Chandrasekhar's perturbation method-oriented theories up to third order - Structure parameters for rotating polytropic stars

1988 ◽  
Vol 66 ◽  
pp. 139 ◽  
Author(s):  
V. S. Geroyannis ◽  
F. N. Valvi
Keyword(s):  
Perturbation Method ◽  
Order Structure ◽  
Structure Parameters ◽  
Third Order

Exact third-order structure functions for two-dimensional turbulence

2018 ◽  
Vol 851 ◽  
pp. 672-686 ◽  
Author(s):  
Jin-Han Xie ◽  
Oliver Bühler
Keyword(s):  
Structure Functions ◽  
Spectral Energy ◽  
Order Structure ◽  
Two Dimensional ◽  
Third Order ◽  
Transfer Rates ◽  
The Third ◽  
Random Forcing ◽  
Asymptotic Expressions ◽  
Special Case

We derive and investigate exact expressions for third-order structure functions in stationary isotropic two-dimensional turbulence, assuming a statistical balance between random forcing and dissipation both at small and large scales. Our results extend previously derived asymptotic expressions in the enstrophy and energy inertial ranges by providing uniformly valid expressions that apply across the entire non-dissipative range, which, importantly, includes the forcing scales. In the special case of white noise in time forcing this leads to explicit predictions for the third-order structure functions, which are successfully tested against previously published high-resolution numerical simulations. We also consider spectral energy transfer rates and suggest and test a simple robust diagnostic formula that is useful when forcing is applied at more than one scale.

A note on Kolmogorov's third-order structure-function law, the local isotropy hypothesis and the pressure–velocity correlation

1996 ◽  
Vol 326 ◽  
pp. 343-356 ◽  
Author(s):  
Erik Lindborg
Keyword(s):  
Structure Function ◽  
Strain Tensor ◽  
Inertial Range ◽  
Order Structure ◽  
Two Dimensional ◽  
Mean Flow ◽  
Velocity Correlation ◽  
Third Order ◽  
The Third ◽  
Point Pressure

We show that Kolmogorov's (1941b) inertial-range law for the third-order structure function can be derived from a dynamical equation including pressure terms and mean flow gradient terms. A new inertial-range law, relating the two-point pressure–velocity correlation to the single-point pressure–strain tensor, is also derived. This law shows that the two-point pressure–velocity correlation, just like the third-order structure function, grows linearly with the separation distance in the inertial range. The physical meaning of both this law and Kolmogorov's law is illustrated by a Fourier analysis. An inertial-range law is also derived for the third-order velocity–enstrophy structure function of two-dimensional turbulence. It is suggested that the second-order vorticity structure function of two-dimensional turbulence is constant and scales with$\epsilon ^{2/3}_\omega$in the enstrophy inertial range, εωbeing the enstrophy dissipation. Owing to the constancy of this law, it does not imply a Fourier-space inertial-range law, and therefore it is not equivalent to thek−1law for the enstrophy spectrum, suggested by Kraichnan (1967) and Batchelor (1969).

scholarly journals Kappa Distributions and Isotropic Turbulence

Entropy ◽  
2019 ◽  
Vol 21 (11) ◽  
pp. 1093 ◽  
Author(s):  
Elias Gravanis ◽  
Evangelos Akylas ◽  
Constantinos Panagiotou ◽  
George Livadiotis
Keyword(s):  
Minimal Model ◽  
Model Comparison ◽  
Isotropic Turbulence ◽  
Order Structure ◽  
Kappa Distribution ◽  
Symmetric Part ◽  
Kappa Index ◽  
Third Order ◽  
Symmetry Properties ◽  
Temperature Parameter

In this work, the two-point probability density function (PDF) for the velocity field of isotropic turbulence is modeled using the kappa distribution and the concept of superstatistics. The PDF consists of a symmetric and an anti-symmetric part, whose symmetry properties follow from the reflection symmetry of isotropic turbulence, and the associated non-trivial conditions are established. The symmetric part is modeled by the kappa distribution. The anti-symmetric part, constructed in the context of superstatistics, is a novel function whose simplest form (called “the minimal model”) is solely dictated by the symmetry conditions. We obtain that the ensemble of eddies of size up to a given length r has a temperature parameter given by the second order structure function and a kappa-index related to the second and the third order structure functions. The latter relationship depends on the inverse temperature parameter (gamma) distribution of the superstatistics and it is not specific to the minimal model. Comparison with data from direct numerical simulations (DNS) of turbulence shows that our model is applicable within the dissipation subrange of scales. Also, the derived PDF of the velocity gradient shows excellent agreement with the DNS in six orders of magnitude. Future developments, in the context of superstatistics, are also discussed.

Some statistical properties of small scale turbulence in an atmospheric boundary layer

1970 ◽  
Vol 41 (1) ◽  
pp. 141-152 ◽  
Author(s):  
R. W. Stewart ◽  
J. R. Wilson ◽  
R. W. Burling
Keyword(s):  
Boundary Layer ◽  
Lognormal Distribution ◽  
A Priori ◽  
Inertial Subrange ◽  
Small Scale ◽  
Order Structure ◽  
Third Order ◽  
The Difference ◽  
Scale Turbulence ◽  
Derivatives Of

Derivatives of velocity signals obtained in a turbulent boundary layer are examined for correspondence to the lognormal distribution. It is found that there is rough agreement but that unlikely events at high values are much less common in the observed fields than would be inferred from the lognormal distribution. The actual distributions correspond more to those obtained from a random walk with a limited number of steps, so the difference between these distributions and the lognormal may be related to the fact that the Reynolds number is finite.The third-order structure function is examined, and found to be roughly consistent with the existence of an inertial subrange of a Kolmogoroff equilibrium reacute;gime over a range of scale which is a priori reasonable but which is far less extensive than the $-\frac{5}{3}$ region of the spectrum.

Structure functions of turbulence in the atmospheric boundary layer over the ocean

1970 ◽  
Vol 44 (1) ◽  
pp. 145-159 ◽  
Author(s):  
C. W. Van Atta ◽  
W. Y. Chen
Keyword(s):  
Boundary Layer ◽  
Atmospheric Boundary Layer ◽  
Fourth Order ◽  
Structure Functions ◽  
Inertial Subrange ◽  
Order Structure ◽  
Third Order ◽  
Wide Range ◽  
Theoretical Predictions ◽  
Order Quantities

Structure functions of turbulent velocity fluctuations up to fourth order have been measured at several heights in the atmospheric boundary layer over the open ocean, and the results are compared with theoretical predictions for separations in the inertial subrange. The behaviour of second- and third-order quantities shows substantial agreement with the predictions of Kolmogorov's original theory over a wide range of separations, but the results of a recent modification of the theory, attempting to account for intermittency in the local dissipation rate, are also consistent with the data over somewhat shorter separation intervals. The behaviour of the measured fourth-order structure function disagrees with that predicted from Kolmogorov's original work, but good agreement is found with the results of the modified theory.

scholarly journals Shallow water wave turbulence

2019 ◽  
Vol 874 ◽  
pp. 1169-1196 ◽  
Author(s):  
Pierre Augier ◽  
Ashwin Vishnu Mohanan ◽  
Erik Lindborg
Keyword(s):  
Reynolds Number ◽  
Structure Function ◽  
Shallow Water ◽  
Water Wave ◽  
Structure Functions ◽  
Wave Turbulence ◽  
Order Structure ◽  
Third Order ◽  
Shallow Water Wave ◽  
The Mean

The dynamics of irrotational shallow water wave turbulence forced at large scales and dissipated at small scales is investigated. First, we derive the shallow water analogue of the ‘four-fifths law’ of Kolmogorov turbulence for a third-order structure function involving velocity and displacement increments. Using this relation and assuming that the flow is dominated by shocks, we develop a simple model predicting that the shock amplitude scales as $(\unicode[STIX]{x1D716}d)^{1/3}$, where $\unicode[STIX]{x1D716}$ is the mean dissipation rate and $d$ the mean distance between the shocks, and that the $p$th-order displacement and velocity structure functions scale as $(\unicode[STIX]{x1D716}d)^{p/3}r/d$, where $r$ is the separation. Then we carry out a series of forced simulations with resolutions up to $7680^{2}$, varying the Froude number, $F_{f}=(\unicode[STIX]{x1D716}L_{f})^{1/3}/c$, where $L_{f}$ is the forcing length scale and $c$ is the wave speed. In all simulations a stationary state is reached in which there is a constant spectral energy flux and equipartition between kinetic and potential energy in the constant flux range. The third-order structure function relation is satisfied with a high degree of accuracy. Mean energy is found to scale approximately as $E\sim \sqrt{\unicode[STIX]{x1D716}L_{f}c}$, and is also dependent on resolution, indicating that shallow water wave turbulence does not fit into the paradigm of a Richardson–Kolmogorov cascade. In all simulations shocks develop, displayed as long thin bands of negative divergence in flow visualisations. The mean distance between the shocks is found to scale as $d\sim F_{f}^{1/2}L_{f}$. Structure functions of second and higher order are found to scale in good agreement with the model. We conclude that in the weak limit, $F_{f}\rightarrow 0$, shocks will become denser and weaker and finally disappear for a finite Reynolds number. On the other hand, for a given $F_{f}$, no matter how small, shocks will prevail if the Reynolds number is sufficiently large.

Applicability of Kolmogorov's and Monin's equations of turbulence

1997 ◽  
Vol 353 ◽  
pp. 67-81 ◽  
Author(s):  
REGINALD J. HILL
Keyword(s):  
Structure Function ◽  
Structure Functions ◽  
Inertial Range ◽  
Homogeneous Turbulence ◽  
Order Structure ◽  
Third Order ◽  
Local Isotropy ◽  
The Third ◽  
Moderate Reynolds Number ◽  
Wide Range

The equation relating second- and third-order velocity structure functions was presented by Kolmogorov; Monin attempted to derive that equation on the basis of local isotropy. Recently, concerns have been raised to the effect that Kolmogorov's equation and an ancillary incompressibility condition governing the third-order structure function were proven only on the restrictive basis of isotropy and that the statistic involving pressure that appears in the derivation of Kolmogorov's equation might not vanish on the basis of local isotropy. These concerns are resolved. In so doing, results are obtained for the second- and third-order statistics on the basis of local homogeneity without use of local isotropy. These results are applicable to future studies of the approach toward local isotropy. Accuracy of Kolmogorov's equation is shown to be more sensitive to anisotropy of the third-order structure function than to anisotropy of the second-order structure function. Kolmogorov's 4/5 law for the inertial range of the third-order structure function is obtained without use of the incompressibility conditions on the second- and third-order structure functions. A generalization of Kolmogorov's 4/5 law, which applies to the inertial range of locally homogeneous turbulence at very large Reynolds numbers, is shown to also apply to the energy-containing range for the more restrictive case of stationary, homogeneous turbulence. The variety of derivations of Kolmogorov's and Monin's equations leads to a wide range of applicability to experimental conditions, including, in some cases, turbulence of moderate Reynolds number.

Exact relationship for third-order structure functions in helical flows

2000 ◽  
Vol 61 (5) ◽  
pp. 5321-5325 ◽  
Author(s):  
T. Gomez ◽  
H. Politano ◽  
A. Pouquet
Keyword(s):  
Structure Functions ◽  
Order Structure ◽  
Third Order ◽  
Exact Relationship
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