The Statistical Dynamics of Turbulence

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It should be of interest to engineers, while the discussion of the underlying tools, namely pdfs, stochastic and statistical equations should also be attractive to applied mathematicians and physicists. The book's emphasis on local pdfs and stochastic Langevin models gives a consistent structure to the book and allows the author to cover almost the whole spectrum of practical modelling in turbulent CFD.

In short, the book is orientated more towards applications than towards turbulence theory; it is written clearly and concisely and should be useful to a large community, interested either in the underlying stochastic formalism or in CFD applications. Considering the book's relatively modest price it would be a useful addition for personal as well as institutional acquisition. Buy this book from :. Taylor [A nice scientific biography of one of the founders of modern mechanics, and of the statistical theory of turbulence; review ] Pierre Berge et al.

Chorin, Vorticity and Turbulence ["This book provides an introduction to turbulence in vortex systems, and to turbulence theory for incompressible flow described in terms of the vorticity field. It is the author's hope that by the end of the book the reader will believe these subjects are identical, and constitute a special case of fairly standard statistical mechanics, with both equilibrium and non-equilibrium aspects.

Kolmogorov [An excellent introduction, very strong on defending Kolmogorov's work from mis-understandings and invalid criticisms. L'vov and Procaccia then go on to describe and extol their particular strategy, which is to try to make field theory work. Assumes little by way of physical or hydrodynamical knowledge, but a good bit of mathematical maturity.

Tennekes and J.

Jayanta Bhattacharjee - Introduction to fluid dynamics and turbulence (5)

Barndorff-Nielsen, J. Jensen and M. Who knew? Chapman, G. Rowlands and Nick W. Crawford, Nicolas Mordant, Andy M. IV Gregory L. Eyink and Katepalli R.

Statistical mechanics of turbulent flows

Sreenivasan, "Onsager and the theory of hydrodynamic turbulence", Reviews of Modern Physics 78 : [ Things like this aren't supposed to happen in real life. Well, they don't do any formal tests of goodness of fit, but by eye it's pretty good.

Shats, H. Xia and H. Tabeling and O. Cardoso eds. Aringazin and M. Arnol'd, Topological Methods in Hydrodynamics A. Bershadskii, J. Niemela, A. | Statistical Mechanics of Turbulent Flows | | Stefan Heinz | Boeken

Praskovsky and K. Bandi, J. Cressman Jr. Goldburg, "Test of the Fluctuation Relation in compressible turbulence on a free surface", nlin. Bandi, W. Goldburg, J. Cressman Jr, "Measurement of entropy production rate in compressible turbulence", nlin. Barenblatt and Alexandre J. Batchelor et al eds. Biferale, G.

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Boffetta, A. Celani, A. Lanotte and F. Toschi "Lagrangian statistics in fully developed turbulence", nlin. Catrakis and Paul E. Equation 5 expresses the dynamics of the fluctuating transported variable in terms of the turbulent oscillations and of some main flow features. Integrating Eq. Equation 9 is called statistical dynamic equation Pantoja, JFAO, private communication for the turbulent transport of a scalar in fluid. Term I of Eq. Therefore, it must be null to obey Galilean relativity.

Term II is dependent on the gradient of the transported variable, and so is directly associated to the gradient diffusion hypothesis. The net turbulent transport correspondent to term II ,. According to Eq. The gradient diffusion hypothesis would be demonstrated if one could prove that term II predominates over the other terms on the right hand side of Eq. However, oppositely, term II can be proved to vanish, as follows.

So, no one-direction second order differential remains in term II , i. As a corollary, the whole term II must be null, because a cross-derivative turns into one-direction second order derivatives by a rotation of axis.

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  7. If a correlation is assumed between the turbulent transport and a cross-derivative of the scalar field, a rotation of axis will produce a correlation between turbulent transport and the one-directional derivatives, which would be contrary to the previous conclusion. Therefore, the unique non-vanishing advective term in Eq. Term IV derives from the viscous transport, and is possibly small compared to the advective term III , since the turbulent transport is dominated by the large eddies, whose dynamic is essentially inviscid, what implies negligibly conductive flows for gases and most liquids, possibly excluding the liquid metals for their very low Prandtl numbers.

    The source related term V can only be discussed in specific situations, it is null in transport of heat and of non-reacting contaminants.

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    Therefore, term III appears to be the most general component of the turbulent transport. If the rigorous gradient term II was dismissed, a gradient model for shear layers may be derived from term III by recourse to the basic ideas of the mixing length model, as follows. Summing up, the dynamic equation approach allows a gradient model to be justified with no need to analogies with molecular transport phenomena, only by recourse to widely accepted approximations, upon an analytic expression that can also be assessed by direct numerical simulations or experimentally.

    Submitting the Navier-Stokes equation to the Reynolds decomposition and taking the mean, one obtains the Reynolds averaged momentum equation Eq. A dynamic equation for the Reynolds stress tensor u j u i will be obtained according to the methodology employed above. It is convenient to depart from the momentum equation using k as dummy index Eq. The Reynolds decomposition employs Eq.

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    The initial developments are analogous to the scalar transport case, from Eq. Analogous to the scalar transport case, term I of Eq. Following a usual procedure, symmetry in each term of the Reynolds stress decomposition is obtained by taking the mean between Eq. It results:. The nil terms I and II were maintained in Eq. Analogous to the scalar case, the viscous transport term IV is possibly negligible compared to term III. Pressure term V poses the major modeling difficulty, and its relevance may be comparable to term III.

    Also in analogy to the scalar case, the mixing length gradient model for shear layers can be recovered by means of term III. Then the term III of Eq. It is clear that the usual extrapolation of the gradient hypothesis to relate shear and strain rate in the form analogous to term II of Eq.

    For steady state mean motions, the kinetic energy transport equation is see, for instance, Hinze :. Dismissing terms I and II , the turbulent transport Eq. It results in Eq. As already recalled, because of the essentially inviscid dynamics of the large eddies, the viscous term IV is probably negligible compared to the advective term III. Also, according to the known energy budgets in wall shear flows, as reported, for instance, by Pope , the pressure fluctuations have minor significance in the energy balance, suggesting term V to be small as well. Term VII is likely to vanish because its integrand is related to dissipation, which occurs at the small scale, high vorticity eddies, and so the integral function does not correlate with the large scale variations of u j that dominate the turbulent transport.

    This conclusion agrees with the recommendation of Younis et al.

    The Statistical Dynamics of Turbulence

    Indeed, both their approach and the present one are based on equations derived, although distinctly, from the same Navier-Stokes equations, so containing corresponding terms. As previously discussed, term III can be associated to the gradient hypothesis by means of the basic ideas of the mixing length model. The best-known gradient model is the isotropic Eq. In order to cope with strongly non-isotropic turbulent flows, Daly and Harlow proposed a generalized gradient model Eq.

    Coefficient C s is found in the literature in the range 0. The direct extension of Daly and Harlow's generalized gradient model for the transport of specific Reynolds stress components is not rotationally invariant. Generalized gradient models for those triple velocity correlations, which do obey the condition of indifference to coordinate frame, were proposed afterwards.

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    The best known is Hanjalic and Launder's model Eq. Clearly, this form introduces gradients of the normal and shear stresses together with gradients of the kinetic energy. This fact prevents the use of Hanjalic and Launder's model to represent gradient part, term III , of the present Statistical Dynamic model, since no dependence on stresses is foreseen in such method. Furthermore, as will be seen in next section, Daly and Harlow's model presents the best results among the gradient models for the strongly anisotropic wall region.