Statistics Of Velocity Increments Research Articles

Adjoint-based variational optimal mixed models for large-eddy simulation of turbulence

An adjoint-based variational optimal mixed model (VOMM) is proposed for subgrid-scale (SGS) closure in large-eddy simulation (LES) of turbulence. The stabilized adjoint LES equations are formulated by introducing a minimal regularization to address the numerical instabilities of the long-term gradient evaluations in chaotic turbulent flows. The VOMM model parameters are optimized by minimizing the discrepancy of energy dissipation spectra between LES calculations and a priori knowledge of direct numerical simulation using the gradient-based optimization. The a posteriori performance of the VOMM model is comprehensively examined in LES of three turbulent flows, including the forced homogeneous isotropic turbulence, decaying homogenous isotropic turbulence, and temporally evolving turbulent mixing layer. The VOMM model outperforms the dynamic Smagorinsky model, dynamic mixed model (DMM), and approximate deconvolution model in predictions of various turbulence statistics, including the velocity spectrum, structure functions, statistics of velocity increments and vorticity, temporal evolutions of the turbulent kinetic energy, dissipation rate, momentum thickness and Reynolds stress, as well as the instantaneous vortex structures at different grid resolutions and times. In addition, the VOMM model only takes up 30% time of the DMM model for all flow scenarios. These results demonstrate that the proposed VOMM model improves the numerical stability of LES and has high a posteriori accuracy and computational efficiency by incorporating the a priori information of turbulence statistics, highlighting that the VOMM model has a great potential to develop advanced SGS models in the LES of turbulence.

Scaling of Acceleration Statistics in High Reynolds Number Turbulence

The scaling of acceleration statistics in turbulence is examined by combining data from the literature with new data from well-resolved direct numerical simulations of isotropic turbulence, significantly extending the Reynolds number range. The acceleration variance at higher Reynolds numbers departs from previous predictions based on multifractal models, which characterize Lagrangian intermittency as an extension of Eulerian intermittency. The disagreement is even more prominent for higher-order moments of the acceleration. Instead, starting from a known exact relation, we relate the scaling of acceleration variance to that of Eulerian fourth-order velocity gradient and velocity increment statistics. This prediction is in excellent agreement with the variance data. Our Letter highlights the need for models that consider Lagrangian intermittency independent of the Eulerian counterpart.

Combined measurement of velocity and temperature in liquid metal convection

Combined measurements of velocity components and temperature in a turbulent Rayleigh–Bénard convection flow at a low Prandtl number of $Pr=0.029$ and Rayleigh numbers of $10^{6}\leqslant Ra\leqslant 6\times 10^{7}$ are conducted in a series of experiments with durations of more than a thousand free-fall time units. Multiple crossing ultrasound beam lines and an array of thermocouples at mid-height allow for a detailed analysis and characterization of the complex three-dimensional dynamics of the single large-scale circulation roll in a cylindrical convection cell of unit aspect ratio which is filled with the liquid metal alloy GaInSn. We measure the internal temporal correlations of the complex large-scale flow and distinguish between short-term oscillations associated with a sloshing motion in the mid-plane as well as varying orientation angles of the velocity close to the top/bottom plates and the slow azimuthal drift of the mean orientation of the roll as a whole that proceeds on a time scale up to a hundred times slower. The coherent large-scale circulation drives a vigorous turbulence in the whole cell that is quantified by direct Reynolds number measurements at different locations in the cell. The velocity increment statistics in the bulk of the cell displays characteristic properties of intermittent small-scale fluid turbulence. We also show that the impact of the symmetry-breaking large-scale flow persists to small-scale velocity fluctuations thus preventing the establishment of fully isotropic turbulence in the cell centre. Reynolds number amplitudes depend sensitively on beam-line position in the cell such that different definitions have to be compared. The global momentum and heat transfer scalings with Rayleigh number are found to agree with those of direct numerical simulations and other laboratory experiments.

Fractal Reconstruction of Sub-Grid Scales for Large Eddy Simulation

In this work, the reconstruction of sub-grid scales in large eddy simulation (LES) of turbulent flows in stratocumulus clouds is addressed. The approach is based on the fractality assumption of turbulent velocity field. The fractal model reconstructs sub-grid velocity field from known filtered values on LES grid, by means of fractal interpolation, proposed by Scotti and Meneveau (Physica D 127, 198–232 1999). The characteristics of the reconstructed signal depend on the stretching parameter d, which is related to the fractal dimension of the signal. In many previous studies, the stretching parameter values were assumed to be constant in space and time. To improve the fractal interpolation approach, we account for the stretching parameter variability. The local stretching parameter is calculated from direct numerical simulation (DNS) data with an algorithm proposed by Mazel and Hayes (IEEE Trans. Signal Process 40(7), 1724–1734, 1992), and its probability density function (PDF) is determined. It is found that the PDFs of d have a universal form when the velocity field is filtered to wave-numbers within the inertial range. In order to investigate Reynolds number (Re) dependence, we compare the inertial-range PDFs of d in DNS and large eddy simulation (LES) of stratocumulus cloud-top and experimental airborne data from Physics of Stratocumulus Top (POST) research campaign. Next, fractal reconstruction of the subgrid velocity is performed and energy spectra and statistics of velocity increments are compared with DNS data. It is assumed that the stretching parameter d is a random variable with the prescribed PDF. We show that the agreement with the DNS is in such case better and the error in mass conservation is smaller compared to the use of constant values of d. The motivation of this study is to reproduce effect of sub-grid scales on a motion of Lagrangian particles (e.g. droplets) in clouds.

Multiscale velocity correlations in turbulence and Burgers turbulence: Fusion rules, Markov processes in scale, and multifractal predictions.

We compare different approaches towards an effective description of multiscale velocity field correlations in turbulence. Predictions made by the operator-product expansion, the so-called fusion rules, are placed in juxtaposition to an approach that interprets the turbulent energy cascade in terms of a Markov process of velocity increments in scale. We explicitly show that the fusion rules are a direct consequence of the Markov property provided that the structure functions exhibit scaling in the inertial range. Furthermore, the limit case of joint velocity gradient and velocity increment statistics is discussed and put into the context of the notion of dissipative anomaly. We generalize a prediction made by the multifractal model derived by Benzi etal. [R. Benzi et al., Phys. Rev. Lett. 80, 3244 (1998)PRLTAO0031-900710.1103/PhysRevLett.80.3244] to correlations among inertial range velocity increment and velocity gradients of any order. We show that for the case of squared velocity gradients such a relation can be derived from first principles in the case of Burgers equations. Our results are benchmarked by intensive direct numerical simulations of Burgers turbulence.

Master equation for She–Leveque scaling and its classification in terms of other Markov models of developed turbulence

The statistics of velocity increments in homogeneous and isotropic turbulence exhibit universal features in the limit of infinite Reynolds numbers. After Kolmogorov’s scaling law from 1941, many turbulence models aim for capturing these universal features, some are known to have an equivalent formulation in terms of Markov processes. We derive the Markov process equivalent to the particularly successful scaling law postulated by She and Leveque. The Markov process is a jump process for velocity increments in scale r in which the jumps occur randomly but with deterministic width in u. From its master equation we establish a prescription to simulate the She–Leveque process and compare it with Kolmogorov scaling. To put the She–Leveque process into the context of other established turbulence models on the Markov level, we derive a diffusion process for using two properties of the Navier–Stokes equation. This diffusion process already includes Kolmogorov scaling, extended self-similarity and a class of random cascade models. The fluctuation theorem of this Markov process implies a ‘second law’ that puts a loose bound on the multipliers of the random cascade models. This bound explicitly allows for instances of inverse cascades, which are necessary to satisfy the fluctuation theorem. By adding a jump process to the diffusion process, we go beyond Kolmogorov scaling and formulate the most general scaling law for the class of Markov processes having both diffusion and jump parts. This Markov scaling law includes She–Leveque scaling and a scaling law derived by Yakhot.

Generation of Lagrangian intermittency in turbulence by a self-similar mechanism

Intermittency, i.e., extreme fluctuations at small scales, causes the deviation of turbulence statistics from Kolmogorov's 1941 theoretical predictions. Intermittency effects are especially strong for Lagrangian statistics. Our understanding of how Lagrangian intermittency manifests, however, is still elusive. Here, we study the Lagrangian intermittency in the framework of an exact, yet unclosed probability density function (PDF) equation. Combining this theoretical approach with data from experiments and simulations, no a priori phenomenological assumptions about the structure or properties of the flow have to be made. In this description, the non-self-similar evolution of the velocity increment PDF is determined at all scales by a single function, which is accessible through data from experiments and simulations. This ‘intermittency generating function’ arises from the dependence of the acceleration of a fluid element on its velocity history, thereby coupling different scales of turbulent motion. Empirically, we find that the intermittency generating function has a simple, approximately self-similar form, which has the surprising implication that Lagrangian intermittency—the absence of self-similarity in the Lagrangian velocity increment statistics—is driven by a self-similar mechanism. The simple form of the intermittency generating function furthermore allows us to formulate a simple model parametrization of the velocity increment PDFs.

Towards a stochastic multi-point description of turbulence

In previous work it was found that the multi-scale statistics of homogeneous isotropic turbulence can be described by a stochastic "cascade" process of the velocity increment from scale to scale, which is governed by a Fokker-Planck equation. We now show how this description for increments can be extended in order to obtain the complete multi-point statistics in real space of the turbulent velocity field (Stresing & Peinke, 2010). We extend the stochastic cascade description by conditioning on an absolute velocity value itself, and find that the corresponding conditioned process is also governed by a Fokker-Planck equation, which contains as a leading term a simple additional velocity-dependent coefficient, d10, in the drift function. Taking the velocity-dependence of the Fokker-Planck equation into account, the multi-point statistics in the inertial range can be expressed by two-scale statistics of velocity increments, which are equivalent to three-point statistics of the velocity field. Thus, we propose a stochastic three-point closure for the velocity field of homogeneous isotropic turbulence. Investigating the coefficient d10 for different flows, we find clear evidence that the multipoint structure of small scale turbulence is not universal but depends on the type of the flow.

Turbulence in the Diffuse Interstellar Medium

AbstractThe diffuse interstellar medium (ISM) hosts the first steps of interstellar chemistry and the seeds of dense structures. Since its turbulent pressure by far exceeds its thermal pressure, turbulence must play a prominent role in its evolution. Fed at galactic scales, turbulent energy cascades down to the dissipation scales, but as in both laboratory and atmospheric turbulence, it does so in an intermittent way : only a tiny fraction of the small-scales is fed by the turbulent cascade, so that dissipation occurs in bursts. In diffuse molecular clouds, where they can be observed, the signatures of intermittency are: (1) the non-Gaussian statistics of velocity increments, and (2) the existence of coherent structures of intense velocity-shear that appear to channel the large-scale turbulent energy down to milliparsec scales. Attempts at modelling the warm chemistry triggered in the diffuse ISM by bursts of turbulent dissipation are promising : in this framework, the so far unexplained molecular richness observed in this medium is naturally understood, in particular its CH+, HCO+ and CO abundances. Turbulent dissipation is also likely at the origin of the H2 rotational line emission of the diffuse ISM and of a significant fraction of its [C II] emission.

Conditional Eulerian and Lagrangian velocity increment statistics of fully developed turbulent flow

Conditional statistics of homogeneous isotropic turbulent flow is investigated by means of high Reynolds number direct numerical simulations performed with 20483 collocation points. Eulerian as well as Lagrangian velocity increment statistics under several conditions are analyzed and compared. In agreement with experimental data, longitudinal probability density functions P(δl||u|εl) conditioned on a scale-averaged energy dissipation rate are close to Gaussian distributions over all scales within the inertial range of scales. Also transverse increments conditioned on either the dissipation rate or the square of the vorticity have quasi-Gaussian probability density functions (PDFs). Concerning Lagrangian statistics we found that conditioning on a trajectory averaged energy dissipation rate ετ significantly reduces the scale dependence of the increment PDFs P(δτui|ετ). By means of dimensional arguments we propose a novel condition for Lagrangian increments which is shown to reduce even more the flatness of the corresponding PDFs and thus intermittency in the inertial range of scales. The conditioned Lagrangian PDF corresponding to the smallest increment considered is reasonably well described by a prediction for the PDF of acceleration of a non-intermittent velocity field. Conditioned structure functions show approximately K41 scaling with a larger scaling range than the unconditioned ones.

Atmospheric wind field conditions generated by active grids

An active grid for turbulence generation of several rotatable axes with surmounted vanes that can be driven via stepper or servo motors is presented. We investigate the impact of different excitation protocols for the grid. Using such protocols that already have the intermittent structure of turbulence, higher intermittent flows can be achieved. This concept can also be used to generate turbulent flows of high turbulence intensities (>25%) exhibiting integral length scales beyond the typical size of the test section of the wind tunnel. Similar two-point correlations measured by the intermittent statistics of velocity increments that are characteristic for flows of high Reynolds number, i.e. in the atmospheric boundary layer, can be reproduced.

Atmospheric turbulence and its influence on the alternating loads on wind turbines

AbstractAnalysis of measurements on atmospheric turbulence with respect to the statistics of velocity increments reveals that the statistics are not Gaussian but highly intermittent. Here, we demonstrate that the higher quantity of extreme events in atmospheric wind fields transfers to alternating loads on the airfoil and on the main shaft in the form of torque fluctuations. For this purpose, alternating loads are discussed with respect to their increment statistics. Our conjecture is that the anomalous wind statistics are responsible for load changes, which may potentially contribute to additional loads and may cause additional fatigue. Our analysis is performed on three different wind field data sets: measured fields, data generated by a standard wind field model and data generated by an alternative model based on continuous time random walks, which grasps the intermittent structure of atmospheric turbulence in a better way. Our findings suggest that fluctuations in the loads might not be reflected properly by the standard wind field models. Copyright © 2010 John Wiley & Sons, Ltd.

Towards a stochastic multi-point description of turbulence

It has been found in previous work that the multi-scale statistics of homogeneous isotropic turbulence can be described by a stochastic ‘cascade’ process of the velocity increment from scale to scale, which is governed by a Fokker–Planck equation. We show in this paper how this description can be extended to obtain the complete multi-point statistics of the velocity field. We extend the stochastic cascade description by conditioning on the velocity value itself and find that the corresponding process is also governed by a Fokker–Planck equation, which contains as a leading term a simple additional velocity-dependent coefficient in the drift function. Taking into account the velocity dependence of the Fokker–Planck equation, the multi-point statistics in the inertial range can be expressed by the two-scale statistics of velocity increments, which are equivalent to the three-point statistics of the velocity field. Thus, we propose a stochastic three-point closure for the velocity field of homogeneous isotropic turbulence.

Turbulence measurements using a nanoscale thermal anemometry probe

A nanoscale thermal anemometry probe (NSTAP) has been developed to measure velocity fluctuations at ultra-small scales. The sensing element is a free-standing platinum nanoscale wire, 100 nm × 2 μm × 60 μm, suspended between two current-carrying contacts and the sensor is an order of magnitude smaller than presently available commercial hot wires. The probe is constructed using standard semiconductor and MEMS manufacturing methods, which enables many probes to be manufactured simultaneously. Measurements were performed in grid-generated turbulence and compared to conventional hot-wire probes with a range of sensor lengths. The results demonstrate that the NSTAP behaves similarly to conventional hot-wire probes but with better spatial resolution and faster temporal response. The results are used to investigate spatial filtering effects, including the impact of spatial filtering on the probability density of velocity and velocity increment statistics.

Lagrangian evolution of velocity increments in rotating turbulence: The effects of rotation on non-Gaussian statistics

The effects of rotation on the evolution of non-Gaussian statistics of velocity increments in rotating turbulence are studied in this paper. Following the Lagrangian evolution of the velocity increments over a fixed distance on an evolving material element, we derive a set of equations for the increments which provides a closed representation for the nonlinear interaction between the increments and the Coriolis force. Applying a restricted-Euler-type closure to the system, we obtain a system of ordinary differential equations which retains the effects of nonlinear interaction between the velocity increments and the Coriolis force. A priori tests using direct numerical simulation data show that the system captures the important dynamics of rotating turbulence. The system is integrated numerically starting from Gaussian initial data. It is shown that the system qualitatively reproduces a number of observations in rotating turbulence. The statistics of the velocity increments tend to Gaussian when strong rotation is imposed. The negative skewness in the longitudinal velocity increments is weakened by rotation. The model also predicts that the transverse velocity increment in the plane perpendicular to the rotation axis will have positive skewness, and that the skewness will depend on the Rossby number in a non-monotonic way. Based on the system, we identify the dynamical mechanisms leading to the observations.

Defining a New Class of Turbulent Flows

We apply a method based on the theory of Markov processes to fractal-generated turbulence and obtain joint probabilities of velocity increments at several scales. From experimental data we extract a Fokker-Planck equation which describes the interscale dynamics of the turbulence. In stark contrast to all documented boundary-free turbulent flows, the multiscale statistics of velocity increments, the coefficients of the Fokker-Planck equation, and dissipation-range intermittency are all independent of Rλ (the characteristic ratio of inertial to viscous forces in the fluid). These properties define a qualitatively new class of turbulence.

Exact relation between Eulerian and Lagrangian velocity increment statistics

We present a formal connection between Lagrangian and Eulerian velocity increment distributions which is applicable to a wide range of turbulent systems ranging from turbulence in incompressible fluids to magnetohydrodynamic turbulence. For the case of the inverse cascade regime of two-dimensional turbulence we numerically estimate the transition probabilities involved in this connection. In this context we are able to directly identify the processes leading to strongly non-Gaussian statistics for the Lagrangian velocity increments.

Observation of Wave Turbulence in Vibrating Plates

The nonlinear interaction of waves in a driven medium may lead to wave turbulence, a state such that energy is transferred from large to small length scales. Here, wave turbulence is observed in experiments on a vibrating plate. The frequency power spectra of the normal velocity of the plate may be rescaled on a single curve, with power-law behaviors that are incompatible with the weak turbulence theory of Düring et al. [Phys. Rev. Lett. 97, 025503 (2006)10.1103/PhysRevLett.97.025503]. Alternative scenarios are suggested to account for this discrepancy -- in particular the occurrence of wave breaking at high frequencies. Finally, the statistics of velocity increments do not display an intermittent behavior.

A multifractal formalism for vector-valued random fields based on wavelet analysis: application to turbulent velocity and vorticity 3D numerical data

Extreme atmospheric events are intimately related to the statistics of atmospheric turbulent velocities. These, in turn, exhibit multifractal scaling, which is determining the nature of the asymptotic behavior of velocities, and whose parameter evaluation is therefore of great interest currently. We combine singular value decomposition techniques and wavelet transform analysis to generalize the multifractal formalism to vector-valued random fields. The so-called Tensorial Wavelet Transform Modulus Maxima (TWTMM) method is calibrated on synthetic self-similar 2D vector-valued multifractal measures and monofractal 3D vector-valued fractional Brownian fields. We report the results of some application of the TWTMM method to turbulent velocity and vorticity fields generated by direct numerical simulations of the incompressible Navier–Stokes equations. This study reveals the existence of an intimate relationship \({D_{{\bf v}}(h+1)=D_{\varvec{\omega}}(h)},\) between the singularity spectra of these two vector fields which are found significantly more intermittent than previously estimated from longitudinal and transverse velocity increment statistics.

Generalizing the wavelet-based multifractal formalism to random vector fields: application to three-dimensional turbulence velocity and vorticity data.

We use singular value decomposition techniques to generalize the wavelet transform modulus maxima method to the multifractal analysis of vector-valued random fields. The method is calibrated on synthetic multifractal 2D vector measures and monofractal 3D fractional Brownian vector fields. We report the results of some application to the velocity and vorticity fields issued from 3D isotropic turbulence simulations. This study reveals the existence of an intimate relationship between the singularity spectra of these two vector fields which are found significantly more intermittent than previously estimated from longitudinal and transverse velocity increment statistics.