The shallow water approximation applied to the aluminium electrolysis process

The industrial production of aluminium is an electrolysis process where two superposed horizontal liquid layers are subjected to a mainly vertical electric current supplied by carbon electrodes. The lower layer consists of molten aluminium and lies on the cathode. The upper layer is the electrolyte and is covered by the anode. The interface between the two layers is often perturbed, leading to oscillations, or waves, similar to the waves on the surface of seas or lakes. The presence of electric currents and the resulting magnetic field are responsible for electromagnetic (Lorentz) forces within the fluid, which can amplify these oscillations and have an adverse influence on the process. The electrolytic bath vertical to horizontal aspect ratio is such, that it is advantageous to use the shallow water equations to model the interface motion. These are the depth-averaging the Navier-Stokes equations so that nonlinear and dispersion terms may be taken into account. Although these terms are essential to the prediction of wave dynamics, they are neglected in most of the literature on interface instabilities in aluminium reduction cells where only the linear theory is usually considered. The unknown variables are the two horizontal components of the fluid velocity, the height of the interface and the electric potential. In this application, a finite volume resolution of the double-layer shallow water equations including the electromagnetic sources has been developed, for incorporation into a generic three-dimensional computational fluid dynamics code that also deals with heat transfer within the cell.

Coupled CFD Shape Optimization for aerodynamic profiles

The present document deals with the optimization of shape of aerodynamic profiles -- The objective is to reduce the drag coefficient on a given profile without penalising the lift coefficient -- A set of control points defining the geometry are passed and parameterized as a B-Spline curve -- These points are modified automatically by means of CFD analysis -- A given shape is defined by an user and a valid volumetric CFD domain is constructed from this planar data and a set of user-defined parameters -- The construction process involves the usage of 2D and 3D meshing algorithms that were coupled into own- code -- The volume of air surrounding the airfoil and mesh quality are also parametrically defined -- Some standard NACA profiles were used by obtaining first its control points in order to test the algorithm -- Navier-Stokes equations were solved for turbulent, steady-state ow of compressible uids using the k-epsilon model and SIMPLE algorithm -- In order to obtain data for the optimization process an utility to extract drag and lift data from the CFD simulation was added -- After a simulation is run drag and lift data are passed to the optimization process -- A gradient-based method using the steepest descent was implemented in order to define the magnitude and direction of the displacement of each control point -- The control points and other parameters defined as the design variables are iteratively modified in order to achieve an optimum -- Preliminary results on conceptual examples show a decrease in drag and a change in geometry that obeys to aerodynamic behavior principles

Formulation and analysis of a diffusion-velocity particle model for transport-dispersion equations

The modelling of diffusive terms in particle methods is a delicate matter and several models were proposed in the literature to take such terms into account. The diffusion velocity method (DVM), originally designed for the diffusion of passive scalars, turns diffusive terms into convective ones by expressing them as a divergence involving a so-called diffusion velocity. In this paper, DVM is extended to the diffusion of vectorial quantities in the three-dimensional Navier–Stokes equations, in their incompressible, velocity–vorticity formulation. The integration of a large eddy simulation (LES) turbulence model is investigated and a DVM general formulation is proposed. Either with or without LES, a novel expression of the diffusion velocity is derived, which makes it easier to approximate and which highlights the analogy with the original formulation for scalar transport. From this statement, DVM is then analysed in one dimension, both analytically and numerically on test cases to point out its good behaviour.

A numerical model for prediction of oil, trajectory due to extraction activity in the north coasts of the Persian Gulf

This thesis considers a three- dimensional numerical model based on 3-D Navier— Stokes and continuity equations involving various wind speeds (North west), water surface levels, horizontal shier stresses, eddy viscosity, densities of oil and gas condensate- water mixture flows. The model is used to simulate the prediction of the surface movement of oil and gas condensate slicks from spill accident in the north coasts of Persian Gulf.

Asymptotic preserving and time diminishing schemes for rarefied gas dynamic

In this work, we introduce a new class of numerical schemes for rarefied gas dynamic problems described by collisional kinetic equations. The idea consists in reformulating the problem using a micro-macro decomposition and successively in solving the microscopic part by using asymptotic preserving Monte Carlo methods. We consider two types of decompositions, the first leading to the Euler system of gas dynamics while the second to the Navier-Stokes equations for the macroscopic part. In addition, the particle method which solves the microscopic part is designed in such a way that the global scheme becomes computationally less expensive as the solution approaches the equilibrium state as opposite to standard methods for kinetic equations which computational cost increases with the number of interactions. At the same time, the statistical error due to the particle part of the solution decreases as the system approach the equilibrium state. This causes the method to degenerate to the sole solution of the macroscopic hydrodynamic equations (Euler or Navier-Stokes) in the limit of infinite number of collisions. In a last part, we will show the behaviors of this new approach in comparisons to standard Monte Carlo techniques for solving the kinetic equation by testing it on different problems which typically arise in rarefied gas dynamic simulations.

Adaptive discontinuous Galerkin methods for eigenvalue problems arising in incompressible fluid flows

In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the hydrodynamic stability problem associated with the incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the eigenvalue problem in channel and pipe geometries. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard Dual-Weighted-Residual approach, originally developed for the estimation of target functionals of the solution, to eigenvalue/stability problems. The underlying analysis consists of constructing both a dual eigenvalue problem and a dual problem for the original base solution. In this way, errors stemming from both the numerical approximation of the original nonlinear flow problem, as well as the underlying linear eigenvalue problem are correctly controlled. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented.

Error estimation and adaptive mesh refinement for aerodynamic flows

This lecture course covers the theory of so-called duality-based a posteriori error estimation of DG finite element methods. In particular, we formulate consistent and adjoint consistent DG methods for the numerical approximation of both the compressible Euler and Navier-Stokes equations; in the latter case, the viscous terms are discretized based on employing an interior penalty method. By exploiting a duality argument, adjoint-based a posteriori error indicators will be established. Moreover, application of these computable bounds within automatic adaptive finite element algorithms will be developed. Here, a variety of isotropic and anisotropic adaptive strategies, as well as $hp$-mesh refinement will be investigated.

Adaptive coatings based on polyaniline for direct 2D observation of diffusion processes in microfluidic systems

[EN] Therefore the understanding and proper evaluation of the flow and mixing behaviour at microscale becomes a very important issue. In this study, the diffusion behaviour of two reacting solutions of HCI and NaOH were directly observed in a glass/polydimethylsiloxane microfluidic device using adaptive coatings based on the conductive polymer polyaniline that are covalently attached to the microchannel walls. The two liquid streams were combined at the junction of a Y-shaped microchannel, and allowed to diffuse into each other and react. The results showed excellent correlation between optical observation of the diffusion process and the numerical results. A numerical model which is based on finite volume method (FVM) discretisation of steady Navier-Stokes (fluid flow) equations and mass transport equations without reactions was used to calculate the flow variables at discrete points in the finite volume mesh element. The high correlation between theory and practical data indicates the potential of such coatings to monitor diffusion processes and mixing behaviour inside microfluidic channels in a dye free environment.

Low cost 3D global instability analysis and flow sensitivity based on dynamic mode decomposition and high-order numerical tools

We explore the recently developed snapshot-based dynamic mode decomposition (DMD) technique, a matrix-free Arnoldi type method, to predict 3D linear global flow instabilities. We apply the DMD technique to flows confined in an L-shaped cavity and compare the resulting modes to their counterparts issued from classic, matrix forming, linear instability analysis (i.e. BiGlobal approach) and direct numerical simulations. Results show that the DMD technique, which uses snapshots generated by a 3D non-linear incompressible discontinuous Galerkin Navier?Stokes solver, provides very similar results to classical linear instability analysis techniques. In addition, we compare DMD results issued from non-linear and linearised Navier?Stokes solvers, showing that linearisation is not necessary (i.e. base flow not required) to obtain linear modes, as long as the analysis is restricted to the exponential growth regime, that is, flow regime governed by the linearised Navier?Stokes equations, and showing the potential of this type of analysis based on snapshots to general purpose CFD codes, without need of modifications. Finally, this work shows that the DMD technique can provide three-dimensional direct and adjoint modes through snapshots provided by the linearised and adjoint linearised Navier?Stokes equations advanced in time. Subsequently, these modes are used to provide structural sensitivity maps and sensitivity to base flow modification information for 3D flows and complex geometries, at an affordable computational cost. The information provided by the sensitivity study is used to modify the L-shaped geometry and control the most unstable 3D mode.

Simulação de fluxo de fluidos em meios porosos desordenados uma análise de efeito de escala na estimativa da permeabilidade e do coeficiente de arrasto

The present study provides a methodology that gives a predictive character the computer simulations based on detailed models of the geometry of a porous medium. We using the software FLUENT to investigate the flow of a viscous Newtonian fluid through a random fractal medium which simplifies a two-dimensional disordered porous medium representing a petroleum reservoir. This fractal model is formed by obstacles of various sizes, whose size distribution function follows a power law where exponent is defined as the fractal dimension of fractionation Dff of the model characterizing the process of fragmentation these obstacles. They are randomly disposed in a rectangular channel. The modeling process incorporates modern concepts, scaling laws, to analyze the influence of heterogeneity found in the fields of the porosity and of the permeability in such a way as to characterize the medium in terms of their fractal properties. This procedure allows numerically analyze the measurements of permeability k and the drag coefficient Cd proposed relationships, like power law, for these properties on various modeling schemes. The purpose of this research is to study the variability provided by these heterogeneities where the velocity field and other details of viscous fluid dynamics are obtained by solving numerically the continuity and Navier-Stokes equations at pore level and observe how the fractal dimension of fractionation of the model can affect their hydrodynamic properties. This study were considered two classes of models, models with constant porosity, MPC, and models with varying porosity, MPV. The results have allowed us to find numerical relationship between the permeability, drag coefficient and the fractal dimension of fractionation of the medium. Based on these numerical results we have proposed scaling relations and algebraic expressions involving the relevant parameters of the phenomenon. In this study analytical equations were determined for Dff depending on the geometrical parameters of the models. We also found a relation between the permeability and the drag coefficient which is inversely proportional to one another. As for the difference in behavior it is most striking in the classes of models MPV. That is, the fact that the porosity vary in these models is an additional factor that plays a significant role in flow analysis. Finally, the results proved satisfactory and consistent, which demonstrates the effectiveness of the referred methodology for all applications analyzed in this study.

Hidden Schrödinger symmetry in FLRW and Bianchi I minisuperspace models of cosmology and black holes Schwarzschild dynamics

A broad sector of literature focuses on the relationship between fluid dynamics and gravitational systems. This thesis presents results that suggest the existence of a new kind of fluid/gravity duality not based on the holographic principle. The goal is to provide tools that allow us to systematically unearth hidden symmetries for reduced models of cosmology. The focus is on the field space of these models, i.e. the superspace. In fact, conformal isometries of the supermetric leave geodesics in the field space unaltered; this leads to symmetries of the models. An innovative aspect is the use of the Eisenhart-Duval’s lift. Using this method, systems constrained by a potential can be treated as free ones. Moreover, charges explicitly dependent on time, i.e. dynamical, can be found. A detailed analysis is carried out on three basic models of homogenous cosmology: i) flat Friedmann-Lemaître-Robertson-Walker’s isotropic universe filled with a massless scalar field; ii) Schwarzschild’s black hole mechanics and its extension to vacuum (A)dS gravity; iii) Bianchi’s anisotropic type I universe with a massless scalar field. The results show the presence of a hidden Schrödinger’s symmetry which, being intrinsic to both Navier-Stokes’ and Schrödinger’s equations, indicates a correspondence between cosmology and hydrodynamics. Furthermore, the central extension of this algebra explicitly relates two concepts. The first is the number of particles coming from the fluid picture; while the second is the ratio between the IR and UV cutoffs that weighs how much a theory has of “classical” over “quantum”. This suggests a spacetime that emerges from an underlying world which is described by quantum building blocks. These quanta statistically conspire to appear as gravitational phenomena from a macroscopic point of view.

URANS calculations for smooth circular cylinder flow in a wide range of Reynolds Numbers: solution verification and validation

The flow around circular smooth fixed cylinder in a large range of Reynolds numbers is considered in this paper. In order to investigate this canonical case, we perform CFD calculations and apply verification & validation (V&V) procedures to draw conclusions regarding numerical error and, afterwards, assess the modeling errors and capabilities of this (U)RANS method to solve the problem. Eight Reynolds numbers between Re = 10 and Re 5 x 10(5) will be presented with, at least, four geometrically similar grids and five discretization in time for each case (when unsteady), together with strict control of iterative and round-off errors, allowing a consistent verification analysis with uncertainty estimation. Two-dimensional RANS, steady or unsteady, laminar or turbulent calculations are performed. The original 1994 k - omega SST turbulence model by Menter is used to model turbulence. The validation procedure is performed by comparing the numerical results with an extensive set of experimental results compiled from the literature. [DOI: 10.1115/1.4007571]

Monte Carlo modeling of polarized light propagation: Stokes vs Jones - Part II

In this second part of our comparative study inspecting the (dis)similarities between “Stokes” and “Jones,” we present simulation results yielded by two independent Monte Carlo programs: (i) one developed in Bern with the Jones formalism and (ii) the other implemented in Ulm with the Stokes notation. The simulated polarimetric experiments involve suspensions of polystyrene spheres with varying size. Reflection and refraction at the sample/air interfaces are also considered. Both programs yield identical results when propagating pure polarization states, yet, with unpolarized illumination, second order statistical differences appear, thereby highlighting the pre-averaged nature of the Stokes parameters. This study serves as a validation for both programs and clarifies the misleading belief according to which “Jones cannot treat depolarizing effects.”

Transverse galloping at low Reynolds numbers

The possibility of transverse galloping of a square cylinder at low Reynolds numbers (Re≤200Re≤200, so that the flow is presumably laminar) is analysed. Transverse galloping is here considered as a one-degree-of-freedom oscillator subjected to fluid forces, which are described by using the quasi-steady hypothesis (time-averaged data are extracted from previous numerical simulations). Approximate solutions are obtained by means of the method of Krylov-Bogoliubov, with two major conclusions: (i) a square cylinder cannot gallop below a Reynolds number of 159 and (ii) in the range 159≤Re≤200159≤Re≤200 the response exhibits no hysteresis.

Reynolds /

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