Four-fifths Law Research Articles

A correspondence between the multifractal model of turbulence and the Navier-Stokes equations.

The multifractal model of turbulence (MFM) and the three-dimensional Navier-Stokes equations are blended together by applying the probabilistic scaling arguments of the former to a hierarchy of weak solutions of the latter. This process imposes a lower bound on both the multifractal spectrum [Formula: see text], which appears naturally in the Large Deviation formulation of the MFM, and on [Formula: see text] the standard scaling parameter. These bounds respectively take the form: (i) [Formula: see text], which is consistent with Kolmogorov's four-fifths law ; and (ii) [Formula: see text]. The latter is significant as it prevents solutions from approaching the Navier-Stokes singular set of Caffarelli, Kohn and Nirenberg. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.

3D Simulations and MLT. I. Renzini’s Critique

Abstract Renzini wrote an influential critique of “overshooting” in mixing-length theory (MLT), as used in stellar evolution codes, and concluded that three-dimensional fluid dynamical simulations were needed. Such simulations are now well tested. Implicit large eddy simulations connect large-scale stellar flow to a turbulent cascade at the grid scale, and allow the simulation of turbulent boundary layers, with essentially no assumptions regarding flow except the number of computational cells. Buoyant driving balances turbulent dissipation for weak stratification, as in MLT, but with the dissipation length replacing the mixing length. The turbulent kinetic energy in our computational domain shows steady pulses after 30 turnovers, with no discernible diminution; these are caused by the necessary lag in turbulent dissipation behind acceleration. Interactions between coherent turbulent structures give multi-modal behavior, which drives intermittency and fluctuations. These cause mixing, which may justify use of the instability criterion of Schwarzschild rather than the Ledoux. Chaotic shear flow of turning material at convective boundaries causes instabilities that generate waves and sculpt the composition gradients and boundary layer structures. The flow is not anelastic; wave generation is necessary at boundaries. A self-consistent approach to boundary layers can remove the need for ad hoc procedures of “convective overshooting” and “semi-convection.” In Paper II, we quantify the adequacy of our numerical resolution in a novel way, determine the length scale of dissipation—the “mixing length”—without astronomical calibration, quantify agreement with the four-fifths law of Kolmogorov for weak stratification, and deal with strong stratification.

Shallow water wave turbulence

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.

Potential landscape and flux field theory for turbulence and nonequilibrium fluid systems

Turbulence is a paradigm for far-from-equilibrium systems without time reversal symmetry. To capture the nonequilibrium irreversible nature of turbulence and investigate its implications, we develop a potential landscape and flux field theory for turbulent flow and more general nonequilibrium fluid systems governed by stochastic Navier–Stokes equations. We find that equilibrium fluid systems with time reversibility are characterized by a detailed balance constraint that quantifies the detailed balance condition. In nonequilibrium fluid systems with nonequilibrium steady states, detailed balance breaking leads directly to a pair of interconnected consequences, namely, the non-Gaussian potential landscape and the irreversible probability flux, forming a ‘nonequilibrium trinity’. The nonequilibrium trinity characterizes the nonequilibrium irreversible essence of fluid systems with intrinsic time irreversibility and is manifested in various aspects of these systems. The nonequilibrium stochastic dynamics of fluid systems including turbulence with detailed balance breaking is shown to be driven by both the non-Gaussian potential landscape gradient and the irreversible probability flux, together with the reversible convective force and the stochastic stirring force. We reveal an underlying connection of the energy flux essential for turbulence energy cascade to the irreversible probability flux and the non-Gaussian potential landscape generated by detailed balance breaking. Using the energy flux as a center of connection, we demonstrate that the four-fifths law in fully developed turbulence is a consequence and reflection of the nonequilibrium trinity. We also show how the nonequilibrium trinity can affect the scaling laws in turbulence.

The inertial subrange in turbulent pipe flow: centreline

The inertial-subrange scaling of the axial velocity component is examined for the centreline of turbulent pipe flow for Reynolds numbers in the range $249\leqslant Re_{{\it\lambda}}\leqslant 986$. Estimates of the dissipation rate are made by both integration of the one-dimensional dissipation spectrum and the third-order moment of the structure function. In neither case does the non-dimensional dissipation rate asymptote to a constant; rather than decreasing, it increases indefinitely with Reynolds number. Complete similarity of the inertial range spectra is not evident: there is little support for the hypotheses of Kolmogorov (Dokl. Akad. Nauk SSSR, vol. 32, 1941a, pp. 16–18; Dokl. Akad. Nauk SSSR, vol. 30, 1941b, pp. 301–305) and the effects of Reynolds number are not well represented by Kolmogorov’s ‘extended similarity hypothesis’ (J. Fluid Mech., vol. 13, 1962, pp. 82–85). The second-order moment of the structure function does not show a constant value, even when compensated by the extended similarity hypothesis. When corrected for the effects of finite Reynolds number, the third-order moments of the structure function accurately support the ‘four-fifths law’, but they do not show a clear plateau. In common with recent work in grid turbulence, non-equilibrium effects can be represented by a heuristic scaling that includes a global Reynolds number as well as a local one. It is likely that non-equilibrium effects appear to be particular to the nature of the boundary conditions. Here, the principal effects of the boundary conditions appear through finite turbulent transport at the pipe centreline, which constitutes a source or a sink at each wavenumber.

Energy cascade and scaling in supersonic isothermal turbulence

AbstractSupersonic turbulence plays an important role in a number of extreme astrophysical and terrestrial environments, yet its understanding remains rudimentary. We use data from a three-dimensional simulation of supersonic isothermal turbulence to reconstruct an exact fourth-order relation derived analytically from the Navier–Stokes equations (Galtier & Banerjee, Phys. Rev. Lett., vol. 107, 2011, p. 134501). Our analysis supports a Kolmogorov-like inertial energy cascade in supersonic turbulence previously discussed on a phenomenological level. We show that two compressible analogues of the four-fifths law exist describing fifth- and fourth-order correlations, but only the fourth-order relation remains ‘universal’ in a wide range of Mach numbers from incompressible to highly compressible regimes. A new approximate relation valid in the strongly supersonic regime is derived and verified. We also briefly discuss the origin of bottleneck bumps in simulations of compressible turbulence.

Examination of the four-fifths law for longitudinal third-order moments in incompressible magnetohydrodynamic turbulence in a periodic box

The four-fifths law for third-order longitudinal moments is examined, using direct numerical simulation (DNS) data on three-dimensional (3D) forced incompressible magnetohydrodynamic (MHD) turbulence without a uniformly imposed magnetic field in a periodic box. The magnetic Prandtl number is set to one, and the number of grid points is 512(3). A generalized Kármán-Howarth-Kolmogorov equation for second-order velocity moments in isotropic MHD turbulence is extended to anisotropic MHD turbulence by means of a spherical average over the direction of r. Here, r is a separation vector. The viscous, forcing, anisotropic and nonstationary terms in the generalized equation are quantified. It is found that the influence of the anisotropic terms on the four-fifths law is negligible at small scales, compared to that of the viscous term. However, the influence of the directional anisotropy, which is measured by the departure of the third-order moments in a particular direction of r from the spherically averaged ones, on the four-fifths law is suggested to be substantial, at least in the case studied here.

Accurate estimation of third-order moments from turbulence measurements

Abstract. Politano and Pouquet's law, a generalization of Kolmogorov's four-fifths law to incompressible MHD, makes it possible to measure the energy cascade rate in incompressible MHD turbulence by means of third-order moments. In hydrodynamics, accurate measurement of third-order moments requires large amounts of data because the probability distributions of velocity-differences are nearly symmetric and the third-order moments are relatively small. Measurements of the energy cascade rate in solar wind turbulence have recently been performed for the first time, but without careful consideration of the accuracy or statistical uncertainty of the required third-order moments. This paper investigates the statistical convergence of third-order moments as a function of the sample size N. It is shown that the accuracy of the third-moment <(δ v||)3> depends on the number of correlation lengths spanned by the data set and a method of estimating the statistical uncertainty of the third-moment is developed. The technique is illustrated using both wind tunnel data and solar wind data.

Laws for third-order moments in homogeneous anisotropic incompressible magnetohydrodynamic turbulence

It is known that Kolmogorov's four-fifths law for statistically homogeneous and isotropic turbulence can be generalized to anisotropic turbulence. This fundamental result for homogeneous anisotropic turbulence says that in the inertial range the divergence of the vector third-order moment 〈|δv(r)|2δv(r)〉 is constant and is equal to -4ϵ, where ϵ is the dissipation rate of the turbulence. This law can be extended to incompressible magnetohydrodyamic (MHD) turbulence where statistical isotropy is often not valid due, for example, to the presence of a large-scale magnetic field. Laws for anisotropic incompressible MHD turbulence were first derived by Politano and Pouquet. In this paper, the laws for vector third-order moments in homogeneous non-isotropic incompressible MHD turbulence are derived by a technique due to Frisch that clarifies the relationship between the energy flux in Fourier space and the vector third-order moments in physical space. This derivation is different from the original derivation of Politano and Pouquet which is based on the Kármán–Howarth equation, and provides some new physical insights. Separate laws are derived for the cascades of energy, cross-helicity and magnetic-helicity, the three ideal invariants of incompressible MHD for flows in three dimensions. These laws are of fundamental importance in the theory of MHD turbulence where non-isotropic turbulence is much more prevalent than isotropic turbulence.

Third-order structure function relations for quasi-geostrophic turbulence

We derive two third-order structure function relations for quasi-geostrophic turbulence, one for the forward cascade of potential enstrophy and one for the inverse cascade of energy. These relations are the counterparts of Kolmovorov's (1941) four-fifths law for the third-order longitudinal structure functions of three-dimensional turbulence.

Energy flux fluctuations in a finite volume of turbulent flow

The flux of turbulent kinetic energy from large to small spatial scales is measured in a small domain of varying size . The probability distribution function of the flux is obtained using a time-local version of Kolmogorov four-fifths law. The measurements, made at a moderate Reynolds number, show frequent events where the flux is backscattered from small to large scales, their frequency increasing as is decreased. The observations are corroborated by a numerical simulation based on the motion of many particles and on an explicit form of the eddy damping.

Does multifractal theory of turbulence have logarithms in the scaling relations?

The multifractal theory of turbulence uses a saddle-point evaluation in determining the power-law behaviour of structure functions. Without suitable precautions, this could lead to the presence of logarithmic corrections, thereby violating known exact relations such as the four-fifths law. Using the theory of large deviations applied to the random multiplicative model of turbulence and calculating subdominant terms, we explain here why such corrections cannot be present.

“Locally homogeneous turbulence”: Is it an inconsistent framework?

In his first 1941 paper Kolmogorov assumed that the velocity has increments that are homogeneous and independent of the velocity at a suitable reference point. This assumption of local homogeneity is consistent with the nonlinear dynamics only in an asymptotic sense when the reference point is far away. This inconsistency is illustrated numerically using the Burgers equation. Kolmogorov’s derivation of the four-fifths law for the third-order structure function and its anisotropic generalization are actually valid only for homogeneous turbulence, but a local version due to Duchon and Robert still holds. A Kolmogorov-Landau approach is proposed to handle the effect of fluctuations in the large-scale velocity on small-scale statistical properties; it is only a mild extension of the 1941 theory and does not incorporate intermittency effects.

Scale-by-scale budget and similarity laws for shear turbulence

Turbulent shear flows, such as those occurring in the wall region of turbulent boundary layers, show a substantial increase of intermittency in comparison with isotropic conditions. This suggests a close link between anisotropy and intermittency. However, a rigorous statistical description of anisotropic flows is, in most cases, hampered by the inhomogeneity of the field. This difficulty is absent for homogeneous shear flow. For this flow the scale-by-scale budget is discussed here by using the appropriate form of the Kámán–Howarth equation, to determine the range of scales where the shear is dominant. The resulting generalization of the four-fifths law is then used to extend to shear-dominated flows the Kolmogorov–Oboukhov theory of intermittency. The procedure leads naturally to the formulation of generalized structure functions, and the description of intermittency thus obtained reduces to the K62 theory for vanishing shear. The intermittency corrections to the scaling exponents are related to the moments of the coarse-grained energy dissipation field. Numerical experiments give indications that the dissipation field is statistically unaffected by the shear, supporting the conjecture that the intermittency corrections are universal. This observation together with the present reformulation of the theory gives a reason for the increased intermittency observed in the classical longitudinal velocity increments.

A Langevin equation for turbulent velocity increments

Recently, Friedrich and Peinke demonstrated empirically that a Fokker–Planck equation describes the scale dependence of probability distribution functions of longitudinal velocity increments vr in fully developed turbulent flows. Thanks to the analysis of an experimental velocity signal, the stochastic process vr is further investigated by examining the related Langevin equation. This process is found to be Markovian in scale because the turbulent velocity field is correlated over distances much larger than the correlation length ρ of its spatial derivative. A Gaussian approximation for the random force yields evolution equations for the structure functions 〈vrn〉. Analytic solutions are obtained, in agreement with experimental data for even-order moments when the scale r is larger than a few times ρ. The third-order moment 〈vr3〉 is found linear in r, as predicted by Kolmogorov’s four-fifths law.

Statistical laws and vortex structures in fully developed turbulence

Fully developed turbulence is structured with a number of intense elongated vortices. Recognizing that statistical laws are related to such structures, the flow field is modeled by an ensemble of strained vortices (i.e. Burgers vortices) distributing randomly in space, from which probability density functions (pdfs) for longitudinal and transversal components of velocity difference are derived by taking statistical averages ensuring isotropy and homogeneity for the velocity field. It is found that the pdfs tend to close-to-exponential forms at small scales, and that there exist two scaling ranges in the structure function of every order, which are identified as the viscous range and inertial range, respectively, with a transition scale between the two ranges being at the order of mean size of Burgers vortices. Velocity structure functions show scaling behaviors in the second interval corresponding to the inertial range with the scaling exponents close to those known in the experiments and direct numerical simulations. It is remarkable that the Kolmogorov's four-fifths law is observed to be valid in a small-scale range. The scaling exponents of higher order structure functions are numerically estimated up to the 25th order. It is found that asymptotic scaling exponents, as the order increases, are in good agreement with the behavior of a recent experiment. The above model analysis is considered to represent successfully the statistical behaviors at small scales (possibly less than the Taylor microscale) and higher orders. The present statistical analysis leads to scale-dependent probability density functions.

Correction to the four-fifths law due to variations of the dissipation

The full Kolmogorov equation is analyzed with respect to Galilean invariance. It is shown that the time-dependent term in typical applications as jets and wakes will cause rather large deviations from the four-fifths law, even at fairly large Reynolds numbers. With the use of the K−ε-model an inertial range model expression is derived for the third-order structure function of freely decaying turbulence at finite Reynolds number. The expression is compared with results from previously reported experiments. It is argued that experiments to test intermittency corrections to the Kolmogorov 1941 theory should be made in flows which are both stationary and homogeneous in the mean flow direction, such as pipe and channel flows.

Turbulence in two-dimensional forced electron-magnetohydrodynamics

2D-EMHD turbulence is studied in the freely decaying case and in the random forced one. The analogue of the Kolmogorov’s four-fifths law is derived under the assumptions of statical homogeneity and isotropy. Direct numerical simulations show that it holds true for different values of the adimensional electron inertial length scale de. In the de ∼ 1 regime, the energy spectrum is found to be close to the Kolmogorov spectrum.

Statistical Laws of Random Strained Vortices in Turbulence

Statistical properties of random distribution of strained vortices (Burgers' vortices) in turbulence are studied, and the scaling behavior of structure functions is investigated. It is found within the scale range of interest (corresponding to the inertial range) that the third-order structure function is negative and the scaling exponent is nearly unity in accordance with Kolmogorov's four-fifths law. The inertial-range scaling exponents are estimated up to the 25th order, which are in good agreement with those obtained from experiments and direct numerical simulations once the probability distribution of the vortex strength is taken into account.