1. Why turbulence occur? Hydrodynamic Instability. Hydrodynamic Instability. Centrifugal Instability: Rayleigh-Benard Instability:

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1 . Why turbulence occur? Hydrodynamc Instablty Hydrodynamc Instablty T Centrfugal Instablty: Ω Raylegh-Benard Instablty: Drvng force: centrfugal force Drvng force: buoyancy flud Dampng force: vscous dsspaton Dampng force: vscous dsspaton Τ T Low Ω: lamnar, concentrc streamlnes Low T: motonless, pure thermal conducton Hgher Ω: steady convecton roll Hgher T: steady convecton roll Even Hgher Ω: unstead turbulent Even Hgher T: unstead turbulent stablty of a heated plate Hydrodynamc Instablty Reynolds experment: crcular ppe flow Low Re: stead axsymmetrc, parallel, parabolc Poseulle flow Re >000: sporadc bursts of turbulence alternate wth lamnar (ntermttence) Hgh Re: fully turbulent Transton from lamnar to turbulent flow need not be smply a functon of the base flow consdered, but also of the ampltude and type of perturbatons. Multple paths to transton through dfferent sequences of lamnar flow nstabltes are extremely possble. Hopf bfurcaton dynamc system: Use polar coordnates: dx = µ x y σx( x y ) x( x y ) dt dy = x µ y σy( x y ) y( x y ) dt fxed pont : perodc orbt : dr 5 = µ r σr r dt dθ = no such r dt c f µ < σ two rc 's f σ < µ < 0 ( x, y) = (0,0) one rc f 0 < µ dr dt 4 = 0 at µ σr c r c = 0

2 Hopf bfurcaton Hopf bfurcaton stable stable unstable stable no rc for µ =.0 < σ = two r c for σ = < µ = 0.8 < 0 stable unstable one r c for σ = < 0 < µ = 0. vortex stree behnd a vertcal plate Hopf bfurcaton Velocty v.s. Vortcty: u = ; u = (ncompressble) = u = ( u) u Vortcty feld s usually more compact than velocty feld. It s therefore more economc to descrbe a flow n terms of ts vortcty feld. velocty sgnals

3 landng Tensor Notaton u = Mxng layer between helum and ntrogen 4 U /U = 0.8; = 7 ; U L µ = 6 0 (L s the wdth of the pcture) δ k f = = 0 otherwse δ = δ = δ δ δ= m ( = ) ± f k = 0 otherwse = mrs = δ r δ s δ s δ r

4 b ( a ) Tensor Notaton b a = b a Physcal Sgnfcance. related to the rotatonal tensor Consder the relatve δx velocty of two nearby flud partcles separated by a dstance : u u k = k = u k k = k= k k k u k k u u u u x x δ = δ = δ δx x x x x δu = S δx Ω S Ω δx = stran tensor (angular deformaton; stranng) = rotatonal tensor = k Ω k ; Ω = k k Physcal Sgnfcance. Twce the average angular velocty Consder an nfntesmal crcle around a pont n space: ds u ds average angular velocty = πr r = ( u) d = d πr πr s r 0, ( n ) n = π πr r The average angular velocty around an nfntesmal crcle = = half the vortcty component n the drecton of the area enclosed by the crcle. Physcal Sgnfcance. Proportonal to the angular momentum of flud partcles whose nerta tensor has sphercal symmetry Consder a flud regon δv centered at x : δu = S δx Ω Ω I km m = = δv kml l δx δx δx m angular momentum about ts centrod : δ = δ = δv k dv = nerta tensor S δx δudv km I m ( δimm I ) If sphercal symmetry I = I δ, δ = I or δ = I ( )

5 Mathematcal Descrpton Incompressble Naver-Stokes flows of Newtonan flud wth constant propertes: u = 0 µ ( u ) u = p g u ( u u) µ u = p g u µ ( u ) ( ) u = p g D = Lagrangan tme change rate of vortcty Mathematcal Descrpton µ RHS = p g vscous dffuson body-force torque pressure toque nonzero when the geometrc centrod s not the mass centrod zero f barotropc,.e. = (p) zero f body force s rrotatonal or conservatve,.e. g = U, where U s some scalar functon, called potental functon nonzero f rotatonal body force, e.g. Corols force µ ( )u vortex stretchng ( u ) = p g barotropc fluds, conservatve body force, nvscd: D = ( )u Consder an nfntesmal materal lne element δl : δ l(t) δl ( t dt) Dδl = δu = ( δl )u = rate at whch two ntally nfntely close-by flud partcles separate

6 vortcty/materal tube: D = ( )u D d = D d = d ( u ) d = u n ds = 0 = stretched (thnnng) rotaton speed up compressed (flatten) rotaton slow down C powerfulness of turbulent stretchng barotropc fluds, conservatve body force: D = ( ) u ν stretchng dffuson nonlnear lnear magntude: vscous dffuson coarse, weaker structures D u ( ) = ν ν x usually > 0, producton (stretchng) always < 0, fner, stronger structures vscous dsspaton

7 Ref: Couder and Basdevant, JFM 7, 5-5 (986) The mage at the rght depcts two counter-rotatng vortces as they nteract wth a rgd crcular cylnder. Ther ntal translaton s caused by ther nteracton and the upward drected et produced n the gap between them. Once they collde wth the cylnder, they produce an unsteady separaton and another par of vortces havng a rotaton whch opposes the orgnal par of vortces. Ths mage was generated by Dr. S. Subramanam and s undoubtedly part of the unsteady separaton work beng conducted by Professor van Dommelen's group n FSU. vortex mergng

8 barotropc fluds, conservatve body force: D = ( ) u ν D ( ) = ν ν x When a balance between the stretchng and the dffuson/dsspaton processes, the fnest scale (Kolmogorov s, η) s reached. When ν s very small (or Reynolds number s extremely large), dffuson won t be sgnfcant untl the gradents have become suffcently large. That s, η decreases wth decreasng ν. Wll η 0 as ν 0? Vortex stretchng does not exst n two-dmensonal flows. u = = ( u( x, t), v( x, t),0) u = ( 0,0, ( x, t) ) z ( ) u = = 0 Turbulence s always three-dmensonal example Townsend, Proc. Roy. Soc. 08, 54 (95) u = U v =, ( αx, βy ( α β) z) v U = 0 (ncompressble) and U = 0 (rrotatonal) = u = v Consder only specal cases when v << U : convecton : ( u ) ( U ) stretchng : ( ) u ( )U D = ( ) u ν ( U ) = ( ) U ν (lnear) model (): U = ( 0, α αz) ( U ) = ( ) U ν Vortex stretchng: U ( U ) = ν z ( ) U = ( 0, α α ), Vortcty wll be compressed n y- and stretched n z-drecton. Expect when stead = ( y) >>, 0 d αy dy d = α ν dy

9 model (): U = ( 0, α αz) Check: = d α d y = α ν dy dy ( y) = ( y = 0) exp y ( ( ).e. vortex sheet structures of thckness ν α (,0, ( y )) v = ( v ( ),0,0) 0 y ~ dffuson rate stretchng rate α ν ( ) v = ( v ) = 0 model (): U = ( αx, α αz) Vortex stretchng: U ( ) = ( α, α, α ) Expect when stead = ( r) >>, 0 U ( U ) = ν z d ν d d αr = α r dr r dr dr model (): U = ( αx, α αz) Check: = ( 0,0, ( r) ) v = ( 0, v ( r),0) d ν αr = α dr r ( r) = ( r = 0) exp d dr d r dr ( r ).e. vortex tube structures of radus θ ν α ~ dffuson rate stretchng rate ( ) v = ( v ) = 0 α ν example Buntne and Pulln, JFM 05, 6 (989) u = U v =, ( αx, βy ( α β) z) v U = 0 (ncompressble) and U = 0 (rrotatonal) = u = v Consder only specal cases when v s two-dmensonal,.e.. v = ( v( x, t), v( x, t),0 ) = 0,0, ( x, t) vortex stretchng: ( ) ( ) u = ( )( U v ) = ( )U convecton: ( u ) = {( U v) } ( U )

10 example T.S. Lundgren, Phys. Fluds 5, 9 (98) U ( U = a( t) r, U = 0, U a( t) z ) = r θ z = ~ same wth those used by Buntne and Pulln ~ analytcal solutons ~ tube core surrounded by spral vortex sheets Nonlnearty Generates Scales u = 0 ( u ) u = p g u If at some nstant tme, the flow has u = cos kx v = B cos mx then, u = cos kx k sn kx v u = cos kx Bmsn mx k Bm = µ = sn( kx) ( sn( m k) x sn( m k) x)