Eulerian Averaging Research Articles

CVFEM for Multiphase Flow with Disperse and Interface Tracking, and Algorithms Performances

A Control-Volume Finite-Element Method (CVFEM) is newly formulated within Eulerian and spatial averaging frameworks for effective simulation of disperse transport, deposit distribution and interface tracking. Their algorithms are implemented alongside an existing continuous phase algorithm. Flow terms are newly implemented for a control volume (CV) fixed in a space, and the CVs' equations are assembled based on a finite element method (FEM). Upon impacting stationary and moving boundaries, the disperse phase changes its phase and the solver triggers identification of CVs with excess deposit and their neighboring CVs for its accommodation in front of an interface. The solver then updates boundary conditions on the moving interface as well as domain conditions on the accumulating deposit. Corroboration of the algorithms' performances is conducted on illustrative simulations with novel and existing Eulerian and Lagrangian solutions, such as (-) other, i. e. external methods with analytical and physical experimental formulations, and (-) characteristics internal to CVFEM.

Effect of wave-induced Stokes drift on the dynamics of ocean mixed layer

The wave-forcing ‘Coriolis-Stokes forcing’ and ‘Stokes-vortex force’ induced by Stokes drift affect the upper ocean jointly. To study the effect of the wave-induced Stokes drift on the dynamics of the ocean mixed layer, a new three-dimensional (3D) numerical model is derived using the primitive basic equations and Eulerian wave averaging. The Princeton Ocean Model (POM), a 3D primitive equation ocean model is used with the upper wave-averaged basic equations. The global ocean circulation is simulated using the POM model, and the Stokes drift is evaluated based on the wave data generated by WAVEWATCH III. We compared simulations with and without the Stokes drift. The results show that the magnitude of the Stokes drift is comparable with the Eulerian mean current. Including the Stokes drift in the ocean model affects both the Eulerian current and the Lagranian drift and causes the vertical mixing coefficients to increase.

On molecular statics and surface-enhanced continuum modeling of nano-structures

Possible links between discrete and continuum formulations have been discussed in the scientific community since several decades. Different atomistic expressions for the continuum fields (most importantly stresses) were proposed. As an example, it is possible to link discrete models to a continuum formulation based on spatial averaging in the Eulerian configuration followed by probability (statistical) averaging. As an alternative to Eulerian averaging, Lagrangian averaging has been recently proposed. Both approaches allow calculation of the local continuum fields from atomistic simulations.From the continuum formulation perspective, the behavior of solids at the nanoscale can be captured based on a surface-enhanced continuum (SEC) theory whereby the surface is equipped with its own constitutive structure. A distinct advantage of continuum models over their atomistic counterparts is the increased computational efficiency.In this contribution we compare atomistic fields obtained from molecular statics (MS) simulations to their counterpart, obtained from numerical approximations to the SEC theory. Bulk elastic parameters for the continuum constitutive model are obtained directly from the atomistic model. A representative numerical simulation of face-centered-cubic (FCC) copper is used to compare the two approaches.The ability of the continuum formulation enhanced with a surface energy to model size effects, as observed in the atomistic simulations, is shown. The local fields evaluated using both the continuum and discrete approach are in a good agreement. The dependence of the results obtained from the atomistic-to-continuum procedure on both the averaging radius and the proximity of the free surface is studied. Eulerian and Lagrangian averaging approaches are shown to give comparable results for the here considered FCC crystal.

Reply to the comment by T. J. McDougall, P. R. Gent and S. Drijfhout

The comment by T.J. McDougall, P.R. Gent and S. Drijfhout (cited as MGD) contains two parts: the first and most relevant one [because of its physical implications to ocean global circulation models (OGCM)] and a second one of a more technical nature. We shall reply in that order. The main physical argument in the comment is that the diapycnal flux Σ that appears in the mean tracer equation can be dispensed with entirely (and thus there is no need to model it) provided one abandons the Eulerian averages in favor of the density averaging suggested by the Temporal Residual Mean (TRM) formalism. Specifically, the authors write that “the appropriate average tracer is the thickness-weighted rather than the Eulerian-mean assumed in DC6”. We reply to this criticism by showing that TRM does not get rid of Σ since we show that Σ reappears in the mean tracer equation via another term. Therefore TRM offers no advantages with respect to the Eulerian averaging since both contain the diapycnal flux Σ. As for the technical parts, we discuss each of them and show that MGD's criticism to our approximations in the treatment of the tracer equation is erroneous and that our approximation is fully justified.

Modelling atmospheric flows

This article demonstrates how numerical methods for atmospheric models can be validated by showing that they give the theoretically predicted rate of convergence to relevant asymptotic limit solutions. This procedure is necessary because the exact solution of the Navier–Stokes equations cannot be resolved by production models. The limit solutions chosen are those most important for weather and climate prediction. While the best numerical algorithms for this purpose largely reflect current practice, some important limit solutions cannot be captured by existing methods. The use of Lagrangian rather than Eulerian averaging may be required in these cases.

On the Lagrangian/Eulerian modeling of dispersed droplet inertia: Internal circulation transition

This article addresses a limitation of Lagrangian methods for droplet tracking, when approaching the transition point of internal circulation within droplets. Laminar multiphase flow with dispersed droplets in a co-flowing airstream is considered. Analytical and numerical formulations of droplet motion are developed based on a Lagrangian finite difference method of droplet tracking. Cases of both high and low relative Reynolds numbers are formulated. The role of interfacial drag in cross-phase momentum exchange increases at higher relative Reynolds numbers. A new transition criterion is developed to characterize conditions leading to shear-driven non-uniformities of velocity within a droplet. This criterion entails a momentum Biot number, in analogy with the Biot number criterion for conjugate heat transfer problems involving conduction and convection. At sufficiently high momentum Biot numbers, appreciable changes of velocity within a droplet imply that Lagrangian methods become unsuitable and transition to Eulerian volume averaging is needed. Predicted results of Lagrangian modeling of droplet motion in a co-flowing airstream are presented and discussed.

Isopycnal Averaging at Constant Height. Part I: The Formulation and a Case Study

Abstract Simple Eulerian averaging of velocities, density, and tracers at constant position is the most natural way of averaging. However, Eulerian averaging gives incorrect watermass distributions and properties as well as spurious diabatic circulations such as the Deacon cell. Instead of averaging at constant height, averaging along isopycnals removes such fictitious mixing and diabatic circulations. Such isopycnal averaging is normally performed at constant latitude, that is, averaging along isopynals as they heave up and down. As a result, height information is lost and the sea surface becomes much warmer (or lighter) than with simple Eulerian averaging. In fact, averaging can be performed along arbitrarily aligned surfaces. This study considers a particular case in which isopycnal averaging is performed at constant height. Thus, this new isopycnal averaging follows isopycnals as they meander horizontally at constant z. Height information is now retained at the cost of losing latitudinal information. The advantage of this averaging is that it avoids the problem of giving a surface that is too warm. Associated with this new isopycnal averaging, a “vertical” transport streamfunction in (ρ, z) space can be defined, in analogy to the conventional meridional overturning streamfunction in (y, ρ) space. Here in Part I, this constant-height isopycnal averaging is explained and illustrated in an idealized zonal channel model. In Part II the relationship between the two different isopycnal averagings and the Eulerian mean eddy flux divergence is explored.

On advection and diffusion in the mesosphere and lower thermosphere: The role of rotational fluxes

A formalism to describe the advective and diffusive eddy transport in terms of the mean tracer is presented. It is based on Eulerian averaging, the flux‐gradient relation, and the decomposition of the eddy flux of a tracer into advective, diffusive, and rotational components. The rotational (nondivergent) flux arises because the conservation equation for the mean tracer contains only a divergence of the eddy flux of the tracer. To provide a closure, a modification to the flux separation technique based on the eddy variance equation is introduced. The “eddy‐induced” advective velocity is represented as the sum of two velocities v1 and v2. Velocity v1 is similar to that in the Transformed Eulerian Mean (TEM) formulation but is generalized to account for both horizontal and vertical eddy fluxes and mean gradients. The velocity v2 depends on the flux of eddy variance of the tracer. The diffusion coefficient is represented as a sum of K1, which may serve as a diagnostic of an irreversible mixing, and K2, which describes up‐ or down‐gradient eddy fluxes of the tracer due to local transformations of the eddy variance. Both v2 and K2 arise from taking account of the rotational fluxes. The scheme is applied to output from a global circulation model of the middle atmosphere. It is shown that in the meridional plane the correction v2 to the TEM velocity is small in the mesosphere and lower thermosphere. For the diffusion coefficient, however, the correction K2 must be accounted for above approximately 110 km.

Regional averaging and scaling in relativistic cosmology

Averaged inhomogeneous cosmologies lie at the forefront of interest, since cosmological parameters such as the rate of expansion or the mass density are to be considered as volume-averaged quantities and only these can be compared with observations. For this reason the relevant parameters are intrinsically scale-dependent and one wishes to control this dependence without restricting the cosmological model by unphysical assumptions. In the latter respect we contrast our way to approach the averaging problem in relativistic cosmology with shortcomings of averaged Newtonian models. Explicitly, we investigate the scale-dependence of Eulerian volume averages of scalar functions on Riemannian three-manifolds. We propose a complementary view of a Lagrangian smoothing of (tensorial) variables as opposed to their Eulerian averaging on spatial domains. This programme is realized with the help of a global Ricci deformation flow for the metric. We explain rigorously the origin of the Ricci flow which, on heuristic grounds, has already been suggested as a possible candidate for smoothing the initial dataset for cosmological spacetimes. The smoothing of geometry implies a renormalization of averaged spatial variables. We discuss the results in terms of effective cosmological parameters that would be assigned to the smoothed cosmological spacetime. In particular, we find that on the smoothed spatial domain evaluated cosmological parameters obey = 1, where and correspond to the standard Friedmannian parameters, while is a remnant of cosmic variance of expansion and shear fluctuations on the averaging domain. All these parameters are ‘dressed’ after smoothing out the geometrical fluctuations, and we give the relations of the ‘dressed’ to the ‘bare’ parameters. While the former provide the framework of interpreting observations with a ‘Friedmannian bias’, the latter determine the actual cosmological model.

A Mean Synoptic View of the Subantarctic Front South of Australia

Abstract A mean synoptic view of the subantarctic front (SAF) is obtained from current meter and hydrographic data by averaging absolute and baroclinic velocity measurements in bins defined by a cross-stream coordinate that moves with the current. The cross-stream coordinate is derived from current meter measurements of vertical shear of the horizontal velocity and is expressed in terms of specific volume anomaly at 780 dbar (δ780). By averaging absolute and baroclinic velocity measurements in stream coordinates, the spatial smoothing that results from Eulerian averaging of measurements of a meandering current is avoided. The mean SAF velocity profile is composed of one central peak and several smaller peaks. In the central peak, the 2-yr mean absolute velocity from current meters reaches 52 cm s−1 at 420 dbar; the mean baroclinic velocity from six CTD sections reaches 34 cm s−1. The SAF flow is coherent at all levels, reaches the seafloor, and is at least 220 km wide. The cross-stream structure of barocl...

Diatom demography in winter – simulated by the Lagrangian Ensemble method

ABSTRACTAccording to Sverdrup's (1953) model of the spring bloom, phytoplankton biomass decreases in winter when the mixed layer depth exceeds the critical depth. We have used a one‐dimensional mathematical model integrated by the Lagrangian Ensemble method to simulate a population of diatoms during the winter between two growing seasons off the Azores. The model allows us to diagnose the demographic changes in the simulated diatom population from a variety of perspectives. The total population falls to a minimum of 70 million diatoms m‐2 at the end of February. The vertical distribution of the population dynamics is first analysed in terms of daily Eulerian averages over 1 m depth intervals. Growth starts in February when the diurnal thermocline becomes shallower than 50 m, but while the mixed layer is still 200 m deep. The natural mortality has a minimum in winter because it is reduced (in the model) with temperature and population density. Eulerian analysis suggests that in winter, diatoms have a life expectancy of more than 3 months, so a significant number will survive the months of December, January and February when there is very little growth. Losses to grazing are negligible in winter. Lagrangian analysis shows how an individual diatom responds to its changing ambient environment caused by variation in depth (due to turbulent mixing) and the diurnal and seasonal changes in the photosynthetically active radiance. The different trajectories followed by the thousands of plankton particles simulated by the model produce diversity in growth rate ranging over several orders of magnitude, so care has to be taken in statistical analysis.The paper ends with a re‐assessment of the value of the critical depth and compensation depth as predictors for onset of the spring bloom. The compensation depth was computed by Eulerian averaging over 1 m depth inter‐vals each day. For 1 month after the vernal equinox the compensation depth follows the ascent of the mixed layer as it rises from a depth of 100 m to 40 m. Lagrangian analysis reveals that this is due to the photo‐adaptation better matching the ambient irradiance experienced by diatoms in the mixed layer compared with those at the same depth in the seasonal thermo‐cline. By mid‐April the spring bloom has already ad‐vanced so far that self shading influences the compensation depth, which then rises into the mixed layer. We conclude that Sverdrup's criterion is not useful for predicting changes in the diatom population simulated by our model.