Eulerian Velocity Research Articles

On the modeling of the 2010 Gulf of Mexico Oil Spill

Abstract Two oil particle trajectory forecasting systems were developed and applied to the 2010 Deepwater Horizon Oil Spill in the Gulf of Mexico. Both systems use ocean current fields from high-resolution numerical ocean circulation model simulations, Lagrangian stochastic models to represent unresolved sub-grid scale variability to advect oil particles, and Monte Carlo-based schemes for representing uncertain biochemical and physical processes. The first system assumes two-dimensional particle motion at the ocean surface, the oil is in one state, and the particle removal is modeled as a Monte Carlo process parameterized by a one number removal rate. Oil particles are seeded using both initial conditions based on observations and particles released at the location of the Maconda well. The initial conditions (ICs) of oil particle location for the two-dimensional surface oil trajectory forecasts are based on a fusing of all available information including satellite-based analyses. The resulting oil map is digitized into a shape file within which a polygon filling software generates longitude and latitude with variable particle density depending on the amount of oil present in the observations for the IC. The more complex system assumes three (light, medium, heavy) states for the oil, each state has a different removal rate in the Monte Carlo process, three-dimensional particle motion, and a particle size-dependent oil mixing model. Simulations from the two-dimensional forecast system produced results that qualitatively agreed with the uncertain “truth” fields. These simulations validated the use of our Monte Carlo scheme for representing oil removal by evaporation and other weathering processes. Eulerian velocity fields for predicting particle motion from data-assimilative models produced better particle trajectory distributions than a free running model with no data assimilation. Monte Carlo simulations of the three-dimensional oil particle trajectory, whose ensembles were generated by perturbing the size of the oil particles and the fraction in a given size range that are released at depth, the two largest unknowns in this problem. 36 realizations of the model were run with only subsurface oil releases. An average of these results yields that after three months, about 25% of the oil remains in the water column and that most of the oil is below 800 m.

Modelling environmentally friendly fairways using Lagrangian trajectories: a case study for the Gulf of Finland, the Baltic Sea

We address possibilities of minimising environmental risks using statistical features of current-driven propagation of adverse impacts to the coast. The recently introduced method for finding the optimum locations of potentially dangerous activities (Soomere et al. in Proc Estonian Acad Sci 59:156–165, 2010) is expanded towards accounting for the spatial distributions of probabilities and times for reaching the coast for passively advecting particles released in different sea areas. These distributions are calculated using large sets of Lagrangian trajectories found from Eulerian velocity fields provided by the Rossby Centre Ocean Model with a horizontal resolution of 2 nautical miles for 1987–1991. The test area is the Gulf of Finland in the northeastern Baltic Sea. The potential gain using the optimum fairways from the Baltic Proper to the eastern part of the gulf is an up to 44% decrease in the probability of coastal pollution and a similar increase in the average time for reaching the coast. The optimum fairways are mostly located to the north of the gulf axis (by 2–8 km on average) and meander substantially in some sections. The robustness of this approach is quantified as the typical root mean square deviation (6–16 km) between the optimum fairways specified from different criteria. Drastic variations in the width of the ‘corridors’ for almost optimal fairways (2–30 km for the average width of 15 km) signifies that the sensitivity of the results with respect to small changes in the environmental criteria largely varies in different parts of the gulf.

Numerical study of flow through and around a circular array of cylinders

This paper describes a study of the local and global effect of an isolated group of cylinders on an incident uniform flow. Using high resolution two-dimensional computations, we analysed the flow through and around a localised circular array of cylinders, where the ratio of array diameter (DG) to cylinder diameter (D) is 21. The number of cylinders varied fromNC= 7 to 133, and they were arranged in a series of concentric rings to allow even distribution within the array with an average void fraction φ =NC(D/DG)2, which varied from 0.016 to 0.30. The characteristic Reynolds number of the array wasReG= 2100. A range of diagnostic tools were applied, including the lift/drag forces on each cylinder (and the whole array), Eulerian and Lagrangian average velocity within the array, and the decay of maximum vorticity with distance downstream. To interpret the flow field, we used vorticity and the dimensionless form of the second invariant of the velocity gradient tensor. A mathematical model, based on representing the bodies as point forces, sources and dipoles, was applied to interpret the results. Three distinct flow regimes were identified. For low void fractions (φ < 0.05), the cylinders have uncoupled individual wake patterns, where the vorticity is rapidly annihilated by wake intermingling downstream and the forces are similar to that of an isolated cylinder. At intermediate void fractions (0.05 < φ < 0.15), a shear layer is generated at the shoulders of the array and the force acting on the cylinders is steady. For high void fractions (φ > 0.15), the array generates a wake in a similar way to a solid body of the same scale. For low void fraction arrays, the mathematical model provides a reasonable assessment of the forces on individual bodies within the array, the Eulerian mean velocity and the upstream velocity field. While it broadly captures the change in the rate of decay of the maximum vorticity magnitude Ωmaxdownstream, the magnitude is underpredicted.

Conditional Eulerian and Lagrangian velocity increment statistics of fully developed turbulent flow

Conditional statistics of homogeneous isotropic turbulent flow is investigated by means of high Reynolds number direct numerical simulations performed with 20483 collocation points. Eulerian as well as Lagrangian velocity increment statistics under several conditions are analyzed and compared. In agreement with experimental data, longitudinal probability density functions P(δl||u|εl) conditioned on a scale-averaged energy dissipation rate are close to Gaussian distributions over all scales within the inertial range of scales. Also transverse increments conditioned on either the dissipation rate or the square of the vorticity have quasi-Gaussian probability density functions (PDFs). Concerning Lagrangian statistics we found that conditioning on a trajectory averaged energy dissipation rate ετ significantly reduces the scale dependence of the increment PDFs P(δτui|ετ). By means of dimensional arguments we propose a novel condition for Lagrangian increments which is shown to reduce even more the flatness of the corresponding PDFs and thus intermittency in the inertial range of scales. The conditioned Lagrangian PDF corresponding to the smallest increment considered is reasonably well described by a prediction for the PDF of acceleration of a non-intermittent velocity field. Conditioned structure functions show approximately K41 scaling with a larger scaling range than the unconditioned ones.

Lagrangian mixing in straight compound channels

Recently Stocchino & Brocchini (J. Fluid Mech., vol. 643, 2010, p. 425 have studied the dynamics of two-dimensional (2D) large-scale vortices with vertical axis evolving in a straight compound channel under quasi-uniform flow conditions. The mixing processes associated with such vortical structures are here analysed through the results of a dedicated experimental campaign. Time-resolved Eulerian surface velocity fields, measured using a 2D particle-image velocimetry system, form the basis for a Lagrangian analysis of the dispersive processes that occur in compound channels when the controlling physical parameters, i.e. the flow depth ratio (rh) and the Froude number (Fr) are changed. Lagrangian mixing is studied by means of various approaches based either on single-particle or multiple-particle statistics (relative and absolute statistics, probability density functions (p.d.f.s) of relative displacements and finite-scale Lyapunov exponents). Absolute statistics reveal that transitional macrovortices, typical of shallow flow conditions, strongly influence the growth in time of the total absolute dispersion, after the initial ballistic regime, leading to a non-monotonic behaviour. In deep flow conditions, on the contrary, the absolute dispersion displays a monotonic growth because the generation of transitional macrovortices does not take place. In all cases an asymptotic diffusive regime is reached.Multiple-particle dynamics is controlled by rh and Fr. Different growth regimes of the relative diffusivity have been found depending on the flow conditions. This behaviour can be associated with different energy transfer processes and it is further confirmed by the p.d.f.s of relative displacements, which show a different asymptotical shape depending on the separation scales and the Froude number. Finally, an equilibrium regime is observed for all the experiments by analysing the decay of the finite-scale Lyapunov exponents with the particle separations.

On Seliger and Whitham’s variational principle for hydrodynamic systems from the point of view of ‘fictitious particles’

According to Miles [9, p. 103], variational principles in fluid dynamics have been associated with important names as, e.g., Finlayson [2], Luke [7], Miloh [10], Seliger and Whitham [12], Serrin [13], Truesdell and Toupin [14]. In the paper by Seliger and Whitham [12], a comprehensive treatment for problems within fluid dynamics, plasma dynamics and elasticity is presented, where general variational principles are stated for a given system of equations. In the context of fluid dynamics, difficulties arisen from the usage of the Eulerian description are overcome by representing the Eulerian velocity through a representation for isentropic flow, first suggested by Clebsh in 1859, and by introducing Lagrange multipliers as variational constraints that enable one to recover the missed identity of the fluid particles. In other words, Seliger and Whitham [12] emphasize that the traditional form of Hamilton’s principle, convenient within the Lagrangian description, must be equipped with side conditions, within the Eulerian approach, if non-trivial flows are studied. As suggested by Lin [6], the identity is preserved through the invariance of the initial coordinates of each particle along its path. Seliger and Whitham [12] showed more: just one initial coordinate is enough. Moreover, such an addition of side conditions enables one to elegantly write the Lagrangian density solely as the pressure field. This is a quite interesting and general result, being valid for rotational and compressible flows. The point to be reinforced is that the paper by Seliger and Whitham bridges the Lagrangian and the Eulerian descriptions through Hamilton’s principle. Whereas the usual form of Hamiltons’s principle is written within a Lagrangian description (see [12], p. 4, Eq. (8)), i.e., considering a “sum over particles” or “volume of the material”, Seliger and Whitham’s version of Hamilton’s principle is written within an Eulerian description, for an arbitrary region of the space. Indeed, in hydrodynamics, many problems are preferably treated by taking open (control) volumes. In the context of marine hydrodynamics, for example, the water impact problem and the oscillating floating body problem are important examples to be mentioned. In the first problem, see, e.g., [5] the splashing jets are cut out and only the ‘bulk’ of the liquid is considered for the sake of formulating the problem mathematically. Thus, the so-called jet root plays the role of a moving control surface, so enabling kinetic energy to flow through it. In the second mentioned problem, when the radiated waves are taken into account, the liquid domain is usually supposed to be bounded by a distant vertical cylindrical surface, a fixed control surface, through which

LAGRANGIAN DROGUE-BASED DRIFTER FOR MONITORING SUSPENDED SEDIMENT TRANSPORT IN INTERTIDAL ENVIRONMENTS

The in-situ Lagrangian-Acoustic Drogue (LAD) presented by Schacht and Lemckert (2007) for monitoring the temporal and spatial distributions of both the current and the suspended sediment concentration within the estuary environments has been modified to operate in the shallow water intertidal regions. The new drogue, called the LAD for Inter-Tidal environments (LAD-IT), is equipped with a Global Positioning System (GPS), a small Acoustic Doppler Current Profiler (ADCP) and nephelometer. The small ADCP, which did not have a bottom tracking facility, was used to maximize the range of depths the LAD-IT could operate over. The accuracy of a vertical current profile measured using the LAD-IT was examined through the laboratory experiments conducted at an outdoor stream pool in Kumamoto, Japan, with uni-directional flow conditions, and through the field experiments conducted within an intertidal zone of Ariake Sound in Kumamot, Japan, with multi-directional flow conditions. Under uni- directional flow conditions the current profile was measured within 7% accuracy by summing the surface current velocity calculated using GPS tracking and the relative current profile measured using ADCP. Under multi-directional flow conditions, such as those of tide and wind-induced wave fields, the current profile agreed within 5% accuracy. This was partly because both Eulerian and Lagrangian mass transport velocities under these conditions were on the order of 1cm/s, and thus the error value was very small. The temporal and spatial distributions of both the current velocity and the suspended sediment concentration were also measured using the LAD-IT on the Brisbane River, Australia under uni-directional flow conditions. The field observation results support the conventional concept of the suspended sediment as a vertical balance between downward suspended sediment settling and upward turbulent diffusion fluxes. The results indicate the LAD-IT is adequate for estimating the sediment settling velocity in the field.

Measurements of velocity and trajectory of water particle for internal waves in two density layers

[1] This article discusses the kinetics of internal waves, which propagate in a two-layer fluid system having a constant water depth, using particle image velocimetry (PIV). The experimental vector field of velocity and vertical distributions of its components were estimated at several phases in one wave cycle and compared with the corresponding predictions on the basis of third-order Stokes internal-wave theory. These attempts proved that the PIV technique enables the measurement of water velocity spatially induced by the nonlinear internal waves with considerable accuracy. This method was applied to trace water particle path. The measured trajectory was compared with the particle positions obtained theoretically by integrating the Eulerian velocity to a higher order in a Taylor series expansion.

Analytical solution for the tidally induced Lagrangian residual current in a narrow bay

In a weakly nonlinear tidal system, the depth-averaged equations for the first-order Lagrangian residual velocity (LRV) are deduced systematically. For the case of a narrow bay, the equations are solved analytically and the results for a specific bottom profile are discussed in detail. According to the pattern of the first-order LRV, the bay can be divided into three parts, namely an inner part, a transitional zone, and an outer part. For the given depth profile, the streamline of the first-order LRV for a shorter bay is a part of that for a longer bay. The first-order LRV depends on a nondimensional parameter that combines the influences of the bottom friction coefficient, the tidal period and the averaged water depth. The form of the bottom friction also has a significant influence on the first-order LRV. The second-order LRV, i.e., the Lagrangian drift, is analytically solved and shows dependence on the initial tidal phase. The LRV differs from the Eulerian residual transport velocity both quantitatively and qualitatively. It is demonstrated that the residual currents obtained according to other definitions may cause misunderstanding of the mass transport in water exchange applications.

MEASUREMENTS OF PARTICLE VELOCITIES AND TRAJECTORIES IN A WAVE-CURRENT MOTION USING PARTICLE IMAGE VELOCIMETRY

This article deals with some physical aspects of a water particle under surface waves, which propagate with or without a current in a constant water depth, using an imaging technique. The use of particle image velocimetry (PIV) made it possible to investigate the velocity and trajectory of each individual water particle. The velocity vector fields and its vertical distributions were estimated at several phases in one wave cycle. The theory of progressive waves based on finite-amplitude approximation was adapted to express the velocity potential, surface displacement and angular frequency. The PIV result showed suitable agreement with a solution solved to the third order by a perturbation method. In addition, the distributions of horizontal and vertical velocity components by the PIV measurement were compared with those by an EC meter. These attempts proved the ability of the PIV technique to accurately measure both temporal and spatial variations of the velocity. This technique was applied to the prediction of particle trajectory in a Eulerian scheme. In the method, the surrounding grid velocities were used to identify a Lagrangian velocity. The measured particle path was compared with the positions found theoretically by integrating the Eulerian velocity to the second order of a Taylor series expansion.

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.

Formula omitted]-Coordinate, an Arbitrary Lagrangian–Eulerian coordinate separating high and low frequency motions

A new geopotential based vertical coordinate, named z˜, is presented, that treats external motions as a z∗-coordinate, internal low frequency motions in a Eulerian way and high frequency ones in a Lagrangian way. To that purpose, a prognostic equation for the low frequency horizontal flux divergence is added in the system in order to discriminate between low and high frequency parts of vertical motions. This coordinate keeps the intrinsic advantages of the z-coordinate while enabling a drastic reduction of the spurious diapycnal mixing induced by vertical and lateral schemes truncation errors associated with high frequency motions in level models. An idealised experiment illustrates this property in an internal wave propagation framework. The reduction of Eulerian vertical velocities enables a proportional reduction of the associated advection schemes truncation errors. Following high frequency motions, the z˜-coordinate lets lateral schemes operate along tracer anomalies created by those motions.

Propagation of water waves over permeable rippled beds

Unsteady two-dimensional Navier–Stokes equations and Navier–Stokes type model equations for porous flow were solved numerically to simulate the propagation of water waves over a permeable rippled bed. A boundary-fitted coordinate system was adopted to make the computational meshes consistent with the rippled bed. The accuracy of the numerical scheme was confirmed by comparing the numerical results concerning the spatial distribution of wave amplitudes over impermeable and permeable rippled beds with the analytical solutions. For periodic incident waves, the flow field over the wavy wall is discussed in terms of the steady Eulerian streaming velocity. The trajectories of the fluid particles that are initially located close to the ripples were also determined. One of the main results herein is that under the action of periodic water waves, fluid particles on an impermeable rippled bed initially moved back and forth around the ripple crest, with increasing vertical distance from the rippled wall. After one or two wave periods, they are then lifted towards the next ripple crest. All of the marked particles on a permeable rippled bed were shifted onshore with a much larger displacement than those on an impermeable bed. Finally, the flow fields and the particle motions close to impermeable and permeable beds induced by a solitary wave are elucidated.

Structure functions of fully developed hydrodynamic turbulence: An analytical approach

We develop a theory of turbulence based on the inviscid Navier-Stokes equation, without using dimensional or phenomenological considerations. The theory allows us to obtain the scaling law and to calculate the scaling exponents of the Lagrangian and Eulerian velocity structure functions in the inertial range. The obtained results are shown to be in very good agreement with numerical simulations and experimental data. The introduction of stochasticity into the equations is discussed in detail. The main equation of the theory obtained directly from the Navier-Stokes equation is analyzed by means of continual integration.

Measurement of Vertical Kinetic Energy and Vertical Velocity Skewness in Oceanic Boundary Layers by Imperfectly Lagrangian Floats

Abstract The effects of upward buoyancy on the accuracy with which Lagrangian floats can measure the Eulerian mean variance 〈ww〉E and skewness SwE of vertical fluid velocity w in the wind-driven upper-ocean boundary layer is investigated using both simulated floats in large-eddy simulation (LES) models and two float datasets. Nearly neutrally buoyant floats are repeatedly advected by the turbulent velocities across the boundary layer. Their vertical position Z is determined from pressure measurements; their W variance 〈WW〉F and skewness SWE are determined from the time series of float W = dZ/dt. If the float buoyancy is small, then the simulated floats measure the Eulerian velocity accurately; that is, δW2 = 〈WW〉F − 〈ww〉E and δSW = SWF − SwE are small compared to 〈ww〉E and SwE respectively. If the floats are buoyant, and thus have an upward vertical velocity Wbias relative to the water, then δW2 and δSW can become significant. Buoyancy causes the floats to oversample both shallow depths and strong vertical velocities, leading to positive values of δW2 and negative values of δSW. The skewness SZ′F of depth measures the degree to which shallow depths are oversampled; it is shown to be a good predictor of Wbias/〈WW〉F1/2, δW2/〈WWF〉, and δSW/SWF over a wide range of float buoyancies and boundary layer forcings. Float data collected during two deployments confirm these results, but averaging times of several float days are typically required to obtain stable statistics. Significant differences in the magnitude of the effect may occur between datasets from different ocean forcing regimes and float designs. Other measures of float buoyancy are also useful predictors. These results can be used to correct existing float measurements of 〈ww〉E for the effects of buoyancy and also can be used as a means to operationally assess and control float buoyancy.

Langevin and diffusion equation of turbulent fluid flow

A derivation of the Langevin and diffusion equations describing the statistics of fluid particle displacement and passive admixture in turbulent flow is presented. Use is made of perturbation expansions. The small parameter is the inverse of the Kolmogorov constant C0, which arises from Lagrangian similarity theory. The value of C0 in high Reynolds number turbulence is 5–6. To achieve sufficient accuracy, formulations are not limited to terms of leading order in C0−1 including terms next to leading order in C0−1 as well. Results of turbulence theory and statistical mechanics are invoked to arrive at the descriptions of the Langevin and diffusion equations, which are unique up to truncated terms of O(C0−2) in displacement statistics. Errors due to truncation are indicated to amount to a few percent. The coefficients of the presented Langevin and diffusion equations are specified by fixed-point averages of the Eulerian velocity field. The equations apply to general turbulent flow in which fixed-point Eulerian velocity statistics are non-Gaussian to a degree of O(C0−1). The equations provide the means to calculate and analyze turbulent dispersion of passive or almost passive admixture such as fumes, smoke, and aerosols in areas ranging from atmospheric fluid motion to flows in engineering devices.

Inertial range Eulerian and Lagrangian statistics from numerical simulations of isotropic turbulence

We present a study of Eulerian and Lagrangian statistics from a high-resolution numerical simulation of isotropic and homogeneous turbulence using the FLASH code, with an estimated Taylor microscale Reynolds number of around 600. Statistics are evaluated over a data set with 18563 spatial grid points and with 2563 = 16.8 million particles, followed for about one large-scale eddy turnover time. We present data for the Eulerian and Lagrangian structure functions up to the tenth order. We analyze the local scaling properties in the inertial range. The Eulerian velocity field results show good agreement with previous data and confirm the puzzling differences previously found between the scaling of the transverse and the longitudinal structure functions. On the other hand, accurate measurements of sixth-and-higher-order Lagrangian structure functions allow us to highlight some discrepancies from earlier experimental and numerical results. We interpret this result in terms of a possible contamination from the viscous scale, which may have affected estimates of the scaling properties in previous studies. We show that a simple bridge relation based on a multifractal theory is able to connect scaling properties of both Eulerian and Lagrangian observables, provided that the small differences between intermittency of transverse and longitudinal Eulerian structure functions are properly considered.

Conditional Relative Acceleration Statistics and Relative Dispersion Modelling

We have used direct numerical simulation results for the Eulerian velocity difference probability density function and the mean acceleration difference conditioned on the velocity difference, to explore some of the assumptions underlying the formulation of Lagrangian stochastic models for relative dispersion. We focussed on the ability of the models to quantitatively represent Richardson’s t 3-law and in particular the value of Richardson’s constant. As a result of intermittency, with decreasing spatial separation and with increasing Reynolds number these Eulerian quantities become more extreme and the model predictions for Richardson’s constant also become more extreme (larger). This is in contrast with recent numerical simulations showing that Richardson’s constant depends only weakly on Reynolds number. We conclude that, at least in the present Lagrangian stochastic modelling framework, in two-particle models (and presumably in multi-particle models) intermittency must be included explicitly in the dissipation rate as well as in the relative velocity statistics.

Lagrangian and Eulerian Velocity Structure Functions in Hydrodynamic Turbulence

The Lagrangian and Eulerian transverse velocity structure functions of fully developed fluid turbulence are found based on the inviscid Navier-Stokes equation. The structure functions are shown to obey the scaling relations S(n)(L)(tau) proportional to tau(xi(n)) and S(n)(E)(l) proportional to l(zeta(n)) inside the inertial range. The scaling exponents zeta(n) and xi(n) are calculated analytically without using dimensional considerations. This result is a significant step toward understanding the nature of hydrodynamic turbulence. The obtained values are in agreement with recent numerical and experimental data.

Sensitivity of Lagrangian coherent structure identification to flow field resolution and random errors

The effect of spatial and temporal resolutions and random errors on identification of Lagrangian coherent structures (LCSs) from Eulerian velocity fields is evaluated using two canonical flows: a two-dimensional vortex pair and a vortex ring formed by transient ejection of a jet from a tube. The flow field for the vortex pair case was steady and obtained analytically while the transient vortex ring flow was simulated using computational fluid dynamics. To evaluate resolution and random error effects, the flow fields were degraded by locally smoothing the flow and sampling it on a sparser grid to reduce spatial resolution, adding Gaussian distributed random noise to provide random errors, and/or subsampling the time series of vector fields to reduce the temporal resolution (the latter applying only for the vortex ring case). The degradation methods were meant to emulate distortions and errors introduced in common flow measurement methods such as digital particle image velocimetry. Comparing the LCS corresponding to the vortex boundary (separatrix) obtained from the degraded velocity fields with the true separatrix (obtained analytically for the vortex pair case or from high resolution, noise-free velocity fields for the vortex ring case) showed that noise levels as low as 5%-10% of the vortex velocity can cause the separatrix to significantly deviate from its true location in a random fashion, but the "mean" location still remained close to the true location. Temporal and spatial resolution degradations were found to primarily affect transient portions of the flow with strong spatial gradients. Significant deviations in the location of the separatrix were observed even for spatial resolutions as high as 2% of the jet diameter for the vortex ring case.