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Taking the dissipation to be quadratic, when expressed in terms of the Fourier coefficients, and constraining the statistics to respect an average balance between forcing and dissipation, he applied the techniques of statistical mechanics and concluded that the dissipation is equally partitioned among the Fourier components.
This conclusion was both interesting and problematic. Equipartition of dissipation leads to an unphysical infinite total dissipation if the phase space of the system is infinite dimensional. Quantum mechanics does not come to the rescue here as it had done earlier when an analogous problem arose in the statistical mechanics of electromagnetic radiation. Despite a series of publications , many of which are reprinted in the memorial volume by Nieuwstadt and Steketee 2 , a completely satisfying solution did not emerge and Burgers finally abandoned the subject.
Several years later Onsager 3 took it up again but decided to pursue a course that is more in line with equilibrium statistical mechanics, as detailed in the review article by Eyink and Sreenivasan 4. It will thus be investigated whether statistical mechanics can be used to deal with forced-dissipative turbulent systems, using as a basic assumption that the statistics is controlled by an average balance between forcing and dissipation.
It deserves attention by the specialists and the beginners as well and has useful applications. Achtergrond Statistical mechanics of turbulent flows Wim Verkley. Stochastic models for small-scale turbulence are addressed in chapter 6. In panel a of Figure 3 we show the vorticity and zonally averaged zonal velocity obtained numerically, averaged over the last days of the integration. 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. The vorticity fields in the consecutive panels vary between
The problem of the infinite dissipation is not resolved but moderated by limiting ourselves to finitely truncated spectral representations of fluid flows. We will phrase the theory in the language of probability theory and the principle of maximum entropy, as advocated by Jaynes 6. The method will be applied to a simple one-layer model of the large-scale atmospheric circulation. The model to be considered describes the motion of a single layer of incompressible fluid on the surface of a rotating sphere.
Orography is taken into account and the flow is assumed to be geostrophically balanced and thus approximately governed by the horizontal advection of quasigeostrophic potential vorticity. The equation that is used, is a somewhat simplified version of an equation discussed by the author 7. The system is forced by relaxation towards a zonally symmetric circulation that consists of jet-streams in both hemispheres, and is damped by a term that has the same structure as the viscosity term in fluid dynamics. By projecting the advection equation of potential vorticity onto the finite set of spherical harmonics, one obtains a dynamical system of quadratically non-linear equations in the Fourier coefficients.
When integrated numerically, this finite-dimensional dynamical system displays chaotic turbulent motion, not unlike what is seen in large-scale atmospheric flow.
To demonstrate this, we show in Figure 1 two snapshots of the vorticity and the zonally averaged zonal velocity, separated by 10 days in time, at the end of an integration of days. This has been shown 5 to work rather well if the statistics is controlled by conservation of energy and enstrophy, i.
Fortunately, the mathematics is similar in both cases because all constraints are quadratic and leads to a probability density function that is a product of normal distributions. Once the probability density function is known, all relevant statistics can be calculated, such as spectra of energy and enstropy and average vorticity fields. The vorticity field at day and day a and b, respectively in a numerical integration of the spectral model of large-scale atmospheric flow.
The colour scale is from blue low values to red high values.
The profiles to the right of the vorticity fields are the zonally averaged zonal velocity in meters per second. The solid dots represent the spectra of energy, the open circles represent the spectra of enstrophy and the solid curves are the theoretical spectra, based on maximum entropy.
In the upper panel a the constraints in the maximization of entropy are energy and enstrophy, in the middle panel b the constraints are the decay rates of energy and enstrophy taken to be zero , and in the lower panel c both energy and enstrophy as well as their decay rates are used as constraints. Zonally averaged zonal velocity profiles in meters per second are displayed to the right of the vorticity fields. Panels b, c and d show the theoretical averages, based on maximization of entropy.
The constraints in b are energy and enstrophy, in c the zero decay rates of energy and enstrophy and in d both energy and enstrophy and their zero decay rates.
The vorticity fields in the consecutive panels vary between We will focus on the numerical integration of days of which two snapshots of vorticity have been shown in Figure 1. We use the last days of the integration to calculate average spectra, vorticity and velocity fields in order to compare these with the theoretical results. Whereas this has been shown to work quite well in the unforced-undamped case, in the forced-damped case that we consider here it does not work at all.
The numerically obtained spectra are very different from the spectra that characterize the unforced-undamped case. In contrast to the case displayed in panel a of Figure 2, we do not need any information from the numerical run except for the fact that it has reached a state of statistical equilibrium.
The resemblance between the theoretical and numerical spectra is nevertheless substantially better that in the case of panel a of Figure 2, in particular for the lower values of the wavenumber n. Journal of Physics A: Mathematical and General , Volume 37 , Number Sign up for new issue notifications. This is a handbook for a computational approach to reacting flows, including background material on statistical mechanics.
In this sense, the title is somewhat misleading with respect to other books dedicated to the statistical theory of turbulence e. In the present book, emphasis is placed on modelling engineering closures for computational fluid dynamics. The probabilistic pdf approach is applied to the local scalar field, motivated first by the nonlinearity of chemical source terms which appear in the transport equations of reacting species. The probabilistic and stochastic approaches are also used for the velocity field and particle position; nevertheless they are essentially limited to Lagrangian models for a local vector, with only single-point statistics, as for the scalar.
Accordingly, conventional techniques, such as single-point closures for RANS Reynolds-averaged Navier-Stokes and subgrid-scale models for LES large-eddy simulations , are described and in some cases reformulated using underlying Langevin models and filtered pdfs. Even if the theoretical approach to turbulence is not discussed in general, the essentials of probabilistic and stochastic-processes methods are described, with a useful reminder concerning statistics at the molecular level.
The book comprises 7 chapters. Chapter 1 briefly states the goals and contents, with a very clear synoptic scheme on page 2. Chapter 3 deals with stochastic processes, pdf transport equations, from Kramer-Moyal to Fokker-Planck for Markov processes , and moments equations. Stochastic differential equations are introduced and their relationship to pdfs described. This chapter ends with a discussion of stochastic modelling. The equations of fluid mechanics and thermodynamics are addressed in chapter 4.
Classical conservation equations mass, velocity, internal energy are derived from their counterparts at the molecular level. In addition, equations are given for multicomponent reacting systems. Chapter 5 is devoted to stochastic models for the large scales of turbulence.
Langevin-type models for velocity and particle position are presented, and their various consequences for second-order single-point corelations Reynolds stress components, Kolmogorov constant are discussed.