Lagrangian Time Research Articles

Lagrangian Time Scale of Passive Rotation for Mesoscale Particles in Turbulence

Turbulence induces rotation in the living and the non-living materials in the ocean. The time scale of rotation for a living organism is important in understanding an organism's feeding efficiency, mating, prey capture rate etc. This time scale is also crucial for understanding the migration of non-living materials such as microplastics. Herein, we investigate the tumbling motion of mesoscale particles that resemble organisms of intermediate size, such as zooplankton that appear in the ocean. Using time-resolved measurements of the orientation of rigid inertial fibers in a turbulence-tank, we analyze the autocorrelation of their tumbling rate. The correlation time ($\tau_d$) is well predicted by Kolmogorov inertial-range scaling based on the fiber length ($L$) when the fiber inertia can be neglected. For inertial fibers, we propose a simple model considering fiber inertia (measured by a tumbling Stokes number) and a viscous torque which accurately predicts both the correlation time and the variance of the tumbling rate. Our measurements and the theoretical model provide a basic understanding of the rotational response of an intermediately sized organism to the surrounding turbulence in its non-active state.

Hydrodynamic Dispersion in Porous Media and the Significance of Lagrangian Time and Space Scales

Transport in porous media is critical for many applications in the environment and in the chemical process industry. A key parameter for modeling this transport is the hydrodynamic dispersion coefficient for particles and scalars in a porous medium, which has been found to depend on properties of the medium structure, on the dispersing compound, and on the flow field characteristics. Previous studies have resulted in suggestions of different equation forms, showing the relationship between the hydrodynamic dispersion coefficient for various types of porous media in various flow regimes and the Peclet number. The Peclet number is calculated based on a Eulerian length scale, such as the diameter of the spheres in packed beds, or the pore diameter. However, the nature of hydrodynamic dispersion is Lagrangian, and it should take the molecular diffusion effects, as well as the convection effects, into account. This work shifts attention to the Lagrangian time and length scales for the definition of the Peclet number. It is focused on the dependence of the longitudinal hydrodynamic dispersion coefficient on the effective Lagrangian Peclet number by using a Lagrangian length scale and the effective molecular diffusivity. The lattice Boltzmann method (LBM) was employed to simulate flow in porous media that were constituted by packed spheres, and Lagrangian particle tracking (LPT) was used to track the movement of individual dispersing particles. It was found that the hydrodynamic dispersion coefficient linearly depends on the effective Lagrangian Peclet number for packed beds with different types of packing. This linear equation describing the dependence of the dispersion coefficient on the effective Lagrangian Peclet number is both simpler and more accurate than the one formed using the effective Eulerian Peclet number. In addition, the slope of the line is a characteristic coefficient for a given medium.

Topology of dynamical reconstructions from Lagrangian data

Branched Manifold Analysis through Homologies (BraMAH) is a technique that computes the state-space topology of a dynamical reconstruction from scalar data. This work introduces the application of this technique to Lagrangian time series. The approach unveils the topological structure underlying the behavior of a fluid particle. When applied to a set of sparse particles, the results of the analysis can be used to classify them according to the dynamics they deploy during a given time window. Topological grids can be constructed to portray the spatial organization of the topological classes. The connection between the topological grids and the transport properties of the flow is examined using streaklines. Even if demonstrated here in the context of kinematic flow models, the generality of the method allows for its potential application to experimental or observational Lagrangian data satisfying the technical requirements for the analysis.

Particle motion within the viscous sublayer of a turbulent shear flow

The motion of isolated heavy particles on the bottom wall of a turbulent shear flow is investigated experimentally (using high-frequency particle tracking) and correlated to that of the fluid (from high-resolution LDV measurements). Numerous particle diameters, particle densities, and flow velocities are considered, where the particles remain within the viscous sublayer (particle Reynolds number lower than 10). It is shown that fluid flow fluctuations within the viscous sublayer (varying Couette flow) induce particle velocity fluctuations of the same magnitude, slightly smaller. The mean particle velocity, standard deviation, and Lagrangian correlation time are found to scale with the viscous scales. Particles roll and slide without takeoff, unlike what happens in viscous laminar flow where the lift force may overcome the immersed particle weight. Arguments are given for this difference. Probability density functions for the streamwise and spanwise velocity are found to follow Gamma and Gaussian laws, respectively, which also scale with the viscous scales.

Eulerian and Lagrangian time scales of the turbulence above staggered arrays of cubical obstacles

We present results from water-channel experiments on neutrally-stable turbulent flows over staggered arrays of cubical obstacles modelling idealised urban canopies. Attention is concentrated on the vertical profiles of the Eulerian (TE) and Lagrangian (TL) time scales of the turbulence above three canopies with different plan area fractions (λP = 0.1, 0.25 and 0.4). The results show that both the streamwise and vertical components of TL increase approximately linearly with height above the obstacles, supporting Raupach’s linear law. The comparisons with the Lagrangian time scales over canyon-type canopies in the skimming flow and wake interference regimes show that the staggered configuration of cubical obstacles increases the streamwise TL, while decreasing its vertical counterpart. A good agreement has also been found between the eddy viscosities (KT) estimated by applying Taylor’s theory and the classical first order closure relating the momentum flux to the velocity gradient. The results show that KT obeys Prandtl’s theory, particularly for λP = 0.25 and 0.4.

On the competition between mixing rate and uniformity in a coaxial hydrodynamic focusing mixer

Fast microfluidic mixers for use with line-of-sight integrating detection schemes pose unique challenges. Such detectors typically cannot discriminate signal from slow moving (e.g. near internal walls) and fast-moving portions of the fluid stream. This convolves reaction rate dynamics with fluid flow residence time dynamics. Further, the small cross sections of typical three-dimensional hydrodynamic focusing devices lead to lower detection signals. The current study focuses on achieving both small time scales of mixing and homogenous residence times. This is achieved by injecting sample through a center capillary and hydrodynamically focusing using a sheath flow within a tapered second capillary. The current design also features a third, larger coaxial capillary. The mixed stream flows into the large cross-section of this third capillary to decelerate and expand the stream by up to 14-fold to improve line-of-sight signal strength of reaction products. Hydrodynamic focusing, mixing, and expansion are studied using analytical and numerical models and also studied experimentally using a fluorescein-iodide quenching reaction. The experimentally validated models are used to explore trade-offs between mixing rate and uniformity. For the first time, this work presents detailed analysis of the Lagrangian time history of species transport during mixing inside coaxial capillaries to measure mixing nonuniformity. The mixing region enables order 100 μs mixing times and residence time widths of the same order (140 μs).

Reconciling Gaussian plume and Computational Fluid Dynamics models of particulate dispersion

Abstract Computational Fluid Dynamics (CFD) is increasingly used to model particulate dispersion in situations where Gaussian Dispersion models are inappropriate or inaccurate. However, there is evidence which indicates that many CFD models under-predict lateral plume spread. This paper aims to address this by implementing a strategy which incorporates wind direction variability into CFD models using a formulation which is also used in the UK-ADMS plume spread module. In the present work, a series of CFD simulations are run at various wind angles. The outputs from these simulations are weighted using a Gaussian probability density function and combined to produce a plume. The standard k − e model has been employed to solve the RANS equations of the flow field for stable, neutral and unstable atmospheric stabilities, coupled with the Lagrangian Particle tracking model to model dispersion. By comparing the CFD accretion profiles to UK-ADMS dry deposition results, it is observed that the proposed modelling methodology produces lateral spreading of the plume which is comparable to that obtained using UK-ADMS. However, the Lagrangian integral time scale constant, c L , which governs the influence turbulence has on the dispersion, must also be modified to bring absolute values of accretion rates in line with those observed in UK-ADMS.

Lagrangian Residence Time in the Bay of Gdańsk, Baltic Sea

The impact of synoptic scale and mesoscale variability on the Lagrangian residence time (LRT) of the surface water in the Bay of Gdansk was investigated using the results from an eddy-resolving model. The computed LRT of 53–60 days was up to four times longer than the estimated flushing time reported by Witek et al. (2003). The highest residence times were those of Puck Bay and near the coast, below 50 m water depth, especially during the winter. These sites also had the highest annual mean in LRT. During the summer, when the level of biological activity is high, the LRT distribution was very heterogeneous and patchy, possibly due to the eddy field and to variable wind forcing. Long-term run tracking of the inflowing water from the Vistula River showed a broad spectrum of tracer distribution. The potential impact of a much higher LRT on the near-coastal nitrogen cycle, coastal filter function and genetic differentiation is discussed, and the consequences for coastal zone management are considered. The incorporation of residence time into the Marine Strategy Framework Directive descriptors would result in an improved, unbiased evaluation of good environmental status.

Turbulent Diffusion and Eddy Scales in Rivers

The relationship between turbulent diffusion with Eulerian and Lagrangian scales of turbulence in natural flows is considered. An estimate of the depth-averaged Lagrangian time scale as a function of Eulerian scale is suggested. The vertical turbulent diffusion in open natural flows is studied. The mean velocity profile is described by a power law. On the assumption of flat flow under incomplete self-similarity of the global Reynolds number, a universal expression for the vertical transfer coefficient is obtained. This expression enables one to estimate the time and length of complete mixing using minimum experimental data. The coefficients of longitudinal and vertical diffusion in different rivers are compared with each other and the universal ratio of these coefficients is suggested.

Inhomogeneous Turbulent Dispersion across the Nearshore Induced by Surfzone Eddies

AbstractIn various oceanic regions, drifter-derived diffusivities reach a temporal maximum and subsequently decrease. Often, these are regions of inhomogeneous eddies, however, the effect of inhomogeneous turbulence on dispersion is poorly understood. The nearshore region (spanning from the surfzone to the inner shelf) also has strong cross-shore inhomogeneous turbulence. Nearshore Lagrangian statistics are estimated from drifter trajectories simulated with a wave-resolving two-dimensional Boussinesq model with random, normally incident, and directionally spread waves. The simulation is idealized and does not include other (wind, tidal, Coriolis) processes. The eddy field cross-shore inhomogeneity affects both the mean cross-shore drift and cross- and alongshore diffusivities. Short-time diffusivities are locally ballistic, and the mean drift is toward the eddy velocity variance maximum. The diffusivities reach a maximum and subsequently decrease, that is, are subdiffusive. The diffusivity maximum and time to maximum are parameterized taking into account the eddy field inhomogeneity. At long times, the cross- and alongshore diffusivities scale as t−1/2 and t−1/4, respectively, which is related to the offshore decay of the eddy intensity. A diffusion equation, with a space-dependent Fickian diffusivity related to the eddy velocity variance, reproduced the short-, intermediate-, and long-time behavior of the mean drift and cross-shore diffusivity. The small Middleton parameter, indicating fixed float dispersion, suggests the Eulerian time scale can parameterize the Lagrangian time scale in this region. Although this idealized simulation had no mean currents, and thus no shear dispersion or mixing suppression, inhomogeneous turbulence effects may be relevant in other regions such as the Antarctic Circumpolar Current (ACC) and western boundary current extensions.

An evaluation of a Lagrangian stochastic model for the assessment of odours

In this work, we propose and evaluate a one-particle Lagrangian stochastic model for the assessment of the concentration-fluctuation variance, which is often used to predict odour hours for example. The main challenge is to provide a suitable parameterisation for the concentration-variance dissipation time scale. While the particle mass is conserved along the particle trajectory, the concentration variance dissipates. The concentration variance in a Lagrangian particle model evolves according to the same Langevin equation used for the simulation of the turbulent dispersion resulting in a mean concentration field. The variance dissipation is taken into account via a decay term with a finite time scale. We herein discuss and compare different parameterisations of the variance-dissipation time scale against observations from a field experiment. It was found that best results are obtained prescribing the ratio between the concentration-variance dissipation time scale and the Lagrangian time scale, which depends on the travel time.

Application of adaptive time marching scheme in exponential basis function method in simulating solitary wave propagation

Solitary wave theories are frequently adopted to simulate the characteristics of tsunami and to predict the resulting run-up in a numerical and experimental manner. A mesh-free method based on exponential basis functions (EBFs) is implemented to simulate the generation and propagation of solitary waves generated by piston-type wave maker. The pressure form of the Euler equations in a Lagrangian formulation is considered in the method. One of the most significant issues in the numerical simulation of solitary wave is to stabilize the wave height during the propagation. A new stable Lagrangian time marching algorithm is introduced which applies an adaptive parameter based on pressure formulation for tracking free surface and to minimize the discrepancies. In order to evaluate the accuracy of this newly introduced scheme, the results of EBF numerical model is applied to wave propagation according to six solitary wave theories. The results indicate the advantage of the introduced adaptive Lagrangian time marching algorithm and the EBF method in simulating solitary wave propagation with pronounced accuracy, convenient performances and the least run time calculation with implications for predicting solitary wave run-up on the beach.

Observation of turbulent dispersion of artificially released SO<sub>2</sub> puffs with UV cameras

Abstract. In atmospheric tracer experiments, a substance is released into the turbulent atmospheric flow to study the dispersion parameters of the atmosphere. That can be done by observing the substance's concentration distribution downwind of the source. Past experiments have suffered from the fact that observations were only made at a few discrete locations and/or at low time resolution. The Comtessa project (Camera Observation and Modelling of 4-D Tracer Dispersion in the Atmosphere) is the first attempt at using ultraviolet (UV) camera observations to sample the three-dimensional (3-D) concentration distribution in the atmospheric boundary layer at high spatial and temporal resolution. For this, during a three-week campaign in Norway in July 2017, sulfur dioxide (SO2), a nearly passive tracer, was artificially released in continuous plumes and nearly instantaneous puffs from a 9 m high tower. Column-integrated SO2 concentrations were observed with six UV SO2 cameras with sampling rates of several hertz and a spatial resolution of a few centimetres. The atmospheric flow was characterised by eddy covariance measurements of heat and momentum fluxes at the release mast and two additional towers. By measuring simultaneously with six UV cameras positioned in a half circle around the release point, we could collect a data set of spatially and temporally resolved tracer column densities from six different directions, allowing a tomographic reconstruction of the 3-D concentration field. However, due to unfavourable cloudy conditions on all measurement days and their restrictive effect on the SO2 camera technique, the presented data set is limited to case studies. In this paper, we present a feasibility study demonstrating that the turbulent dispersion parameters can be retrieved from images of artificially released puffs, although the presented data set does not allow for an in-depth analysis of the obtained parameters. The 3-D trajectories of the centre of mass of the puffs were reconstructed enabling both a direct determination of the centre of mass meandering and a scaling of the image pixel dimension to the position of the puff. The latter made it possible to retrieve the temporal evolution of the puff spread projected to the image plane. The puff spread is a direct measure of the relative dispersion process. Combining meandering and relative dispersion, the absolute dispersion could be retrieved. The turbulent dispersion in the vertical is then used to estimate the effective source size, source timescale and the Lagrangian integral time. In principle, the Richardson–Obukhov constant of relative dispersion in the inertial subrange could be also obtained, but the observation time was not sufficiently long in comparison to the source timescale to allow an observation of this dispersion range. While the feasibility of the methodology to measure turbulent dispersion could be demonstrated, a larger data set with a larger number of cloud-free puff releases and longer observation times of each puff will be recorded in future studies to give a solid estimate for the turbulent dispersion under a variety of stability conditions.

Quasi-Experimental Determination of Turbulent Dispersion Parameters for Different Stability Conditions from a Tall Micrometeorological Tower

AbstractTurbulent wind data measured by sonic anemometers installed at various heights on a 140-m-tall micrometeorological tower located at a coastal site are used to obtain vertical profiles of the velocity standard deviations σi, Lagrangian decorrelation local time scales TLi, and eddy diffusivities Kα for distinct stability conditions. The novelty of the study lies in the use of turbulent data directly measured over the extension of the atmospheric surface layer at a coastal site for that purpose. Furthermore, the approach employs the Hilbert–Huang transform to determine the wind energy spectral peak frequencies. These are applied to the asymptotic spectral equation from Taylor statistical diffusion theory to obtain the turbulent dispersion parameters, which are shown to generally agree well with those provided by a classical autocorrelation approach. For neutral and stable situations the vertical profiles of momentum eddy diffusivities agree well with those derived from the spectral and autocorrelation method. Additionally, the turbulent integral time scales and eddy diffusivities determined by the method at a coastal location are found to overestimate those predicted from analytical expressions based on continental field observations. The turbulence parameters found are suitable to be employed in air pollution dispersion models.

Eddy Lifetime, Number, and Diffusivity and the Suppression of Eddy Kinetic Energy in Midwinter

Abstract The wintertime evolution of the North Pacific storm track appears to challenge classical theories of baroclinic instability, which predict deeper extratropical cyclones when baroclinicity is highest. Although the surface baroclinicity peaks during midwinter, and the jet is strongest, eddy kinetic energy (EKE) and baroclinic conversion rates have a midwinter minimum over the North Pacific. This study investigates how the reduction in EKE translates into a reduction in eddy potential vorticity (PV) and heat fluxes via changes in eddy diffusivity. Additionally, it augments previous observations of the midwinter storm-track evolution in both hemispheres using climatologies of tracked surface cyclones. In the North Pacific, the number of surface cyclones is highest during midwinter, while the mean EKE per cyclone and the eddy lifetime are reduced. The midwinter reduction in upper-level eddy activity hence is not associated with a reduction in surface cyclone numbers. North Pacific eddy diffusivities exhibit a midwinter reduction at upper levels, where the Lagrangian decorrelation time is shortest (consistent with reduced eddy lifetimes) and the meridional parcel velocity variance is reduced (consistent with reduced EKE). The resulting midwinter reduction in North Pacific eddy diffusivities translates into an eddy PV flux suppression. In contrast, in the North Atlantic, a milder reduction in the decorrelation time is offset by a maximum in velocity variance, preventing a midwinter diffusivity minimum. The results suggest that a focus on causes of the wintertime evolution of Lagrangian decorrelation times and parcel velocity variance will be fruitful for understanding causes of seasonal storm-track variations.

Quantification of Lagrangian travel time statistics under nonstationary random groundwater flow conditions

AbstractThis study presents a Lagrangian description of advection dominated field‐scale transport of inert solutes, namely, the statistics of advective solute travel time and trajectory, in nonstationary random groundwater flow fields. The vertical flow field is generated by the effects of changes in total stress applied in fluid‐saturated heterogeneous deformable porous media. The Lagrangian statistics are developed directly from the velocity statistics based on the application of stochastic methodology such as the representation theorem and Fourier–Stieltjes integral representation. The focus of our numerical evaluation is placed on the influence of the soil compressibility on the Lagrangian statistics. The prediction on the statistics of solute travel time and trajectory may serve as the basic input to the environmental risk assessment.

Numerical computation of the transport matrix in toroidal plasma with a stochastic magnetic field

A new numerical method, based on integrating along the full orbit of guiding centers, to compute the transport matrix is realized. The method is successfully applied to compute the phase-space diffusion tensor of passing electrons in a tokamak with a stochastic magnetic field. The new method also computes the Lagrangian correlation function, which can be used to evaluate the Lagrangian correlation time and the turbulence correlation length. For the case of the stochastic magnetic field, we find that the order of magnitude of the parallel correlation length can be estimated by qR0, as expected previously.

Single-particle dispersion in stably stratified turbulence

We present models for single-particle dispersion in vertical and horizontal directions of stably stratified flows. The model in the vertical direction is based on the observed Lagrangian spectrum of the vertical velocity, while the model in the horizontal direction is a combination of a continuous-time eddy-constrained random walk process with a contribution to transport from horizontal winds. Transport at times larger than the Lagrangian turnover time is not universal and dependent on these winds. The models yield results in good agreement with direct numerical simulations of stratified turbulence, for which single-particle dispersion differs from the well studied case of homogeneous and isotropic turbulence.

Verification of a One-Dimensional Model of $$\hbox {CO}_{2}$$ CO 2 Atmospheric Transport Inside and Above a Forest Canopy Using Observations at the Norunda Research Station

A model of (Formula presented.) atmospheric transport in vegetated canopies is tested against measurements of the flow, as well as (Formula presented.) concentrations at the Norunda research station located inside a mixed pine–spruce forest. We present the results of simulations of wind-speed profiles and (Formula presented.) concentrations inside and above the forest canopy with a one-dimensional model of profiles of the turbulent diffusion coefficient above the canopy accounting for the influence of the roughness sub-layer on turbulent mixing according to Harman and Finnigan (Boundary-Layer Meteorol 129:323–351, 2008; hereafter HF08). Different modelling approaches are used to define the turbulent exchange coefficients for momentum and concentration inside the canopy: (1) the modified HF08 theory—numerical solution of the momentum and concentration equations with a non-constant distribution of leaf area per unit volume; (2) empirical parametrization of the turbulent diffusion coefficient using empirical data concerning the vertical profiles of the Lagrangian time scale and root-mean-square deviation of the vertical velocity component. For neutral, daytime conditions, the second-order turbulence model is also used. The flexibility of the empirical model enables the best fit of the simulated (Formula presented.) concentrations inside the canopy to the observations, with the results of simulations for daytime conditions inside the canopy layer only successful provided the respiration fluxes are properly considered. The application of the developed model for radiocarbon atmospheric transport released in the form of (Formula presented.) is presented and discussed. (Less)

A transient model for nozzle clogging

This model has been developed for transient simulation of clogging (also called as fouling) in submerged entry nozzle (SEN) during continuous casting. Three major steps of the clogging have been taken into account: (a) transport of non-metallic inclusions (NMIs) by turbulent melt flow towards the SEN wall; (b) interactions between melt and wall, and the adhesion of the NMI on the wall; (c) formation and growth of the clog by NMI deposition. The computational domain is divided into bulk and near-wall regions. An Eulerian-Lagrangian approach is employed to calculate the transport of NMIs by the turbulent flow; a stochastic near-wall model is adopted to trace particles in the turbulent boundary layer (near-wall region). The early stage of clogging is modeled by the dynamical change in wall roughness, while the late stage of the clogging is modeled by building a layer of porous clog region from the wall. This layer is called as ‘clog’, and it continues to grow by attaching more NMI particles. To evaluate the model, a laboratory experiment, which was designed to study the clogging of SEN during steel continuous casting, is simulated. It is verified that the model can reproduce the experiment: the calculated clogged section of the nozzle is qualitatively comparable with as-clogged sections in laboratory experiments; the calculated mass flow rate through the nozzle agrees with the experimentally-monitored result as well. New knowledge is obtained. (1) Clogging is a transient process interacting with the melt flow, and it includes the initial coverage of the nozzle wall with deposited particles, the evolution of a bulged clog front, and then the development of a branched structure. (2) Clogging is a stochastic and self-accelerating process. Finally, model capabilities/limitations, uncertainties for choosing the modeling parameters such as mesh size, Lagrangian time scale, the correction factor in the interpolation of clog permeability are studied and discussed.