Taylor Scale Research Articles

Anisotropy of the Taylor scale and the correlation scale in plasma sheet and solar wind magnetic field fluctuations

Magnetic field data from nine spacecraft in the magnetospheric plasma sheet and the solar wind are employed to determine the correlation scale and the magnetic Taylor microscale from simultaneous multiple‐point measurements for multiple intervals with a range of mean magnetic field directions. We have determined that in the solar wind the Taylor scale is independent of direction relative to the mean magnetic field, but the correlation scale along the mean magnetic field (2.7 × 106 ± 0.2 × 106 km) is longer than along the perpendicular direction (1.5 × 106 ± 0.1 × 106 km). Within the plasma sheet we found that the correlation scale varies from 16,400 ± 1000 km along the mean magnetic field direction to 9200 ± 600 km in the perpendicular direction. The Taylor scale is also longer parallel to the magnetic field (2900 ± 100 km) than perpendicular to it (1100 ± 100 km). In the solar wind the ratio of the parallel correlation scale to the perpendicular correlation scale is 2.62 ± 0.79; in the plasma sheet the ratio is 1.78 ± 0.16, which indicates that the turbulence in both regions is anisotropic. The correlation and Taylor scales may be used to estimate effective magnetic Reynolds numbers separately for each angular channel. Reynolds numbers were found to be approximately independent of the angle relative to the mean magnetic field. These results may be useful in magnetohydrodynamic modeling of the solar wind and the magnetosphere and can contribute to our understanding of solar and galactic cosmic ray diffusion in the heliosphere.

Why we need experiments at high Reynolds numbers

This brief review focuses on the effect of Reynolds number on turbulence small-scale anisotropy and intermittency, on Lagrangian experiments of fluid and inertial particles and on complex turbulent flows. Although Taylor scale Reynolds numbers of 103 (equivalent to turbulence Reynolds number based on the integral scale of 105) can be achieved in the laboratory, it is argued that there is still a need to do experiments (and computation) at even higher Reynolds numbers to resolve outstanding basic and applied issues in turbulence.

Assessment Measures for Engineering LES Applications

Abstract Anticipating that large eddy simulations will increasingly become the future engineering tool for research, development, and design, it is deemed necessary to formulate some quality assessment measures that can be used to judge the resolution of turbulent scales and the accuracy of predictions. In this context some new and refined measures are proposed and compared with those already published by the authors in the common literature. These measures involve (a) fraction of the total turbulent kinetic energy, (b) relative grid size with respect to Kolmogorov or Taylor scales, and (c) relative effective subgrid/numerical viscosity with respect to molecular viscosity. In addition, an attempt is made to segregate the contributions from numerical and modeling errors. Proposed measures are applied to various test cases and validated against fully resolved large eddy simulation and/or direct numerical simulation whenever possible.

A study of differentiation errors in large-eddy simulations based on the EDQNM theory

This paper is concerned with the investigation of numerical errors in large-eddy simulations by means of two-point turbulence modeling. Based on the eddy-damped quasi-normal Markovian (EDQNM) theory, a stochastic model is developed in order to predict the time evolution of the kinetic energy spectrum obtained by a large-eddy simulation (LES), including the effects of the numerics. Using this framework, the influence of the accuracy of the approximate space differencing schemes on LES quality is studied, for decaying homogeneous isotropic incompressible turbulence, with Reynolds numbers Re λ based on the transverse Taylor scale equal to 780, 2500 and 8000. The results show that the discretization of the filtered Navier–Stokes equations leads to differentiation and aliasing errors. Error spectra are also presented, and indicate that the numerical errors are mainly originating from the approximate differentiation. In addition, increasing the order of accuracy of the differencing schemes or using algorithms optimized in the Fourier space is found to widen the range of well-resolved scales. Unfortunately, for all the schemes, the smaller scales with wavenumbers close to the grid cut-off wavenumber, are badly calculated and generate differentiation errors over the whole energy spectrum. The eventual use of explicit filtering to remove spurious motions with short wavelength is finally shown to significantly improve LES accuracy.

Interplanetary Magnetic Taylor Microscale and Implications for Plasma Dissipation

The Taylor microscale, a measure of mean square spatial derivatives, is evaluated for interplanetary magnetic field fluctuations from single- and multiple-point data using Cluster and ACE spacecraft data. The Taylor scale is compared to the measured inner scale, which for hydrodynamics would correspond to the Kolmogorov scale. The results are not consistent with dissipation of the hydrodynamic type, and indicate that solar wind dissipation involves kinetic plasma physics at both proton and electron scales.

Dissipation and enstrophy in isotropic turbulence: Resolution effects and scaling in direct numerical simulations

Existing experimental and numerical data suggest that the turbulence energy dissipation and enstrophy (i.e., the square of vorticity) possess different scaling properties, while available theory suggests that there should be no differences at sufficiently high Reynolds numbers. We have performed a series of direct numerical simulations with up to 20483 grid points where advanced computational power is used to increase the Reynolds number (up to 650 on the Taylor scale) or to resolve the small scales better (down to 1∕4 of a Kolmogorov scale). Our primary goal is to assess the differences and similarities between dissipation and enstrophy. Special attention is paid to the effects of small-scale resolution on the quality and reliability of the data, in view of recent theoretical work [V. Yakhot and K. R. Sreenivasan, “Anomalous scaling of structure functions and dynamic constraints on turbulence simulations,” J. Stat. Phys. 121, 823 (2005)] which stipulates the resolution needed to obtain a moment of a given order. We also provide error estimates as a function of small-scale resolution. Probability density functions of dissipation and enstrophy at high Reynolds number reveal the presence of extreme events several thousands times of the mean. The extreme events in dissipation and enstrophy fields appear to scale alike, substantially overlap in space, and are nearly statistically isotropic, while fluctuations of moderate amplitudes, at least for the present Reynolds numbers, show persistent differences. Conditional sampling shows that intense dissipation is likely to be accompanied by similarly intense enstrophy, but intense enstrophy is not always accompanied by intense dissipation.

Taylor scale and effective magnetic Reynolds number determination from plasma sheet and solar wind magnetic field fluctuations

Cluster data from many different intervals in the magnetospheric plasmas sheet and the solar wind are employed to determine the magnetic Taylor microscale from simultaneous multiple point measurements. For this study we define the Taylor scale as the square root of the ratio of the mean square magnetic field (or velocity) fluctuations to the mean square spatial derivatives of their fluctuations. The Taylor scale may be used, in the assumption of a classical Ohmic dissipation function, to estimate effective magnetic Reynolds numbers, as well as other properties of the small scale turbulence. Using solar wind magnetic field data, we have determined a Taylor scale value of 2400 ± 100 km, which is used to obtain an effective magnetic Reynolds number of about 260,000 ± 20,000, and in the plasma sheet we calculated a Taylor scale of 1900 ± 100 km, which allowed us to obtain effective magnetic Reynolds numbers in the range of about 7 to 110. The present determination makes use of a novel extrapolation technique to derive a statistically stable estimate from a range of small scale measurements. These results may be useful in magnetohydrodynamic modeling of the solar wind and the magnetosphere and may provide constraints on kinetic theories of dissipation in space plasmas.

An age extended progress variable for conditioning reaction rates

An aging progress variable (APV) is proposed as a convenient tool for conditioning quantities used to calculate reaction rates in premixed turbulent combustion. The APV is defined to obey an advection-diffusion-reaction equation where the source term is linearly related to the fuel consumption rate when the APV is less than a threshold representative of the trailing edge of the fuel consumption layer. Above this threshold, the APV has a constant source term. To test the proposal, three-dimensional fully compressible direct numerical simulation (DNS) is performed in an inflow-outflow configuration with inlet turbulence forcing using a modified version of the DNS code SENGA. A model chemical mechanism involving two steps and four species is used which has an order of magnitude difference between the time scales associated with the two steps. The inlet Taylor scale Reynolds number is 51 and the Damköhler numbers are 2.0 and 0.29 for the two steps. When conditionally averaged on the proposed APV the scalar fluctuations about the conditional average are negligible. Further, it is shown that the probability density function of the APV can be reasonably approximated based on the first two moments of the APV and the fuel mass fraction. The APV probability density function (PDF) is approximated on rectangular slabs of cells normal to the flow direction using only these moments as a test case. Convolution of the PDF so approximated with conditional mean reaction rates—calculated from conditionally averaged scalar fields where the averaging was carried out over the entire domain—leads to an approximation for the unconditional mean reaction rates on each of these slabs typically within 10% of the true value for both steps. That the correlation between the reaction rates and the APV is strong, and that the PDF can be approximated for a situation where approximating the PDF for a product based progress variable is nontrivial, makes the proposed APV a strong alternative to traditional progress variables for both flamelet models and premixed conditional moment closure (CMC) approaches.

PIV Measurements in the Atmospheric Boundary Layer within and above a Mature Corn Canopy. Part I: Statistics and Energy Flux

Particle image velocimetry (PIV) measurements just within and above a mature corn canopy have been performed to clarify the small-scale spatial structure of the turbulence. The smallest resolved scales are about 15 times the Kolmogorov length scale (η ≈ 0.4 mm), the Taylor microscales are about 100η, and the Taylor scale Reynolds numbers range between Rλ = 2000 and 3000. The vertical profiles of mean flow and turbulence parameters match those found in previous studies. Frequency spectra, obtained using the data as time series, are combined with instantaneous spatial spectra to resolve more than five orders of magnitude of length scales. They display an inertial range spanning three decades. However, the small-scale turbulence in the dissipation range exhibits anisotropy at all measurement heights, in spite of apparent agreement with conditions for reaching local isotropy, following a high-Reynolds-number wind tunnel study. Directly calculated subgrid-scale (SGS) energy flux, determined by spatially filtering the PIV data, increases significantly with decreasing filter size, providing support for the existence of a spectral shortcut that bypasses the cascading process and injects energy directly into small scales. The highest measured SGS flux is about 40% of the estimated energy cascading rate as determined from a −5/3 fit to the spectra. Terms appearing in the turbulent kinetic energy budget that can be calculated from the PIV data are in agreement with previous results. Evidence of a very strong correlation between dissipation rate and out-of-plane component of the vorticity is demonstrated by a striking similarity between their time series.

Rapidly oscillating asymptotic solutions of the Navier-Stokes equations, coherent structures, Fomenko invariants, Kolmogorov spectrum, and flicker noise

Asymptotic solutions of the Navier-Stokes equations describing three-phase coherent structures of the Taylor scale are constructed. These equations are related to topological invariants of the Liouville foliations on the three-dimensional torus into two-dimensional tori. The solutions turn out to be unstable with respect to small-scale perturbations, which finally leads to the appearance of perturbations of Kolmogorov scale. The flicker noise occurs at the corresponding frequencies.

Potential for interpretation disparities of Halstead–Reitan neuropsychological battery performances in a litigating sample

The performances of 110 litigants on seven variables from the Halstead–Reitan neuropsychological battery (HRNB) were used to compare Heaton, Miller, Taylor, and Grant's (2004) Deficit Scale (DS) and Reitan and Wolfson's (1993) Neuropsychological Deficit Scale (NDS). Additional comparisons were made for people who passed or failed the Test of Memory Malingering (TOMM) to determine effects of effort on scores generated by either scoring system. Wilcoxon signed-rank tests revealed that all seven comparisons were significantly different for the full sample ( p ≤ 0.001). The NDS indicated greater levels of impairment compared to DS across all variables. These findings were also obtained when considering effort, though TOMM failure was related to non-significant differences for two variables. These findings suggest that the two scoring systems are not equivalent, with Heaton et al.'s DS resulting in consistently higher identification rates of normal brain functioning compared to those generated from Reitan and Wolfson's NDS system.

The length-scale distribution function of the distance between extremal points in passive scalar turbulence

In order to extract small-scale statistical information from passive scalar fields obtained by direct numerical simulation (DNS) a new method of analysis is introduced. It consists of determining local minimum and maximum points of the fluctuating scalar field via gradient trajectories starting from every grid point in the directions of ascending and descending scalar gradients. The ensemble of grid cells from which the same pair of extremal points is reached determines a spatial region which is called a ‘dissipation element’. This region may be highly convoluted but on average it has an elongated shape with, on average, a nearly constant diameter of a few Kolmogorov scales and a variable length that has the mean of a Taylor scale. We parameterize the geometry of these elements by the linear distance between their extremal points and their scalar structure by the absolute value of the scalar difference at these points.The joint p.d.f. of these two parameters contains most of the information needed to reconstruct the statistics of the scalar field. It is decomposed into a marginal p.d.f. of the linear distance and a conditional p.d.f. of the scalar difference. It is found that the conditional mean of the scalar difference follows the 1/3 inertial-range Kolmogorov scaling over a large range of length-scales even for the relatively small Reynolds number of the present simulations. This surprising result is explained by the additional conditioning on minima and maxima points.A stochastic evolution equation for the marginal p.d.f. of the linear distance is derived and solved numerically. The stochastic problem that we consider consists of a Poisson process for the cutting of linear elements and a reconnection process due to molecular diffusion. The resulting length-scale distribution compares well with those obtained from the DNS.

High-resolution direct numerical simulation of turbulence

We performed high-resolution direct numerical simulations (DNS) of incompressible turbulence in a periodic box by using a Fourier spectral method with the number of grid points up to 40963. The simulations consist of two series: one with k maxη∼ 1 (series 1), and the other with k maxη∼ 2 (series 2), where k max is the highest wavenumber in each simulation, and η is the Kolmogorov length scale. In the 40963 DNS, the Taylor scale Reynolds number R λ ∼ 1200 and the ratio of L/η of the integral length scale L to η is approximately 2200, in series 1. The DNS data analysis reveals simple scaling of various spectra, and also sheds some light on (i) the energy spectrum at high R λ, (ii) the asymptotic R λ-dependence of the normalized energy dissipation rate, (iii) the anomalous scaling of the spectra of energy dissipation and enstrophy, and so on. After some preliminary remarks on the methods and limitations of the DNS, this paper presents a review of the DNS data analysis. Discussions are also made on some questions invoked by the DNS.

On the Taylor hypothesis in forced unsteady wall flows

Experiments on the modulation characteristics of the wall shear stress τ′-longitudinal velocity u′ and u′−u′ space–time correlations are reported in a forced turbulent channel flow in a wide range of imposed frequencies. The resulting integral and Taylor scale properties are discussed in detail in the low buffer layer under steady and unsteady flow conditions. It is shown that the small-scale turbulence is sensitive to the imposed unsteadiness since the amplitude and phase of the Taylor length scale vary considerably in the imposed frequency range investigated here. The Taylor hypothesis is acceptably valid in steady and unsteady wall layers just above the low buffer layer. Production and instantaneous pressure gradients are mostly responsible for the deviation of the frozen turbulence-state in the viscous and low buffer sublayers.

Decay of magnetohydrodynamic turbulence from power-law initial conditions.

We derive relations for the decay of the kinetic and magnetic energies and the growth of the Taylor and integral scales in unforced, incompressible, homogeneous, and isotropic three-dimensional magnetohydrodynamic (3DMHD) turbulence with power-law initial energy spectra. We also derive bounds for the decay of the cross and magnetic helicities. We then present results from systematic numerical studies of such decay both within the context of a MHD shell model and direct numerical simulations of 3DMHD. We show explicitly that our results about the power-law decay of the energies hold for times t< t*, where t* is the time at which the integral scales become comparable to the system size. For t< t*, our numerical results are consistent with those predicted by the principle of "permanence of large eddies."

Direct numerical simulation of passive scalar in decaying compressible turbulence

The passive scalars in the decaying compressible turbulence with the initial Reynolds number (defined by Taylor scale and RMS velocity) Re=72, the initial turbulent Mach numbers (defined by RMS velocity and mean sound speed) Mt=0.2-0.9, and the Schmidt numbers of passive scalar Sc=2-10 are numerically simulated by using a 7th order upwind difference scheme and 8th order group velocity control scheme. The computed results are validated with different numerical methods and different mesh sizes. The Batchelor scaling with k(-1) range is found in scalar spectra. The passive scalar spectra decay faster with the increasing turbulent Mach number. The extended self-similarity (ESS) is found in the passive scalar of compressible turbulence.

Extension of the Kida law in turbulence

We extend the validity range of Kida's log-stable law of stability index α=1.65 and intermittency parameter μ=0.2 to a new range of Reynolds number. This law describes intermittencies in fully developed turbulent flows or more precisely the p.d.f. of turbulence dissipation. Former measurements of the hyper-flatness factors of order 4, 5, 6 of turbulent velocity increments, coming from both experimental works and numerical simulations are used. We show that the power-law variation of these hyper-flatness factors with Taylor scale based Reynolds numbers Re λ can be fitted, for Re λ ranging from 35 to 750, by a log-stable law of stability index α=1.65 and intermittency parameter μ=0.21. To cite this article: N. Rimbert, O. Séro-Guillaume, C. R. Mecanique 331 (2003).

Generation of PDFS for flame curvature and for flame stretch rate in premixed turbulent combustion

Experimentally derived pdfs of turbulent, premixed, flame curvatures from a variety of sources, for a wide range of conditions are surveyed and a suitable expression sought to generalize these. This proves to be one based on the Damköhler number, Da. This is tantamount to normalizing the curvature by multiplying it by the Taylor scale of turbulence. It enables the distribution of flame curvature when normalized by the laminar flame thickness, to be expressed in terms of the Karlovitz stretch factor, K, and the turbulent Reynolds number, R l. The value of the pdf at zero curvature is linearly related to Da 1/2. The pdf expressions of Yeung et al. [3] obtained from numerical simulations are used for the strain rate distribution and, on the assumption that these and that for flame curvature are statistically independent, values of flame stretch rate pdfs are generated numerically. It is necessary to define an appropriate surface to define the burning velocity, flame stretch rate, and appropriate Markstein numbers. Two surfaces are considered and employed in the computations, one located at the start of the preheat zone, the other at the start of the reaction zone. The latter seems more rational and gives the better generalisation of the pdfs of flame stretch rate. An assumed linearity of laminar burning velocity with flame stretch rate, extending over both positive and negative stretch rates, enables flame stretch rate pdfs to be generated. It is concluded that negative values of burning velocity are unlikely and that burning velocities should tend to zero rather than attain negative values. This modifies the derivation of flame stretch rate pdfs. These depend on the Markstein number, Karlovitz stretch factor and turbulent Reynolds number. Computations suggest that, for values of K above 0.1 and of R l above 100, the pdf of stretch rate is similar to that of strain rate. At very low values of K and negative values of Markstein number, pronounced flamelet instability might be anticipated.

Possible alternative to Rλ-scaling of small-scale turbulence statistics

Traditionally, trends of universal turbulence statistics are presented versus Rλ, which is the Reynolds number based on Taylor's scale λ and the root-mean-square (rms) of a component of velocity urms. λ and urms, and hence Rλ, do not have the attribute of universality. The ratio of rms fluid-particle acceleration to the rms of the acceleration caused by the viscous force, Ra, is an alternative to Rλ. Ra is a Reynolds number. It is composed of statistics of the small scales of turbulence. It can be evaluated with single-wire hot-wire anemometry. Like Rλ, it can be partially evaluated by means of flow similarity. Direct measurement of Ra is challenging; therefore, Ra is not a replacement for Rλ. For isotropic turbulence the relationship between Ra and Rλ is given. Anisotropic turbulence is discussed.

Scalar dissipation rate measurements in flames: A method to improve spatial resolution by using nitric oxide PLIF

Scalar dissipation rate (χ) previously has been identified as one of the most important parameters inturbulent non-premixed flames. A new method to image χ is demonstrated which is a variation of the method of Strner, Bilger, Frank, and Long; nitric oxide (NO) is added to the jet fluid to mark the fuel, replacing previous fuel markers such as acetone. Planar laser-induced fluorescence of NO is combined with temperature images from Rayleigh scattering to measure mixture fraction and scalar dissipation rate with greatly improved signal-to-noise ratio (SNR) and spatial resolution. Calibration measurements of mixture fraction (f) and χ in a laminar flame and qualitative images of a turbulent flame are presented. The method has three advantages. (1) Differential diffusion problems (that have limited the use of acetone) are eliminated, since the diffusivity of NO is equivalent to that of the methane/nitrogen fuel mixture. (2) NO is an excellent marker of the fuel (unlike acetone) since it properly disappears from centerline to the flame boundary due to rapid reactions with H and CH radicals. A small correction (typically 5% of the centerline value) must be applied to account for slight leakage of NO through the CH layer, but this correction can be made entirely from the experimental data. (3) NO can be added in large amounts to achieve ecceptional SNR values exceeding 150 when the spatial resolution equals a typical Taylor scale of 400 μm. Radial profiles of mixture fraction and scalar dissipation rate in the calibration flame compare favorably with previous measurements.