Introduction to Lie Symmetries and Invariance in Fluid ...

Introduction to Lie Symmetries

and Invariance in

Fluid Dynamics and Turbulence

15.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 1

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


Concept of symmetries

§  Analogy between symmetric object and differential equation §  Symmetric Object (under rotation)

§  Symmetry of differential equation

§  Differential equation

§  Symmetry transformation

§  Form invariance


Symmetry of differential equation

§  Example 1: 1D heat equation

§  Symmetry transformation 1: scaling of space-time

§  Into heat equation:




§  Symmetry transformation 2: scaling of

§  Into heat equation:




§  Symmetry transformation 3: “Galilean” transformation

§  Into heat equation: The heat equation admits 6 symmetry transformations


Generation of solutions from symmetries

§ Recall: symmetry transformation and form invariance

§ Very important:

If a symmetry transformation maps a DE to itself is also maps a given solutions to a (new) solutions!



§ Existing solutions:

and using:

§ Implementing (3) into (2):



§ Example 1 heat equation: Translation symmetry

§ Given solution:

§ New solution from :



§ Example 2 heat equation: Space-time scaling

§ Given solution:

§ New solution:

§ Key idea of invariant solutions (e.g. self-similar solutions):

Solution is invariant under a symmety group i.e. indepedent of group parameter (here )


§  independent of :

§ Note:

If a DE admits a symmetry transformation for the independent

variables, this always allows for an invariant solution or

symmetry reduction (of the independent variables)



§ Once two or more symmetries are known, they may be combined to a multi-parameter symmetry according to

§ Example: Heat equation

is a multi-parameter symmetry group of the heat equation

Combination of symmetry groups

S( 1) , . . . , S(n )


Invariants

§ An invariant does not change its functional form under a given symmetry transformation

§ Example 1: Space-time scaling of 1D heat equation

§ Invariants

1)

2)


Invariants

§ Example 2: Combined scalings of 1D heat equation

§ Invariants

1)

2)


§ Implementing the invariants into the DE leads to a (Symmetry) reduction invariant solution

§ Example: 1D heat equation and combined scaling group

§ Employed as independent and dependent variables into the heat equation

§ If scaling used: similarity solution

Invariant solutions


§ Example 1: 1D heat equation

§ 2 parameter scaling group:

§ Boundary condition

§ Combined with the scaling symmetry

Symmetry breaking


§ Preserving the form of the BC

remains arbitrary

§  is symmetry breaking

§ Reduced form of invariants

§ Reduced consistent with BC:

Symmetry breaking


§ Computing and using symmetries is always in infinitesimal form:

§ The infinitesimal and are fully equivalent to and

§ The invariance condition

§ rewrites to

Infinitesimal transformations

with


Symmetries of Boundary Layer equations §  Prandtl boundary layer (BL) equations:

§  BL equations admit 5 symmetry transformations

§  All known analytic solutions arise from these symmetries


§  Translation in : direction:

§  Pressure invariance:

Symmetries of Boundary Layer equations

x1


§  Scaling 1:

§  Scaling 2:

§  Blasius, Falkner-Skan etc. emerge from these groups.



§  Generalized translation group (infinite dimensional):

§  This symmetry is the basis for the triple-deck theory etc.

§  Note: In order to be consistent with the BL asymptotic we need



§  In the following we consider combination of two scaling groups:

§  In contrast to Navier-Stokes and Euler equations an

inhomogeneous scaling in admitted, i.e. different scaling in ,

and directions

Invariants of BL equations


§ Independent variables:

§ Dependent variables:

Ansatz:




§ Note: §  und only appear as ratio

The more symmetries are admitted the better we can fit to BCs.

§ 

in Bl PDE:



§ Results have been derived without any dimensional reasoning

§ The number of possible solutions may become very large – even more if unsteady BLs are considered

§ Blasius solution is recoverd for


Summing up

§ We have learned about: § Symmetries

§ Combination of symmetries

§ Invariants with respect to symmetries

§ Symmetry breaking

§ Invariant solutions

§ Infinitesimal transformations

§ Next step: what do we know about Navier-Stokes and Euler

equations with respect to their symmetry properties?


Navier-Stokes equations

§ Continuity equation:

§ Momentum equation:

§ Note: §  consitutes frame rotation rate

§ The constant density and the centrifugal force has been absorbed into the pressure


Symmetries of the Euler and Navier-Stokes equations

§  Scaling of space:

§ Scaling of time:

for (Euler)


Scaling symmetries of the Navier-Stokes equations

§ Invoking the two scaling groups into the NaSt equations:

§ Scaling group of the Navier-Stokes equations:

The Navier-Stokes equations admits only one scaling group Viscosity is symmetry breaking


§  Translation in time:

§  Finite rotation:

Note:

Symmetries of the Euler and Navier-Stokes equations II


Symmetries of the Euler and Navier-Stokes equations III

§  Generalized Galilean invariance:

§  Comprises two classical cases:

§  Translation in space:

§  Classical Galilean invariance:

§  A few more for special flows we only present one


§  Instantaneous value

§  Mean value

§  Fluctuating quantities

§  Two-point correlation tensor (TPCT)

§  Classical definition TPCT

§  Relation TPCT – Reynolds stress tensor

Notations in turbulence


Multi-point correlation: fluctuation approach (classical)

... §  Definition multi-point tensor

§  Constitues a infinite set of nonlinear and non-local PDEs


Multi-point correlation: Instantaneous approach

... §  Definition multi-point tensor based on instantaneous quantities

§  Constitutes a infinite set of linear PDEs (compare Hopf and Lundgren-Novikov-Monin equations)


§  Non-linear bijective relations among tensor , and

Relations between Correlation

...


Symmetries of correlation Equations

§ All symmetries of Euler and Navier-Stokes equations transfer to the correlation equations

§ The correlation equation admits more symmetries (of statistical nature only)

§  Important for scaling laws such as the log-law and many others

§ One of them has been used in turbulence models for many decades – though has not been mentioned explicitly


New Statistical Symmetry I

§ Translation in function space:

with and arbitrary constant tensors

§ Key ingredient for log-law, etc. and turbulence models

§ Formulation transfers to classical notation


New Statistical Symmetry II

§ Scaling of correlations:

§ Important for deriving higher order correlations

§ Very difficult to implement into turbulence models

§ Formulation transfers to classical notation


Examples of scaling laws

§ 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

§ Depending on the values of the , five different solutions exist for mean velocity (Oberlack JFM 2001)

§ With new statistical symmetries we derive higher order scaling laws

Solutions for parallel shear flows


§ 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)



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


Symmetries and Turbulence Modeling

§ Essentially all RANS models capture the proper symmetries of Euler and Navier-Stokes equations

§ From the new statistical symmetries the following is capture by almost all models:

§ Note: this is not Galilean invariance and was probably first discovered by Kraichnan (1965)

§ Kraichnan modified DIA in order to capture (*), which in turn led to the proper Kolmogorov scaling law

(*)


Conclusions

§ New sets of symmetries (beyond Navier-Stokes) have been

derived for the infinite set of multi-point correlation equations

§  Turbulence not only restores but also creates symmetries

§ They are the key ingredients for classical and new scaling laws

here: wall turbulence, rotating, decaying turbulence, …

§ Most important: scaling for higher moments have be derived

Scaling laws are solutions of the multi-point equations

New Symmetries should be part of turbulence models


Open questions

§ Are there more statistical symmetries - completeness?

§ How do we determine the values for the parameters such as

in the “old” log-law - if not just fitted?

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

§ Reynolds number dependence?

§ …


Thank you for your

attention!

Introduction to Lie Symmetries and Invariance in Fluid ...

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