Introduction to Lie Symmetries and Invariance in Fluid ...
Transcript of 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
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
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Symmetry of differential equation
§ Example 1: 1D heat equation
§ Symmetry transformation 1: scaling of space-time
§ Into heat equation:
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Symmetry of differential equation
§ Example 2: 1D heat equation
§ Symmetry transformation 2: scaling of
§ Into heat equation:
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Symmetry of differential equation
§ Example 3: 1D heat equation
§ Symmetry transformation 3: “Galilean” transformation
§ Into heat equation: The heat equation admits 6 symmetry transformations
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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!
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Generation of solutions from symmetries
§ Existing solutions:
and using:
§ Implementing (3) into (2):
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Generation of solutions from symmetries
§ Example 1 heat equation: Translation symmetry
§ Given solution:
§ New solution from :
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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 )
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§ 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
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§ 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 )
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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)
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Invariants
§ Example 2: Combined scalings of 1D heat equation
§ Invariants
1)
2)
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§ 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
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§ Example 1: 1D heat equation
§ 2 parameter scaling group:
§ Boundary condition
§ Combined with the scaling symmetry
Symmetry breaking
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§ Preserving the form of the BC
remains arbitrary
§ is symmetry breaking
§ Reduced form of invariants
§ Reduced consistent with BC:
Symmetry breaking
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§ 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
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Symmetries of Boundary Layer equations § Prandtl boundary layer (BL) equations:
§ BL equations admit 5 symmetry transformations
§ All known analytic solutions arise from these symmetries
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§ Translation in : direction:
§ Pressure invariance:
Symmetries of Boundary Layer equations
x1
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§ Scaling 1:
§ Scaling 2:
§ Blasius, Falkner-Skan etc. emerge from these groups.
Symmetries of Boundary Layer equations
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§ 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
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§ 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
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§ Independent variables:
§ Dependent variables:
Ansatz:
Invariants of BL equations
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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:
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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
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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?
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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
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Symmetries of the Euler and Navier-Stokes equations
§ Scaling of space:
§ Scaling of time:
for (Euler)
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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
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§ Translation in time:
§ Finite rotation:
Note:
Symmetries of the Euler and Navier-Stokes equations II
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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
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§ Instantaneous value
§ Mean value
§ Fluctuating quantities
§ Two-point correlation tensor (TPCT)
§ Classical definition TPCT
§ Relation TPCT – Reynolds stress tensor
Notations in turbulence
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Multi-point correlation: fluctuation approach (classical)
... § Definition multi-point tensor
§ Constitues a infinite set of nonlinear and non-local PDEs
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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)
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§ Non-linear bijective relations among tensor , and
Relations between Correlation
...
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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
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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
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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
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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
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§ 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
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§ Symmetry breaking: wall-friction velocity
§ Scale invariance
§ Mean velocity
Classical log law
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§ Invariance condition
§ Solutions for the stresses
Classical log law
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§ Data from channel (Jimenez & Hoyas) at Reτ = 2006a
Classical log law
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§ No obvious symmetry breaking
§ Scale invariance
§ Solution
Centre region of a channel flow
x2
x1
x3
u1
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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
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§ 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
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§ Symmetry breaking: rotation rate
§ Scale invariance
§ Solution
Rotating channel flow
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Rotating channel flow
Channel DNS
(Kristoffersen
& Andersson 1993)
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Rotating channel flow
Channel DNS
(Kristoffersen
& Andersson 1993)
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Rotating channel flow
§ Non-linear variation of mass flux
Ω2x2
x1
x3
u1
u3
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Rotating channel flow
§ New scaling law of Ekman type channel flow
§ DNS (Mehdizadeh, Oberlack, Phys. Fluids 2010)
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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
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Ω2x2
x1
x3
u1
u3
Rotating channel flow
DNS at
Reτ = 360
Ro = 0.072
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§ Symmetry suggested solution for the centre region
Channel flow with transpiration
x2
x1
x3
u1
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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
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x2
x1
x3
u1
Centre region of a channel flow
§ Turbulent stresses (skip formula) § Reτ=250 § V0
+=0.16
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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
(*)
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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
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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?
§ …
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Thank you for your
attention!