Structure Function Equations Research Articles

Most-likely DCF Estimates of Magnetic Field Strength

The Davis–Chandrasekhar–Fermi (DCF) method is widely used to evaluate magnetic fields in star-forming regions. Yet it remains unclear how well DCF equations estimate the mean plane-of-the-sky field strength in a map region. To address this question, five DCF equations are applied to an idealized cloud map. Its polarization angles have a normal distribution with dispersion σ θ , and its density and velocity dispersion have negligible variation. Each DCF equation specifies a global field strength B DCF and a distribution of local DCF estimates. The “most-likely” DCF field strength B ml is the distribution mode, for which a correction factor β ml ≡ B ml/B DCF is calculated analytically. For each equation, β ml < 1, indicating that B DCF is a biased estimator of B ml. The values of β ml are β ml ≈ 0.7 when BDCF∝σθ−1 due to turbulent excitation of Alfvénic MHD waves, and β ml ≈ 0.9 when BDCF∝σθ−1/2 due to non-Alfvénic MHD waves. These statistical correction factors β ml have partial agreement with correction factors βsim obtained from MHD simulations. The relative importance of the statistical correction is estimated by assuming that each simulation correction has both a statistical and a physical component. Then the standard, structure function, and original DCF equations appear most accurate because they require the least physical correction. Their relative physical correction factors are 0.1, 0.3, and 0.4 on a scale from 0 to 1. In contrast, the large-angle and parallel-δ B equations have physical correction factors 0.6 and 0.7. These results may be useful in selecting DCF equations, within model limitations.

Analysis of structure function equations up to the seventh order

ABSTRACTWe examine balances of structure function equations up to the seventh order N = 7 for longitudinal, mixed and transverse components. Similarly, we examine the traces of the structure function equations, which are of interest because they contain invariant scaling parameters. The trace equations are found to be qualitatively similar to the individual component's equations. In the even-order equations, the source terms proportional to the correlation between velocity increments and the pseudo-dissipation tensor are significant, while for odd N, source terms proportional to the correlation of velocity increments and pressure gradients are dominant. Regarding the component equations, one finds under the inertial range assumptions as many equations as unknown structure functions for even N, i.e. can solve for them as function of the source terms. On the other hand, there are more structure functions than equations for odd N under the inertial range assumptions. Similarly, there are not enough linearly independent equations in the viscous range r → 0 for orders N > 3.

High-order structure functions for passive scalar fed by a mean gradient

Transport equations for even-order structure functions are written for a passive scalar mixing fed by a mean scalar gradient, with a Schmidt number Sc=1. Direct numerical simulations (DNS), in a range of Reynolds numbers Rλ ∈ [88, 529] are used to assess the validity of these equations, for the particular cases of second-and fourth-order moments. The involved terms pertain to molecular diffusion, transport, production, and dissipative-fluxes. The latter term, present at all scales, is equal to: i) the mean scalar variance dissipation rate, ⟨χ⟩, for the second-order moments transport equation; ii) non-linear correlations between χ and second-order moments of the scalar increment, for the fourth-order moments transport equation.The equations are further analysed to show that the similarity scales (i.e., variables which allow for perfect collapse of the normalised terms in the equations) are, for second-order moments, fully consistent with Kolmogorov–Obukhov–Corrsin (KOC) theory. However, for higher-order moments, adequate similarity scales are built from ⟨χn⟩. The similarity is tenable for the dissipative range, and the beginning of the scaling range.

Longitudinal and transverse structure functions in high Reynolds-number magneto-hydrodynamic turbulence

We investigate the scaling behavior of longitudinal and transverse structure functions in homogeneous and isotropic magneto-hydrodynamic (MHD) turbulence by means of an exact hierarchy of structure function equations as well as by direct numerical simulations of two- and three-dimensional MHD turbulence. In particular, rescaling relations between longitudinal and transverse structure functions are derived and utilized in order to compare different scaling behavior in the inertial range. It is found that there are no substantial differences between longitudinal and transverse structure functions in MHD turbulence. This finding stands in contrast to the case of hydrodynamic turbulence which shows persistent differences even at high Reynolds numbers. We propose a physical picture that is based on an effective reduction of pressure contributions due to local regions of same magnitude and alignment of velocity and magnetic field fluctuations. Finally, our findings underline the importance of the pressure term for the actually observed scaling differences in hydrodynamic turbulence.

Finite Reynolds number corrections of the 4/5 law for decaying turbulence

We examine finite Reynolds number contributions to the inertial range solution of the third order structure functions D3,0 and D1,2 stemming from the unsteady and viscous terms. Under the assumption that the second order correlations f and g are self-similar under a coordinate change, we are able to rewrite the exact second order equations as a function of a normalized scale r̃ only. We close the resulting system of equations using a power law and an eddy-viscosity ansatz. If we further assume K41 scaling, we find the same Reynolds number dependence as previously in the literature. We proceed to extrapolate towards higher Reynolds numbers to examine the unsteady and viscous terms in more detail. We find that the intersection between the two terms, where their contribution to the solution of the structure function equations is relatively small, scales with the Taylor scale λ.2 MoreReceived 29 February 2016DOI:https://doi.org/10.1103/PhysRevFluids.1.064403Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasTurbulenceFluid Dynamics

Higher-order dissipation in the theory of homogeneous isotropic turbulence

The two-point theory of homogeneous isotropic turbulence is extended to source terms appearing in the equations for higher-order structure functions. For this, transport equations for these source terms are derived. We focus on the trace of the resulting equations, which is of particular interest because it is invariant and therefore independent of the coordinate system. In the trace of the even-order source term equation, we discover the higher-order moments of the dissipation distribution, and the individual even-order source term equations contain the higher-order moments of the longitudinal, transverse and mixed dissipation distribution functions. This shows for the first time that dissipation fluctuations, on which most of the phenomenological intermittency models are based, are contained in the Navier–Stokes equations. Noticeably, we also find the volume-averaged dissipation $\unicode[STIX]{x1D700}_{r}$ used by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85) in the resulting system of equations, because it is related to dissipation correlations.

Very large-scale computation of very small-scale motions in a fully developed pipe flow

Local isotropy theory is examined using direct numerical simulation in a fully developed pipe flow at two Reynolds numbers Reτ=1285.6 and 684.8. The approach to local isotropy is assessed with reference to the two Kolmogorov classical equations for longitudinal and transverse velocity structure functions. The results for the second-order longitudinal structure functions in both the dissipative and inertial ranges indicate an improved agreement with the local isotropy hypothesis as the centreline is approached. However, the transverse structure functions satisfy isotropy neither in the dissipative range or in the inertial range. The distribution of the longitudinal and transverse structure functions also shows a substantial Reynolds number dependance in the logarithmic region of the flow and beyond. The results for the third-order longitudinal structure function demonstrate an increased Reynolds number influence, and a deteriorating tendency to local isotropy for large separations. Contour images of axial velocity differences in the dissipative and inertial ranges have exhibited interesting patterns in relation to those of the instantaneous axial velocity. Finally, the results obtained in this investigation are in very good agreement with other published experimental and numerical data on channel and duct flows. Copyright © 2010 John Wiley & Sons, Ltd.

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.

Towards a dynamical theory of multifractals in turbulence

Making use of the exact equations for structure functions, supplemented by the equations for dissipa tive anomaly as well as an estimate for the Lagrangian acceleration of fluid particles, we obtain a main result of the multifractal theory of turbulence. The central element of the theory is a dissipation cut-off that depends on the order of the structure function. An expression obtained for the exponents $s_{n}$ in the scaling relations $\bar{(\frac{\partial u}{\partial x})^{n}}/\bar{(\frac{\partial u}{\partial x})^{2}}^{\frac{n}{2}} \propto Re^{s_{n}}$, between the velocity gradients $\frac{\partial u}{\partial x}$ and the Reynolds number $Re$, agrees well with experimental data.

Towards a dynamical theory of multifractals in turbulence

Making use of the exact equations for structure functions, supplemented by the equations for dissipative anomaly as well as an estimate for the Lagrangian acceleration of fluid particles, we obtain a main result of the multifractal theory of turbulence. The central element of the theory is a dissipation cut-off that depends on the order of the structure function. An expression obtained for the exponents s n in the scaling relations ∂ u ∂ x n ¯ ∂ u ∂ x 2 n / 2 ¯ ∝ Re s n , between the velocity gradients ∂ u / ∂ x and the Reynolds number Re, agrees well with experimental data.

Structure‐function‐based approach to analyzing received signals for spaced antenna radars

This paper describes an alternative approach to analyzing multiple received signals for spaced antenna radars. The approach is based on computation and analysis of the different order auto and cross structure functions (SF) for the received signal's power. SF of order p ≥ 2 is the p‐th order statistical moment of the difference between signals. Equations for cross and auto SF at p = 2 and 4 are derived for a spaced antenna radar with a vertically directed transmitted beam. SF parameters in these equations are related to the mean horizontal velocity components and turbulence characteristics of a scattering medium. Theoretical limitations on applying the derived equations to a specific radar are presented. Advantages and shortcomings of the proposed approach with respect to the correlation function and spectra‐based approaches are discussed as well. The major advantage of the proposed approach is that SF of any order p ≥ 2 can be derived and applied to practical measurements; the higher‐order SF provide additional information about the scattering medium.

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. Case (iii) introduces the average over orientations of r into the structure-function equations. The energy dissipation rate ε appears in the exact trace equation without averaging, whereas in previous formulations ε 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.

Structure-function equations for scalars

An exact equation is obtained which relates the products of two-point differences of a scalar quantity and those differences with the difference of velocity and other quantities. The averages of such products are structure functions. Equations that follow from the scalar equation but with no other approximations are called “exact” here. Exact equations for structure functions are obtained. Ensemble, temporal, and spatial averages are all considered because each one produces different statistical equations from the others and because they differ in application to theoretical purposes, experiment, and numerical simulation of turbulence. The midpoint and the difference of the two points at which the hydrodynamic quantities are obtained are X and r, respectively. Use of X and r as independent variables combined with Reynolds decomposition reveals quantifiable effects of inhomogeneity and large-scale structures. Exact equations obtained by averaging over a sphere in r space have a particularly simple form. The case of a numerical simulation that has periodic boundary conditions leads to particularly simple equations. For the case of grid-generated wind-tunnel turbulence, approximate equations are derived relating structure functions of fluctuations. Applications of the equations to experiments and direct numerical simulation are discussed.

Dynamical equations for high-order structure functions, and a comparison of a mean-field theory with experiments in three-dimensional turbulence.

Two recent papers [V. Yakhot, Phys. Rev. E 63, 026307, (2001) and R. J. Hill, J. Fluid Mech. 434, 379, (2001)] derive, through two different approaches that have the Navier-Stokes equations as the common starting point, a set of dynamic equations for structure functions of arbitrary order in turbulence. These equations are not closed. Yakhot proposed a "mean-field theory" to close the equations for locally isotropic turbulence, and obtained scaling exponents of structure functions and expressions for the peak in the probability density function of transverse velocity increments, and for its behavior for intermediate amplitudes. At high Reynolds numbers, some relevant experimental data on pressure gradient and dissipation terms are presented that are needed to provide closure, as well as on other aspects predicted by the theory. Comparison between the theory and the data shows varying levels of agreement, and reveals gaps inherent to the implementation of the theory.

Equations relating structure functions of all orders

Exact equations are given 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. The uses of these equations for investigating the approach to local homogeneity as well as to local isotropy and the balance of the equations and identification of scaling ranges are discussed. The implications for scaling exponents and investigation of intermittency are briefly discussed.

Non-singlet splitting functions in QED

Iterative solution of QED evolution equations for non-singlet electron structure functions is considered. Analytical expressions in the fourth and fifth orders are presented in terms of splitting functions. Relation to the existing exponentiated solution is discussed.

Investigations into the BFKL mechanism with a running QCD coupling

We present approximations of varying degree of sophistication to the integral equations for the (gluon) structure functions of a hadron (``the partonic flux factor'') in a model valid in the Leading Log Approximation with a running coupling constant. The results are all of the BFKL-type, i.e. a power in the Bjorken variable x_B^{-\lambda} with the parameter \lambda determined from the size \alpha_0 of the ``effective'' running coupling \bar{\alpha}\equiv 3\alpha_s/\pi= \alpha_0/\log(k_{\perp}^2) and varying depending upon the treatment of the transverse momentum pole. We also consider the implications for the transverse momentum (k_{\perp}) fluctuations along the emission chains and we obtain an exponential falloff in the relevant \kappa\equiv \log(k_{\perp}^2)-variable, i.e. an inverse power (k_{\perp}^2)^{-(2+\lambda)} with the same parameter \lambda. This is different from the BFKL-result for a fixed coupling, where the distributions are Gaussian in the \kappa-variable with a width as in a Brownian motion determined by ``the length'' of the emission chains, i.e. \log(1/x_B). The results are verified by a realistic Monte Carlo simulation and we provide a simple physics motivation for the change.

Evolution paths for small-x structure functions

We compare the solutions of the “one-loop” (summing αS ln″ Q2) and “all-loop” (including also α″S ln″ x) evolution equations for small-x structure functions. In the semi-classical approximation one can define evolution paths in the (ln x, ln Q2)-plane, which are helpful in understanding many features of the solutions. We compute the asymptotic form of the mean path and the dispersion of paths for both the one-loop and all-loop cases, and compare the results with those of Monte Carlo simulations. We also use the Monte Carlo simulations to study the infrared stability of small-x evolution. Our results explain why the one loop and all-loop solutions are similar, over a wide range of x and Q2, especially when the input structure function at low Q2 is a rapidly increasing function of 1x.

Isotropic forms of vorticity and velocity structure function equations in several turbulent shear flows

Isotropic forms of the vorticity and velocity structure function equations are valid in the logarithmic regions of a fully developed turbulent duct flow and a boundary layer over a wide range of Reynolds numbers. These equations are also valid in a low Reynolds number turbulent wake and a high Reynolds number atmospheric surface layer. However, these equations are not satisfied in the near-wall region of duct and boundary layer flows where the production of vorticity is significantly larger than the dissipation. The appropriateness of the equations as indicators of local isotropy is discussed in the context of the present and other published measurements.