Eulerian Velocity Research Articles

CFD-DEM Investigation on van der Waals Force in Gas-Solid Bubbling Fluidized Beds

Effect of interparticle force on the hydrodynamics of gas-solid fluidized beds was investigated using the combined method of computational fluid dynamics and discrete element method (CFD-DEM). The cohesive force between particles was considered to follow the van der Waals equation form. The model was validated by experimental results from literature in terms of bed voidage distribution and Eulerian solid velocity field. The results revealed that the incorporated model can satisfactorily predict the hydrodynamics of the fluidized bed in the presence of interparticle forces. Effect of interparticle force on the bubble rise characteristics, such as bubble stability, bubble diameter and bubble velocity, was investigated. It was shown that the emulsion voidage increases with increasing the interparticle force in the bed and it can hold more gas inside its structure. In addition, by increasing the interparticle force, size of bubbles and rise velocity of bubbles increase while the average velocity of particles decreases.

Evaluation of surface Lagrangian transport barriers in the Gulf of Trieste

The present work aims to detect Lagrangian transport barriers in the Gulf of Trieste by means of Lyapunov-exponent approach and tensorlines of the Cauchy-Green tensor. Lagrangian Coherent Structures (LCSs) are calculated employing 2D surface velocity fields measured by the coastal radars of the TOSCA EU research project (Tracking Oil Spills & Coastal Awareness Network). Moreover, surface drifters were deployed during the project. Comparisons between Eulerian velocity of HF-radar fields and Lagrangian velocity of drifters are carried out alongside single-particle tracking reliability. In particular, the possible influence of the data gaps in the HF-radar fields have been carefully considered. LCSs have proven to be robust against the quality of the starting HF-radar fields, leading to helpful insights in drifter positions. Indeed, after 24-hour integration the observed position of the drifter is approximately 1.5 km far from the nearest LCS, while a standard approach based on single-particle computations leads to larger errors (up to 5–7 km). However, such result must be properly interpreted taking into account the elongated nature of LCSs. A comparison between two common diagnostic tools of Lagrangian barriers is performed: Finite-Time and Finite-Size Lyapunov Exponent fields are compared in order to assess whether the patterns detected by the two measures are comparable. Finally, a joint analysis between LCSs and single-particle tracking is carried out and the results suggest that it would be desirable to couple these two approaches in real applications.

Multimaterial Eulerian finite element formulation for pressure‐sensitive adhesives

SummaryA temperature‐dependent visco‐hyperelastic formulation is proposed based on a Eulerian finite element method for large‐deformation and multimaterial problems, which is pertinent in the application to pressure‐sensitive adhesives. All the basic equations are computed in the Eulerian description because it allows arbitrarily large deformations. This formulation employs Simo's finite‐strain viscoelastic model, where hyperelasticity is modeled as a novel strain‐energy function of the left Cauchy‐Green deformation tensor. The left Cauchy‐Green deformation tensor is temporally updated from the Eulerian velocity field. Temperature dependence is described with the time‐temperature superposition principle. To validate the proposed approach, we simulated tests of uniaxial tension with various tensile speeds and temperature conditions and performed a steel‐ball‐drop test on an acrylic pressure‐sensitive adhesive.

Mechanisms of dispersion in a porous medium

This paper studies the mechanisms of dispersion in the laminar flow through the pore space of a three-dimensional porous medium. We focus on preasymptotic transport prior to the asymptotic hydrodynamic dispersion regime, in which solute motion may be described by the average flow velocity and a hydrodynamic dispersion coefficient. High-performance numerical flow and transport simulations of solute breakthrough at the outlet of a sand-like porous medium evidence marked deviations from the hydrodynamic dispersion paradigm and identify two distinct regimes. The first regime is characterised by a broad distribution of advective residence times in single pores. The second regime is characterised by diffusive mass transfer into low-velocity regions in the wake of solid grains. These mechanisms are quantified systematically in the framework of a time-domain random walk for the motion of marked elements (particles) of the transported material quantity. Particle transitions occur over the length scale imprinted in the pore structure at random times given by heterogeneous advection and diffusion. Under globally advection-dominated conditions, i.e., Péclet numbers larger than 1, particles sample the intrapore velocities by diffusion and the interpore velocities through advection. Thus, for a single transition, particle velocities are approximated by the mean pore velocity. In order to quantify this advection mechanism, we develop a model for the statistics of the Eulerian velocity magnitude based on Poiseuille’s law for flow through a single pore and for the distribution of mean pore velocities, both of which are linked to the distribution of pore diameters. Diffusion across streamlines through immobile zones in the wake of solid grains gives rise to exponentially distributed residence times that decay on the diffusion time over the pore length. The trapping rate is determined by the inverse diffusion time. This trapping mechanism is represented by a compound Poisson process conditioned on the advective residence time in the proposed time-domain random walk approach. The model is parameterised with the characteristics of the porous medium under consideration and captures both preasymptotic regimes. Macroscale transport is described by an integro-differential equation for solute concentration, whose memory kernels are given in terms of the distribution of mean pore velocities and trapping times. This approach quantifies the physical non-equilibrium caused by a broad distribution of mass transfer time scales, both advective and diffusive, on the representative elementary volume (REV). Thus, while the REV indicates the scale at which medium properties like porosity can be uniquely defined, this does not imply that transport can be characterised by hydrodynamic dispersion.

Energy dissipation statistics along the Lagrangian trajectories in three-dimensional turbulent flows

Energy dissipation rate is relevant in the turbulent phenomenology theory, such as the classical Kolmogorov 1941 and 1962 refined similarity hypothesis. However, it is extremely difficult to retrieve experimentally or numerically. In this paper, the full energy dissipation, its proxy and the pseudo-energy dissipation rate along the Lagrangian trajectories in the three-dimensional turbulent flows are examined by using a state-of-art high resolution direct numerical simulation database with a Reynolds number Re λ = 400. It is found that the energy dissipation proxy e P is more correlated with the full energy dissipation rate e. The corresponding correlation coefficient ρ between the velocity gradient and e shows a Gaussian distribution. Furthermore, the coarse-grained dissipation rate is considered. The cross correlation ρ is found to be increased with the increasing of the scale τ. Finally, the hierarchical structure is extracted for the full energy dissipation rate, its proxy and the pseudo one. The results show a power-law behavior in the inertial range 10 ≤τ/τ η ≤ 100. The experimental scaling exponent of the full energy dissipation rate is found to be h L =0.69, agrees very well with the one found for the Eulerian velocity. The experimental values for e P and e S are around h L = 0.78, implying a more intermittent Lagrangian turbulence. Therefore, the intermittency parameter provided by e P and e S will be biased.

Impact of local diffusion on macroscopic dispersion in three-dimensional porous media

While macroscopic longitudinal and transverse dispersion in three-dimensional porous media has been simulated previously mostly under purely advective conditions, the impact of diffusion on macroscopic dispersion in 3D remains an open question. Furthermore, both in 2D and 3D, recurring difficulties have been encountered due to computer limitation or analytical approximation. In this work, we use the Lagrangian velocity covariance function and the temporal derivative of second-order moments to study the influence of diffusion on dispersion in highly heterogeneous 2D and 3D porous media. The first approach characterizes the correlation between the values of Eulerian velocity components sampled by particles undergoing diffusion at two times. The second approach allows the estimation of dispersion coefficients and the analysis of their behaviours as functions of diffusion. These two approaches allowed us to reach new results. The influence of diffusion on dispersion seems to be globally similar between highly heterogeneous 2D and 3D porous media. Diffusion induces a decrease in the dispersion in the direction parallel to the flow direction and an increase in the dispersion in the direction perpendicular to the flow direction. However, the amplification of these two effects with the permeability variance is clearly different between 2D and 3D. For the direction parallel to the flow direction, the amplification is more important in 3D than in 2D. It is reversed in the direction perpendicular to the flow direction.

Kullback-Leibler divergence measure of intermittency: Application to turbulence.

For generic systems exhibiting power law behaviors, and hence multiscale dependencies, we propose a simple tool to analyze multifractality and intermittency, after noticing that these concepts are directly related to the deformation of a probability density function from Gaussian at large scales to non-Gaussian at smaller scales. Our framework is based on information theory and uses Shannon entropy and Kullback-Leibler divergence. We provide an extensive application to three-dimensional fully developed turbulence, seen here as a paradigmatic complex system where intermittency was historically defined and the concepts of scale invariance and multifractality were extensively studied and benchmarked. We compute our quantity on experimental Eulerian velocity measurements, as well as on synthetic processes and phenomenological models of fluid turbulence. Our approach is very general and does not require any underlying model of the system, although it can probe the relevance of such a model.

Simulation of Finite-Size Particles in Turbulent Flows Using the Lattice Boltzmann Method

Particle laden turbulent flows occur in a variety of industrial applications. While the numerical simulation of such flows has seen significant advances in recent years, it still remains a challenging problem. Many studies investigated the rheology of dense suspensions in laminar flows as well as the dynamics of point-particles in turbulence. Here we will present results on the development of numerical algorithms, based on the lattice Boltzmann method, suitable for the study of suspensions of finite-size particles under turbulent flow conditions. The turbulent flow is modeled by the lattice Boltzmann method, and the interaction between particles and carrier fluid is modeled using the bounce-back rule. Direct contact and lubrication force models for particle-particle interactions and particle-wall interaction are taken into account to allow for a full four-way coupled interaction. The accuracy and robustness of the method is discussed by validating the velocity profile in turbulent pipe flow, the sedimentation velocity of spheres in duct flow and the resistance functions of approaching particles. Preliminary results from the turbulent pipe flow simulations with particles show that the angular and axial velocities of the particles are scattered around values of mean axial velocity and shear rate obtained from the Eulerian velocity field.

Motion estimation under location uncertainty for turbulent fluid flows

In this paper, we propose a novel optical flow formulation for estimating two-dimensional velocity fields from an image sequence depicting the evolution of a passive scalar transported by a fluid flow. This motion estimator relies on a stochastic representation of the flow allowing to incorporate naturally a notion of uncertainty in the flow measurement. In this context, the Eulerian fluid flow velocity field is decomposed into two components: a large-scale motion field and a small-scale uncertainty component. We define the small-scale component as a random field. Subsequently , the data term of the optical flow formulation is based on a stochastic transport equation, derived from the formalism under location uncertainty proposed in Memin (2014) and Resseguier et al. (2017a). In addition, a specific regulariza-tion term built from the assumption of constant kinetic energy involves the very same diffusion tensor as the one appearing in the data transport term. Opposite to the classical motion estimators, this enables us to devise an optical flow method dedicated to fluid flows in which the regularization parameter has now a clear physical interpretation and can be easily estimated. Experimental evaluations are presented on both synthetic and real world image sequences. Results and comparisons indicate very good performance of the proposed formulation for turbulent flow motion estimation.

Laboratory experiment on the 3D tide-induced Lagrangian residual current using the PIV technique

The 3D structure of the tide-induced Lagrangian residual current was studied using the particle image velocimetry (PIV) technique in a long shallow narrow tank in the laboratory. At the mouth of the tank, a wave generator was used to make periodic wave which represents the tide movement, and at the head of the tank, a laterally sloping topography with the length of one fifth of the water tank was installed, above which the tide-induced Lagrangian residual current was studied. Under the weakly nonlinear condition in the present experiment setup, the results show that the Lagrangian residual velocity (LRV) field has a three-layer structure. The residual current flows inwards (towards the head) in the bottom layer and flows outwards in the middle layer, while in the surface layer, it flows inwards along the shallow side of the sloping topography and outwards along the deep side. The depth-averaged and breadth-averaged LRV are also analyzed based on the 3D LRV observations. Our results are in good agreement with the previous experiment studies, the analytical solutions with similar conditions and the observational results in real bays. Moreover, the volume flux comparison between the Lagrangian and Eulerian residual currents shows that the Eulerian residual velocity violates the mass conservation law while the LRV truly represents the inter-tidal water transport. This work enriches the laboratory studies of the LRV and offers valuable references for the LRV studies in real bays.

A multi-time-step noise reduction method for measuring velocity statistics from particle tracking velocimetry

We present a method to improve the accuracy of velocity measurements for fluid flow or particles immersed in it, based on a multi-time-step approach that allows for cancellation of noise in the velocity measurements. Improved velocity statistics, a critical element in turbulent flow measurements, can be computed from the combination of the velocity moments computed using standard particle tracking velocimetry (PTV) or particle image velocimetry (PIV) techniques for data sets that have been collected over different values of time intervals between images. This method produces Eulerian velocity fields and Lagrangian velocity statistics with much lower noise levels compared to standard PIV or PTV measurements, without the need of filtering and/or windowing. Particle displacement between two frames is computed for multiple different time-step values between frames in a canonical experiment of homogeneous isotropic turbulence. The second order velocity structure function of the flow is computed with the new method and compared to results from traditional measurement techniques in the literature. Increased accuracy is also demonstrated by comparing the dissipation rate of turbulent kinetic energy measured from this function against previously validated measurements.

A highly scalable particle tracking algorithm using partitioned global address space (PGAS) programming for extreme-scale turbulence simulations

A new parallel algorithm utilizing a partitioned global address space (PGAS) programming model to achieve high scalability is reported for particle tracking in direct numerical simulations of turbulent fluid flow. The work is motivated by the desire to obtain Lagrangian information necessary for the study of turbulent dispersion at the largest problem sizes feasible on current and next-generation multi-petaflop supercomputers. A large population of fluid particles is distributed among parallel processes dynamically, based on instantaneous particle positions such that all of the interpolation information needed for each particle is available either locally on its host process or neighboring processes holding adjacent sub-domains of the velocity field. With cubic splines as the preferred interpolation method, the new algorithm is designed to minimize the need for communication, by transferring between adjacent processes only those spline coefficients determined to be necessary for specific particles. This transfer is implemented very efficiently as a one-sided communication, using Co-Array Fortran (CAF) features which facilitate small data movements between different local partitions of a large global array. The cost of monitoring transfer of particle properties between adjacent processes for particles migrating across sub-domain boundaries is found to be small. Detailed benchmarks are obtained on the Cray petascale supercomputer Blue Waters at the University of Illinois, Urbana–Champaign. For operations on the particles in a 81923 simulation (0.55 trillion grid points) on 262,144 Cray XE6 cores, the new algorithm is found to be orders of magnitude faster relative to a prior algorithm in which each particle is tracked by the same parallel process at all times. This large speedup reduces the additional cost of tracking of order 300 million particles to just over 50% of the cost of computing the Eulerian velocity field at this scale. Improving support of PGAS models on major compilers suggests that this algorithm will be of wider applicability on most upcoming supercomputers.

Responses of water environment to tidal flat reduction in Xiangshan Bay: Part II locally re-suspended sediment dynamics

Xiangshan Bay is a semi-enclosed bay in China, in which tidal flats have been substantially reclaimed to support the development of local economies and society over previous decades. The loss of tidal flats has led to changes of tides and locally suspended sediment in the bay. The effects of tidal flat reduction on locally suspended sediment dynamics was investigated using a numerical model forced by tidal data and calibrated by observed tidal elevation and currents. The model satisfactorily reproduces observed water levels, currents, and suspended sediment concentration in the estuary, and therefore is subsequently applied to analyze the impact of tidal flat reclamation on locally suspended sediment transport. After the loss of the tidal flats from 1963 to 2010, the suspended sediment concentrations (SSC) at the bottom boundary layer were reduced/increased in the outer bay/tidal flat areas due to weakened tidal currents. In the inner bay, the SSC values near the bottom level increased from 1963 to 2003 due to the narrowed bathymetry, and then decreased from 2003 to 2010 because of the reduced tidal prism. The model scenarios suggest that: (1) a reduction of tidal flat areas appears to be the main factor for enhancing the transport of sediments up-estuary, due to the increased Eulerian velocity and tidal pumping; (2) A reduction of tidal flat areas impacts on spatial and temporal SSC distribution: reducing the SSC values in the water areas due to the reduced current; and (3) a tidal flat reduction influences the net sediment fluxes: lessening the erosion and inducing higher/lower landward/seaward sediment transportation.

Eulerian spatial and temporal autocorrelations: assessment of Taylor's hypothesis and a model

ABSTRACTWe present measurements of Eulerian longitudinal velocity autocorrelations in homogeneous, isotropic, high-intensity (∼9%) free-stream turbulence behind an active grid. Spatial correlations are measured using particle image velocimetry as well as with two-point hot-wire anemometry (HWA), while temporal correlations are measured using HWA. The temporal correlations are transformed into spatial correlations by using Taylor's ‘frozen’ hypothesis with both the mean as well as instantaneous velocities. A model relating Eulerian spatial and temporal autocorrelations is also used for this purpose. The differences from the measured spatial correlation resulting from the use of Taylor's hypothesis on the temporal correlation is quantified; even at this moderately high level of turbulent intensity, the result from the use of the instantaneous velocity as convection velocity is practically indistinguishable from that obtained using the mean velocity. Use of the model produces a good agreement between the estimates of the spatial correlation function. A relation between Eulerian spatial and temporal integral scales is also derived.

Contaminant attenuation by shallow aquifer systems under steady flow

We present a framework for analyzing advection-dominated solute transport and transformation in aquifer systems of boreal catchments that are typically shallow and rest on crystalline bedrock. A methodology is presented for estimating tracer discharge based on particle trajectories from recharge to discharge locations and computing their first passage times assuming that the flow pattern is approximately steady-state. Transformation processes can be included by solving one-dimensional reactive transport with randomized water travel time as the independent variable; the distribution of the travel times incorporates morphological dispersion (due to catchment geometry/topography) as well as macro-dispersion (due to heterogeneity of underlying hydraulic properties). The implementation of the framework is illustrated for the well characterized coastal catchment of Forsmark (Sweden). We find that macro-dispersion has a notable effect on attenuation even though the morphological dispersion is significantly larger. Preferential flow on the catchment scale is found to be considerable with only 5% of the Eulerian velocities contributing to transport over the simulation period of 375 years. Natural attenuation is illustrated as a simple (linear decay) transformation process. Simulated natural attenuation can be estimated analytically reasonably well by using basic hydrological and structural information, the latter being the pathway length distribution and average aquifer depth to the bedrock.

Concentration and velocity statistics of inertial particles in upward and downward pipe flow

Three-dimensional particle tracking velocimetry is applied to particle-laden turbulent pipe flows at a Reynolds number of 10 300, based on the bulk velocity and the pipe diameter, for developed fluid flow and not fully developed flow of inertial particles, which favours assessment of the radial migration of the inertial particles. Inertial particles with Stokes number ranging from 0.35 to 1.11, based on the particle relaxation time and the radial-dependent Kolmogorov time scale, and a ratio of the root-mean-square fluid velocity to the terminal velocity of order 1 have been used. Core peaking of the concentration of inertial particles in up-flow and wall peaking in down-flow have been found. The difference in mean particle and Eulerian mean liquid velocity is found to decrease to approximately zero near the wall in both flow directions. Although the carrier fluid has all of the characteristics of the corresponding turbulent single-phase flow, the Reynolds stress of the inertial particles is different near the wall in up-flow. These findings are explained from the preferential location of the inertial particles with the aid of direct numerical simulations with the point-particle approach.

Geophysical flows under location uncertainty, Part I Random transport and general models

A stochastic flow representation is considered with the Eulerian velocity decomposed between a smooth large scale component and a rough small-scale turbulent component. The latter is specified as a random field uncorrelated in time. Subsequently, the material derivative is modified and leads to a stochastic version of the material derivative to include a drift correction, an inhomogeneous and anisotropic diffusion, and a multiplicative noise. As derived, this stochastic transport exhibits a remarkable energy conservation property for any realizations. As demonstrated, this pivotal operator further provides elegant means to derive stochastic formulations of classical representations of geophysical flow dynamics.

Unsteadily manipulating internal flow barriers

Typical flows contain internal flow barriers: specialised time-moving Lagrangian entities which demarcate distinct motions. Examples include the boundary between an oceanic eddy and a nearby jet, the edge of the Antarctic circumpolar vortex or the interface between two fluids which are to be mixed together in an microfluidic assay. The ability to control the locations of these barriers in a user-specified time-varying (unsteady) way can profoundly impact fluid transport between the coherent structures which are separated by the barriers. A condition on the unsteady Eulerian velocity required to achieve this objective is explicitly derived, thereby solving an ‘inverse Lagrangian coherent structure’ problem. This is an important first step in developing flow-barrier control in realistic flows, and in providing a postprocessing tool for observational/experimental velocity data. The excellent accuracy of the method is demonstrated using the Kelvin–Stuart cats-eyes flow and the unsteady double gyre, utilising finite-time Lyapunov exponents.

Anomalous transport in disordered fracture networks: Spatial Markov model for dispersion with variable injection modes

We investigate tracer transport on random discrete fracture networks that are characterized by the statistics of the fracture geometry and hydraulic conductivity. While it is well known that tracer transport through fractured media can be anomalous and particle injection modes can have major impact on dispersion, the incorporation of injection modes into effective transport modeling has remained an open issue. The fundamental reason behind this challenge is that—even if the Eulerian fluid velocity is steady—the Lagrangian velocity distribution experienced by tracer particles evolves with time from its initial distribution, which is dictated by the injection mode, to a stationary velocity distribution. We quantify this evolution by a Markov model for particle velocities that are equidistantly sampled along trajectories. This stochastic approach allows for the systematic incorporation of the initial velocity distribution and quantifies the interplay between velocity distribution and spatial and temporal correlation. The proposed spatial Markov model is characterized by the initial velocity distribution, which is determined by the particle injection mode, the stationary Lagrangian velocity distribution, which is derived from the Eulerian velocity distribution, and the spatial velocity correlation length, which is related to the characteristic fracture length. This effective model leads to a time-domain random walk for the evolution of particle positions and velocities, whose joint distribution follows a Boltzmann equation. Finally, we demonstrate that the proposed model can successfully predict anomalous transport through discrete fracture networks with different levels of heterogeneity and arbitrary tracer injection modes.

On the topology of wall turbulence in physical space

The topological features dominating the different sublayers of the near wall turbulence are revisited through direct numerical simulations performed in large computational domains up to the Karman number of 1100. The averages of the invariants of the velocity gradient tensor, conditioned by fixed magnitudes of the velocity components, are adequately performed. In the physical space, the dominating topology in the viscous sublayer is the stable focus stretching cohabiting with the unstable node/saddle/saddle points. The stable and unstable focuses mostly populate the inner layer with equal importance, while the former become predominant in the meso and outer layers. Eddies within the low speed streaks with intense streamwise negative velocity of very high shear stress are located near the zero-discriminant facets. The enstrophy and straining prevails the stable focus stretching and the unstable compressing, respectively. The Eulerian velocity field conditioned by a given topology seen by an observer moving with the local velocity is also considered. It is found that the wall normal velocity marks the stretching line of the unstable saddles and the compressing line of the stable nodes/saddles/saddles. The conditional velocity fields linked to the unstable focuses compressing and stable focuses stretching are closely similar.