stochastic models for turbulence and turbulent...

Center for Wind Energy Research

stochastic models for turbulence and turbulent driven systems

Dank an Mitarbeiter

J. GottschallM. Hölling,M.R. LuhurSt LückP. MIlanA. NawrothPh. RinnN. ReinkeCh. RennerR. StresingM. Wächter

Koopertionen

R. FriedrichD. KleinhansM. KühnR. TabarCh. Vassilicos

Mittwoch, 14. November 12


HWK 2012

synergetic approach to turbulence

stochastic cascade model - n-point statistics of turbulence

deeper insights into turbulence and turbulent driven systems



HWK 2012

synergetic approach

order parameter

complex systeminteracting subsystems

} slaving tolow dimensional

dynamics



HWK 2012

synergetics and hierarchical structures

order parameter

}hierarchical

cascade structure--allows this a

simplification, too?order parameters?



HWK 2012


turbulence

• cascade structure of interacting vorticities

L

η

r

integral length

dissipation length



HWK 2012


turbulence

• cascade structure of interacting vorticities

• Rudolf - look at the interacting structures on different scales

L

η

r

p(ur, r|ur0 , r0)

@rp(ur, r)process evolving in the cascade parameter r



HWK 2012



deeper insights to turbulence and turbulent driven systems

L

r2

η

r1



HWK 2012

turbulence

p(u(x1), ..., u(xn+1))

comprehensive description by n-point statistics

p(u(x1), . . . , u(xn+1)) = p(ur1 , . . . , urn , u(x1))

uri = u(x + ri)� u(xi)using velocity increments:

ri


http://dict.leo.org/ende?lp=ende&p=DOKJAA&search=comprehensive&trestr=0x8004

http://dict.leo.org/ende?lp=ende&p=DOKJAA&search=comprehensive&trestr=0x8004


HWK 2012

n-point statistics

Bayes theorem - joint pdf by cond. pdf

p(u(x1), ..., u(xn+1)) = p(ur1 , ..., urn , u(x1))= p(ur1 , ..., urn |u(x1)) · p(u(x1))

= p(ur1 |..., urnu(x1))p(ur2 |..., urnu(x1))... · p(u(x1))

= p(ur1 |ur2 , u(x1)) ... p(urn�1 |urn , u(x1)) · p(u(x1))

L

r2

η

r1



HWK 2012

n-point statistics

Bayes theorem - joint pdf by cond. pdf

p(u(x1), ..., u(xn+1)) = p(ur1 , ..., urn , u(x1))= p(ur1 , ..., urn |u(x1)) · p(u(x1))

= p(ur1 |..., urnu(x1))p(ur2 |..., urnu(x1))... · p(u(x1))

= p(ur1 |ur2 , u(x1)) ... p(urn�1 |urn , u(x1)) · p(u(x1))

can this be simplified?

or even one increment statistics? p(ur1) . . . p(urn)



HWK 2012

increment statistics measurable

time signals, u(t) ,

measured increments ur for different r

11,5

22,5

33,5

0 0,5 1 1,5 2 2,5 3 3,5 4

v [m

/s]

-2-1,5

-1-0,5

00,5

11,5

2

0 0,5 1 1,5 2 2,5 3 3,5 4

δ vL [

m/s

]

-0,8-0,6-0,4-0,2

00,20,40,60,8

0 0,5 1 1,5 2 2,5 3 3,5 4

δvλ [

m/s

]

t [s]

p(ur1 |. . . , urn , u(x1))



HWK 2012

statistics of turbulence

simplification(1)

(2)

experimental test

experimental result:

p(u1|u2,u3) = p(u1|u2)

(1) holds

(2) not

€

p( u 1,r1 u 2,r2; ... ;

u n ,rn )= p( u 1,r1 u 2,r2)

€

p( u 1,r1 u 2,r2; ... ;

u n,rn )= p( u 1,r1)

-2 -1 0 1 2-3

-2

-1

0

u /1 σ-3 -2 -1 0 1 2

-3

-2

-1

0

u /1σ

u /1σ

-4 -2 0 2 4-4

-2

0

2

4

u /2 σ



HWK 2012

n-point statistics

p(u(x1), ..., u(xn+1))= p(ur1 |ur2 , u(x1)) ... p(urn�1 |urn , u(x1)) · p(u(x1))

L

new view of cascade process : three point closure

ηr....

-2 -1 0 1 2-3

-2

-1

0

u /1 σ-3 -2 -1 0 1 2

-3

-2

-1

0

u /1σ

u /1σ

-4 -2 0 2 4-4

-2

0

2

4

u /2 σ



HWK 2012

x1

n-point statistics - three point closure

p(ur1 |ur2 , u(x1))

three point quantity

x2

x3

r1

r2



HWK 2012

n-point statistics


new view of cascade process:three point closurelocal in the cascade means no memoryor Markow process in r

L ηr....



HWK 2012


Markow prop & cascade with Fokker-Planck Equ.

L ηr....

�rj@

@rjp(urj |urk , u(x1)) = {� @

@urj

D

(1)(urj , rj , u(x1)) +@

2

@u

2rj

D

(2)(urj , rj , u(x1))} p(urj |urk , u(x1))

n-point statistics



HWK 2012

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shi$%of%dri$%func-on, % %no%u/dependence%of%diffusion%func-on%Stresing et.al. New Journal of Physics 12 (2010)

urur

�rj@

@rjp(urj |urk , u(x1)) = {� @

@urj

D

(1)(urj , rj , u(x1)) +@

2

@u

2rj

D


D(n)(ur, r) =1

n! · �lim�!0

Z(ur � ur)n p(ur+�, r + �|ur, r)dur

n-point statistics

Markow prop. => Fokker-Planck Equ. measured by KM coefficients



HWK 2012

n-point statistics

3. Verification of the measured Fokker-Planck equation

- numerical solution compared with experimental results

- => n-scale statistics

10 -6

10 -4

10 -2

100

102

104

-4 -2 0 2 4u / σ∞

r

u0/σ∞-6 -4 -2 0 2 4 6-6

-4

-2 0

2

4

6

8

Journal of Fluid Mechanics 433 (2001)

€

p(ur,r ur0 ,r0)



HWK 2012



• stochastic process in r - tremendous reduction in complexity

L

r2

η

r1

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shi$%of%dri$%func-on, % %no%u/dependence%of%diffusion%func-on%urur



HWK 2012



• stochastic process in r - tremendous reduction in complexity

deeper insights to turbulence and turbulent driven systems

L

r2

η

r1

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shi$%of%dri$%func-on, % %no%u/dependence%of%diffusion%func-on%urur



HWK 2012

reconstruction of time series

D(1)(ur, r, u) = d10(r, u)� d11(r)ur

D(2)(ur, r, u) = D(2)(ur, r)

no u dependence

with u dependence

blow upNawroth et al Phys. Lett. (2006)



HWK 2012

turbulence - classical theory

Friedrich JP PRL (1996)

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shi$%of%dri$%func-on, % %no%u/dependence%of%diffusion%func-on%ur ur

�rj@

@rjp(urj |urk , u(x1)) = {� @

@urj

D

(1)(urj , rj , u(x1)) +@

2

@u

2rj

D


D(1)(ur) =13ur

@

@rur =

13

ur

r ur / r1/3

< unr >/ rn/3

< unr >/ rn/3+µ(n)

D(1)(ur, ) = �ur

D(2)(ur) = �u2r

Kolmogorov 41

Langevin equation

Kolmogorov 62

RF & JP PRL 78 (1997)



HWK 2012

finance

Functional form of the coefficients D(1) and D(2) is presented

),(),(),(),( )2(2

2)1( ττ

∂∂

τ∂∂

τ∂τ∂ RpRD

RRD

RRp ⎥

⎦

⎤⎢⎣

⎡+−=

Example: Volkswagen, τ = 10 min

-2.10-3 0 2.10-3-0.01

0.00

0.01

-1.10-3 0 1.10-30

2.10-7

4.10-7

R /σ∞ R /σ∞



HWK 2012

multi-scale statistics

additive term in the diffusion term: -> additive noise

D(1)1 (ur, r) = du

1 (r)ur

D(2)(ur, r) = d2(r) + du2 (r)ur + duu

2 (r)u2r

10 -6

10 -4

10 -2

100

102

104

-4 -2 0 2 4u / σ∞

r

Gaussian tip

-0.5 0.0 0.510-7

10-5

10-3

10-1

101

103

105

R / σ

p(R

) [a

.u.]

12 h4 h1 h15 min4 min

comparison of data withnumerical solution of theKolmogorov equation



HWK 2012

wake flow behind a cylinder - turbulent structures

drift term as function of r



HWK 2012

wake flow behind a cylinder - turbulent structures

drift term as function of r

phase transition to isotropic turbulence



HWK 2012

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)



HWK 2012

turbulent driven systems



HWK 2012


open question - what is the corresponding dynamics

x =??

x(t + ø) =??



HWK 2012

synergetic approach

order parameter

complex systeminteracting subsystems


dynamics

•deterministic part •noisy part

x(t) = D

(1)(x(tj), tj) +q

D

(2)(x(tj)°0(tj)



HWK 2012

Markov process - Langevin Equation

the basic element is the

transition probability

from this we can get

x(t) = D

(1)(x(tj), tj) +q

D

(2)(x(tj)°0(tj)

p(x, t + ø |x, t)

D

(1)(x) = limø!0

1ø

DX(t + ø)° x

EØØØX(t)=x

D

(2)(x) = limø!0

1ø

D(X(t + ø)° x)x)(X(t + ø)° x)T

EØØØX(t)=x

x3

x1

x2

x

p(x,t+o | x,t)~x~

Siegert et.al. Phys. Lett. A 243, 275 (1998)



HWK 2012

Rayleigh Benard Experiment

max. temperature difference 20°CRa < 9*109

measurements with ultrasonic dopper Anemometer DOP200ß



HWK 2012


DOP2000 - profile measurements

red rising,; blue sinking

time

h

h



HWK 2012


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

Potential

Ra = 1010



HWK 2012


Analysis as stochastic process in time -Langevin Equation Potential

PDF and its reconstruction

Ra = 1010

x(t) = D

(1)(x(tj), tj) +q

D

(2)(x(tj)°0(tj)



HWK 2012


tilting - by less than 1°

Potential

MA ThesisM. Peters M. Langner



HWK 2012

turbulent wind energy




HWK 2012

dynamics of power conversion

PWT =12cp(�) ⇢ u3

wind · A



HWK 2012

working conditions for wind turbine



HWK 2012

stochastic motion in a potential

P = D(1)(P |u) +q

D(2)(P |u) · �

D(1)(P |u)



HWK 2012

conversion of wind power a stoch. process

P = D(1)(P |u) +q

D(2)(P |u) · �



HWK 2012

power production

summed up difference in the power production with the same

measured wind data as input

Zeit in sec

5%difference

0 5 10 150%

50%

100%

u [m/s]

P(u)/Prated

<P(u)> (10%)

<P(u)> (20%)<P(u)> (30%)

D(1)(u,P)=0

P(u)real



HWK 2012

material fatigue



HWK 2012

characterization of dynamic stall with turb inflow

dcL(t)dt

= �⇤⇥(cL(t), u) + g(cL(t), u) · �(t)noise term

J. Schneemenn et al, EWEC 2010;



HWK 2012

time dependent complexity

synergetics for complexity

} hierarchicalcascade structure--

allows this a simplification, too


dynamics

scale dependent complexity

stochastic equation are measurable

comprehensive description of complex systems

deeper insights

high accuracy


SA March 2010

and all best wishesMittwoch, 14. November 12

stochastic models for turbulence and turbulent...

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