52 4 2013 8 ) JournalofFudanUniversityNaturalScience) Vol52No4 Aug2013 0427-71042013)04-0547-11 谢锡麟, 陈瑜, 史倩, 200433), Euclid ), Riemann ),,,,, ; ; ; ; O33;O35 A 1 11, Bernouli,,,, [1-4], WuCJ 2009,2010) [3,5,6] CFDComputationalFluidDynamics, ) 3D ), ;,, Karman DongGJ 2007) [7], 4 [8] [9] DuG 2008) [10], ) LuX Y 2005) [11],, WuCJ 2003) [12], fluidrolerbearing),, WuCJ 2007) [13] 2013-05-08 11172069) ; 2011 1974 ),,,,E-mailxiexilin@fudaneducn
548 ) 52, Karman,,,, 12 1,, X 1 =r η, ζ,t)+ξ R η, ζ )-r η, 烄 ζ,t))) cosη, 烅 X 2 =ζ, 烆 X 3 =r η, ζ,t)+ξ R η, ζ )-r η, ζ,t))) sinη η, ζ ) ) ;r η, ζ,t) R η, ζ ),, 1 Fig1 Sketchofcurvilinearcoordinatesincludingtimeexplicitlythatissuitableto flowsaroundanairfoilwithdeformableboundaries,,, 2 珝 V 珝 V=V j k ) jv i 珝 gi=v j x,t) Vi +Γ i jkv x j x,t) 珝 gix,t),v=v i x,t) 珝 gix,t), Γjkx,t) Christofel i, 2 Fig2 Sketchofgeneralcurvilinearcoordinateswithitslocalinducedbasis
4 549, 1 PartialDiferentialEquations,PDE),, ;2, Christofel, Christofel, ;3, ) [14] S1 S2 ; [15],, ), 2012) [16] 13 [17], [18], 4 1) ; 2) ;3) 1 ;2 ;3 ;4 ), 3 4 ;4), 珝 V 珤 X t ξ, t)= 珤 X x i x,t) xi t ξ, t)+ 珤 X t x,t)= xi t ξ, t) 珝 gix,t)+ 珤 X t x,t) ) ), 珤 X t x,t)= 珤 X t, i 珝 g gix,t)= 珝 珤 i X x,t) gix,t), 珝 3 瓗 t,, Φ Φ t ξ, t)= Φ t x,t)+ Φ x i x,t) xi t ξ, t)= Φ t x,t)+ 珝 V, - X t x,t) ) Φ- 珤 X t x,t) Φ Φ, Φ= 珗 g l l Φ x 珗 g l Φ x l x,t) Φ Euclid F xi ξ A ξ, t)gix,t) G A ξ ), ) df dt = V 珝 x,t ) F= L F, d i dt detf=θdetf, detf 槡 g x det 槡 G ξ A ξ, t ), θ = V 珝 x,t), ;,, Euclid, 14 [18,19] [13],, 3 550 )
550 ) 52 3 15) ), ) Re=400 Fig3 Suppressionofvortexstreetbytravelingwavegeneratedonthesurfacesofcircularandelipticalcylinders theratioofthelongandshortaxesis15)leftsubplot)vorticitydistribution,rightsubplot)stream functiondistributionre=400 2 21,, ),, twodimensional [20] flows),, ) Aris1964) [21],, [22], 22,, ); 4 ; [18], Stokes [18,22] Stokes, 4 t τ 珒珗 ) n -Φ = t -Φ+H 珗 n -Φ))dσ Φ,H =b s s=g st b ts,gij= gi, 珝 gj 珝 ) 3 瓗 x ) x i ) 3 瓗,b ij = 珗 gj x ), n, gj 珝 x ) ; -
4 551 4 Stokes Fig4 SketchoftheintrinsicgeneralizedStokesformulaofthesecondkind,, τ 珒珗 n τ 珒珗 n Stokes, t=t i j 珝 gi 珝 g j T 2 T), Stokes, τ 珒 n) t= 珗 t t+h n t))dσ= 珗 tdσ, t t= i t i j 珝 g j +b j it i j n, 珗 = 珝 g l ; i Riemann ) x l, ρ d 珝 V = dt f =f j 珝 gj+f 3 珗 n t it ij j +f ) gj 珝 + b j it i j 3 +f ) n, 珗 t, f dσ; [21] Aris, ) Stokes,, τ 珒珗 n Stokes, 23 231, ω珗 =ε st3 sv t n = 珗 ω 3 n, 珗 Eddington ε st3 = est3, 槡 g 槡 g = g1, 珝 g2, 珝 n) 珗 ρ+ρθ=0, ρ, θ = i V i = 1 槡槡 g x i gv i ) x,t) θ = 0, V s =ε st3 ψ x t x,t) t= -p+ γi+μ ) jv i + iv j ) 珝 g i 珝 g j T 2 T ),p,γ, μ,i=gij 珝 g i 珝 g j ; [23]
552 ) 52 烄 ψ =g ij 2 ψ x i x j x,t)-γ ij k ψ x k x,t ) =-ω 3, 烅 ω 3 = ω3 t x, t) +V s ω3 x, x s t) = μ s sω 3 +2ε kl3 k K GV ) l ρ + 1 烆 ρ εkl3 kfsur,l, Gauss 5, [20], ;,, 5 ), ) Re=500 Fig5 Theflowaroundacircularcylinderonafixedsurfaceleftsubplot)threedimensionalview ofstreamfunctiondistribution;rightsubplot)planformofvorticitydistributionre=500 6, ),Reynolds 300 ),, ), ω 3 2 45 135 [23] 7, ; 6 ), 2 ; ) Gauss Re=300 Fig6 leftsubplot)threedimensionalviewofatwodimensionalincompressibleflowonafixedundulated helicoidalsurfacespatialdistributionofthevorticitytwolinesegmentsmaketheintervalofthe innerboundarywithnegativevorticityrightsubplot)projectiveviewsofthespatialdistributionsof thegaussiancurvatureoftheundulatedhelicoidalsurfacere=300
4 553 7 ) 2-2 ) 45 135 Fig7 Eigen-problemanalysisofthestraintensorontheinnerboundaryleftsubplot)Distributionoftheratioofthe vorticityandthemaximumeigenvalueofthestraintensoralongtheinnerboundaryoftheundulatedhelicoidal surfaceremarkthetheoreticalindicatedvalueoftheratiois2or-2rightsubplot)distributionofthe anglebetweentheeigenvectorwithrespecttothemaximumeigenvalueofthestraintensorandthetangentvector oftheboundaryalongtheinnerboundaryoftheundulatedhelicoidalsurfaceremarkthetheoreticalindicated valueoftheangleis45or135degree,, ), [24] [23], 232, 8, x = x ξ, t) 瓗 2 珝 V = 珝 t x,t)+ x s 珝 gs, x = s xs t, ξ t) V 3 ρ t x,t)+ x s ρ x,t)+ρ s V s -HV 3 ) =0, x s 8 ), ) Fig8 Sketchoftheconstructionsofconfigurationswithrespecttooildifusionsonseasurfaces leftsubplot)physicalconfigurations,rightsubplot)parametricconfigurations gs 珝 ) 3 +2 xs 瓗 t l x,t), 珝 g ) 瓗 ) ρ x l, 珝 ξ t)+γ l pq x p x q + 2 t l x 2,t), 珝 g 3 =
554 ) 52 - p x x l,t)+ρf l +μ l s V s + s sv l +K GV l ls V3-3b - 2 x s s l b s ) V 3 -b l sb s tv ) gs 珝 ) 3 +2 xs 瓗 t x,t), n ) 瓗 ) 珝 ρb pq x p x q + 2 t x 2,t), n γ-p)h +ρf 3 + μ s s V 3 +3b s t sv t + 3 = qb ) s q Vs -2b st b stv 3 ) Lagrange, x= x 2 ξ,t) 瓗,, 9 10, zx,y,t)= 01sin 2π λ x-ω t, ) λ=10,ω=40 t, 11 12, zx,y,t) =01sin 2π x λ1 ) sin 2π y sinωt, λ2 ) λ=10,ω=40,,
4 555 3, Euclid Riemann,,,,,,,, ),
556 ) 52 Euclid, Riemann,Euclid ) Riemann ),,, ; Lagrange [1] Triantafylou M S,TriantafylouGS,YueD K PHydrodynamicsoffishlikeswimming[J]Annu Rev Fluid Mech,2000,3233-53 [2] FishFE,LauderG VPassiveandactiveflowcontrolbyswimmingfishesandmammals[J]Annu Rev Fluid Mech,2006,38193-224 [3] WuCJ,WangLAdaptiveoptimalcontroloftheflappingruleofafixedflappingplate[J]Advancesin Applied Mathematicsand Mechanics,2009,13)402-414 [4] LuX Y,Yin X Z,YangJ M,etalStudiesofhydrodynamicsinfishlikeswimmingpropulsion[J] Journalof Hydrodynamics,2010,225)12-22 [5] WuCJ,WangLNumericalsimulationsofself-propeledswimmingof3Dbionicfishschool[J]Science in ChinaE),2009,523)658-669 [6] WuCJ,Wang LWhereistherudderofafish -The mechanism ofswimmingandcontrolofself- propeledfishschool[j]acta Mechanica Sinica,2010,261)45-65 [7] DongGJ,LuX YCharacteristicsofflowovertraveling wavyfoilsinaside-by-sidearrangement[j] Physicsof Fluids,2007,19057107 [8] WiliamsonC H KEvolutionofasinglewakebehindapairofblufbodies[J]J Fluid Mech,1985,1591-18 [9] GovardhanR,WiliamsonC H KModesofvortexformationandfrequencyresponseofafreelyvibrating cylinder[j]j Fluid Mech,2000,4201)85-130 [10] DuG,Sum MEfectsofunsteadydeformationofflappingwingonitsaerodynamicsforces[J]Applied Mathematicsand Mechanics,2008,296)731-743 [11] LuX Y,YinXZPropulsiveperformanceofafish-liketravelingwavywal[J]Acta Mechanica,2005, 1751-4)197-215 [12] WuCJ,XieY Q,WuJZ Fluidrolerbearing efectandflowcontrol[j]acta Mechanica Sinca, 2003,195)476-484 [13] WuCJ,WangL,WuJZSuppressionofthevonKarmanvortexstreetbehindacircularcylinderbya travelingwavegeneratedbyaflexiblesurface[j]j Fluid Mech,2007,574365-391 [14] [M],1976 [15], [M],2004 [16] [C],, 110,2012224-236 [17] [M],1980 [18] XieX L,Chen Y,ShiQSomestudieson mechanicsofcontinuous mediumsviewedasdiferential manifolds[j]scichina-phys Mech Astron,2013,562)432-456 [19],, [C],, 110,2012212-223 [20] BofetaG,EckeRETwo-dimensionalturbulence[J]Annu Rev Fluid Mech,2012,44427-451
4 557 [21] Aris R Vectors,tensors,and the basic equations offluid mechanics[m ] New York Dover Publications,INC,1989 [22] XieXLOntwokindsofdiferentialoperatorsongeneralsmoothsurfaces[J]arXiv13063671v1 [physicsflu-dyn],2013 [23] XieX LA theoreticalframework ofvorticitydynamicsfortwodimensionalflowsonfixedsmooth surfaces[j]arxiv13045145v1 [physicsflu-dyn],2013 [24] WuJZ,MaH Y,Zhou M DVorticityandvortexdynamics[M]New YorkSpring-Verlag,2005 SomeDevelopmentsofFiniteDeformationTheorieswith SomeApplicationsinFluid Mechanics XIEXi-lin,CHENYu,SHIQian Departmentof Mechanicsand Engineering Science,Fudan University,Shanghai200433,China) AbstractSomerecentdevelopmentsoffinitedeformationtheoriestermedas finitedeformationtheorywithrespect tocurvilinearcoordinatescorrespondingtocurrentphysicalconfigurationsincludingtimeexplicitly and finite deformationtheory withrespecttocontinuous mediums whosegeometricalconfigurationsaretwodimensional surfaces arenarratedtheformerandthelatertheoriescorrespondtocontinuousmediumswhosegeometrical configurationsareeuclidianmanifoldsbulkstatus)andriemannianmanifoldssurfacestatus),respectivelyas comparedtogeneraltheories,bothnewlydevelopedtheoriesincludeconstructionsofphysicalandparametric configurations,definitionofdeformationgradienttensor withitsprimaryproperties,deformationdescriptions, transporttheoriesandgoverningequationsofconservationlawsasapplications,stream function & vorticity algorithm withrespecttocurvilinearcoordinatesincludingtimeexplicitly,streamfunction & vorticityalgorithm fortwodimensionalincompressibleflowsonfixedsurfaces,andgoverningequationsforoildifusiononseasurfaces arepresentedwithsomeresultsoftentativenumericalstudies Keywordsfinitedeformationtheory;streamfunction & vorticityalgorithm;respecttocurvilinearcoordinates includingtimeexplicitly;two dimensionalflows onfixed surfaces;flowsaround cylinders with deformable boundaries;oildifusiononthesea