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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
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 2
Concept of symmetries
§ Analogy between symmetric object and differential equation § Symmetric Object (under rotation)
§ Symmetry of differential equation
§ Differential equation
§ Symmetry transformation
§ Form invariance
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 3
Symmetry of differential equation
§ Example 1: 1D heat equation
§ Symmetry transformation 1: scaling of space-time
§ Into heat equation:
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 4
Symmetry of differential equation
§ Example 2: 1D heat equation
§ Symmetry transformation 2: scaling of
§ Into heat equation:
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 5
Symmetry of differential equation
§ Example 3: 1D heat equation
§ Symmetry transformation 3: “Galilean” transformation
§ Into heat equation: The heat equation admits 6 symmetry transformations
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 6
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!
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 7
Generation of solutions from symmetries
§ Existing solutions:
and using:
§ Implementing (3) into (2):
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 8
Generation of solutions from symmetries
§ Example 1 heat equation: Translation symmetry
§ Given solution:
§ New solution from :
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 9
Generation of solutions from symmetries
§ 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 )
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 10
§ 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)
Generation of solutions from symmetries
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 11
§ 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 )
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 12
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)
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 13
Invariants
§ Example 2: Combined scalings of 1D heat equation
§ Invariants
1)
2)
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 14
§ 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
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 15
§ Example 1: 1D heat equation
§ 2 parameter scaling group:
§ Boundary condition
§ Combined with the scaling symmetry
Symmetry breaking
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 16
§ Preserving the form of the BC
remains arbitrary
§ is symmetry breaking
§ Reduced form of invariants
§ Reduced consistent with BC:
Symmetry breaking
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 17
§ 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
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 18
Symmetries of Boundary Layer equations § Prandtl boundary layer (BL) equations:
§ BL equations admit 5 symmetry transformations
§ All known analytic solutions arise from these symmetries
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 19
§ Translation in : direction:
§ Pressure invariance:
Symmetries of Boundary Layer equations
x1
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 20
§ Scaling 1:
§ Scaling 2:
§ Blasius, Falkner-Skan etc. emerge from these groups.
Symmetries of Boundary Layer equations
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 21
§ 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
Symmetries of Boundary Layer equations
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 22
§ 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
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 23
§ Independent variables:
§ Dependent variables:
Ansatz:
Invariants of BL equations
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 24
Invariants of BL equations
§ Note: § und only appear as ratio
The more symmetries are admitted the better we can fit to BCs.
§
in Bl PDE:
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 25
Invariants of BL equations
§ 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
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 26
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?
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 27
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
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 28
Symmetries of the Euler and Navier-Stokes equations
§ Scaling of space:
§ Scaling of time:
for (Euler)
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 29
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
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 30
§ Translation in time:
§ Finite rotation:
Note:
Symmetries of the Euler and Navier-Stokes equations II
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 31
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
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 32
§ Instantaneous value
§ Mean value
§ Fluctuating quantities
§ Two-point correlation tensor (TPCT)
§ Classical definition TPCT
§ Relation TPCT – Reynolds stress tensor
Notations in turbulence
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 33
Multi-point correlation: fluctuation approach (classical)
... § Definition multi-point tensor
§ Constitues a infinite set of nonlinear and non-local PDEs
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 34
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)
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 35
§ Non-linear bijective relations among tensor , and
Relations between Correlation
...
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 36
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
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 37
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
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 38
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
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 39
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
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 40
§ 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
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 41
§ Symmetry breaking: wall-friction velocity
§ Scale invariance
§ Mean velocity
Classical log law
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 42
§ Invariance condition
§ Solutions for the stresses
Classical log law
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 43
§ Data from channel (Jimenez & Hoyas) at Reτ = 2006a
Classical log law
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 44
§ No obvious symmetry breaking
§ Scale invariance
§ Solution
Centre region of a channel flow
x2
x1
x3
u1
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 45
Centre region of a channel flow
10 3 10 2 10 1 100
10 4
10 2
100
Channel DNS
Reτ = 2006
(Hoyas & Jimenez)
x2
x1
x3
u1
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 46
§ Invariance condition
§ Solutions for the stresses
Centre region of a channel flow
x2
x1
x3
u1
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 47
Centre region of a channel flow
Channel DNS
Reτ = 2006
(Hoyas & Jimenez)
x2
x1
x3
u1
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 48
§ Symmetry breaking: rotation rate
§ Scale invariance
§ Solution
Rotating channel flow
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 49
Rotating channel flow
Channel DNS
(Kristoffersen
& Andersson 1993)
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 50
Rotating channel flow
Channel DNS
(Kristoffersen
& Andersson 1993)
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 51
Rotating channel flow
§ Non-linear variation of mass flux
Ω2x2
x1
x3
u1
u3
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 52
Rotating channel flow
§ New scaling law of Ekman type channel flow
§ DNS (Mehdizadeh, Oberlack, Phys. Fluids 2010)
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 53
Rotating channel flow
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
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 54
Ω2x2
x1
x3
u1
u3
Rotating channel flow
DNS at
Reτ = 360
Ro = 0.072
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 55
§ Symmetry suggested solution for the centre region
Channel flow with transpiration
x2
x1
x3
u1
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 56
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
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 57
x2
x1
x3
u1
Centre region of a channel flow
§ Turbulent stresses (skip formula) § Reτ=250 § V0
+=0.16
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 58
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
(*)
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 59
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
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 60
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?
§ …
16.10.2013 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack | 61
Thank you for your
attention!