On logarithmic extensions of local scale-invariance

Ageing phenomena far from equilibrium naturally present dynamical scaling and in many situations this may be generalised to local scale-invariance. Generically, the absence of time-translation-invariance implies that each scaling operator is characterised by two independent scaling dimensions. Building on analogies with logarithmic conformal invariance and logarithmic Schrödinger-invariance, this work proposes a logarithmic extension of local scale-invariance, without time-translation-invariance. Carrying this out requires in general to replace both scaling dimensions of each scaling operator by Jordan cells. Co-variant two-point functions are derived for the most simple case of a two-dimensional logarithmic extension. Their form is compared to simulational data for autoresponse functions in several universality classes of non-equilibrium ageing phenomena.     

Small scale intermittency and bursting in a turbulent channel flow

The statistical properties of the streamwise velocity fluctuations in a fully developed turbulent channel flow are studied experimentally by means of single hot wire measurements. The intermittency features, studied through the scaling of the moments of the velocity structure function computed using the extended self- similarity and through the probability density function of the wavelet coefficients, are found to be dependent on the distance from the wall. The maximum intermittency effects are observed in the region between the buffer layer and the inner part of the logarithmic region where it is known that the bursting phenomenon, related to coherent structures such as low speed streaks and streamwise vortices, is the dominant dynamical feature. An eduction technique based on wavelet transform for identification of organized motion is developed and used to analyze the turbulent signals. Streamwise velocity conditional averages computed on events educed with the proposed method are reported. Events responsible for intermittency are found to consist of regions of high velocity gradients and are directly correlated with the observed increase of intermittency close to the wall.     

Moments of the Wigner distribution of rotationally symmetric partially coherent light

The Wigner distribution of rotationally symmetric partially coherent light is considered, and the constraints for its moments are derived. Although all odd-order moments vanish, these constraints lead to a drastic reduction in the number of parameters that we need to describe all even-order moments: whereas in general we have (N + 1)(N + 2)(N + 3)/6 different moments of order N, this number reduces to (1 + N/2)2 in the case of rotational symmetry. A way to measure the moments as intensity moments in the output planes of (generally anamorphic) fractional Fourier-transform systems is presented.     

Experimental study on the local similarity scaling of the turbulence spectrum in the turbulent boundary layer

The streamwise fluctuating velocity in the turbulent boundary layer is measured under approximately medium Reynolds Number by hot wire in order to investigate the scaling properties of the overlapped turbulent spectrum among energy-containing area, inertial subrange and dissipation range based on FFT analysis. The experiment indicates that the high Reynolds flow reported before is not indispensable to produce −1 scaling. So far as the measured position is provided with much higher spatial resolution and enough closing to the wall, −1 scaling is determinate to exist when approaching medium Reynolds. The scaling ranges are supposed to begin at inner scale and end in outer scale, which reveals the local similarity of the energy spectrum over the energy-containing eddies near the wall. In the logarithmic area (y ⁺ > 130), −5/3 scaling occurs in the energy spectrum, while moving away from the wall with Reynolds numbers increasing, the inertial subrange extends to the lower wavenumbers. On the condition k ₁ η ≫ 0.1, the curves of the turbulence spectrum in the logarithmic layer are superposed, which expresses the similarity of turbulence energy distributed in Komogorov scaling area and exhibits local isotropy characteristics by virtue of the viscous dissipation. Supported by the National Natural Science Foundation of China (Grant Nos. 10832001 and 10872145), the Program for New Century Excellent Talents in Universities of Education Ministry of China, and the Plan of Tianjin Science and Technology Development (Grant No. 06TXTJJC13800)     

Analysis of turbulent axisymmetric inner near wake

An analysis is presented describing the characteristics of mean velocity profile in the axisymmetric turbulent inner near-wake flow behind a body of revolution. The near wake is developing under zero streamwise pressure gradient and the upstream turbulent boundary layer is fully developed. It is shown that the boundary layer condition that exists at the trailing edge can be used to describe the mean velocity profiles in the inner near wake. It is shown that the logarithmic layer of the upstream turbulent boundary layer continues to be valid for some more distance in the near wake, and as the streamwise distance is increased, the logarithmic layer is slowly getting destroyed. It is also shown that the central line velocity exhibits a logarithmic behaviour for large streamwise distance. Results of the analysis have been validated using available experimental data.     

Convection and correlation of coherent structure in turbulent boundary layer using tomographic particle image velocimetry

The present experimental work focuses on a new model for space–time correlation and the scale-dependencies of convection velocity and sweep velocity in turbulent boundary layer over a flat wall. A turbulent boundary layer flow at Reθ= 2460 is measured by tomographic particle image velocimetry(tomographic PIV). It is demonstrated that arch, cane,and hairpin vortices are dominant in the logarithmic layer. Hairpins and hairpin packets are responsible for the elongated low-momentum zones observed in the instantaneous flow field. The conditionally-averaged coherent structures systemically illustrate the key roles of hairpin vortice in the turbulence dynamic events, such as ejection and sweep events and energy transport. The space–time correlations of instantaneous streamwise fluctuation velocity are calculated and confirm the new elliptic model for the space–time correlation instead of Taylor hypothesis. The convection velocities derived from the space–time correlation and conditionally-averaged method both suggest the scaling with the local mean velocity in the logarithmic layer. Convection velocity result based on Fourier decomposition(FD) shows stronger scale- dependency in the spanwise direction than in streamwise direction. Compared with FD, the proper orthogonal decomposition(POD) has a distinct distribution of convection velocity for the large- and small-scales which are separated in light of their contributions of turbulent kinetic energy.     

Mean velocity behaviour in the near wake of long slender cylinder with a sharp trailing edge at a high Reynolds number

A simple eddy viscosity model is applied to the governing equations to establish the behaviour of the mean velocity in the turbulent axisymmetric near wake. The near wake develops from a long slender cylinder which is kept parallel to the flow and is developing under zero streamwise pressure gradient. The upstream turbulent boundary layer on the body of revolution is fully developed. In the inner layer of the flow downstream of the trailing edge, the turbulent inner layer of the upstream boundary layer grows into the initial logarithmic layer, and as a consequence, the centreline velocity in the near wake is shown to increase logarithmically with streamwise distances for large streamwise distances. The analysis further leads to two regions of the near wake flow (the inner near wake and the outer near wake), similar to that of a fully developed turbulent boundary layer, for which the governing equations have been derived. The matching between these two regions leads to a logarithmic variation in the normal direction. Also shown is the variation of the square of the wake width which varies logarithmically with streamwise distance in the near wake. These features are validated by comparison with available experimental data.     

Turbulent plane wall jets over smooth and rough surfaces

Large-eddy simulations were carried out to study the effects of surface roughness on a plane wall-jet using the Lagrangian dynamic eddy-viscosity subgrid-scale model, at Re = 7500 (based on the jet bulk velocity and height). Results over both smooth and rough surfaces were validated by experimental data at the same Reynolds number. As the jet is injected into the still environment, large-scale rollers are generated in the shear layer between the high-momentum fluid of the jet and the surrounding and are convected downstream with the flow. To understand the extent to which the outer-layer structures modify the flow in the inner layer and the extent to which the effect of roughness spreads away from the wall, both instantaneous and mean flow fields were investigated. The results revealed that, for the Reynolds number and roughness height considered in this study, the effect of roughness is mostly confined to the near-wall region of the wall jet. There is no structural difference between the outer layer of the wall jet over the smooth and rough surfaces. Roughness does not affect the size of the outer-layer structures or the scaling of the profiles of Reynolds stresses in the outer layer. However, in the inner layer, roughness redistributes stresses from streamwise to wall-normal and spanwise directions toward isotropy. Contours of joint probability-density function of the streamwise and wall-normal velocity fluctuations at the bottom of the logarithmic region match those of the turbulent boundary layer at the same height; while the traces of the outer-layer structure were detected at the top of the logarithmic region, indicating that they do not affect the flow very close to the wall, but still modify a major portion of the inner layer. This modification must be taken into consideration when the inner layer of a wall jet is compared with the conventional turbulent boundary layer.     

Reynolds number dependence of streamwise velocity spectra in turbulent pipe flow

Spectra of the streamwise velocity component in fully developed turbulent pipe flow are presented for Reynolds numbers up to 5.7x10(6). Even at the highest Reynolds number, streamwise velocity spectra exhibit incomplete similarity only: while spectra collapse with both classical inner and outer scaling for limited ranges of wave number, these ranges do not overlap. Thus similarity may not be described as complete, and a region varying with the inverse of the streamwise wave number, k(1), is not expected, and any apparent k(-1)(1) range does not attract any special significance and does not involve a universal constant. Reasons for this are suggested.     

Hermite-DG methods for pdf equations modelling particle transport and deposition in turbulent boundary layers

A novel methodology is presented for the numerical treatment of multi-dimensional pdf (probability density function) models used to study particle transport in turbulent boundary layers. A system of coupled Fokker–Planck type equations is constructed to describe the transport of phase-space conditioned moments of particle and fluid velocities, both streamwise and wall-normal. This system, unlike conventional moment-based transport equations, allows for an exact treatment of particle deposition at the flow boundary and provides an efficient way to handle the 5-dimensional phase-space domain. Moreover, the equations in the system are linear and can be solved in a sequential fashion; there is no closure problem to address.A hybrid Hermite-Discontinuous Galerkin scheme is developed to treat the system. The choice of Hermite basis functions in combination with an iterative scaling approach permits the efficient computation of solutions to high accuracy. Results demonstrate the effectiveness of the methodology in resolving the extreme gradients characteristic of distributions near an absorbing boundary.     

Decaying magnetohydrodynamics: effects of initial conditions

We study the effects of homogenous and isotropic initial conditions on decaying magnetohydrodynamics (MHD). We show that for an initial distribution of velocity and magnetic-field fluctuations, appropriately defined structure functions decay as a power law in time. We also show that for a suitable choice of initial cross correlations between velocity and magnetic fields even-order structure functions acquire anomalous scaling in time where as scaling exponents of the odd-order structure functions remain unchanged. We discuss our results in the context of fully developed MHD turbulence.     

The four-dimensional site-diluted Ising model: A finite-size scaling study

Using finite-size scaling techniques, we study the critical properties of the site-diluted Ising model in four dimensions. We carry out a high-statistics Monte Carlo simulation for several values of the dilution. The results support the perturbative scenario: there is only the Ising fixed point with large logarithmic scaling corrections. We obtain, using the Perturbative Renormalization Group, functional forms for the scaling of several observables that are in agreement with the numerical data.     

Cascade-like and scaling behavior of wind velocity increments in the atmospheric surface layer

By using a large amount of data collected in the atmospheric surface layer, we analyze the probability density functions (PDFs), the probability of return and the moments of wind velocity increments. Results show that the PDFs change from the non-Gaussian long-tailed distributions to Gaussian with the increase of time scales. This is similar to what has been observed and interpreted as an indication of cascade in the fully developed homogeneous and isotropic turbulence. Besides, both the probability of return and the moments are found to be scaling with time scales. We then compare above results with the truncated Lévy flights and the log-normal PDF model. It is found that although both models show the cascade-like behavior in the PDFs and the scaling behavior in the probability of return and the moments under some conditions, they are not good enough for quantitatively describing the random process of wind velocity increments.     

Local multifractal thermodynamics of 3D turbulence

Using results of a direct numerical simulation (DNS) of 3D turbulence we show that the observed generalized scaling (i.e. scaling moments versus moments of different orders) is consistent with a lognormal-like distribution of turbulent energy dissipation fluctuations with moderate amplitudes for all space scales available in this DNS (beginning from the molecular viscosity scale up to largest ones). Local multifractal thermodynamics has been developed to interpret the data obtained using the generalized scaling, and a new interval of space scales with inverse cascade of generalized energy has been found between dissipative and inertial intervals of scales for sufficiently large values of the Reynolds number. Received 21 July 2000     

Angular structure of QCD jets

We derive the angular correlation functions between an arbitary number of partons inside a quark or gluon jet emerging from a high energy hard collision. As an application results for the correlation in the relative angle between two partons as well as the multiplicity moments of any order for partons in a sidewise angular region are calculated. At asymptotic energies the distributions of rescaled angular variables approach a scaling limit of a new type with two redundant quantities. All observables reveal the power behaviour characteristic to the fully developed selfsimilar cascade in appropriate regions of phase space. We present analytical results from the double logarithmic approximation as well as Monte Carlo results.     

The Kosterlitz-Thouless universality class

We examine the Kosterlitz-Thouless universality class and show that essential scaling at this type of phase transition is not self-consistent unless multiplicative logarithmic corrections are included. In the case of specific heat these logarithmic corrections are identified analytically. To identify those corresponding to the susceptibility we set up a numerical method involving the finite-size scaling of Lee-Yang zeroes. We also study the density of zeroes and introduce a new concept called index scaling. We apply the method to the XY model and the c]osely related step model in two dimensions. The critical parameters (including logarithmic corrections) of the step model are compatible with those of the XY model indicating that both models belong to the same universality class. This result then raises questions over how a vortex binding scenario can be the driving mechanism for the phase transition. Furthermore, the logarithmic corrections identified numerically by our methods of fitting are not in agreement with the renormalization group predictions of Kosterlitz and Thouless.     

The generalised vector dominance/colour-dipole picture of deep-inelastic scattering at low x

We give a detailed account of the recently formulated generalised vector dominance/colour-dipole picture (GVD/CDP) of deep-inelastic scattering at low , including photoproduction. The approach, based on transitions, propagation and diffractive scattering via the generic structure of the two-gluon exchange, provides a unique and quantitatively successful theory for the total cross section, , at low x. The GVD/CDP is shown to imply the empirical low-x scaling law, with , that was established by a model-independent analysis of the experimental data. Received: 16 February 2001 / Published online: 3 May 2001     

Extensive air showers and Feynman scaling above 1000 GeV

We describe a method for using extensive air shower data to determine whether a Feynman scaling limit exists in the central region for inclusive cross sections. Preliminary analysis suggests that the shower muon to electron ratio is more consistent with a power law increase of pion multiplicity with energy than with the logarithmic multiplicity increase characteristic of a scaling limit, but an analysis of the intra-nuclear cascade is required to reach a firm conclusion.     

Longitudinal and transverse structure functions in decaying nearly homogeneous and isotropic turbulence

Streamwise evolution of longitudinal and transverse velocity structure functions in a decaying homogeneous and nearly isotropic turbulence is reported for Reynolds numbers Reλ up to 720. First, two theoretical relations between longitudinal and transverse structure functions are examined in the light of recently derived relations and the results show that the low-order transverse structure functions can be well approximated by longitudinal ones within the sub-inertial range. Reconstruction of fourth-order transverse structure functions with a recently proposed relation by Grauer et al. is comparatively less valid than the relation already proposed by Antonia et al. Secondly, extended self-similarity methods are used to measure the scaling exponents up to order eight and the streamwise evolution of scaling exponents is explored. The scaling exponents of longitudinal structure functions are, at first location, close to Zybin’s model, and at the fourth location, close to She–Leveque model. No obvious trend is found for the streamwise evolution of longitudinal scaling exponents, whereas, on the contrary, transverse scaling exponents become slightly smaller with the development of a steamwise direction. Finally, the stremwise variation of the order-dependent isotropy ratio indicates the turbulence at the last location is closer to isotropic than the other three locations.