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Kolmogorov’s Theory of Turbulence

It is impractical in climate models to give sufficient resolution in order to capture all of the inertial dynamics in the system. Inertial dynamics will dominate at large scales while on small scales viscous forces  will despite the energy associated with these dynamics.

The Reynolds number gives the ratio of inertial forces to viscous forces.  When viscous forces dominate it is known as laminar flow. Laminar flow occurs when Reynolds number is less then 10. If velocities is held constant and the length scale decreases then so does reynolds number and for this small region the flow will approach Laminar flow and average shear forces will be dominated by the viscosity.

When viscous forces dominate the shear forces are dominated by the momentum exchange though partial diffusion, when viscous forces don’t dominate then the bulk flow of fluid can give a momentum exchange that acts like an effective increase in viscosity.  Therefore when making large scale approximations it is important to know how the sub scale dynamics effect large scale properties such as the average shear forces.

One way we might do this is though Kolmogrov’s Theory of Turbulence:

A turbulent flow is characterized by a hierarchy of scales through which the energy cascade takes place. Dissipation of kinetic energy takes place at scales of the order of Kolmogorov length η, while the input of energy into the cascade comes from the decay of the large scales, of order L. These two scales at the extremes of the cascade can differ by several orders of magnitude at high Reynolds numbers. In between there is a range of scales (each one with its own characteristic length r) that has formed at the expense of the energy of the large ones. These scales are very large compared with the Kolmogorov length, but still very small compared with the large scale of the flow (i.e. \eta \ll r \ll L). Since eddies in this range are much larger than the dissipative eddies that exist at Kolmogorov scales, kinetic energy is essentially not dissipated in this range, and it is merely transferred to smaller scales until viscous effects become important as the order of the Kolmogorov scale is approached. Within this range inertial effects are still much larger than viscous effects, and it is possible to assume that viscosity does not play a role in their internal dynamics (for this reason this range is called “inertial range”).

http://en.wikipedia.org/wiki/Turbulence#Kolmogorov.27s_Theory_of_1941

Some other relevant links:

http://www.sjsu.edu/faculty/watkins/kolmo.htm
http://en.wikipedia.org/wiki/Kolmogorov_microscales
http://scienceworld.wolfram.com/physics/Turbulence.html

On a related curiosity, I am wondering how knowing the flow conditions at a discrete number of points effects the range of possible solutions to Navier Stokes Equations. I’ve posted a bit on this in the next two links:

http://www.climateaudit.org/phpBB3/viewtopic.php?f=4&t=760
http://www.physicsforums.com/showthread.php?t=331713

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August 24, 2009 - Posted by | Turbulence

14 Comments »

  1. “We were delighted to read Alexander Melnikov’s splendid
    account, in the last issue of π in the Sky, of the life of the great Russian mathematician Alexander Nikolaevich Kolmogorov. The main purpose of this short article is to describe how a study in B.C. waters provided the first and strongest evidence n support of Kolmogorov’s famous 1941 “Two-Thirds Law.”

    As nicely summarised by Melnikov, this law predicts that in every turbulent flow (such as we see in rushing rivers or vigorously boiling pans of water, for example), “the mean square difference of the velocities at two points a distance r apart is proportional to r^(2/3) (Figure 1).” Quite apart from the elegance of this result, it provides a key to understanding and predicting how pollutants and other substances are
    dispersed.”
    http://maelstrom.seos.uvic.ca/people/garrett/abstracts/page22-23.pdf

    Comment by s243a | August 24, 2009 | Reply

  2. “RG Analysis of the Kolmogorov Cascade in Isotropic Turbulence
    David Benjamin
    Department of Physics, Harvard University
    (Dated: May 12, 2008)

    We review the application of renormalization group (RG) methods to the problem of isotropic turbulence, with attention to Kolmogorov’s phenomenological scaling law. We see that RG reproduces the form of the 5/3 law and can also be used to obtain estimates of the Kolmogorov constant that
    agree with experiment and numerical simulation. The articles reviewed also confirm the universality of the scaling law and the constant. Finally, we posit a new form for correlations in Fourier space motivated by Kolmogorov’s theory and use it to apply RG to a modified Navier-Stokes equation.”
    http://web.mit.edu/8.334/www/grades/projects/DavidBenjamin.pdf

    Comment by s243a | August 24, 2009 | Reply

  3. As a side note I want to make a general observation here. A particle traveling in a circle or ellipse will have periodic components for it’s position and velocity when expressed in Cartesian components.

    Comment by s243a | August 24, 2009 | Reply

  4. “Weak Turbulent Kolmogorov Spectrum for Surface GravityWaves
    A. I. Dyachenko,1,* A.O. Korotkevich,1,† and V. E. Zakharov1,2,3

    We study the long-time evolution of surface gravity waves on deep water excited by a stochastic external force concentrated in moderately small wave numbers. We numerically implemented the primitive Euler equations for the potential flow of an ideal fluid with free surface written in Hamiltonian canonical variables, using the expansion of the Hamiltonian in powers of nonlinearity of terms up to fourth order.We show that because of nonlinear interaction processes a stationary Fourier spectrum of a surface elevation close to hjkj2i  k7=2 is formed. The observed spectrum can be interpreted as a weak-turbulent Kolmogorov spectrum for a direct cascade of energy.”
    http://math.arizona.edu/~akorotkevich/textfiles/PRLKolmogorov.pdf

    Comment by s243a | August 24, 2009 | Reply

  5. “Examination of hypotheses in
    the Kolmogorov re ned turbulence theory
    through high-resolution simulations.
    Part 2. Passive scalar field

    By LIAN-PING WANG1y, SHIYI CHEN2
    AND JAMES G. BRASSEUR3

    Using direct numerical simulations (DNS) and large-eddy simulations (LES) of velocity and passive scalar in isotropic turbulence (up to 5123 grid points), we examine directly and quantitatively the re ned similarity hypotheses as applied to passive scalar fi elds (RSHP) with Prandtl number of order one. Unlike previous experimental investigations,
    exact energy and scalar dissipation rates were used and scaling exponents were quanti ed as a function of local Reynolds number. We rst demonstrate that the forced DNS and LES scalar elds exhibit realistic inertial-range dynamics and that the statistical characteristics compare well with other numerical, theoretical and experimental studies. The Obukhov{Corrsin constant for the k−5=3 scalar variance
    spectrum obtained from the 5123 mesh is 0:87  0:10. Various statistics indicated that the scalar eld is more intermittent than the velocity eld. The joint probability distribution of locally-averaged energy dissipation r and scalar dissipation r is close to log-normal with a correlation coecient of 0:25  0:01 between the logarithmic
    dissipations in the inertial subrange. The intermittency parameter for scalar dissipation is estimated to be in the range 0:43  0:77, based on direct calculations of the variance of ln r . The scaling exponents of the conditional scalar increment rjr ;r suggest a tendency to follow RSHP. Most signi cantly, the scaling exponent of rjr ;r over r was shown to be approximately −16 in the inertial subrange, con-
    rming a dynamical aspect of RSHP. In agreement with recent experimental results (Zhu et al. 1995; Stolovitzky et al. 1995), the probability distributions of the random variable s = rjr ;r=(1=2
    r −1=6 r r1=3) were found to be nearly Gaussian. However,
    contrary to the experimental results, we nd that the moments of s are almost identical to those for the velocity eld found in Part 1 of this study (Wang et al. 1996) and are insensitive to Reynolds number, large-scale forcing, and subgrid modelling. y Author to whom

    http://www.mne.psu.edu/BrasseurLab/Publications/Turbulence_Publications/Wang,Chen,Brasseur-JFM1999.pdf

    Comment by s243a | August 24, 2009 | Reply

  6. The following link is on Taylor theory of turbulent diffusion:

    http://envhydro.yonsei.ac.kr/lecture/Environmental%20Hydrodynamics/turbulence%20diffusion.pdf

    Comment by s243a | August 24, 2009 | Reply

  7. “Turbulence research at large Reynolds numbers using
    high resolution DNS
    Toshiyuki Gotoh∗
    Department of Systems Engineering, Nagoya Institute of Technology

    Introduction
    Wind, water flowing in a river, ocean currents, and air motion behind a moving car are very familiar to us, and we know that they are always turbulent flows, because the velocity and pressure of the fluid fluctuate with time and position and the flow pattern is random. Humans recognized fluid motion in antiquity; they utilized this motion and recorded it in various ways, such as in paintings by Hokusai Katsusika and Leonard da Vinci, and in literature. Nowadays, the importance of turbulence in scientific, engineering, and environmental contexts, such as in weather prediction, aircraft design, and air pollution, is widely recognized.

    Although turbulence is routinely encountered in everyday life, our understanding of it from a scientific perspective is still limited, and turbulence is a notorious problem in physics. The most successful theory of turbulence is that of Kolmogorov.1,2) He studied the statistical laws of the velocity field at small scale in turbulence at very high Reynolds numbers under two hypotheses: 1) local isotropy and homogeneity, and 2) the existence of a range independent of viscosity and large-scale properties at sufficiently large Reynolds numbers. The results were the scaling laws for the structure functions of velocity increments”
    http://www.riken.go.jp/lab-www/library/publication/review/pdf/No_40/40_003.pdf

    Comment by s243a | August 24, 2009 | Reply

  8. “KOLMOGOROV CASCADE
    The velocity fluctuations of a high Reynolds number
    flow in a three-dimensional velocity field are typically
    dispersed over all possible wavelengths of the system,
    from the smallest scales, where viscosity dominates the
    advection and dissipates the energy of fluid motion, to
    the effective size of the system. This is not so bizarre:
    our everyday experience tells us it is so. On the corner
    of a city street, one might watch the fluttering and
    whirling of a discarded tram ticket as it is swept by an
    updraught, driven by localized thermal gradients from
    traffic or air-conditioning units; later, on the television
    news, one might see reports or predictions of storms on
    the city or district scale, and a weather map with isobars
    spanning whole continents. If you are a sailor you will
    know how to sail, or not, the multi-scaled surface of
    a turbulent ocean (Figure 1). The mechanism for this
    dispersal is vortex stretching and tilting: a conservative
    process whereby interactions between vorticity and
    velocity gradients create smaller and smaller eddies
    with amplified vorticity, until viscosity takes over
    (Tennekes & Lumley, 1972; Chorin, 1994).”

    http://wwwrsphysse.anu.edu.au/~rxb105/cascade.pdf

    Comment by s243a | August 24, 2009 | Reply

  9. “A Quasi-Normal Scale Elimination (QNSE) theory of turbulent flows with
    stable stratification
    Sukoriansky Semion & Galperin Boris

    Abstract :
    A new spectral model of turbulent flows with stable stratification is presented. The theory is based upon a mapping of the actual velocity field to a quasi-Gaussian field using the Langevin equation. The parameters of the mapping are calculated using a systematic process of successive averaging over small shells of velocity and temperature modes that eliminates them from the equations of motion. This procedure does not differentiate between turbulence and internal waves and accounts for their combined effect. This approach offers a powerful mathematical tool for dealing with previously nearly intractable
    aspects of anisotropic turbulence; among these aspects are the threshold criterion for generation of internal waves and the modification of their dispersion relation by turbulence. The process of successive
    small scales elimination results in a model describing the largest scales of a flow. Partial scale elimination yields subgrid-scale viscosities and diffusivities that can be used in large eddy simulations. The elimination of all fluctuating scales results in RANS models. The model predicts various important characteristics of stably stratified flows, such as the dependence of the vertical turbulent Prandtl number on Froude and Richardson numbers, anisotropization of the flow filed, and decay of vertical diffusivity under strong stratification, all in good agreement with computational and observational data. The theory also yields analytical expressions for various 1D and 3D kinetic and potential energy spectra that reflect the effects of waves and anisotropy. The model’s results are suitable for immediate use in practical applications.”
    http://documents.irevues.inist.fr/bitstream/2042/15586/1/CFM2007-0624.pdf

    Comment by s243a | August 24, 2009 | Reply

  10. “Introduction and
    Overview of Turbulence

    Introduction
    In this chapter we first briefly recall, in Section 1, the derivation of the Navier–Stokes equations (NSE)starting from the basic conservation principles in mechanics: conservation of mass and momentum. Section 2 contains some general remarks on turbulence, and it alludes to some developments not presented in the book. For the benefit
    of the mathematically oriented reader (and perhaps others), Section 3 provides a fairly detailed account of the Kolmogorov theory of turbulence, which underlies many parts
    of Chapters III–V. For the physics-oriented reader, Section 4 gives an intuitive introduction to the mathematical perspective and the necessary tools. A more rigorous
    presentation appears in the first half of Chapter II and thereafter as needed. For each of the aspects that we develop, the present chapter should prove more useful for the nonspecialist than for the specialist.

    http://assets.cambridge.org/97805213/60326/excerpt/9780521360326_excerpt.pdf

    Comment by s243a | August 24, 2009 | Reply

  11. “Singularities in Multifractal Turbulence—Dissipation
    Networks and Their Degeneration
    A. Bershadskii¤ and C. H. Gibson¤¤

    Abstract
    We suggest that large-scale turbulence dissipation is concentrated along caustic networks (that appear due to vortex sheet instability in three-dimensional space),
    leading to an effective fractal dimension Deff = 5/3 of the network backbone and a turbulence intermittency exponent μ = 1/6. Actually, Deff 1/6 due to singularities on these caustic networks. It is shown (using the theory of caustic singularities) that the strongest (however, stable on the backbone) singularities lead
    to Deff = 4/3 (an elastic backbone) and to μ = 1/3. Thus, there is a restriction of the network fractal variability: 4/3 < Deff < 5/3, and consequently: 1/6 < μ < 1/3.
    Degeneration of these networks into a system of smooth vortex filaments: Deff = 1, leads to μ = 1/2. After degeneration, the strongest singularities of the dissipation
    field, ", lose their power-law form, while the smoother field ln" takes it. It is shown (using the method of multifractal asymptotics) that the probability distribution of the dissipation changes its form from exponential-like to log-normal-like with this degeneration, and that the multifractal asymptote of the field ln" is related to the
    multifractal asymptote of the energy field. Finally, a phenomenon of acceleration of large-scale turbulent diffusion of passive scalar by the singularities is briefly discussed. All results are based on experimental data."
    http://sdcc3.ucsd.edu/~ir118/BG.pdf

    Comment by s243a | August 24, 2009 | Reply

  12. For reference an old post I made about fractals and turbulence:

    http://www.climateaudit.org/phpBB3/viewtopic.php?f=4&t=112

    The thread doesn’t have much posts in it but has a useful link to a lot of intense mathematical papers:

    http://www.me.jhu.edu/meneveau/pubs-fractals.html

    Comment by s243a | August 27, 2009 | Reply

  13. […] now this is just a place holder to discuss topics about fractals. For related topics see my post on Kolmogorov’s Turbulance. I would like to present the quote though to illustrate the difficulty of using numerical methods […]

    Pingback by Fractal Modeling of Turbulence « Earth Cubed | August 27, 2009 | Reply

    • Lol, I pinged myself. That just sounds funny.

      Comment by s243a | August 28, 2009 | Reply


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