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.

Two-dimensional isotropic inertia–gravity wave turbulence

2019 ◽  
Vol 872 ◽  
pp. 752-783 ◽  
Author(s):  
Jin-Han Xie ◽  
Oliver Bühler
Keyword(s):  
Total Energy ◽  
Isotropic Turbulence ◽  
Vertical Plane ◽  
Boussinesq Equations ◽  
Structure Functions ◽  
Wave Turbulence ◽  
Spectral Energy ◽  
Order Structure ◽  
Two Dimensional ◽  
Third Order

We present an idealized study of rotating stratified wave turbulence in a two-dimensional vertical slice model of the Boussinesq equations, focusing on the peculiar case of equal Coriolis and buoyancy frequencies. In this case the fully nonlinear fluid dynamics can be shown to be isotropic in the vertical plane, which allows the classical methods of isotropic turbulence to be applied. Contrary to ordinary two-dimensional turbulence, here a robust downscale flux of total energy is observed in numerical simulations that span the full parameter regime between Ozmidov and forcing scales. Notably, this robust downscale flux of the total energy does not hold separately for its various kinetic and potential components, which can exhibit both upscale and downscale fluxes, depending on the parameter regime. Using a suitable extension of the classical Kármán–Howarth–Monin equation, exact expressions that link third-order structure functions and the spectral energy flux are derived and tested against numerical results. These expressions make obvious that even though the total energy is robustly transferred downscale, the third-order structure functions are sign indefinite, which illustrates that the sign and the form of measured third-order structure functions are both crucially important in determining the direction of the spectral energy transfer.

scholarly journals Third-order structure functions for isotropic turbulence with bidirectional energy transfer

2019 ◽  
Vol 877 ◽  
Author(s):  
Jin-Han Xie ◽  
Oliver Bühler
Keyword(s):  
Energy Transfer ◽  
Isotropic Turbulence ◽  
Full Range ◽  
Structure Functions ◽  
Spectral Energy ◽  
Order Structure ◽  
Stratified Turbulence ◽  
Mhd Turbulence ◽  
Third Order ◽  
Transfer Rates

We derive and test a new heuristic theory for third-order structure functions that resolves the forcing scale in the scenario of simultaneous spectral energy transfer to both small and large scales, which can occur naturally, for example, in rotating stratified turbulence or magnetohydrodynamical (MHD) turbulence. The theory has three parameters – namely the upscale/downscale energy transfer rates and the forcing scale – and it includes the classic inertial-range theories as local limits. When applied to measured data, our global-in-scale theory can deduce the energy transfer rates using the full range of data, therefore it has broader applications compared with the local theories, especially in situations where the data is imperfect. In addition, because of the resolution of forcing scales, the new theory can detect the scales of energy input, which was impossible before. We test our new theory with a two-dimensional simulation of MHD turbulence.

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.

scholarly journals On the Determination of Kappa Distribution Functions from Space Plasma Observations

Entropy ◽  
2020 ◽  
Vol 22 (2) ◽  
pp. 212 ◽  
Author(s):  
Georgios Nicolaou ◽  
George Livadiotis ◽  
Robert T. Wicks
Keyword(s):  
Distribution Function ◽  
Velocity Distribution ◽  
Space Plasma ◽  
Distribution Functions ◽  
Specific Method ◽  
Kappa Distribution ◽  
Kappa Index ◽  
Velocity Distribution Functions ◽  
Bulk Parameters ◽  
Plasma Bulk

The velocities of space plasma particles, often follow kappa distribution functions. The kappa index, which labels and governs these distributions, is an important parameter in understanding the plasma dynamics. Space science missions often carry plasma instruments on board which observe the plasma particles and construct their velocity distribution functions. A proper analysis of the velocity distribution functions derives the plasma bulk parameters, such as the plasma density, speed, temperature, and kappa index. Commonly, the plasma bulk density, velocity, and temperature are determined from the velocity moments of the observed distribution function. Interestingly, recent studies demonstrated the calculation of the kappa index from the speed (kinetic energy) moments of the distribution function. Such a novel calculation could be very useful in future analyses and applications. This study examines the accuracy of the specific method using synthetic plasma proton observations by a typical electrostatic analyzer. We analyze the modeled observations in order to derive the plasma bulk parameters, which we compare with the parameters we used to model the observations in the first place. Through this comparison, we quantify the systematic and statistical errors in the derived moments, and we discuss their possible sources.

scholarly journals Higher-order dissipation in the theory of homogeneous isotropic turbulence

2016 ◽  
Vol 803 ◽  
pp. 250-274 ◽  
Author(s):  
Norbert Peters ◽  
Jonas Boschung ◽  
Michael Gauding ◽  
Jens Henrik Goebbert ◽  
Reginald J. Hill ◽  
...  
Keyword(s):  
Source Term ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Higher Order ◽  
Distribution Functions ◽  
Order Structure ◽  
Source Terms ◽  
Homogeneous Isotropic Turbulence ◽  
Higher Order Moments ◽  
Even Order

The two-point theory of homogeneous isotropic turbulence is extended to source terms appearing in the equations for higher-order structure functions. For this, transport equations for these source terms are derived. We focus on the trace of the resulting equations, which is of particular interest because it is invariant and therefore independent of the coordinate system. In the trace of the even-order source term equation, we discover the higher-order moments of the dissipation distribution, and the individual even-order source term equations contain the higher-order moments of the longitudinal, transverse and mixed dissipation distribution functions. This shows for the first time that dissipation fluctuations, on which most of the phenomenological intermittency models are based, are contained in the Navier–Stokes equations. Noticeably, we also find the volume-averaged dissipation $\unicode[STIX]{x1D700}_{r}$ used by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85) in the resulting system of equations, because it is related to dissipation correlations.

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 Damage Tensors and Tensors of Continuity in Stress Analysis

1983 ◽  
Vol 38 (12) ◽  
pp. 1383-1390 ◽  
Author(s):  
J. Betten
Keyword(s):  
Linear Operator ◽  
Coordinate System ◽  
Stress Tensor ◽  
Constitutive Equations ◽  
Symmetric Tensor ◽  
Diagonal Form ◽  
Symmetric Part ◽  
Third Order ◽  
Symmetric Property ◽  
Damage Tensor

Abstract Starting from a third order skew-symmetric tensor of continuity to represent area vectors (bivectors) of Cauchy's tetrahedron in a damaged state, a second order damage tensor is found which has the diagonal form with respect to the considered coordinate system. The second part of the paper is concerned with the stresses in a damaged continuum. Introducing a linear operator of rank four a net-stress tensor is formulated. This tensor can be decomposed into a symmetric part and into an antisymmetric one, where only the symmetric part is equal to the net-stress tensor introduced by Rabotnov [7].In view of the formulation of constitutive equations the non-symmetric property of the actual net-stress tensor is a disadvantage. Therefore, a pseudo-net-stress tensor is introduced, which is symmetric.

scholarly journals Turbulence dynamics in separated flows: the generalised Kolmogorov equation for inhomogeneous anisotropic conditions

2018 ◽  
Vol 841 ◽  
pp. 1012-1039 ◽  
Author(s):  
J.-P. Mollicone ◽  
F. Battista ◽  
P. Gualtieri ◽  
C. M. Casciola
Keyword(s):  
Isotropic Turbulence ◽  
Energy Content ◽  
Separation Bubble ◽  
Turbulent Channel Flow ◽  
Kolmogorov Equation ◽  
Cyclic Process ◽  
Order Structure ◽  
Scale Production ◽  
Turbulent Structures ◽  
Scale Turbulence

The generalised Kolmogorov equation is used to describe the scale-by-scale turbulence dynamics in the shear layer and in the separation bubble generated by a bulge at one of the walls in a turbulent channel flow. The second-order structure function, which is the basis of such an equation, is used as a proxy to define a scale-energy content, that is an interpretation of the energy associated with a given scale. Production and dissipation regions and the flux interchange between them, in both physical and separation space, are identified. Results show how the generalised Kolmogorov equation, a five-dimensional equation in our anisotropic and strongly inhomogeneous flow, can describe the turbulent flow behaviour and related energy mechanisms. Such complex statistical observables are linked to a visual inspection of instantaneous turbulent structures detected by means of the Q-criterion. Part of these turbulent structures are trapped in the recirculation where they undergo a pseudo-cyclic process of disruption and reformation. The rest are convected downstream, grow and tend to larger streamwise scales in an inverse cascade. The classical picture of homogeneous isotropic turbulence in which energy is fed at large scales and transferred to dissipate at small scales does not simply apply to this flow where the energy dynamics strongly depends on position, orientation and length scale.

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