Fluid Particle Acceleration Research Articles

Velocity and temperature derivatives in high-Reynolds-number turbulent flows in the atmospheric surface layer. Part 1. Facilities, methods and some general results

This is a report on a field experiment in an atmospheric surface layer at heights between 0.8 and 10m with the Taylor micro-scale Reynolds number in the rangeReλ = 1.6−6.6 ×103. Explicit information is obtained on the full set of velocity and temperature derivatives both spatial and temporal, i.e. no use of Taylor hypothesis is made. The report consists of three parts. Part 1 is devoted to the description of facilities, methods and some general results. Certain results are similar to those reported before and give us confidence in both old and new data, since this is the first repetition of this kind of experiment at better data quality. Other results were not obtained before, the typical example being the so-called tear-dropR-Qplot and several others. Part 2 concerns accelerations and related matters. Part 3 is devoted to issues concerning temperature, with the emphasis on joint statistics of temperature and velocity derivatives. The results obtained in this work are similar to those obtained in experiments in laboratory turbulent grid flow and in direct numerical simulations of Navier–Stokes equations at much smaller Reynolds numbersReλ ~ 102, and this similarity is not only qualitative, but to a large extent quantitative. This is true of such basic processes as enstrophy and strain production, geometrical statistics, the role of concentrated vorticity and strain, reduction of nonlinearity and non-local effects. The present experiments went far beyond the previous ones in two main respects. (i) All the data were obtained without invoking the Taylor hypothesis, and therefore a variety of results on fluid particle accelerations became possible. (ii) Simultaneous measurements of temperature and its gradients with the emphasis on joint statistics of temperature and velocity derivatives. These are reported in Parts 2 and 3.

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.

Multifractal PDF analysis for intermittent systems

The formula for probability density functions (PDFs) has been extended to include PDF for energy dissipation rates in addition to other PDFs such as for velocity fluctuations, velocity derivatives, fluid particle accelerations, energy transfer rates, etc., and it is shown that the formula actually explains various PDFs extracted from direct numerical simulations and experiments performed in a wind tunnel. It is also shown that the formula with appropriate zooming increment corresponding to experimental situation gives a new route to obtain the scaling exponents of velocity structure function, including intermittency exponent, out of PDFs of velocity fluctuations.

Lagrangian statistics for fluid particles and bubbles in turbulence

The dispersion of bubbles in homogeneous and isotropic turbulence is numerically examined. The fluid velocity field evolves according to the Navier–Stokes equations that are solved by direct numerical simulation. The bubble paths are followed by Lagrangian tracking. The aim of the work is to quantify dispersion properties of bubbles in a regime ranging from low- to high-turbulence velocity fluctuations as compared to the bubble velocity scale, and to compare them to those of fluid particles. Moreover, the forces that are relevant for the bubble dispersion are analysed and their probability density functions, as well as their intermittency characteristics, are compared to those of fluid particle accelerations.

Towards a dynamical theory of multifractals in turbulence

Making use of the exact equations for structure functions, supplemented by the equations for dissipa tive anomaly as well as an estimate for the Lagrangian acceleration of fluid particles, we obtain a main result of the multifractal theory of turbulence. The central element of the theory is a dissipation cut-off that depends on the order of the structure function. An expression obtained for the exponents $s_{n}$ in the scaling relations $\bar{(\frac{\partial u}{\partial x})^{n}}/\bar{(\frac{\partial u}{\partial x})^{2}}^{\frac{n}{2}} \propto Re^{s_{n}}$, between the velocity gradients $\frac{\partial u}{\partial x}$ and the Reynolds number $Re$, agrees well with experimental data.

Towards a dynamical theory of multifractals in turbulence

Making use of the exact equations for structure functions, supplemented by the equations for dissipative anomaly as well as an estimate for the Lagrangian acceleration of fluid particles, we obtain a main result of the multifractal theory of turbulence. The central element of the theory is a dissipation cut-off that depends on the order of the structure function. An expression obtained for the exponents s n in the scaling relations ∂ u ∂ x n ¯ ∂ u ∂ x 2 n / 2 ¯ ∝ Re s n , between the velocity gradients ∂ u / ∂ x and the Reynolds number Re, agrees well with experimental data.

Turbulence and Tsallis statistics

Fully developed incompressible Navier–Stokes turbulence in three dimensions is a dissipative dynamical system that exhibits strong departure from absolute equilibrium. Nevertheless, several kinds of representation by Tsallis equilibria have been proposed. The symmetry of the contributions to the kinetic energy from the degrees of freedom of the flow must be broken in order to construct turbulence applications. Tsallis representations of turbulence involve the extrapolation of results for systems with a single degree of freedom, a procedure that requires critical examination. Theories of Tsallis statistics for acceleration of fluid particles are compared with computer simulations of the pressure gradient field in the present paper. Applications of Tsallis formulations to statistics of longitudinal velocity differences are analyzed. No flavor of equilibrium statistics based on the kinetic energy can encompass the dynamically fundamental turbulent energy cascade, which requires correlation of triads of degrees of freedom at different scales.

Multifractal analysis of fluid particle accelerations in turbulence

The probability density function (PDF) of accelerations in turbulence is derived analytically with the help of the multifractal analysis based on generalized entropy, i.e., the Tsallis or the Rényi entropy. It is shown that the derived PDF explains quite well the one obtained by Bodenschatz et al. in the measurement of fluid particle accelerations in the Lagrangian frame at R λ =690, and the one by Gotoh et al. in the DNS with the mesh size 1024 3 at R λ =380.

One-dimensional Langevin models of fluid particle acceleration in developed turbulence.

We make a comparative analysis of some recent one-dimensional Langevin models of the acceleration of a Lagrangian fluid particle in developed turbulent flow. The class of models characterized by random intensities of noises (RIN models) provides a fit to the recent experimental data on the acceleration statistics. We review the model by Laval, Dubrulle, and Nazarenko (LDN) formulated in terms of temporal velocity derivative in the rapid distortion theory approach, and propose its extension due to the RIN framework. The fit of the contribution to fourth-order moment of the acceleration is found to be better than in the other stochastic models. We study the acceleration probability density function conditional on velocity fluctuations implied by the RIN approach to the LDN-type model. The shapes of the conditional distributions and the conditional acceleration variance have been found in a good agreement with the recent experimental data by Mordant, Crawford, and Bodenschatz [Physica D (to be published), e-print physics/0303003].

Lagrangian statistics in turbulent channel flow

The Lagrangian dispersion of fluid particles in inhomogeneous turbulence is investigated by a direct numerical simulation of turbulent channel flow. Lagrangian velocity and acceleration along a particle trajectory are computed by employing several interpolation schemes. Among the schemes tested, the four-point Hermite interpolation in the homogeneous directions combined with Chebyshev polynomials in the wall-normal direction seems to produce most reliable Lagrangian statistics. Inhomogeneity of Lagrangian statistics in turbulent boundary layer is investigated by releasing many particles at several different wall-normal locations and tracking those particles. The fluid particle dispersion and Lagrangian structure function of velocity are investigated for the Kolmogorov similarity. The behavior of the Lagrangian integral time scales, Kolmogorov constants a0 and C0 of the velocity structure function near the wall are discussed. The intermittent behavior of the fluid particle acceleration is also examined by kurtosis of the Lagrangian structure function. Finally, the effect of the initial particle location on the dispersion is analyzed by the probability density function of particle position at several downstream locations.

Third-order Lagrangian stochastic modeling

The formulation of third-order Lagrangian stochastic models for the hyper-accelerations (time derivative of acceleration) of fluid particles in turbulence is presented. Agreement between model predictions and data from direct numerical simulations [Yeung and Pope, J. Fluid Mech. 207, 531 (1989)] for the Lagrangian velocity structure function and Lagrangian acceleration autocorrelation function is excellent and significantly better than that obtained with a second-order Lagrangian stochastic model. The effects of third-order processes on the turbulent dispersion of fluid particles are found to be significant at Reynolds numbers typical of laboratory-scale experiments and direct numerical simulations and comparable in magnitude to the effects of second-order processes.

난류채널유동의 라그란지안 해석 (I)- 입자추적 알고리듬 평가 -

The Lagrangian dispserion of fluid particles in inhomogeneous turbulence is investigated by a direct numerical simulation of turbulent channel flow. Fluid particle velocity and acceleration along a particle trajectory are computed by employing several interpolation schemes such as linear interpolation, high-order Lagrange polynomial interpolation and the Hermite interpolation schemes. The performances of the schemes are evaluated through comparison of errors in computed particle positions, velocities and accelerations against spectral interpolation. Adopting the four-point Hermite interpolation in the homogeneous directions and Chebyshev polynomials in the wall-normal direction appears to produce most reliable Lagrangian statistics including acceleration correlations with a reasonable amount of computational overhead.<br/>

The PDF of fluid particle acceleration in turbulent flow with underlying normal distribution of velocity fluctuations

We describe a formal procedure to obtain and specify the general form of a marginal distribution for the Lagrangian acceleration of fluid particle in developed turbulent flow using Langevin type equation and the assumption that velocity fluctuation u follows a normal distribution with zero mean, in accord to the Heisenberg–Yaglom picture. For a particular representation, β=exp[ u], of the fluctuating parameter β, we reproduce the underlying log-normal distribution and the associated marginal distribution, which was found to be in a very good agreement with the new experimental data by Crawford, Mordant, and Bodenschatz on the acceleration statistics. We discuss on arising possibilities to make refinements of the log-normal model.

Length scales of acceleration for locally isotropic turbulence.

Length scales are determined that govern the behavior at small separations of the correlations of fluid-particle acceleration, viscous force, and pressure gradient. The length scales and an associated universal constant are quantified on the basis of published data. The length scale governing pressure spectra at high wave numbers is discussed. Fluid-particle acceleration correlation is governed by two length scales: one arises from the pressure gradient, the other from the viscous force.

Possible alternative to Rλ-scaling of small-scale turbulence statistics

Traditionally, trends of universal turbulence statistics are presented versus Rλ, which is the Reynolds number based on Taylor's scale λ and the root-mean-square (rms) of a component of velocity urms. λ and urms, and hence Rλ, do not have the attribute of universality. The ratio of rms fluid-particle acceleration to the rms of the acceleration caused by the viscous force, Ra, is an alternative to Rλ. Ra is a Reynolds number. It is composed of statistics of the small scales of turbulence. It can be evaluated with single-wire hot-wire anemometry. Like Rλ, it can be partially evaluated by means of flow similarity. Direct measurement of Ra is challenging; therefore, Ra is not a replacement for Rλ. For isotropic turbulence the relationship between Ra and Rλ is given. Anisotropic turbulence is discussed.

Scaling of acceleration in locally isotropic turbulence

The variances of the fluid-particle acceleration and of the pressure-gradient and viscous force are given. The scaling parameters for these variances are velocity statistics measureable with a single-wire anemometer. For both high and low Reynolds numbers, asymptotic scaling formulas are given; these agree quantitatively with DNS data. Thus, the scaling can be presumed known for all Reynolds numbers. Fluid-particle acceleration variance does not obey K41 scaling at any Reynolds number; this is consistent with recent experimental data. The non-dimensional pressure-gradient variance named λT/λP is shown to be obsolete.

Fluid particle accelerations in fully developed turbulence.

The motion of fluid particles as they are pushed along erratic trajectories by fluctuating pressure gradients is fundamental to transport and mixing in turbulence. It is essential in cloud formation and atmospheric transport, processes in stirred chemical reactors and combustion systems, and in the industrial production of nanoparticles. The concept of particle trajectories has been used successfully to describe mixing and transport in turbulence, but issues of fundamental importance remain unresolved. One such issue is the Heisenberg-Yaglom prediction of fluid particle accelerations, based on the 1941 scaling theory of Kolmogorov. Here we report acceleration measurements using a detector adapted from high-energy physics to track particles in a laboratory water flow at Reynolds numbers up to 63,000. We find that, within experimental errors, Kolmogorov scaling of the acceleration variance is attained at high Reynolds numbers. Our data indicate that the acceleration is an extremely intermittent variable--particles are observed with accelerations of up to 1,500 times the acceleration of gravity (equivalent to 40 times the root mean square acceleration). We find that the acceleration data reflect the anisotropy of the large-scale flow at all Reynolds numbers studied.

Conditional Fluid-Particle Accelerations in Turbulence

Simple closures for average fluid-particle accelerations, conditional on fixed local fluid velocity, are considered in isotropic, homogeneous and stationary turbulence using exact probability density transport equations and are compared with direct numerical simulations (DNS). Such accelerations are common ingredients in Lagrangian stochastic models for fluid-particle trajectories in turbulence. One-particle accelerations are essentially trivial, so the focus is on two-particle relative accelerations, which are important in the relative dispersion process. The closure is simply a quadratic form in the velocity variable and this special form also defines the Eulerian velocity probability density function (pdf), and comparisons with DNS (for grids up to 5123) of both the acceleration closure and velocity pdf's are encouraging.

Velocity, acceleration and vorticity under a breaking wave

The fluid particle velocities in the overturning jet of a breaking wave have been measured by the Particle Image Velocimetry (PIV) technique. Monochromatic waves with wave height of 14.5 cm and wavelength of 121 cm were generated in the water depth of 20 cm. The measured fluid particle velocity at the tip of the overturning jet reached 1.68 times of the phase velocity calculated from the linear wave theory. Fluid particle accelerations were estimated from the velocity data with the following results: The overturning jet enters the horizontal water surface with an acceleration of 1.1 g at an angle of 88° downward. The PIV technique was also used to measure the instantaneous vertical vorticities generated by breaking waves. The number and locations of the vortices on the horizontal plane appear to be random. The maximum instantaneous vorticity was in the order of magnitude of 20–30 s−1, whereas the ensemble-averaged vorticity was quite small.

One- and two-particle Lagrangian acceleration correlations in numerically simulated homogeneous turbulence

Lagrangian statistics of the fluid particle acceleration are studied by direct numerical simulation, in stationary isotropic turbulence (at three different Reynolds numbers) and homogeneous shear flow with uniform mean shear rate. The one-particle acceleration autocorrelation decays rapidly with time, with a zero crossing just over two Kolmogorov time scales. In contrast, two-particle correlations are relatively persistent, especially for particle pairs of small initial separation distance. Results for intermediate times at a Taylor-scale Reynolds number of 140 resemble a t−1 inertial-range scaling suggested in the literature, but even higher Reynolds numbers are needed for more definitive comparisons. Use of a theoretical argument and conditional sampling indicates that the two-particle correlation is determined by a coupling between a correlation localized in space and a particle-pair separation probability density of positive skewness. The scenario which emerges is that whereas some fluid particles accelerate rapidly away from each other, the majority of particle pairs can still be relatively close together and hence help maintain the two-particle correlation at significant levels. Acceleration correlations in homogeneous shear flow are found to display a tendency toward local isotropy.