arXiv subjects

Reginald J. Hill

Publications and source records attributed to Reginald J. Hill.

9 recordsLinked to original sources

arXiv2016

Equations relating structure functions of all orders

The hierarchy of exact equations is given that relates two-spatial-point velocity structure functions of arbitrary order with other statistics. Because no assumption is used, the exact statistical equations can apply to any flow for which the Navier-Stokes equations are accurate, and they apply no matter how small the number of samples in the ensemble. The exact statistical equations can be used to verify DNS computations and to detect their limitations. For example,if DNS data are used to evaluate the exact statistical equations, then the equations should balance to within numerical precision, otherwise a computational problem is indicated. The equations allow quantification of the approach to local homogeneity and to local isotropy. Testing the balance of the equations allows detection of scaling ranges for quantification of scaling-range exponents. The second-order equations lead to Kolmogorov's equation. All higher-order equations contain a statistic composed of one factor of the two-point difference of the pressure gradient multiplied by factors of velocity difference. Investigation of this pressure-gradient-difference statistic can reveal much about two issues: 1) whether or not different components of the velocity structure function of given order have differing exponents in the inertial range, and 2) the increasing deviation of those exponents from Kolmogorov scaling as the order increases. Full disclosure of the mathematical methods is in xxx.lanl.gov/list/physics.flu-dyn/0102055.

physics.flu-dyn↗

arXiv2016

Mathematics of Structure-Function Equations of All Orders

Exact equations are derived that relate velocity structure functions of arbitrary order with other statistics. "Exact" means that no approximations are used except that the Navier-Stokes equation and incompressibility condition are assumed to be accurate. The exact equations are used to determine the structure-function equations of all orders for locally homogeneous but anisotropic turbulence as well as for the locally isotropic case. These equations can be used for investigating the approach to local homogeneity and to local isotropy as well as the balance of the equations and identification of scaling ranges.

physics.flu-dyn↗

arXiv2008

Methods for measuring energy dissipation rate in anisotropic turbulence

Energy dissipation rate is an important parameter for nearly every experiment on turbulent flow. Mathematically precise relationships between energy dissipation rate and other measurable statistics for the case of anisotropic turbulence are useful to experimentalists. Such relationships are obtained for which the measurable statistics are the 3rd-order and 2nd-order velocity structure functions as well as the acceleration-velocity structure function. The relationships are derived using the Navier-Stokes equation without approximation. Approximate versions are obtained on the basis of local stationarity and local homogeneity. The latter are valid for arbitrary Reynolds numbers for the case of stationary, homogeneous turbulence. Precise use of the mathematics requires care noted in the Discussion section.

physics.flu-dyn↗

arXiv2005

Opportunities for use of exact statistical equations

Exact structure function equations are an efficient means of obtaining asymptotic laws such as inertial range laws, as well as all measurable effects of inhomogeneity and anisotropy that cause deviations from such laws. "Exact" means that the equations are obtained from the Navier-Stokes equation or other hydrodynamic equations without any approximation. A pragmatic definition of local homogeneity lies within the exact equations because terms that explicitly depend on the rate of change of measurement location appear within the exact equations; an analogous statement is true for local stationarity. An exact definition of averaging operations is required for the exact equations. Careful derivations of several inertial range laws have appeared in the literature recently in the form of theorems. These theorems give the relationships of the energy dissipation rate to the structure function of acceleration increment multiplied by velocity increment and to both the trace of and the components of the third-order velocity structure functions. These laws are efficiently derived from the exact velocity structure function equations. In some respects, the results obtained herein differ from the previous theorems. The acceleration-velocity structure function is useful for obtaining the energy dissipation rate in particle tracking experiments provided that the effects of inhomogeneity are estimated by means of displacing the measurement location.

physics.flu-dyn↗

arXiv2002

Exact Second-Order Structure-Function Relationships

Equations that follow from the Navier-Stokes equation and incompressibility but with no other approximations are "exact.". Exact equations relating second- and third-order structure functions are studied, as is an exact incompressibility condition on the second-order velocity structure function. Opportunities for investigations using these equations are discussed. Precisely defined averaging operations are required to obtain exact averaged equations. Ensemble, temporal, and spatial averages are all considered because they produce different statistical equations and because they apply to theoretical purposes, experiment, and numerical simulation of turbulence. Particularly simple exact equations are obtained for the following cases: i) the trace of the structure functions, ii) DNS that has periodic boundary conditions, and iii) an average over a sphere in r-space. The last case (iii) introduces the average over orientations of r into the structure function equations. The energy dissipation rate appears in the exact trace equation without averaging, whereas in previous formulations energy dissipation rate appears after averaging and use of local isotropy. The trace mitigates the effect of anisotropy in the equations, thereby revealing that the trace of the third-order structure function is expected to be superior for quantifying asymptotic scaling laws. The orientation average has the same property.

physics.flu-dyn↗

arXiv2002

Length Scales of Acceleration for Locally Isotropic Turbulence

Length scales are determined that govern the behavior at small separations of the correlations of fluid-particle acceleration, viscous force, and pressure gradient. The length scales and an associated universal constant are quantified on the basis of published data. The length scale governing pressure spectra at high wave numbers is discussed. Fluid-particle acceleration correlation is governed by two length scales; one arises from the pressure gradient, the other from the viscous force.

physics.flu-dyn↗

arXiv2002

The Approach of Turbulence to the Locally Homogeneous Asymptote as Studied using Exact Structure-Function Equations

Equations that follow from the Navier-Stokes equation and incompressibility but with no other approximations are called "exact" here. Exact equations relating 2nd and 3rd-order structure functions are obtained, as is an exact incompressibility condition on the 2nd-order velocity structure function. Ensemble, temporal, and spatial averages are all considered because they produce different statistical equations. Exact equations have particularly simple forms for the cases of an average over a sphere in r-space as well as DNS that has periodic boundary conditions. The trace mitigates the effect of anisotropy in the equations, thereby revealing that the trace of the third-order structure function is expected to be superior for quantifying asymptotic scaling laws. The midpoint and the difference of the two points at which the hydrodynamic quantities are obtained are X and r; t is time. Dependencies on X and on the orientation of r and on t fade as the asymptotic statistical states of local homogeneity, local isotropy, and local stationarity, respectively, are approached. The exact equations are thus applicable to study of the approach toward those asymptotic states. A new definition of local homogeneity is contrasted with previous definitions. The approach toward the asymptotic state of local homogeneity is studied by using scale analysis to determine the required approximations and the approximate equations pertaining to experiments and simulations of the small-scale structure of high-Reynolds-number turbulence, but without invoking local isotropy. Those equations differ from equations for homogeneous turbulence.

physics.flu-dyn↗

arXiv2001

Scaling of acceleration in locally isotropic turbulence

The variances of the fluid-particle acceleration and of the pressure-gradient and viscous force are given. The scaling parameters for these variances are velocity statistics measureable with a single-wire anemometer. For both high and low Reynolds numbers, asymptotic scaling formulas are given; these agree quantitatively with DNS data. Thus, the scaling can be presumed known for all Reynolds numbers. Fluid-particle acceleration variance does not obey K41 scaling at any Reynolds number; this is consistent with recent experimental data. The non-dimensional pressure-gradient variance named lambda-sub{T} /lambda-sub{P} is shown to be obsolete.

physics.flu-dyn↗

arXiv2001

Alternative to R_lambda-scaling of Small-Scale Turbulence Statistics

Traditionally, trends of universal turbulence statistics are presented versus R-lambda, which is the Reynolds number based on Taylor's scale, lambda, and the root-mean-squared (rms) velocity component, u'. Taylor's scale and u', and hence R-lambda, do not have the attribute of universality. The ratio of rms fluid-particle acceleration to rms viscous acceleration is an alternative to R-lambda which has the advantage of being determined by the small scales of turbulence. This ratio, denoted R-a, has the following attributes: it 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, and like R-lambda, can be partially evaluated by means of flow similarity. For isotropic turbulence the relationship between R-a and R-lambda is shown. Velocity derivative flatness measured between counter-rotating blades appears monotonic when graphed versus R-a, unlike when graphed versus R-lambda.

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