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Isaac Newton Institute for Mathematical Sciences, Cambridge , UK Workshop Inertial-Range Dynamics and Mixing , 29 September-3 October, 2008. Kolmogorov 4/5 law, nonlocality and sweeping decorrelation hypothesis and the role of kinematic relations.
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Isaac Newton Institute for Mathematical Sciences, Cambridge , UK Workshop Inertial-Range Dynamics and Mixing, 29 September-3 October, 2008 Kolmogorov 4/5 law, nonlocality and sweeping decorrelation hypothesisand the role of kinematic relations Michael Kholmyansky Tel-Aviv University, Tel-Aviv, Israel Vladimir SabelnikovONERA/DEFA/EFCA,Fort de Palaiseau, France Arkady TsinoberImperial College, London, UK; Tel-Aviv University, Tel-Aviv, Israel; 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 decorrelation hypothesis. Phys. Fluids, 20, 041704. M. Kholmyansky, V. Sabelnikov and A. Tsinober (2008) New developments in field experiments in ASL: Kolmogorov 4/5 law and nonlocality, 5B.1, 18th AMS Symposium on Boundary Layers and Turbulence, June 9-12, Stockholm, http://ams.confex.com/ams/pdfpapers/139408.pdf.
Acknowledgments The experimental data were obtained with partial support from: • The US-Israel Binational Science Foundation (BSF) • The German-Israel Science Foundation (GIF) • The Israel Science Foundation, Israel Academy of Science (ISF) • The Swiss National Foundation
The main points There is an important aspect of non-locality of turbulent flows, understood as direct and bidirectional interaction of ‘large’ and ‘small scales’. The Kolmogorov 4/5 law points to this important aspect of non-locality of turbulent flows. An essential role in the nonlocal(ity) interpretation of the 4/5 law is played by purely kinematic relations, the role of which goes far beyond their use in this interpretation of the Kolmogorov 4/5 law. We report results of field and airborne experiments at large Reynolds numbers supporting the above.
Motivation We start with a well-known relation for the 3-d order velocity structure function, S3, obtained by A. Kolmogorov 1941b*: S3 = <(Δu)3> = −(4/5)<ε>r. Here: Δu= u2 − u1; u1 = u(x), u2 = u(x+r) - are the longitudinal velocity component fluctuations; <ε> – is the mean dissipation. This is the famous Kolmogorov 4/5 law. *Kolmogorov, A.N. (1941) Dissipation of energy in locally isotropic turbulence, Dokl. Acad. Nauk SSSR, 32, 19–21; for English translation see: Selected works of A. N. Kolmogorov, I, ed. V.M. Tikhomirov, pp. 324–327, Kluwer, 1991.
Motivation In a recent paper by Iwao Hosokawa 2007* a several purely kinematic relations were obtained in terms of u+ = (u2 + u1)/2 and u− = Δu/2= (u2 – u1)/2. The most important of them is: 3<u+2u−> + <u−3> = 0, (1) Thus, together with the 4/5 law (1) results in <u+2u−> = <ε>r/30, (2) This 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.
Motivation <u+2u−> = <ε>r/30, (2) u+ = (u2 + u1)/2 and u− = Δu/2= (u2 – u1)/2. The relation (2) is a clear indication of the absence of statistical independence between u+ and u− , i.e. between ‘large’and ‘small’ scales. Otherwise,<u+2u−> = 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.
Motivation <u+2u−> = <ε>r/30(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. M. Kholmyansky and A. Tsinober (2008) Kolmogorov 4/5 law, nonlocality, and sweeping decorrelation hypothesis. Phys. Fluids, 20, 041704.
Motivation The Kolmogorov 4/5 law S3=<(Δu)3> = −(4/5)<ε>r and the Hosokawa relation <u+2u−> = <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.
The experiments More in: G. Gulitski, M. Kholmyansky, W. Kinzelbach, B. Lüthi, A. Tsinober and S. Yorish (2007) Velocity and temperature derivatives in high-Reynolds-number turbulent flows in the atmospheric surface layer. Part 1. Facilities, methods and some general results. Part 2. Accelerations and related matters. Part 3 . Temperature and joint statistics of temperature and velocity derivatives. Journal of Fluid Mechanics, 589, 57−123. And references therein
The experiments Field experiment – 1999 Field station near kibbutz Kfar Glikson, Israel, with 10 m mast. The data from one of the runs, obtained at this station, are marked 102.
Airborne experiment – 2000-1 The experiments In collaboration with DLR, Oberpfaffenhoffen, Germany The data from one of the runs, obtained in the airborne experiment, are marked Falcon.
Field experiment – 2003-4 The experiments In collaboration with ETH, Zurich, Switzerland, Sils-Maria site The data from two of the runs, obtained in this experiment, are marked SNM11 and SNM12.
What was measured: The experiments • all three components of velocity fluctuations vector, ui ; • all nine components of spatial velocity gradients tensor, ui/xj (without invoking the Taylor hypothesis); • temporal velocity derivatives, ui/t. With synchronous data on: • fluctuations of temperature,θ; • spatial gradient of temperature, θ/xj ; • temporal derivative of temperature, θ/t. • profiles of mean velocity and temperature
The probe The experiments 10 mm 5-channel thermometer unit, 20-channel anemometer unit, signal limitation unit, signal conditioning interface, and data acquisition are all connected to 3 mm point.
The experiments Three-dimensional calibration unit Air flow The calibration unit (in laboratory, with cover removed)
The experiments Three-dimensional calibration unit The calibration procedure enables to measure velocity in the presence of temperature variations. Heating element at the entrance to the nozzle Schematic of the flow in calibration unit
The experiments The experiments More in: G. Gulitski, M. Kholmyansky, W. Kinzelbach, B. Lüthi, A. Tsinober and S. Yorish (2007) Velocity and temperature derivatives in high-Reynolds-number turbulent flows in the atmospheric surface layer. Part 1. Facilities, methods and some general results. Part 2. Accelerations and related matters. Part 3 . Temperature and joint statistics of temperature and velocity derivatives. Journal of Fluid Mechanics, 589, 57−123. And references therein
Local versus nonlocal contributions
Local versus nonlocal contributions 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 u22 + ½ u12 − u−2, i.e. u+2u− = ½ (u22 + u12) u− − u−3. Averaging of the first two terms (nonlocal interaction) gives: ½< (u22 + u12) u−>=−(1/15)<ε>r, (3) since both <u22u− >=<u12u− >= −(1/15)<ε>r. Averaging of the third term (local interaction) gives: <−u−3 >= (1/10)<ε>r, (4) The sum of (3) and (4) leads to the Hosokawa relation (2). <u+2u−> = <ε>r/30, (2)
Local versus nonlocal contributions It is noteworthy that the 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+2u−> , but rather with u+2u−
Local versus nonlocal contributions In the figure below we show the experimental verification of the relation (3): ½< (u22 + u12) u−>=−(1/15)<ε>r. (3)
On the role of kinematic relations
On the role of kinematic relations 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 relations 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 Isotropic turbulence. Some asymmetric relations in terms of u1 and Δu or u2 and Δu.
On the role of kinematic relations n=4 <(Δu)4>= 4<u13Δu> +6<u12(Δu)2> <(Δu)4>= –4<u23Δu> + 6<u22(Δu)2> Any n≥2 <(Δu)n>= –2<u1(Δu)n–1> (7) <(Δu)n>= 2<u2(Δu)n–1> (8) Isotropic turbulence. Some asymmetric relations in terms of u1 and Δu or u2 and Δu.
On 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.
Sweeping decorrelation hypothesis Sweeping decorrelation hypothesis
Sweeping decorrelation hypothesis 〈(Δu)2〉=–2〈u1Δu〉=2〈u2Δu〉, 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 r2/3. 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)
Sweeping decorrelation hypothesis FollowingKraichnan 1964and Tennekes 1975there are two main ingredients in the (Eulerian) decorrelation: the sweeping of microstucture by the large scale motions (and associated kinematic nonlocality) and the local straining (which is roughly pure Lagrangian). It appears that this kind of “decomposition” is insufficient as it is missing an essential dynamical aspect - the interaction between the two as it is clearly demonstrated by the nonlocal version of the Kolmogorov 4/5 law. The random Taylor hypothesis (and, of course, the conventional Taylor hypothesis) lack/discard this aspect at the outset (this does not mean that these hypotheses are useless): both are “too kinematic”.
The main result is that purely kinematic exact relations demonstrate one of important aspects of non-locality of turbulent flows. Though purely kinematic they have important dynamical consequences. There is no exaggeration in saying that without nonlocality (understood as direct and bidirectionalcoupling of large and small scales) there is no turbulence. Though in some limited sense one can speak about separating the local and nonlocal contributions, generally such a separation seems impossible and even in some sense meaningless, since "what is local is also nonlocal". The kinematic relations stand in contradiction with the sweeping decorrelation hypothesis understood as statistical independence 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