-+Two-dimensionalowsinslowlydeformingdomainsAdiabaticinvarianceandgeo - PDF document

�-+Two-dimensionalowsinslowlydeformingdomainsAdiabaticinvarianceandgeometricangleJ.VannesteSchoolofMathematicsUniversityofEdinburgh,UKwww.maths.ed.ac.uk/˜vannestewithD.WirosoetisnoUniversityofDurham,UKTwo-dimensionalowsinslowlydeformingdomains–p.1/20 �-+FlowinslowlydeformingdomainMotivation:effectofslow,conservativeperturbationsonperfectuidsSimplestmodel:2D,incompressibleuidslowlydeformingdomain(almost)steadyow����DD0LTwoissues:Eulerianowu(x;t)(Lagrangian)uid-particlepositionsforO(1)deformationsofthedomain.Two-dimensionalowsinslowlydeformingdomains–p.2/20 �-+FlowinslowlydeformingdomainMotivation:effectofslow,conservativeperturbationsonperfectuidsSimplestmodel:2D,incompressibleuidslowlydeformingdomain(almost)steadyow����DD0LTwoissues:Eulerianowu(x;t)(Lagrangian)uid-particlepositionsforO(1)deformationsofthedomain.Two-dimensionalowsinslowlydeformingdomains–p.2/20 �-+FlowinslowlydeformingdomainMotivation:effectofslow,conservativeperturbationsonperfectuidsSimplestmodel:2D,incompressibleuidslowlydeformingdomain(almost)steadyow����DD0LTwoissues:Eulerianowu(x;t)(Lagrangian)uid-particlepositionsforO(1)deformationsofthedomain.Two-dimensionalowsinslowlydeformingdomains–p.2/20 �-+FlowinslowlydeformingdomainMotivation:effectofslow,conservativeperturbationsonperfectuidsSimplestmodel:2D,incompressibleuidslowlydeformingdomain(almost)steadyow����DD0LTwoissues:Eulerianowu(x;t)(Lagrangian)uid-particlepositionsforO(1)deformationsofthedomain.Two-dimensionalowsinslowlydeformingdomains–p.2/20 �-+FlowinslowlydeformingdomainMotivation:effectofslow,conservativeperturbationsonperfectuidsSimplestmodel:2D,incompressibleuidslowlydeformingdomain(almost)steadyow����DD0LTwoissues:Eulerianowu(x;t)(Lagrangian)uid-particlepositionsforO(1)deformationsofthedomain.Two-dimensionalowsinslowlydeformingdomains–p.2/20 �-+FlowinslowlydeformingdomainFinite-dimensionalanalogue:Hamiltoniansystemwithslowlyvariableparameters,e.g.pendulumwithlength(t);1.LAmplitude(energy):determinedbytheadiabaticinvarianceofactionI,
I=O().adiabaticinvarianceofIstemsfrominvarianceofpdq.energydependsoninstantaneously.Anglehi:dynamicalangleR!dt+Hannay±Berry(geometric)angledependsonpathinparameterspaceTwo-dimensionalowsinslowlydeformingdomains–p.3/20 �-+FlowinslowlydeformingdomainFinite-dimensionalanalogue:Hamiltoniansystemwithslowlyvariableparameters,e.g.pendulumwithlength(t);1.LAmplitude(energy):determinedbytheadiabaticinvarianceofactionI,
I=O().adiabaticinvarianceofIstemsfrominvarianceofpdq.energydependsoninstantaneously.Anglehi:dynamicalangleR!dt+Hannay±Berry(geometric)angledependsonpathinparameterspaceTwo-dimensionalowsinslowlydeformingdomains–p.3/20 �-+FlowinslowlydeformingdomainFinite-dimensionalanalogue:Hamiltoniansystemwithslowlyvariableparameters,e.g.pendulumwithlength(t);1.LAmplitude(energy):determinedbytheadiabaticinvarianceofactionI,
I=O().adiabaticinvarianceofIstemsfrominvarianceofpdq.energydependsoninstantaneously.Anglehi:dynamicalangleR!dt+Hannay±Berry(geometric)angledependsonpathinparameterspaceTwo-dimensionalowsinslowlydeformingdomains–p.3/20 �-+FlowinslowlydeformingdomainsLe0t�Geometricangledependsonlyonthecurveinparameterspace,notonthespeedatwhichthiscurveistraced.Analogyf-dsystem/uid:f-dsystemuidamplitudeEulerianowangleparticlepositionTwo-dimensionalowsinslowlydeformingdomains–p.4/20 �-+FlowinslowlydeformingdomainsLe0t�Geometricangledependsonlyonthecurveinparameterspace,notonthespeedatwhichthiscurveistraced.Analogyf-dsystem/uid:f-dsystemuidamplitudeEulerianowangleparticlepositionTwo-dimensionalowsinslowlydeformingdomains–p.4/20 �-+FormulationDomaindenedbyparameters:=(t);1.D0DLL-spaceL0(t)����Perfect,incompressibleowin2D:@tu+u
ru=rp;divu=0.Vorticityformulation:!=curlu,u=r? =(@y ;@x ),@t!+[ ;!]=0;
=!;where[ ;!]=@x1 @x2!@x2 @x1!.Two-dimensionalowsinslowlydeformingdomains–p.5/20 �-+FormulationDomaindenedbyparameters:=(t);1.D0DLL-spaceL0(t)����Perfect,incompressibleowin2D:@tu+u
ru=rp;divu=0.Vorticityformulation:!=curlu,u=r? =(@y ;@x ),@t!+[ ;!]=0;
=!;where[ ;!]=@x1 @x2!@x2 @x1!.Two-dimensionalowsinslowlydeformingdomains–p.5/20 �-+FormulationBoundarycondition:@DisamaterialcurveThevorticityisrearranged:!(x;t)=!0(g1tx),i.e.!=!0g1t;wheregtisanarea-preservingdiffeomorphism,with_gt=v.SteadyFlows:[ ;!]=0) =F(!)inD:Vorticityandstreamfunctionarefunctionalyrelated.Two-dimensionalowsinslowlydeformingdomains–p.6/20 �-+FormulationBoundarycondition:@DisamaterialcurveThevorticityisrearranged:!(x;t)=!0(g1tx),i.e.!=!0g1t;wheregtisanarea-preservingdiffeomorphism,with_gt=v.SteadyFlows:[ ;!]=0) =F(!)inD:Vorticityandstreamfunctionarefunctionalyrelated.Two-dimensionalowsinslowlydeformingdomains–p.6/20 �-+PerturbationexpansionExpandinpowerseries:!=!(0)+!(1)+


; = (0)+ (1)+


andsubstituteinto2DEuler.Assumingallcoefcientsdependont,wend:Atleadingorder,[!(0); (0)]=0(instantaneously)steadyow, (0)=F(!(0)):Atthenextorder,@t!(0)+[ (1);!(0)]+[ (0);!(1)]=0.Rewritingas@t!(0)+[;!(0)]=0;with= (1)F0(!(0))!(1)showsthat!(0)isrearrangedbyvelocityr?.Two-dimensionalowsinslowlydeformingdomains–p.7/20 �-+PerturbationexpansionExpandinpowerseries:!=!(0)+!(1)+


; = (0)+ (1)+


andsubstituteinto2DEuler.Assumingallcoefcientsdependont,wend:Atleadingorder,[!(0); (0)]=0(instantaneously)steadyow, (0)=F(!(0)):Atthenextorder,@t!(0)+[ (1);!(0)]+[ (0);!(1)]=0.Rewritingas@t!(0)+[;!(0)]=0;with= (1)F0(!(0))!(1)showsthat!(0)isrearrangedbyvelocityr?.Two-dimensionalowsinslowlydeformingdomains–p.7/20 �-+PerturbationexpansionExpandinpowerseries:!=!(0)+!(1)+


; = (0)+ (1)+


andsubstituteinto2DEuler.Assumingallcoefcientsdependont,wend:Atleadingorder,[!(0); (0)]=0(instantaneously)steadyow, (0)=F(!(0)):Atthenextorder,@t!(0)+[ (1);!(0)]+[ (0);!(1)]=0.Rewritingas@t!(0)+[;!(0)]=0;with= (1)F0(!(0))!(1)showsthat!(0)isrearrangedbyvelocityr?.Two-dimensionalowsinslowlydeformingdomains–p.7/20 �-+EulerianowTheleading-orderow!(0)canbefoundbyimposingthatitis:steady, (0)=
1!(0)=F(!(0))forsomeF(
).arearrangement,!(0)(x;t)=!0(g1x)foranarea-preservingdiffeomorphismg.����DD0LgLThemapgsatisesdetg=1,maps@D0to@D.dependsontonlythroughTwo-dimensionalowsinslowlydeformingdomains–p.8/20 �-+EulerianowTheleading-orderow!(0)canbefoundbyimposingthatitis:steady, (0)=
1!(0)=F(!(0))forsomeF(
).arearrangement,!(0)(x;t)=!0(g1x)foranarea-preservingdiffeomorphismg.����DD0LgLThemapgsatisesdetg=1,maps@D0to@D.dependsontonlythroughTwo-dimensionalowsinslowlydeformingdomains–p.8/20 �-+EulerianowTheleading-orderow!(0)canbefoundbyimposingthatitis:steady, (0)=
1!(0)=F(!(0))forsomeF(
).arearrangement,!(0)(x;t)=!0(g1x)foranarea-preservingdiffeomorphismg.����DD0LgLThemapgsatisesdetg=1,maps@D0to@D.dependsontonlythroughTwo-dimensionalowsinslowlydeformingdomains–p.8/20 �-+Eulerianow(continued)Theexistenceofgalsoanswersthequestionofrobustnessofsteadyowstodomaindeformation:givenasteadyowinthedomainD0,doesitpersistwhenthedomainisdeformedtoD?(Wirosoetisno&V,2005)TheproblemiswrittenentirelyintermsofgandF:gsatisesthenonlinearPDE,!0=
(F!0g1)g;Fisdeterminedbyasolvabilitycondition.Two-dimensionalowsinslowlydeformingdomains–p.9/20 �-+Eulerianow(continued)Theexistenceofgalsoanswersthequestionofrobustnessofsteadyowstodomaindeformation:givenasteadyowinthedomainD0,doesitpersistwhenthedomainisdeformedtoD?(Wirosoetisno&V,2005)TheproblemiswrittenentirelyintermsofgandF:gsatisesthenonlinearPDE,!0=
(F!0g1)g;Fisdeterminedbyasolvabilitycondition.Two-dimensionalowsinslowlydeformingdomains–p.9/20 �-+Eulerianow(continued)Useanimplicit-function-theoremargumenttoshow:asolutionexistsforsmalldomaindeformation,providedthatrotationperiodT=Hd=jr (0)j1(+technicalhypotheses),thesolutionisuniquemodulotranslationalongconstant!(0)(gaugefreedom).Thisleadstounique!(0)and (0)thatdependontonlythroughForadiabaticdeformations,theconstraintT1isarequirementofslowness.Two-dimensionalowsinslowlydeformingdomains–p.10/20 �-+Eulerianow(continued)Useanimplicit-function-theoremargumenttoshow:asolutionexistsforsmalldomaindeformation,providedthatrotationperiodT=Hd=jr (0)j1(+technicalhypotheses),thesolutionisuniquemodulotranslationalongconstant!(0)(gaugefreedom).Thisleadstounique!(0)and (0)thatdependontonlythroughForadiabaticdeformations,theconstraintT1isarequirementofslowness.Two-dimensionalowsinslowlydeformingdomains–p.10/20 �-+Eulerianow(continued)Useanimplicit-function-theoremargumenttoshow:asolutionexistsforsmalldomaindeformation,providedthatrotationperiodT=Hd=jr (0)j1(+technicalhypotheses),thesolutionisuniquemodulotranslationalongconstant!(0)(gaugefreedom).Thisleadstounique!(0)and (0)thatdependontonlythroughForadiabaticdeformations,theconstraintT1isarequirementofslowness.Two-dimensionalowsinslowlydeformingdomains–p.10/20 �-+Eulerianow(continued)Useanimplicit-function-theoremargumenttoshow:asolutionexistsforsmalldomaindeformation,providedthatrotationperiodT=Hd=jr (0)j1(+technicalhypotheses),thesolutionisuniquemodulotranslationalongconstant!(0)(gaugefreedom).Thisleadstounique!(0)and (0)thatdependontonlythroughForadiabaticdeformations,theconstraintT1isarequirementofslowness.Two-dimensionalowsinslowlydeformingdomains–p.10/20 �-+Eulerianow(continued)Howcanwendg?ThePDEmaybesolvednumerically,e.g.usinganiterativescheme.ThisissimplerifD0isachanneloradisc,butconvergenceappearslimitedtoverysmallboundarydeformations.Aperturbativeschemecanbedevelopedforsmallboundarydeformations1,usingLieseries.Lieseries:writegastheowatofvectoreldr?'(),andexpand'tondf(gx)=f(x)+['1;f](x)+22(['1;['1;f]](x)+['2;f](x))+
Thisleadstoasequenceoflinearproblemsforthe'iandF=F0+F1+


.Two-dimensionalowsinslowlydeformingdomains–p.11/20 �-+Eulerianow(continued)Howcanwendg?ThePDEmaybesolvednumerically,e.g.usinganiterativescheme.ThisissimplerifD0isachanneloradisc,butconvergenceappearslimitedtoverysmallboundarydeformations.Aperturbativeschemecanbedevelopedforsmallboundarydeformations1,usingLieseries.Lieseries:writegastheowatofvectoreldr?'(),andexpand'tondf(gx)=f(x)+['1;f](x)+22(['1;['1;f]](x)+['2;f](x))+
Thisleadstoasequenceoflinearproblemsforthe'iandF=F0+F1+


.Two-dimensionalowsinslowlydeformingdomains–p.11/20 �-+Eulerianow(continued)Example:deformationofanaxisymmetricowinadisc.TakeD0tobetheunitdisc,and (0)= (r;0)=r1=2.ThedeformeddomainDisdenedbyr=1+Xmmexp(im)+O(2)Wendthat:'1=Xmimrmmexp(im);withm=pm23=4+3=2andF1=0:Two-dimensionalowsinslowlydeformingdomains–p.12/20 �-+Eulerianow(continued)Example:deformationofanaxisymmetricowinadisc.TakeD0tobetheunitdisc,and (0)= (r;0)=r1=2.ThedeformeddomainDisdenedbyr=1+Xmmexp(im)+O(2)Wendthat:'1=Xmimrmmexp(im);withm=pm23=4+3=2andF1=0:Two-dimensionalowsinslowlydeformingdomains–p.12/20 �-+Eulerianow(continued)Example:deformationofanaxisymmetricowinadisc.TakeD0tobetheunitdisc,and (0)= (r;0)=r1=2.ThedeformeddomainDisdenedbyr=1+Xmmexp(im)+O(2)Wendthat:'1=Xmimrmmexp(im);withm=pm23=4+3=2andF1=0:Two-dimensionalowsinslowlydeformingdomains–p.12/20 �-+Eulerianow(continued)Example:deformationofanaxisymmetricowinadisc.TakeD0tobetheunitdisc,and (0)= (r;0)=r1=2.ThedeformeddomainDisdenedbyr=1+Xmmexp(im)+O(2)Wendthat:'1=Xmimrmmexp(im);withm=pm23=4+3=2andF1=0:Two-dimensionalowsinslowlydeformingdomains–p.12/20 �-+Eulerianow(continued)Example:deformationofanaxisymmetricowinadisc.TakeD0tobetheunitdisc,and (0)= (r;0)=r1=2.ThedeformeddomainDisdenedbyr=1+Xmmexp(im)+O(2)Wendthat:'1=Xmimrmmexp(im);withm=pm23=4+3=2andF1=0:Two-dimensionalowsinslowlydeformingdomains–p.12/20 �-+LagrangiantrajectoriesFort=O(1),position(x;y)(t)ofparticlesisgovernedby:dxdt=@@y (0)+ (1)
+O(2);dydt=@@x (0)+ (1)
+O(2):Thisisa`doublyperturbed'HamiltoniansystemwithHamiltonianH(x;y;)= (0)(x;y;(t))+ (1)(x;y;(t)).Determinationof (1):recall@t!(0)+[;!(0)]=0;with= (1)F0(!(0))!(1):Since!(0)isknown,canbedetermineduptoanarbitraryfunctionof!(0):r?=ddtgx=dg
_:Two-dimensionalowsinslowlydeformingdomains–p.13/20 �-+LagrangiantrajectoriesFort=O(1),position(x;y)(t)ofparticlesisgovernedby:dxdt=@@y (0)+ (1)
+O(2);dydt=@@x (0)+ (1)
+O(2):Thisisa`doublyperturbed'HamiltoniansystemwithHamiltonianH(x;y;)= (0)(x;y;(t))+ (1)(x;y;(t)).Determinationof (1):recall@t!(0)+[;!(0)]=0;with= (1)F0(!(0))!(1):Since!(0)isknown,canbedetermineduptoanarbitraryfunctionof!(0):r?=ddtgx=dg
_:Two-dimensionalowsinslowlydeformingdomains–p.13/20 �-+LagrangiantrajectoriesFort=O(1),position(x;y)(t)ofparticlesisgovernedby:dxdt=@@y (0)+ (1)
+O(2);dydt=@@x (0)+ (1)
+O(2):Thisisa`doublyperturbed'HamiltoniansystemwithHamiltonianH(x;y;)= (0)(x;y;(t))+ (1)(x;y;(t)).Determinationof (1):recall@t!(0)+[;!(0)]=0;with= (1)F0(!(0))!(1):Since!(0)isknown,canbedetermineduptoanarbitraryfunctionof!(0):r?=ddtgx=dg
_:Two-dimensionalowsinslowlydeformingdomains–p.13/20 �-+Lagrangiantrajectories(continued)Hence, (1)=
1!(1)isfoundbysolving (1)F0(!(0))!(1)=;Thegaugefreedominisxedbytheconditionthatthetotalvorticityisrearranged:ZZ!(0)+!(1)=\ndxZZ!(0)=\ndx=O(2)=)I!(0)=\n!(1)ds=0;whereds=dl=jr!(0)j.AconsequenceisthatI!(0)=\n (1)ds=I!(0)=\nds:Two-dimensionalowsinslowlydeformingdomains–p.14/20 �-+Lagrangiantrajectories(continued)Tondparticletrajectories,weuseaction±anglecoordinates:I=A(!(0)),areainsidecontour,isanadiabaticinvariant,,conjugatetoI,givespositionalong!-contours.��DLqNote:2ds=A0(!(0))d.Useageneratingfunction:x2=@x1S(x1;I);=@IS(x1;I),withthenewHamiltonianH(I;;)= (0)(I;;)+ (1)(I;;)+@tSTwo-dimensionalowsinslowlydeformingdomains–p.15/20 �-+Lagrangiantrajectories(continued)Tondparticletrajectories,weuseaction±anglecoordinates:I=A(!(0)),areainsidecontour,isanadiabaticinvariant,,conjugatetoI,givespositionalong!-contours.��DLqNote:2ds=A0(!(0))d.Useageneratingfunction:x2=@x1S(x1;I);=@IS(x1;I),withthenewHamiltonianH(I;;)= (0)(I;;)+ (1)(I;;)+@tSTwo-dimensionalowsinslowlydeformingdomains–p.15/20 �-+Lagrangiantrajectories(continued)With@tS=@tSx1@tx2,wendtheevolutionequationfortheangle:_=+@Ih@tSx1@tx2+ (1)i;where=@I (0)isthefrequency.�����DLDq�D0D0DCLLL-spaceConsidercyclicdeformationofthedomain,anduseaveraging:
ddIZt0h (0)hx1@tx2+ (1)iidt0;whereh
i=12Z
d:Two-dimensionalowsinslowlydeformingdomains–p.16/20 �-+Lagrangiantrajectories(continued)With@tS=@tSx1@tx2,wendtheevolutionequationfortheangle:_=+@Ih@tSx1@tx2+ (1)i;where=@I (0)isthefrequency.�����DLDq�D0D0DCLLL-spaceConsidercyclicdeformationofthedomain,anduseaveraging:
ddIZt0h (0)hx1@tx2+ (1)iidt0;whereh
i=12Z
d:Two-dimensionalowsinslowlydeformingdomains–p.16/20 �-+Lagrangiantrajectories(continued)Now,use:R (1)d=Rd=R
_d,where =Pm mdmisafunction-valueone-form(connection),withr? =dg,x(I;;)=gx(I;;0),Stokes'theorem.tond
(t)ddIZt0 (0)((s))ds+ddIZDd 12[ ; ]with[ ; ]=Pm;n[ m; n]dm^dndynamicalanglegeometricangleTwo-dimensionalowsinslowlydeformingdomains–p.17/20 �-+Lagrangiantrajectories(continued)Now,use:R (1)d=Rd=R
_d,where =Pm mdmisafunction-valueone-form(connection),withr? =dg,x(I;;)=gx(I;;0),Stokes'theorem.tond
(t)ddIZt0 (0)((s))ds+ddIZDd 12[ ; ]dynamicalanglegeometricangleTwo-dimensionalowsinslowlydeformingdomains–p.17/20 �-+Lagrangiantrajectories(continued)Geometricangle:determinedcompletelyfromtheconnection denedbygwithsuitablegauge,has2parts:d ,dueto (1)[ ; ]=2,Hannay±Berryangle,duetot-dependenceof (0)curvatureoftheconnectionform Smalldeformations:f(gx)=f(x)+['1;f](x)+


,leadsto =d'1+22(d'2+[d'1;'1])+


Two-dimensionalowsinslowlydeformingdomains–p.18/20 �-+Lagrangiantrajectories(continued)Geometricangle:determinedcompletelyfromtheconnection denedbygwithsuitablegauge,has2parts:d ,dueto (1)[ ; ]=2,Hannay±Berryangle,duetot-dependenceof (0)curvatureoftheconnectionform Smalldeformations:f(gx)=f(x)+['1;f](x)+


,leadsto =d'1+22(d'2+[d'1;'1])+


Two-dimensionalowsinslowlydeformingdomains–p.18/20 �-+Lagrangiantrajectories(continued)Geometricangle:determinedcompletelyfromtheconnection denedbygwithsuitablegauge,has2parts:d ,dueto (1)[ ; ]=2,Hannay±Berryangle,duetot-dependenceof (0)curvatureoftheconnectionform Smalldeformations:f(gx)=f(x)+['1;f](x)+


,leadsto =d'1+22(d'2+[d'1;'1])+


Two-dimensionalowsinslowlydeformingdomains–p.18/20 �-+Lagrangiantrajectories(continued)Geometricangle:determinedcompletelyfromtheconnection denedbygwithsuitablegauge,has2parts:d ,dueto (1)[ ; ]=2,Hannay±Berryangle,duetot-dependenceof (0)curvatureoftheconnectionform Smalldeformations:f(gx)=f(x)+['1;f](x)+


,leadsto =d'1+22(d'2+[d'1;'1])+


Two-dimensionalowsinslowlydeformingdomains–p.18/20 �-+Lagrangiantrajectories(continued)Geometricangle:determinedcompletelyfromtheconnection denedbygwithsuitablegauge,has2parts:d ,dueto (1)[ ; ]=2,Hannay±Berryangle,duetot-dependenceof (0)curvatureoftheconnectionform Smalldeformations:f(gx)=f(x)+['1;f](x)+


,leadsto =d'1+22(d'2+[d'1;'1])+


Two-dimensionalowsinslowlydeformingdomains–p.18/20 �-+Lagrangiantrajectories(continued)Geometricangle:determinedcompletelyfromtheconnection denedbygwithsuitablegauge,has2parts:d ,dueto (1)[ ; ]=2,Hannay±Berryangle,duetot-dependenceof (0)curvatureoftheconnectionform Smalldeformations:f(gx)=f(x)+['1;f](x)+


,leadsto =d'1+22(d'2+[d'1;'1])+


Two-dimensionalowsinslowlydeformingdomains–p.18/20 �-+Lagrangiantrajectories(continued)Example:deformedaxisymmetricow:r=1+Pmmexp(im)+O(2)giveshigeom=2X�m0fm(r)Am+O(3);whereAmistheareaenclosedbypathofminthecomplexplane.Re�ImLLmm0510150.20.40.60.81rrfm34562=mFor (0)=r1=2,fm(r)isasumof2powersofr.Two-dimensionalowsinslowlydeformingdomains–p.19/20 �-+ConclusionsProcedureforcomputingowsinslowlydeformingdomainsEulerianowisquasi-steadyEulerianowdependsonlyontheinitialandnaldomainshapesParticlepositionsdenedbytheiranglealongvorticitycontoursAngledependsonthehistoryofthedomain,inageometricmannerExtensions:rotationperiodTunbounded(non-parallelcriticallevels)3DowsTwo-dimensionalowsinslowlydeformingdomains–p.20/20