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 marta.waclawczyk@igf.fuw.edu.pl

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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