Inertial Range Scaling Research Articles

Effects of fluctuating energy input on the small scales in turbulence

AbstractIn the standard cascade picture of three-dimensional turbulent fluid flows, energy is input at a constant rate at large scales. Energy is then transferred to smaller scales by an intermittent process that has been the focus of a vast literature. However, the energy input at large scales is not constant in most real turbulent flows. We explore the signatures of these fluctuations of large-scale energy input on small-scale turbulence statistics. Measurements were made in a flow between oscillating grids, with ${R}_{\lambda } $ up to 262, in which temporal variations in the large-scale energy input can be introduced by modulating the oscillating grid frequency. We find that the Kolmogorov constant for second-order longitudinal structure functions depends on the magnitude of the fluctuations in the large-scale energy input. We can quantitatively predict the measured change with a model based on Kolmogorov’s refined similarity theory. The effects of fluctuations of the energy input can also be observed using structure functions conditioned on the instantaneous large-scale velocity. A linear parametrization using the curvature of the conditional structure functions provides a fairly good match with the measured changes in the Kolmogorov constant. Conditional structure functions are found to provide a more sensitive measure of the presence of fluctuations in the large-scale energy input than inertial range scaling coefficients.

On the universality of supersonic turbulence

Compressible turbulence shapes the structure of the interstellar medium of our Galaxy and likely plays an important role also during structure formation in the early Universe. The density PDF and the power spectrum of such compressible, supersonic turbulence are the key ingredients for theories of star formation. However, both the PDF and the spectrum are still a matter of debate, because theoretical predictions are limited and simulations of supersonic turbulence require enormous resolutions to capture the inertial-range scaling. To advance our limited knowledge of compressible turbulence, we here present and analyse the world's largest simulations of supersonic turbulence. We compare hydrodynamic models with numerical resolutions of 256^3-4096^3 mesh points and with two distinct driving mechanisms, solenoidal (divergence-free) driving and compressive (curl-free) driving. We find convergence of the density PDF, with compressive driving exhibiting a much wider and more intermittent density distribution than solenoidal driving. Analysing the power spectrum of the turbulence, we find a pure velocity scaling close to Burgers turbulence with P(v) k^(-2) for both driving modes in our hydrodynamical simulations with Mach = 17. The spectrum of the density-weighted velocity rho^(1/3)v, however, does not provide the previously suggested universal scaling for supersonic turbulence. We find that the power spectrum P(rho^(1/3)v) scales with wavenumber as k^(-1.74) for solenoidal driving, close to incompressible Kolmogorov turbulence, k^(-5/3), but is significantly steeper with k^(-2.10) for compressive driving. We show that this is consistent with a recent theoretical model for compressible turbulence that predicts P(rho^(1/3)v) k^(-19/9) in the presence of a strong div(v) component as is produced by compressive driving and remains remarkably constant throughout the supersonic turbulent cascade.

On the strength of the nonlinearity in isotropic turbulence

AbstractTurbulence governed by the Navier–Stokes equations shows a tendency to evolve towards a state in which the nonlinearity is diminished. In fully developed turbulence, this tendency can be measured by comparing the variance of the nonlinear term to the variance of the same quantity measured in a Gaussian field with the same energy distribution. In order to study this phenomenon at high Reynolds numbers, a version of the direct interaction approximation is used to obtain a closed expression for the statistical average of the mean-square nonlinearity. The wavenumber spectrum of the mean-square nonlinear term is evaluated and its scaling in the inertial range is investigated as a function of the Reynolds number. Its scaling is dominated by the sweeping by the energetic scales, but this sweeping is weaker than predicted by a random sweeping estimate. At inertial range scales, the depletion of nonlinearity as a function of the wavenumber is observed to be constant. At large scales it is observed that the mean-square nonlinearity is larger than its Gaussian estimate, which is shown to be related to the non-Gaussianity of the Reynolds-stress fluctuations at these scales.

Advection of passive magnetic field by the Gaussian velocity field with finite correlations in time and spatial parity violation

Using the field theoretic renormalization group technique the model of passively advected weak magnetic field by an incompressible isotropic helical turbulent flow is investigated up to the second order of the perturbation theory (two-loop approximation) in the framework of an extended Kazantsev-Kraichnan model of kinematic magnetohydrodynamics. Statistical fluctuations of the velocity field are taken in the form of a Gaussian distribution with zero mean and defined noise with finite correlations in time. The two-loop analysis of all possible scaling regimes is done and the influence of helicity on the stability of scaling regimes is discussed and shown in the plane of exponents ɛ − η, where ɛ characterizes the energy spectrum of the velocity field in the inertial range E ∞ k 1 − 2e, and η is related to the correlation time at the wave number k which is scaled as k −2 + η. It is shown that in non-helical case the scaling regimes of the present vector model are completely identical and have also the same properties as those obtained in the corresponding model of passively advected scalar field. Besides, it is also shown that when the turbulent environment under consideration is helical then the properties of the scaling regimes in models of passively advected scalar and vector (magnetic) fields are essentially different. The results demonstrate the importance of the presence of a symmetry breaking in a given turbulent environment for investigation of the influence of an internal tensor structure of the advected field on the inertial range scaling properties of the model under consideration and will be used in the analysis of the influence of helicity on the anomalous scaling of correlation functions of passively advected magnetic field.

Field Measurements of Wind Turbine Wakes with Lidars

AbstractField measurements of the wake flow produced from a 2-MW Enercon E-70 wind turbine were performed using three scanning Doppler wind lidars. A GPS-based technique was used to determine the position of the wind turbine and the wind lidar locations, as well as the direction of the laser beams. The lidars used in this study are characterized by a high spatial resolution of 18 m, which allows the detailed characterization of the wind turbine wake. Two-dimensional measurements of wind speed were carried out by scanning a single lidar over the vertical symmetry plane of the wake. The mean axial velocity field was then retrieved by averaging 2D scans performed consecutively. To investigate wake turbulence, single lidar measurements were performed by staring the laser beam at fixed directions and using the maximum sampling frequency. From these tests, peaks in the velocity variance are detected within the wake in correspondence of the turbine top tip height; this enhanced turbulence could represent a source of dangerous fatigue loads for downstream turbines. The spectral density of the measured velocity fluctuations shows a clear inertial-range scaling behavior. Then, simultaneous measurements with two lidars were performed in order to characterize both the axial and the vertical velocity components. For this setup, the two velocity components were retrieved only for measurement points for which the two laser beams crossed nearly at a right angle. Statistics were computed over the sample set for both velocity components, and they showed strong flow fluctuations in the near-wake region at turbine top tip height, with a turbulence intensity of about 30%.

On the Relation between Small-scale Intermittency and Shocks in Turbulent Flows

High Reynolds number turbulence is characterized by extreme fluctuations of velocity gradients which can interact with shock waves in compressible flows. While these processes are traditionally thought to happen at very disparate range of scales, both turbulence gradients as well as shock gradients become stronger as the Reynolds number increases. Our interest here is to in- vestigate their relation in the high-Reynolds number limit. Our conclusion is that for intermittent turbulence with inertial range scaling exponents which grow more slowly than linear at asymptotically high orders, small-scale intermittency produces gradients which are commensurate with shocks. This result is interpreted in the context of shock-turbulence interactions where intermittency appears to be responsible, in part, for the holes observed in shocks from simulations and experiments. This effect is aided by the correlation between strong gradients and flow retardation ahead of the shock which is observed from analysis of our direct numerical simulation database of incompressible and compressible turbulence.

Structure Function Analysis of Two-Scale Scalar Ramps. Part I: Theory and Modelling

Structure functions are used to study the dissipation and inertial range scales of turbulent energy, to parametrize remote turbulence measurements, and to characterize ramp features in the turbulent field. Ramp features are associated with turbulent coherent structures, which dominate energy and mass fluxes in the atmospheric surface layer. The analysis of structure functions to identify ramp characteristics is used in surface renewal methods for estimating fluxes. It is unclear how commonly observed different scales of ramp-like shapes (i.e., smaller ramps and spikes embedded in larger ramps) influence structure function analysis. Here, we examine the impact of two ramp-like scales on structure function analysis using artificially generated data. The range of time lags in structure function analysis was extended to include time lags typically associated with isotropic turbulence to those larger than the ramp durations. The Van Atta procedure (Arch Mech 29:161–171, 1977) has been expanded here to resolve the characteristics of two-scale ramp models. This new method accurately, and in some cases, exactly determines the amplitude and duration of both ramp scales. Spectral analysis was applied to the structure functions for a broad range of time lags to provide qualitative support for the expanded Van Atta procedure results. The theory reported here forms the foundation for novel methods of analyzing turbulent coherent structures.

On energetics and inertial-range scaling laws of two-dimensional magnetohydrodynamic turbulence

AbstractWe study two-dimensional magnetohydrodynamic turbulence, with an emphasis on its energetics and inertial-range scaling laws. A detailed spectral analysis shows that dynamo triads (those converting kinetic into magnetic energy) are associated with a direct magnetic energy flux while anti-dynamo triads (those converting magnetic into kinetic energy) are associated with an inverse magnetic energy flux. As both dynamo and anti-dynamo interacting triads are integral parts of the direct energy transfer, the anti-dynamo inverse flux partially neutralizes the dynamo direct flux, arguably resulting in relatively weak direct energy transfer and giving rise to dynamo saturation. This result is consistent with a qualitative prediction of energy transfer reduction due to Alfvén wave effects by the Iroshnikov–Kraichnan theory (which was originally formulated for magnetohydrodynamic turbulence in three dimensions). We numerically confirm the correlation between dynamo action and direct magnetic energy flux and investigate the applicability of quantitative aspects of the Iroshnikov–Kraichnan theory to the present case, particularly its predictions of energy equipartition and ${k}^{\ensuremath{-} 3/ 2} $ spectra in the energy inertial range. It is found that for turbulence satisfying the Kraichnan condition of magnetic energy at large scales exceeding total energy in the inertial range, the kinetic energy spectrum, which is significantly shallower than ${k}^{\ensuremath{-} 3/ 2} $, is shallower than its magnetic counterpart. This result suggests no energy equipartition. The total energy spectrum appears to depend on the energy composition of the turbulence but is clearly shallower than ${k}^{\ensuremath{-} 3/ 2} $ for $r\approx 2$, even at moderate resolutions. Here $r\approx 2$ is the magnetic-to-kinetic energy ratio during the stage when the turbulence can be considered fully developed. The implication of the present findings is discussed in conjunction with further numerical results on the dependence of the energy dissipation rate on resolution.

Large Eddy Simulation of Turbulent Reacting Mixing of Liquids in a Coaxial Jet Mixer

AbstractLarge eddy simulation is used to investigate passive and reactive scalar mixing at high Reynolds Re and Schmidt Sc numbers in order to prove capability of the LES‐SGS micromixing approaches based on the eddy dissipation and DQMOM‐IEM models properly simulating liquid reacting flows. Simulations were performed for a fast neutralization reaction in a confined jet reactor. The mean profiles for passive scalar agree well with measurements. It was shown that the most contribution to the scalar variance is made by large scale motions whereas the contribution of fine scales smaller than typical inertial range scales is negligible. Thus, the existing LES models are capable of predicting the scalar variance at large Sc numbers. The results obtained for reactive transport revealed discrepancies in the determination of micromixing rate and product concentration. A special study was performed to investigate the dynamics of fine structures using locally refined box embedded into global grid. Typical statistical properties of fine structures were reproduced numerically.

Reynolds number effect on the velocity increment skewness in isotropic turbulence

Second and third order longitudinal structure functions and wavenumber spectra of isotropic turbulence are computed using the eddy-damped quasi-normal Markovian model (EDQNM) and compared to results of the multifractal formalism. It is shown that both the multifractal model and EDQNM give power-law corrections to the inertial range scaling of the velocity increment skewness. For the multifractal formalism, this is an intermittency correction that persists at any high Reynolds number. For EDQNM, this correction is a finite Reynolds number effect, and it is shown that very high Reynolds numbers are needed for this correction to become insignificant with respect to intermittency corrections. Furthermore, the two approaches yield realistic behavior of second and third order statistics of the velocity fluctuations in the dissipative and near-dissipative ranges. Similarities and differences are highlighted, in particular, the Reynolds number dependence.

Implementation and Evaluation of Dynamic Subfilter-Scale Stress Models for Large-Eddy Simulation Using WRF*

Abstract The performance of a range of simple to moderately-complex subfilter-scale (SFS) stress models implemented in the Weather Research and Forecasting (WRF) model is evaluated in large-eddy simulations of neutral atmospheric boundary layer flow over both a flat terrain and a two-dimensional symmetrical transverse ridge. Two recently developed dynamic SFS stress models, the Lagrangian-averaged scale-dependent (LASD) dynamic model and the dynamic reconstruction model (DRM), are compared with the WRF model’s existing constant-coefficient linear eddy-viscosity and (as of version 3.2) nonlinear SFS stress models to evaluate the benefits of more sophisticated and accurate, but also more computationally expensive approaches. Simulation results using the different SFS stress models are compared among each other, as well as against the Monin–Obukhov similarity theory. For the flat terrain case, vertical profiles of mean wind speed from the newly implemented dynamic models show the best agreement with the similarity solution, improving even upon the nonlinear model, which likewise yields a significant improvement compared to the Smagorinsky model. The more sophisticated SFS stress models more successfully predict the expected production and inertial range scaling of power spectra, especially near the surface, with the dynamic models achieving the best scaling overall. For the transverse ridge case, the nonlinear model predicts the greatest amount of reverse flow in the lee of the ridge, and also demonstrates the greatest ability to duplicate qualitative features of the highest-resolution simulations at coarser resolutions. The dynamic models’ flow distributions in the lee of the ridge did not differ significantly from the constant-coefficient Smagorinsky model.

TURBULENT CLUSTERING OF PROTOPLANETARY DUST AND PLANETESIMAL FORMATION

We study clustering of inertial particles in turbulent flows and discuss its applications to dust particles in protoplanetary disks. Using numerical simulations, we compute the radial distribution function (RDF), which measures the probability of finding particle pairs at given distances, and the probability density function of the particle concentration. The clustering statistics depend on the Stokes number, $St$, defined as the ratio of the particle friction timescale, $\tau_{\rm p} $, to the Kolmogorov timescale in the flow. In the dissipation range, the clustering intensity strongly peaks at $St \simeq 1$, and the RDF for $St \sim 1$ shows a fast power-law increase toward small scales, suggesting that turbulent clustering may considerably enhance the particle collision rate. Clustering at inertial-range scales is of particular interest to the problem of planetesimal formation. At these scales, the strongest clustering is from particles with $\tau_{\rm p}$ in the inertial range. Clustering of these particles occurs primarily around a scale where the eddy turnover time is $\sim\tau_{\rm p}$. Particles of different sizes tend to cluster at different locations, leading to flat RDFs between different particles at small scales. In the presence of multiple particle sizes, the overall clustering strength decreases as the particle size distribution broadens. We discuss particle clustering in recent models for planetesimal formation. We point out that, in the model based on turbulent clustering of chondrule-size particles, the probability of finding strong clusters that can seed planetesimals may have been significantly overestimated. We discuss various clustering mechanisms in simulations of planetesimal formation by gravitational collapse of dense clumps of meter-size particles, in particular the contribution from turbulent clustering due to the limited numerical resolution.

EXTENDED SCALING LAWS IN NUMERICAL SIMULATIONS OF MAGNETOHYDRODYNAMIC TURBULENCE

Magnetised turbulence is ubiquitous in astrophysical systems, where it notoriously spans a broad range of spatial scales. Phenomenological theories of MHD turbulence describe the self-similar dynamics of turbulent fluctuations in the inertial range of scales. Numerical simulations serve to guide and test these theories. However, the computational power that is currently available restricts the simulations to Reynolds numbers that are significantly smaller than those in astrophysical settings. In order to increase computational efficiency and, therefore, probe a larger range of scales, one often takes into account the fundamental anisotropy of field-guided MHD turbulence, with gradients being much slower in the field-parallel direction. The simulations are then optimised by employing the reduced MHD equations and relaxing the field-parallel numerical resolution. In this work we explore a different possibility. We propose that there exist certain quantities that are remarkably stable with respect to the Reynolds number. As an illustration, we study the alignment angle between the magnetic and velocity fluctuations in MHD turbulence, measured as the ratio of two specially constructed structure functions. We find that the scaling of this ratio can be extended surprisingly well into the regime of relatively low Reynolds number. However, the extended scaling becomes easily spoiled when the dissipation range in the simulations is under-resolved. Thus, taking the numerical optimisation methods too far can lead to spurious numerical effects and erroneous representation of the physics of MHD turbulence, which in turn can affect our ability to correctly identify the physical mechanisms that are operating astrophysical systems.

Multi-particle and tetrad statistics in numerical simulations of turbulent relative dispersion

The evolution in size and shape of three and four-particle clusters (triangles and tetrads, respectively) in isotropic turbulence is studied using direct numerical simulations at grid resolution up to 40963 and Taylor-scale Reynolds numbers from 140 to 1000. A key issue is the attainment of inertial range behavior at high Reynolds number, while the small- and large-time limits of ballistic and diffusive regimes, respectively, are also considered in some detail. Tetrad size expressed by the volume (V) and (more appropriately) the gyration radius (R) is shown to display inertial range scaling consistent with a Richardson constant close to 0.56 for two-particle relative dispersion. For tetrads of initial size in a suitable range moments of shape parameters, including the ratio V2/3/R2 and normalized eigenvalues of a moment-of-inertia-like dispersion tensor, show a regime of near-constancy which is identified with inertial-range scaling. Sheet-like structures are dominant in this period, while pancakes and needles are more prevalent at later times. For triangles taken from different faces of each tetrad effects of the initial shape (isosceles right-angled or equilateral) are retained only for about one Batchelor time scale. In the inertial range there is a prevalence of nearly isosceles triangles of two long sides and one short side, representing one particle moving away from the other two which are still close together. In general, measures of shape display asymptotic scaling ranges more readily than measures of size. With some caveats, the simulation results are also compared with the limited literature available for multiparticle cluster dispersion in turbulent flow.

Conditional Eulerian and Lagrangian velocity increment statistics of fully developed turbulent flow

Conditional statistics of homogeneous isotropic turbulent flow is investigated by means of high Reynolds number direct numerical simulations performed with 20483 collocation points. Eulerian as well as Lagrangian velocity increment statistics under several conditions are analyzed and compared. In agreement with experimental data, longitudinal probability density functions P(δl||u|εl) conditioned on a scale-averaged energy dissipation rate are close to Gaussian distributions over all scales within the inertial range of scales. Also transverse increments conditioned on either the dissipation rate or the square of the vorticity have quasi-Gaussian probability density functions (PDFs). Concerning Lagrangian statistics we found that conditioning on a trajectory averaged energy dissipation rate ετ significantly reduces the scale dependence of the increment PDFs P(δτui|ετ). By means of dimensional arguments we propose a novel condition for Lagrangian increments which is shown to reduce even more the flatness of the corresponding PDFs and thus intermittency in the inertial range of scales. The conditioned Lagrangian PDF corresponding to the smallest increment considered is reasonably well described by a prediction for the PDF of acceleration of a non-intermittent velocity field. Conditioned structure functions show approximately K41 scaling with a larger scaling range than the unconditioned ones.

Second-order relative exponent of isotropic turbulence

Theoretical results on the scaling properties of turbulent velocity fields are reported in this letter. Based on the Kolmogorov equation and typical models of the second-order statistical moments (energy spectrum and the second-order structure function), we have studied the relative scaling using the ESS method. It is found that the relative EES scaling exponent S2 is greater than the real or theoretical inertial range scaling exponent ξ2, which is attributed to an evident bump in the ESS range.

Lagrangian Frequency Spectrum as a Diagnostic for Magnetohydrodynamic Turbulence Dynamics

For the phenomenological description of magnetohydrodynamic turbulence competing models exist, e.g., Boldyrev [Phys. Rev. Lett. 96, 115002 (2006)] and Gogoberidze [Phys. Plasmas 14, 022304 (2007)], which predict the same Eulerian inertial-range scaling of the turbulent energy spectrum although they employ fundamentally different basic interaction mechanisms. A relation is found that links the Lagrangian frequency spectrum with the autocorrelation time scale of the turbulent fluctuations τ(ac) and the associated cascade time scale τ(cas). Thus, the Lagrangian energy spectrum can serve to identify weak (τ(ac) ≪ τ(cas)) and strong (τ(ac) ∼ τ(cas)) interaction mechanisms providing insight into the turbulent energy cascade. The new approach is illustrated by results from direct numerical simulations of two- and three-dimensional incompressible MHD turbulence.

Statistics of particle pair relative velocity in the homogeneous shear flow

Small scale clustering of inertial particles and relative velocity of particle pairs have been fully characterized for statistically steady homogeneous isotropic flows. Depending on the particle Stokes relaxation time, the spatial distribution of the disperse phase results in a multi-scale manifold characterized by local particle concentration and voids and, because of finite inertia, the two nearby particles have high probability to exhibit large relative velocities. Both effects might explain the speed-up of particle collision rate in turbulent flows. Recently it has been shown that the large scale geometry of the flow plays a crucial role in organizing small scale particle clusters. For instance, a mean shear preferentially orients particle patterns. In this case, depending on the Stokes time, anisotropic clustering may occur even in the inertial range of scales where the turbulent fluctuations which drive the particles have already recovered isotropy. Here we consider the statistics of particle pair relative velocity in the homogeneous shear flow, the prototypical flow which manifests anisotropic clustering at small scales. We show that the mean shear, by imprinting anisotropy on the large scale velocity fluctuations, dramatically affects the particle relative velocity distribution even in the range of small scales where the anisotropic mechanisms of turbulent kinetic energy production are sub-dominant with respect to the inertial energy transfer which drives the carrier fluid velocity towards isotropy. We find that the particles’ populations which manifest strong anisotropy in their relative velocities are the same which exhibit small scale clustering. In contrast to any Kolmogorov-like picture of turbulent transport these phenomena may persist even below the smallest dissipative scales where the residual level of anisotropy may eventually blow-up. The observed anisotropy of particle relative velocity and spatial configuration is suggested to influence the directionality of the collision probability, as inferred on the basis of the so-called “ghost collision” model.

Relative velocity of inertial particles in turbulent flows

We present a model for the relative velocity of inertial particles in turbulent flows that provides new physical insight into this problem. Our general formulation shows that the relative velocity has contributions from two terms, referred to as the ‘generalized acceleration’ and ‘generalized shear’, because they reduce to the well-known acceleration and shear terms in the Saffman–Turner limit. The generalized shear term represents particles' memory of the flow velocity difference along their trajectories and depends on the inertial particle pair dispersion backward in time. The importance of this backward dispersion in determining the particle relative velocity is emphasized. We find that our model with a two-phase separation behaviour, an early ballistic phase and a later tracer-like phase, as found by recent simulations for the forward (in time) dispersion of inertial particle pairs, gives good fits to the measured relative speeds from simulations at low Reynolds numbers. In the monodisperse case with identical particles, the generalized acceleration term vanishes and the relative velocity is determined by the generalized shear term. At large Reynolds numbers, our model gives a St1/2-dependence of the relative velocity on the Stokes number St in the inertial range for both the ballistic behaviour and the Richardson separation law. This leads to the same inertial-range scaling for the two-phase separation that well fits the simulation results. Our calculations for the bidisperse case show that, with the friction timescale of one particle fixed, the relative speed as a function of the other particle's friction time has a dip when the two timescales are similar. This indicates that similar-size particles tend to have stronger velocity correlation than different ones. We find that the primary contribution at the dip, i.e. for similar particles, is from the generalized shear term, while the generalized acceleration term is dominant for particles of very different sizes. Future numerical studies are motivated to check the accuracy of the assumptions made in our model and to investigate the backward-in-time dispersion of inertial particle pairs in turbulent flows.

Dissipation-energy flux correlations as evidence for the Lagrangian energy cascade in turbulence

We study the spatial and temporal evolution of kinetic energy flux at different scales using direct numerical simulations of isotropic turbulence. The correlation coefficients at different times, between the molecular energy dissipation and local energy fluxes across inertial-range scales, are computed in both Eulerian and Lagrangian frames. The Eulerian correlation coefficients are found to decay monotonically backward in time. However, the Lagrangian correlation coefficients peak after a certain time delay. The peak time delay is found to be proportional to the local eddy turnover time (it scales with wave number k according to k−2/3), consistent with Kolmogorov’s theory. Conditional sampling is used to isolate effects of strong rotation. The results presented provide strong evidence of the Lagrangian nature of turbulent energy cascade.