On the scaling of three-dimensional homogeneous and isotropic turbulence (original) (raw)
Scaling laws and intermittency in homogeneous shear flow
Physics of Fluids, 2002
In this paper we discuss the dynamical features of intermittent fluctuations in homogeneous shear flow turbulence. In this flow the energy cascade is strongly modified by the production of turbulent kinetic energy related to the presence of vortical structures induced by the shear. By using direct numerical simulations, we show that the refined Kolmogorov similarity is broken and a new form of similarity is observed, in agreement to previous results obtained in turbulent boundary layers. As a consequence, the intermittency of velocity fluctuations increases with respect to homogeneous and isotropic turbulence. We find here that the statistical properties of the energy dissipation are practically unchanged with respect to homogeneous isotropic conditions, while the increased intermittency is entirely captured in terms of the new similarity law.
Generalized scaling in fully developed turbulence
Physica D-nonlinear Phenomena, 1996
In this paper we report numerical and experimental results on the scaling properties of the velocity turbulent fields in several flows. The limits of a new form of scaling, named Extended Self Similarity(ESS), are discussed. We show that, when a mean shear is absent, the self scaling exponents are universal and they do not depend on the specific flow (3D homogeneous turbulence, thermal convection , MHD). In contrast, ESS is not observed when a strong shear is present. We propose a generalized version of self scaling which extends down to the smallest resolvable scales even in cases where ESS is not present. This new scaling is checked in several laboratory and numerical experiment. A possible theoretical interpretation is also proposed. A synthetic turbulent signal having most of the properties of a real one has been generated.
Helical shell models for three-dimensional turbulence
Physical Review E, 1996
In this paper we study a new class of shell models, defined in terms of two complex dynamical variables per shell, transporting positive and negative helicity respectively. The dynamical equations are derived from a decomposition into helical modes of the velocity Fourier components of Navier-Stokes equations (F. Waleffe, Phys. Fluids A {\bf 4}, 350 (1992)). This decomposition leads to four different types of shell models, according to the possible non-equivalent combinations of helicities of the three interacting modes in each triad. Free parameters are fixed by imposing the conservation of energy and of a ``generalized helicity'' HalphaH_\alphaHalpha in the inviscid and unforced limit. For alpha=1\alpha=1alpha=1 this non-positive invariant looks exactly like helicity in the Fourier-helical decomposition of the Navier-Stokes equations. Long numerical integrations are performed, allowing the computation of the scaling exponents of the velocity increments and energy flux moments. The dependence of the models on the generalized helicity parameter alpha\alphaalpha and on the scale parameter lambda\lambdalambda is also studied. PDEs are finally derived in the limit when the ratio between shells goes to one.
The Multifractal Structure of Contrast Changes in Natural Images: From Sharp Edges to Textures
Neural Computation, 2000
We present a formalism that leads naturally to a hierarchical description of the different contrast structures in images, providing precise definitions of sharp edges and other texture components. Within this formalism, we achieve a decomposition of pixels of the image in sets, the fractal components of the image, such that each set contains only points characterized by a fixed strength of the singularity of the contrast gradient in its neighborhood. A crucial role in this description of images is played by the behavior of contrast differences under changes in scale. Contrary to naive scaling ideas where the image is thought to have uniform transformation properties , each of these fractal components has its own transformation law and scaling exponents. A conjecture on their biological relevance is also given.
Different intermittency for longitudinal and transversal turbulent fluctuations
Physics of Fluids, 1997
Scaling exponents of the longitudinal and transversal velocity structure functions in numerical Navier-Stokes turbulence simulations with Taylor-Reynolds numbers up to Re ϭ110 are determined by the extended self similarity method. We find significant differences in the degree of intermittency: For the sixth moments the scaling corrections to the classical Kolmogorov expectations are ␦ 6 L ϭϪ0.21Ϯ0.01 and ␦ 6 T ϭϪ0.43Ϯ0.01, respectively, independent of Re . Also the generalized extended self similarity exponents p,q ϭ␦ p /␦ q differ significantly for the longitudinal and transversal structure functions. Within the She-Leveque model this means that longitudinal and transversal fluctuations obey different types of hierarchies of the moments. Moreover, the She-Leveque model hierarchy parameters  L and  T show small but significant dependences on the order of the moment.
Scaling Properties in the Production Range of Shear Dominated Flows
Physical Review Letters, 2005
Recent developments in turbulence are focused on the effect of large scale anisotropy on the small scale statistics of velocity increments. According to Kolmogorov, isotropy is recovered in the large Reynolds number limit as the scale is reduced and, in the so-called inertial range, universal featuresnamely the scaling exponents of structure functions -emerge clearly. However this picture is violated in a number of cases, typically in the high shear region of wall bounded flows. The common opinion ascribes this effect to the contamination of the inertial range by the larger anisotropic scales, i.e. the residual anisotropy is assumed as a weak perturbation of an otherwise isotropic dynamics. In this case, given the rotational invariance of the Navier-Stokes equations, the isotropic component of the structure functions keeps the same exponents of isotropic turbulence. This kind of reasoning fails when the anisotropic effects are strong as in the production range of shear dominated flows. This regime is analyzed here by means of both numerical and experimental data for a homogeneous shear flow. A well defined scaling behavior is found to exist, with exponents which differ substantially from those of classical isotropic turbulence. Contrary to what predicted by the perturbation approach, such a deep alteration concerns the isotropic sector itself. The general validity of these results is discussed in the context of turbulence near solid walls, where more appropriate closure models for the coarse grained Navier-Stokes equations would be advisable.
Double scaling and intermittency in shear dominated flows
Physical Review E, 2001
The Refined Kolmogorov Similarity Hypothesis is a valuable tool for the description of intermittency in isotropic conditions. For flows in presence of a substantial mean shear, the nature of intermittency changes since the process of energy transfer is affected by the turbulent kinetic energy production associated with the Reynolds stresses. In these conditions a new form of refined similarity law has been found able to describe the increased level of intermittency which characterizes shear dominated flows. Ideally a length scale associated with the mean shear separates the two ranges, i.e. the classical Kolmogorov-like inertial range, below, and the shear dominated range, above. However, the data analyzed in previous papers correspond to conditions where the two scaling regimes can only be observed individually.
Physics of Fluids, 1997
A class of dynamical models of turbulence living on a one-dimensional dyadic-tree structure is introduced and studied. The models are obtained as a natural generalization of the popular GOY shell model of turbulence. These models are found to be chaotic and intermittent. They represent the first example of (1+1)-dimensional dynamical systems possessing non trivial multifractal properties. The dyadic structure allows to study spatial and temporal fluctuations. Energy dissipation statistics and its scaling properties are studied. Refined Kolmogorov Hypothesis is found to hold.
Lagrangian Velocity Statistics in Turbulent Flows: Effects of Dissipation
Physical Review Letters, 2003
We use the multifractal formalism to describe the effects of dissipation on Lagrangian velocity statistics in turbulent flows. We analyze high Reynolds number experiments and direct numerical simulation (DNS) data. We show that this approach reproduces the shape evolution of velocity increment probability density functions (PDF) from Gaussian to stretched exponentials as the time lag decreases from integral to dissipative time scales. A quantitative understanding of the departure from scaling exhibited by the magnitude cumulants, early in the inertial range, is obtained with a free parameter function D(h) which plays the role of the singularity spectrum in the asymptotic limit of infinite Reynolds number. We observe that numerical and experimental data are accurately described by a unique quadratic D(h) spectrum which is found to extend from hmin ≈ 0.18 to hmax ≈ 1, as the signature of the highly intermittent nature of Lagrangian velocity fluctuations.
Intermittency of near-bottom turbulence in tidal flow on a shallow shelf
Journal of Geophysical Research, 2010
1] The higher-order structure functions of vertical velocity fluctuations (transverse structure functions (TSF)) were employed to study the characteristics of turbulence intermittency in a reversing tidal flow on a 19 m deep shallow shelf of the East China Sea. Measurements from a downward-looking, bottom-mounted Acoustic Doppler Velocimeter, positioned 0.45 m above the seafloor, which spanned two semidiurnal tidal cycles, were analyzed. A classical lognormal single-parameter (m) model for intermittency and the universal multifractal approach (specifically, the two-parameter (C 1 and a) log-Levy model) were employed to analyze the TSF exponent x(q) in tidally driven turbulent boundary layer and to estimate m, C 1 , and a. During the energetic flooding tidal phases, the parameters of intermittency models approached the mean values of e ≈ 0.24, e C 1 ≈ 0.15, and e ≈ 1.5, which are accepted as the universal values for fully developed turbulence at high Reynolds numbers. With the decrease of advection velocity, m and C 1 increased up to m ≈ 0.5-0.6 and C 1 ≈ 0.25-0.35, but a decreased to about 1.4. The results explain the reported disparities between the smaller "universal" values of intermittency parameters m and C 1 (mostly measured in laboratory and atmospheric high Reynolds number flows) and those (m = 0.4-0.5) reported for oceanic stratified turbulence in the pycnocline, which is associated with relatively low local Reynolds numbers R lw . The scaling exponents x(2) of the second-order TSF, relative to the third-order structure function, was also found to be a decreasing function of R lw , approaching the classical value of 2/3 only at very high R lw . A larger departure from the universal turbulent regime at lower Reynolds numbers could be attributed to the higher anisotropy and associated intermittency of underdeveloped turbulence.
Spatial properties of velocity structure functions in turbulent wake flows
Physical Review E, 1998
In this paper we present experimental evidence that the scaling laws for the velocity structure functions S n (r)ϭ͓͗V(xϩr)ϪV(x)͔ n ͘ nϭ2,4,6,8 hold in various parts of the flow domain. The exponents that characterize the scaling are, however, a function of the position in the wake that is the local strength and ubiquity of coherent structures. This variation is shown to be systematic and considerably exceeds the inaccuracy involved in the determination of the exponents. This is an objective indication of the influence that the organized flow structures and inhomogeneity may have on intermittency. In the analysis we invoke the concept of the extended self-similarity ͑ESS͒. ͓S1063-651X͑98͒50101-8͔ PACS number͑s͒: 47.27.Jv, 47.27.Nz, 47.27.Vf One of the most intriguing features of fully developed turbulent flows is the phenomenon of intermittency ͓1͔. It can be defined as significant departures from the mean exhibited by certain flow quantities ͓2͔. In the analysis of intermittent aspects of turbulent flows attention is most often focused on the scaling properties of structure functions S n (r). The scaling S n ϳr n is usually examined, where n is the scaling exponent. Its deviation from the linear K41 prediction ͓3͔ may be regarded as a measure of the intermittency effects.
Extended self-similarity in kinetic surface roughening
Physical Review E, 1998
We show from simulations that a limited mobility solid-on-solid model of kinetically rough surface growth exhibits extended self-similarity analogous to that found in fluid turbulence. The range over which scale-independent power-law behavior is observed is significantly enhanced if two correlation functions of different order, such as those representing two different moments of the difference in height between two points, are plotted against each other. This behavior, found in both one and two dimensions, suggests that the "relative" exponents may be more fundamental than the "absolute" ones.
Intermittency and rough-pipe turbulence
Physical Review E, 2008
Recently, by analyzing the measurement data of Nikuradze, it has been proposed (N. Goldenfeld, Phys. Rev. Lett. 96, 044503, 2006) that the friction factor, f , of rough pipe flow obeys a scaling law in the turbulent regime. Here, we provide a phenomenological scaling argument to explain this law and demonstrate how intermittency modifies the scaling form, thereby relating f to the intermittency exponent, η. By statistically analyzing the measurement data of f , we infer a satisfactory estimate for η (≈ 0.02), the inclusion of which is shown to improve the data-collapse curve. This provides empirical evidence for intermittency other than the direct measurement of velocity fluctuations.
Experimental assessment of a new form of scaling law for near-wall turbulence
Physics of Fluids, 2002
Scaling laws and intermittency in the wall region of a turbulent flow are addressed by analyzing moderate Reynolds number data obtained by single component hot wire anemometry in the boundary layer of a flat plate. The paper aims in particular at the experimental validation of a new form of refined similarity recently proposed for the shear dominated range of turbulence, where the classical Kolmogorov-Oboukhov inertial range theory is inappropriate. An approach inspired to the extended self-similarity allows for the extraction of the different power laws for the longitudinal structure functions at several wall normal distances. A double scaling regime is found in the logarithmic region, confirming previous experimental results. Approaching the wall, the scaling range corresponding to the classical cascade-dominated range tends to disappear and, in the buffer layer, a single power law is found to describe the available range of scales. The double scaling is shown to be associated with two different forms of refined similarity. The classical form holds below the shear scale L s . The other, originally introduced on the basis of DNS data for a turbulent channel, is experimentally confirmed to set up above L s . Given the experimental difficulties in the evaluation of the instantaneous dissipation rate, some care is devoted to check that its one-dimensional surrogate does not bias the results. The increased intermittency as the wall is approached is experimentally found entirely consistent with the failure of the refined Kolmogorov-Oboukhov similarity and the establishment of its new form near the wall.
Multifractal modeling of anomalous scaling laws in rainfall
Water Resources Research, 1999
The coupling of hydrological distributed models to numerical weather prediction outputs is an important issue for hydrological applications such as forecasting of flood events. Downscaling meteorological predictions to the hydrological scales requires the resolution of two fundamental issues regarding precipitation, namely, (1) understanding the statistical properties and scaling laws of rainfall fields and (2) validation of downscaling models that are able to preserve statistical characteristics observed in real precipitation. In this paper we discuss the first issue by introducing a new multifractal model that appears particularly suitable for random generation of synthetic rainfall. We argue that the results presented in this paper may be also useful for the solution of the second question. Statistical behavior of rainfall in time is investigated through a highresolution time series recorded in Genova (Italy). The multifractal analysis shows the presence of a temporal threshold, localized around 15-20 hours, which separates two ranges of anomalous scaling laws. Synthetic time series, characterized by very similar scaling laws to the observed one, are generated with the multifractal model. The potential of the model for extreme rainfall event distributions is also discussed. The multifractal analysis of Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment (GATE) radar fields have shown that statistical properties of rainfall in space depend on time durations over which precipitation is accumulated. Further analysis of some rainfall fields produced with a meteorological limited area model exhibited the same anomalous scaling as the GATE fields.
Scaling property of turbulent flows
Physical Review E, 1996
We discuss a possible theoretical interpretation of the self scaling property of turbulent flows (Extended Self Similarity). Our interpretation predicts that, even in cases when ESS is not observed, a generalized self scaling, must be observed. This prediction is checked on a number of laboratory experiments and direct numerical simulations.
Self-scaling properties of velocity circulation in shear flows
Physical Review E, 1997
We investigate the scaling properties of the velocity circulation of a turbulent shear flow. We evaluate, using extended self-similarity, the circulation scaling exponents both at maximum and minimum shear regions. We show that the anomalous component of the velocity circulation and the anomalous component of the velocity structure functions are equal. ͓S1063-651X͑97͒11202-8͔
Temporal intermittency and cascades in shell models of turbulence
Physical Review E, 1996
The two-dimensional-͑2D͒ and three-dimensional-͑3D͒ like Gletzer, Okhitani, and Yamada shell models are examined. The 2D-like model shows a transition from statistical quasiequilibrium to cascade of enstrophy as a function of the spectral ratio of energy to enstrophy. The transition is related to the ratio of time scales, corresponding to eddy turnover times, between shells. The anomalous scaling, giving rise to nonlinear scaling functions, is also connected to the ratio of eddy turnover times. This is illustrated in a simple stochastic model, where the structure function (q) becomes independent of q. In the 3D-like model the multiscaling is also influenced by the existence of a second nonpositive-definite inviscid invariant, the helicity.
Transverse structure functions in high-Reynolds-number turbulence
Physical Review E, 1997
Transverse structure functions are obtained at high Reynolds numbers in atmospheric turbulence ͑Taylor microscale Reynolds numbers between 10 000 and 15 000͒. These measurements confirm that their scaling exponents are different from those for longitudinal structure functions. Implications of this conclusion are discussed briefly. ͓S1063-651X͑97͒50511-3͔
Scaling exponents saturate in three-dimensional isotropic turbulence
Physical Review Fluids, 2020
From a database of direct numerical simulations of homogeneous and isotropic turbulence, generated in periodic boxes of various sizes, we extract the spherically symmetric part of moments of velocity increments and first verify the following (somewhat contested) results: the 4/5-ths law holds in an intermediate range of scales and that the second order exponent over the same range of scales is anomalous, departing from the self-similar value of 2/3 and approaching a constant of 0.72 at high Reynolds numbers. We compare with some typical theories the dependence of longitudinal exponents as well as their derivatives with respect to the moment order n, and estimate the most probable value of the Hölder exponent. We demonstrate that the transverse scaling exponents saturate for large n, and trace this trend to the presence of large localized jumps in the signal. The saturation value of about 2 at the highest Reynolds number suggests, when interpreted in the spirit of fractals, the presence of vortex sheets rather than more complex singularities. In general, the scaling concept in hydrodynamic turbulence appears to be more complex than even the multifractal description.
Local properties of extended self-similarity in three-dimensional turbulence
Physical Review E, 2001
Using a generalization of extended self-similarity we have studied local scaling properties of incompressible homogeneous isotropic 3D turbulence in a direct numerical simulation. We have found that these properties are consistent with lognormal-like behavior of the velocity increments with moderate amplitudes for space scales r beginning from Kolmogorov length η up to the largest scales, and in the whole range of the Reynolds numbers: 50 ≤ R λ ≤ 459. The locally determined intermittency exponent µ(r) varies with r; it has a maximum at scale r = 14η, independent of R λ .
Scaling law of resolved-scale isotropic turbulence and its application in large-eddy simulation
Acta Mechanica Sinica, 2014
Eddy-damping quasinormal Markovian (EDQNM) theory is employed to calculate the resolved-scale spectrum and transfer spectrum, based on which we investigate the resolved-scale scaling law. Results show that the scaling law of the resolved-scale turbulence, which is affected by several factors, is far from that of the full-scale turbulence and should be corrected. These results are then applied to an existing subgrid model to improve its performance. A series of simulations are performed to verify the necessity of a fixed scaling law in the subgrid modeling. Keywords Scaling law • Large-eddy simulation • CZZS model 1 Introduction Kolmogorov introduced the 2/3 scaling law for the secondorder structure function in isotropic turbulence, which is
Physical review, 1996
Elements of the analytic structure of anomalous scaling and intermittency in fully developed hydrodynamic turbulence are described. We focus here on the structure functions of velocity differences that satisfy inertial range scaling laws Sn(R) ∼ R ζn , and the correlation of energy dissipation Kǫǫ(R) ∼ R -µ . The goal is to understand the exponents ζn and µ from first principles. In paper II of this series it was shown that the existence of an ultraviolet scale (the dissipation scale η) is associated with a spectrum of anomalous exponents that characterize the ultraviolet divergences of correlations of gradient fields. The leading scaling exponent in this family was denoted ∆. The exact resummation of ladder diagrams resulted in the calculation of ∆ which satisfies the scaling relation ∆ = 2 -ζ2. In this paper we continue our analysis and show that nonperturbative effects may introduce multiscaling (i.e. ζn not linear in n) with the renormalization scale being the infrared outer scale of turbulence L. It is shown that deviations from K41 scaling of Sn(R) (ζn = n/3) must appear if the correlation of dissipation is mixing (i.e. µ > 0). We derive an exact scaling relation µ = 2ζ2 -ζ4. We present analytic expressions for ζn for all n and discuss their relation to experimental data. One surprising prediction is that the time decay constant τn(R) ∝ R zn of Sn(R) scales independently of n: the dynamic scaling exponent zn is the same for all n-order quantities, zn = ζ2.