reduced-, low- and least-order models

Reduced-, low- and least-order models— emerging insights from nonlinear dynamics,statistical physics, and turbulence theories

0

E

Ej

i j

i k

kj

k

(E + E )/2

(E + E )/2

(E + E )/2

iE

E

j

k

i

n( E )/3Σ

Bernd R. Noack & friends

Berlin & elsewhere

— supported by DFG, CNRS, US NSF & USAFA —

WallTurb Workshop, Lille, France, 21–23 April 2009

Bernd’s friends — small subset contributing to this talk

Michael

Schlegel

Joel

Delville

Marek

Morzynski

Gilead

Tadmor

Overview

1. Introduction. . . . . . . . . . . . . . . . . physics & cybernetics dreams revisited

2. Reduced-order modelling. . . . . . . . . . . . . . . . . . . . . . . . transient + attractor dynamics

3. Finite-time thermodynamics closure. . . . . . . . . . . . . . . . . . . . . . . . . . system reduction of dynamics

4. Towards model-based turbulence control. . . . . . . . . . . . . . . . . . . . . . . . . . . perspectives of FTT closure

5. Summary and outlook

Overview




4. Towards turbulence control. . . . . . . . . . . . . . . . . . . . . . . . . . . perspectives of FTT closure


Turbulence control problem

Decrease drag

FD = ex ·∮

dS p︸︷︷︸

pressure drag

+ ex ·∮

dS · ν∇u

︸︷︷︸

viscous drag

⇒ Change mean flow via Reynolds stress

∇ · u ⊗ u + ∇p − νu = −∇ · u′ ⊗ u′

u′ ⊗ u′ =N∑

i=1

2Ei ui ⊗ ui

⇒ Tickle the right instabilities

∂tu′ + ∇ · u ⊗ u

′ + ∇ · u′ ⊗ u

+ ∇ · u′ ⊗ u′ −∇ · u′ ⊗ u′

+ ∇p′ − νu′ = f =unsteady force

Reynolds decomp.

F =1

T

∫ T

0dt F (t)

u = u + u′

p = p + p′

POD

(u,v)Ω :=∫

ΩdV u · v

u′ =

N∑

i=1

ai ui

(ui,uj)Ω = δij

ai = 0

ai aj = 2 Ei δij

⇒ Challenges at high Reynolds number

(1) Stabilization of steady solution not possible ⇒ no linearization

(2) Highly nonlinear control, infinite horizon problem (turbulence cascade)

Closure for first (u) and second moments (u′ ⊗ u′ ) with actuation needed!

Turbulence control = attractor control

Phase space

natural

forced

attractor

attractor

actuation

drag reduction

lift increase

more happiness

example:

mean-field model

Dream #1: Instabilities 7→ turbulence

8

Landau−Hopf scenario

Lev D. Landau

(1902−1983)Eberhard Hopf

(1908−1968)

Paul Manneville

(1903−1987)KolmogorovAndrei N.

(1901−1976)

WernerHeisenberg

Edward Lorenz (1917−2008)

Ruelle−Takens

David Ruelle(1935−)

FeigenbaumMitchell

(1944−)

(1946−)

torus

3T

2Ttorus

strange attractor

limit cycle

fixed point

lu

br

uT e nc e

Hopf

Hopf

Hopf

Ruelle−Takens

Feigenbaumscenario

intermittencyscenario

Hopf

Newhouse s.~

scenario

Dream #2: Statistical physics 7→ turbulence

Ludwig Boltzmann

(1840–1906)

Equivalent

subsystems:

1877: Entropy

S = k lnW

Lars Onsager

(1903–1976)

Particle/vortex

picture:

1949: point vortices

in 2D flows

= thermodyn. degree of freedom

Hans W Liepmann (1914-)’s

WARNING:

Don’t

forget →

Robert H Kraichnan

(1928–2008)

Wave/Galerkin picture:

1955: Fourier modes

= thermodyn. degrees of freedom

(absolute equilibrium ensemble)

How to partition the flow in equivalent subsystems (atoms)

(= thermodynamic degrees of freedom)???

Dream #3: Control 7→ turbulence

DazumalAnno Ott, Grebogi, Yorke

1990 PRLMaxwell 1867Wiener 1948

S = k ln Wda/dt = f (a,b)linear dynamics

linear control chaos control

strange attractor statistical physicsda/dt = A a + B b

b =K aMaxwell’s deamon

Overview






Flow phenomenologies

Milton van Dyke (1975): Album of Fluid Motion

von Karman vortex street in nature

Rear side of the island Guadalupe (20 Aug. 1999)

von Karman vortex street in technology

Damaged tanker — oil visualization

Van Dyke (1982), page 100

von Karman vortex street in technology

Tacoma Narrows Bridge (7 Nov. 1940)

Phenomenogram of cylinder wake

Reynolds number Re = UDν

Re < 4 2D steady flow

without vortex pair

Re < 47 2D steady flow

with vortex pair

Re < 180 2D vortex shedding

180 < Re 2D vortex shedding

superimposed by 3D

modes / fluctuations

’Traditional’ Galerkin method— Fletcher 1984 Computational Galerkin Methods, Springer —

Galerkin method

u(x, t) → ∂tu = 1Reu −∇(uu) −∇p

↓ ↓

u[N]=N∑

i=0ai(t)ui(x) →

dai

dt= ci +

N∑

j=1lij aj +

N∑

j,k=1

qijk aj ak

uu u u1 2 i N

ai

infinitely

hardware

(music)

many keys

software

(piano)

Transient dynamics of wake— ≡ Noack, Afanasiev, Morzynski, Tadmor & Thiele (2003) JFM —

Operating

condition II

on attractor

↑

transient

on paraboloid

u3

↑

averaged flow

shift mode

invariantmanifold

attractor

steady solution

POD modes

stability eigenmodes

Operating

condition I

near fixed point

us u1 u2

ddta1 = σa1 − ωa2ddta2 = σa2 + ωa1ddta3 = −σ3a3 + cA2

σ = σ1 − βa3ω = ω1 + γa3

A2 = a21 + a2

2

Landau equationddtA = σ1A − β⋆A3

Generalized mean-field model≡ Morzynski, Stankiewicz, Noack, Thiele & Tadmor (2006) AIAA

3-dim. Galerkin approx.:

u = uB + u′

uB = us + a∆u∆

u′ = aκ1u

κ1 + aκ

2uκ2

Galerkin system

daκ1/dt = σaκ

1 − ωaκ2

daκ2/dt = σaκ

2 + ωaκ1

Fluctuation energy

for transient1

0

-1

-2

-3 0 5 10 15 20 25 30 35 40 45 50

t/T

log(K)

DNSGM (POD-6)GM (POD-2)

GM (interpolated modes)

da∆/dt = σ∆a∆ + c(

(aκ1)

2 + (aκ2)

2)

σ = σ1 − βa∆, ω = ω1 + γa∆, κ = a∆/a∞∆

Generalized 3-dim. model ∼ 10% error.

POD Galerkin model≡ Noack, Afanasiev, Morzynski, Tadmor & Thiele (2003) JFM

POD at Re = 100 ≡ Deane et al (1991) PF Galerkin solution

u =8∑

i=0aiui

dai

dt= ν

∑

jlijaj +

∑

j,kqijajak

i=1 i=2

i=3 i=4

i=5 i=6

i=7 i=8

-2 -1 0 2a1

-2

-1

0

2

a2

8-dim. POD model reproduces DNS.

POD Galerkin model of cylinder wake≡ Noack, Schlegel, Morzynski, & Tadmor (2009) IJNMF (to appear)

a1(t)

0 1 2 3 4 6t

-2

-1

0

1

a1

0 1 2 3 4 6t

-2

-1

0

1

a2 a2(t)

a3(t)

0 1 2 3 4 6t

-0.4

-0.2

0.0

0.2

a3

0 1 2 3 4 6t

-0.4

-0.2

0.0

0.2

a4 a4(t)

a5(t)

0 1 2 3 4 6t

-0.2

-0.1

0.0

0.1

a5

0 1 2 3 4 6t

-0.2

-0.1

0.0

0.1

a6 a6(t)

a7(t)

0 1 2 3 4 6t

-0.06

-0.04

-0.02

0.0

0.02

0.06

a7

0 1 2 3 4 6t

-0.06

-0.04

-0.02

0.0

0.02

0.06

a8 a8(t)

POD Galerkin system of cylinder wake≡ Noack, Afanasiev, Morzynski, Thiele & Tadmor (2003) JFM

Dynamical system: Modal energy: Ei = a2i /2

da1/dt = σa1 −ωa2 +h1da2/dt = σa2 +ωa1 +h2da3/dt = σ2a3 −2ωa4 +h3da4/dt = σ2a4 +2ωa3 +h4da5/dt = σ3a5 −3ωa6 +h5da6/dt = σ3a6 +3ωa5 +h6da7/dt = σ4a7 −4ωa8 +h7da8/dt = σ4a8 +4ωa7 +h8

hi =8∑

j=1

8∑

k=1qijkajak

0 = 2σE1 +T10 = 2σE2 +T20 = 2σ2E3 +T30 = 2σ2E4 +T40 = 2σ3E5 +T50 = 2σ3E6 +T60 = 2σ4E7 +T70 = 2σ4E8 +T8

Ti =8∑

j=1

8∑

k=1qijkaiajak

Dynamical system = harmonically related oscillators

with growth rates σ > 0 > σ2 > σ3 > σ4.

Quadratic coupling is energy preserving:∑

Ti = 0

POD Galerkin system of cylinder wake II≡ Noack, Afanasiev, Morzynski, Thiele & Tadmor (2003) JFM

Modal

energetics:

0 = Qi + Ti

Qi = qiEi

10log Ei

−3

−2

−1

0

1

i

i i

i

Q = q Ei

1 4 5 72 3 6 8

T

σ = 0.007

i Σ ijk ii,jquadratic (internal) interactions

ω

2

ω4

ω3

ω

σ =−0.296

σ

σ

4

3

2=−0.083

=−0.137

linear (external) interactions

Q

kT = q a a aj

Hamiltonized Galerkin system of cylinder wake≡ Noack, Schlegel, Morzynski, & Tadmor (2009) IJNMF (to appear)

a1(t)

0 20 40 60 80 100 120 160t-2

-1

0

2

a1

0 20 40 60 80 100 120 160t-2

-1

0

2

a2a2(t)

a3(t)

0 20 40 60 80 100 120 160t-2

-1

0

2

a3

0 20 40 60 80 100 120 160t-2

-1

0

2

a4a4(t)

a5(t)

0 20 40 60 80 100 120 160t-2

-1

0

2

a5

0 20 40 60 80 100 120 160t-2

-1

0

2

a6a6(t)

a7(t)

0 20 40 60 80 100 120 160t-2

-1

0

2

a7

0 20 40 60 80 100 120 160t-2

-1

0

2

a8a8(t)

Thermal equilibrium (TE) limit of wake model≡ Noack, Schlegel, Morzynski, & Tadmor (2009) IJNMF (to appear)

1 2 3 4 5 6 8i

-3

-2

-1

0

1

log E i

Approximate thermal equilibrium (E1 = E2 = . . . = EN)

if energy sources are turned off (Qi = 0, Hamiltonian dynamics)

Overview






’Traditional’ Galerkin method— Fletcher 1984 Computational Galerkin Methods, Springer —

Galerkin method

u(x, t) → ∂tu = 1Reu −∇(uu) −∇p

↓ ↓

u[N]=N∑

i=0ai(t)ui(x) →

dai

dt= ci +

N∑

j=1lij aj +

N∑

j,k=1

qijk aj ak

uu u u1 2 i N

ai

infinitely

hardware

(music)

many keys

software

(piano)

Finite-time thermodynamics formalism≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski, Comte & Tadmor (2008) JNET

Dynamical system

(1) ai = fi = ci +N∑

j=1lijaj +

N∑

j,k=1qijkajak

Prerequisite #1: Ergodic property is fulfilled.

All trajectories converge to one and only one attractor.

(Underlying assumption of statistical fluid mechanics).

⇒ ∇ · f =N∑

i=1

∂fi

∂ai=

N∑

i=1

lii < 0

Prerequisite #2: Quadratic term is energy preserving.

qijk + qikj + qjik + qjki + qkij + qkji = 0

≡ Kraichnan 1989 Phys. D


Dynamics & approximations ⇒ FTT formalism

(1) ai = ci +∑

jlijaj +∑

j,kqijkajak (2) ai = mi +a′i, Ei = (a′i)2/2



(1) ai = ci +∑

jlijaj +∑


Mean-field equation . . . . . . . . . . .

...from averaging (1)

(3)

mi = ci +∑

lijmj +∑

qijkmj mk

+∑

j,k qijk a′ja′k = 0



(1) ai = ci +∑

jlijaj +∑



Assumption #1: a′ja′k = 2Ejδjk

(3)

mi = ci +∑

lijmj +∑

qijkmj ak

+∑

j 2 qijj Ej = 0



(1) ai = ci +∑

jlijaj +∑




(3)

mi = ci +∑

lijmj +∑

qijkmj mk

+∑

j 2 qijj Ej = 0

Modal e-flow balance . . . . . . . . .

...from averaging a′i × (1)

(4) Ei = Qi + Ti

• External interactions . . . . . . . .

(mean flow + molecular chaos)

• Qi :=∑

j qij a′ia′j

qij := lij +∑

k(qijk + qikj)mk

• Internal interactions . . . . . . . . .

(3 modes interacting)

• Ti :=∑

j,kTijk

Tijk := qijk a′ia′ja

′k



(1) ai = ci +∑

jlijaj +∑




(3)

mi = ci +∑

lijmj +∑

qijkmj mk

+∑

j 2 qijj Ej = 0

Modal e-flow balance . . . . . . . . . (4) Ei = Qi + Ti

Assumption #1 a′ja′k = 2Ejδjk Qi := qi Ei

qi := 2lii + 2∑

k(qiik + qiki)mk

Ti :=∑

j,kTijk

Tijk := qijk a′ia′ja

′k



(1) ai = ci +∑

jlijaj +∑




(3)

mi = ci +∑

lijmj +∑

qijkmj mk

+∑

j 2 qijj Ej = 0

Modal e-flow balance . . . . . . . . . (4) Ei = Qi + Ti

Qi := qi Ei

qi := 2 lii + 2∑

k(qiik + qiki)mk

Assumption #2:

Tijk = T (qijk, Ei, Ej, Ek)

Properties from NSE

Ti :=∑

j,kTijk

Tijk := α χijk ×

×√

EiEjEk

[

1 −3Ei

Ei + Ej + Ek

]



(1) ai = ci +∑

jlijaj +∑

j,kqijkajak (2) ai = ai +a′i, Ei = (a′i)2/2



(3) ai = ci +∑

lijaj +∑

qijkaj ak

+∑

j 2 qijj Ej = 0

Modal e-flow balance . . . . . . . . . (4) Ei =

Qi︷︸︸︷

qi Ei +

Ti︷︸︸︷∑

j,k

Tijk

Assumption #2:

Tijk = T (qijk, Ei, Ej, Ek)

Properties from NSE

qi := 2 lii + 2∑

k(qiik + qiki)ak

Tijk := α χijk ×

×√

EiEjEk

[

1 −3Ei

Ei + Ej + Ek

]

(3), (4) = 2N equations for 2N quantities ai, Ei.

System closed!

Fick’s law for triadic interactions≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski, Comte & Tadmor (2008) JNET

Ansatz Tijk = Tijk(Ei, Ej, Ek)

Properties from analysis of Tijk = qijk aiajak

(1) Homogeneity . . . . . . . . . . . . .Tijk(λEi, λEj, λEk) = λ3/2 Tijk(Ei, Ej, Ek)

(2) Zeros . . . . . . . . . . Tijk(Ei, Ej,0) = Tijk(Ei,0, Ek) = Tijk(0, Ej, Ek) = 0

(3) Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tijk = Tikj

(4) Monotonicity . . . . . . . . . . . . Ei < minEj, Ek ⇒ Tijk(Ei, Ej, Ek) > 0

(5) Energy preservation . . . . . . . Tijk + Tikj + Tjik + Tjki + Tkij + Tkji = 0

(6) Realizability (strictly: |Tijk| ≤ |qijk| |ai|max |aj|max |ak|max )

|Tijk| . |qijk|√

Ei Ej Ek

Solution

Tijk = αχijk

√

EiEjEk

[

1 −3Ei

Ei + Ej + Ek

]

with the totally symmetric triadic structure function

χijk := 16

(

|qijk| + |qikj| + |qjik| + |qjki| + |qkij| + |qkji|)

and α determined

from energy flow consistency between donor and recipient modes.

Fick’s law for triadic interactions≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski, Comte & Tadmor (2008) JNET

Ansatz Tijk = Tijk(Ei, Ej, Ek)

Properties from analysis of Tijk = qijk aiajak

(1) Homogeneity . . . . . . . . . . . . .Tijk(λEi, λEj, λEk) = λ3/2 Tijk(Ei, Ej, Ek)

(2) Zeros . . . . . . . . . . Tijk(Ei, Ej,0) = Tijk(Ei,0, Ek) = Tijk(0, Ej, Ek) = 0

(3) Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tijk = Tikj

(4) Monotonicity . . . . . . . . . . . . Ei < minEj, Ek ⇒ Tijk(Ei, Ej, Ek) > 0

(5) Energy preservation . . . . . . . Tijk + Tikj + Tjik + Tjki + Tkij + Tkji = 0

(6) Realizability |Tijk| . |qijk|√

Ei Ej Ek

SolutionTijk = αχijk

√

EiEjEk

[

1 −3Ei

Ei + Ej + Ek

]

Kraichnan’ DIA . . . . . . . . . . . . . . . . . .

violates (1) homogeneity

Tijk =∑

l,m∈i,j,kχlm

ijk El Em

Aupoix’s (2009) DIA correction .

matches all FTT axioms

Tijk =∑

l,m∈i,j,kχlm

ijkElEm

(EiEjEk)1/6

Fick’s law of triadic interactions≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski, Comte & Tadmor (2008) JNET

Tijk = σijk

[

1 −3Ei

Ei + Ej + Ek

]

, where σijk = α χijk

√

EiEjEk

0

0

0

i

k

j

dE /dtdE /dtdE /dt ji kE

i j k3

E + E +E

0

iE

kE

Ej

Tijk(loss)

T jik

(gain)Tkij

(gain)

Finite-time thermodynamics — limits≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski, Comte & Tadmor (2008) JNET

PartialLinear dynamics thermal equilibrium thermal equilibrium

Q

E E

QQ

.i iE = Q

N

2

2

1

1

external interactions

internal interactions

E

Q

E E

T T T

QQ

E = Q + T.i i i

1

N

N

N

2

2

2

1

1



EE E

T T T

.i iE = T

1

N

N

2

2

1



Time-scale Time-scale Time-scalefor E-growth for E-growth for E-growth

≪ ∼ ≫time-scale time-scale time-scale

for E-redistribution for E-redistribution for E-redistribution

≡ Andresen, Salamon & Berry 1977 JCP: Thermodynamics in finite time...

Finite-time thermodynamics — limits≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski, Comte & Tadmor (2008) JNET

Linear dynamics Partial TE TE

Q

E E

QQ

.i iE = Q

N

2

2

1

1



−−− capitalism −−−

−−− communism −−−

E

Q

E E

T T T

QQ

E = Q + T.i i i

1

N

N

N

2

2

2

1

1





EE E

T T T

.i iE = T

1

N

N

2

2

1





— Economics analogy —Neo liberalism Social market e.∼ Communism

G.W. Bush (1946–) & Co Luwdig Erhard (1897–1977) Karl Marx (1818–1883)

Periodic cylinder wake (Re = 100)≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski, Comte & Tadmor (2008) JNET

2D flow around circular

cylinder (DNS)

10-dim. Galerkin model

u =N∑

i=0aiui (POD modes)

ai = ci +N∑

j=1lij aj

+N∑

j,k=1qijk aj ak

Energy distribution (com-

puted and FTT predicted)

0 2 4 6 8 10i

-4

-3

-2

-1

1

log E i

•: DNS; : FTT

Good agreement between DNS and FTT prediction!

Thermal equilibrium (TE) limit of wake model≡ Noack, Schlegel, Morzynski, & Tadmor (2009) IJNMF (to appear)

1 2 3 4 5 6 8i

-3

-2

-1

0

1

log E i

Approximate thermal equilibrium (E1 = E2 = . . . = EN)

if energy sources are turned off (Qi = 0, Hamiltonian dynamics)

Dynamic FTT: idea≡ Noack, Schlegel, Morzynski & Tadmor (2009) IJNMF (to appear)

ω

base flowvariations

fluctuationssmall−scale

coherent structures

frequency ωhighdominantlow

fluct

uatio

n en

ergy

E(

)

0

Dynamic FTT: evolution equations≡ Noack, Schlegel, Morzynski & Tadmor (2009) IJNMF (to appear)

thermodyn. modes

Ei > 0 , mi = 0

ai = 0

high-frequency, small scale

Ei = qiEi +∑

Tijk

where qi = χi +∑

χijmj

dynamic modes

t 7→ ai(t)

mi = ai,

Ei = (a′i)2/2

dominant frequency, coh. structures

ai =∑

lijaj +∑

qijkajak + αiai

where 2αEi + T>i = 0,

and T>i :=

∑

j or k = thermodyn. Tijk

mean-field modes

mi 6= 0 , Ei = 0

ai = mi

low frequency, base flow

mi =∑

lijmj+∑

qijkmjmk+∑

2qijjEj

E

mi

i

i

Coupling termsderived from FTT

Dynamic FTT: system reduction≡ Noack, Schlegel, Morzynski & Tadmor (2009) IJNMF (to appear)

Reduced-

order model

Low-order

model

Least-order

model

Stochastic

model

1

2

3

4

5

7

8

0

4 ω

3 ω

2 ω

1 ω

0 ω

6

1

2

3

4

5

7

8

0

4 ω

3 ω

2 ω

1 ω

0 ω

6

1

2

3

4

5

7

8

0

4 ω

3 ω

2 ω

1 ω

0 ω

6

1

2

3

4

5

7

8

0

4 ω

3 ω

2 ω

1 ω

0 ω

6

Dynamic FTT: results for wake model≡ Noack, Schlegel, Morzynski & Tadmor (2009) IJNMF (to appear)

1 2 3 4 5 6 8i

-3

-2

-1

0

1

log E i

Overview






Homogeneous shear turbulence (Re = 1000)≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski, Comte & Tadmor (2008) JNET

3D flow

_

1

0

11

01

01__

x

z_

y

1459-dim. Galerkin model

u =N∑

i=0aiui (Stokes modes)

ai = ci +N∑

j=1lij aj

+N∑

j,k=1qijk aj ak

Cumulative transfer term

(GM and FTT)

0 200 400 600 800 1000 1400I-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

0.0

T[1..I]

T [1..I] := T1 + . . . + TI

: GM; — : FTT

Good agreement between GM and FTT prediction!

FTT modeling of truncated Euler solutions≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski, Comte & Tadmor (2008) JNET

0 200 400 600 800 1000 1400I0.0

0.2

0.4

0.6

1.0

E[1..I]

0 200 400 600 800 1000 1400I-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

0.0

T[1..I]

The systems approaches local thermal equilibrium

(E1 = E2 = . . . = EN) without external interactions, i.e. Qi ≡ 0.

FTT applications≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski, Comte & Tadmor (2008) JNET

Noack, Schlegel, Morzynski & Tadmor (2009) IJNMF preprint & ≡ ≡

Instabilities and turbulence

• FTT generalizes the Landau equation dA/dt = σA − βA3

• rigorous system reduction of evolution equation

u(x, t) =∑

i∈Itd

ai(t) ui(x) +∑

i∈Imf

ai(t) ui(x) +∑

i∈Idyn

ai(t) ui(x)

– thermodynamic modes . . . . . . . . . . . . . . . . . . . . . statistical treatment

– mean-field (shift) modes) . . . . . . . . . . . . . . . . . . . algebraic equations

– oscillatory modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .dynamical system

• derivation of nonlinear subgrid turbulence model

• unified description of normal and inverse turbulence cascade

Variational principles

• statistical mechanics & definition of entropy

• MaxEnt principle for attractor

Attractor control

• E-based control 7→ manipulation of the turbulence cascade

ai = ci +∑

j cij aj +∑

j,k cijk aj ak + gi b .

Control design in FTT — an example≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski, Comte & Tadmor (2008) JNET

Goal functional (e.g. lift or drag) Z = z0 +N∑

i=1zi ai

Galerkin system (e.g. with single volume force)

dai

dt= ci +

N∑

j=1lijaj +

N∑

j,k=1qijkajak + gi b

Control ansatz b =N∑

i=1ki ai

Controlled dynamics with lcij = lij + gikj

dai

dt= ci +

N∑

j=1lcijaj +

N∑

j,k=1qijkajak

⇒ FTT applicable ⇒ ai = ai(k1, . . . , kN)

Control problem: Z = Z(k1, . . . , kN) = max

⇒ Fully nonlinear, infinite horizon control!

Configurations

3D flow over a step 3D mixing layer jet noise

Ahmed body airfoilwake

channel flow

combustor

cavity flow

. . .

Toolbox and configurations

Overview






Conclusions≡ Noack, Morzynski & Tadmor (2010) Springer

Galerkin modelling for flow control is a doable art!

u =N∑

i=0ai ui, ai = ci +

N∑

j=1lij aj +

N∑

j,k=1qijk aj ak + gi b

http://BerndNoack.com

Turbulence control = attractor control

Physics mechanisms are strongly nonlinear.

• drag reduction of D-shaped body

• lift increase of high-lift configuration

• ...

Model for natural and controlled attractor needed!

⇒ Upgrade Galerkin model with ergodic measure

Conclusions≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski, Comte & Tadmor (2008) JNET

Finite-time thermodynamics model builds on GM

u =N∑

i=0ai ui, ai = ci +

N∑

j=1lij aj +

N∑

j,k=1qijk aj ak

⇒ first and second moments of unsteady flows

• 1D Burgers’ eq., • 2D wake, • 3D shear turbulence.

FTT 7→ Statistical physics (economics) link

• ui . . . . .person /thermodyn. degrees of freedom)

• Ei . . . . . . . . . . . . . . . . . . . . . wealth /order parameter

•∑

lijaj ⇒ Qi . . . . . . pure capitalism /lin. instability

•∑

qijkajak ⇒ Ti . . . . . . . . . . .pure communism /LTE

• Both terms . . . . . . . . . social market /partial LTE

FTT 7→ energy-based and nonlinear control design

More information

Call

+1-617-373.5277 +48-61-665.2778 +49-30-314.24732

... or read

≡ Noack, Afanasiev, Morzynski, Tadmor & Thiele

(2003) JFM . . . . . . . . . . . . .generalized mean-field model

≡ Noack, Papas & Monkewitz (2005) JFM

. . . . . . . . . . . . . . . . . . . . . . . . . . . . modal energy flow analysis

≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski,

Comte & Tadmor (2008) JNET

. . . . . . . . . . . . . . . . . . . . . . . . . . .Finite-time thermodynamics

... or ask now!

reduced-, low- and least-order models

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