Conservation of Circulations in Turbulent Flow

Transcription

1 (D) Conservation of Circuations in Turbuent Fow We have emphasized the importance of deveoping a better understanding of the dynamica & statistica origin of the positivity of vortex-stretching rate ω S ω > 0. For the fine-grained vorticity ( 0), this is important in Tayor s proposed mechanism for production of energy dissipation ν ω 2. For in the inertia-range, we have seen that the fux of energy to sma-scaes is proportiona to the stretching rate: Π 2 ω S ω, to a first approximation, oca in space and in scae. A deeper understanding of the vortexstretching process might hep us to better predict and contro turbuent fow. As we reviewed at the beginning of this chapter, most arguments for positivity of vortexstretching use in some form or another the conservation of circuations. Here we give, just as one instance, the foowing expanation of T&L: Vorticity ampification is a resut of the kinematics of turbuence. As an exampe, take a situation in which the principa axes of the instantaneous strain rate are aigned with the coordinate system, so that S ij has ony diagona components (S 11,S 22,and S 33 ). Let us assume for simpicity that S 22 = S 33, so that, by virtue of continuity, S 11 = 2S 22. The term ω i ω j S ij becomes, if we aso assume that ω 2 2 = ω2 3, ω 2 1 S 11 + ω 2 22 S 22 + ω 2 3 S 33 = S 11 (ω 2 1 ω2 2 ) If S 11 > 0, ω 2 1 is ampified (see Figure 3.5), but ω2 2 and ω2 3 are attenuated because S 22 and S 33 are negative. Thus, ω 2 1 ω2 2 tends to become positive if S 11 is positive. Again, if S 11 < 0, ω 2 1 decreases, but ω2 2 and ω2 3 increase, so that ω1 2 ω2 2 < 0, making the stretching term positive again. 28

2 H. Tennekes & J. L. Lumey, A First Course in Turbuence, (MIT press, Cambridge, MA, 1972), Section 3.3. p.92 This argument assumes that, as the vortex engthens aong a particuar direction the 1- direction, say the corresponding component ω 1 of vorticity increases. This resut depends, however, on the conservation of circuations. T&L discuss this expicity in their expanation of their Fig. 3.5, which iustrates vortex-stretching in a wind-tunne contraction: To expain this figure, T&L wrote: The change of vorticity by vortex stretching is a consequence of the conservation of anguar momentum. The anguar momentum of a materia voume eement woud remain constant if viscous effects were absent; if the fuid eement is stretched so that its cross-sectiona area and moment of inertia become smaer, the component of the anguar veocity in the direction of the stretching must increase in order to conserve anguar momentum. 29

3 - H. Tennekes & J. L. Lumey, A First Course in Turbuence, (MIT press, Cambridge, MA, 1972), Section 3.3. pp As usua, T&L use the term conservation of anguar momentum as a more eementary substitute for the more proper term conservation of circuations. The argument is essentiay that of G. I. Tayor (1937, 1938) that we discussed earier. If we et ω 1, ω 1 represent the vorticity magnitude in the 1-direction before and after stretching, respectivey, and ikewise A 1, A 1 the cross-sectiona area of the vortex tube before and after, conservation of circuations impies that ω 1 A 1 = ω 1 A 1. If we et 1, 1 represent the ength of the tube before and after stretching, then incompressibiity impies 1 A 1 = 1 A 1. It therefore foows that ω 1 / 1 = ω 1 / 1 or, equivaenty, ω 1 /ω 1 = 1 / 1. Thus, the vorticity magnitude grows in direct propotion to ine ength. Large-Scae Circuation Baance The arguments of Tayor, Tennekes & Lumey, etc. assume that circuations wi be conserved if viscous effects were absent. The negigibiity of the viscous terms may indeed be expected in the arge scaes, with fixed as ν 0. Thus, et us consider in detai the baance of circuations in the scaes >. We have seen that the momentum baance in the arge scaes takes the form D t ū = p + f s ν ω + f B = p + f v ν ω + f B (32) with p = p + k. Let us now consider the Lagrangian circuation 30

4 K (t) ū (t) dx (33) C (t) where C (t) is the oop C advected by the arge-scae veocity ū to time t. One can see from a coupe of points of view that this is the interesting quantity to consider. First, mathematicay, we note that d dt K (t) [f s,v ν ω + f B C (t) ] dx (34) Secondy, from the physica point of view, K (t) is the quantity that an experimentaist woud consider who wanted to test the vaidity of Kevin s Theorem by means of fuid measurements at space resoution. Measuring ū (x,t) [e.g. by hoographic PIV or other techniques], he woud then be abe to construct C (t) by soving d dt C (θ i,t)=ū (C (θ i,t),t) for a set of fuid markers θ i [0, 2π],i=0,,N 1. Finay, he coud take K (t) = N 1 i=0 ū (C (θ i,t),t) [C (θ i+1,t) C (θ i,t)] (35) or some other discrete approximation to the integra, converging in the imit N. By then taking ν as sma as possibe and measurements at finer resoution 0, the experimentaist coud attempt to verify the vaidity of conservation of circuations. Now et us consider the order of magnitude of various terms in the circuation baance. First, as we have seen before, so that f s,v = O( δu2 () ) C (t) f s,v (t) dx = O( δu2 () L(C (t))) (36) where L(C) is the ength of the rectifiabe curve C. Note that C (t) is rectifiabe when the starting oop C is so, since ū is smooth and generates a fow of voume-preserving diffeomorphisms, which carries a rectifiabe oop to another rectifiabe oop. 31

5 Now consider the viscous term with Re δu()/ν. Thus, ν ω = ν ū = O( νδu() 2 ν C (t) ( ω ) dx = O( δu2 () )=O( δu2 () Re 1 ) L(C (t)) Re 1 ) (37) which is much smaer than the turbuent force term for Re 1. Thus, we have some theoretica support to the idea that the viscous term is negigibe. In particuar, it vanishes in the imit as ν 0 with fixed. Now consider the contribution from the body force f B. To get the best estimate, it is hepfu to transform this term using Stokes Theorem: B f (t) dx = C (t) ( f B ) da minima surface spanning C (t) (38) We have used the freedom in choosing the surface which spans the oop C (t) to seect the one of minima area. It foows from the work of J. Dougas, Soution of the probem of Pateau, Trans. Ann. Math. Soc (1931) that such a minima spanning surface S exists for any cosed simpe curve C not necessariy even rectifiabe. [For this work, Dougas won the first Fieds Meda in 1936!] Thus, C (t) f B (t) dx = O( f B A(S min (t))) (39) Note that f B = ( f B ) so that f B f B. Let us assume further that f B (x,t)= k < 2π L ˆf B (k,t)e ik x (40) so that f B L 2π L ˆf B L 1 = O( 1 L ) (41) 32

6 for some arge L of the order of the integra ength. Thus, we see that f B is bounded, independent of, and quite sma. On the contrary, the term Γ (C, t) C (t) f v,b (t) dx = O( δu2 () L(C (t))) (42) may diverge in the imit as 0! This term represents the torque around the oop C (t) imposed by the subscae force f s [or f v ]. For exampe, in K41 theory, δu() (ε) 1/3 = δu2 () (ε) 2/3 1/3, as 0 More generay, if u has Höder exponent h δu() h = δu2 () (const.) 2h 1 which ony needs to vanish if h> 1 2. This seems to have been first observed by G. L. Eyink, Turbuent cascade of circuations, C. R. Physique, (2006) Since there seem to be many points in a turbuent fow with h<1/2 (in particuar, the mostprobabe vaue h = 1/3),we see that f s diverges as 0 for a arge number of points! Furthermore, it is beieved that a oop C(t) advected by such a rough veocity fied becomes fracta with (Hansdorff) dimension D>1. For exampe, see F. C. G. A. Nicoeau & A. Emaihy, Study of the deveopment of threedimensiona sets of fuid partices and iso-concentration fieds using kinematic simuations, J. Fuid Mech (2004) and many references therein. Note that N (C(t)) = L(C (t)) where N (C(t)) is the number of bas of radius required to cover C(t). thus, and N (C(t)) ( L 0 ) D, as 0 33

7 L(C (t)) ( L 0 ) D 1 D, as 0 This aso diverges as 0 since a fracta curve with D>1 is non-rectifiabe and has infinite ength. Note, however, that S min (t) has finite area, so that A(S min (t)) is constant for L 0. We thus see that the dominant term in the circuation baance is Γ (C, t) = f v,b (t) dx = O(δu 2 ()( L 0 C (t) )D ) (43) as 0, which eads to the distinct possibiity that Γ (C, t) diverges in the imit. Of course, the RHS above is just a big-o bound or upper bound. There is the possibiity of arge canceations in the integra over C (t), which coud prevent divergence or even in principe aow Γ (C, t) to vanish in the imit 0. To address this issue, we consider numerica resuts from S. Chen et a., Is the Kevin Theorem Vaid for High Reynods Number Turbuence? PRL 97: (2006): (a) PDF of the circuation fux for oops with radius R = 64 and for cutoff wave numbers k c = π/ with < R. (b) The rms vaue of the circuation fux as a function of k c for various oop sizes R. The inset pots the pateau rms vaue versus R. Thus, we see that Γ (C, t) 0 as 0! The effects of the subscae force are persistent as 0. There is no conservation of circuations for fixed as ν 0, athough the effects of viscosity indeed become negigibe in that imit. 34

8 Even as 0 after having taken ν 0 first, the effects of the subscae force term does not disappear! This casts doubt on the arguments of Tayor, Tennekes & Lumey, and others that appea to conservation of circuations in expaining turbuent dynamics. Simiar resuts hod for the fine-grained circuation baance: d u(t) dx = [ ν ω + f B ] dx (44) dt C(t) C(t) There is no subscae force, but now the viscous force f vis ν = ν ω is arge! This may be estimated as f vis ν = O(ν δu(η h) )=O( νu 0 ( η L 2 h L ) h 2 )=O( νu 2 h 0 L Re 1+h ) since η h /L (Re) 1/(1+h) in the mutifracta phenomenoogy. Then η 2 h f vis ν = O( u2 0 L Re 1 2h 1+h ) at a point with Höder exponent h. This diverges as Re uness h> 1 2!! We see thus aso that the fine-grained circuations wi not be conserved in a turbuent fow, even in the imit as ν 0 (at east not in the conventiona sense.) This was appreciated, to some extent, by G. I. Tayor. For exampe, he wrote when ω 2 has increased to some vaue which depends on viscosity, it is no onger possibe to negect the effects of viscosity in the equation for the convervation of circuation. - G. I. Tayor & A. E. Green (1937) In a aminar fow, these effects wi became negigibe in the imit as ν 0, but in a turbuent fow they persist even in the inviscid imit. 35

9 The concusion is that there is no ength-scae in a turbuent fow at which circuations are conserved, for individua oops. In the inertia-range, the circuations are not conserved because of the subscae force terms f v,s. In the dissipation range, the circuations are not conserved because of the viscous force fν vis. At every ength-scae, the circuations must be expected to change substantiay in time and not to be conserved, as has often been assumed. This casts doubt on the standard arguments of Tayor, Tennekes & Lumey, etc. for growth of ω 2 (t), for positivity of ω i S ij ω j, etc. which depend on conservation of circuations. Is there any way to see how these arguments might be somehow vaid? Recent fundamenta progress on this question has come from an important mathematica resut: P. Constantin & G. Iyer, A stochastic Lagrangian representation of the threedimensiona incompressibe Navier-Stokes equations, Commun. Pure App. Math. Vo. LXI, (2008) These authors have shown that there is a beautifu generaization of the Kevin Theorem on conservation of circuations which appies to the incompressibe Navier-Stokes equation. They show that circuations are conserved even for ν > 0, not deterministicay but in a precise stochastic sense! To state their resut, one must consider an ensembe of stochastic Lagrangian fows x(τ a, t), generated by SDE s d x = u( x,τ)dτ + 2νd W(τ), x(t) =a (45) where W(τ) is a d-dimensiona vector Brownian motion. For a given veocity fied u(x,t) in spacetime, this stochastic equation may be soved both forward and backward in time. Suppose that we consider a specific cose, rectifiabe oop C at time t and generate the stochastic Lagrangian fows backward in time. Define the ensembe of oops generated by advecting this fixed oop C backward in time by C(τ) = x(c, τ), τ < t. (46) 36

10 The figure beow represents this infinite ensembe of oops with three typica sampes: ndquist formua. ackward in time nd bue. Starting ored arrows, are rotated to the ive the resutant rrow. icay from the magnetic fied ocations ã(t ). FIG. 4. (Coor) Iustration of the stochastic Afvén theorem. Shown are three members (red, green, and bue) of the infinite ensembe of oops obtained by stochastic advection of a oop C (back) at time t backward in time to t 0. The average of the magnetic fux through the ensembe of oops is equa to the magnetic fux through C. FIGURE. Shown are three members (red, green, and bue) of the infinite ensembe of oops obtained by stochastic advection of a oop C (back) at time t backward in time to τ < t. Constantin & Iyer (2008) have proved the foowing beautifu resut: the spacetime veocity fied u(x,t) is a smooth soution of the incompressibe Navier-Stokes equation if and ony if dx u(x,t)= dx u(x,τ) (47) mutivariabe cacuus C manipuations ec(τ) convert this into a surface integra over ã(s,t), with the surface S randomy advected backward in time to the initia time t 0. As before, ã(,t) = x 1 (,t) is the back-to-abe map for the stochastic forward fow. The resut is the foowing stochastic Afvén theorem: B(x,t) da(x) = B 0 (a) da(a), t > t 0. (33) for a rectifiabe oops C and times τ < t, where the overine ( ) denotes average over the ensembe of Brownian motions. Thus, circuations are statisticay conserved backward in time 3! Note that this resut impies an arrow of time. The statistica conservation aw woud hod forward in time instead for soutions of the negative-viscosity Navier-Stokes equation (which is we-posed ony soved backward in time for given fina conditions). S ã(s,t) Formay, the stochastic Kevin Theorem for incompressibe Navier-Stokes equations reduces This resut generaizes a previous theorem [45] to compressibe pasmas. Equation (33) expresses the conservation of magnetic fux on average, as iustrated in Fig. 4. An initia oop C, boundary of the surface S, is shown there in back. This is stochasticay advected backward in time to give an infinite ensembe of oops at the initia time t 0. These are represented by the three coored oops. The ensembe average of the magnetic fux through the coection of oops at the initia to the standard Kevin Theorem for incompressibe Euer equations in the imit as ν 0. This is rigorousy true if the Euer soution remains smooth in the imit. We sha return to the question how this conservation aw behaves in high-reynods-number turbuent fow when we study turbuent Lagrangian dynamics in the next chapter. 3 In probabiistic terminoogy, the stochastic process of circuations is a backward martingae. time t 0 is equa to the magnetic fux through the oop C at the 37 fina time t. The stochastic Afvén theorem is an exampe of what