Eulerian Data Research Articles

Topology of two-dimensional turbulent flows of dust and gas

We perform direct numerical simulations (DNS) of passive heavy inertial particles (dust) in homogeneous and isotropic two-dimensional turbulent flows (gas) for a range of Stokes number, ${\rm St} < 1$, using both Lagrangian and Eulerian approach (with a shock-capturing scheme). We find that: The dust-density field in our Eulerian simulations have the same correlation dimension $d_2$ as obtained from the clustering of particles in the Lagrangian simulations for ${\rm St} < 1$; The cumulative probability distribution function of the dust-density coarse-grained over a scale $r$ in the inertial range has a left-tail with a power-law fall-off indicating presence of voids; The energy spectrum of the dust-velocity has a power-law range with an exponent that is same as the gas-velocity spectrum except at very high Fourier modes; The compressibility of the dust-velocity field is proportional to ${\rm St}^2$. We quantify the topological properties of the dust-velocity and the gas-velocity through their gradient matrices, called $\mathcal{A}$ and $\mathcal{B}$, respectively. The topological properties of $\mathcal{B}$ are the same in Eulerian and Lagrangian frames only if the Eulerian data are weighed by the dust-density -- a correspondence that we use to study Lagrangian properties of $\mathcal{A}$. In the Lagrangian frame, the mean value of the trace of $\mathcal{A} \sim - \exp(-C/{\rm St}$, with a constant $C\approx 0.1$. The topology of the dust-velocity fields shows that as ${\rm St} increases the contribution to negative divergence comes mostly from saddles and the contribution to positive divergence comes from both vortices and saddles. Compared to the Eulerian case, the density-weighed Eulerian case has less inward spirals and more converging saddles. Outward spirals are the least probable topological structures in both cases.

Characterizing Ventilation and Exposure in Street Canyons Using Lagrangian Particles

AbstractThe residence time measures the rate at which a pollutant escapes from a region of interest. Previous studies of urban ventilation have estimated the mean residence time from Eulerian data by assuming a spatially homogeneous pollutant field. Using a large-eddy simulation and a Lagrangian particle model, the residence and exposure times are calculated for an idealized street canyon in the skimming-flow region and a deep street canyon within a realistic urban area. For both domains, the mean residence time is on the order of a canyon circulation time scale, while the mean exposure time, which includes re-entrainment and characterizes the total time spent by a pollutant in a region of interest, is about 20% longer. Intensive quantities such as the Lagrangian visitation factor and return coefficient indicate that re-entrainment is modest. Probability distribution functions of the exposure and residence times are nearly exponential for both domains, in accord with pure diffusion and single-time-scale, vertical-exchange parameterizations. It is argued that, by analogy with Brownian motion, the mean residence and exposure times are set primarily by the mean circulation rather than the turbulence when the flow approximates that within a two-dimensional street canyon.

Traffic state estimation based on Eulerian and Lagrangian observations in a mesoscopic modeling framework

The paper proposes a model-based framework for estimating traffic states from Eulerian (loop) and/or Lagrangian (probe) data. Lagrangian-Space formulation of the LWR model adopted as the underlying traffic model provides suitable properties for receiving both Eulerian and Lagrangian external information. Three independent methods are proposed to address Eulerian data, Lagrangian data and the combination of both, respectively. These methods are defined in a consistent framework so as to be implemented simultaneously. The proposed framework has been verified on the synthetic data derived from the same underlying traffic flow model. Strength and weakness of both data sources are discussed. Next, the proposed framework has been applied to a freeway corridor. The validity has been tested using the data from a microscopic simulator, and the performance is satisfactory even for low rate of probe vehicles around 5%.

Assimilating Eulerian and Lagrangian data in traffic-flow models

Data assimilation of traffic flow remains a challenging problem. One difficulty is that data come from different sources ranging from stationary sensors and camera data to GPS and cell phone data from moving cars. Sensors and cameras give information about traffic density, while GPS data provide information about the positions and velocities of individual cars. Previous methods for assimilating Lagrangian data collected from individual cars relied on specific properties of the underlying computational model or its reformulation in Lagrangian coordinates. These approaches make it hard to assimilate both Eulerian density and Lagrangian positional data simultaneously. In this paper, we propose an alternative approach that allows us to assimilate both Eulerian and Lagrangian data. We show that the proposed algorithm is accurate and works well in different traffic scenarios and regardless of whether ensemble Kalman or particle filters are used. We also show that the algorithm is capable of estimating parameters and assimilating real traffic observations and synthetic observations obtained from microscopic models.

Turbulence: Does Energy Cascade Exist?

To answer the question whether a cascade of energy exists or not in turbulence, we propose a set of correlation functions able to test if there is an irreversible transfert of energy, step by step, from large to small structures. These tests are applied to real Eulerian data of a turbulent velocity flow, taken in the wind grid tunnel of Modane, and also to a prototype model equation for wave turbulence. First we demonstrate the irreversible character of the flow by using multi-time correlation function at a given point of space. Moreover the unexpected behavior of the test function leads us to connect irreversibility and finite time singularities (intermittency). Secondly we show that turbulent cascade exists, and is a dynamical process, by using a test function depending on time and frequency. The cascade shows up only in the inertial domain where the kinetic energy is transferred more rapidly (on average) from the wavenumber $k_{1}$ to $k_{2}$ than from $k_{1}$ to $k'_{2}$ larger than $k_{2}$.

Resolving Temporal Variations in Data-Driven Flow Models Constructed by Motion Tomography

Abstract: Modeling and predicting ocean flow are great challenges in physical oceanography. To answer such challenges, mobile sensing platforms have been an effective tool for providing Lagrangian flow information. Such information is typically incorporated into ocean models using Lagrangian data assimilation which requires significant amount of computing power and time. Motion tomography (MT) constructs generic environmental models (GEMs) that combine computational ocean models with real-time data collected from mobile platforms to provide high-resolution predictions near the mobile platforms. MT employs Lagrangian data from mobile platforms to create a spatial map of flow in the region traversed by the mobile platforms. This paper extends the MT method to resolve the coupling between temporal variations and spatial variations in flow modeling. Along with Lagrangian data from a mobile sensor, Eulerian data are collected from a stationary sensor deployed in the region where the mobile sensor collects data. Assimilation of these two data sets into GEMs introduces a nonlinear filtering problem. This paper presents the formulation of such nonlinear filtering problem and derives a filtering method for estimating flow model parameters. We analyze observability for the derived filters and demonstrate that the resulting method improves navigation accuracy for mobile platforms.

Variational Lagrangian data assimilation in open channel networks

AbstractThis article presents a data assimilation method in a tidal system, where data from both Lagrangian drifters and Eulerian flow sensors were fused to estimate water velocity. The system is modeled by first‐order, hyperbolic partial differential equations subject to periodic forcing. The estimation problem can then be formulated as the minimization of the difference between the observed variables and model outputs, and eventually provide the velocity and water stage of the hydrodynamic system. The governing equations are linearized and discretized using an implicit discretization scheme, resulting in linear equality constraints in the optimization program. Thus, the flow estimation can be formed as an optimization problem and efficiently solved. The effectiveness of the proposed method was substantiated by a large‐scale field experiment in the Sacramento‐San Joaquin River Delta in California. A fleet of 100 sensors developed at the University of California, Berkeley, were deployed in Walnut Grove, CA, to collect a set of Lagrangian data, a time series of positions as the sensors moved through the water. Measurements were also taken from Eulerian sensors in the region, provided by the United States Geological Survey. It is shown that the proposed method can effectively integrate Lagrangian and Eulerian measurement data, resulting in a suited estimation of the flow variables within the hydraulic system.

Prediction slurry reactor design under uncertainty using CFD model

Mechanically agitated reactors find wide range of applications for solid suspension and mixing in the chemical, biochemical, and mineral processing industries. Understanding the solids dynamics in these reactors is necessary to improve the design and operation of such reactors. Computational fluid dynamic (CFD) models are often useful in this regard, as it can provide significant insights into the flow and mixing of the phases involved. However, the model predictions need extensive evaluation with experimental results before they can be confidently used for the scale-up and optimization of large scale reactors. In the present work the predictive capabilities of CFD techniques as applied to solid-liquid stirred vessels are investigated. Suspensions of sand (diameter equal to 327 m) in water are studied. Eulerian-Granular multiphase model was simulated using FLUNT 6.3.26 to predict the slurry reactor design under uncertainty. The profiles obtained with the Eulerian-Granular approach coupled with the Eulerian, Granular models, and experimental data for comparison purposes. The present model provides an improvement of the predictions in the lower part of the vessel, with respect to the Eulerian model; while the same results can be observed in the rest of the tank where the solid concentration is lower. It seems that the interaction phenomena between the solid and the liquid phases and those among the solid particles do not vary appreciably for low solid concentrations, while at higher concentration some effects become noticeable. It was found that the quasi-steady state behavior of the sand in the mixing tank reached after 20 sec, also after 10 sec were the free surface of the static pressure will be growth until reach peremptory shape after 20 sec. The present model provides a proper representation for the solid distribution, by adopting particle drag coefficient.

A new assessment of the second-order moment of Lagrangian velocity increments in turbulence

The behaviour of the second-order Lagrangian structure functions on state-of-the-art numerical data both in two and three dimensions is studied. On the basis of a phenomenological connection between Eulerian space-fluctuations and the Lagrangian time-fluctuations, it is possible to rephrase the Kolmogorov 4/5-law into a relation predicting the linear (in time) scaling for the second-order Lagrangian structure function. When such a function is directly observed on current experimental or numerical data, it does not clearly display a scaling regime. A parameterisation of the Lagrangian structure functions based on Batchelor model is introduced and tested on data for 3d turbulence, and for 2d turbulence in the inverse cascade regime. Such parameterisation supports the idea, previously suggested, that both Eulerian and Lagrangian data are consistent with a linear scaling plus finite-Reynolds number effects affecting the small- and large timescales. When large-time saturation effects are properly accounted for, compensated plots show a detectable plateau already at the available Reynolds number. Furthermore, this parameterisation allows us to make quantitative predictions on the Reynolds number value for which Lagrangian structure functions are expected to display a scaling region. Finally, we show that this is also sufficient to predict the anomalous dependency of the normalised root mean squared acceleration as a function of the Reynolds number, without fitting parameters.

Real-Time Lagrangian Traffic State Estimator for Freeways

Freeway traffic state estimation and prediction are central components in real-time traffic management and information applications. Model-based traffic state estimators consist of a dynamic model for the state variables (e.g., a first- or second-order macroscopic traffic flow model), a set of observation equations relating sensor observations to the system state (e.g., the fundamental diagrams), and a data-assimilation technique to combine the model predictions with the sensor observations [e.g., the extended Kalman filter (EKF)]. Commonly, both process and observation models are formulated in Eulerian (space-time) coordinates. Recent studies have shown that this model can be formulated and solved more efficiently and accurately in Lagrangian (vehicle number-time) coordinates. In this paper, we propose a new model-based state estimator based on the EKF technique, in which the discretized Lagrangian Lighthill-Whitham and Richards (LWR) model is used as the process equation, and in which observation models for both Eulerian and Lagrangian sensor data (from loop detectors and vehicle trajectories, respectively) are incorporated. This Lagrangian state estimator is validated and compared with a Eulerian state estimator based on the same LWR model using an empirical microscopic traffic data set from the U.K. The results indicate that the Lagrangian estimator is significantly more accurate and offers computational and theoretical benefits over the Eulerian approach.

Lagrangian velocity autocorrelations in statistically steady rotating turbulence

Lagrangian statistics of passive tracers in rotating turbulence is investigated by particle tracking velocimetry. A confined and steadily forced turbulent flow is subjected to five different rotation rates. The PDFs of the velocity components clearly reveal the anisotropy induced by background rotation. Although the statistical properties of the horizontal turbulent flow field are approximately isotropic, in agreement with previously reported results by van Bokhoven and coworkers [Phys. Fluids 21, 096601 (2009)], the velocity component parallel to the (vertical) rotation axis gets strongly reduced (compared to the horizontal ones) while the rotation is increased. The auto-correlation coefficients of all three components are progressively enhanced for increasing rotation rates, although the vertical one shows a tendency to decrease for slow rotation rates. The decorrelation is approximately exponential. Lagrangian data compare favorably with previously reported Eulerian data for horizontal velocity components but show a different behavior for the vertical velocity components at higher rotation rates.

Lagrangian tools for the analysis of mixing in single‐phase and multiphase flow systems

AbstractMixing is intrinsically a Lagrangian process and, while Eulerian data are very important, Lagrangian information is necessary for its complete description. Lagrangian data can, in principle, be generated by using numerical simulations or experimental techniques based on Lagrangian tracking to provide the trajectories of fluid elements or particles. We present a set of theoretical and computational tools specifically developed for the analysis and validation of single‐phase and multiphase Lagrangian flow data obtained from tracer trajectories in mechanically agitated vessels. The implementation and power of these tools are demonstrated by analyzing a wide range of measurements acquired using the technique of positron emission particle tracking during mixing of Newtonian and non‐Newtonian fluids as well as mixing of concentrated solid–liquid suspensions. Finally, the accuracy of the raw Lagrangian data treatment developed here is illustrated by comparing with an Eulerian technique and showing very precise mass continuity for an agitated solid–liquid system. © 2011 American Institute of Chemical Engineers AIChE J, 2012

Closing the loop – Approaches to monitoring the state of the Arctic Mediterranean during the International Polar Year 2007–2008

During the 4th International Polar Year 2007–2009 (IPY), it has become increasingly obvious that we need to prepare for a new era in the Arctic. IPY occurred during the time of the largest retreat of Arctic sea ice since satellite observations started in 1979. This minimum in September sea ice coverage was accompanied by other signs of a changing Arctic, including the unexpectedly rapid transpolar drift of the Tara schooner, a general thinning of Arctic sea ice and a double-dip minimum of the Arctic Oscillation at the end of 2009. Thanks to the lucky timing of the IPY, those recent phenomena are well documented as they have been scrutinized by the international research community, taking advantage of the dedicated observing systems that were deployed during IPY. However, understanding changes in the Arctic System likely requires monitoring over decades, not years. Many IPY projects have contributed to the pilot phase of a future, sustained, observing system for the Arctic. We now know that many of the technical challenges can be overcome. The Norwegian projects iAOOS-Norway, POLEWARD and MEOP were significant ocean monitoring/research contributions during the IPY. A large variety of techniques were used in these programs, ranging from oceanographic cruises to animal-borne platforms, autonomous gliders, helicopter surveys, surface drifters and current meter arrays. Our research approach was interdisciplinary from the outset, merging ocean dynamics, hydrography, biology, sea ice studies, as well as forecasting. The datasets are tremendously rich, and they will surely yield numerous findings in the years to come. Here, we present a status report at the end of the official period for IPY. Highlights of the research include: a quantification of the Meridional Overturning Circulation in the Nordic Seas (“ the loop”) in thermal space, based on a set of up to 15-year-long series of current measurements; a detailed map of the surface circulation as well as characterization of eddy dispersion based on drifter data; transport monitoring of Atlantic Water using gliders; a view of the water mass exchanges in the Norwegian Atlantic Current from both Eulerian and Lagrangian data; an integrated physical–biological view of the ice-influenced ecosystem in the East Greenland Current, showing for instance nutrient-limited primary production as a consequence of decreasing ice cover for larger regions of the Arctic Ocean. Our sea ice studies show that the albedo of snow on ice is lower when snow cover is thinner and suggest that reductions in sea ice thickness, without changes in sea ice extent, will have a significant impact on the arctic atmosphere. We present up-to-date freshwater transport numbers for the East Greenland Current in the Fram Strait, as well as the first map of the annual cycle of freshwater layer thickness in the East Greenland Current along the east coast of Greenland, from data obtained by CTDs mounted on seals that traveled back and forth across the Nordic Seas. We have taken advantage of the real-time transmission of some of these platforms and demonstrate the use of ice-tethered profilers in validating satellite products of sea ice motion, as well as the use of Seagliders in validating ocean forecasts, and we present a sea ice drift product – significantly improved both in space and time – for use in operational ice-forecasting applications. We consider real-time acquisition of data from the ocean interior to be a vital component of a sustained Arctic Ocean Observing System, and we conclude by presenting an outline for an observing system for the European sector of the Arctic Ocean.

Lagrangian Prediction and Correlation Analysis with Eulerian Data

A velocity field obtained from the ocean surface by high-frequency radar is used to test Lagrangian prediction algorithms designed to evaluate the position of a particle given its initial position and observations of several other simultaneously released particles. The problem is motivated by oceanographic applications such as search and rescue operations and spreading pollutants, especially in coastal regions. The prediction skill is essentially determined by temporal and spatial covariances of the underlying velocity field. For this reason correlation analysis of both Lagrangian and Eulerian velocities was carried out. Space covariance functions and spectra of the velocity field are also presented to better illustrate statistical environments for the predictability studies. The results show that the regression prediction algorithm performs quite well on scales comparable with and higher than the velocity correlation scales.

Myocardial trajectory estimation for 2D cine DENSE MRI using thin plate splines

Background A critical stage in the quantification of myocardial strain from raw cine DENSE images is the transformation of Eulerian to Lagrangian displacement. Eulerian displacement, directly measured by DENSE, indicates the prior location of a myocardial element. Lagrangian displacement, generally easier to interpret by cardiologists and radiologists, tracks a myocardial element forward in time. Estimating Lagrangian displacement proves challenging, given the non-uniformly spaced displacement origins and the possibly low signal-to-noise ratio (SNR) of the Eulerian data. In this study, we investigated using the TPAS to accurately estimate myocardial strain from raw DENSE displacement fields.

Alternative Reference Frames in River System Science

Understanding environmental processes begins with mental conceptualizations of system components and interactions. Conceptualizing rivers begins with adopting one of two reference frames for observing movement: Eulerian, wherein the flux of objects is observed in a spatially bounded area, or Lagrangian, wherein specific objects are tracked through time. Mechanistic studies include Eulerian and Lagrangian data, with some negotiation of how much Eulerian and Lagrangian information may be needed to maximize the accuracy of understanding processes and the efficiency of data collection. Most studies rely on a presupposed reference frame, yet we suspect breakthroughs lurk in explicit alterations of presupposed reference frames. We analyze the importance of reference frames by contrasting the extent to which alternative reference frames have been used and combined in studies of sediment transport, fish migration, and river biogeochemistry. We show how adopting alternative or nonintuitive reference frames can fac...

Analytical approaches to investigating seabird–environment interactions: a review

A goal of seabird ecology is to relate the physiology, population dynamics, distribution and behaviour of seabirds with their biotic and abiotic environments. One of the most challenging aspects is to understand how seabirds interact with their environment when direct observations are not always possible or practical. In the present paper, we reviewed 218 published studies that exam- ined associations between seabird distribution, behaviour and their environment, in order to assess current trends, weaknesses and the future directions of research. Based on the number of publica- tions, it is evident that the field is growing rapidly and that methods for evaluating seabird distribu- tion are becoming increasingly more sophisticated and are changing from Eulerian (grid-like) to Lagrangian (particle-like) data types. This has been accompanied by a reduction in the spatial and temporal scale of observation, where, in most cases, no behavioural information is inferred from Lagrangian data; instead they are often used as if they were Eulerian data. In parallel, environmen- tal remote sensing is becoming more common; however, we did not record significant changes in the statistical approaches used to describe seabird distributions and used to link them with oceano- graphic variables. In particular, despite the spatially explicit nature of the data, spatial statistics have rarely been used. The vast majority of studies used environmental variables that described water masses (descriptive approach), whereas a few studies determined oceanographic features that enhance prey availability to seabirds (process-based approach). Future studies could enhance their ecological interpretation of seabird-environment interactions by making greater use of ad hoc statis- tical approaches that facilitate appropriate pattern detection (e.g. area-restricted searching pattern for birds, mesoscale patterns for environment). Furthermore, appropriate hypothesis testing and modelling that accounts for the spatially explicit, multiscale and multivariate nature of the interaction between seabirds and their habitats is recommended. Although quantitative methods currently exist (but are rarely used), further application could greatly improve our understanding of the processes linking seabird distribution to their environment.

Predicting dense nonaqueous phase liquid dissolution using a simplified source depletion model parameterized with partitioning tracers

Simulations of nonpartitioning and partitioning tracer tests were used to parameterize the equilibrium stream tube model (ESM) that predicts the dissolution dynamics of dense nonaqueous phase liquids (DNAPLs) as a function of the Lagrangian properties of DNAPL source zones. Lagrangian, or stream‐tube‐based, approaches characterize source zones with as few as two trajectory‐integrated parameters, in contrast to the potentially thousands of parameters required to describe the point‐by‐point variability in permeability and DNAPL in traditional Eulerian modeling approaches. The spill and subsequent dissolution of DNAPLs were simulated in two‐dimensional domains having different hydrologic characteristics (variance of the log conductivity field = 0.2, 1, and 3) using the multiphase flow and transport simulator UTCHEM. Nonpartitioning and partitioning tracers were used to characterize the Lagrangian properties (travel time and trajectory‐integrated DNAPL content statistics) of DNAPL source zones, which were in turn shown to be sufficient for accurate prediction of source dissolution behavior using the ESM throughout the relatively broad range of hydraulic conductivity variances tested here. The results were found to be relatively insensitive to travel time variability, suggesting that dissolution could be accurately predicted even if the travel time variance was only coarsely estimated. Estimation of the ESM parameters was also demonstrated using an approximate technique based on Eulerian data in the absence of tracer data; however, determining the minimum amount of such data required remains for future work. Finally, the stream tube model was shown to be a more unique predictor of dissolution behavior than approaches based on the ganglia‐to‐pool model for source zone characterization.

Simulated Lagrangian pathways between the Leeuwin Current System and the upper-ocean circulation of the southeast Indian Ocean

Abstract The Leeuwin Current System, along the west Australian coast (22°S–34°S), forms a unique but poorly understood eastern boundary regime in which tropical waters flow poleward. Here we depict the three-dimensional paths connecting this eastern boundary system with the upper-ocean large-scale circulation around Australia based on selected trajectories from an online numerical particle tracking performed during the 1993/1997 integration of the 0.28° Los Alamos National Laboratory Parallel Ocean Program model. The simulated trajectories reveal a wealth of details about the regional circulation that are difficult to understand from observed and model Eulerian data alone. They reveal links between the Leeuwin Current, Leeuwin Undercurrent, Eastern Gyral Current, and zonal flows within the Subtropical Gyre. New findings include: a remote tropical source of the Leeuwin Current in the equatorial Indian Ocean, via the South Java Current; inshore (along the southern part of the North West Shelf) and offshore routes in the Indo-Australian Basin feeding the Leeuwin Current; strong exchange between the Leeuwin Undercurrent and adjacent Subtropical Gyre through a series of near surface eastward jets and deeper westward jets; and the tropical origin of the Eastern Gyral Current as a recirculation of the South Equatorial Current. We propose a current schematic summarising the links between the meridional boundary flows off Western Australia and the larger-scale circulation.

Extinction and reignition in a diffusion flame: a direct numerical simulation study

The goal of this study is to provide a window into the physics of extinction and reignition via three-dimensional simulations of non-premixed combustion in isotropic decaying turbulence using one-step global reaction and neglecting density variations. Initially non-premixed fields of fuel and oxidant are developing in a turbulent field. Due to straining, the scalar dissipation rate is initially increasing and its fluctuations create extinguished regions on the stoichiometric surface. Later in the process, the stoichiometric surface again becomes uniformly hot. Besides using Eulerian data, this research applies flame element tracking and investigates the time history of individual points (‘flame elements’) along the stoichiometric surface. The main focus of the study is the discussion of the different scenarios of reignition. This paper identifies three major scenarios: independent flamelet scenario, reignition via edge (triple) flame propagation, and reignition through engulfment by a hot neighbourhood. The results give insight into the role different scenarios play in the reignition process, reveal the physical processes associated with each scenario, and provide the relative frequency of reignition for each scenario.