Turbulence and symmetries - University of...

13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack

Turbulence and symmetries Statistical symmetries of the infinite-dimensional correlation system of turbulence, new invariant solutions and its validation using large-scale turbulence simulations

Martin Oberlack Andreas Rosteck Marta Waclawczyk Victor Avsarkisov Amirfarhang Mehdizadeh Chair of Fluid Dynamics Technische Universität Darmstadt Darmstadt/Germany


§ Introduction to turbulence

§ Probability density equation

§ Multi-point correlation equations

§ Symmetries of Euler and Navier-Stokes equations

§ Statistical symmetries

§ Turbulent scaling laws for higher order moments

§ Statistical symmetries of the probability density equation

§ Conclusion and open questions

Content

Turbulence and Navier-Stokes eqn.

§ General belief (physicists, engineers, …)


Navier-Stokes equations model laminar, transitional and turbulent flows

Large scale simulation of NaSt

§ Example: turbulent channel flow with transpiration


x2

x1

x3

u1

§  Beyond our capabilities for techn. applications at large

§  Degrees of freedom grow as

Statistical turbulence description

§ Turbulence details are unnecessary! determine only turbulence statistics

§ Three approaches to determine “full” turbulence statistics

1)  Multi-point probability density function (PDF) equation (Lundgren-Monin-Novikov eqn.)

2)  Multi-point correlation/moment equation (MPC) (Friedmann-Keller eqn.)

3)  Hopf functional differential equation for characteristic functional


PDF approach


§ 1-point PDF: probability of random variable u within v and :

§ 2-point PDF: analogous for 2 points

§ n-point PDF: analogous for n points

PDF equation


§ PDF equation from Navier-Stokes

§ Infinite set of linear (Integro-)PDEs

§ Weak coupling only between neighbours and

§ If truncated at some n: closure problem of turbulence

Correlation/moment approach

§ For most engineering application only interest in correlations

§ 1-point moment (mean velocity)

§ 2-point moment

§ n-point moment


Multi-point correlation equation

§ Infinite set of linear PDEs

§ Weak coupling: Eqns of order are only coupled to order

Crucial for computation of symmetries (proof by Rosteck)


§  Translation in time:

§  Finite rotation:


Symmetries of the Euler and Navier-Stokes equations I

Note:


Symmetries of the Euler and Navier-Stokes equations II

§  Scaling of space:

§ Scaling of time:

in the Euler limit


All symmetries transfer to the statistical equations!

§  Generalized Galilean invariance:

§  Comprises two classical cases:

§  Translation in space:

§  Classical Galilean invariance:

§  A few more ... not relevant here ...

Symmetries of the Euler and Navier-Stokes equations II


New Statistical Symmetry I

§ Translation in function space:

§ For the mean velocity: (not Galilean group)

§ Since the 30th part of eng. turbulence models § Key ingredient of famous logarithmic law-of-the-wall

§ Implies non-gaussianity of turbulence statistics § A symmetry but not a Lie group in PDF formulation


New Statistical Symmetry II

§ Scaling of correlations/moments:

§ Independent of scaling of Euler or Navier-Stokes

§ Implies intermittency of turbulence statistics

§ A symmetry but not a Lie group in PDF formulation

§ Invariant solutions for wall-bounded shear flows:


Examples canonical shear flows

§ Near-wall scaling laws

Ω2x2

x1

x3

u1

u3

§  Rotating channel flows §  Transpirating channel flow

§ Channel flow x2

x1

x3

u1

x2

x1

x3

u1


§ Invariance condition

§ Parameter of new statistical symmetries

§ Invariant solution (turbulent scaling laws) are only valid in certain restricted regions of the flow

§ Classical well known logarithmic law of the wall is included

§ In principles arbitrary correlations can be computed

Solutions for parallel shear flows


§ Common notation in turbulence: Reynolds decomposition § Instantaneous velocity: § Mean velocity: § Fluctuation velocity:

§ Correlations:

§ Reynolds stress tensor:

Notation in turbulence

§ Symmetry breaking: wall-friction velocity

§ Scale invariance

§ Mean velocity


Classical log law


§ Solutions for the stresses


Classical log law

§ Data from channel (Jimenez & Hoyas) at Reτ = 2006a


Classical log law

§ No obvious symmetry breaking


§ Solution

Centre region of a channel flow


x2

x1

x3

u1



10−3 10−2 10−1 100

10−4

10−2

100

Channel DNS

Reτ = 2006

(Hoyas & Jimenez)

x2

x1

x3

u1


§ Solutions for the stresses



x2

x1

x3

u1



Channel DNS

Reτ = 2006

(Hoyas & Jimenez)

x2

x1

x3

u1

§ Symmetry breaking: rotation rate


§ Solution

Rotating channel flow




Channel DNS

(Kristoffersen

& Andersson 1993)


§ Non-linear variation of mass flux


Ω2x2

x1

x3

u1

u3


§ New scaling law of Ekman type channel flow

§ DNS (Mehdizadeh, Oberlack, Phys. Fluids 2010)



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025

0.03

Ω2x2

x1

x3

u1

u3



DNS at

Reτ = 360

Ro = 0.072

§ Symmetry suggested solution for the centre region

Channel flow with transpiration


x2

x1

x3

u1

Channel flow with transpiration


0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.20

50

100

150

200

250

300

350

400

10−1 1000

50

100

150

200

250

300

350

400

Increasing transpiration

x2

x1

x3

u1



§ Turbulent stresses (skip formula) § Reτ=250 § V0

+=0.16

Transfer stat. symmetries to pdf

§ Short reminder:

§ 1-point moment (mean velocity)

§ 2-point moment



Statistical Symmetry – PDF I

§ Translation in correlation/moment space:

§ PDF “equivalent”

§ 

§ Note: both and are non-negative functions additional constraint on not a Lie group but still a symmetry of the PDF equation

with

Statistical Symmetry – PDF I

§ “Shape” symmetry

§ Non-gaussianity and hence non-zero higher order moments (skewness, flatness, etc.) of the pdf are induced



Statistical Symmetry – PDF II

§ Scaling of correlations/moments:

§ PDF “equivalent”

§ Consequences § Mathematically: since and are non-negative functions

(semi-group) § Physically: measure of intermittency


Conclusions & open questions

§ Navier-Stokes (Burgers, …) admit statistical symmetries

§ They are the key ingredients for invariant solutions (turbulent scaling laws) here: wall turbulence, rotating turbulence, …

§ Group properties of some statistical symmetries are unknown

§ Are there more statistical symmetries - completeness?

§ How to determine the values for the parameters such as

in the log-law - if not just fitted?

§ How do layers of scaling laws match e.g. in wall bounded flows?

§ …


Thank you for your attention!

Turbulence and symmetries - University of...

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