Kolmogorov Phenomenology Research Articles

Solar Cycle Dependence of the Turbulence Cascade Rate at 1 au

We study the solar cycle dependence of various turbulence cascade rates based on the methodology developed by Adhikari et al. that utilizes Kolmogorov phenomenology. This approach is extended to derive the heating rates for an Iroshnikov–Kriachnan (IK) phenomenology. The observed turbulence cascade rates corresponding to the total turbulence energy, fluctuating magnetic energy density, fluctuating kinetic energy, and the normalized cross helicity are derived from WIND spacecraft plasma and magnetometer data from 1995 through 2020. We find that (i) the turbulence cascade rate derived from a Kolmogorov phenomenology and an IK phenomenology changes with solar cycle, such that the cascade rate is largest during solar maximum and smallest during solar minimum; (ii) the turbulence energy Kolmogorov cascade rate increases from θ UB (angle between mean magnetic field and velocity) = 0° to 90° and peaks near θ UB = 90°, and then decreases as θ UB tends to 180°; (iii) the 2D turbulence heating rate is larger than the slab heating rate; (iv) the 2D and slab fluctuating magnetic energy density cascade rates are larger than the corresponding cascade rates of the fluctuating kinetic energy; and (v) the total turbulence energy cascade rate is positively correlated with the solar wind speed and temperature and the normalized cross-helicity cascade rate. Finally, we find that the total turbulent energy Kolmogorov cascade rate is larger than the IK cascade rate.

Small-scale intermittency of premixed turbulent flames

Premixed turbulent flames, encountered in power generation and propulsion engines, are an archetype of a randomly advected, self-propagating surface. While such a flame is known to exhibit large-scale intermittent flapping, the possible intermittency of its small-scale fluctuations has been largely disregarded. Here, we experimentally reveal the inner intermittency of a premixed turbulent V-flame, while clearly distinguishing this small-scale feature from large-scale outer intermittency. From temporal measurements of the fluctuations of the flame, we find a frequency spectrum that has a power-law subrange with an exponent close to $-2$ , which is shown to follow from Kolmogorov phenomenology. Crucially, however, the moments of the temporal increment of the flame position are found to scale anomalously, with exponents that saturate at higher orders. This signature of small-scale inner intermittency is shown to originate from high-curvature, cusp-like structures on the flame surface, which have significance for modelling the heat release rate and other key properties of premixed turbulent flames.

Two-time Lagrangian velocity correlation function for particle pairs in two-dimensional inverse energy-cascade turbulence

We numerically investigate a two-time Lagrangian velocity correlation function (TTLVCF) for particle pairs in two-dimensional energy inverse-cascade turbulence. We consider self similarity of the correlation function by means of incomplete similarity. In this framework, we propose a self-similar form of the correlation function, whose scaling exponents cannot be determined by only using the dimensional analysis based on the Kolmogorov's phenomenology. As a result, the scaling laws of the correlation function can depend on the initial separation. This initial-separation dependency is frequently observed in laboratory experiments and direct numerical simulations of the relative dispersion, which is directly related to the correlation function, at moderate Reynolds numbers. We numerically verify the self-similar form by direct numerical simulations of two-dimensional energy inverse-cascade turbulence. The involved scaling exponents and the dependencies on finite Reynolds number effects are determined empirically. Then, we consider implication of the scaling laws of the correlation function on the relative dispersion, i.e. the Richardson-Obukhov $t^3$ law. Our results suggest a possibility not to recover the Richardson-Obukhov $t^3$ law at infinite Reynolds number.

A differential approximation model for passive scalar turbulence

Two-dimensional passive scalar turbulence is studied by means of a k-space diffusion model based on a third order differential approximation. This simple description of local nonlinear interactions in Fourier space is shown to provide a general expression, in line with previous seminal works, and appears to be suitable for various 2D turbulence problems. Steady state solutions for the spectral energy density of the flow are shown to recover the Kraichnan–Kolmogorov phenomenology of the dual cascade, while various passive scalar spectra, such as Batchelor or Obukhov–Corssin spectra are recovered as steady state solutions of the spectral energy density of the passive scalar. These analytical results are then corroborated by numerical solutions of the time evolving problem with energy and passive scalar injection and dissipation on a logarithmic wavenumber space grid over a large range of scales. The particular power law spectra that are obtained are found to depend mainly on the location of the kinetic and passive scalar energy injections. In particular, it is shown that by injecting the energy simultaneously at large and small scales, wave number spectra consistent with those of Nastrom–Gage can be obtained.

Non-Kolmogorov scaling for two-particle relative velocity in two-dimensional inverse energy-cascade turbulence

Herein,we numerically examine the relative dispersion of Lagrangian particle pairs in two-dimensional inverse energy-cascade turbulence. Behind the Richardson-Obukhov $t^3$ law of relative separation, we discover that the second-order moment of the relative velocity have a temporal scaling exponent different from the prediction based on the Kolmogorov's phenomenology. The results also indicate that time evolution of the probability distribution function of the relative velocity is self-similar. The findings are obtained by enforcing Richardson-Obukhov law either by considering a special initial separation or by conditional sampling. In particular, we demonstrate that the conditional sampling removes the initial-separation dependence of the statistics of the separation and relative velocity. Furthermore, we demonstrate that the conditional statistics are robust with respect to the change in the parameters involved, and that the number of the removed pairs from the sampling decreases when the Reynolds number increases. We also discuss the insights gained as a result of conditional sampling.

Damping Functions correct over‐dissipation of the Smagorinsky Model

This paper studies the time‐averaged energy dissipation rate ⟨εSMD(u)⟩ for the combination of the Smagorinsky model and damping function. The Smagorinsky model is well known to over‐damp. One common correction is to include damping functions that reduce the effects of model viscosity near walls. Mathematical analysis is given here that allows evaluation of ⟨εSMD(u)⟩ for any damping function. Moreover, the analysis motivates a modified van Driest damping. It is proven that the combination of the Smagorinsky with this modified damping function does not over‐dissipate and is also consistent with Kolmogorov phenomenology. Copyright © 2017 John Wiley & Sons, Ltd.

Scaling laws and bounds for the turbulent G.O. Roberts dynamo

Exact bounds are derived for the total energy in magnetohydrodynamic flows generating a magnetic field, and they conform with the Kolmogorov phenomenology of turbulence. Rigorous bounds interpreted as equalities agree to within an order of magnitude with numerical simulations of dynamos with a helical flow field.

Phenomenological Blasius-type friction equation for turbulent power-law fluid flows.

We propose a friction formula for turbulent power-law fluid flows, a class of purely viscous non-Newtonian fluids commonly found in applications. Our model is derived through an extension of the friction factor analysis based on Kolmogorov's phenomenology, recently proposed by Gioia and Chakraborty. Tests against classical empirical data show excellent agreement over a significant range of Reynolds number. Limits of the model are also discussed.

Transition of the scaling law in inverse energy cascade range caused by a nonlocal excitation of coherent structures observed in two-dimensional turbulent fields

We numerically investigate the inverse energy cascade range of two-dimensional Navier-Stokes turbulence. Our focus is on the universality of the Kolmogorov's phenomenology. In our direct numerical simulations, two types of forcing processes, the random forcing and the deterministic forcing, are employed besides the systematically varied numerical parameters. We first calculate the two-dimensional Navier-Stokes equations and confirm that results in the quasi steady state are consistent with the classical phenomenology for both types of forcing processes. It is also found that the difference in forcing process appears after the inverse energy cascade range reaches the system size; the dipole coherent vortices emerge and grow only when the random forcing is adopted. Then we add a large-scale drag term to the Navier-Stokes equations to obtain the statistically stationary state. When the random forcing is used, the scaling exponent of the energy spectrum in the stationary state starts to differ from the predicted -5/3 in the inverse energy cascade range as the infrared Reynolds number Re(d) increases, where Re(d) is defined as k(f)/k(d) with the forcing wave number k(f) and the large-scale drag wave number k(d). That can be interpreted as a transition phenomenon in which the local maximum vorticity grows like an order parameter caused by excitation of strong coherent vortices. Strong coherent vortices emerge and grow after the quasi steady state and destroy the scaling law when Re(d) is over a critical value. These coherent vortices are not due to the finite-size effect, unlike the dipole coherent vortices. On the other hand, when the deterministic forcing is adopted, strong coherent vortices are hardly seen and the -5/3 scaling law holds independently of Re(d). We examine the cases of the combination of both types of forcing processes and find that formation of such coherent vortices is sensitive to the mechanism of the external forcing process as well as the numerical parameters. Several types of large-scale drag terms are also tested and their insignificant influence on these qualitative properties is revealed.

A dynamical systems approach to fluid turbulence

This study presents a theoretical approach to fluid turbulence as an alternative to Kolmogorov's phenomenology. The new approach uses the basic elements and concepts of dynamical systems theory and applies them to a variety of fluid models, allowing us to recover key predictions made by the classical method. These include the number of degrees of freedom, the dissipation wavenumber and the exponent of the power-law energy spectrum of the inertial range. The two-dimensional magnetohydrodynamic system at unity magnetic Prandtl number is used as an illustrative example, with the theoretical predictions corroborated by numerical results.

Universal scaling laws for fully-developed magnetic field turbulence near and far upstream of the Earth’s bow shock

We analyze the multifractal scaling of the modulus of the interplanetary magnetic field near and far upstream of the Earth’s bow shock, measured by Cluster and ACE, respectively, from 1 to 3 February 2002. The maximum order of the structure function is carefully estimated for each time series using two different techniques, to ensure the validity of our high-order statistics. The first technique consists of plotting the integrand of the pth order structure function, and the second technique is a quantitative method which relies on the power-law scaling of the extreme events. We compare the scaling exponents computed from the structure functions of magnetic field differences with the predictions obtained by the She–Lévêque model of intermittency in anisotropic magnetohydrodynamic turbulence. Our results show a good agreement between the model and the observations near and far upstream of the Earth’s bow shock, rendering support for the modelling of universal scaling laws based on the Kolmogorov phenomenology in the presence of sheet-like dissipative structures.

Vortex events in Euler and Navier–Stokes simulations with smooth initial conditions

AbstractWe present high-resolution numerical simulations of the Euler and Navier–Stokes equations for a pair of colliding dipoles. We study the possible approach to a finite-time singularity for the Euler equations, and contrast it with the formation of developed turbulence for the Navier–Stokes equations. We present numerical evidence that seems to suggest the existence of a blow-up of the inviscid velocity field at a finite time (${t}_{s} $) with scaling $\vert u\vert _{\infty } \ensuremath{\sim} \mathop{ ({t}_{s} \ensuremath{-} t)}\nolimits ^{\ensuremath{-} 1/ 2} $, $\vert \omega \vert _{\infty } \ensuremath{\sim} \mathop{ ({t}_{s} \ensuremath{-} t)}\nolimits ^{\ensuremath{-} 1} $. This blow-up is associated with the formation of a ${k}^{\ensuremath{-} 3} $ spectral range, at least for the finite range of wavenumbers that are resolved by our computation. In the evolution toward ${t}_{s} $, the total enstrophy is observed to increase at a slower rate, $\Omega \ensuremath{\sim} \mathop{ ({t}_{s} \ensuremath{-} t)}\nolimits ^{\ensuremath{-} 3/ 4} $, than would naively be expected given the behaviour of the maximum vorticity, ${\omega }_{\infty } \ensuremath{\sim} \mathop{ ({t}_{s} \ensuremath{-} t)}\nolimits ^{\ensuremath{-} 1} $. This indicates that the blow-up would be concentrated in narrow regions of the flow field. We show that these regions have sheet-like structure. Viscous simulations, performed at various $\mathit{Re}$, support the conclusion that any non-zero viscosity prevents blow-up in finite time and results in the formation of a dissipative exponential range in a time interval around the estimated inviscid ${t}_{s} $. In this case the total enstrophy saturates, and the energy spectrum becomes less steep, approaching ${k}^{\ensuremath{-} 5/ 3} $. The simulations show that the peak value of the enstrophy scales as ${\mathit{Re}}^{3/ 2} $, which is in accord with Kolmogorov phenomenology. During the short time interval leading to the formation of an inertial range, the total energy dissipation rate shows a clear tendency to become independent of $\mathit{Re}$, supporting the validity of Kolmogorov’s law of finite energy dissipation. At later times the kinetic energy shows a ${t}^{\ensuremath{-} 1. 2} $ decay for all $\mathit{Re}$, in agreement with experimental results for grid turbulence. Visualization of the vortical structures associated with the stages of vorticity amplification and saturation show that, prior to ${t}_{s} $, large-scale and the small-scale vortical structures are well separated. This suggests that, during this stage, the energy transfer mechanism is non-local both in wavenumber and in physical space. On the other hand, as the spectrum becomes shallower and a ${k}^{\ensuremath{-} 5/ 3} $ range appears, the energy-containing eddies and the small-scale vortices tend to be concentrated in the same regions, and structures with a wide range of sizes are observed, suggesting that the formation of an inertial range is accompanied by transfer of energy that is local in both physical and spectral space.

Simulating supersonic turbulence in magnetized molecular clouds

We present results of large-scale three-dimensional weakly magnetized supersonic turbulence simulations with an isothermal equation of state at grid resolutions up to 10243 cells with the Piecewise Parabolic Method on a Local Stencil. The turbulence is driven by a large-scale isotropic solenoidal force in a periodic computational domain and fully develops in a few flow crossing times. We then evolve the flow for a number of flow crossing times and analyze various statistical properties of the saturated turbulent state. We show that the energy transfer rate in the inertial range of scales is surprisingly close to a constant, indicating that Kolmogorov's phenomenology for incompressible turbulence can be extended to magnetized supersonic flows. We also discuss numerical dissipation effects and convergence of different turbulence diagnostics as grid resolution refines from 2563 to 10243 cells.

A simple model for the streamwise fluctuations in the log-law region of a boundary layer

We build on the work of Davidson et al. [Phys. Fluids 18, 065112 (2006)] and propose an elementary model for the log-law region of a boundary layer. The model is remarkably simple, contains only one free parameter (which we set equal to unity) and, unlike Davidson et al., assumes very little about the shape of the boundary-layer eddies. The physical content of the model is simple: we assume that the two-point statistics of the streamwise velocity fluctuations know about the presence of the wall only to the extent that, over a range of eddy sizes, it imposes a kinetic energy scale proportional to the square of the shear velocity. Little else is assumed, other than classic Kolmogorov phenomenology for the small scales. Despite its naivety, and the minimal number of free parameters, the model is a good fit to experimental data for the k−1 law of the one-dimensional, longitudinal spectrum, Φuu(k), and also to Φuu(k) in the inertial range. It is also an excellent fit to experimental data for the real-space analog of the k−1 law; that is, the logarithmic law for the longitudinal structure function. In addition, the model predicts the cross-stream variation of the variance of the streamwise velocity fluctuations, ⟨ux2⟩, and, to within an additive constant, it too is a reasonable fit to the data. Our model differs from the classic analyses of Townsend [J. Fluid Mech. 165, 163 (1986)] and Perry et al. [J. Fluid Mech. 165, 163 (1986)] in three respects. First, we do not invoke the attached eddy hypothesis, which assumes that the vortical structures in the log-layer extend to the wall. Second, we predict the precise shape of Φuu(k) for all values of k and not just the k−1 region. Third, there are three adjustable parameters in the theory of Perry et al., whereas we have only one. It is instructive that a model which contains very little information about the morphology of the vorticity field, and has only one free parameter, is able to reproduce both the structural form for, and magnitude of, many features of the one-dimensional spectrum and the second-order structure function. This suggests that these two measures of the scale-by-scale energy distribution are insensitive to the detailed structure of the vorticity field. We argue that this is indeed the case.

Energy Cascades in a Partially Ionized Astrophysical Plasma

A local turbulence model is developed to study energy cascades in the interstellar medium (ISM) based on self-consistent two-dimensional fluid simulations. The model describes a partially ionized magnetofluid interstellar medium (ISM) that couples a neutral hydrogen fluid with a plasma primarily through charge exchange interactions. Charge exchange interactions are ubiquitous in warm ISM plasma and the strength of the interaction depends largely on the relative speed between the plasma and the neutral fluid. Unlike small length-scale linear collisional dissipation in a single fluid, charge exchange processes introduce channels that can be effective on a variety of length-scales that depend on the neutral and plasma densities, temperature, relative velocities, charge exchange cross section and the characteristic length scales. We find, from scaling arguments and nonlinear coupled fluid simulations, that charge exchange interactions modify spectral transfer associated with large scale energy containing eddies. Consequently, the warm ISM turbulent cascade rate prolongs spectral transfer amongst inertial range turbulent modes. Turbulent spectra associated with the neutral and plasma ISM fluids are therefore steeper than those predicted by Kolmogorov's phenomenology.

The Statistics of Supersonic Isothermal Turbulence

We present results of large-scale three-dimensional simulations of supersonic Euler turbulence with the piecewise parabolic method and multiple grid resolutions up to 20483 points. Our numerical experiments describe nonmagnetized driven turbulent flows with an isothermal equation of state and an rms Mach number of 6. We discuss numerical resolution issues and demonstrate convergence, in a statistical sense, of the inertial range dynamics in simulations on grids larger than 5123 points. The simulations allowed us to measure the absolute velocity scaling exponents for the first time. The inertial range velocity scaling in this strongly compressible regime deviates substantially from the incompressible Kolmogorov laws. The slope of the velocity power spectrum, for instance, is -1.95 compared to -5/3 in the incompressible case. The exponent of the third-order velocity structure function is 1.28, while in incompressible turbulence it is known to be unity. We propose a natural extension of Kolmogorov's phenomenology that takes into account compressibility by mixing the velocity and density statistics and preserves the Kolmogorov scaling of the power spectrum and structure functions of the density-weighted velocity υ ≡ ρ1/3u. The low-order statistics of υ appear to be invariant with respect to changes in the Mach number. For instance, at Mach 6 the slope of the power spectrum of υ is -1.69, and the exponent of the third-order structure function of υ is unity. We also directly measure the mass dimension of the "fractal" density distribution in the inertial subrange, Dm ≈ 2.4, which is similar to the observed fractal dimension of molecular clouds and agrees well with the cascade phenomenology.

Cluster–cluster aggregation as an analogue of a turbulent cascade: Kolmogorov phenomenology, scaling laws and the breakdown of self-similarity

We present a detailed study of the statistical properties of a system of diffusing aggregating particles in the presence of a steady source of monomers. We emphasize the case of low spatial dimensions where strong diffusive fluctuations invalidate the mean-field description provided by standard Smoluchowski kinetic theory. The presence of a source of monomers allows the system to reach a statistically stationary state at large times. This state is characterized by a constant flux of mass directed from small to large masses. It therefore admits a phenomenological description based on the assumption of self-similarity and constant mass flux analogous to the Kolmogorov’s 1941 theory of turbulence. Unlike turbulence, the aggregation problem is analytically tractable using powerful methods of statistical field theory. We explain in detail how these methods should be adapted to study the far-from-equilibrium, flux-dominated states characteristic of turbulent systems. We consider multipoint correlation functions of the mass density. By an exact evaluation of the scaling exponents for the one and two-point correlation functions, we show that the assumption of self-similiarity breaks down at large masses for spatial dimensions, d ≤ 2 . We calculate non-rigorously the exponents of the higher order correlation functions as an ϵ -expansion where ϵ = 2 − d . We show that the mass distribution exhibits non-trivial multiscaling. An analogy can be drawn with the case of hydrodynamic turbulence. The physical origin of this multiscaling is traced to the presence of strong correlations between particles participating in large mass aggregation events. These correlations stem from the recurrence of diffusion processes in d ≤ 2 . The analytic methods developed here will have more general applicability beyond the study of this specific problem.

Cascade time scales for energy and helicity in homogeneous isotropic turbulence.

We extend the Kolmogorov phenomenology for the scaling of energy spectra in high-Reynolds-number turbulence, to explicitly include the effect of helicity. There exists a time scale tau(H) for helicity transfer in homogeneous, isotropic turbulence with helicity. We arrive at this time scale using the phenomenological arguments used by Kraichnan to derive the time scale tau(E) for energy transfer [J. Fluid Mech. 47, 525 (1971)]]. We show that in general tau(H) may not be neglected compared to tau(E), even for rather low relative helicity. We then deduce an inertial range joint cascade of energy and helicity in which the dynamics are dominated by tau(E) in the low wave numbers with both energy and helicity spectra scaling as k(-5/3); and by tau(H) at larger wave numbers with spectra scaling as k(-4/3). We demonstrate how, within this phenomenology, the commonly observed "bottleneck" in the energy spectrum might be explained. We derive a wave number k(h) which is less than the Kolmogorov dissipation wave number, at which both energy and helicity cascades terminate due to dissipation effects. Data from direct numerical simulations are used to check our predictions.

Self-Similar Energy Decay in Magnetohydrodynamic Turbulence

The self-similar decay of energy in magnetohydrodynamic flows is examined in the spirit of the Kolmogorov phenomenology while incorporating in the expression for the energy transfer to small scales the interactions between turbulent eddies and Alfv\'en waves due to a large-scale magnetic field. The model is parameter-free and does not rely on the existence of the several invariants of ideal MHD except energy. High-resolution numerical simulations in two dimensions confirm its validity in a way that leads as expected to a decay significantly slower than for neutral fluids.

Symmetry and Scaling of Turbulent Mixing.

The stationary condition (Hopf equation) for the $(n+1)$-point correlation function of a passive scalar advected by turbulent flow is argued to have an approximate $\mathrm{SL}\left(n,R\right)$ symmetry which provides a starting point for our phenomenological theory in which less symmetric terms are treated perturbatively. The large scale anisotropy is found to be a relevant field, in contradiction with Kolmogorov phenomenology, but in agreement with the large scalar skewness observed in shear flows. Exponents are not universal, yet quantitative predictions for experiments to test the $\mathrm{SL}\left(n,R\right)$ symmetry can be formulated in terms of the correlation functions.