Turbulence and symmetries - University of...
Transcript of 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
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
§ 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, …)
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
Navier-Stokes equations model laminar, transitional and turbulent flows
Large scale simulation of NaSt
§ Example: turbulent channel flow with transpiration
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
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
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
PDF approach
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
§ 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
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
§ 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
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
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)
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
§ Translation in time:
§ Finite rotation:
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
Symmetries of the Euler and Navier-Stokes equations I
Note:
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
Symmetries of the Euler and Navier-Stokes equations II
§ Scaling of space:
§ Scaling of time:
in the Euler limit
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
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
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
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
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
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:
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
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
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
§ 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
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
§ 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
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
Classical log law
§ Invariance condition
§ Solutions for the stresses
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
Classical log law
§ Data from channel (Jimenez & Hoyas) at Reτ = 2006a
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
Classical log law
§ No obvious symmetry breaking
§ Scale invariance
§ Solution
Centre region of a channel flow
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
x2
x1
x3
u1
Centre region of a channel flow
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
10−3 10−2 10−1 100
10−4
10−2
100
Channel DNS
Reτ = 2006
(Hoyas & Jimenez)
x2
x1
x3
u1
§ Invariance condition
§ Solutions for the stresses
Centre region of a channel flow
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
x2
x1
x3
u1
Centre region of a channel flow
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
Channel DNS
Reτ = 2006
(Hoyas & Jimenez)
x2
x1
x3
u1
§ Symmetry breaking: rotation rate
§ Scale invariance
§ Solution
Rotating channel flow
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
Rotating channel flow
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
Channel DNS
(Kristoffersen
& Andersson 1993)
Rotating channel flow
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
Channel DNS
(Kristoffersen
& Andersson 1993)
Rotating channel flow
§ Non-linear variation of mass flux
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
Ω2x2
x1
x3
u1
u3
Rotating channel flow
§ New scaling law of Ekman type channel flow
§ DNS (Mehdizadeh, Oberlack, Phys. Fluids 2010)
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
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
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
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
Rotating channel flow
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
DNS at
Reτ = 360
Ro = 0.072
§ Symmetry suggested solution for the centre region
Channel flow with transpiration
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
x2
x1
x3
u1
Channel flow with transpiration
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
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
Centre region of a channel flow
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
§ Turbulent stresses (skip formula) § Reτ=250 § V0
+=0.16
Transfer stat. symmetries to pdf
§ Short reminder:
§ 1-point moment (mean velocity)
§ 2-point moment
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
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
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
Statistical Symmetry – PDF II
§ Scaling of correlations/moments:
§ PDF “equivalent”
§ Consequences § Mathematically: since and are non-negative functions
(semi-group) § Physically: measure of intermittency
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
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?
§ …
13.-16. May 2014 | Dep. Mech. Eng. | Chair of Fluid Dynamics | Prof. M. Oberlack
Thank you for your attention!