stochastic models for turbulence and turbulent driven systems

Transcription

1 stochastic models for turbulence and turbulent driven systems Dank an Mitarbeiter J. Gottschall M. Hölling, M.R. Luhur St Lück P. MIlan A. Nawroth Ph. Rinn N. Reinke Ch. Renner R. Stresing M. Wächter Koopertionen R. Friedrich D. Kleinhans M. Kühn R. Tabar Ch. Vassilicos

2 synergetic approach to turbulence stochastic cascade model - n-point statistics of turbulence deeper insights into turbulence and turbulent driven systems

3 synergetic approach order parameter } slaving to low dimensional dynamics complex system interacting subsystems

4 synergetics and hierarchical structures order parameter } hierarchical cascade structure-- allows this a simplification, too? order parameters?

5 synergetics and hierarchical structures turbulence cascade structure of interacting vorticities L integral length r η dissipation length

6 synergetics and hierarchical structures turbulence cascade structure of interacting vorticities Rudolf - look at the interacting structures on different scales L r η p(u r,r u r 0,r 0 r p(u r,r) process evolving in the cascade parameter r

7 synergetic approach to turbulence L r 2 stochastic cascade model - n-point statistics of turbulence r 1 η deeper insights to turbulence and turbulent driven systems

8 turbulence comprehensive description by n-point statistics r i p(u(x 1 ),...,u(x n+1 )) using velocity increments: u ri = u(x + r i ) u(x i ) p(u(x 1 ),...,u(x n+1 )) = p(u r1,...,u rn,u(x 1 ))

9 n-point statistics p(u(x 1 ),...,u(x n+1 )) = p(u r1,...,u rn,u(x 1 )) = p(u r1,...,u rn u(x 1 )) p(u(x 1 )) Bayes theorem - joint pdf by cond. pdf L = p(u r1..., u rn u(x 1 ))p(u r2..., u rn u(x 1 ))... p(u(x 1 )) r 2 r 1 = p(u r1 u r2,u(x 1 ))... p(u rn 1 u rn,u(x 1 )) p(u(x 1 )) η

10 n-point statistics p(u(x 1 ),...,u(x n+1 )) = p(u r1,...,u rn,u(x 1 )) = p(u r1,...,u rn u(x 1 )) p(u(x 1 )) Bayes theorem - joint pdf by cond. pdf = p(u r1..., u rn u(x 1 ))p(u r2..., u rn u(x 1 ))... p(u(x 1 )) can this be simplified? = p(u r1 u r2,u(x 1 ))... p(u rn 1 u rn,u(x 1 )) p(u(x 1 )) or even one increment statistics? p(u r1 )... p(u rn )

11 increment statistics measurable time signals, u(t), measured increments ur for different r p(u r1...,u rn,u(x 1 )) v [m/s] δv λ [m/s] δv L [m/s] 3,5 3 2,5 2 1, ,5 1 1,5 2 2,5 3 3,5 4 0,8 0,6 0,4 0,2-0,2 0-0,4-0,6-0,8 1, ,5 1 1,5 2 2,5 3 3,5 4 0,5 1-0,5 0-1, ,5 1 1,5 2 2,5 3 3,5 4 t [s]

12 statistics of turbulence simplification (1) p( u 1,r 1 u 2,r 2 ;...; u n,r n )= p( u 1,r 1 (2) p( u 1,r 1 experimental test u 2,r 2 ;...; u n,r n )= p( u 1,r 1 ) u 2,r 2 ) σ / 1 u u / 2 σ experimental result: p(u 1 u 2,u 3 ) = p(u 1 u 2 ) (1) holds (2) not u / 1 σ u / 1 σ

13 n-point statistics p(u(x 1 ),...,u(x n+1 )) = p(u r1 u r2,u(x 1 ))... p(u rn 1 u rn,u(x 1 )) p(u(x 1 )) 4 2 L r... η σ/ 1 u u 2/ σ new view of cascade process : three point closure u / σ u 1/ σ

14 n-point statistics - three point closure three point quantity p(u r1 u r2,u(x 1 )) x 1 r 2 x 3 r 1 x 2

15 n-point statistics p(u(x 1 ),...,u(x n+1 )) = p(u r1 u r2,u(x 1 ))... p(u rn 1 u rn,u(x 1 )) p(u(x 1 )) L r... η new view of cascade process: three point closure local in the cascade means no memory or Markow process in r

16 n-point statistics p(u(x 1 ),...,u(x n+1 )) = p(u r1 u r2,u(x 1 ))... p(u rn 1 u rn,u(x 1 )) p(u(x 1 )) L r... η Markow prop & cascade with Fokker-Planck Equ. j p(u rj u rk,u(x 1 )) rj D (1) (u rj,r j,u(x 1 2 r j D (2) (u rj,r j,u(x 1 ))} p(u rj u rk,u(x 1 ))

17 n-point statistics j p(u rj u rk,u(x 1 )) rj D (1) (u rj,r j,u(x 1 2 r j D (2) (u rj,r j,u(x 1 ))} p(u rj u rk,u(x 1 )) Markow prop. => Fokker-Planck Equ. measured by KM coefficients D (n) (u r,r)= 1 n! lim!0 Z (ũ r u r ) n p(ũ r+,r+ u r,r)dũ r ' (#)!#$ $ #$!%#%*%+%*%# +%,%%%$%%%-%" + &. +%,%!" + %-%" + &. +%,%%%" + %-%" + &.!"!# $ # "!%&%" # u r shi$%of%dri$%func-on, % * +", $%$ $%" $%& $%'!(#(-(.(-(#.(/((($(((0(". )1.(/(!". (0(". )'.(/(((". (0(". )'!"!# $ # "!()(" # %no%u/dependence%of%diffusion%func-on% Stresing et.al. New Journal of Physics 12 (2010) u r

18 n-point statistics 3. Verification of the measured Fokker-Planck equation - numerical solution compared with experimental results - => n-scale statistics r p(u r,r u r0,r 0 ) u 0 /σ u / σ Journal of Fluid Mechanics 433 (2001)

19 synergetic approach to turbulence L r 2 stochastic cascade model - n-point statistics of turbulence stochastic process in r - tremendous reduction in complexity ' (#)!#$ $ #$!%#%*%+%*%# +%,%%%$%%%-%" + &. +%,%!" + %-%" + &. +%,%%%" + %-%" + &.!"!# $ # "!%&%" # u r shi$%of%dri$%func-on, % * +", $%$ $%" $%& $%'!(#(-(.(-(#.(/((($(((0(". )1.(/(!". (0(". )'.(/(((". (0(". )'!"!# $ # "!()(" # u r %no%u/dependence%of%diffusion%func-on% η r 1

20 synergetic approach to turbulence L r 2 stochastic cascade model - n-point statistics of turbulence stochastic process in r - tremendous reduction in complexity ' (#)!#$ $ #$!%#%*%+%*%# +%,%%%$%%%-%" + &. +%,%!" + %-%" + &. +%,%%%" + %-%" + &.!"!# $ # "!%&%" # u r shi$%of%dri$%func-on, % * +", $%$ $%" $%& $%'!(#(-(.(-(#.(/((($(((0(". )1.(/(!". (0(". )'.(/(((". (0(". )'!"!# $ # "!()(" # u r %no%u/dependence%of%diffusion%func-on% η r 1 deeper insights to turbulence and turbulent driven systems

21 reconstruction of time series no u dependence D (1) (u r,r,u) = d 10 (r, u) d 11 (r)u r D (2) (u r,r,u) = D (2) (u r,r) with u dependence Nawroth et al Phys. Lett. (2006) blow up

22 turbulence - classical theory j p(u rj u rk,u(x 1 )) rj D (1) (u rj,r j,u(x 1 2 r j D (2) (u rj,r j,u(x 1 ))} p(u rj u rk,u(x 1 )) ' (#)!#$ $ #$!%#%*%+%*%# +%,%%%$%%%-%" + &. +%,%!" + %-%" + &. +%,%%%" + %-%" + &.!"!# $ # "!%&%" # u r shi$%of%dri$%func-on, % * +", $%$ $%" $%& $%'!(#(-(.(-(#.(/((($(((0(". )1.(/(!". (0(". )'.(/(((". (0(". )'!"!# $ # "!()(" # u r %no%u/dependence%of%diffusion%func-on% Kolmogorov 41 D (1) (u r )= 1 3 u r Kolmogorov 62 u r = 1 3 u r r u r / r 1/3 <u n r >/ r n/3 <u n r >/ r n/3+µ(n) Friedrich JP PRL (1996) D (1) (u r, )= D (2) (u r )= u r u 2 r RF & JP PRL 78 (1997)

23 finance Functional form of the coefficients D(1) and D(2) is presented R /σ R /σ (1) 2 ( 2) p ( R,τ ) = D ( R,τ ) + 2 D ( R,τ ) p ( R,τ ) τ R R Example: Volkswagen, τ = 10 min

24 multi-scale statistics additive term in the diffusion term: -> additive noise D (1) 1 (u r,r) = d u 1(r)u r D (2) (u r,r) = d 2 (r)+d u 2(r)u r + d uu 2 (r)u 2 r Gaussian tip r p(r) [a.u.] comparison of data wi numerical solution of t Kolmogorov equation u / σ h 4 h 1 h 15 min 4 min R / σ

25 wake flow behind a cylinder - turbulent structures drift term as function of r

26 wake flow behind a cylinder - turbulent structures drift term as function of r phase transition to isotropic turbulence

27 turbulence: new insights Markov-length - a coherence length statistics of longitudinal and transversal increments universality of turbulence: fractal grid turbulence role of transfered energy er: fusion rules ri => ri+1 (Davoudi, Tabar 2000; L vov, Procaccia 1996) passive scalar (Tutkun, Mydlarski 2004) Lagrangian turbulence (Friedrich 2003,2008)

28 turbulent driven systems

29 turbulent driven systems open question - what is the corresponding dynamics ẋ =?? x(t + ø) =??

30 synergetic approach order parameter } slaving to low dimensional dynamics complex system interacting subsystems deterministic part ẋ(t) = D (1) (x(t j ), t j ) + noisy part q D (2) (x(t j ) 0 (t j )

31 Markov process - Langevin Equation ẋ(t) = D (1) (x(t j ), t j ) + q D (2) (x(t j ) 0 (t j ) the basic element is the transition probability p( x, t + ø x, t) x 3 from this we can get x 2 x ~ ~ p(x,t+ x,t) D (1) 1 D EØ (x) = lim X(t + ø) x ø!0 ø ØØX(t)=x x 1 x D (2) 1 D (x) = lim (X(t + ø) x)x)(x(t + ø) x) TEØ Ø ØX(t)=x ø!0 ø Siegert et.al. Phys. Lett. A 243, 275 (1998)

32 Rayleigh Benard Experiment max. temperature difference 20 C Ra < 9*10 9 measurements with ultrasonic dopper Anemometer DOP200ß

33 Rayleigh Benard Experiment DOP profile measurements h red rising,; blue sinking time h

34 Rayleigh Benard Experiment Ra = Potential model for bistability Sreenivasan K. R., Bershadskii A., and Niemela J. J., Mean wind and its reversal in thermal convection, Phys. Rev. E, 65:056306, 2002

35 Rayleigh Benard Experiment PDF and its reconstruction Ra = Analysis as stochastic process in time - Langevin Equation ẋ(t) = D (1) (x(t j ), t j ) + q D (2) (x(t j ) 0 (t j ) Potential

36 Rayleigh Benard Experiment tilting - by less than 1 Potential MA Thesis M. Peters M. Langner

37 turbulent driven systems turbulent wind energy

38 dynamics of power conversion P WT = 1 2 c p( ) u 3 wind A

39 working conditions for wind turbine

40 stochastic motion in a potential D (1) (P u) P = D (1) (P u)+ q D (2) (P u)

41 conversion of wind power a stoch. process P = D (1) (P u)+ q D (2) (P u)

42 power production summed up difference in the power production with the same measured wind data as input P(u)/Prated 100% 50% P(u) real <P(u)> (10%) <P(u)> (20%) <P(u)> (30%) D (1) (u,p)=0 0% u [m/s] 5% Zeit in sec difference

43 material fatigue

44 characterization of dynamic stall with turb inflow dc L (t) dt noise term = (c L (t),u)+g(c L (t),u) (t) J. Schneemenn et al, EWEC 2010;

45 synergetics for complexity time dependent complexity } slaving to low dimensional dynamics } scale dependent complexity hierarchical cascade structure-- allows this a simplification, too stochastic equation are measurable comprehensive description of complex systems deeper insights high accuracy

46 and all best wishes SA March 2010