Geometry and statistics in turbulence

Transcription

1 Geomety and statistics in tubulence Auoe Naso, Univesity of Twente, Misha Chetkov, Los Alamos, Bois Shaiman, Santa Babaa, Alain Pumi, Nice.

2 Tubulent fluctuations obey a complex dynamics, involving subtle nonlinea, nonlocal inteactions, leading to enegy tansfe between scales. Objective : obtain a desciption of the fluctuating velocity field that captues both the scaling and stuctual aspects of the flow. To popely chaacteize the flow, focus on the full velocity gadient tenso : m ab = " a u b O its coase gained genealization : M ab = 1 V R " V R # m ab d d

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5 The evolution of the tetahedon and M can be modelled by a stochastic diffeential equation (Chetkov et al, 1999) which we ae studying diectly. Potential pay-offs : Fundamental infomation about the nonlinea pocesses in the Navie-Stokes equations. Invitation to think about multipoint coelation. Get insight about the tansfe pocess between scales (Pumi et al, 2001, Bandi et al, 2006). Potentially, paticle based LES (Shaiman et al, 2003).

6 M as a diagnostic of flow topology * The eigenvalues of M chaacteize the local topology of the flow. * The depend (Cayley- Hamilton) on the two invaiants : Votex dominated Q = " 1 2 T(m2 ) R = " 1 3 T(m3 ) Stain dominated

7 Outline of the pesentation The stochastic M-model : deivation and definition. Semi-classical solutions of the model. Numeical solutions and compaisons with DNS with an isotopic focing. Numeical solutions in the pesence of a lage shea flow. Conclusions and pespectives.

8 The stochastic model : Deivation and definition

9 The stochastic M-model : deivation and definition (1) Wite the Navie-Stokes equation fo the velocity gadient tenso : dm ab 2 + m ab = "# ab p + viscosity + focing dt Cucial ingedient : the pessue hessian Isotopic appoximation (esticted Eule dynamics, cf Vieillefosse, Cantwell) : " ab p = # 1 3 T(m2 )$ ab The esulting system can be completely solved, with the help of the invaiants Q and R (Q = -t(m 2 )/2; R= -t(m 3 )/3) : -> Finite time singulaity!

10 The stochastic M-model : deivation and definition (2) To go beyond the Vieillefosse singulaity, one needs to intoduce the geomety of the Lagangian set of points. Equation fo the geomety, deived fom : Whee : " i a = ( " i ) a ".M " d" dt = v = ".M + # = coheent component of the velocity field (k~1/r) = apidly fluctuation component (k >> 1/R). = set of educed coodinates, paametizing the tetad. Intoduce the moment of inetia tenso : g = " t "

11 The stochastic M-model : deivation and definition (3) Equation fo the coase-gained velocity gadient tenso (obtained fom an appoximation of the pessue Hessian, based on analytical and numeical esults) : dm dt (M 2 " #T(M 2 )) "(M 2 # $T(M 2 )) " $ g #1 ' &" = ) % T(g #1 )( + (M 2 " #T(M 2 )) = $(M 2 " #T(M 2 )) + % «local» component of the pessue «non local» component of the pessue. fluctuating component Reduction of the nonlineaity though the pessue Hessian : the impotance of this effect is measued by α

12 The stochastic M-model : deivation and definition (4) One finally obatains the following system of stochastic diffeential equations : ( dm * + (1"#)(M 2 " $T(M 2 )) = % dt ) * d& dt " g.m " M t.g " ' T(MM t )(g " T(g)Id) = 0 + The effect of the noise in the g-equation is assumed to (mostly) estoe the isotopy of the g-tenso. It is substituted hee by the β-tem. The noise η is modelled by a Gaussian white noise tem, obeying the K41-scaling (ρ 2 = T(g)) : ' " ab (#,t)." cd (0,0) = $ % ac % bd & 1 3 % * ) ( ab% cd, - + # %(t) 2

13 The stochastic M-model : deivation and definition (5) Summay : the model thus educes to a set of nonlinea, stochastic diffeential equations, with 3 dimensionless paametes : Reduction of nonlineaity by the paamete α. Stength of the mechanism that estoes isotopy of the g tenso, β. Intensity of the fluctuations in the M-equation, γ.

14 Enegy balance Define the enegy at scale ρ by E=T(VV t )/2 with : V ia =ρ ia M ba Equation of evolution of the enegy : " t E(#) = $ " V a a i t(vv T ) + % t(vv T M) + (coupling with small scales) "# # # i Physical intepetation : " # V a : lage scale enegy flux a i t(vv T ) #$ $ i " t(vv T M) # : eddy-damping tem (see Boue and Oszag,1998, Meneveau and Katz 2000, )

15 The model povides a way to compute the statistical popeties of the M-tenso as a function of scale! What is the qualitative behavio of the solutions of this system of equations? N.b. : it depends on the thee paametes : α, β and γ.

16 Methods of esolution of the system

17 The equation satisfied by the Euleian PDF A Fokke-Planck equation fo the Euleian PDF can be deived fom this stochastic system : " t P(M,g,t) = L.P(M,g,t) The statitionay solutions must satisfy the system : L.P = 0 " dmp(m,g) =1 & P(M,g = L 2 Id) # exp $ T(MM ) ( T ) + ( (%L $2 ) 2 / 3 + ' * (Gaussian distibution at the integal scale)

18 and its solution in tems of path integals The system can be solved using Geen s functions methods : " t P(M,g) = $ dm' $ dt G #T (M,g M',g')P(M',g') (G : Geen s function; P(M,g ) : bounday condition) With : G "T (M,g M',g') = # [ DM'' ][ Dg'' ] exp "S(M'';g'') [ ] Hence : P(M,g) = [ ] [ Dg'' ] [ ] " dm' " dt " DM'' " exp #S(M'';g'') + T(M' M' t ) /($L #2 ) 2 / 3 (Geen s function) (bounday condition)

19 Stating fom an initial condition at the integal scale, one integates the system up to a fixed scale (in the inetial ange). In pinciple, one has to integate ove all tajectoies in phase space.

20 (Appoximate) method of esolution (1) One could use a staightfowad Monte-Calo method (exact in pinciple) Difficulty : the method is extemely inefficient, since one has to deal with tajectoies with widely diffeent statistical weight (by odes of magnitude!). Obtaining eliable numeical esults equies pohibitively lage compute time. Look fo deteministic solutions (γ=0) -> encouaging esults when compaed with DNS (Chetkov et al, 1999)

21 (Appoximate) method of esolution (2) One uses hee the semiclassical appoximation (saddle point appoximation of the path integal) Method : one consides only the tajectoy fo which the action is minimal (the one with the lagest statistical weight). Hope : The method should povide impotant infomation, especially since many tajectoies do not contibute vey much. Dawback : the method is not igoous; it is difficult to contol the eos made. => A bette algoithm has to be implemented to undestand the effect of fluctuations (~Monte-Calo), and to eally estimate the eos made by using the semi-classical appoximation.

22 Numeical solutions of the system in the semiclassical appoximation with isotopic focing. Compaison with DNS data A. Naso and A. Pumi, Phys. Rev. E 72, (2005)

23 Scaling laws of the 2nd and 3d ode moments of M : DNS solutions (R λ =130; ) Accoding to the K41 scaling laws, "u() # 1/ 3 so M() " #2 / 3 and " 2, T(S 2 ) # $4 / 3 $T(M 2 M t ) # $2 DNS esults : these thee quantities follow the expected Kolmogoov scaling

24 Evolution of P(R,Q) as a function of scale; DNS solutions (R λ =130; ) "u() # 1/ 3 /L =1 /L = 1 2 /L = 1 4 /L = 1 8

25 P(R,Q) measued in expeiments (inetial ange) F. van de Bos, B. Tao, C. Meneveau and J. Katz, Phys. Fluids 14, 2456 (2002)

26 Model pedictions The paamete that has the most impotant effect on the solution is α (eduction of the nonlineaity). The pedictions of the model agee with DNS esults povided α is in a naow inteval aound α ~ 0.5.

27 Scaling popeties of the matix M : model esults (1) L = 1 2 L = 1 4 L = 1 8 L = 1 16 The second moment of M has the ight scaling povided α is not too small!

28 Scaling popeties of the matix M : model esults (2) L = 1 2 L = 1 4 L = 1 8 L = 1 16 At small value of α, the stain gows with a powe that diffes significantly fom 4/3 The eduction of nonlineaity should not be too small!

29 Scaling popeties of the matix M : model esults (3) L = 1 4 L = 1 16 The sign of <t(m 2 M t )> is negative, as it should, fo small values of α. The eduction of nonlineaity should not be too lage!

30 Scaling popeties of the matix M : model esults (4) Influence of the paamete β : Not much effect povided β is lage enough. Influence of the paamete γ : Main effect : change the numeical value of " 2 # 4 / 3

31 Evolution of P(R,Q) as a function of scale; semiclassical solutions of the model (1) Paametes : α=0.45; β=0.4; γ=0.25 L = 1 2 L = 1 4 L = 1 8 L = 1 16

32 Evolution of P(R,Q) as a function of scale; semiclassical solutions of the model (2) Paametes : α=0.6; β=0.4; γ=0.25 L = 1 2 L = 1 4 L = 1 8 L = 1 16

33 Scale dependence of the enegy tansfe density : DNS L = 1 2 L = 1 8 L = 1 4 L = 1 2 L = 1 16

34 Scale dependence of the enegy tansfe density : semiclassical solution L = 1 2 L = 1 8

35 Summay : acceptable values of α The solution is acceptable povided a is in a naow inteval aound α ~ !

36 Semiclassical solution : a What we have done : detemine the optimal solution, and ignoe the contibutions of othe neaby tajectoies. Effect of vaying voticity aound the optimum : a bette calculation may be necessay. caveat

37 Numeical solutions of the system in the semiclassical appoximation with a lage scale shea. A. Naso, M. Chetkov and A. Pumi, J. Tub. (2006)

38 The issue of etun to isotopy One of the postulates of tubulence theoy is the univesality of small scale velocity fluctuations, which implies that as the scale diminishes, the flow popeties should estoe isotopy. Study hee an homogeneous shea flow. Nb : Expeimental data (Shen and Wahaft, 2000) and numeical data (Pumi&Shaiman, 1995,1996) suggest that the etun to isotopy is much slowe than naively expected.

39 The poblem studied hee The tetad model can be used to study seveal kinds of focing, simply by changing the lage scale condition. -> impose a lage scale shea, and calculate the scale dependence of P(R,Q), and othe quantities. Same equations as in the isotopic case; simply change the lage scale bounday condition : %"T[(M " #)(M " P(M.g = L 2 #)t ]( Id) ~ exp ' * &' ($L "2 ) 2 / 3 )* Whee : # 0 s 0& % ( " = % ( $ % 0 0 0' ( ; s measues the shea intensity

40 Scale dependence of P(R,Q) : semiclassical solutions with s=0,1,6 L = 1 2 L = 1 8 L = 1 4 L = 1 16 Paametes : α=0.6, β=0.4; γ=0.25

41 Scale dependence of <ω 2 > at diffeent values of s

42 Scale dependence of <T(S) 2 > at diffeent values of s

43 Scale dependence of the enegy tansfe at diffeent values of s

44 The issue of etun to isotopy Ou esults ae consistent with the accepted view that the effects of lage scale anisotopy decease when the scale deceases. New finding : diffeence of behavio between voticity dominated and stain dominated stuctues. The anisotopy effects decease faste fo voticity dominated quantities (enstophy) athe than fo stain dominated objects (stain, enegy tansfe). Faste elaxation of voticity dominated quantities towads isotopy may be consistent with the facts that voticity is found to be moe intense, hence less sensitive to the lage scale focing.

45 Conclusions and pespectives.

46 Conclusions and pespectives (1) Ou wok is based on a dynamical model of tubulent velocity fluctuations, that contains seveal key fluid mechanical ingedients. The model is fomulated in tems of a stochastic diffeential equations, that depend on 3 dimensionless paametes. The solutions have been obtained in the semiclassical limit, in two cases. - isotopic focing : compaison with DNS esults shows the impotant ole of the nonlineaity eduction (ole of the paamete α). - anisotopic focing : diffeence in the popeties of etun to isotopy between voticity dominated and stain dominated stuctues.

47 Conclusions and pespectives (2) Easy to study the influence of bounday conditions at lage scales on small scales. In pogess : development of an hybid method that incopoates moe pecisely the fluctuations in the dynamics ( beyond the semiclassical appoximation). Expected output : find out about the impotance of the fluctuations as a function of the flow stuctues.

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