Tsinober Enstrophy Production&Dissipation

8/3/2019 Tsinober Enstrophy Production&Dissipation

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ON BALANCE OFON BALANCE OFENSTROPHYENSTROPHY

PRODUCTION AND ITS DISSIPATIONPRODUCTION AND ITS DISSIPATIONArkadyArkadyTsinoberTsinober

Professor andProfessor and

Marie CurieMarie Curie

Chair in Fundamental and Conceptual Aspects of Turbulent FlowsChair in Fundamental and Conceptual Aspects of Turbulent Flows

Institute for Mathematical SciencesInstitute for Mathematical Sciences and Department of Aeronautics, Imperial College Londonand Department of Aeronautics, Imperial College London

Does the Tennekes & Lumley balance hold onlyDoes the Tennekes & Lumley balance hold only##

in the mean?*in the mean?*## at sufficiently large Reynolds numbers?*at sufficiently large Reynolds numbers?*## inin statistically stationary turbulencestatistically stationary turbulence?

*See fig 6.6 in Tsinober 2001, Kluwer


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DNS:DNS:B.B. GALANTIGALANTIN.N.SANDHAMSANDHAM

Experiments:Experiments:

G.G. GULITSKIIGULITSKII,,M.M. KHOLMYANSKYKHOLMYANSKY

S.S.YORISHYORISH


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for statistically stationary turbulent shear flowfor statistically stationary turbulent shear flow

TTEE

NNNN

EE

KK

EE

SS

&&

LL

UUMM

LLEE

YY

11

9977

22

pp. 86-91


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EnstrophyEnstrophy production is approximately balanced in theproduction is approximately balanced in the meanmean by viscousby viscousterms atterms at sufficientlysufficiently highhigh Reynolds numbersReynolds numbers** inin statisticallystatisticallystationarystationary turbulent shear flowturbulent shear flow Three questions:Three questions:# Is it only/just in the mean?Is it only/just in the mean?# Does this happen only at sufficiently large Reynolds numbers?Does this happen only at sufficiently large Reynolds numbers?# Is this balance violated in statistically nonIs this balance violated in statistically non--stationary turbulencestationary turbulence?____________________________________________________________________________________________________________________________________________*In this sense - but not only in this - turbulence is not slightly viscous at

whatever large Reynolds number. In this context the question: whathappens with enstrophy/strain production as 0 is of special interest

TENNEKES AND LUMLEY BALANCE(1972,TENNEKES AND LUMLEY BALANCE(1972, P.91P.91):):


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(()D)D/Dt =/Dt = ii

jjss

ijij++

ii

ii ++

ijkijkii FFkk//xxjj

(()Ds)Ds//DtDt == -- ssijijssjkjksskiki (1/4)(1/4)

iijjssijij ss

ijijpp//xx

iixx

jj++

ssijijssijij ++ ssijijFFijij

SELFSELF--

AMPLIFICATIONAMPLIFICATION

OF VORTICITYOF VORTICITY

ANDANDSTRAINSTRAIN

The property of self amplification of vorticity and strain is responsible for the fact the neitherenstrophy nor the total strain s are inviscid invariants as is the kinetic energy u


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SELFSELF--

AMPLIFICATIONAMPLIFICATIONSELFSELF--RANDOMIZATION/INTRINSIC STOCHASTICITY: NO SOURCE OFRANDOMIZATION/INTRINSIC STOCHASTICITY: NO SOURCE OF

RANDOMNESS IS NEEDED, THE FORCING CAN BE CONSTANT IN TIMERANDOMNESS IS NEEDED, THE FORCING CAN BE CONSTANT IN TIME

AT THE LEVEL OF VELOCITYAT THE LEVEL OF VELOCITYDERIVATIVES: VORTICITY ANDDERIVATIVES: VORTICITY AND

STRAIN (DISSIPATION)STRAIN (DISSIPATION)THE EXTERNAL FORCING ISTHE EXTERNAL FORCING ISIRRELEVANTIRRELEVANT

0 5000-5000

0.5

-0.5

0

5

-5

-10

0

VELOCITYVELOCITY DERIVATIVESDERIVATIVES

FF

OO

RR

CC

II

NN

GGThree cases:

1. DNS in a periodic box, Re=102

2. DNS in a channel flow, Re=56003. Atmospheric SL, Re=104; Re=108

OF VORTICITYOF VORTICITY

ANDANDSTRAINSTRAIN


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DNS IN A CHANNEL FLOWRE=5600


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y+ y/d

EnstrophyEnstrophybalancebalance


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y+ y/d

Strain balanceStrain balance


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ATMOSPHERIC SURFACEATMOSPHERIC SURFACELAYERLAYER

RREE=10=1044; R; REE=10=1088


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THE MARIATHE MARIASILSSILSSITE, SWITZERLANDSITE, SWITZERLAND


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FIELD EXPERIMENT SUMMER 2004 SILSMARIA, SWITZERLAND

TheIsraeli

team

The calibration unit at 3 min the field

Height 1850 mHeight 1850 mExperiment wasExperiment was

performed inperformed incollaboration ofcollaboration ofInstituteInstituteof Hydromechanics andof Hydromechanics and

Water ResourcesWater ResourcesManagement, ETH ZurichManagement, ETH Zurich


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THE PROBETHE PROBE

Manganin is used as amaterial for the sensor

prongs instead oftungsten because thetemperature coefficient of

the electrical resistance ofmanganin is 400 timessmaller than that of

tungsten.

cold wiresold wires

hot wiresot wires

The tip of the probe with prongs made ofThe tip of the probe with prongs made ofmanganinmanganin

3 mm3 mm


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DNS IN A PERIODIC BOX, RE~102

DNS IN A PERIODIC BOX

RE~102


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T

256

STATIONARYSTATIONARY.


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ddEE//dtdt PP

----DD

FF

TEMPORAL EVOLUTION OFSPATIAL INTEGRALS

IN THE ENSTROPHY BALANCE EQUATION

NoteNote i)i) approximate balance betweenapproximate balance between PP

andand ----DD

andand ii)ii) irrelevance of the forcing termirrelevance of the forcing term FF

SimilarSimilarbehavourbehavouris observed withis observed with hyperviscosityhyperviscosity

Re= 2x102

(()D)D/Dt =/Dt = iijjssijij ++ iiii ++ ijkijkii FFkk//xxjj


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DOES THE T & L BALANCEHOLD ONLY/JUST IN THE

MEAN?

DOESDOES THE T & L BALANCEHOLD ONLY ATHOLD ONLY AT

SUFFICIENTLY LARGESUFFICIENTLY LARGEREYNOLDS NUMBERSREYNOLDS NUMBERS??


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DOESDOES THE T & L BALANCE

HOLD ONLYHOLD ONLY INSTATISTICALLYSTATISTICALLYSTATIONARY TURBULENCESTATIONARY TURBULENCE??


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PERIODICALLY FORCEDPERIODICALLY FORCED

TURBULENCETURBULENCE

The simulation parameters:The simulation parameters:Resolution 128Resolution 128

Forcing: ABC multiplied byForcing: ABC multiplied by (1 + A(1 + Att coscos t)t) withwithAAtt = 0.5= 0.5 andand = 0, 6= 0, 6 andand 3030Velocity field parameters:Velocity field parameters:

Eddy turn over timeEddy turn over time 5050

Taylor Reynolds numberTaylor Reynolds number ReRe

= 50= 50


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From now onFrom now on meansmeans dVdV


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EnstrophyEnstrophybalancebalance

(()D)D/Dt =/Dt = iijjssijij ++ iiii ++ijkijkii FFkk//xxjj


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1.1 1.12 1.14 1.16 1.18 1.2 1.22 1.24x 10

4

0.02

0

0.02

0.04

0.06

0.08

0.1

Time

1/2

=0


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1.08 1.1 1.12 1.14 1.16 1.18 1.2 1.22

x 104

0.01

0

0.01

0.02

0.03

0.04

0.05

Time

1/2

=6


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1.24 1.26 1.28 1.3 1.32 1.34 1.36x 10

4

0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Time

1/2

=30


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StrainStrain balancebalance

(()Ds)Ds//DtDt == -- ssijijssjkjksskiki (1/4)(1/4)iijjssijij ssijijpp//xxiixxjj ++

ssijijssijij ++ ssijijFFijij


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1.1 1.12 1.14 1.16 1.18 1.2 1.22 1.24

x 104

0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Time

1/2

1/4

=0

0 025


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1.08 1.1 1.12 1.14 1.16 1.18 1.2 1.22x 10

4

0.005

0

0.005

0.01

0.015

0.02

0.025

Time

1/2

1/4

=6

3


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1.24 1.26 1.28 1.3 1.32 1.34 1.36x 10

4

0

5

10

15

20x 10

3

Time

1/2

1/4

=30


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TheThe T&Lenstrophyenstrophy balance holdsbalance holds notot onlyonly## in the meanin the mean

## at sufficiently large Reynolds numbersat sufficiently large Reynolds numbers## inin statistically stationary turbulencestatistically stationary turbulence

## A similar balance holds for the total strainA similar balance holds for the total strain ssikikssikik

CONCLUDINGCONCLUDING

Tsinober Enstrophy Production&Dissipation

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