Lagrangian Autocorrelation Research Articles

Lagrangian correlations and scales in uniformly sheared turbulence

The present study focuses on the relationship between dispersion and Lagrangian statistics in a flow with uniform shear and with turbulent stresses and scales that are nearly transversely homogeneous but grow exponentially in the streamwise direction. Based on the assumption that the Lagrangian velocity field is self-preserving, an expression is derived for the Lagrangian autocoreelation coefficient in terms of the mean-square particle dispersion. Estimates of the mean-square particle dispersion, obtained from measurements of the mean temperature field downstream of a thin heated line source in uniformly sheared turbulence, are subsequently used to evaluate the Lagrangian autocorrelation coefficient and the corresponding integral scale and microscale. The ratios of the estimated Lagrangian scales and the Eulerian scales measured in a convected frame are found comparable to the estimates of previous investigators in other flows.

Quantitative properties of shelfbreak eddies

We present a quantitative analysis of properties of the eddy field associated with the shelfbreak front south of New England off the U.S. east coast. We employ records from current meters, drifting buoys, and hydrography obtained during June 1984. Mean currents during the study period showed the across‐shelf cyclonic shear in the alongshelf velocity to be similar to that found in earlier studies when eddies were probably absent most of the time. Our mean across‐shelf currents, however, differed from earlier findings and showed a marked divergence of the across‐shelf flow for the upper 65 m of water that implied high rates of upwelling from beneath. This across‐shelf divergence was also consistent with observed spreading of surface frontal isotherms in the study area. Records from both current meters and drifting buoys enabled us to compute both the lagged Eulerian and Lagrangian velocity autocorrelations. The former showed a decorrelation time of about 4 days and lacked any negative lobe, but the latter showed a strong negative lobe. Using a simple model for a band‐limited process in frequency, we inferred a rotational period for fluid particles in trajectories within eddies of 4 days and a Lagrangian decorrelation time of 2.6 days, comparable to the Eulerian time. The corresponding estimate of horizontal diffusivity was zero. These results were consistent with the generation close upstream of discrete, coherent vortices with dominantly deterministic character. We could find no evidence that across‐shelf exchange of heat or salt by the eddies was enhanced above estimates of climatic mean exchange, despite having a detailed alongshelf hydrographic transect across the strongest eddy features that we found.

EXPERIMENTAL AND NUMERICAL INVESTIGATIONS OF SOLIDS MIXING IN A GAS FLUIDIZED BED

ABSTRACT Solids mixing in a bubbling gas-fluidized bed was Investigated using two Independent methods. In the first method, distributions of solids mean velocity and bulk density, and dispersion coefficients were measured, employing the single particle mode of the particle tracking technique. This technique tracks the position of a single radioactive particle, dynamically identical to bed particles, by using a bank of scintillation detectors around the bed. Time-differentiation of the instantaneous tracer positions then yields the local velocities which were ensemble-averaged locally after long experimental runs to yield the mean velocity distribution. Lagrangian autocorrelation functions were obtained to determine the radial and axial dispersion coefficients, which were found to differ by an order of magnitude. These experimental data formed the basis of a numerical model of convective-diffusive mixing, formulated to predict the time-dependent concentration distribution for a given set of initial conditi...

A random-walk model for dispersion of heavy particles in turbulent air flow

A random-walk model is presented for calculating the dispersion of heavy particles in a turbulent air flow when only air turbulence statistics and the drag characteristics of the particle are known. Algebraic expressions for the modification of air velocity variance Σ2 and Lagrangian autocorrelation tune-scale T L,due to particle inertia effects, are derived. These expressions introduce only a very small computational overhead on the random-walk models for inertia-less particles of Wilson et al. (1983). Measurements of T Land Σ by Snyder and Lumley (1971) for four different particles are used to determine constants in the heavy-particle model. It is shown that the agreement between the model, for a single set of constants, and the dispersion measurements is good for the 47 Μm hollow glass, 87 Μm glass, and 47 Μm copper particles. The predictions for the 87 Μm corn pollen particles show less satisfactory agreement by underestimating dispersion measurements by 15% after 0.4s. Finally, some aspects of the model's application to spray dispersion in and above a crop canopy are considered.

Statistical prediction of iceberg trajectories

The unpredictability of low frequency currents in the ocean reduces the practicality of iceberg trajectory prediction using deterministic models. However, an optimum statistical model may be developed in which the future velocity of an iceberg (other than the predictable part due to wind, mean flow or tides) is a weighted sum of previous (measured) velocities. The weights are related to the Lagrangian velocity autocorrelation function. The scheme may be used to predict error bars as well as future positions. The theory is outlined in this paper and extended to allow for noise and inertial waves. For appropriate parameter values it is found that there is little value in using more than a one-term predictor, i.e. one based solely on the most recently measured velocity. It is shown how the scheme could also be used to predict, approximately, the probability that a given iceberg will enter a circle of specified radius around a wellsite within a specified time.

Turbulent diffusion in two-dimensional, strongly magnetized plasmas

Particle diffusion is investigated in a strictly two-dimensional collisionless guiding-centre model for a strongly magnetized plasma. An analytical expression is presented for the entire time variation of the mean square test-particle displacement in the limit of low-frequency, strongly turbulent, electric field fluctuations. The analysis relies on an explicit integral expression for the Lagrangian autocorrelation function in terms of the Eulerian wavenumber spectrum and a time-varying weight function. Bohm diffusion is discussed by means of a simple model spectrum. The analysis applies for turbulent transport associated with electrostatic convective cells, magnetostatic cells and drift wave turbulence with the assumption of local homogeneity and isotropy in two dimensions.

Stirring and transport of tracer fields by geostrophic turbulence

We investigate the interaction of concentration fields of passive tracer with velocity fields characterizing geostrophic turbulence. We develop and compare results from equilibrium statistical mechanics, from turbulence-closure theory and from numerical simulation. A consistent account emerges. Among the results we show (1) that velocity fields efficiently scatter tracer variance to all scales, (2) that tracer variance evolves toward an equilibrium spectrum which is different from the equilibrium spectrum for vorticity variance, and (3) that intermittency of the tracer field is characteristic of a cascade of tracer variance across wavenumber space. The greater efficiency of the cascade of tracer variance relative to a vorticity cascade is due to wavenumber-local advective terms which affect tracer but not vorticity. We suggest that the more efficient tracer cascade results in shorter Lagrangian autocorrelation times for tracer than for vorticity.We investigate the spatial flux of tracer when a uniform gradient of average tracer concentration is imposed. We show (1) that the spatial flux has dominant contributions from fluctuations on scales slightly larger than the dominant energetic scales, (2) that an effective eddy-diffusivity formulation is valid and that the diffusivity agrees with simple mixing-length estimates, and (3) that eddy diffusivity is significantly anisotropic if Rossby-wave propagation occurs. Meridional diffusivity is suppressed relative to zonal diffusivity.We complement the study of stirring down from a uniform gradient with a numerical investigation of the stirring out of an initially concentrated spot. We see that eddy diffusivity can be a dangerous concept for such problems.

Thermal diffusion of aerosol particles: lagrangian autocorrelation as an alternative to langevin's equation

As an alternative to Langevin's derivation of an equation for σ 2(t), the mean square displacement of particles by thermal diffusion, a derivation is given, which seems to be more stringent and obvious. It makes use only of the Lagrangian autocorrelation of particle velocities and of the equipartition law.

The dispersion of discrete particles in a turbulent fluid field

AbstractThe degree of radial dispersion of medium size particles (ηK < dp < lf) emanating from a point source, is measured photographically in the turbulent core of a fully developed vertical pipe flow of water. From the classical theoretical results of G. Taylor, it was then possible to calculate statistical parameters which characterize the turbulent particle motion. A comparison is made between a tracer and neutrally dense, buoyant and heavy particles. Results indicate that the effect of particle intertia is negligible, and that the crossing trajectories effect dominates the dispersion process. A “wake effect” causes a buoyant particle to disperse to a greater degree than an equivalent heavy particle. An exponential‐cosine form of the Lagrangian autocorrelation coefficient provides a best fit for the dispersion data, although a purely exponential form is equally adequate for practical calculations.

A Relation Between the Lagrangian and Eulerian Turbulent Velocity Autocorrelations

Direct computation of the Lagrangian autocorrelation is not feasible since generally one cannot measure the turbulent velocity of each fluid particle. A model whereby the Lagrangian autocorrelation is determined in terms of a domain integral of a set of regular Eulerian autocorrelations is advanced. The Eulerian autocorrelations are to be acquired concurrently at all positions in the flow field. Three novel averaging procedures are utilized for obtaining the relationship between the Lagrangian and Eulerian autocorrelations. This relationship is not constrained to either homogeneous or isotropic turbulence.

An exercise on diffusion by continuous movements

Se proponen expresiones analíticas que se comprenden mejor con los valores experimentales evaluados por Paul Frenzen, en 1963, para la Funcion Lagrangiana de Autocorrelación R (ᶓ) acerca de campos turbulentos generales en fluidos de diversas estabilidades gravitacionales. Se demuestra que las expresiones analíticas aquí propuestas, representan correspondencia en los datos experimentales, comparadas con las propuestas por Frenzen. Se incluye una descripción referente a la curva de correspondencia para cada grupo de valores experimentales considerados. Se ensayan soluciones generales para la ecuación de difusión de Taylor, empleando las expresiones analíticas propuestas para la Función Lagrangiana de autocorrelación.

A vortex model to relate eulerian and lagrangian “ turbulent” velocity fields

AbstractThe phenomenon of turbulent diffusion was studied by using computed models. Free pseudo‐turbulent fluid flow was simulated on a digital computer by ensembles of randomly positioned moving and interacting vortices to represent the eddy structure of the turbulence. A vortex model, which had statistical values corresponding to a measured real flow in an Eulerian frame of reference, was then obtained. By considering the motion of a particular vortex or a fluid particle in the computed pseudo‐turbulent flow it was possible to determine measurements in a Lagrangian frame of reference. Estimates of the mean square lateral excursions of a marked particle were obtained and finally the Lagrangian auto correlation was compared with the auto correlation in a moving frame of reference.

A measurement of Lagrangian velocity autocorrelation in approximately isotropic turbulence

By measuring the heat dispersion behind a heated wire stretched across a wind tunnel (Taylor 1921, 1935), the Lagrangian velocity autocorrelation was determined in an approximately isotropic, grid-generated turbulent flow. The techniques were similar to previous ones, but the scatter is less. Assuming self-preservation of the Lagrangian velocity statistics in a form consistent with recent measurements of decay in this flow (Comte-Bellot & Corrsin 1966, 1971), a stationary and an approximately self-preserving form for the dispersion were derived and approximately verified over the range of the experiment.Possibly the most important aspect of this experiment is that data were available in the same flow on the simplest Eulerian velocity autocorrelation in time, the correlation at a fixed spatial point translating with the mean flow (Comte-Bellot & Corrsin 1971). Thus, the Lagrangian velocity autocorrelation coefficient function calculated from the dispersion data could be compared with this corresponding Eulerian function. It was found that the Lagrangian Taylor micro-scale is very much larger than the analogous Eulerian microscale (76 ms compared with 6.2ms), contrary to an estimate of Corrsin (1963). The Lagrangian integral time scale is roughly equal to the Eulerian one, being larger by about 25 %.

Longitudinal dispersion within a two-dimensional turbulent shear flow

This paper describes some laboratory and numerical experiments made on the longitudinal dispersion in an open channel flow. Particular attention has been paid to the initial stages of the process.Physical arguments suggest that the streamwise dispersion of a line of marked fluid elements across a two-dimensional turbulent shear flow occurs in three distinct stages. These stages are identified by a change in the form of the distribution of marked fluid elements in the flow direction. The skewed distribution of the first stage is readily identified by a constant value (approximately 1·1) for the ratio of the peak velocity (V1) of the distribution to the mean-flow velocity ; experiments using dyed fluid, made at this stage of the process, have revealed six identifiable features of the suggested distribution. The distributions suggested for the second and the third stage are consistent with the experimental findings of Elder (1959) for the second stage and Taylor (1954) for the third stage.An attempt has been made to simulate the process numerically using a Markovian model. The results of the simulation confirm features suggested by physical arguments and are in agreement with the open channel experiments.The Lagrangian autocorrelation function is found to be related to the Lagrangian velocity-history of marked fluid released from extreme positions on the flow cross-section. The correlation function, as expressed in terms of the velocity-history function provided by the numerical simulation, is \begin{eqnarray*} && R(t^{\prime}) = \exp (-bt^{\prime})\int_0^{1}U^{+2}dy^{\prime};\\ && t^{\prime} = tu_{*}/d,\quad U^{+} = \frac{U(y^{\prime})-\overline{U}}{u_{*}}, \end{eqnarray*} where u* is the friction velocity and U(y′) is the temporal mean velocity at a (non-dimensionalized) distance y′ from the flow boundary. In an open channel flow at a Reynolds number (based on friction velocity and channel depth) of 500, the numerical simulation provides the value of b = 0·536.The results of an experiment, in which the three-dimensional motion of small neutrally buoyant spheres was recorded in many small discrete time intervals, corroborate the theoretical suggestions and simulation results.

Lagrangian Statistics from Numerically Integrated Turbulent Shear Flow

Lagrangian turbulence statistics were obtained from a three-dimensional numerical model of plane Poiseuille flow at large Reynolds number. Single-particle Lagrangian correlations were computed and compared with Eulerian space correlations. Although the curves were not self-similar, a dimensionless scale ratio defined at the e-folding time was comparable to Eulerian-Lagrangian scale ratios found in the literature. The numerically obtained mean-square particle displacements exhibited correct short- and long-time behavior. Mean-square particle separations were analyzed, and two-particle Lagrangian velocity correlations taken at the same time were more persistent than Lagrangian autocorrelations. A semiempirical functional form was constructed for the two-particle velocity correlations which yielded two-particle distance correlations in good agreement with those of the numerical model. The effect of mean shear on downstream separation was examined. Results indicate a t3 or steeper dependence for downstream mean-square separation 〈(xa − xb)2〉, with strong shear and Reynolds stress. Batchelor's similarity law, namely, that 〈(xaj − xbj)2〉 ∝ εt3 in directions not controlled by shear, is postulated for the direction of shear when shear generates ε. This postulate was tested numerically and found to be consistent.

Some Lagrangian Properties of Turbulence Deduced from Atmospheric Diffusion Experiments

Abstract Selected Prairie Grass diffusion experiments have been analyzed to determine the so-called Hay-Pasquill scale factor relating Lagrangian and Eulerian scales of turbulence. It was found that an average value of the scale factor equal to four as suggested by Hay and Pasquill is obtained only under conditions closely approximating stationary processes. When experiments conducted under non-stationary as well as stationary conditions are considered, simple regression techniques are more efficient than the Hay-Pasquill technique for predicting the lateral spread of the diffusing plume. Lagrangian autocorrelations and eddy-wind variances for the crosswind velocity component deduced from the data using Taylor's diffusion equation are compared with corresponding Eulerian quantities for experiments conducted under thermally stable conditions. It is shown that the Lagrangian integral scale of turbulence exceeds the Eulerian scale and that the Lagrangian and Eulerian variances are approximately equal.

Some aspects of the large‐scale turbulence and diffusion in the atmosphere

AbstractThe Eulerian and Lagrangian autocorrelations and energy spectra of the large‐scale turbulent motion in the atmosphere are computed and analysed. It is found that the distributions of the Eulerian and Lagrangian autocorrelations of the zonal velocities show the characteristic of an exponentially decreasing function, whereas those of the meridional velocities exhibit the behaviour of a sinusoidal function with a damping amplitude. The energy spectra appear to be proportional to the minus second power of the frequency in the high frequencies.Analyses are also made of the energy spectra, autocorrelations, and the Eulerian‐Lagrangian time scale transformation in a planetary wave model. The ratio of the Eulerian and Lagrangian frequencies for the large‐scale atmospheric turbulence is found to be a function of the mean zonal velocity, latitude, and the angular wave number of the planetary wave. Comparisons are made of the energy spectra and autocorrelations derived from the planetary wave model and those found from observations.

Turbulent Diffusion in a Stratified Fluid

Abstract This paper examines the dynamical effects on a particle drifting in a homogeneous field of turbulence, stationary and uniform except for a constant vertical density gradient. The ordinary “mechanical” impulses due to random normal pressures and shear stresses are assumed to have the same statistical properties as in the absence of density stratification. The buoyancy forces arising from the displacement of the particle are, on the other hand, seen to be more nearly “organized” forces and their presence in the equations of motion modifies Lagrangian velocity autocorrelations significantly. The results show that under unstable stratification the velocity history of a diffusing particle is not stationary to second order, the variance increasing exponentially with time. When the stratification is stable, on the other hand, the standard deviation of vertical displacement tends asymptotically to a constant value. These results are in qualitative agreement with observations of the Prairie Grass project ...

Lagrangian and eulerian correlations and energy spectra of geostrophic velocities

AbstractAn analysis is made of the dispersion of the end‐points of geostrophic trajectories initiated daily at 0000Z for the International Geophysical Year, 1958, at 500 mb over Seattle, Washington. Computations of the yearly mean of the Lagrangian autocorrelation coefficients and eddy diffusivities show that the Lagrangian integral time scale, ·, is equal to 5·0 × 104 sec. It is found that the zonal, x, and meridional, y, components of eddy diffusivity are respectively Kx = 5·90 × 1010 cm2 sec−1 and Ky = 2·30 × 1010 cm2 sec−1. Spectral analysis of the kinetic energy of the Lagrangian geostrophic velocities shows that the kinetic energy densities of both the zonal and meridional components of the geostrophic winds generally increase from high to low frequency with a peak of the meridional component occurring at the frequency of 0·01 cycles hr−1. Analyses are also made of the Eulerian autocorrelation coefficients and energy spectra of the geostrophic velocities. It is found that the Eulerian integral time‐scale, ϵ−1, is equal to 9·4 × 104 sec, making the ratio of the Lagrangian and Eulerian time‐scales near 0·5.

A Reappraisal of Sutton's Parametern

From dimensional arguments Sutton in 1932 proposed a power law form for the Lagrangian autocorrelation, R(ξ). The resulting diffusion equations have been widely used. There have been very few experimental determinations of R(ξ) in the last 30 years. However, an examination of available data suggests that Sutton's proposal may be a satisfactory approximation for a usefully broad range. Extrapolation to long travel distances is not justified on the basis of published curves for R(ξ) and on theoretical grounds (largely because the Lagrangian integral scale of turbulence becomes infinite).