Karman-Howarth Equation Research Articles

Two-point stress–strain-rate correlation structure and non-local eddy viscosity in turbulent flows

Abstract

Isotropic turbulence correlation functions modelling on the basis of Karman-Howarth equation

A closure model for the von Karman-Howarth equation is considered. The offered algebraic closure model is based on a single spatial-point third-order correlation function, unlike the widely used models based on a distance between two separated points. The model holds for homogeneous isotropic turbulence theory. Numerical solution of the von Karman-Howarth equation for the flow behind a bi-plane grid of circular cylinders in comparison to experimental data is presented.

Scale-by-scale energy transfer in a dual-plane jet flow

We numerically investigate the scale-by-scale energy transfer along all three directions (i.e., streamwise, vertical, and spanwise directions) at three specifically selected centerline locations (i.e., X/L0 = 7, 12, and 20) in a dual-plane jet flow by using the Karman–Howarth–Monin–Hill (KHMH) equation, where X and L0 are the streamwise distance from the inlet and the initial jet width, respectively. Unlike the well-known Karman–Howarth equation, the fully generalized KHMH equation allows us to study the scale-by-scale energy balance without any assumptions (i.e., homogeneity and isotropy). We calculate each term in the KHMH equation by using the data from a direct numerical simulation [Zhou et al., “Dual-plane turbulent jets and their non-Gaussian velocity fluctuations,” Phys. Rev. Fluids 3, 124604 (2018)]. At X/L0 = 7, where the flow is inhomogeneous and anisotropic, the scale-by-scale energy transfer is quite different in the streamwise, vertical, and spanwise directions. One interesting finding is that a negative production term in the vertical direction can be found in the reversal flow region. Unlike most flows previously investigated, the linear energy cascade plays an important role in the energy cascade. The linear energy transfer is heavily dependent on the direction: in the streamwise direction, it is forward (from the large scale to the small scale), but in the vertical direction, it is backward (from the small scale to the large scale). A physical model is proposed and also verified, which suggests that the forward linear energy cascade corresponds to fluid compression, and the backward linear energy cascade corresponds to the fluid stretching. At X/L0 = 12, where the energy spectrum exhibits a well-defined −5/3 scaling, there is no equilibrium energy cascade. The linear energy cascade is still dominant and shows a backward cascade in the streamwise direction and a forward cascade in the vertical direction, while the non-linear energy cascade remains forward. At X/L0 = 20, where the flow becomes much more homogeneous and isotropic, within a short scale-range around the Taylor microscale, the dissipation term can be balanced by the combination of the non-linear energy transfer term and advection term. This observation to some extent echoes Kolmogorov’s hypothesis, but limited only to length scales around the Taylor microscale, and the persistence of the advection term can find its root in the low local Reynolds number at that length scale.

Cascade of turbulent energy and scalar variance in DC electrokinetic turbulence

Abstract Turbulent flow can be generated by DC electrokinetic (EK) mechanism resulted from the electric conductivity and permittivity variation in fluids, as has been demonstrated by Varshney et al (2016), where a −1.4 slope of velocity power spectrum is observed. Here, we theoretically investigate the scaling exponents of both velocity and scalar structure functions in DC EK turbulence. The scaling exponents of the second order structure functions of velocity and scalar in the so-called electric body force (EBF) dominant subrange are 2/5 (equivalent to the −7/5 slope of velocity power spectrum) and 4/5 respectively. The theory directly explains the experimental results of Varshney et al. (2016). Based on Karman–Howarth equation with forcing terms, the third order structure function of velocity which is the representative of energy transport process is investigated in the DC EK turbulence. Compared with the conventional hydrodynamic turbulence, the energy transport in DC EK turbulence has an additional scaling subrange and is governed by additional small length scales.

On the Local Structure of Two-Dimensional Turbulence

Possible consequences of the application of the Kolmogorov theory of local isotropy to two-dimensional turbulence are discussed. In this regard, the cases of both freely decaying and stationary turbulence are considered. Formulae for the third-order structure functions in the inertial interval are obtained as the solutions of the Karman—Howarth equation with the use of the well-known relation between the velocity and vorticity correlators. These formulae differ from those given by other authors; that is, the variable term linear in space turns out to be negative in the corresponding formula, as in the three-dimensional case. The possible relation of the results to the form of the energy spectrum is briefly mentioned.

On the Construction of Turbulent Transfer Spectral Models Using 3D Numerical Simulation of Taylor–Green Vortex Decay

In the present paper we recall the classical Taylor–Green vortex turbulent decay in terms of isotropic turbulence theory. In particular, we calculate the spectral turbulent kinetic energy transfer $$T(k)$$ and the spectral flux $${{\Pi }_{E}}(k)$$ basing on the longitudinal triple correlation function (the two-point third-order velocity moment). These functions can be also obtained in other way via the generalized Karman–Howarth equation for homogeneous turbulence using integral of triple modes interaction. In both cases, the spectral transfer and the flux appear to be different from the characteristics of isotropic or homogeneous turbulence due to the peculiarities of the problem statement. The results obtained outline the range of obstacles associated with the construction of spectral turbulent models in complex heat and mass transfer problems arising due to periodicity effects, closeness of integral turbulent scales to modeling box size and lack of spectral resolution.

Isotropic turbulence in compact space

Isotropic turbulence is typically studied numerically through direct numerical simulations (DNS). The DNS flows are described by the Navier–Stokes equation in a ‘box’, defined through periodic boundary conditions. Ideal isotropic turbulence lives in infinite space. The DNS flows live in a compact space and they are not isotropic in their large scales. Hence, the investigation of important phenomena of isotropic turbulence, such as anomalous scaling, through DNS is affected by large-scale effects in the currently available Reynolds numbers. In this work, we put isotropic turbulence – or better, the associated formal theory – in a ‘box’, by imposing periodicity at the level of the correlation functions. This is an attempt to offer a framework where one may investigate isotropic theories/models through the data of DNS in a manner as consistent with them as possible. We work at the lowest level of the hierarchy, which involves the two-point correlation functions and the Karman–Howarth equation. Periodicity immediately gives us the discrete wavenumber space of the theory. The wavenumbers start from 1.835, 2.896, 3.923, and progressively approach integer values, in an interesting correspondence with the DNS wavenumber shells. Unlike the Navier–Stokes equation, infinitely smooth periodicity is obstructed in this theory, a fact expressed by a sequence of relations obeyed by the normal modes of the Karman–Howarth equation at the endpoints of a unit period interval. Similar relations are imparted to the two-point functions under the condition that the energy spectrum and energy transfer function are realizable. Hence, these relations are necessary conditions for realizability in this theory. Naturally constructed closure schemes for the Karman–Howarth equation do not conform to such relations, thereby destroying realizability. A closure can be made to conform to a finite number of them by adding corrective terms, in a procedure that possesses certain analogies with the renormalization of quantum field theory. Perhaps the most important one is that we can let the spectrum be unphysical (through sign-changing oscillations of decreasing amplitude) for the infinitely large wavenumbers, as long as we can controllably extend the regime where the spectrum remains physical, deep enough in the dissipation subrange so as to be realistically adequate. Indeed, we show that one or two such ‘regularity relations’ are needed at most for comparisons of the predictions of the theory with the current resolution level results of the DNS. For the implementation of our arguments, we use a simple closure scheme previously proposed by Oberlack and Peters. The applicability of our ideas to more complex closures is also discussed.

On the numerical modeling of the dynamics of homogeneous isotropic turbulence

Using a series of mathematical models based on the closed Karman-Howarth equation, the numerical modeling of the dynamics of homogeneous isotropic turbulence was carried out. Computational results agree with known experimental data. Mathematical models are compared.

The effect of velocity field forcing techniques on the Karman–Howarth equation

Many velocity field forcing methods exist to simulate isotropic turbulence, but no in-depth analysis of the effects that these methods have on the generated turbulence has been performed. This work contains such a detailed study. It focuses on Lundgren’s linear and Alvelius’ spectral velocity forcing methods. Based on the constraints imposed on their associated forcing terms, these two are representative of the numerous forcing methods available in the literature. This study is conducted in the context of the Karman–Howarth equation, which, as a consequence of velocity forcing, has a source term appended to it. The expressions for the forcing method-specific Karman–Howarth source terms are derived, and their effect on key turbulent metrics, e.g. structure functions and spectra, is investigated. The behaviour of these source terms determines the state to which all turbulent metrics evolve, allowing for the differences noted between linearly and spectrally forced turbulent fields to be explained.

One-dimensional energy spectra in three-dimensional incompressible homogeneous isotropic turbulence

The paper investigates the detailed features of one-dimensional energy spectra in three-dimensional isotropic turbulence, based on the exact solution of Karman-Howarth equation. Particular interest will be paid on the degree to which spectral scaling can lead the spectral data to be collapsed. The theory appears to be consistent with the wealth of experimental data (Comte-Bellot and Corrsin, 1971) at least in low Taylor microscale Reynolds number.

Approximation of the third two-point moments of the velocity field in isotropic turbulence

Different methods for approximating the third two-point moments of the velocity fields entering in the von Karman-Howarth equation are compared using the known experimental results of A. Townsend and R. Stewart, together with the model form of the energy spectrum of turbulence, which makes it possible to approximate the experimentally determined second moments of the velocity field. The latter moments are used for calculating the third two-point moments following the methods proposed by various authors. The calculated results are compared with the experimental data which makes it possible to quantitatively estimate the approximation accuracy. For several models the second-order structure function is calculated from the Kolmogorov equation for the inertial interval. In this case, for the models of K. Hasselmann and Yu.M. Lytkin the power-law dependence D LL (r) ∼ r 2/3 expected for the inertial interval is obtained for the structure function D LL (r).

A geometry of the correlation space and a nonlocal degenerate parabolic equation from isotropic turbulence

AbstractConsidering the metric tensor ds2(t), associated with the two‐point velocity correlation tensor field (parametrized by the time variable t) in the space 𝒦3of correlation vectors, at the first part of the paper we construct the Lagrangian system (Mt,ds2(t)) in the extended space 𝒦3 × R+ for homogeneous isotropic turbulence. This allows to introduce into the consideration common concept and technics of Lagrangian mechanics for the application in turbulence. Dynamics in time of (Mt,ds2(t)) (a singled out fluid volume equipped with a family of pseudo‐Riemannian metrics) is described in the frame of the geometry generated by the 1‐parameter family of metrics ds2(t) whose components are the correlation functions that evolve according to the von Kármán‐Howarth equation. This is the first step one needs to get in a future analysis the physical realization of the evolution of this volume. It means that we have to construct isometric embedding of the manifold Mt equipped with metric ds2(t) into R3 with the Euclidean metric. In order to specify the correlation functions, at the second part of this paper we study in details an initial‐boundary value problem to the closure model [19,26] for the von Kármán‐Howarth equation in the case of large Reynolds numbers limit.

Self-similarity in fully developed homogeneous isotropic turbulence using the lyapunov analysis

In this work, we calculate the self-similar longitudinal velocity correlation function and the statistical properties of velocity difference using the results of the Lyapunov analysis of the fully developed isotropic homogeneous turbulence just presented by the author in a previous work (arXiv:0911.1463). There, a closure of the von Karman-Howarth equation is proposed and the statistics of velocity difference is determined through a specific analysis of the Fourier-transformed Navier-Stokes equations. The correlation functions correspond to steady-state solutions of the von Karman-Howarth equation under the self-similarity hypothesis introduced by von Karman. These solutions are numerically determined with the statistics of velocity difference. The obtained results adequately describe the several properties of the fully developed isotropic turbulence.

New Sedov-Type Solution of Isotropic Turbulence

The starting point lies in the results obtained by Sedov (1944) for isotropic turbulence with a self-preserving hypothesis. A careful consideration of the mathematical structure of the Karman–Howarth equation leads to an exact analysis of all cases possible and to all admissible solutions of the problem. I study this interesting problem from a new point of view. New solutions are obtained. Based on these exact solutions, some physical significant consequences of recent advances in the theory of self-preserved homogeneous statistical solution of the Navier–Stokes equations are presented.

A Geometric Interpretation of the Second-Order Structure Function Arising in Turbulence

We primarily deal with homogeneous isotropic turbulence and use a closure model for the von Karman-Howarth equation to study several geometric properties of turbulent fluid dynamics. We focus our attention on the application of Riemannian geometry methods in turbulence. Some advantage of this approach consists in exploring the specific form of a closure model for the von Karman-Howarth equation that enables to equip a model manifold (a cylindrical domain in the correlation space) by a family of inner metrics (length scales of turbulent motion) which depends on time. We show that for large Reynolds numbers (in the limit of large Reynolds numbers) the radius of this manifold can be evaluated in terms of the second-order structure function and the correlation distance. This model manifold presents a shrinking cylindrical domain as time evolves. This result is derived by using a selfsimilar solution of the closure model for the von Karman-Howarth equation under consideration. We demonstrate that in the new variables the selfsimilar solution obtained coincides with the element of Beltrami surface (or pseudo-sphere): a canonical surface of the constant sectional curvature equals − 1.

On the properties of decaying two-dimensional turbulence: Characteristic features of the spectra and coherent structures

The main results of the theory of two-dimensional turbulence are presented. The conventional approach based on the Karman-Howarth equation is used to describe anomalous properties (sharply differing from those inherent in usual three-dimensional turbulence) of two-dimensional turbulent motion of an incompressible fluid, in particular, energy transfer across the spectrum from larger toward smaller wave numbers and the formation of coherent structures, whose origin is associated with the fact that the spectrum attains the form of a δ-function. A uniform method is proposed for obtaining self-similar spectra of two-dimensional turbulence in inertial ranges. The problem of turbulent diffusion of a passive tracer in the two-dimensional case is also considered. It turns out that the corresponding quantities, as well as those related to the dynamic characteristics of motion, can exhibit anomalous properties under certain conditions. The relationship between the results and experimental data is discussed. In particular, the experimentally observed inversion of the spectra of two-dimensional turbulence in the atmosphere is explained.

Exact Statistical Theory Of Isotropic Turbulence

The starting point for this paper lies in the results obtained by Sedov (1944) for isotropic turbulence with the self-preserving hypothesis. A careful consideration of the mathematical structure of the Karman-Howarth equation leads to an exact analysis of all possible cases and to all admissible solutions of the problem. This kind of appropriate manipulation escaped the attention of a number of scientists who developed the theory of turbulence and processed the experimental data for a long time. This paper revisits this interesting problem from a new point of view. Firstly, a new complete set of solutions are obtained, and Sedov’s solution is one special case of this set of solutions. Based on these exact solutions, some physically significant consequences of recent advances in the theory of selfpreserved homogenous statistical solution of the Navier-Stokes equations are presented. New results could be obtained for the analysis on turbulence features, such as the scaling behaviour, the energy spectra, and also the large scale dynamics.

Homogeneous isotropic turbulence in dilute polymers

The modification of the turbulent cascade by polymeric additives is addressed by direct numerical simulations of homogeneous isotropic turbulence of a FENE-P fluid. According to the appropriate form of the Karman-Howarth equation, two kinds of energy fluxes exist, namely the classical transfer term and the coupling with the polymers. Depending on the Deborah number, the response of the flow may result either in a pure damping or in the depletion of the small scales accompanied by increased fluctuations at large scale. The latter behaviour corresponds to an overall reduction of the dissipation rate with respect to an equivalent Newtonian flow with identical fluctuation intensity. The relevance of the position of the crossover scale between the two components of the energy flux with respect to the Taylor microscale of the system is discussed.

Kolmogorov two-thirds law by matched asymptotic expansion

The Kolmogorov two-thirds law is derived for large Reynolds number isotropic turbulence by the method of matched asymptotic expansions. Inner and outer variables are derived from the Karman–Howarth equation by using the von Karman self-preservation hypothesis. Matching the resulting large Reynolds number asymptotic expansions yields the Kolmogorov law. The Kolmogorov similarity hypotheses are not assumed; only the Navier–Stokes equation is employed and the assumption that dissipation is finite. This indicates that the Kolmogorov results are a direct consequence of the Navier–Stokes equations.

Axisymmetric form of Kármán-Howarth equation and its limiting forms

Kinematics and dynamics of homogeneous axisymmetric turbulence have been derived with the assumption that the properties of the turbulence are invariant with respect to rotation about a preferred direction λ. In particular, the "axisymmetric" equivalent of Karman-Howarth "isotropic" equation is derived using Lindborg's representation of the two-point correlation tensors of homogeneous axisymmetric turbulence. When the more constraining assumption of isotropy is made, this equation reduces to the well-known Karman-Howarth equation. There are two interesting limiting forms of the axisymmetric Karman-Howarth equation: the axisymmetric form of the energy balance equation and the axisymmetric form of the vorticity balance equation.