Scaling Of Structure Functions Research Articles

Scaling Effects of the Weissenberg Number in Electrokinetic Oldroyd-B Fluid Flow Within a Microchannel.

This study attempts to extend previous research on electrokinetic turbulence (EKT) in Oldroyd-B fluid by investigating the relationship between the Weissenberg number ( ) and the second-order velocity structure function ( ) under applied electric fields. Inspired by Sasmal's demonstration in Sasmal (2022) of how heterogeneous zeta potentials induce turbulence above a critical , we develop a mathematical framework linking to turbulent phenomena. Our analysis incorporates recent findings on AC (Zhao & Wang, 2017) and DC (Zhao & Wang 2019) EKT, which have defined scaling laws for velocity and scalar structure functions in the forced cascade region. Our finding shows that and , for a length scale , and , where is a velocity fluctuations quantity and denotes the time relaxation parameter. This work establishes a positive correlation between and turbulent flow phenomena through a rigorous analysis of velocity structure functions, thereby offering a mathematical foundation for building the design and optimization of EKT-based microfluidicdevices.

Scale dependence of local shearing motion in decaying turbulence generated by multiple-jet interaction

Scale dependence of local shearing motion is investigated experimentally in decaying homogeneous isotropic turbulence generated through multiple-jet interaction. The turbulent Reynolds number, based on the Taylor microscale, is between approximately 900 and 400. Velocity fields, measured using particle image velocimetry, are analysed through the triple decomposition of a low-pass filtered velocity gradient tensor, which quantifies the intensities of shear and rigid-body rotation at a given scale. These motions manifest predominantly as layer and tubular vortical structures, respectively. The scale dependence of the moments of velocity increments, associated with shear and rigid-body rotation, exhibits power-law behaviours. The scaling exponents for shear are in quantitative alignment with the anomalous scaling of the velocity structure functions, suggesting that velocity increments are influenced predominantly by shearing motion. In contrast, the exponents for rigid-body rotation are markedly smaller than those predicted by Kolmogorov scaling, reflecting the high intermittency of rigid-body rotation. The mean flow structure associated with shear at intermediate scales is investigated with conditional averages around locally intense shear regions in the filtered velocity field. The averaged flow field exhibits a shear layer structure with aspect ratio approximately 4.5, surrounded by rotating motion. The analysis at different scales reveals the existence of self-similar structures of shearing motion across various scales. The mean velocity jump across the shear layer increases with the layer thickness. This relationship is well predicted by Kolmogorov's second similarity hypothesis, which is useful in predicting the mean characteristics of shear layers across a wide range of scales.

Towards synthetic magnetic turbulence with coherent structures

Synthetic turbulence is a relevant tool to study complex astrophysical and space plasma environments inaccessible by direct simulation. However, conventional models lack intermittent coherent structures, which are essential in realistic turbulence. We present a novel method featuring coherent structures, conditional structure function scaling and fieldline curvature statistics comparable to magnetohydrodynamic turbulence. Enhanced transport of charged particles is investigated as well. This method presents significant progress towards physically faithful synthetic turbulence.

Universal relative scaling of longitudinal structure functions in shear-dominated turbulence

Shear significantly influences turbulence in the energy-containing range of shear-dominated flows, and the longitudinal structure functions do not have a universal form as they do in homogeneous isotropic turbulence. Despite this, the relative scaling of structure functions exhibits universal sub-Gaussian behaviour in shear-dominated flows, in particular for turbulent boundary layers, channels and Taylor–Couette flows. Our investigation of a turbulent vertical buoyancy layer at $Pr = 0.71$ using direct numerical simulation shows this universality even in moderate-Reynolds-number buoyancy-driven but shear-dominated boundary layers. It is demonstrated that the universality is related to the energy density of the eddies, which attains a hierarchical equilibrium in the energy-containing range of shear-dominated turbulence. We conjecture that the universal sub-Gaussian behaviour of the energy density of the energy-containing range, which was considered to be non-trivial in prior studies, is related to the universal anomalous scaling exponents of the inertial subrange turbulence. Based on this conjecture, we propose a hypothesis that relates large-scale eddies and the intermittent dissipation field in shear-dominated turbulence, highlighting a relationship between large and small scales. A phenomenological model is also developed to predict the scaling, which is verified using data from a turbulent boundary layer, half-channel and vertical buoyancy layer at friction Reynolds numbers spanning four orders of magnitude. Excellent agreement is observed.

Dual scaling and the n-thirds law in grid turbulence

A dual scaling of the turbulent longitudinal velocity structure function $\overline {{(\delta u)}^n}$ , i.e. a scaling based on the Kolmogorov scales ( $u_K$ , $\eta$ ) and another based on ( $u'$ , $L$ ) representative of the large scale motion, is examined in the context of both the Kármán–Howarth equation and experimental grid turbulence data over a significant range of the Taylor microscale Reynolds number $Re_\lambda$ . As $Re_\lambda$ increases, the scaling based on ( $u'$ , $L$ ) extends to increasingly smaller values of $r/L$ while the scaling based on ( $u_K$ , $\eta$ ) extends to increasingly larger values of $r/\eta$ . The implication is that both scalings should eventually overlap in the so-called inertial range as $Re_\lambda$ continues to increase, thus leading to a power-law relation $\overline {{(\delta u)}^n} \sim r^{n/3}$ when the inertial range is rigorously established. The latter is likely to occur only when $Re_\lambda \to \infty$ . The use of an empirical model for $\overline {{(\delta u)}^n}$ , which complies with $\overline {{(\delta u)}^n} \sim r^{n/3}$ as $Re_\lambda \to \infty$ , shows that the finite Reynolds number effect may differ between even- and odd-orders of $\overline {{(\delta u)}^n}$ . This suggests that different values of $Re_\lambda$ may be required between even and odd values of $n$ for compliance with $\overline {{(\delta u)}^n} \sim r^{n/3}$ . The model describes adequately the dependence on $Re_\lambda$ of the available experimental data for $\overline {{(\delta u)}^n}$ and supports indirectly the extrapolation of these data to infinitely large $Re_\lambda$ .

Characteristics of clustered particle relative velocity in homogeneous and isotropic turbulence

Particle collisions are mainly governed by the preferential concentration of inertia particles and the formation of fold caustics. By fold caustics, we mean that relative velocity does not go smoothly to zero when the particle separations decrease due to inertia. Despite the importance of the second-order relative velocity structure function, there has been relatively little experimental research on the formation of caustics due to the high accuracy requirements for the particle relative velocity measurements. In the dissipation range, an obvious departure between the second-order structure function of particles normalized by the square of the Kolmogorov velocity and the Kolmogorov turbulent scaling of r2 was observed for all four experimental conditions. In the inertial range, the second-order structure function normalized by the square of the Kolmogorov velocity was consistent with the scale form of r2/3 in all cases. The conditional second-order relative velocity structure function of particles in clusters differs from that of the arithmetically averaged particle statistics in both the dissipation range and the inertial subrange. In the dissipation range, caustics are present for all categories of particles. In the inertial range, the scaling of the second-order velocity structure function is almost identical for all categorized particle types.

Scaling Anisotropy with Stationary Background Field in Near-Sun Solar Wind Turbulence

The scaling of magnetic fluctuations provides crucial information for the understanding of solar wind turbulence. However, the observed magnetic fluctuations contain not only turbulence but also magnetic structures, leading to the violation of time stationarity. This violation would conceal the true scaling and influence the determination of the sampling angle with respect to the local background magnetic field. Here, to investigate scaling anisotropy, we utilize a simple but effective criterion ϕ < 10° to ensure the time stationarity of the magnetic field, where ϕ is the angle between the two averaged magnetic fields after cutting the interval into two halves. We study the scaling anisotropy using higher-order statistics of structure functions under the condition of stationarity for near-Sun solar wind turbulence for the first time based on measurements obtained from the Parker Solar Probe at 0.17 au. We find that the scaling indices ξ of the magnetic field show a linear dependence on the order p close to ξ(p) = p/4. The multifractal scaling of the magnetic-trace structure functions becomes monoscaling close to ξ(p) = p/3 with the local magnetic field perpendicular to the sampling direction and close to ξ(p) = p/4 with the local magnetic field parallel to the sampling direction when measured with the stationary background magnetic field. The scaling of the velocity-trace structure functions has similar but less significant changes. The near-Sun solar wind turbulence displays different scaling anisotropies from the near-Earth solar wind turbulence, suggesting the evolution of the nonlinear interaction process during solar wind expansion.

On the Statistics of Elsasser Increments in Solar Wind and Magnetohydrodynamic Turbulence

In this Letter we investigate the dependency with scale of the empirical probability distribution functions (PDF) of Elsasser increments using large sets of WIND data (collected between 1995 and 2017) near 1 au. The empirical PDF are compared to the ones obtained from high-resolution numerical simulations of steadily driven, homogeneous reduced MHD turbulence on a 20483 rectangular mesh. A large statistical sample of Alfvénic increments is obtained by using conditional analysis based on the solar wind average properties. The PDF tails obtained from observations and numerical simulations are found to have exponential behavior in the inertial range, with an exponential decrement that satisfies power laws of the form α l ∝ l −μ , where l is the scale size, with μ between 0.17 and 0.25 for observations and 0.43 for simulations. PDF tails were extrapolated assuming their exponential behavior extends to arbitrarily large increments in order to determine structure function scaling laws at very high orders. Our results point to potentially universal scaling laws governing the PDF of Elsasser increments and to an alternative approach to investigate high-order statistics in solar wind observations.

Investigation of the anisotropy and scaling of the phase structure function of a spatially coherent light beam propagating through convective air turbulence.

We report on applications of moiré deflectometry in measurements of the anisotropy and scaling of the phase structure function (PSF), obtained after passing a laser beam through an indoor enclosure containing convective air turbulence. We combine the use of two telescopes, with a two-channel wavefront sensor based on moiré deflectometry, to attain high sensitivity and resolution to fluctuations in the wavefront phase, caused by turbulent fluctuations in the enclosure. The measurements of the wavefront PSF along two directions perpendicular to the direction of the light beam propagation at different heater temperatures show that the convective air turbulence is anisotropic turbulence, where the value of the anisotropy increases with increasing temperature gradient. Various models are fitted to the measured PSFs, and we find that the turbulent is also non-Kolmogorov, in which, for the separation distances of two points on the wavefront less than 10 cm, the von Kármán PSF is the best fit to the experimental data. For higher values of separations, the experimental data do not fit with existing models. By fitting the von Kármán PSF on the data, we estimate values of the refractive index structure constant, Cn2, as well as the outer scale of the turbulence. The value of the outer scale decreases with increasing temperature of the heater up to approximately 50°C, where it saturates, while the value of Cn2 monotonically increases. Over the complete range of heater temperatures, from 40°C to 160°C, the Rayleigh number, Ra, for the enclosed air flow varied from 5.80×108<Ra<5.89×109 so that all measurements were conducted in a state of developed convective turbulence.

Modelling the transport equation of the scalar structure function

A model for the third-order mixed velocity–scalar structure function $-\overline {(\delta u)( \delta \phi )^{2}}$ ( $\delta a$ represents the spatial increment of the quantity $a$ ; $u$ is the velocity and $\phi$ is a scalar) is proposed for closing the transport equation of the second-order moment of the scalar increment $\overline {( \delta \phi )^{2}}$ . The closure model is based on a gradient-type hypothesis with an eddy-viscosity model which exploits the analogy between the turbulent kinetic energy and a passive scalar when the Prandtl number, $Pr$ , is equal to 1. The solutions of the closed transport equation agree well with both measurements and numerical simulations, when $Pr$ is not too different from 1. However, the agreement deteriorates when $Pr$ differs significantly from 1. Nevertheless, the calculations capture the effect expected when $Pr$ varies. For example, as $Pr$ increases, a range of scales emerges where $\overline {( \delta \phi )^{2}}$ remains constant. This emerging scaling range should correspond to the $k^{-1}$ spectral range in the three-dimensional scalar spectrum commonly denoted the viscous–convective range. Also when $Pr$ decreases below 1, the calculations reproduce the emergence of an expected inertial–diffusive range for scales larger than the Kolmogorov length scale.

Hidden spatiotemporal symmetries and intermittency in turbulence

We consider general infinite-dimensional dynamical systems with the Galilean and spatiotemporal scaling symmetry groups. Introducing the equivalence relation with respect to temporal scalings and Galilean transformations, we define a representative set containing a single element within each equivalence class. Temporal scalings and Galilean transformations do not commute with the evolution operator (flow) and, hence, the equivalence relation is not invariant. Despite of that, we prove that a normalized flow with an invariant probability measure can be introduced on the representative set, such that symmetries are preserved in the statistical sense. We focus on hidden symmetries, which are broken in the original system but restored in the normalized system. The central motivation and application of this construction is the intermittency phenomenon in turbulence. We show that hidden symmetries yield power law scaling for structure functions, and derive formulas for their exponents in terms of normalized measures. The use of Galilean transformation in the equivalence relation leads to the quasi-Lagrangian description, making the developed theory applicable to the Euler and Navier–Stokes systems.

Fractal Dimension Analysis of Earth Magnetic Field during 26 August 2018 Geomagnetic Storm

We analyse the fractal nature of geomagnetic field northward and eastward horizontal components with 1 min resolution measured by the four stations Belsk, Hel, Sodankylä and Hornsund during the period of 22 August–1 September, when the 26 August 2018 geomagnetic storm appeared. To reveal and to quantitatively describe the fractal scaling of the considered data, three selected methods, structure function scaling, Higuchi, and detrended fluctuation analysis are applied. The obtained results show temporal variation of the fractal dimension of geomagnetic field components, revealing differences between their irregularity (complexity). The values of fractal dimension seem to be sensitive to the physical conditions connected with the interplanetary shock, the coronal mass ejection, the corotating interaction region, and the high-speed stream passage during the storm development. Especially, just after interplanetary shock occurrence, a decrease in the fractal dimension for all stations is observed, not straightforwardly visible in the geomagnetic field components data.

Solvable Intermittent Shell Model of Turbulence

We introduce a shell model of turbulence featuring intermittent behaviour with anomalous power-law scaling of structure functions. This model is solved analytically with the explicit derivation of anomalous exponents. The solution associates the intermittency with the hidden symmetry for Kolmogorov multipliers, making our approach relevant for real turbulence.

Conditional and unconditional second-order structure functions in bubbly channel flows of power-law fluids

The influence of non-Newtonian fluid behavior and the Eötvös number on conditional and unconditional second-order structure functions of bubbly channel flows has been investigated by conducting a series of direct numerical simulations at a friction Reynolds number of 127.3. Two Eötvös numbers have been considered (Eo = 0.3125 and Eo = 3.75) together with three different power-law indexes representing shear-thinning (n = 0.7), Newtonian (n = 1.0), and shear-thickening (n = 1.3) fluid behavior. The scaling of the second-order structure functions (SFs) can be translated into an inertial range scaling of the turbulent kinetic energy spectrum. However, because of the discontinuous character of the fluid properties in bubbly flows, SFs are more easily accessible than turbulence spectra, which are based on Fourier transform. It has been found that the different parameters (i.e., Eo, n) have an influence on the energy content as well as the peak location of the compensated second-order SFs (i.e., the dimensions of the large scales). However, after appropriate scaling, the curves nearly collapse. To confirm and further explain the above findings, directional length scales have been evaluated and discussed in detail. Finally, the anisotropy of the Reynolds stress tensor and dissipation tensor has been analyzed in terms of the Lumley triangle, showing that bubbly channel flows are less isotropic than their single-phase counterpart, although they are more homogeneous in the channel center. While the dissipation tensor is slightly more isotropic than the Reynolds stress tensor in the bulk region of the channel flow, overall, a very similar behavior is observed.

Multifractal Characteristics of Geomagnetic Field Fluctuations for the Northern and Southern Hemispheres at Swarm Altitude.

This paper explores the spatial variations of the statistical scaling features of low to high latitude geomagnetic field fluctuations at Swarm altitude. The data for this study comes from the vector field magnetometer onboard Swarm A satellite, measured at low resolution (1 Hz) for one year (from 9 March 2016, to 9 March 2017). We estimated the structure-function scaling exponents using the p-leaders discrete wavelet multifractal technique, from which we obtained the singularity spectrum related to the magnetic fluctuations in the North-East-Center (NEC) coordinate system. From this estimation, we retain just the maximal fractal subset, associated with the Hurst exponent H. Here we present thresholding for two levels of the Auroral Electrojet index and almost the whole northern and southern hemispheres, the Hurst exponent, the structure-function scaling exponent of order 2, and the multifractal p-exponent width for the geomagnetic fluctuations. The latter quantifies the relevance of the multifractal property. Sometimes, we found negative values of H, suggesting a behavior similar to wave breaking or shocklet-like propagating front. Furthermore, we found some asymmetries in the magnetic field turbulence between the northern and southern hemispheres. These estimations suggest that different turbulent regimes of the geomagnetic field fluctuations exist along the Swarm path.

Spectroscopic imaging of sub-kilometer spatial structure in lower-tropospheric water vapor

Abstract. The subgrid spatial variability of water vapor is an important geophysical parameter for modeling tropical convention and cloud processes in atmospheric models. This study maps sub-kilometer spatial structures in total atmospheric column water vapor with visible to shortwave infrared (VSWIR) imaging spectroscopy. We describe our inversion approach and validate its accuracy with coincident measurements by airborne imaging spectrometers and the AERONET ground-based observation network. Next, data from NASA's AVIRIS-NG spectrometer enable the highest-resolution measurement to date of water vapor's spatial variability and scaling properties. We find second-order structure function scaling exponents consistent with prior studies of convective atmospheres. Airborne lidar data show that this total column measurement provides information about variability in the lower troposphere. We conclude by discussing the implications of these measurements and paths toward future campaigns to build upon these results.

On small-scale and large-scale intermittency of Lagrangian statistics in canopy flow

Abstract

Turbulence is an Ineffective Mixer when Schmidt Numbers Are Large.

We solve the advection-diffusion equation for a stochastically stationary passive scalar θ, in conjunction with forced 3D Navier-Stokes equations, using direct numerical simulations in periodic domains of various sizes, the largest being 8192^{3}. The Taylor-scale Reynolds number varies in the range 140-650 and the Schmidt number Sc≡ν/D in the range 1-512, where ν is the kinematic viscosity of the fluid and D is the molecular diffusivity of θ. Our results show that turbulence becomes an ineffective mixer when Sc is large. First, the mean scalar dissipation rate ⟨χ⟩=2D⟨|∇θ|^{2}⟩, when suitably nondimensionalized, decreases as 1/logSc. Second, 1D cuts through the scalar field indicate increasing density of sharp fronts on larger scales, oscillating with large excursions leading to reduced mixing, and additionally suggesting weakening of scalar variance flux across the scales. The scaling exponents of the scalar structure functions in the inertial-convective range appear to saturate with respect to the moment order and the saturation exponent approaches unity as Sc increases, qualitatively consistent with 1D cuts of the scalar.

On Yaglom's Law for the Interplanetary Proton Density and Temperature Fluctuations in Solar Wind Turbulence.

In the past decades, there has been an increasing literature on the presence of an inertial energy cascade in interplanetary space plasma, being interpreted as the signature of Magnetohydrodynamic turbulence (MHD) for both fields and passive scalars. Here, we investigate the passive scalar nature of the solar wind proton density and temperature by looking for scaling features in the mixed-scalar third-order structure functions using measurements on-board the Ulysses spacecraft during two different periods, i.e., an equatorial slow solar wind and a high-latitude fast solar wind, respectively. We find a linear scaling of the mixed third-order structure function as predicted by Yaglom’s law for passive scalars in the case of slow solar wind, while the results for fast solar wind suggest that the mixed fourth-order structure function displays a linear scaling. A simple empirical explanation of the observed difference is proposed and discussed.

On the combined effect of internal and external intermittency in turbulent non-premixed jet flames

This paper analyzes the combined effect of internal intermittency and external intermittency on the dynamics of small-scale turbulent mixing in a turbulent non-premixed jet flame. The phenomenon of external intermittency in turbulent jet flames originates from a very thin layer, known as turbulent/non-turbulent interface, that separates the inner turbulent core from the outer irrotational surrounding fluid. The impact of external intermittency on turbulence is evaluated across the jet flame by the self-similarity scaling of higher-order structure functions of the mixture fraction. It is shown that the scaling of structure functions exhibits a growing departure from the prediction of classical scaling laws toward the edge of the flame. Empirical evidence is provided that this departure is primarily due to external intermittency and the associated break down of self-similarity. External intermittency creates local gradients that are significantly more intense compared to those created by internal intermittency alone. As chemical reactions usually take place in thin layers in the vicinity of the turbulent/non-turbulent interface, an accurate statistical description of these intense events is necessary to predict the turbulence-chemistry interaction. The study is based on data from a highly-resolved direct numerical simulation of a temporally evolving planar jet flame.