Statistical Theory Of Turbulence Research Articles

On the structure and statistical theory of turbulence of extended magnetohydrodynamics

Recent progress regarding the noncanonical Hamiltonian formulation of extended magnetohydrodynamics (XMHD), a model with Hall drift and electron inertia, is summarized. The advantages of the Hamiltonian approach are invoked to study some general properties of XMHD turbulence, and to compare them against their ideal MHD counterparts. For instance, the helicity flux transfer rates for XMHD are computed, and Liouville’s theorem for this model is also verified. The latter is used, in conjunction with the absolute equilibrium states, to arrive at the spectra for the invariants, and to determine the direction of the cascades, e.g., generalizations of the well-known ideal MHD inverse cascade of magnetic helicity. After a similar analysis is conducted for XMHD by inspecting second order structure functions and absolute equilibrium states, a couple of interesting results emerge. When cross helicity is taken to be ignorable, the inverse cascade of injected magnetic helicity also occurs in the Hall MHD range—this is shown to be consistent with previous results in the literature. In contrast, in the inertial MHD range, viz at scales smaller than the electron skin depth, all spectral quantities are expected to undergo direct cascading. The consequences and relevance of our results in space and astrophysical plasmas are also briefly discussed.

ANÁLISE DA PERFORMANCE DE UM MODELO DE COLUNA SIMPLES UTILIZANDO DIFERENTES PARAMETRIZAÇÕES PARA A TAXA DISSIPAÇÃO VISCOSA DE ENERGIA CINÉTICA TURBULENTA

The proper description of the atmospheric flow in the stable atmospheric boundary layer is one of the most complex tasks to be performed by numerical models of weather and climate prediction. Most of the parameterizations used in these models are based on the statistical theory of turbulence in their conception. However, this theory is valid only for a flow in which turbulence is homogeneous, isotropic and stationary, a conditions that are not commonly found overnight. Basically, the nighttime flow can be classified in two major regimes: very stable, where turbulence is almost entirely suppressed; and weakly stable regime, with intense turbulent mixing. The transition between these regimes is known as atmospheric coupling, and it can happens a lot of times along the same night. In this work, we implemented a single column model with turbulence closure 1.5, in three different configurations and three different turbulence formulations. In general, the model results show that, all the configurations are able to reproduce the average characteristics of the flow in the weakly stable conditions.

Hydrodynamics of sediment threshold

A novel hydrodynamic model for the threshold of cohesionless sediment particle motion under a steady unidirectional streamflow is presented. The hydrodynamic forces (drag and lift) acting on a solitary sediment particle resting over a closely packed bed formed by the identical sediment particles are the primary motivating forces. The drag force comprises of the form drag and form induced drag. The lift force includes the Saffman lift, Magnus lift, centrifugal lift, and turbulent lift. The points of action of the force system are appropriately obtained, for the first time, from the basics of micro-mechanics. The sediment threshold is envisioned as the rolling mode, which is the plausible mode to initiate a particle motion on the bed. The moment balance of the force system on the solitary particle about the pivoting point of rolling yields the governing equation. The conditions of sediment threshold under the hydraulically smooth, transitional, and rough flow regimes are examined. The effects of velocity fluctuations are addressed by applying the statistical theory of turbulence. This study shows that for a hindrance coefficient of 0.3, the threshold curve (threshold Shields parameter versus shear Reynolds number) has an excellent agreement with the experimental data of uniform sediments. However, most of the experimental data are bounded by the upper and lower limiting threshold curves, corresponding to the hindrance coefficients of 0.2 and 0.4, respectively. The threshold curve of this study is compared with those of previous researchers. The present model also agrees satisfactorily with the experimental data of nonuniform sediments.

Probability Density Functions of Vorticities in Turbulent Channels with Effects of Blowing and Suction

Abstract This paper presents direct numerical simulation (DNS) result of the Navier–Stokes equations for turbulent channel flows with blowing and suction effects. The friction Reynolds number is R e τ = 394 ${\rm{R}}{{\rm{e}}_\tau} = 394$ and a range of blowing and suction conditions is covered with different perturbation strengths, i. e. A = 0.05 , $A = 0.05, $ 0.1, 0.2. While the mean velocity profile has been severely altered, the probability density function (PDF) for (spanwise) vorticity – depending on wall distance ( y + ) $({y^ +})$ and blowing/suction strength (A) – satisfies the generalized hyperbolic distribution (GHD) of Birnir [The Kolmogorov-Obukhov statistical theory of turbulence, J. Nonlinear Sci. (2013a), doi: 10.1007/s00332-012-9164–z; The Kolmogorov-Obukhov theory of turbulence, Springer, New York, 2013b] in the bulk of the flow. The latter leads to accurate descriptions of all PDFs (at y + = 40 , ${y^ +} = 40, $ 200, 390 and A = 0.05 , $A = 0.05, $ 0.2, for instance) with only four parameters. The result indicates that GHD is a general tool to quantify PDF for turbulent flows under various wall surface conditions.

A review of the statistical turbulence theory required extending the population balance closure models to the entire spectrum of turbulence

The models for fluid particle breakage and coalescence due to turbulence are generally limited to the inertial subrange of isotropic turbulence and infinite Reynolds numbers. A rigorous procedure for extending the fluid particle breakage and coalescence closures to the entire spectrum of isotropic turbulence and for a wider range of Reynolds numbers can be established based on statistical turbulence theory. The modeling procedure consists of a three‐dimensional (3‐D) literature model energy spectrum for the dissipation, inertial, and energy containing subranges of isotropic turbulence, and an exact literature integral relation for determining the second‐order longitudinal structure function from the 3‐D energy spectrum. A review of the requisite statistical turbulence theory and the use of the model energy spectrum is provided, because the necessary details are not easily accessible in the chemical engineering literature and misconceptions are found in the publications of the previous modeling attempts. © 2016 American Institute of Chemical Engineers AIChE J, 62: 1795–1820, 2016

Turbulence question: How do view "the homogenous and isotropic turbulence"?

In the statistical theory of turbulence, the clear-air turbulence in the atmosphere, the wind tunnel flow state in the core zone of the work area behind the grids, can be seen as a homogeneous and isotropic turbulence. Kolmogorovs study pointed out that the inertial set-range of spectrum function corresponds to homogeneous and isotropic turbulence; therefore, the scaling law is universal. From this, Kolmogorov deduced famous (-5/3) power law of turbulent energy cascade. But in the case observed intermittent and coherent structure in the experiments, it is needed to conduct comparative study between the energy cascade models and its corresponding scaling laws, the focus is to explore and study the physical mechanism of their statistical methods. Because there is a nonlinear term in N-S equation, especially the closure issues, which make the results of theoretical studies and numerical experiments have diversity and complexity, on this problem, there is no effective mathematical approach at present. In this paper, atmospheric turbulence as the central subject under discussion, several major issues related to this are discussed, namely: what is the problem of turbulence, turbulence theory and boundary layer theory, the nature of the turbulence problem, atmospheric turbulence, and the definition of the turbulence problem. We, especially, discuss the three different kinds of scaling laws, and look to prospects of turbulence research.

Lamb vector in isotropic turbulence

The Lamb vector is known to be of profound importance for fluid dynamics generally and for the dynamics of turbulence in particular. Based on the new developments of the statistical theory of isotropic turbulence, some new analytical indications are found that the Lamb vector has a contribution in the potential and solenoidal parts, which are almost equal for some typical parameters.

Модель вытягивающихся вихрей и обоснование статистических свойств турбулентности

Модель вытягивающихся вихрей и обоснование статистических свойств турбулентности

Symmetries and their importance for statistical turbulence theory

The special importance of turbulence maybe comprehended by its ubiquity in innumerable natural and technical systems. Examples for natural turbulent flows are the atmospheric flow and the oceanic current which to calculate is a crucial point in climate research. Classical engineering application involving turbulence are the flows around airplanes or cars or turbulence within jet or reciprocating engines. Though supposed for more than hundred years, only with the advent of super computers it became apparent that the Navier-Stokes equations provide an excellent continuummechanical model for turbulent flows. Still, the exclusive and direct application of the Navier-Stokes equations to practical flow problems at very high Reynolds numbers without invoking any additional assumptions is still several decades away. However, in most applications it is not at all necessary to know all the detailed fluctuations of velocity and pressure present in turbulent flows but for the most part statistical measures are sufficient. This was in fact the key idea of O. Reynolds who was the first to suggest a statistical description of turbulence. The Navier-Stokes equations, however, constitute a non-linear and, due to the pressure Poisson equation, a non-local set of equations. As an immediate consequence of this the equations for the mean or expectation values for velocity and pressure lead to an infinite set of statistical equations, or, if truncated at some level of statistics, an un-closed system is generated.

The Kolmogorov-Obukhov-She-Leveque scaling in turbulence

We construct the 1962 Kolmogorov-Obukhov statistical theory of turbulence from the stochastic Navier-Stokes equations driven by generic noise. The intermittency corrections to the scaling exponents of the structure functions of turbulence are given by the She-Leveque intermittency corrections. We show how they are produced byShe-Waymire log-Poisson processes, that are generated by the Feynmann-Kac formula from the stochastic Navier-Stokes equation. We find the Kolmogorov-Hopf equations and compute the invariant measures of turbulence for 1-point and 2-point statistics. Then projecting these measures we find the formulas for the probability distribution functions (PDFs) of the velocity differences in the structure functions. In the limit of zero intermittency, these PDFs reduce to the Generalized Hyperbolic Distributions of Barndorff-Nilsen.

Global invariants in ideal magnetohydrodynamic turbulence

Magnetohydrodynamic (MHD) turbulence is an important though incompletely understood factor affecting the dynamics of many astrophysical, geophysical, and technological plasmas. As an approximation, viscosity and resistivity may be ignored, and ideal MHD turbulence may be investigated by statistical methods. Incompressibility is also assumed and finite Fourier series are used to represent the turbulent velocity and magnetic field. The resulting model dynamical system consists of a set of independent Fourier coefficients that form a canonical ensemble described by a Gaussian probability density function (PDF). This PDF is similar in form to that of Boltzmann, except that its argument may contain not just the energy multiplied by an inverse temperature, but also two other invariant integrals, the cross helicity and magnetic helicity, each multiplied by its own inverse temperature. However, the cross and magnetic helicities, as usually defined, are not invariant in the presence of overall rotation or a mean magnetic field, respectively. Although the generalized form of the magnetic helicity is known, a generalized cross helicity may also be found, by adding terms that are linear in the mean magnetic field and angular rotation vectors, respectively. These general forms are invariant even in the presence of overall rotation and a mean magnetic field. We derive these general forms, explore their properties, examine how they extend the statistical theory of ideal MHD turbulence, and discuss how our results may be affected by dissipation and forcing.

The Kolmogorov–Obukhov Statistical Theory of Turbulence

In 1941 Kolmogorov and Obukhov postulated the existence of a statistical theory of turbulence, which allows the computation of statistical quantities that can be simulated and measured in a turbulent system. 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 1962 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.

A theory of 3-4-5-wave resonance and conservation for ocean surface waves

With respect to the whole of two elemental waves in ocean surface, capillary and gravity waves, a deterministic, rich and fundamental theory of 3-4-5-wave resonance and conservation for ocean surface waves in a finite depth is developed, which presents fully the preciseness, symmetry and completeness as compared with a variety of the classical, modern results, and provides an indispensable basis for the succeeding and universal statistical theory of ocean wave turbulence.

Statistical Theory of Turbulence by the Late Lamented Dr. Shunichi Tsugé Case Study on Flow through a Grid in Wind Tunnel

This paper is concerned with statistical theory of turbulence by the late lamented Dr. Shunichi Tsuge. The theory has been applied to the primary flow through a grid fixed vertically with respect to the horizontal axis of the wind tunnel. The first analytical solution has been obtained and explained the well-known “the inverse-linear decay law” of the turbulent intensity. It is believed that the present result is the first exact solution in the theory of turbulence.

Statistical theories versus statistical mechanics of homogeneous isotropic turbulence

The present paper is concerned with the “closure problem of turbulence,” which constituted the main subject of the Session on the “Energy transfer in homogeneous turbulence” in the last Turbulence Colloquium Marseille (TCM) 1961. It is usual in statistical theory of turbulence to express the random velocity field of turbulence in terms of the velocity correlations of various orders and deal with the set of dynamical equations governing these velocity correlations. This way is, however, associated with the difficulty of unclosedness since each equation involves two velocity correlations of different orders as unknowns according to the nonlinearity of the Navier–Stokes equation, and hence we have to introduce some relationship between the velocity correlations of different orders in order to make the set of equations closed. Among various closuretheories proposed so far, the quasi-normal closure by Proudman and Reid (Proudman and W.H. Reid, Phil. Trans. R. Soc. Lond. A 247 (1954), pp. 163–189) and Tatsumi (T. Tatsumi, Proc. Roy. Soc. Lond. A 239 (1957), p. 16.) and the direct-interaction closure by Kraichnan (R.H. Kraichnan, J. Fluid Mech. 5 (1959), pp. 497–543.) have been taken up in the Session and critical survey and evaluation have been made on these theories. Apart from particular merits and demerits of the theories, it has been recognized that they still remain in approximate levels with either partial or no fulfillment of Kolmogorov’s nonzero inertial energy-dissipation hypothesis. More recently, Tatsumi (T. Tatsumi, in Geometry and Statistics of Turbulence, T. Kambe ed., Kluwer, 2001, pp. 3–12.) formulated statistical mechanics of fluid turbulence in terms of the Lundgren–Monin equations (T.S. Lundgren, Phys. Fluids, 10 (1967), pp. 969–975; A.S. Monin, PMM J. Appl. Math. Mech. 31 (1967), pp. 1057–1068.) for the multipoint velocity distributions of turbulence and the cross-independence closure of these equations. It has been shown by Tatsumi (T. Tatsumi, J. Fluid Mech. 670 (2011), pp. 365–403.) that this closure provides us with exactclosure of the Lundgren–Monin equations and thus exact solution to any problem concerning the mean velocity products.

Applications of the new symmetries of the multi-point correlation equations

We presently show that the infinite set of multi-point correlation equations, which are direct statistical consequences of the Navier-Stokes equations, admit a rather large set of Lie symmetry groups. Additional to the symmetries stemming from the Navier-Stokes equations a new scaling group and translational groups of the correlation vectors and all independent variables have been discovered. These new statistical groups have important consequences on our understanding of turbulent scaling laws. Exemplarily, we consider one of the key foundations of statistical turbulence theory, the universal law of the wall, and show that the log-law fundamentally relies on one of the new translational groups. Furthermore, we present rotating channel flows, where different rotational axes result in very different scaling laws.

Scale-wise coherent vorticity extraction for conditional statistical modeling of homogeneous isotropic two-dimensional turbulence

Classical statistical theories of turbulence have shown their limitations, in that they cannot predict much more than the energy spectrum in an idealized setting of statistical homogeneity and stationarity. We explore the applicability of a conditional statistical modeling approach: can we sort out what part of the information should be kept, and what part should be modeled statistically, or, in other words, “dissipated”? Our mathematical framework is the initial value problem for the two-dimensional (2D) Euler equations, which we approximate numerically by solving the 2D Navier–Stokes equations in the vanishing viscosity limit. In order to obtain a good approximation of the inviscid dynamics, we use a spectral method and a resolution going up to 8192 2. We introduce a macroscopic concept of dissipation, relying on a split of the flow between coherent and incoherent contributions: the coherent flow is constructed from the large wavelet coefficients of the vorticity field, and the incoherent flow from the small ones. In previous work, a unique threshold was applied to all wavelet coefficients, while here we also consider the effect of a scale by scale thresholding algorithm, called scale-wise coherent vorticity extraction. We study the statistical properties of the coherent and incoherent vorticity fields, and the transfers of enstrophy between them, and then use these results to propose, within a maximum entropy framework, a simple model for the incoherent vorticity. In the framework of this model, we show that the flow velocity can be predicted accurately in the L 2 norm for about 10 eddy turnover times.

Owen Martin Phillips

Owen Martin Phillips

Assessment of the modulated gradient model in decaying isotropic turbulence

A recently introduced nonlinear model undergoes evaluations based on two isotropic turbulent cases: a University of Wiscosion-Madison case at a moderate Reynolds number and a Johns Hopkins University case at a high Reynolds number. The model uses an estimation of the subgrid-scale (SGS) kinetic energy to model the magnitude of the SGS stress tensor, and uses the normalized velocity gradient tensor to model the structure of the SGS stress tensor. Testing is performed for the first case through a comparison between direct numerical simulation (DNS) results and large eddy simulation (LES) results regarding resolved kinetic energy and energy spectrum. In the second case, we examine the resolved kinetic energy, the energy spectrum, as well as other key statistics including the probability density functions of velocities and velocity gradients, the skewness factors, and the flatness factors. Simulations using the model are numerically stable, and results are satisfactorily compared with DNS results and consistent with statistical theories of turbulence.

LIE ALGEBRA OF THE SYMMETRIES OF THE MULTI-POINT EQUATIONS IN STATISTICAL TURBULENCE THEORY

We briefly derive the infinite set of multi-point correlation equations based on the Navier–Stokes equations for an incompressible fluid. From this we reconsider the previously derived set of Lie symmetries, i.e. those directly induced by the ones from classical mechanics and also new symmetries. The latter are denoted statistical symmetries and have no direct counterpart in classical mechanics. Finally, we considerably extend the set of symmetries by Lie algebra methods and give the corresponding commutator tables. Due to the infinite dimensionality of the multi-point correlation equations completeness of its symmetries is not proven yet and is still an open question.