Kolmogorov 4/5 law, nonlocality and Kolmogorov 4/5 law, nonlocality and sweeping decorrelation hypothesissweeping decorrelation hypothesisand the role of kinematic relationsand the role of kinematic relations

References:References:Hosokawa I. (2007) A Paradox concerning the refined similarity hypothesis of Kolmogorov for isotropic turbulence, Prog. Theor. Phys., 118, 169–173.M. Kholmyansky and A. Tsinober (2008) Kolmogorov 4/5 law, nonlocality, and sweeping M. Kholmyansky and A. Tsinober (2008) Kolmogorov 4/5 law, nonlocality, and sweeping decorrelation hypothesis.decorrelation hypothesis. Phys. Fluids Phys. Fluids, , 2020, 041704., 041704.M. Kholmyansky, V. Sabelnikov and A. Tsinober (2008) M. Kholmyansky, V. Sabelnikov and A. Tsinober (2008) New developments in field New developments in field experiments in ASL: Kolmogorov 4/5 law and nonlocality, experiments in ASL: Kolmogorov 4/5 law and nonlocality, 5B.1, 18th AMS Symposium on Boundary Layers and Turbulence, June 9-12, 5B.1, 18th AMS Symposium on Boundary Layers and Turbulence, June 9-12, Stockholm, Stockholm, http://ams.confex.com/ams/pdfpapers/139408.pdf.

Michael Kholmyansky Tel-Aviv University, Tel-Aviv, Israel

Vladimir Sabelnikov ONERA/DEFA/EFCA,Fort de Palaiseau, France

Arkady Tsinober Imperial College, London, UK; Tel-Aviv University, Tel-Aviv, Israel;

Isaac Newton Institute for Mathematical Sciences, Cambridge , UKWorkshop Inertial-Range Dynamics and Mixing, 29 September-3 October, 2008

The experimental data were The experimental data were obtained obtained

with partial support from:with partial support from:

• The US-Israel Binational Science Foundation (BSF) The US-Israel Binational Science Foundation (BSF)

•The German-Israel Science Foundation (GIF) The German-Israel Science Foundation (GIF)

•The Israel Science Foundation, Israel Academy of Science The Israel Science Foundation, Israel Academy of Science (ISF)(ISF)

•The Swiss National FoundationThe Swiss National Foundation

AcknowledgmAcknowledgmentsents

There is an important aspect of non-locality of There is an important aspect of non-locality of turbulent flows, understood as direct and turbulent flows, understood as direct and bidirectional interaction of ‘large’ and ‘small bidirectional interaction of ‘large’ and ‘small scales’. scales’. The Kolmogorov 4/5 law points to this important The Kolmogorov 4/5 law points to this important aspect of non-locality of turbulent flows. An aspect of non-locality of turbulent flows. An essential role in the nonlocal(ity) interpretation essential role in the nonlocal(ity) interpretation of the 4/5 law is played by purely kinematic of the 4/5 law is played by purely kinematic relations, the role of which goes far beyond their relations, the role of which goes far beyond their use in this interpretation of the Kolmogorov 4/5 use in this interpretation of the Kolmogorov 4/5 lawlaw.. We report results of field and airborne We report results of field and airborne experiments at large Reynolds numbers experiments at large Reynolds numbers supporting the above.supporting the above.

The main pointsThe main points

We start with a well-known relation for the 3-d order velocity structure function, S3, obtained by A. Kolmogorov 1941b*:

SS3 3 = <(= <(ΔΔuu))33> = −(4/5)<> = −(4/5)<εε>>rr.Here:ΔΔuu = = uu2 2 − − uu11;;uu1 1 = = uu((xx), ), uu2 2 = = uu((xx ++ rr)) - are the longitudinal velocity

component fluctuations;<<εε>> – is the mean dissipation.This is the famous Kolmogorov 4/5 law.

MotivationMotivation

*Kolmogorov, A.N. (1941) Dissipation of energy in locally isotropic turbulence, Dokl. Acad. NaukSSSR, 32, 19–21; for English translation see: Selected works of A. N. Kolmogorov, I, ed. V.M. Tikhomirov, pp. 324–327, Kluwer, 1991.

In a recent paper by Iwao Hosokawa 2007* a several purely kinematic relations were obtained in terms of

uu+ + = (= (uu2 2 + + uu11)/2 and )/2 and uu− − = =

ΔΔu/u/22 = (= (uu2 2 – – uu11)/2)/2..The most important of them is:

33<u<u++22 uu−−> + > + <u<u−−

33> = 0, > = 0, (1)(1)Thus, together with the 4/5 law (1) results in

<u<u++22 uu−−> = <> = <εε>>rr/30, /30, (2)(2)

This relation is equivalent, to the 4/5 lawThis relation is equivalent, to the 4/5 law..

*Hosokawa I. (2007) A Paradox concerning the refined similarity hypothesis of Kolmogorov for isotropic turbulence, Prog. Theor. Phys. 118, 169–173.

MotivationMotivation

<u<u++22 uu−−> = <> = <εε>>rr/30, /30, (2)(2)

uu+ + = (= (uu2 2 + + uu11)/2 and )/2 and uu− − = = ΔΔu/2 = (u2 – u1)/2.

The relation (2) is a clear indication of the The relation (2) is a clear indication of the absence of statistical independence between absence of statistical independence between uu++ and and uu−− , i.e. between , i.e. between ‘large‘large’’ and and ‘small’ ‘small’

scalesscales.. OtherwiseOtherwise,, <u<u++

22 uu−−> = 0> = 0

As formulated by Hosokawa himself,

it is proven that the famous third-order structure function of the velocity in homogeneous isotropic turbulence derived by Kolmogorov implies the statistical interdependence of the difference and sum of the velocities at two points separated by a distance r.

MotivationMotivation

<u<u++22 uu−−> = <> = <εε>>rr/30/30 (2)(2)

The 4/5 law (a) and its equivalent (b), as displayed by the relation (2), normalized on <ε>r, are shown in this slide.

Both hold for about 2.5 decades for the field experiments and more than for 3.5 decades in the airborne experiment.

MotivationMotivation

M. Kholmyansky and A. Tsinober (2008) Kolmogorov 4/5 M. Kholmyansky and A. Tsinober (2008) Kolmogorov 4/5 law, nonlocality, and sweeping decorrelation hypothesis.law, nonlocality, and sweeping decorrelation hypothesis. Phys. FluidsPhys. Fluids, , 2020, 041704., 041704.

The Kolmogorov 4/5 law S3=<(Δu)3> = −(4/5)<ε>r and the Hosokawa relation <u+

2 u−> = <ε>r/30 are equivalent.But the latter has an important advantage for the experimentalists:

it is linear in velocity increments u−=Δu/2, while the 4/5 law is cubic. Therefore the Hosokawa relation holds much better than the 4/5 law, especially in the case of lower quality data as in the airborne experiment.

MotivationMotivation

The experimentsThe experimentsMore in:More in:

G. Gulitski, M. Kholmyansky, W. Kinzelbach, B. Lüthi, A. G. Gulitski, M. Kholmyansky, W. Kinzelbach, B. Lüthi, A. Tsinober and S. Yorish (2007) Velocity and temperature Tsinober and S. Yorish (2007) Velocity and temperature derivatives in high-Reynolds-number turbulent flows in the derivatives in high-Reynolds-number turbulent flows in the atmospheric surface layer.atmospheric surface layer. Part 1. Facilities, methods and some general results.Part 1. Facilities, methods and some general results. Part 2. Accelerations and related matters.Part 2. Accelerations and related matters. Part 3 . Temperature and joint statistics of temperature Part 3 . Temperature and joint statistics of temperature and velocity derivativesand velocity derivatives.. Journal of Fluid MechanicsJournal of Fluid Mechanics, , 589589, 57−123., 57−123. And references thereinAnd references therein

Field experiment – 1999Field experiment – 1999

Field stationField stationnear kibbutz Kfar Glikson, Israel,near kibbutz Kfar Glikson, Israel,with 10 m mast.with 10 m mast.

The data from one of the runs,The data from one of the runs,obtained at this station, areobtained at this station, aremarked marked 102102..

The experimentsThe experiments

Airborne experiment – Airborne experiment – 2000-12000-1

In collaboration with DLR, Oberpfaffenhoffen, GermanyIn collaboration with DLR, Oberpfaffenhoffen, Germany

The experimentsThe experiments

The data from one of the runs, obtained in the airborne experiment, are marked The data from one of the runs, obtained in the airborne experiment, are marked FalconFalcon..

Field experiment – 2003-4Field experiment – 2003-4

In collaboration with ETH, Zurich, Switzerland, Sils-Maria siteIn collaboration with ETH, Zurich, Switzerland, Sils-Maria site

The experimentsThe experiments

The data from two of the runs,The data from two of the runs,obtained in this experiment,obtained in this experiment,are markedare markedSNM11 SNM11 andand SNM12 SNM12..

What was measured:What was measured:• all three components of velocity fluctuations vector, all three components of velocity fluctuations vector, uuii ; ;• all nine components of spatial velocity gradients tensor, all nine components of spatial velocity gradients tensor,

uuii //xxj j (without invoking the Taylor hypothesis);(without invoking the Taylor hypothesis);

• temporal velocity derivatives, temporal velocity derivatives, uuii//tt..

With synchronous data on:With synchronous data on:• fluctuations of temperature,fluctuations of temperature, θθ;;• spatial gradient of temperature, spatial gradient of temperature, θθ//xxj j ;;• temporal derivative of temperature, temporal derivative of temperature, θθ//tt ..

• profiles of mean velocity and temperatureprofiles of mean velocity and temperature

The experimentsThe experiments

The The probeprobe

10 mm

5-channel thermometer unit,5-channel thermometer unit,20-channel anemometer unit,20-channel anemometer unit,signal limitation unit,signal limitation unit,signal conditioning interface,signal conditioning interface,and data acquisitionand data acquisitionare all connected to 3 mm point.are all connected to 3 mm point.

The experimentsThe experiments

Three-dimensional calibration unitThree-dimensional calibration unit

The calibration unitThe calibration unit(in laboratory,(in laboratory,with cover removed)with cover removed)

Air flowAir flow

The experimentsThe experiments

Three-dimensional calibration unitThree-dimensional calibration unit

The calibration procedure enables to measure velocity in the presence of temperature

variations.

Heating element at the entrance Heating element at the entrance to the nozzleto the nozzle

Schematic of the flow in calibration unitSchematic of the flow in calibration unit

The experimentsThe experiments

The experimentsThe experimentsMore in:More in:

G. Gulitski, M. Kholmyansky, W. Kinzelbach, B. Lüthi, A. G. Gulitski, M. Kholmyansky, W. Kinzelbach, B. Lüthi, A. Tsinober and S. Yorish (2007) Velocity and temperature Tsinober and S. Yorish (2007) Velocity and temperature derivatives in high-Reynolds-number turbulent flows in the derivatives in high-Reynolds-number turbulent flows in the atmospheric surface layer.atmospheric surface layer. Part 1. Facilities, methods and some general results.Part 1. Facilities, methods and some general results. Part 2. Accelerations and related matters.Part 2. Accelerations and related matters. Part 3 . Temperature and joint statistics of temperature Part 3 . Temperature and joint statistics of temperature and velocity derivativesand velocity derivatives.. Journal of Fluid MechanicsJournal of Fluid Mechanics, , 589589, 57−123., 57−123. And references thereinAnd references therein

The experimentsThe experiments

Local versus nonlocalLocal versus nonlocal

contributionscontributions

Though the Hosokawa version of the 4/5 law clearly points to the nonlocality, it contains both nonlocal and local contributions. This can be seen looking at correlations of a different kind, involving u−

and u1 (or equivalently u2), which is one-point quantity (u+ − is a

two-point quantity): u+2 = ½ u2

2 + ½ u12 − u−

2, i.e.

uu++22 uu−− = ½ ( = ½ (uu22

2 2 + + uu1122) ) uu−− − − uu−−

33..

Averaging of the first two terms (nonlocal interaction) gives:

½< (½< (uu2222 + + uu11

22) ) uu−−>=−(1/15)<>=−(1/15)<εε>>rr,, (3)(3)since both <u2

2 u− >=<u12 u− >= −(1/15)<ε>r.

Averaging of the third term (local interaction) gives:

<−u<−u−−33 >= >= (1/10)<(1/10)<εε>>r, r, (4)(4)

The sum of (3) and (4) leads to the Hosokawa relation (2).<u<u++

22 uu−−> = <> = <εε>>rr/30, (2)/30, (2)

Local versus nonlocal contributionsLocal versus nonlocal contributions

Local versus nonlocal contributionsLocal versus nonlocal contributions

It is noteworthy thatthe contributions from nonlocal and local effects are of opposite signs, i.e. the nonlocal effects strongly reduce the local ones,as concerns the Hosokawa’s version of the 4/5 law.

Note that the above interpretation is possible due to the use of simple algebra mostly before the averaging, i.e. not directly with <u<u++

22 uu−−>> , but rather

with uu++22 uu−−

In the figure below we show the experimental verification of the relation (3):

½< (u22 + u1

2) u−>=−(1/15)<ε>r. (3)

Local versus nonlocal contributionsLocal versus nonlocal contributions

On the role On the role

of kinematic of kinematic relationsrelations

The role of kinematic relations in the issue of nonlocality goes far beyond their use in the nonlocal interpretation of the Kolmogorov 4/5 law.

The structure functions Sn(r)<(u2–u1)n > are expressed via terms,

all of which have the form of correlations between large- and small-scale quantities.

In other words, in the absence of nonlocal interactions – as manifested by correlations between large scale (velocity) and small scale (velocity increments) – all structure functions vanish.

Hence the utmost dynamical importance of purely kinematics relations.

On the role of kinematic relationsOn the role of kinematic relations

Isotropic turbulence. Some asymmetric relationsin terms of u1 and Δu or u2 and Δu.

n=2 <(Δu)2> = –2<u1Δu> (5)

<(Δu)2> = 2<u2Δu> (6)

n=3 <(Δu)3> = 6<u12Δu> = – (4/5)<ε>r

<(Δu)3> = 6<u22Δu> = – (4/5)<ε>r

<(Δu)3> = –2<u1(Δu)2> = –(4/5)<ε>r

<(Δu)3> = 2<u2(Δu)2> = –(4/5)<ε>r

On the role of kinematic relationsOn the role of kinematic relations

Isotropic turbulence. Some asymmetric relationsin terms of u1 and Δu or u2 and Δu.

n=4 <(Δu)4>= 4<u13Δu> +6<u1

2(Δu)2>

<(Δu)4>= –4<u23Δu> + 6<u2

2(Δu)2>

Any n≥2 <(Δu)n>= –2<u1(Δu)n–1> (7)

<(Δu)n>= 2<u2(Δu)n–1> (8)

On the role of kinematic relationsOn the role of kinematic relations

In the figures below examples of experimental verification of kinematic relations (5, 6)

⟨(Δu) n =⟩ –2⟨u1(Δu) n-1 =2⟩ ⟨u2(Δu) n-1 , ⟩ n≥2

are shown. Of the variety of the relations, the asymmetric versions are chosen to emphasize the nonlocal aspects.

On the role of kinematic relationsOn the role of kinematic relations

Sweeping Sweeping decorrelation decorrelation hypothesishypothesis

Sweeping decorrelation hypothesisSweeping decorrelation hypothesis

Another point is that all kinematic relations under consideration (such as shown above) stand in contradiction with the so-called sweeping decorrelation hypothesis (SDH), understood as statistical independence between large and small scales. This is seen from many of the discussed relations. The simplest are the relations (3, 4)

the right-hand side of which vanishes assuming the sweeping decorrelation hypothesis to be valid, whereas in reality the left-hand side is well known to scale as rr2/32/3.

⟨(Δu)2 =–2⟩ ⟨u1Δu =2⟩ ⟨u2Δu ,⟩

Sweeping decorrelation hypothesisSweeping decorrelation hypothesis

FollowingFollowing Kraichnan 1964 and and Tennekes 1975 there are two main ingredients in there are two main ingredients in the (Eulerian) decorrelation: the the (Eulerian) decorrelation: the sweeping of microstucture by the large sweeping of microstucture by the large scale motions (and associated scale motions (and associated kinematickinematic nonlocality) and the local nonlocality) and the local straining (which is roughly pure straining (which is roughly pure Lagrangian). It appears that this kind of Lagrangian). It appears that this kind of “decomposition” is insufficient as it is “decomposition” is insufficient as it is missing an essential dynamical aspect - missing an essential dynamical aspect - the interaction between the two as it is the interaction between the two as it is clearly demonstrated by the nonlocal clearly demonstrated by the nonlocal version of the Kolmogorov 4/5 law. The version of the Kolmogorov 4/5 law. The random Taylor hypothesis (and, of random Taylor hypothesis (and, of course, the conventional Taylor course, the conventional Taylor hypothesis) lack/discard this aspect at hypothesis) lack/discard this aspect at the outset (this does not mean that the outset (this does not mean that these hypotheses are useless): both are these hypotheses are useless): both are “too kinematic”“too kinematic”..

Sweeping decorrelation hypothesisSweeping decorrelation hypothesis

ConcludingConcluding

The main result is that purely kinematic exact relations The main result is that purely kinematic exact relations demonstrate one of important aspects of non-locality of demonstrate one of important aspects of non-locality of turbulent flowsturbulent flows. Though purely kinematic they have Though purely kinematic they have important dynamical important dynamical consequencesconsequences..

There is no exaggeration in saying that without nonlocality There is no exaggeration in saying that without nonlocality (understood as direct and bidirectional(understood as direct and bidirectional coupling of large coupling of large and small scales) there is no turbulence. and small scales) there is no turbulence.

Though in some limited sense one can speak about Though in some limited sense one can speak about separating the local and nonlocal contributions, generally separating the local and nonlocal contributions, generally such a separation seems impossible and even in some sense such a separation seems impossible and even in some sense meaningless, since "what is local is also nonlocal".meaningless, since "what is local is also nonlocal".

The kinematic relations stand in contradiction with the The kinematic relations stand in contradiction with the sweeping decorrelation hypothesis understood as statistical sweeping decorrelation hypothesis understood as statistical independence between large and small scalesindependence between large and small scales.

Future work

The main effort should be directed in looking at relations/properties beyond correlations,

e.g. PDFs, etc. and sub-Kolmogorov resolution

Thank you for your Thank you for your patiencepatience