Download - 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
1. Introduction. . . . . . . . . . . . . . . . . physics & cybernetics dreams revisited
2. Reduced-order modelling. . . . . . . . . . . . . . . . . . . . . . . . transient + attractor dynamics
3. Finite-time thermodynamics closure. . . . . . . . . . . . . . . . . . . . . . . . . . system reduction of dynamics
4. Towards turbulence control. . . . . . . . . . . . . . . . . . . . . . . . . . . perspectives of FTT closure
5. Summary and outlook
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
1. Introduction. . . . . . . . . . . . . . . . . physics & cybernetics dreams revisited
2. Reduced-order modelling. . . . . . . . . . . . . . . . . . . . . . . . transient + attractor dynamics
3. Finite-time thermodynamics closure. . . . . . . . . . . . . . . . . . . . . . . . . . system reduction of dynamics
4. Towards turbulence control. . . . . . . . . . . . . . . . . . . . . . . . . . . perspectives of FTT closure
5. Summary and outlook
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
1. Introduction. . . . . . . . . . . . . . . . . physics & cybernetics dreams revisited
2. Reduced-order modelling. . . . . . . . . . . . . . . . . . . . . . . . transient + attractor dynamics
3. Finite-time thermodynamics closure. . . . . . . . . . . . . . . . . . . . . . . . . . system reduction of dynamics
4. Towards turbulence control. . . . . . . . . . . . . . . . . . . . . . . . . . . perspectives of FTT closure
5. Summary and outlook
’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
Finite-time thermodynamics formalism≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski, Comte & Tadmor (2008) JNET
Dynamics & approximations ⇒ FTT formalism
(1) ai = ci +∑
jlijaj +∑
j,kqijkajak (2) ai = mi +a′i, Ei = (a′i)2/2
Finite-time thermodynamics formalism≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski, Comte & Tadmor (2008) JNET
Dynamics & approximations ⇒ FTT formalism
(1) ai = ci +∑
jlijaj +∑
j,kqijkajak (2) ai = mi +a′i, Ei = (a′i)2/2
Mean-field equation . . . . . . . . . . .
...from averaging (1)
(3)
mi = ci +∑
lijmj +∑
qijkmj mk
+∑
j,k qijk a′ja′k = 0
Finite-time thermodynamics formalism≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski, Comte & Tadmor (2008) JNET
Dynamics & approximations ⇒ FTT formalism
(1) ai = ci +∑
jlijaj +∑
j,kqijkajak (2) ai = mi +a′i, Ei = (a′i)2/2
Mean-field equation . . . . . . . . . . .
Assumption #1: a′ja′k = 2Ejδjk
(3)
mi = ci +∑
lijmj +∑
qijkmj ak
+∑
j 2 qijj Ej = 0
Finite-time thermodynamics formalism≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski, Comte & Tadmor (2008) JNET
Dynamics & approximations ⇒ FTT formalism
(1) ai = ci +∑
jlijaj +∑
j,kqijkajak (2) ai = mi +a′i, Ei = (a′i)2/2
Mean-field equation . . . . . . . . . . .
Assumption #1: a′ja′k = 2Ejδjk
(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
Finite-time thermodynamics formalism≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski, Comte & Tadmor (2008) JNET
Dynamics & approximations ⇒ FTT formalism
(1) ai = ci +∑
jlijaj +∑
j,kqijkajak (2) ai = mi +a′i, Ei = (a′i)2/2
Mean-field equation . . . . . . . . . . .
Assumption #1: a′ja′k = 2Ejδjk
(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
Finite-time thermodynamics formalism≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski, Comte & Tadmor (2008) JNET
Dynamics & approximations ⇒ FTT formalism
(1) ai = ci +∑
jlijaj +∑
j,kqijkajak (2) ai = mi +a′i, Ei = (a′i)2/2
Mean-field equation . . . . . . . . . . .
Assumption #1: a′ja′k = 2Ejδjk
(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
]
Finite-time thermodynamics formalism≡ Noack, Schlegel, Ahlborn, Mutschke, Morzynski, Comte & Tadmor (2008) JNET
Dynamics & approximations ⇒ FTT formalism
(1) ai = ci +∑
jlijaj +∑
j,kqijkajak (2) ai = ai +a′i, Ei = (a′i)2/2
Mean-field equation . . . . . . . . . . .
Assumption #1: a′ja′k = 2Ejδjk
(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
.i iE = Q
N
2
2
1
1
external interactions
internal interactions
E
Q
E E
T T T
E = Q + T.i i i
1
N
N
N
2
2
2
1
1
external interactions
internal interactions
EE E
T T T
.i iE = T
1
N
N
2
2
1
external interactions
internal interactions
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
.i iE = Q
N
2
2
1
1
external interactions
internal interactions
−−− capitalism −−−
−−− communism −−−
E
Q
E E
T T T
E = Q + T.i i i
1
N
N
N
2
2
2
1
1
external interactions
internal interactions
−−− capitalism −−−
−−− communism −−−
EE E
T T T
.i iE = T
1
N
N
2
2
1
external interactions
internal interactions
−−− capitalism −−−
−−− communism −−−
— 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
1. Introduction. . . . . . . . . . . . . . . . . physics & cybernetics dreams revisited
2. Reduced-order modelling. . . . . . . . . . . . . . . . . . . . . . . . transient + attractor dynamics
3. Finite-time thermodynamics closure. . . . . . . . . . . . . . . . . . . . . . . . . . system reduction of dynamics
4. Towards turbulence control. . . . . . . . . . . . . . . . . . . . . . . . . . . perspectives of FTT closure
5. Summary and outlook
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
1. Introduction. . . . . . . . . . . . . . . . . physics & cybernetics dreams revisited
2. Reduced-order modelling. . . . . . . . . . . . . . . . . . . . . . . . transient + attractor dynamics
3. Finite-time thermodynamics closure. . . . . . . . . . . . . . . . . . . . . . . . . . system reduction of dynamics
4. Towards turbulence control. . . . . . . . . . . . . . . . . . . . . . . . . . . perspectives of FTT closure
5. Summary and outlook
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!