Small-scale Intermittency Research Articles

Co-spectrum and mean velocity in turbulent boundary layers

Connections are explored between spectral descriptions of turbulence and the mean velocity profile in the equilibrium layer of wall-bounded flows using a modeled budget for the co-spectral density. Using a standard model for the wall normal velocity variance and a Rotta-like return-to-isotropy closure for the pressure-strain effects, the co-spectrum is derived. The approach establishes a relation between the von Kármán (κ), one-dimensional Kolmogorov (\documentclass[12pt]{minimal}\begin{document}$C^\prime _K$\end{document}CK′), and Rotta (A) constants, namely, \documentclass[12pt]{minimal}\begin{document}$\kappa = (4 A/7C^\prime _K)^{-3/4}$\end{document}κ=(4A/7CK′)−3/4. Depending on the choices made about small-scale intermittency corrections, the logarithmic mean velocity profile or a power-law profile with an exponent that depends on the intermittency correction are derived thereby offering a new perspective on a long standing debate.

Small-scale and lateral intermittency of oceanic microstructure in the pycnocline

The intermittency correction factor μc for the Obukhov–Corrsin ‘ − 5/3’ inertial-convective subrange spectral law is estimated using conductivity (temperature) fluctuation measurements conducted within the microstructure patches of oceanic pycnocline in Monterey Bay. The value of μc was found to be in the range of 0.46–0.51, depending on the accuracy of calculation, which is applicable to stratified, low-Reynolds-number oceanic turbulence. The intermittency factor μ Mc for mesoscale (up to 1 km) lateral variations of scalar dissipation is estimated as 0.43 ± 0.02. The breakdown mechanisms of small-scale locally isotropic and mesoscale non-isotropic (lateral) turbulent temperature fluctuations in oceanic turbulence are discussed in the light of the above observations.

Probing Small-Scale Intermittency with a Fluctuation Theorem

We characterize statistical properties of the flow field in developed turbulence using concepts from stochastic thermodynamics. On the basis of data from a free air-jet experiment, we demonstrate how the dynamic fluctuations induced by small-scale intermittency generate analogs of entropy-consuming trajectories with sufficient weight to make fluctuation theorems observable at the macroscopic scale. We propose an integral fluctuation theorem for the entropy production associated with the stochastic evolution of velocity increments along the eddy hierarchy and demonstrate its extreme sensitivity to the accurate description of the tails of the velocity distributions.

On the Relation between Small-scale Intermittency and Shocks in Turbulent Flows

High Reynolds number turbulence is characterized by extreme fluctuations of velocity gradients which can interact with shock waves in compressible flows. While these processes are traditionally thought to happen at very disparate range of scales, both turbulence gradients as well as shock gradients become stronger as the Reynolds number increases. Our interest here is to in- vestigate their relation in the high-Reynolds number limit. Our conclusion is that for intermittent turbulence with inertial range scaling exponents which grow more slowly than linear at asymptotically high orders, small-scale intermittency produces gradients which are commensurate with shocks. This result is interpreted in the context of shock-turbulence interactions where intermittency appears to be responsible, in part, for the holes observed in shocks from simulations and experiments. This effect is aided by the correlation between strong gradients and flow retardation ahead of the shock which is observed from analysis of our direct numerical simulation database of incompressible and compressible turbulence.

Influence of polymer additives on turbulent energy cascading in forced homogeneous isotropic turbulence studied by direct numerical simulations

Direct numerical simulations (DNS) were performed for the forced homogeneous isotropic turbulence (FHIT) with/without polymer additives in order to elaborate the characteristics of the turbulent energy cascading influenced by drag-reducing effects. The finite elastic non-linear extensibility-Peterlin model (FENE-P) was used as the conformation tensor equation for the viscoelastic polymer solution. Detailed analyses of DNS data were carried out in this paper for the turbulence scaling law and the topological dynamics of FHIT as well as the important turbulent parameters, including turbulent kinetic energy spectra, enstrophy and strain, velocity structure function, small-scale intermittency, etc. A natural and straightforward definition for the drag reduction rate was also proposed for the drag-reducing FHIT based on the decrease degree of the turbulent kinetic energy. It was found that the turbulent energy cascading in the FHIT was greatly modified by the drag-reducing polymer additives. The enstrophy and the strain fields in the FHIT of the polymer solution were remarkably weakened as compared with their Newtonian counterparts. The small-scale vortices and the small-scale intermittency were all inhibited by the viscoelastic effects in the FHIT of the polymer solution. However, the scaling law in a fashion of extended self-similarity for the FHIT of the polymer solution, within the presently simulated range of Weissenberg numbers, had no distinct differences compared with that of the Newtonian fluid case.

Inverse Energy Cascade in Three-Dimensional Isotropic Turbulence

We study the statistical properties of homogeneous and isotropic three-dimensional (3D) turbulent flows. By introducing a novel way to make numerical investigations of Navier-Stokes equations, we show that all 3D flows in nature possess a subset of nonlinear evolution leading to a reverse energy transfer: from small to large scales. Up to now, such an inverse cascade was only observed in flows under strong rotation and in quasi-two-dimensional geometries under strong confinement. We show here that energy flux is always reversed when mirror symmetry is broken, leading to a distribution of helicity in the system with a well-defined sign at all wave numbers. Our findings broaden the range of flows where the inverse energy cascade may be detected and rationalize the role played by helicity in the energy transfer process, showing that both 2D and 3D properties naturally coexist in all flows in nature. The unconventional numerical methodology here proposed, based on a Galerkin decimation of helical Fourier modes, paves the road for future studies on the influence of helicity on small-scale intermittency and the nature of the nonlinear interaction in magnetohydrodynamics.

Rapid growth of cloud droplets by turbulence

Assuming perfect collision efficiency, we demonstrate that turbulence can initiate and sustain the rapid growth of very small water droplets in air even when these droplets are too small to cluster, and even without having to take gravity and small-scale intermittency into account. This is because the range of local Stokes numbers of identical droplets in the turbulent flow field is broad enough even when small-scale intermittency is neglected. This demonstration is given for turbulence which is one order of magnitude less intense than is typical in warm clouds but with a volume fraction which, even though small, is nevertheless large enough for an estimated a priori frequency of collisions to be ten times larger than in warm clouds. However, the time of growth in these conditions turns out to be one order of magnitude smaller than in warm clouds.

Small-scale intermittency and local anisotropy in turbulent mixing with rotation

The statistics of the velocity and passive scalar gradients in rotating turbulence are studied using Lagrangian stochastic models. Models for the velocity gradients are derived generalizing the approach proposed by Chevillard and Meneveau (Lagrangian dynamics and statistical geometric structure of turbulence, Phys. Rev. Lett. 97 (2006), p. 174501), whereas the scalar gradients are described using the model proposed by Gonzalez (Kinematic properties of passive scalar gradient predicted by a stochastic Lagrangian model, Phys. Fluids 21 (2009), p. 055104). The non-Gaussian and anisotropic statistics of the gradients are analyzed, and compared with available results in the literature. It is found that the models reproduce the observation that rotation tends to reduce small-scale intermittency for both velocity and scalar gradients. The models predict the skewness of the transverse velocity gradient components in the perpendicular plane and its non-monotonic dependence on the rotation rate. The models also reproduce the anisotropy in the scalar gradient at intermediate Rossby numbers. Furthermore, we show that the anisotropy is reached at an intermediate rotation rate, and the maximum coincides with a transition in the relative importance of the self- and cross-production terms for the scalar gradient.

Multiscale edge detection and imperfect phase synchronization of the wall turbulence

Multiscale edge detection wavelet analysis is applied to the fluctuating velocity field obtained by direct numerical simulations of a fully developed turbulent channel flow at the Reynolds number based on the shear velocity, Re τ=180. The wavelet coefficients are rewritten using analytic signal approach to sort out their local amplitudes and wavenumbers. Large zones of approximate constant wavenumbers have been identified at different scales, and a parallelism is constructed between these observations and stochastic synchronization phenomena. The results that we analyze strongly suggest that the wall turbulence is chaotically synchronized with the force induced by convecting coherent vortices near the wall, thus comforting our earlier results based on experimental velocity and wall shear stress time series. The spatial extend of phase-locked, synchronized zones feature a clear type-I intermittency behavior. The local amplitude intermittency in the synchronized zones is low, and the small-scale amplitude intermittency increases significantly when they are suppressed. The type-I intermittency disappears in the viscous and log layers.

Why we need experiments at high Reynolds numbers

This brief review focuses on the effect of Reynolds number on turbulence small-scale anisotropy and intermittency, on Lagrangian experiments of fluid and inertial particles and on complex turbulent flows. Although Taylor scale Reynolds numbers of 103 (equivalent to turbulence Reynolds number based on the integral scale of 105) can be achieved in the laboratory, it is argued that there is still a need to do experiments (and computation) at even higher Reynolds numbers to resolve outstanding basic and applied issues in turbulence.

Fine-scale statistics of temperature and its derivatives in convective turbulence

We study the fine-scale statistics of temperature and its derivatives in turbulent Rayleigh–Bénard convection. Direct numerical simulations are carried out in a cylindrical cell with unit aspect ratio filled with a fluid with Prandtl number equal to 0.7 for Rayleigh numbers between 107 and 109. The probability density function of the temperature or its fluctuations is found to be always non-Gaussian. The asymmetry and strength of deviations from the Gaussian distribution are quantified as a function of the cell height. The deviations of the temperature fluctuations from the local isotropy, as measured by the skewness of the vertical derivative of the temperature fluctuations, decrease in the bulk, but increase in the thermal boundary layer for growing Rayleigh number, respectively. Similarly to the passive scalar mixing, the probability density function of the thermal dissipation rate deviates significantly from a log-normal distribution. The distribution is fitted well by a stretched exponential form. The tails become more extended with increasing Rayleigh number which displays an increasing degree of small-scale intermittency of the thermal dissipation field for both the bulk and the thermal boundary layer. We find that the thermal dissipation rate due to the temperature fluctuations is not only dominant in the bulk of the convection cell, but also yields a significant contribution to the total thermal dissipation in the thermal boundary layer. This is in contrast to the ansatz used in scaling theories and can explain the differences in the scaling of the total thermal dissipation rate with respect to the Rayleigh number.

Study of scalar macro- and microstructures in a confined jet

The scalar structures in a confined jet are studied at high Reynolds and Schmidt numbers. Both flow modes without the recirculation zone (jet mode) and with the massive separation and creation of the recirculation zone (r-mode) are considered. Despite of the big difference in the flow modes, the fine scale scalar structures gained from highly resolved PLIF measurements possess similar statistical properties such as the normalized cumulative distributions and probability densities of the dissipation rate. The fine scalar structures are distributed nearly uniformly in space with the scalar gradient vector having a slight preference to align with the most compressive mean strain axis. The scalar field exhibits small-scale intermittency which is strongly dependent on the flow mode. The intermittency is most pronounced in the front of the recirculation zone and becomes weaker on the centerline and downstream. The most contribution to the scalar variance is made by large scale motions whereas the contribution of fine scales smaller than typical inertial range scales is negligible. Examination of the multiplier distributions has not supported the concept of the multifractal nature of the scalar dissipation field.

Backscattering and reflection of acoustic waves in the stable atmospheric boundary layer

Among different applications of sodars in studies of the atmospheric boundary layer (ABL), the remote sensing of the temperature structure parameter CT2 and the closely associated structure parameter of refraction index Cn2 occupies an important place. These parameters are calculated from sodar return signal with the help of the theory of Bragg volumetric scattering of sound by small-scale turbulent irregularities of temperature. According to the Kolmogorov-Oboukhov conception, the irregularities are assumed to be locally homogeneous and isotropic. The theory was confirmed by experiments and universally recognized as a classical one. However, detailed comparisons are still being made between CT2 values derived from sodar data and from aircraft and meteorological-tower measurements. The results of such comparisons performed by different researchers under different conditions contradict to each other. In this paper, the results of comparisons between remote and in situ measurements of CT2 in the ABL are briefly reviewed. Some examples of a systematic overestimation of CT2 calculated from sodar data obtained under stable stratification of the ABL are given. Technical and physical causes of these discrepancies are discussed. The following factors are considered as possible physical factors: contribution of a partial (Fresnel) reflection of sound from layers with a sharp temperature jump, anisotropy of small-scale turbulence, intermittency, and contribution of quasi-wave air-density inhomogeneities. It is shown that these factors can explain some short-lived increase of the sodar return signal, but not a systematic overestimation of CT2. The role of sound-beam refraction (which regularly occurs in the stable ABL due to strong air-temperature and wind-speed gradients) is discussed. Due to a refraction turn of sodar beam, the angle of scattering changes, and scattering by turbulent wind-speed irregularities could permanently contribute to return signal of a monostatic sodar.

Lagrangian particle statistics in turbulent flows from a simple vortex model

The statistics of Lagrangian particles in turbulent flows is considered in the framework of a simple vortex model. Here, the turbulent velocity field is represented by a temporal sequence of Burgers vortices of different circulation, strain, and orientation. Based on suitable assumptions about the vortices' statistical properties, the statistics of the velocity increments is derived. In particular, the origin and nature of small-scale intermittency in this model is investigated both numerically and analytically. We critically compare our results to experimental studies.

Dissipative structures of diffuse molecular gas

We further characterize the structures tentatively identified on thermal and chemical grounds as the sites of dissipation of turbulence in molecular clouds (Papers I and II). Our study is based on two-point statistics of line centroid velocities (CV), computed from three large 12CO maps of two fields. Probability density functions (PDF) of the CO line centroid velocity increments (CVI) over lags varying by an order of magnitude and structure functions of the line CV, up to the 6th order, are computed. We show that the line CV bear the three signatures of intermittency in a turbulent velocity field: (1) the non-Gaussian tails in the CVI PDF grow as the lag decreases, (2) the departure from Kolmogorov scaling of the high-order structure functions is more pronounced in the more turbulent field, (3) the positions contributing to the CVI PDF tails delineate narrow filamentary structures (thickness ~ 0.02 pc), uncorrelated to dense gas structures and spatially coherent with thicker ones (~0.18 pc) observed on larger scales. The confrontation with theoretical predictions leads us to identify these small-scale filamentary structures with extrema of velocity-shears associated with gas warmer than the bulk. Last, their average direction is parallel (or close) to that of the local magnetic field projection. Turbulence in these translucent fields exhibits the statistical and structural signatures of small-scale and inertial-range intermittency. The more turbulent field on the 30 pc-scale is also the more intermittent on small scales. The small-scale intermittent structures coincide with those formerly identified as sites of enhanced dissipation. They are organized into parsec-scale coherent structures, coupling a broad range of scales.

On the spatial length scales of scalar dissipation in turbulent jet flames

Spatial length scales of the rate of dissipation, χ, of fluctuations of a conserved scalar, Z, are inferred numerically using a DNS database of a turbulent planar jet flame. The Taylor-scale Reynolds numbers lie in the range of 38 to 58 along the centreline of the simulated jet flame. Three different methods are used to study the spatial length scales associated with χ. First, analysis of the one-dimensional dissipation spectra shows an expected Reδ−3/4 (Kolmogorov) scaling with the outer-scale Reynolds number, Reδ. Secondly, thin sheet-like three-dimensional scalar dissipation structures have been investigated directly. Such structures were identified within the computational domain using level-sets of the χ-field, and their thicknesses were subsequently computed. The study shows, in accordance with experimental studies, that the captured dissipation-layer thickness also shows a Kolmogorov scaling with Reδ. Finally, spatial filters of varying widths were applied to the instantaneous Z field in order to model the averaging effect that takes place with some experimental measurement techniques. The filtered scalar dissipation rate was then calculated from the filtered scalar field. The peaks in the instantaneous filtered χ-profiles are observed to decrease exponentially with increasing filter width, yielding estimates of the true value of χ. Unlike the dissipation length scales obtained from the spectral analysis and the level-set method, the length-scale estimates from the spatial-filtering method are found to be proportional to Reδ−1. This is consistent with the small-scale intermittency of χ which cannot be captured by techniques that just resolve the conventional Batchelor/Obukhov–Corrsin scale. These results have implications when considering resolution requirements for measuring scalar dissipation length scales in experimental flows.

A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence

A public database system archiving a direct numerical simulation (DNS) data set of isotropic, forced turbulence is described in this paper. The data set consists of the DNS output on 10243 spatial points and 1024 time samples spanning about one large-scale turnover time. This complete 10244 spacetime history of turbulence is accessible to users remotely through an interface that is based on the Web-services model. Users may write and execute analysis programs on their host computers, while the programs make subroutine-like calls that request desired parts of the data over the network. The users are thus able to perform numerical experiments by accessing the 27 terabytes (TB) of DNS data using regular platforms such as laptops. The architecture of the database is explained, as are some of the locally defined functions, such as differentiation and interpolation. Test calculations are performed to illustrate the usage of the system and to verify the accuracy of the methods. The database is then used to analyze a dynamical model for small-scale intermittency in turbulence. Specifically, the dynamical effects of pressure and viscous terms on the Lagrangian evolution of velocity increments are evaluated using conditional averages calculated from the DNS data in the database. It is shown that these effects differ considerably among themselves and thus require different modeling strategies in Lagrangian models of velocity increments and intermittency.

Small-scale intermittency in anisotropic turbulence

Isotropic, rotating, and stratified turbulent flows are analyzed using a scale- and direction-dependent flatness. The anisotropy of the spatial fluctuations of the energy distribution can hereby be quantified for different length scales. This measure allows one to distinguish between longitudinal and transversal intermittency as well as between horizontal and vertical intermittency. The difference between longitudinal and transversal intermittency is argued to be related to the incompressiblity constraint. A large difference between horizontal and vertical intermittency for stratified turbulence can be explained by an energy depletion of the horizontal plane in Fourier space.

A conditionally cubic-Gaussian stochastic Lagrangian model for acceleration in isotropic turbulence

The modelling of fluid-particle acceleration in homogeneous isotropic turbulence in terms of stochastic models for the Lagrangian velocity, acceleration and a dissipation rate variable is considered. The basis for the Reynolds model (A. M. Reynolds, Phys. Rev. Lett. vol. 91, 2003, 084503) is reviewed and examined by reference to direct numerical simulations (DNS) of isotropic turbulence at Taylor-scale Reynolds number (Rλ) up to about 650. In particular, we show DNS data that support stochastic modelling of the logarithm of pseudo-dissipation as an Ornstein–Uhlenbeck process and reveal non-Gaussianity of the acceleration conditioned on fluctuations of the pseudo-dissipation rate. The DNS data are used to construct a new stochastic model that is exactly consistent with Gaussian velocity and conditionally cubic-Gaussian acceleration statistics. This model captures the effects of small-scale intermittency on acceleration and the conditional dependence of acceleration on pseudo-dissipation (which differs from that predicted by the refined Kolmogorov hypotheses). Non-Gaussianity of the conditionally standardized acceleration probability density function (PDF) is accounted for in terms of model nonlinearity. The large-time behaviour of the new model is that of a velocity-dissipation model that can be matched with DNS data for conditional second-order Lagrangian velocity structure functions. As a result, the diffusion coefficient for the new model incorporates two-time information and its Reynolds-number dependence as observed in DNS. The resulting model predictions for conditional and unconditional velocity autocorrelations and time scales are shown to be in very good agreement with DNS.

Lagrangian conditional statistics, acceleration and local relative motion in numerically simulated isotropic turbulence

Lagrangian statistics of fluid-particle velocity and acceleration conditioned on fluctuations of dissipation, enstrophy and pseudo-dissipation representing different characteristics of local relative motion are extracted from a direct numerical simulation database of stationary (forced) homogeneous isotropic turbulence. The grid resolution in the simulations is up to 20483, and the Taylor-scale Reynolds number ranges from about 40 to 650, where characteristics of small-scale intermittency in the Eulerian flow field are well developed. A key joint statistic of the conditioning variables is the dissipation-enstrophy cross-correlation, which is asymmetric, but becomes less so at high Reynolds number. Conditional velocity autocorrelations are consistent with rapid changes in the velocity of fluid particles moving in regions of large velocity gradients. Examination of statistics conditioned upon enstrophy, especially in a local coordinate frame moving with the vorticity vector, and of the centripetal acceleration suggests the presence of vortex-trapping effects which persist for several Kolmogorov time scales. Further results on acceleration statistics and joint velocity-acceleration autocorrelations are also presented to help characterize in detail the properties of a joint stochastic process of velocity, acceleration and the pseudo-dissipation. Together with recent work on Eulerian conditional acceleration and Reynolds-number dependence of basic Lagrangian quantities, the present results are directly useful for the development of a new stochastic model formulated to account for intermittency and Reynolds-number effects as described in detail in a companion paper.