Eulerian Velocity Research Articles

Using Lagrangian coherent structures to analyze fluid mixing by cilia

Motivated by the desire to understand the fluid flow within the airway surface liquid of the lung, we consider the flow generated by a computational model of a motile, internally actuated cilium. The cilium, along with a mucus layer modeled by linear elastic elements, is coupled to a viscous, incompressible fluid. The evolution of this coupled system is captured using an immersed boundary method. The Eulerian velocity field computed on a grid is used to compute finite-time Lyapunov exponent fields, whose maximal ridges identify Lagrangian coherent structures (LCSs). The computed LCS uncovers a barrier that separates a recirculation region of fluid that remains near the beating cilium from fluid that is advected downstream. Moreover, periodic stretching and folding of this region gives rise to complex mixing. Flow structures around a cilium propelling a mucus layer are compared to flow structures around a cilium with no mucus load.

Scale-similarity model for Lagrangian velocity correlations in isotropic and stationary turbulence

A scale-similarity model for Lagrangian two-point, two-time velocity correlations (LVCs) in isotropic turbulence is developed from the Kolmogorov similarity hypothesis. It is a second approximation to the isocontours of LVCs, while the Smith-Hay model is only a first approximation. This model expresses the LVC by its space correlation and a dispersion velocity. We derive the analytical expression for the dispersion velocity from the Navier-Stokes equations using the quasinormality assumption. The dispersion velocity is dependent on enstrophy spectra and shown to be smaller than the sweeping velocity for the Eulerian velocity correlation. Therefore, the Lagrangian decorrelation process is slower than the Eulerian decorrelation process. The data from direct numerical simulation of isotropic turbulence support the scale-similarity model: the LVCs for different space separations collapse into a universal form when plotted against the separation axis defined by the model.

Improving SCIPUFF Dispersion Forecasts with NWP Ensembles

Abstract The relationships between atmospheric transport and dispersion (AT&D) plume uncertainty and uncertainties in the transporting wind fields are investigated using the Second-Order Closure, Integrated Puff (SCIPUFF) AT&D model driven by numerical weather prediction (NWP) meteorological fields. Modeled contaminant concentrations for episode 1 of the 1983 Cross-Appalachian Tracer Experiment (CAPTEX-83) are compared with recorded ground-level concentrations of the inert tracer gas C7F14. This study evaluates a Taylor-diffusion-based parameterization of dispersion uncertainty for SCIPUFF that uses Eulerian meteorological ensemble velocity statistics and a Lagrangian integral time scale as input. These values are diagnosed from NWP ensemble data. Individual simulations of the tracer release fail to reproduce some of the monitored surface concentrations of the tracer. The plumes that are predicted using the uncertainty model in SCIPUFF are broader, improving the overlap between the predicted and observed results. Augmenting the meteorological input to SCIPUFF with meteorological ensemble-uncertainty parameters therefore provides both a better estimate of the expected plume location and the relative uncertainties in the predicted concentrations than single deterministic forecasts. These results suggest that this new parameterization of NWP wind field uncertainty for dispersion may provide more sophisticated information that may benefit emergency response and decision making.

Eulerian–Lagrangian multiscale methods for solving scalar equations – Application to incompressible two-phase flows

The present article proposes a new hybrid Eulerian–Lagrangian numerical method, based on a volume particle meshing of the Eulerian grid, for solving transport equations. The approach, called Volume Of Fluid Sub-Mesh method (VOF-SM), has the advantage of being able to deal with interface tracking as well as advection–diffusion transport equations of scalar quantities. The Eulerian evolutions of a scalar field could be obtained on any orthogonal curvilinear grid thanks to the Lagrangian advection and a redistribution of particles on the Eulerian grid. The Eulerian concentrations result from the projection of the volume and scalar informations handled by the particles. The particle velocities are interpolated from the Eulerian velocity field. The VOF-SM method is validated on several scalar interface tracking and transport problems and is compared to existing schemes within the literature. It is finally coupled to a Navier–Stokes solver and applied to the simulation of two free-surface flows, i.e. the two-dimensional buckling of a viscous jet during the filling of a square mold and the three-dimensional dam-break flow in a tank.

Euler's fluid equations: Optimal control vs optimization

An optimization method used in image-processing (metamorphosis) is found to imply Euler's equations for incompressible flow of an inviscid fluid, without requiring that the Lagrangian particle labels exactly follow the flow lines of the Eulerian velocity vector field. Thus, an optimal control problem and an optimization problem for incompressible ideal fluid flow both yield the same Euler fluid equations, although their Lagrangian parcel dynamics are different. This is a result of the gauge freedom in the definition of the fluid pressure for an incompressible flow, in combination with the symmetry of fluid dynamics under relabeling of their Lagrangian coordinates. Similar ideas are also illustrated for SO ( N ) rigid body motion.

Resolution dependent relative dispersion statistics in a hierarchy of ocean models

In order to determine the effect of Eulerian spatial resolution on the two particle statistics of synthetic drifter trajectories, we examine a hierarchy of ocean models, starting from 2D turbulence simulations, progressing to idealized simulations of a buoyant coastal jet with ROMS, and finally to realistic HYCOM simulations of the Gulf Stream. In each case, particle dispersion at large time and space scales is found to be controlled by energetic meso-scale features of the flow that are relatively insensitive to the resolution of finer scale motions. In all cases, time-distance graphs given in terms of computed Finite Scale Lyapunov Exponents show an expected increase in the extent of exponential scaling with increasing spatial smoothing of the velocity field. The limiting value of the FSLE at small separation distances is found to scale remarkably well with the resolution of Eulerian velocity gradients as given by the average of positive Okubo–Weiss parameter values.

Investigation of penetrative convection in stratified fluids through 3D-PTV

Testing existing theories of flow and transport in stably stratified fluids when “penetrative convection” occurs requires experimental methods capable of fully three-dimensional descriptions of both the Eulerian velocity field and Lagrangian particle trajectories. Boundary layer experiments utilizing a photogrammetric three-camera three-dimensional Particle Tracking Velocimetry (3D-PTV) technique were performed on a laboratory model which simulated the evolution of the mixing layer in a stably stratified fluid body by heating a tank filled with water from below. A combination of image- and object space-based information was employed to establish the spatio-temporal correspondences between particle positions at consecutive time steps. An appropriate calibration procedure for the stereoscopic system provides the intrinsic and extrinsic parameters of the stereoscopic system to be used for determining the correspondence of points in the object or world reference frame and in the image reference frame. The photogrammetric principles used by 3D-PTV are described. First, the fundamental mathematical model of the collinearity condition and its extensions are explained. Then, the epipolar line intersection method built upon multicamera correspondences (structure-from-stereo) is discussed as well as the procedure of reconstruction of particle trajectories in a single step by conjunctly establishing the spatial and temporal correspondences between particle images. Tests on simulated data were performed as well as experimental data in order to ensure the method’s operability and robustness. The reconstructed velocity fields are compared to other laboratory investigations and field data.

Measuring segregation of inertial particles in turbulence by a full Lagrangian approach

Preferential concentration of inertial particles in turbulence is studied numerically by evaluating the Lagrangian compressibility of the particle velocity field using the "full Lagrangian method." This is compared with the "mesoscopic Eulerian particle velocity field" both in a direct numerical simulation of turbulence and in a synthetic flow field. We demonstrate that the Lagrangian method, in contrast to the Eulerian, accurately predicts the compressibility of the particle velocity field even when the latter is characterized by singularities. In particular we use the method to evaluate the growth rates of spatial moments of the particle number density which reflect the fractal structure of segregation and the occurrence of singularities.

Flow and travel time statistics in highly heterogeneous porous media

In this paper we present flow and travel time ensemble statistics based on a new simulation methodology, the adaptive Fup Monte Carlo method (AFMCM). As a benchmark case, we considered two‐dimensional steady flow in a rectangular domain characterized by multi‐Gaussian heterogeneity structure with an isotropic exponential correlation and lnK variance σY2 up to 8. Advective transport is investigated using the travel time framework where Lagrangian variables (e.g., velocity, transverse displacement, or travel time) depend on space rather than on time. We find that Eulerian and Lagrangian velocity distributions diverge for increasing lnK variance due to enhanced channeling. Transverse displacement is a nonnormal for all σY2 and control planes close to the injection area, but after xIY = 20 was found to be nearly normal even for high σY2. Travel time distribution deviates from the Fickian model for large lnK variance and exhibits increasing skewness and a power law tail for large lnK variance, the slope of which decreases for increasing distance from the source; no anomalous features are found. Second moment of advective transport is analyzed with respect to the covariance of two Lagrangian velocity variables: slowness and slope which are directly related to the travel time and transverse displacement variance, which are subsequently related to the longitudinal and transverse dispersion. We provide simple estimators for the Eulerian velocity variance, travel time variance, slowness, and longitudinal dispersivity as a practical contribution of this analysis. Both two‐parameter models considered (the advection‐dispersion equation and the lognormal model) provide relatively poor representations of the initial part of the travel time probability density function in highly heterogeneous porous media. We identify the need for further theoretical and experimental scrutiny of early arrival times, and the need for computing higher‐order moments for a more accurate characterization of the travel time probability density function. A brief discussion is presented on the challenges and extensions for which AFMCM is suggested as a suitable approach.

Exact relation between Eulerian and Lagrangian velocity increment statistics

We present a formal connection between Lagrangian and Eulerian velocity increment distributions which is applicable to a wide range of turbulent systems ranging from turbulence in incompressible fluids to magnetohydrodynamic turbulence. For the case of the inverse cascade regime of two-dimensional turbulence we numerically estimate the transition probabilities involved in this connection. In this context we are able to directly identify the processes leading to strongly non-Gaussian statistics for the Lagrangian velocity increments.

Two-fluid approach for direct numerical simulation of particle-laden turbulent flows at small Stokes numbers

A two-fluid approach is proposed for direct numerical simulation of particle-laden turbulent flows in two-way coupling where the particle Stokes number is small. An Eulerian velocity field is calculated for the particle phase through a truncated series expansion in terms of the velocity and acceleration of the fluid phase [M. R. Maxey, J. Fluid Mech. 174, 441 (1987)]. This expansion is valid for particles with a sufficiently small Stokes number defined as the ratio of particle time constant to the Kolmogorov time scale. The transport equation of the Eulerian concentration field of particles (particle volume fraction) is solved along with the fluid-phase equations for which the effect of the particles on the fluid phase is taken into account through source terms in the momentum equations. For the assessment purposes, particle-laden decaying isotropic turbulence is studied. The results obtained through the proposed two-fluid approach are compared against those obtained by the trajectory approach in which the particle equations are solved in the Lagrangian framework. It is shown that there is a good agreement between various fluid-phase statistics obtained by these approaches for different small Stokes numbers and mean particle concentrations.

Statistics of a mixed Eulerian–Lagrangian velocity increment in fully developed turbulence

We investigate the relationship between Eulerian and Lagrangian probability density functions obtained from numerical simulations of two-dimensional as well as three-dimensional turbulence. We show that in contrast to the structure functions of the Lagrangian velocity increment δτv(y)=u(x(y, τ), τ)- u(y, 0), where u(x, t) denotes the Eulerian velocity and x(y, t) the particle path initially starting at x(y, 0)=y, the structure functions of the velocity increment δτw(y)=u(x(y, τ), τ)- u(y, τ) exhibit a wide range of scaling behavior. Similar scaling indices are detected for the structure functions for particles diffusing in frozen turbulent fields. Furthermore, we discuss a connection to the scaling of Eulerian transversal structure functions.

Wave-induced setup of the mean surface over a sloping beach

A new theoretical approach for the wave-induced setup over a sloping beach is presented that takes into consideration the explicit variations of the surface waves due to bottom slope and viscosity. In this way, the wave forcing of the mean Lagrangian volume fluxes is calculated without assuming that the local depth is constant. The analysis is valid in the region outside the surf zone and is based on the shallow-water assumption. A novel approach for separating the viscous damping of the waves from the frictional damping of the mean flow is introduced, where the mean Eulerian velocity is applied in the bottom stress for the mean fluxes. In the case where the onshore Lagrangian mean transport is zero, a new formula is derived for the Eulerian mean free surface slope, in which the effects of bottom slope, viscous wave damping and frictional bottom drag on the mean flow are clearly identified. The analysis suggests that viscous damping of the waves and frictional dissipation of the Eulerian near-bed return flow could lead to setup outside the surf zone.

Mass transport in the Stokes edge wave

The Lagrangian mass transport in the Stokes progressive edge wave is obtained from the vertically integrated equations of momentum and mass, correct to second order in wave steepness. The cross-shore momentum balance is between the mean pressure at the sloping bottom, the radiation stress, and the pressure gradient due to the mean surface slope. In the alongshore direction, the effect of viscosity leads to a wave attenuation, and hence a radiation stress component. The frictional effect on the mean Eulerian motion is modeled through a turbulent bottom drag. The alongshore momentum balance is between the mean pressure gradient due to the surface slope, the radiation stress, and the turbulent drag on the mean Eulerian flow. It is shown that E/ y, where E is the total mean energy density for waves along the y-axis, is the wave-forcing term for the total mean Lagrangian momentum in the trapping region. This result is independent of the bottom slope angle. Vertically-averaged drift velocity components are obtained from the fluxes, divided by the local depth. Utilizing physical parameters relevant for field conditions, it appears the traditional Stokes drift in the Stokes edge wave is negligible compared to the mean Eulerian velocity component. The importance of this drift for the near-shore transport of effluents and suspended light sediments is discussed.

Stochastic analysis of transport in a combined heterogeneous vadose zone–groundwater flow system

First‐order analysis was used to investigate the effect of a few characteristics of the vadose zone and the solute source at the soil surface on the expected breakthrough curves (BTCs) of tracer solutes in a combined unsaturated‐saturated spatially heterogeneous formation with a three‐dimensional, statistically anisotropic structure. The analysis is based on a stochastic continuum representation of the stationary and nonstationary Eulerian velocity vectors in the unsaturated and saturated domains, respectively, and a general Lagrangian description of the solute mass flux in steady state flow. Results of the present analysis suggest that the length scales and the flow characteristics of the spatially heterogeneous, unsaturated zone that affect the travel time probability density function between the soil surface and the water table also affect the expected solute BTC at a control plane located downstream in the groundwater. The flow in the unsaturated zone might enhance the early arrival time and the spreading of the expected solute BTC in the groundwater, especially when the vertical extent of the unsaturated zone is relatively large; when the unsaturated zone consists of relatively dry, coarse‐textured, stratified soil material; and when the horizontal extent of the solute source in the direction of the mean groundwater flow is relatively large. The effect of the characteristics of the vadose zone and the solute source at the soil surface on the expected solute BTC in the groundwater decreases as the distance to the control plane increases and as the process of the solute breakthrough is more advanced. The relatively good agreement between the results of the present first‐order, stochastic analysis with the results of detailed, numerical analyses of flow and transport in a three‐dimensional heterogeneous, combined vadose zone–groundwater flow domain suggests that the first‐order stochastic analyses may be sufficiently reliable to show appropriate trends which might occur in realistic combined vadose zone–groundwater flow systems.

A closed model of fluctuating particle motion in a turbulent flow

Random particle motion in a turbulent and molecular velocity fluctuation field is considered. Using a spectral representation of the carrier-phase Eulerian velocity fluctuation correlations, a closed system of integral equations for calculating the carrier-phase velocity correlation along the particle trajectory and the particle Lagrangian velocity fluctuation correlation is obtained. Based on this system, the fluctuations of the particle parameters are analyzed. In the limiting case of a passive admixture, an estimate is found for the ratio of the integral Lagrangian and Eulerian time scales and the Kolmogorov constant for the Lagrangian structure function of the carrier-phase velocity fluctuations.

Structure and variability of the West Greenland Current in Summer derived from 6 repeat standard sections

Six historical sections across the West Greenland Current are examined. Three sections have been regularly occupied since the late 1950s, while the three southern ones have been taken since 1984. Significant variability is observed for the freshwater core of the coastal current on the shelf, with salinity varying by over 3 units between years. There is also significant variability in the shape and offshore position of the main shelf break front, leading to large variability in Eulerian velocities. Significant presence of Irminger Water is seen during the 1960s and the 2000s, being found right across the sections in recent years. Maximum mean transport relative to the 34.8 isohaline of 5.5 ± 3.9 Sv , relative to 700 db, for 1984–2005, is observed at the Cape Desolation section. Transports decrease to the north, with the majority of the exchange with the interior of the Labrador Sea occurring between Cape Desolation and Fylla Bank. Inter-decadal transport variability is observed at Fylla Bank while a decline in transports since peaks in the early 1990s is seen at Cape Farewell and Cape Desolation. Freshwater transport is largest at Cape Desolation, with a mean Summer transport of 60.2 ± 20.5 mSv . Freshwater transport increases slightly between Cape Farewell and Cape Desolation and we suggest it is related to local discharge by glaciers into Juliannehaab Bight, as well as the melting of sea-ice. We also find that years of high Greenland ice cap melt are consistently associated with years of high freshwater transport at Cape Desolation, suggesting a portion of the freshwater transport of the West Greenland Current may be associated with melt from the Greenland ice sheet. Finally, significantly enhanced freshwater transports (33 mSv at Cape Desolation compared to the long term mean) are seen in 2008, probably a signature of the record Arctic Ocean ice melt and export in 2007.

The Lagrangian stochastic model for estimating footprint and water vapor fluxes over inhomogeneous surfaces

This study investigated a two-dimensional Lagrangian stochastic dispersion model for estimating water vapor fluxes and footprint over homogeneous and inhomogeneous surfaces. Over the homogeneous surface, particle trajectories were computed from a 2-D Lagrangian model forced by Eulerian velocity statistics determined by Monin-Obukhov similarity theory (MOST). For an inhomogeneous surface, the velocity and atmospheric stability profiles were computed using a second-order Eulerian closure model, and these local profiles were then used to drive the Lagrangian model. The model simulations were compared with water vapor flux measurements carried out above an irrigated bare soil site and an irrigated potato site. The inhomogeneity involved a step change in surface roughness, humidity, and temperature. Good agreement between eddy-correlation-measured and Lagrangian-model-predicted water vapor fluxes was found for both sites. Hence, this analysis demonstrates the practical utility of second-order closure models in conjunction with Lagrangian analysis to estimate the scalar footprint in planar inhomogeneous flows.

Observations of Cross-Shelf Flow Driven by Cross-Shelf Winds on the Inner Continental Shelf

Abstract Six-yr-long time series of winds, waves, and water velocity from a cabled coastal observatory in 12 m of water reveal the separate dependence of the cross-shelf velocity profile on cross-shelf and along-shelf winds, waves, and tides. During small waves, cross-shelf wind is the dominant mechanism driving the cross-shelf circulation after tides and tidal residual motions are removed. The along-shelf wind does not drive a substantial cross-shelf circulation. During offshore winds, the cross-shelf circulation is offshore in the upper water column and onshore in the lower water column, with roughly equal and opposite volume transports in the surface and bottom layers. During onshore winds, the circulation is nearly the reverse. The observed profiles and cross-shelf transport in the surface layer during winter agree with a simple two-dimensional unstratified model of cross-shelf wind stress forcing. The cross-shelf velocity profile is more vertically sheared and the surface layer transport is stronger in summer than in winter for a given offshore wind stress. During large waves, the cross-shelf circulation is no longer roughly symmetric in the wind direction. For onshore winds, the cross-shelf velocity profile is nearly vertically uniform, because the wind- and wave-driven shears cancel; for offshore winds, the profile is strongly vertically sheared because the wind- and wave-driven shears have the same sign. The Lagrangian velocity profile in winter is similar to the part of the Eulerian velocity profile due to cross-shelf wind stress alone, because the contribution of Stokes drift to the Lagrangian velocity approximately cancels the contribution of waves to the Eulerian velocity.

A two-dimensional horizontal wave propagation and mud mass transport model

The present study offers a two-dimensional horizontal wave propagation and morphodynamic model for muddy coasts. The model can be applied on a general three-dimensional bathymetry of a soft muddy coast to calculate wave damping, fluid mud mass transport and resulting bathymetry change under wave actions. The wave propagation model is based on time-dependent mild slope equations including the wave energy dissipation due to the wave-mud interaction of bottom mud layers as well as the combined effects of the wave refraction, diffraction and breaking. The constitutive equations of the visco-elastic–plastic model are adopted for the rheological behavior of fluid mud. The mass transport velocity within the fluid mud layer is calculated combining the Stokes’ drift, the mean Eulerian velocity and the gravity-driven mud flow. The results of the numerical model are compared against a series of conducted wave basin experiments, wave flume experiments and field observations. Comparisons between the computed results with both the field and laboratory data reveal the capability of the proposed model to predict the wave transformation and mud mass transport.