Scaling Exponents Of Structure Functions Research Articles

Fluctuations of a passive scalar in a turbulent mixing layer

The turbulent flow originating downstream of the Kelvin-Helmholtz instability in a mixing layer has great relevance in many applications, ranging from atmospheric physics to combustion in technical devices. The mixing of a substance by the turbulent velocity field is usually involved. In this paper, a detailed statistical analysis of fluctuations of a passive scalar in the fully developed region of a turbulent mixing layer from a direct numerical simulation is presented. Passive scalar spectra show inertial ranges characterized by scaling exponents -4/3 and -3/2 in the streamwise and spanwise directions, in agreement with a recent theoretical analysis of passive scalar scaling in shear flows [Celani et al., J. Fluid Mech. 523, 99 (2005)]. Scaling exponents of high-order structure functions in the streamwise direction show saturation of intermittency with an asymptotic exponent ζ_{∞}=0.4 at large orders. Saturation of intermittency is confirmed by the self-similarity of the tails of the probability density functions of the scalar increments at different scales r with the scaling factor r^{-ζ_{∞}} and by the analysis of the cumulative probability of large fluctuations. Conversely, intermittency saturation is not observed for the spanwise increments and the relative scaling exponents agree with recent results for homogeneous isotropic turbulence with mean scalar gradient. Probability density functions of the scalar increments in the three directions are compared to assess anisotropy.

Finite-Péclet-number effects on the scaling exponents of high-order passive scalar structure functions

AbstractThe effect of scalar-field (temperature) boundary conditions on the inertial-convective-range scaling exponents of the high-order passive scalar structure functions is studied in the turbulent, heated wake downstream of a circular cylinder. The temperature field is generated two ways: using (i) a heating element embedded within the cylinder that generates the hydrodynamic wake (thus creating a heated cylinder) and (ii) a mandoline (an array of fine, heated wires) installed downstream of the cylinder. The hydrodynamic field is independent of the scalar-field boundary conditions/injection methods, and the same in both flows. Using the two heat injection mechanisms outlined above, the inertial-convective-range scaling exponents of the high-order passive scalar structure functions were measured. It is observed that the different scalar-field boundary conditions yield significantly different scaling exponents (with the magnitude of the difference increasing with structure function order). Moreover, the exponents obtained from the mandoline experiment are smaller than the analogous exponents from the heated cylinder experiment (both of which exhibit a significant departure from the Kolmogorov prediction). Since the observed deviation from the Kolmogorov $n/ 3$ prediction arises due to the effects of internal intermittency, the typical interpretation of this result would be that the scalar field downstream of the mandoline is more internally intermittent than that generated by the heated cylinder. However, additional measures of internal intermittency (namely the inertial-convective-range scaling exponents of the mixed, sixth-order, velocity–temperature structure functions and the non-centred autocorrelations of the dissipation rate of scalar variance) suggest that both scalar fields possess similar levels of internal intermittency – a distinctly different conclusion. Examination of the normalized high-order moments reveals that the smaller scaling exponents (of the high-order passive scalar structure functions) obtained for the mandoline experiment arise due to the smaller thermal integral length scale of the flow (i.e. the narrower inertial-convective subrange) and are not solely the result of a more intermittent scalar field. The difference in the passive scalar structure function scaling exponents can therefore be interpreted as an artifact of the different, finite Péclet numbers of the flows under consideration – an effect that is notably less prominent in the other measures of internal intermittency.

Stationary two-dimensional turbulence statistics using a Markovian forcing scheme

In this study we investigate the statistics of two-dimensional stationary turbulence using a Markovian forcing scheme, which correlates the forcing process in the current time step to the previous time step according to a defined memory coefficient. In addition to the Markovian forcing mechanism, the hyperviscous dissipation mechanism for small scales and the Ekman friction type of linear damping mechanism for the large scales are included in the model. We examine the effects of various dissipation and forcing parameters on the turbulence statistics in both wave space and physical space. Our analysis includes the effects of the effective forcing scale, the bandwidth of the forcing, the memory correlation coefficient, and the forcing amplitude, along with the large scale friction and small scale dissipation coefficients. Scaling exponents of structure functions and energy spectra are calculated, and the role of the parameters associated with the Markovian forcing is discussed. We found that the scaling exponents are approximately invariant and show a universal behavior for the various forms of forcing schemes used. We found, however, that the final states strongly depend on the large scale friction mechanism considered. When the large scale friction mechanism is included in the model with a high friction coefficient, we demonstrate that the behavior is no longer universal. Our analysis also shows that the second-order vorticity structure function has an asymptotic scaling exponent for larger dissipation. Additionally, we confirmed that vorticity behaves as a passive scalar when the dissipation mechanism becomes less effective. Finally, although turbulence is not believed to have a separation of time scales in the dynamics of the velocity field, we conjectured that a separation of time scales exists in the dynamics of the energy spectrum.

First-Order Structure Function Analysis of Statistical Scale Invariance in the AIRS-Observed Water Vapor Field

AbstractThe power-law scale dependence, or scaling, of first-order structure functions of the tropospheric water vapor field between 58°S and 58°N is investigated using observations from the Atmospheric Infrared Sounder (AIRS). Power-law scale dependence of the first-order structure function would indicate that the water vapor field exhibits statistical scale invariance. Directional and directionally independent first-order structure functions are computed to assess the directional dependence of derived first-order structure function scaling exponents (H) for a range of scales from 50 to 500 km. In comparison to other methods of assessing statistical scale invariance, the methodology used here requires minimal assumptions regarding the homogeneity of the spatial distribution of data within regions of analysis. Additionally, the methodology facilitates the evaluation of anisotropy and quantifies the extent to which the structure functions exhibit scale invariance.The spatial and seasonal dependence of the computed scaling exponents are explored. Minimum scaling exponents at all levels are shown to occur proximate to the equator, while the global maximum is shown to occur in the middle troposphere near the tropical–subtropical margin of the winter hemisphere.From a detailed analysis of AIRS maritime scaling exponents, it is concluded that the AIRS observations suggest the existence of two scaling regimes in the extratropics. One of these regimes characterizes the statistical scale invariance the free troposphere with H approximately = 0.55 and a second that characterizes the statistical scale invariance of the boundary layer with H approximately = ⅓.

Universality and anisotropy in passive scalar fluctuations in turbulence with uniform mean gradient

Passive scalar and velocity fluctuations in homogeneous isotropic turbulence with or without mean scalar gradient are studied by using high-resolution direct numerical simulation (DNS). The local scaling exponents of the velocity structure functions are newly computed at larger Reynolds number, and the values at their plateau in the inertial range are found to be consistent with the previous DNS and experimental data. The 4/3 law for the velocity–scalar mixed correlation and the high-order structure functions of the passive scalar increment are analyzed in terms of the Legendre polynomial expansion. It is shown theoretically that the contributions from the second order in the expansion can be removed by taking the average over three directions parallel and perpendicular to the mean gradient, and it is found numerically that the contributions higher than the fourth order are negligible. The scaling exponents of the isotropic part of the scalar structure functions under the mean gradient are found to have the well-developed scaling range and are smaller than those in the isotropic case. The different behavior in the local scaling exponents with or without the mean gradient is examined and the universality of the scaling exponents is discussed.

Lagrangian Structure Function's Scaling Exponents in Turbulent Channel Flow

The Lagrangian structure function's scaling exponents and intermittency of three-dimensional incompressible turbulent channel flow are investigated by using direct numerical simulation. The intermittency in streamwise velocity increments is found to increase in the near-wall region, which can be attributed to the presence of strong mean shear and organized motions in the near-wall region. It is found that the intermittency of transverse velocity increments is weaker than that of longitudinal ones. The present ESS evaluation of ζL(q) for the structure function of the streamwise velocity component in the channel centre is fairly close to experimental estimates of isotropic turbulence.

Application of a GOY model to atmospheric boundary layer data

Abstract. This article analyzes the possibility of applying a GOY theoretical model to atmospheric boundary layer data. Bearing this in mind, relative scaling exponents of velocity structure functions are used to compare the model with the data under study. In the model, these exponents are set based on two parameters (q and δ), which are appropriate to define the model that better features a certain atmospheric state. From these scaling exponents, the gap between 2-D and 3-D turbulence is observed in the model, depending on the fact that δ is higher or lower than unity, respectively. Atmospheric data corresponding to very different states of stratification stability have been analyzed. For convective or near-neutral situations (usually associated to 3-D turbulence), it is possible to find parameters q and δ to define a model that fits the measured data. More stable situations can be featured by GOY models with higher values of δ. However, it is clear that it is impossible to represent nocturnal situations of strong stable stratification (with a more similar behaviour to 2-D) with this type of model.

Structure and decay of rotating homogeneous turbulence

Navier–Stokes turbulence subject to solid-body rotation is studied by high-resolution direct numerical simulations (DNS) of freely decaying and stationary flows. Setups characterized by different Rossby numbers are considered. In agreement with experimental results strong rotation is found to lead to anisotropy of the direct nonlinear energy flux, which is attenuated primarily in the direction of the rotation axis. In decaying turbulence the evolution of kinetic energy follows an approximate power law with a distinct dependence of the decay exponent on the rotation frequency. A simple phenomenological relation between exponent and rotation rate reproduces this observation. Stationary turbulence driven by large-scale forcing exhibits k⊥−2-scaling in the rotation-dominated inertial range of the one-dimensional energy spectrum taken perpendicularly to the rotation axis. The self-similar scaling is shown to be the cumulative result of individual spectral contributions which, for low rotation rate, display k⊥−3-scaling near the k∥ = 0 plane. A phenomenology which incorporates the modification of the energy cascade by rotation is proposed. In the observed regime the nonlinear turbulent interactions are strongly influenced by rotation but not suppressed. Longitudinal two-point velocity structure functions taken perpendicularly to the axis of rotation indicate weak intermittency of the k∥ = 0 (2D) component of the flow while the intermittent scaling of k∥ ≠ 0 (3D) fluctuations is well captured by a modified She–Lévêque intermittency model which yields the expression ζp = p/6 + 2(1 − (2/3)p/2) for the structure function scaling exponents.

Effect of the Scalar Injection Mechanism on Passive Scalar Structure Functions in a Turbulent Flow

Higher-order passive scalar (temperature) structure functions are measured in the turbulent wake of a circular cylinder at a Taylor-microscale Reynolds number (Rlambda) of 370. The scalar is injected by two different means: (i) heating of the cylinder and (ii) use of a mandoline. Even though the second-order statistics (e.g., power spectra, second-order structure functions) of the scalar field are experimentally indistinguishable in the inertial and dissipative ranges, we observe notable differences in the inertial-range scaling exponents (xin) of the scalar structure functions at higher orders. The implication is therefore that the variations in previous estimates of xin may be attributable to differences in the scalar field initial conditions (and may not be deemed characteristic of a universal nature of the small-scale statistics of turbulent passive scalars).

Reconnection dynamics for quantized vortices

By analyzing trajectories of solid hydrogen tracers in superfluid 4He, we identify tens of thousands of individual reconnection events between quantized vortices. We characterize the dynamics by the minimum separation distance δ ( t ) between the two reconnecting vortices both before and after the events. Applying dimensional arguments, this separation has been predicted to behave asymptotically as δ ( t ) ≈ A ( κ | t − t 0 | ) 1 / 2 , where κ = h / m is the quantum of circulation. The major finding of the experiments and their analysis is strong support for this asymptotic form with κ as the dominant controlling feature, although there are significant event to event fluctuations. At the three-parameter level the dynamics may be about equally well-fit by two modified expressions: (a) an arbitrary power-law expression of the form δ ( t ) = B | t − t 0 | α and (b) a correction-factor expression δ ( t ) = A ( κ | t − t 0 | ) 1 / 2 ( 1 + c | t − t 0 | ) . The measured frequency distribution of α is peaked at the predicted value α = 0.5 , although the half-height values are α = 0.35 and 0.80 and there is marked variation in all fitted quantities. Accepting (b) the amplitude A has mean values of 1.24 ± 0.01 and half height values of 0.8 and 1.6 while the c distribution is peaked close to c = 0 with a half-height range of −0.9 s −1 to 1.5 s −1. In light of possible physical interpretations we regard the correction-factor expression (b), which attributes the observed deviations from the predicted asymptotic form to fluctuations in the local environment and in boundary conditions, as best describing our experimental data. The observed dynamics appear statistically time-reversible, which suggests that an effective equilibrium has been established in quantum turbulence on the time scales (≤0.25 s) investigated. We discuss the impact of reconnection on velocity statistics in quantum turbulence and, as regards classical turbulence, we argue that forms analogous to (b) could well provide an alternative interpretation of the observed deviations from Kolmogorov scaling exponents of the longitudinal structure functions.

Structure function analysis and intermittency in the atmospheric boundary layer

Abstract. Data from the SABLES98 experimental campaign (Cuxart et al., 2000) have been used in order to study the relationship of the probability distribution of velocity increments (PDFs) to the scale and the degree of stability. This connection is demonstrated by means of the velocity structure functions and the PDFs of the velocity increments. Using the hypothesis of local similarity, so that the third order structure function scaling exponent is one, the inertial range in the Kolmogorov sense has been identified for different conditions, obtaining the velocity structure function scaling exponents for several orders. The degree of intermittency in the energy cascade is measured through these exponents and compared with the forcing intermittency revealed through the evolution of flatness with scale. The role of non-homogeneity in the turbulence structure is further analysed using Extended Self Similarity (ESS). A criterion to identify the inertial range and to show the scale independence of the relative exponents is described. Finally, using least-squares fits, the values of some parameters have been obtained which are able to characterize intermittency according to different models.

Turbulent mixing: the roots of intermittency

Mixing in fully developed incompressible turbulent flows is known to lead to a cascade of discontinuity fronts of passive scalar fields; these fronts make the tracer fields strongly intermittent and give rise to an anomalous scaling of structure functions. Here, a simple model, essentially a one-dimensional (1D) variant of Baker's map, is developed, which captures the main mechanism responsible for the emergence of the discontinuities. The model is studied both numerically and analytically. In particular, the structure function scaling exponent ζp is derived; for the Kolmogorov turbulence, . The analytical findings are consistent with simulations, and explain the results of a series of numerical and experimental studies.

Scaling laws in the central region of confined turbulent thermal convection

In confined turbulent thermal convection, the velocity is separated into two parts: one that is correlated with some function of the temperature fluctuations, and thus associated with the plume velocity, and the other part, the background velocity, which is uncorrelated with any function of the temperature fluctuations. As a result, one should focus on the plume velocity, and not the whole velocity, and the temperature when studying the scaling behavior. In this paper, a phenomenological theory for the scaling behavior in the central region of confined turbulent thermal convection is presented. The spatial (temporal) plume velocity structure functions are found to have the same scaling behavior as the spatial (temporal) temperature structure functions. For tau> or = taub, where the buoyant scale taub is determined in terms of measurable quantities, the scaling exponents of the temporal temperature structure functions and hence those of the temporal plume velocity structure functions are obtained. These results are checked against experimental measurements, and good agreement is found.

Asymptotic exponents from low-Reynolds-number flows

The high-order statistics of fluctuations in velocity gradients in the crossover range from the inertial to the Kolmogorov and sub-Kolmogorov scales are studied by direct numerical simulations (DNS) of homogeneous isotropic turbulence with vastly improved resolution. The derivative moments for orders 0 ⩽ n ⩽ 8 are represented well as powers of the Reynolds number, Re, in the range 380 ⩽ Re ⩽ 5275, where Re is based on the periodic box length Lx. These low-Reynolds-number flows give no hint of scaling in the inertial range even when extended self-similarity is applied. Yet, the DNS scaling exponents of velocity gradients agree well with those deduced, using a recent theory of anomalous scaling, from the scaling exponents of the longitudinal structure functions at infinitely high Reynolds numbers. This suggests that the asymptotic state of turbulence is attained for the velocity gradients at far lower Reynolds numbers than those required for the inertial range to appear. We discuss these findings in the light of multifractal formalism. Our numerical studies also resolve the crossover of the velocity gradient statistics from Gaussian to non-Gaussian behaviour that occurs as the Reynolds number is increased.

Hierarchical structures in a turbulent free shear flow

We have conducted a detailed analysis of scaling for longitudinal and transverse velocity structure functions in a turbulent free shear flow. The free shear flow is generated via a mixing layer under varying conditions of upstream flow disturbances. Two velocity components are simultaneously measured with a pair of cross-wires at two spanwise locations, with varying positions of the second cross-wire, which allows us to study the statistics of two longitudinal and four transverse velocity increments. Spectra, probability density functions of the velocity increments, and scaling exponents are measured and discussed in relation to flow structures such as streamwise and spanwise vortices. Scaling exponents of the velocity structure functions are interpreted in the phenomenological framework of the hierarchical structure (HS) model of She & Leveque (Phys. Rev. Lett. vol. 72, 1994, p. 336). One HS parameter () related to the singularity index of the so-called most intermittent structures shows strong dependence on flow structures. The strongest intermittency occurs in the form of streamwise vortices. The results confirm that coherent small-scale flow structures are responsible for intermittency effects and anomalous scaling, and a complete set of measurements of longitudinal and transverse velocity variations are required to derive flow structural information.

Inversion formula of multifractal energy dissipation in three-dimensional fully developed turbulence

The concept of inverse statistics in turbulence has attracted much attention in recent years. It is argued that the scaling exponents of the direct structure functions and the inverse structure functions satisfy an inversion formula. This proposition has already been verified by numerical data using the shell model. However, no direct evidence was reported for experimental three-dimensional turbulence. We propose to test the inversion formula using experimental data of three-dimensional fully developed turbulence by considering the energy dissipation rates instead of the usual efforts on the structure functions. The moments of the exit distances are shown to exhibit nice multifractality. The inversion formula between the direct and inverse exponents is then verified.

Multifractal Dimension of Lagrangian Turbulence

We report experimental measurements of the Lagrangian multifractal dimension spectrum in an intensely turbulent laboratory water flow by the optical tracking of tracer particles. The Legendre transform of the measured spectrum is compared with measurements of the scaling exponents of the Lagrangian velocity structure functions, and excellent agreement between the two measurements is found, in support of the multifractal picture of turbulence. These measurements are compared with three model dimension spectra. When the nonexistence of structure functions of order less than -1 is accounted for, the models are shown to agree well with the measured spectrum.

Review Article

The problem of anisotropy and its effects on the statistical theory of high Reynolds number ( Re ) turbulence (and turbulent transport) is intimately related and intermingled with the problem of the universality of the (anomalous) scaling exponents of structure functions. Both problems had seen tremendous progress in the last 5 years. In this review we present a detailed description of the new tools that allow effective data analysis and systematic theoretical studies such as to separate isotropic from anisotropic aspects of turbulent statistical fluctuations. Employing the invariance of the equations of fluid mechanics to all rotations, we show how to decompose the (tensorial) statistical objects in terms of the irreducible representation of the SO ( d ) symmetry group (with d being the dimension, d = 2 or 3). This device allows a discussion of the scaling properties of the statistical objects in well-defined sectors of the symmetry group, each of which is determined by the “angular momenta” sector numbers ( j , m ) . For the case of turbulent advection of passive scalar or vector fields, this decomposition allows rigorous statements to be made: (i) the scaling exponents are universal, (ii) the isotropic scaling exponents are always leading, (iii) the anisotropic scaling exponents form a discrete spectrum which is strictly increasing as a function of j . This emerging picture offers a complete understanding of the decay of anisotropy upon going to smaller and smaller scales. Next, we explain how to apply the SO ( 3 ) decomposition to the statistical Navier–Stokes theory. We show how to extract information about the scaling behavior in the isotropic sector. Doing so furnishes a systematic way to assess the universality of the scaling exponents in this sector, clarifying the anisotropic origin of the many measurements that claimed the opposite. A systematic analysis of direct numerical simulations (DNS) of the Navier–Stokes equations and of experiments provides a strong support to the proposition that also for the non-linear problem there exists foliation of the statistical theory into sectors of the symmetry group. The exponents appear universal in each sector, and again strictly increasing as a function of j . An approximate calculation of the anisotropic exponents based on a closure theory is reviewed. The conflicting experimental measurements on the rate of decay of anisotropy upon reducing the scales are explained and systematized, showing that isotropy is eventually recovered at small scales.

Inverse structure functions

While the ordinary structure function in turbulence is concerned with the statistical moments of the velocity increment Deltau measured over a distance r , the inverse structure function is related to the distance r where the turbulent velocity exits the interval Deltau. We study inverse structure functions of wind-tunnel turbulence which covers a range of Reynolds numbers Re(lambda) = 400-1100. We test a recently proposed relation between the scaling exponents of the ordinary structure functions and those of the inverse structure functions [S. Roux and M. H. Jensen, Phys. Rev. E 69, 16309 (2004)]. The relatively large range of Reynolds numbers in our experiment also enables us to address the scaling with Reynolds number that is expected to highlight the intermediate dissipative range. While we firmly establish the (relative) scaling of inverse structure functions, our experimental results fail both predictions. Therefore, the question of the significance of inverse structure functions remains open.

The scaling properties of dissipation in incompressible isotropic three-dimensional magnetohydrodynamic turbulence

The statistical properties of the dissipation process constrain the analysis of large scale numerical simulations of three-dimensional incompressible magnetohydrodynamic (MHD) turbulence, such as those of Biskamp and Müller [Phys. Plasmas 7, 4889 (2000)]. The structure functions of the turbulent flow are expected to display statistical self-similarity, but the relatively low Reynolds numbers attainable by direct numerical simulation, combined with the finite size of the system, make this difficult to measure directly. However, it is known that extended self-similarity, which constrains the ratio of scaling exponents of structure functions of different orders, is well satisfied. This implies the extension of physical scaling arguments beyond the inertial range into the dissipation range. The present work focuses on the scaling properties of the dissipation process itself. This provides an important consistency check in that we find that the ratio of dissipation structure function exponents is that predicted by the She and Leveque [Phys. Rev. Lett 72, 336 (1994)] theory proposed by Biskamp and Müller. This supplies further evidence that the cascade mechanism in three-dimensional MHD turbulence is nonlinear random eddy scrambling, with the level of intermittency determined by dissipation through the formation of current sheets.