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Lie point symmetries and invariantsolutions of equations for turbulencestatistics
Marta Wacławczyk1
1Institute of Geophysics, Faculty of Physics, University of Warsaw
Warszawa, 26.10.2016
www.igf.fuw.edu.pl [email protected]
Outline
� Randomness vs. symmetries in turbulence
� Complete statistical description of turbulence in terms ofmultipoint correlations
� Group analysis and invariant solutions
� Invariant turbulence modelling
� Conclusions and perspectives
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Introduction
� Randomness of turbulence follows from its dependence on smallvariations in the initial and boundary conditions
Figure: Whirlpools of water. Image: Leonardo da Vinci,RL 12660, Windsor, Royal Library.
Figure: Laminar flow turns turbulent. Creative Commonscredit, Image: James Marvin Phelps.
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Introduction
Symmetries summarize regularities of the laws of nature, provide aroute between abstract theory and experimental observations
Figure: Fractal structure from www.fractalzone.beFigure: Top: wake behind rotating cylinders, S. Kumar, B.Gonzalez [Phys. Fluids 23, 014102 (2011)].Bottom: Santa Cristina della Fondazza, University ofBologna.
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Introduction
Symmetry breaking in the wake behind two rotating cylinders
Figure: Wake behind rotating cylinders, S. Kumar, B. Gonzalez [Phys. Fluids 23, 014102 (2011)]www.aps.org - image gallery, http://dx.doi.org/10.1063/1.3528260
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Introduction
Breaking of symmetry after the transi-tion to turbulence
Mean solution - reflection symmetry isrecovered
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Statistical description of turbulence
� After the transition to turbulence we usually observe breaking ofsymmetries, however, the symmetries may be recovered in thestatistical sense.
� "To identify underlying symmetries is a central problem of thestatistical physics of infinite-dimensional strongly fluctuatingsystems."Bernard et al. Inverse Turbulent Cascades and Conformally Invariant Curves, PRL 024501 (2007)
� Aim: Investigate symmetries of a system of equations describingturbulence statistics
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Probability density functions (pdf’s) of velocity
The probability that a random variable U is contained within v andv + dv : P(v ≤ U ≤ v + dv) = f (v)dv
Pdf of velocity U(x , t) at point x and at time t : f (v ;x , t)
Ensemble average:
〈G(U(x , t))〉 =∫
G(v)f (v ;x , t)dv
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Multipoint correlations (MPC)
Reynolds-averaged Navier-Stokes equations
∂〈U i 〉∂x i = 0,
∂〈U i 〉∂t
+∂〈U i U j 〉∂x j = −1
%
∂〈P〉∂x i + ν
∂2〈U i 〉∂x j∂x j .
Equations for the two-point correlations: 〈U i (x1, t)U j (x2, t)〉
∂〈U i (x1, t)U j (x2, t)〉∂t
+2∑
n=1
∂〈U i (x1, t)U j (x2, t)Uk (xn, t)〉∂x k
n=
−1ρ
(∂〈U i (x1, t)P(x2, t)〉
∂x j2
+∂〈P(x1, t)U j (x2, t)〉
∂x i1
)+ ν
2∑n=1
∂〈U i (x1, t)U j (x2, t)〉∂x k
n ∂x kn
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Multipoint correlations (MPC)
� Equation for the n-th point velocity correlation contains anunclosed correlation of the order n + 1
� This forms an infinite hierarchy of MPC equations - theFriedman-Keller hierarchy[L. Keller and A. Friedmann, Proc. First. Int. Congr. Appl. Mech. (Technische Boekhandel en Drukkerij, Delft, 1924]
� Analogously, an infinite hierarchy for the multipoint pdf’s can bederived where the n-th equation contains an unknown n + 1-pointpdf.[T. S. Lundgren, Phys. Fluids 10, 1967]
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LMN hierarchy
Infinite Lundgren-Monin-Novikov hierarchy for n-point pdf:
∂f1∂t
+ v i1∂fn∂x i
1=
∂
∂v i1
[F i
1 ([f2])]
∂fn∂t
+2∑
k=1
v ik∂fn∂x i
k=
2∑k=1
∂
∂v ik
[F i
k ([f3])]
· · ·∂fn∂t
+n∑
k=1
v ik∂fn∂x i
k=
n∑k=1
∂
∂v ik
[F i
k ([fn+1])]
· · ·
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Statistical description of turbulence
Different levels of statistical description of turbulence
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Lie symmetries
Transformations of variables which do not change the form of aconsidered equation
x∗ = φx (x , t ,U, ε), t∗ = φt(x , t ,U, ε), U∗ = φU (x , t ,U, ε)
Example: scaling group of Euler equation: ∂U∂t + U · ∇U = − 1
ρ∇P,
t∗ = t , x∗ = eεx , U∗ = eεU, P∗ = e2εP
Transformations satisfy group properties:
1. closure x∗ = eε1+ε2x = eε3x2. associativity x∗ = e(ε1+ε2)+ε3x = eε1+(ε2+ε3)x3. identity element x∗ = e0x = x4. inverse element x∗ = eεe−εx = x
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Lie symmetries - time translation
Time translation
T1 : t∗ = t + k1, x∗ = x ,U∗ = U, P∗ = P,
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Lie symmetries - scaling
T2 : x∗ = ek2x , U∗ = ek2U, P∗ = e2k2P, =⇒ x i∗/x i = v i∗/v i
Figure: Scale invariance. From: G. Falkovich and K. R.Sreenivasan Lessons from Hydrodynamic Turbulence,Phys. Today 59, 43-49, 2006
Two-dimensional passivescalar field. "The four panelsin the sequence a-d show in-creasingly magnified views of afixed location at a specific time.Clearly the four images are notidentical. But they are similar ina statistical sense" (statisticalself-similarity).
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Lie symmetries - rotational invariance
Rotational invariance
T4 − T6 : t∗ = t , x∗ = a · x ,U∗ = a · U, P∗ = P,
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Lie symmetries - generalised Galilean invariance
Galilei invariance - laws of mo-tion do not change in a movingframe.
T7 − T9 : t∗ = t , x∗ = x + f (t),
U∗ = U +dfdt, P∗ = P − x · d2f
dt2 ,
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Lie symmetries
Symmetries of the Euler equations with ρ = const
T1 : t∗ = t + k1, x∗ = x , U∗ = U, P∗ = P,
T2 : t∗ = t , x∗ = ek2x , U∗ = ek2U, P∗ = e2k2P,
T3 : t∗ = ek3 t , x∗ = x , U∗ = e−k3U, P∗ = e−2k3P,T4 − T6 : t∗ = t , x∗ = a · x , U∗ = a · U, P∗ = P,
T7 − T9 : t∗ = t , x∗ = x + f (t), U∗ = U +dfdt, P∗ = P − x · d2f
dt2 ,
T10 : t∗ = t , x∗ = x , U∗ = U, P∗ = P + f4(t) ,
For the Navier-Stokes equations instead of T2 and T3 we have TNaSt
TNaSt : t∗ = e2k4 t , x∗ = ek4x , U∗ = e−k4U, P∗ = e−2k4P,
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Symmetries of the MPC equations
MPC equations have symmetries which follow from the symmetries ofthe Navier-Stokes equations+ additional scaling and translational invariance.[Rosteck & Oberlack, J. Nonlinear Math. Phys. 18, 2011]
New symmetries transform a turbulent signal into a laminar orintermittent laminar-turbulent signal [Wacławczyk et al. 2014. PRE, 90, 013022]
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Symmetries of LMN hierarchy
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Lie symmetries
Invariant function f (x , t ,U) = f (x∗, t∗,U∗)
� Invariant functions describe scaling laws - as attractors of theinstantaneous, fluctuating solutions of the Navier-Stokesequations.
Invariant modelling� To properly describe a physical phenomenon (e.g. turbulence), a
model should be invariant under the same set of symmetries asthe original equations (here: MPC equations).
� Additional symmetries of a model, not seen in the originalequations lead to unphysical results (example: k − ε model whichis additionally frame-rotation invariant).
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Invariant functions
Logarithmic profile
Rona A. et al., Aeronautical Journal -New Series- 116(1180) · June 2012
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Invariant functions
New scaling and translation symmetries:
〈U〉∗ = eks〈U〉, 〈U〉∗ = 〈U〉+ C1
(1− y2
H2
)+
Classical scaling symmetry: 〈U〉∗ = e−kNS 〈U〉, y∗ = ekNS y
Lead to the characteristic system: d〈U〉(ks−kNS)〈U〉+C1
(1− y2
H2
) = dykNSy
which for kNS = ks has the solution
〈U〉 = C1
ksln(y) +
C1
2ks
(1− y2
H2
)+ C
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Invariant functions
Exponential tails:� Based on the new symmetries it is possible to derive solutions for
pdf’s, e.g. exponential tails
Pictures from http://www.uni-oldenburg.de/en/physics/research/twist/research/turbulent-flows/atmospheric-turbulence
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Symmetry analysis - current work
Symmetry analysis of infinite systems - cannot be performed withcomputer algebra systems. New symmetries - guessed (notcalculated), hence it is possible that the set of symmetries is notcomplete.Lie group analysis of equations for pdf’s:
� Lie group methods for integro-differential equations� Infinite system - additional difficulty (look for some recurrence
relations)� Requires tedious algebra - but results are very promising
Wacławczyk M., Grebenev V. N., Oberlack M., submitted to Journal of Physics A: Mathematical and Theoretical, 2016
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Invariant modelling
Modelling of external intermittency: air-water flowWacławczyk M. & Oberlack M., Int. J. Multiphase Flow, 37, 2011
Wacławczyk M. & Wacławczyk T., Int. J. Heat Fluid Flow, 52, 2015
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Invariant modelling
Figure: Two eddies reaching and deforming the surface. [ Wacławczyk M. & Wacławczyk T., Int. J. Heat Fluid Flow, 52, 2015]
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Invariant modelling
Figure: Evolution of the intermittency region. [ Wacławczyk M. & Wacławczyk T., Int. J. Heat Fluid Flow, 52, 2015],Development of numerical methods: [ Wacławczyk T., J. Comp. Phys. 299 July 2015]
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Invariant modelling
Figure: Evolution of the intermittency region. Symbols: values of 〈u · n〉sΣ from numerical experiment, Lines: model∇ · (Dt∇α) +∇ · (Ctα(1− α)n) [ Wacławczyk M. & Wacławczyk T., Int. J. Heat Fluid Flow, 52, 2015]
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Conclusions and perspectives
� Lie symmetry analysis provides connection between theory andexperimental observations
� New statistical symmetries reflect the fact that a flow may have differentcharacter (laminar or turbulent)
� Lie symmetry analysis of LMN hierarchy was performed - first resultsobtained.
Perspectives
� Invariant solutions for pdf’s
� Invariant turbulence modelling - internal intermittency in atmosphericflows, external intermittency
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Acknowledgements
The financial support of the National Science Centre, Poland (project No.2014/15/B/ST8/00180) is gratefully ackowledged.
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