Statistical Hydromechanics and Functional Calculus

Hopf, Eberhard

doi:10.1512/iumj.1952.1.51004

Cited by 164 publications

(246 citation statements)

References 0 publications

Supporting

Mentioning

238

Contrasting

Unclassified

Order By: Relevance

“…We then built on Hopf's [18] ideas to compute the invariant measure of turbulent flow. This measure can be computed because it solves a linear functional differential equation, see [33].…”

Section: Resultsmentioning

confidence: 99%

“…We used the lower estimate in (18) and this makes the estimate in (22) be an overestimate. The measure of the discrete process must be written as…”

Section: Estimates Of the Structure Functionsmentioning

confidence: 99%

“…This quantity determines all the statistical properties of the turbulent velocity field, see [33], and in distinction to the nonlinear Navier-Stokes equation, the invariant measure satisfies a linear but a functional differential equation, see [33]. In fact Hopf [18] …”

Section: The Invariant Measure Of Turbulencementioning

confidence: 99%

“…The statistical theory of turbulence is determined, including the scaling of the structure functions of turbulence, by the invariant measure of the Navier-Stokes equation and the PDFs for the various statistics (one-point, two-point,· · · , N-point) can be obtained by taking the trace of the corresponding invariant measures. Hopf [18] derived a functional equation for the characteristic function (Fourier transform) of the invariant measure. In distinction to the nonlinear Navier-Stokes equation, this is a linear functional differential equation.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Kolmogorov–Obukhov Statistical Theory of Turbulence

Birnir

2013

J Nonlinear Sci

View full text Add to dashboard Cite

In 1941 Kolmogorov and Obukhov proposed that there exists a statistical theory of turbulence that should allow the computation of all the statistical quantities that can be computed and measured in turbulent systems. These are quantities such as the moments, the structure functions and the probability density functions (PDFs) of the turbulent velocity field. In this paper we will outline how to construct this statistical theory from the stochastic Navier-Stokes equation. The additive noise in the stochastic Navier-Stokes equation is generic noise given by the central limit theorem and the large deviation principle. The multiplicative noise consists of jumps multiplying the velocity, modeling jumps in the velocity gradient. We first estimate the structure functions of turbulence and establish the Kolmogorov-Obukhov '62 scaling hypothesis with the She-Leveque intermittency corrections. Then we compute the invariant measure of turbulence writing the stochastic Navier-Stokes equation as an infinite-dimensional Ito process and solving the linear Kolmogorov-Hopf functional differential equation for the invariant measure. Finally we project the invariant measure onto the PDF. The PDFs turn out to be the normalized inverse Gaussian (NIG) distributions of Barndorff-Nilsen, and compare well with PDFs from simulations and experiments.

show abstract

“…We then built on Hopf's [18] ideas to compute the invariant measure of turbulent flow. This measure can be computed because it solves a linear functional differential equation, see [33].…”

Section: Resultsmentioning

confidence: 99%

“…We used the lower estimate in (18) and this makes the estimate in (22) be an overestimate. The measure of the discrete process must be written as…”

Section: Estimates Of the Structure Functionsmentioning

confidence: 99%

Section: The Invariant Measure Of Turbulencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Kolmogorov–Obukhov Statistical Theory of Turbulence

Birnir

2013

J Nonlinear Sci

View full text Add to dashboard Cite

show abstract

“…since again, according to (6), the transformations of x and u induce the complementary transformationsk = k andv = e −c * 2 k ϵv , which then, according to (10) with its Hermitian form (12), results toz = e c 2 k ϵ z as given in (2). Although this symmetry has been correctly recognized in Ref.…”

Section: The New Statistical Symmetries and Their Deterministic Ormentioning

confidence: 93%

Comment on “Application of the extended Lie group analysis to the Hopf functional formulation of the Burgers equation” [J. Math. Phys. 54, 072901 (2013)]

Frewer¹,

Khujadze

Foysi

2016

Journal of Mathematical Physics

View full text Add to dashboard Cite

The quest to find new statistical symmetries in the theory of turbulence is an ongoing research endeavor which is still in its beginning and exploratory stage. In our comment we show that the recently performed study of Wacławczyk and Oberlack [J. Math. Phys. 54, 072901 (2013)] failed to present such new statistical symmetries. Despite their existence within a functional Fourier space of the statistical Burgers equation, they all can be reduced to the classical and well-known symmetries of the underlying deterministic Burgers equation itself, except for one symmetry, but which, as we will demonstrate, is only a mathematical artefact without any physical meaning. Moreover, we show that the proposed connection between the translation invariance of the multi-point moments and a symmetry transformation associated to a certain invariant solution of the inviscid functional Burgers equation is invalid. In general, their study constructs and discusses new particular solutions of the functional Burgers equation without referring them to the well-established general solution. Finally, we also see a shortcoming in the presented methodology as being too restricted to construct a complete set of Lie point symmetries for functional equations. In particular, for the considered Burgers equation essential symmetries are not captured.

show abstract

Ülber die Hopfsehe Gleichung für eine mikropolare Flüssigkeit

Förste¹

1984

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

Statistical Hydromechanics and Functional Calculus

Cited by 164 publications

References 0 publications

The Kolmogorov–Obukhov Statistical Theory of Turbulence

The Kolmogorov–Obukhov Statistical Theory of Turbulence

Comment on “Application of the extended Lie group analysis to the Hopf functional formulation of the Burgers equation” [J. Math. Phys. 54, 072901 (2013)]

Ülber die Hopfsehe Gleichung für eine mikropolare Flüssigkeit

Contact Info

Product

Resources

About