Model Order Reduction for Determining Bubble Parameters to Attain a Desired Fluid Surface Shape

In this paper, a new methodology for predicting fluid free surface shape using Model Order Reduction (MOR) is presented. Proper Orthogonal Decomposition combined with a linear interpolation procedure for its coefficient is applied to a problem involving bubble dynamics near to a free surface. A model is developed to accurately and efficiently capture the variation of the free surface shape with different bubble parameters. In addition, a systematic approach is developed within the MOR framework to find the best initial locations and pressures for a set of bubbles beneath the quiescent free surface such that the resultant free surface attained is close to a desired shape. Predictions of the free surface in two-dimensions and three-dimensions are presented.

Dynamics of convectively driven banded jets in the laboratory

The banded organization of clouds and zonal winds in the atmospheres of the outer planets has long fascinated observers. Several recent studies in the theory and idealized modeling of geostrophic turbulence have suggested possible explanations for the emergence of such organized patterns, typically involving highly anisotropic exchanges of kinetic energy and vorticity within the dissipationless inertial ranges of turbulent flows dominated (at least at large scales) by ensembles of propagating Rossby waves. The results from an attempt to reproduce such conditions in the laboratory are presented here. Achievement of a distinct inertial range turns out to require an experiment on the largest feasible scale. Deep, rotating convection on small horizontal scales was induced by gently and continuously spraying dense, salty water onto the free surface of the 13-m-diameter cylindrical tank on the Coriolis platform in Grenoble, France. A “planetary vorticity gradient” or “β effect” was obtained by use of a conically sloping bottom and the whole tank rotated at angular speeds up to 0.15 rad s−1. Over a period of several hours, a highly barotropic, zonally banded large-scale flow pattern was seen to emerge with up to 5–6 narrow, alternating, zonally aligned jets across the tank, indicating the development of an anisotropic field of geostrophic turbulence. Using particle image velocimetry (PIV) techniques, zonal jets are shown to have arisen from nonlinear interactions between barotropic eddies on a scale comparable to either a Rhines or “frictional” wavelength, which scales roughly as (β/Urms)−1/2. This resulted in an anisotropic kinetic energy spectrum with a significantly steeper slope with wavenumber k for the zonal flow than for the nonzonal eddies, which largely follows the classical Kolmogorov k−5/3 inertial range. Potential vorticity fields show evidence of Rossby wave breaking and the presence of a “hyperstaircase” with radius, indicating instantaneous flows that are supercritical with respect to the Rayleigh–Kuo instability criterion and in a state of “barotropic adjustment.” The implications of these results are discussed in light of zonal jets observed in planetary atmospheres and, most recently, in the terrestrial oceans.

On the existence of extreme waves and the Stokes conjecture with vorticity

This is a study of singular solutions of the problem of traveling gravity water waves on flows with vorticity. We show that, for a certain class of vorticity functions, a sequence of regular waves converges to an extreme wave with stagnation points at its crests. We also show that, for any vorticity function, the profile of an extreme wave must have either a corner of 120° or a horizontal tangent at any stagnation point about which it is supposed symmetric. Moreover, the profile necessarily has a corner of 120° if the vorticity is nonnegative near the free surface.

On the distortion of turbulence by a progressive surface wave

A rapid-distortion model is developed to investigate the interaction of weak turbulence with a monochromatic irrotational surface water wave. The model is applicable when the orbital velocity of the wave is larger than the turbulence intensity, and when the slope of the wave is sufficiently high that the straining of the turbulence by the wave dominates over the straining of the turbulence by itself. The turbulence suffers two distortions. Firstly, vorticity in the turbulence is modulated by the wave orbital motions, which leads to the streamwise Reynolds stress attaining maxima at the wave crests and minima at the wave troughs; the Reynolds stress normal to the free surface develops minima at the wave crests and maxima at the troughs. Secondly, over several wave cycles the Stokes drift associated with the wave tilts vertical vorticity into the horizontal direction, subsequently stretching it into elongated streamwise vortices, which come to dominate the flow. These results are shown to be strikingly different from turbulence distorted by a mean shear flow, when `streaky structures' of high and low streamwise velocity fluctuations develop. It is shown that, in the case of distortion by a mean shear flow, the tendency for the mean shear to produce streamwise vortices by distortion of the turbulent vorticity is largely cancelled by a distortion of the mean vorticity by the turbulent fluctuations. This latter process is absent in distortion by Stokes drift, since there is then no mean vorticity. The components of the Reynolds stress and the integral length scales computed from turbulence distorted by Stokes drift show the same behaviour as in the simulations of Langmuir turbulence reported by McWilliams, Sullivan & Moeng (1997). Hence we suggest that turbulent vorticity in the upper ocean, such as produced by breaking waves, may help to provide the initial seeds for Langmuir circulations, thereby complementing the shear-flow instability mechanism developed by Craik & Leibovich (1976). The tilting of the vertical vorticity into the horizontal by the Stokes drift tends also to produce a shear stress that does work against the mean straining associated with the wave orbital motions. The turbulent kinetic energy then increases at the expense of energy in the wave. Hence the wave decays. An expression for the wave attenuation rate is obtained by scaling the equation for the wave energy, and is found to be broadly consistent with available laboratory data.

Dissipation of shear-free turbulence near boundaries

The rapid-distortion model of Hunt & Graham (1978) for the initial distortion of turbulence by a flat boundary is extended to account fully for viscous processes. Two types of boundary are considered: a solid wall and a free surface. The model is shown to be formally valid provided two conditions are satisfied. The first condition is that time is short compared with the decorrelation time of the energy-containing eddies, so that nonlinear processes can be neglected. The second condition is that the viscous layer near the boundary, where tangential motions adjust to the boundary condition, is thin compared with the scales of the smallest eddies. The viscous layer can then be treated using thin-boundary-layer methods. Given these conditions, the distorted turbulence near the boundary is related to the undistorted turbulence, and thence profiles of turbulence dissipation rate near the two types of boundary are calculated and shown to agree extremely well with profiles obtained by Perot & Moin (1993) by direct numerical simulation. The dissipation rates are higher near a solid wall than in the bulk of the flow because the no-slip boundary condition leads to large velocity gradients across the viscous layer. In contrast, the weaker constraint of no stress at a free surface leads to the dissipation rate close to a free surface actually being smaller than in the bulk of the flow. This explains why tangential velocity fluctuations parallel to a free surface are so large. In addition we show that it is the adjustment of the large energy-containing eddies across the viscous layer that controls the dissipation rate, which explains why rapid-distortion theory can give quantitatively accurate values for the dissipation rate. We also find that the dissipation rate obtained from the model evaluated at the time when the model is expected to fail actually yields useful estimates of the dissipation obtained from the direct numerical simulation at times when the nonlinear processes are significant. We conclude that the main role of nonlinear processes is to arrest growth by linear processes of the viscous layer after about one large-eddy turnover time.

On the variability of the surface environment response to synoptic forcing over complex terrain: a multivariate data analysis approach

Synoptic climatology relates the atmospheric circulation with the surface environment. The aim of this study is to examine the variability of the surface meteorological patterns, which are developing under different synoptic scale categories over a suburban area with complex topography. Multivariate Data Analysis techniques were performed to a data set with surface meteorological elements. Three principal components related to the thermodynamic status of the surface environment and the two components of the wind speed were found. The variability of the surface flows was related with atmospheric circulation categories by applying Correspondence Analysis. Similar surface thermodynamic fields develop under cyclonic categories, which are contrasted with the anti-cyclonic category. A strong, steady wind flow characterized by high shear values develops under the cyclonic Closed Low and the anticyclonic H–L categories, in contrast to the variable weak flow under the anticyclonic Open Anticyclone category.

Morphology induced spinodal decomposition at the surface of symmetric diblock copolymer films

Atomic force microscopy is used to study the ordering dynamics of symmetric diblock copolymer films. The films order to form a lamellar structure which results in a frustration when the film thickness is incommensurate with the lamellae. By probing the morphology of incommensurate films in the early ordering stages, we discover an intermediate phase of lamellae arranged perpendicular to the film surface. This morphology is accompanied by a continuous growth in amplitude of the film surface topography with a characteristic wavelength, indicative of a spinodal process. Using selfconsistent field theory, we show that the observation of perpendicular lamellae suggests an intermediate state with parallel lamellae at the substrate and perpendicular lamellae at the free surface. The calculations confirm that the intermediate state is unstable to thickness fluctuations, thereby driving the spinodal growth of surface structures.

A singular integro-differential equation model for dryout in LMFBR boiler tubes

A 2D steady model for the annular two-phase flow of water and steam in the steam-generating boiler pipes of a liquid metal fast breeder reactor is proposed The model is based on thin-layer lubrication theory and thin aerofoil theory. The exchange of mass between the vapour core and the liquid film due to evaporation of the liquid film is accounted for using some simple thermodynamics models, and the resultant change of phase is modelled by proposing a suitable Stefan problem Appropriate boundary conditions for the now are discussed The resulting non-lineal singular integro-differential equation for the shape of the liquid film free surface is solved both asymptotically and numerically (using some regularization techniques) Predictions for the length to the dryout point from the entry of the annular regime are made The influence of both the traction tau provided by the fast-flowing vapour core on the liquid layer and the mass transfer parameter eta on the dryout length is investigated

Front tracking with moving-least-squares surfaces

The representation of interfaces by means of the algebraic moving-least-squares (AMLS) technique is addressed. This technique, in which the interface is represented by an unconnected set of points, is interesting for evolving fluid interfaces since there is]to surface connectivity. The position of the surface points can thus be updated without concerns about the quality of any surface triangulation. We introduce a novel AMLS technique especially designed for evolving-interfaces applications that we denote RAMLS (for Robust AMLS). The main advantages with respect to previous AMLS techniques are: increased robustness, computational efficiency, and being free of user-tuned parameters. Further, we propose a new front-tracking method based on the Lagrangian advection of the unconnected point set that defines the RAMLS surface. We assume that a background Eulerian grid is defined with some grid spacing h. The advection of the point set makes the surface evolve in time. The point cloud can be regenerated at any time (in particular, we regenerate it each time step) by intersecting the gridlines with the evolved surface, which guarantees that the density of points on the surface is always well balanced. The intersection algorithm is essentially a ray-tracing algorithm, well-studied in computer graphics, in which a line (ray) is traced so as to detect all intersections with a surface. Also, the tracing of each gridline is independent and can thus be performed in parallel. Several tests are reported assessing first the accuracy of the proposed RAMLS technique, and then of the front-tracking method based on it. Comparison with previous Eulerian, Lagrangian and hybrid techniques encourage further development of the proposed method for fluid mechanics applications. (C) 2008 Elsevier Inc. All rights reserved.

Remarks on Parametric Surface Waves in a Nonlinear and Non-Ideally Excited Tank

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Effects of a temperature dependent viscosity in surface nonlinear waves propagating in a shallow fluid heated from below

The effects of a temperature dependent viscosity in surface nonlinear waves propagating in a shallow fluid heated from below are investigated. It is shown that the (2+1)-dimensional Burgers equation may appear as the equation governing the upper free surface perturbations of a Bénard system, even when the viscosity is assumed to depend on temperature. The critical Rayleigh number for the appearance of waves governed by the Kadomtsev-Petviashvili equation, however, will be smaller than R=30, which is the critical number obtained for a constant viscosity. © 1992.

Mitigação de riscos e catástrofes naturais: análise numérico-experimental de roll waves evoluindo em canais inclinados

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Abordagem matemática de roll waves em escoamentos hiperconcentrados com superfície livre

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Um panorama sobre roll waves em escoamentos laminares e turbulentos com superfície livre

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

An integrable evolution equation for surface waves in deep water

In order to describe the dynamics of monochromatic surface waves in deep water, we derive a nonlinear and dispersive system of equations for the free surface elevation and the free surface velocity from the Euler equations in infinite depth. From it, and using a multiscale perturbative method, an asymptotic model for small wave steepness ratio is derived. The model is shown to be completely integrable. The Lax pair, the first conserved quantities as well as the symmetries are exhibited. Theoretical and numerical studies reveal that it supports periodic progressive Stokes waves which peak and break in finite time. Comparison between the limiting wave solution of the asymptotic model and classical results is performed.