Contractions in the 2-wasserstein lenght space and thermalization of granular media

An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow-through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite-dimensional manifold. An abstract theory is developed for gradient flows in length spaces, which shows how degenerate convexity (or even non-convexity) | if uniformly controlled | will quantify contractivity (limit expansivity) of the flow.

A mixed finite element method for nonlinear diffusion equations

We propose a mixed finite element method for a class of nonlinear diffusion equations, which is based on their interpretation as gradient flows in optimal transportation metrics. We introduce an appropriate linearization of the optimal transport problem, which leads to a mixed symmetric formulation. This formulation preserves the maximum principle in case of the semi-discrete scheme as well as the fully discrete scheme for a certain class of problems. In addition solutions of the mixed formulation maintain exponential convergence in the relative entropy towards the steady state in case of a nonlinear Fokker-Planck equation with uniformly convex potential. We demonstrate the behavior of the proposed scheme with 2D simulations of the porous medium equations and blow-up questions in the Patlak-Keller-Segel model.

A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Analysis of Self-Organization

The dissertation is devoted to the study of problems in calculus of variation, free boundary problems and gradient flows with respect to the Wasserstein metric. More concretely, we consider the problem of characterizing the regularity of minimizers to a certain interaction energy. Minimizers of the interaction energy have a somewhat surprising relationship with solutions to obstacle problems. Here we prove and exploit this relationship to obtain novel regularity results. Another problem we tackle is describing the asymptotic behavior of the Cahn-Hilliard equation with degenerate mobility. By framing the Cahn-Hilliard equation with degenerate mobility as a gradient flow in Wasserstein metric, in one space dimension, we prove its convergence to a degenerate parabolic equation under the framework recently developed by Sandier-Serfaty.

Bed shear stress measurements in dam break and swash flows

A novel shear plate was used to make direct bed shear stress measurements in laboratory dam break and swash flows on smooth, fixed, impermeable beds. The pressure gradient due to the slope of the fluid free-surface across the plate was measured using pressure transducers. Surface elevation was measured at five locations using acoustic displacement sensors. Flow velocity was measured using an Acoustic-Doppler Velocimeter and calculated using the ANUGA inundation model. The measured bed shear stress at the dam break fluid tip for an initially dry, horizontal bed was close to twice that estimated using steady flow theory. The temporal variation of swash bed shear stress showed a large peak in landward directed stress at the uprush tip, followed by a rapid decay throughout the uprush flow interior. The peak seaward directed stress during the backwash phase was less than half that measured in the uprush. Close to the still water line, in the region of bore collapse and at the time of initial uprush, favourable pressure gradients were measured. In the lower swash region predominately weak adverse pressure gradients were measured.

Morphostructural analysis of an alpine debris flows catchment: implication for debris supply

Rock slope instabilities are implicitly linked to the supply of sediment and debris recharging channels prone to debris flow. Hence, the incorporation of bedrock structure and terrain morphology can be relevant in the analysis of sediment budget and debris flow hazard assessment. Here, the mode of debris production of the Manival catchment (northern French Alps) is documented by the study of its morphostructural aspects extracted from high resolution DEM. Terrain implication in the process of debris supply is evaluated by: a) A systematic classification of the major morphological units based on the slope gradient that enables a spatial analysis of zones of debris production and deposition. b) A detailed structural analysis performed on DEM in order to identify potential unstable slopes. c) An analysis of the gullies orientation that informs in term of structural control of the sources zones. d) Localisation of high density joints sets that document about whether sources of continuous debris production are controlled by the structural setting of the catchment. These DEM-based indicators can be used as proxies for assessing the influences of the current topography and enable to quantify a degree of susceptibility to mass wasting and hillslope erosion activity. This present contribution suggests some directions for characterizing sediment flux dynamic in small alpine catchment.

Wind-direction effects on urban-type flows

Practically all extant work on flows over obstacle arrays, whether laboratory experiments or numerical modelling, is for cases where the oncoming wind is normal to salient faces of the obstacles. In the field, however, this is rarely the case. Here, simulations of flows at various directions over arrays of cubes representing typical urban canopy regions are presented and discussed. The computations are of both direct numerical simulation and large-eddy simulation type. Attention is concentrated on the differences in the mean flow within the canopy region arising from the different wind directions and the consequent effects on global properties such as the total surface drag, which can change very significantly—by up to a factor of three in some circumstances. It is shown that for a given Reynolds number the typical viscous forces are generally a rather larger fraction of the pressure forces (principally the drag) for non-normal than for normal wind directions and that, dependent on the surface morphology, the average flow direction deep within the canopy can be largely independent of the oncoming wind direction. Even for regular arrays of regular obstacles, a wind direction not normal to the obstacle faces can in general generate a lateral lift force (in the direction normal to the oncoming flow). The results demonstrate this and it is shown how computations in a finite domain with the oncoming flow generated by an appropriate forcing term (e.g. a pressure gradient) then lead inevitably to an oncoming wind direction aloft that is not aligned with the forcing term vector.

The dependence of high-latitude dayside ionospheric flows on the North-South component of the IMF: A high time resolution correlation analysis using EISCAT “Polar” and AMPTE UKS and IRM data

In 1984 and 1985 a series of experiments was undertaken in which dayside ionospheric flows were measured by the EISCAT “Polar” experiment, while observations of the solar wind and interplanetary magnetic field (IMF) were made by the AMPTE UKS and IRM spacecraft upstream from the Earth's bow shock. As a result, 40 h of simultaneous data were acquired, which are analysed in this paper to investigate the relationship between the ionospheric flow and the North-South (Bz) component of the IMF. The ionospheric flow data have 2.5 min resolution, and cover the dayside local time sector from ∼ 09:30 to ∼ 18:30 M.L.T. and the latitude range from 70.8° to 74.3°. Using cross-correlation analysis it is shown that clear relationships do exist between the ionospheric flow and IMF Bz, but that the form of the relations depends strongly on latitude and local time. These dependencies are readily interpreted in terms of a twinvortex flow pattern in which the magnitude and latitudinal extent of the flows become successively larger as Bz becomes successively more negative. Detailed maps of the flow are derived for a range of Bz values (between ± 4 nT) which clearly demonstrate the presence of these effects in the data. The data also suggest that the morning reversal in the East-West component of flow moves to earlier local times as Bz, declines in value and becomes negative. The correlation analysis also provides information on the ionospheric response time to changes in IMF Bz, it being found that the response is very rapid indeed. The most rapid response occurs in the noon to mid-afternoon sector, where the westward flows of the dusk cell respond with a delay of 3.9 ± 2.2 min to changes in the North-South field at the subsolar magnetopause. The flows appear to evolve in form over the subsequent ~ 5 min interval, however, as indicated by the longer response times found for the northward component of flow in this sector (6.7 ±2.2 min), and in data from earlier and later local times. No evidence is found for a latitudinal gradient in response time; changes in flow take place coherently in time across the entire radar field-of-view.

Seasonality of submesoscale flows in the ocean surface boundary layer

A signature of submesoscale flows in the upper ocean is skewness in the distribution of relative vorticity. Expected to result for high Rossby-number flows, such skewness has implications for mixing, dissipation and stratification within the upper ocean. An array of moorings deployed in the Northeast Atlantic for one year as part of the OSMOSIS experiment reveals that relative vorticity is positively skewed during winter even though the scale of the Rossby number is less than 0.5. Furthermore, this skewness is reduced to zero during spring and autumn. There is also evidence of modest seasonal variations in the gradient Rossby number. The proposed mechanism by which relative vorticity is skewed is that the ratio of lateral to vertical buoyancy gradients, as summarized by the inverse gradient Richardson number, restricts its range during winter but less so at other times of the year. These results support recent observations and model simulations suggesting the upper ocean is host to a seasonal cycle in submesoscale turbulence.

An unstructured CVFEM and moving interface algorithm for non-Newtonian Hele-Shaw flows in injection molding

Purpose - The purpose of this paper is to develop a novel unstructured simulation approach for injection molding processes described by the Hele-Shaw model. Design/methodology/approach - The scheme involves dual dynamic meshes with active and inactive cells determined from an initial background pointset. The quasi-static pressure solution in each timestep for this evolving unstructured mesh system is approximated using a control volume finite element method formulation coupled to a corresponding modified volume of fluid method. The flow is considered to be isothermal and non-Newtonian. Findings - Supporting numerical tests and performance studies for polystyrene described by Carreau, Cross, Ellis and Power-law fluid models are conducted. Results for the present method are shown to be comparable to those from other methods for both Newtonian fluid and polystyrene fluid injected in different mold geometries. Research limitations/implications - With respect to the methodology, the background pointset infers a mesh that is dynamically reconstructed here, and there are a number of efficiency issues and improvements that would be relevant to industrial applications. For instance, one can use the pointset to construct special bases and invoke a so-called ""meshless"" scheme using the basis. This would require some interesting strategies to deal with the dynamic point enrichment of the moving front that could benefit from the present front treatment strategy. There are also issues related to mass conservation and fill-time errors that might be addressed by introducing suitable projections. The general question of ""rate of convergence"" of these schemes requires analysis. Numerical results here suggest first-order accuracy and are consistent with the approximations made, but theoretical results are not available yet for these methods. Originality/value - This novel unstructured simulation approach involves dual meshes with active and inactive cells determined from an initial background pointset: local active dual patches are constructed ""on-the-fly"" for each ""active point"" to form a dynamic virtual mesh of active elements that evolves with the moving interface.

Roll waves evolution in high gradient channels to a non-Newtonian rheology

The criteria for the occurrence of roll wave phenomenon in the supercritical and turbulent Newtonian and non-Newtonian flows from the engineering point of view was analyzed. Imposing a constant discharge at the upstream of the canal and superposing a small perturbation, it was observed that roll waves can be developed more easily for small wave numbers and for high cohesions. Moreover, from the mathematical model used, it was demonstrated that the numerical viscosity was 10 times the physical viscosity.

A new enrichment space for the treatment of discontinuous pressures in multi-fluid flows

In this work, a new enrichment space to accommodate jumps in the pressure field at immersed interfaces in finite element formulations, is proposed. The new enrichment adds two degrees of freedom per element that can be eliminated by means of static condensation. The new space is tested and compared with the classical P1 space and to the space proposed by Ausas et al (Comp. Meth. Appl. Mech. Eng., Vol. 199, 10191031, 2010) in several problems involving jumps in the viscosity and/or the presence of singular forces at interfaces not conforming with the element edges. The combination of this enrichment space with another enrichment that accommodates discontinuities in the pressure gradient has also been explored, exhibiting excellent results in problems involving jumps in the density or the volume forces. Copyright (c) 2011 John Wiley & Sons, Ltd.

A numerical study of temperature effects on hot wire measurements inside turbulent channel flows

A way to investigate turbulence is through experiments where hot wire measurements are performed. Analysis of the in turbulence of a temperature gradient on hot wire measurements is the aim of this thesis work. Actually - to author's knowledge - this investigation is the first attempt to document, understand and ultimately correct the effect of temperature gradients on turbulence statistics. However a numerical approach is used since instantaneous temperature and streamwise velocity fields are required to evaluate this effect. A channel flow simulation at Re_tau = 180 is analyzed to make a first evaluation of the amount of error introduced by temperature gradient inside the domain. Hot wire data field is obtained processing the numerical flow field through the application of a proper version of the King's law, which connect voltage, velocity and temperature. A drift in mean streamwise velocity profile and rms is observed when temperature correction is performed by means of centerline temperature. A correct mean velocity pro�le is achieved correcting temperature through its mean value at each wall normal position, but a not negligible error is still present into rms. The key point to correct properly the sensed velocity from the hot wire is the knowledge of the instantaneous temperature field. For this purpose three correction methods are proposed. At the end a numerical simulation at Re_tau =590 is also evaluated in order to confirm the results discussed earlier.

The 1987 debris flows in Switzerland: documentation and analysis

A great number of debris flows occurred during the flood catastrophes of the summer of 1987 in the Swiss Alps. Aerial photography, field investigations and eyewitness accounts documented and analysed the events. As an example of the reconstructed major events, the large debris flow in the Varuna valley involved an estimated peak discharge between 400 and 800 m3/s and an event magnitude of 200,000 m3. Several single pulses were observed; the duration of each of them appeared to be not more than a few minutes. Apart from incision into weak bedrock, the maximum erosion depth seemed to depend on the channel gradient. Based on approximately 600 events, typical starting zones and rainfall conditions are discussed with regard to the triggering conditions. Existing and new empirical formulae are proposed to estimate the most important flow parameters. These values are compared to debris flow data from Canada and Japan.

A regular perturbation approach to surface tension driven flows

Surface tension induced convection in a liquid bridge held between two parallel, coaxial, solid disks is considered. The surface tension gradient is produced by a small temperature gradient parallel Co the undisturbed surface. The study is performed by using a mathematical regular perturbation approach based on a small parameter, e, which measures the deviation of the imposed temperature field from its mean value. The first order velocity field is given by a Stokes-type problem (viscous terms are dominant) with relatively simple boundary conditions. The first order temperature field is that imposed from the end disks on a liquid bridge immersed in a non-conductive fluid. Radiative effects are supposed to be negligible. The second order temperature field, which accounts for convective effects, is split into three components, one due to the bulk motion, and the other two to the distortion of the free surface. The relative importance of these components in terms of the heat transfer to or from the end disks is assessed