetric flow

• only one ring shown Loss of symmetry

• up/down symmetry can be lost:

• also experiments reveal instabilities More general geometry

• Bustamante & Kerr 2008 Conclusions

• vorticity perhaps best way to understand nearly inviscid flows

• many challenges both for mathematics and analysing physical processes

• such as stretching and reconnection

• with links to outstanding theoretical issues such as the finite-time singularity question ….

• … and the nature of turbulence.