The two-phase flow displacement of a viscous fluid by a less viscous one in a confined environment leads to a viscous fingering instability commonly encountered in natural systems, for example, in flows through porous media or pulmonary airways. The classical study of viscous fingering has been conducted in rectangular channels of high aspect ratio (large channel width/height), known as Hele-Shaw channels where a unique, steady symmetric, semi-infinite bubble (finger) emerges. In this Journal Format thesis, the propagation of semi-infinite (open) and finite (closed) air bubbles is considered in Hele-Shaw channels where thin, axially-uniform occlusions are introduced. This configuration is known to generate symmetric, asymmetric and oscillatory modes with complex interactions and rich behaviour. Numerical results of finger propagation using a depth-averaged model in these constricted channels are found to be in quantitative agreement with experimental results once the aspect ratio reaches a value of $\alpha\geq40$ and capillary numbers below $Ca\leq 0.012$. The same evolution of the bifurcation scenario between multiple modes is found, however, it occurs for decreasing values of occlusion height as the value of aspect ratio is increased that the system exhibits sensitivity to small but finite depth-variations. The numerical simulations reveal multiple-tipped unstable symmetric solutions which interact with the single symmetric mode at vanishing occlusion heights resulting in stabilisation of the asymmetric and oscillatory modes. Moreover, deviations from the single symmetric mode are predicted when depth-variations of order of the roughness of the channel walls ($\sim 1$ $\mu$m) are introduced for larger aspect ratios of $\alpha\geq 155$. The propagation of finite bubbles is studied in a channel with constant aspect ratio of $\alpha=30$ and where the height of the occlusion, termed rail, is $1/40$ of the channel height. For bubble diameters of the order of the rail width, a tongue-shaped stability boundary for symmetric (on-rail) propagation is encountered so that for flow rates marginally larger than a critical value, a narrow band of bubble sizes can propagate (stably) over the rail while bubbles of other sizes segregate to the side of the rail. The numerical depth-averaged model is adapted for bubble propagation and captures in qualitative agreement the experimental observations. Time-dependent calculations are additionally performed, showing that on-rail bubble propagation is the result of a non-trivial dynamical interaction between capillary and viscous forces.

The intent of this thesis is to simulate the effect of a background magnetic field on Rayleigh-Taylor (RT) instability morphology and evolution in support of a Discovery Science campaign at the National Ignition Facility. The RT instability is relevant in High Energy Density (HED) systems including supernova remnants such as the Crab Nebula and inertial fusion confinement (ICF). Magnetic fields affect RT evolution and can suppress small-scale fluid motion. Thus far no experimental work has quantified the effect of a B-field on RT evolution morphology. RT evolution under a B-field was examined in three-dimensional magnetohydrodynamic (MHD) simulations using the hydrocode ARES, developed by Lawrence Livermore National Laboratory. The parameter space of the experiment is explored to determine the parameters that yield a visible effect on RT evolution. The effect of resistive MHD and conductivity is examined to further establish the desired parameter space to observe the suppression of RT morphology.

Cette thèse est consacrée à l’étude de la turbulence homogène stratifiée instable, un écoulement idéalisé décrivant l’évolution de la turbulence au sein d’une zone de mélange de type Rayleigh-Taylor. Cette approche se concentre sur l’évolution des quantités fluctuantes ;l’influence de l’écoulement moyen est prise en compte au travers d’un gradient moyen de densité. Un modèle spectral est utilisé pour étudier cette turbulence, conjointement à des simulations numériques directes. En comparaison avec ces simulations, l’étape de validation du modèle met en lumière le rôle des termes de stratification sur la dynamique du transfert d’énergie. Une première étude montre l’établissement, dans l’état autosemblable, de lois d’échelles ainsi que l’influence de la distribution initiale d’énergie sur l’état asymptotique et sur l’anisotropie de l’écoulement. Dans une seconde étude, la rétroaction de la turbulence sur le gradient moyen est introduite, dans un premier temps, afin de rapprocher la dynamique autosemblable de la turbulence homogène stratifiée instable de celle observée en turbulence Rayleigh-Taylor. Dans un second temps, l’influence d’un renversement de la stratification sur la dynamique du mélange est étudiée au travers d’un profil d’accélération variable. / This thesis is dedicated to the study of unstably stratified homogeneous turbulence.This flow is an idealized framework introduced to investigate the turbulence developing at the centerline of a Rayleigh-Taylor mixing zone. This approach focuses on turbulent quantities, when the mean flow acts on the turbulent field through a mean density gradient.A spectral model and direct numerical simulations are used to study this turbulent flow.The validation step reveals the role of stratification terms on the energy transfer dynamic.Then, a first study shows the emergence of scaling laws in the self-similar state, together with the large scale energy distribution impact on the asymptotic state and on the flow anisotropy. In a second study, the turbulent retroaction on the mean density gradient is introduced in order to bring unstably stratified homogeneous turbulence closer to theRayleigh-Taylor turbulence dynamics. This step leads to investigate the consequences of a stratification inversion on the mixing dynamics through a variable acceleration profile.

There are two coupled primary objectives for this study of buoyancy-driven turbulence. The first objective is to create a new diagnostic for collection of measurements to capture the physics of Rayleigh-Taylor (RT) mixing. The second objective is to use the new diagnostic to specifically elucidate the physics of large Atwood number, ( )( )2 1 2 1 / ρ ρ ρ ρ + − = t A , RT mixing. Both of these objectives have been satisfied through the development of a new hot-wire diagnostic to study buoyancy-driven turbulence in a statistically steady gas channel of helium and air ( 6 . 0 03 . 0 ≤ ≤ t A ). The capability of the diagnostic to simultaneously and instantaneously measure turbulent velocity and density fluctuations allows for a unique investigation into the dynamics of Rayleigh-Taylor mixing layers at large At, through measurements of turbulence and mixing statistics. The new hot-wire diagnostic uses temperature as a fluid marker for helium and air, which is possible due to the Lewis number ~ 1 (Le = ratio of thermal diffusivity to mass diffusivity) for helium and air, and the new diagnostic has been validated in an At = 0.03 mixing layer. The energy density spectrum of v′ ′ ρ , measured experimentally for the first time in RT mixing, is found to closely follow the energy distribution of v′ , up to the Reynolds numbers investigated ( ( ) mix t h gA h υ 6 2 Re 2 / 3 = ~ 1450). Large At experiments, with At = 0.6, have also been achieved for the first time in a miscible RT mixing layer. An asymmetric penetration of the bubbles (rising fluid) and spikes (falling fluid) has been observed, resulting in measured self similar growth parameters αb = 0.060 and αs = 0.088 for the bubbles and spikes, respectively. The first experimental measurements of turbulent velocity and density fluctuations for the large At case, show a strong similarity to lower At behaviors when normalized. However conditional statistics, which separate the bubble (light fluid) and spike (heavy fluid) dynamics, has highlighted differences in v′ ′ ρ and rms v′ in the bubbles and spikes. Larger values of v′ ′ ρ and rms v′ were found in the downward falling spikes, which is consistent with the larger growth rates and momentum of the spikes compared to the bubbles. These conditional statistics are a first in RT driven turbulence.

Experiments and simulations were performed to examine the complex processes that occur in Rayleigh�Taylor driven mixing. A water channel facility was used to examine a buoyancy-driven Rayleigh�Taylor mixing layer. Measurements of �uctuating den- sity statistics and the molecular mixing parameter were made for Pr = 7 (hot/cold water) and Sc 103 (salt/fresh water) cases. For the hot/cold water case, a high- resolution thermocouple was used to measure instantaneous temperature values that were related to the density �eld via an equation of state. For the Sc 103 case, the degree of molecular mixing was measured by monitoring a di�usion-limited chemical reaction between the two �uid streams. The degree of molecular mixing was quanti- �ed by developing a new mathematical relationship between the amount of chemical product formed and the density variance 02. Comparisons between the Sc = 7 and Sc 103 cases are used to elucidate the dependence of on the Schmidt number. To further examine the turbulent mixing processes, a direct numerical simu- lation (DNS) model of the Sc = 7 water channel experiment was constructed to provide statistics that could not be experimentally measured. To determine the key physical mechanisms that in�uence the growth of turbulent Rayleigh�Taylor mixing layers, the budgets of the exact mean mass fraction em1, turbulent kinetic energy fE00, turbulent kinetic energy dissipation rate e 00, mass fraction variance gm002 1 , and mass fraction variance dissipation rate f 00 equations were examined. The budgets of the unclosed turbulent transport equations were used to quantitatively assess the relative magnitudes of di�erent production, dissipation, transport, and mixing processes. Finally, three-equation (fE00-e 00-gm002 1 ) and four-equation (fE00-e 00-gm002 1 -f 00) turbulent mixing models were developed and calibrated to predict the degree of molecular mix- ing within a Rayleigh�Taylor mixing layer. The DNS data sets were used to assess the validity of and calibrate the turbulent viscosity, gradient-di�usion, and scale- similarity closures a priori. The modeled transport equations were implemented in a one-dimensional numerical simulation code and were shown to accurately reproduce the experimental and DNS results a posteriori. The calibrated model parameters from the Sc = 7 case were used as the starting point for determining the appropri- ate model constants for the mass fraction variance gm002 1 transport equation for the Sc 103 case.

The Richtmyer-Meshkov instability occurs when density-stratified interfaces are impulsively accelerated, typically by a shock wave. We present a numerical method to simulate the Richtmyer-Meshkov instability in cylindrical geometry. The ideal MHD equations are linearized about a time-dependent base state to yield linear partial differential equations governing the perturbed quantities. Convergence tests demonstrate that second order accuracy is achieved for smooth flows, and the order of accuracy is between first and second order for flows with discontinuities. Numerical results are presented for cases of interfaces with positive Atwood number and purely azimuthal perturbations. In hydrodynamics, the Richtmyer-Meshkov instability growth of perturbations is followed by a Rayleigh-Taylor growth phase. In MHD, numerical results indicate that the perturbations can be suppressed for sufficiently large perturbation wavenumbers and magnetic fields.

We investigate the Richtmyer-Meshkov instability (RMI) that occurs when an incident shock impulsively accelerates the interface between two different fluids. RMI is important in many technological applications such as Inertial Confinement Fusion (ICF) and astrophysical phenomena such as supernovae. We consider RMI in the presence of the magnetic field in converging geometry through both simulations and analytical means in the framework of ideal magnetohydrodynamics (MHD). In this thesis, we perform linear stability analyses via simulations in the cylindrical geometry, which is of relevance to ICF. In converging geometry, RMI is usually followed by the Rayleigh-Taylor instability (RTI). We show that the presence of a magnetic field suppresses the instabilities. We study the influence of the strength of the magnetic field, perturbation wavenumbers and other relevant parameters on the evolution of the RM and RT instabilities. First, we perform linear stability simulations for a single interface between two different fluids in which the magnetic field is normal to the direction of the average motion of the density interface. The suppression of the instabilities is most evident for large wavenumbers and relatively strong magnetic fields strengths. The mechanism of suppression is the transport of vorticity away from the density interface by two Alfv ́en fronts. Second, we examine the case of an azimuthal magnetic field at the density interface. The most evident suppression of the instability at the interface is for large wavenumbers and relatively strong magnetic fields strengths. After the shock interacts with the interface, the emerging vorticity breaks up into waves traveling parallel and anti-parallel to the magnetic field. The interference as these waves propagate with alternating phase causing the perturbation growth rate of the interface to oscillate in time. Finally, we propose incompressible models for MHD RMI in the presence of normal or azimuthal magnetic field. The linearized equations are solved numerically using inverse Laplace transform. The incompressible models show that the magnetic field suppresses the RMI, and the mechanism of this suppression depends on the orientation of the initially applied magnetic field. The incompressible model agrees reasonably well with compressible linear simulations.

Experimental results and simulations of imploding fusion concepts have identified the Rayleigh-Taylor (RT) instability as one of the largest inhibitors to achieving fusion. Understanding the origin and development of the RT instability will allow for the development of mitigating measures to dampen the instability growth, thus improving the chance that fusion concepts such as inertial confinement fusion (ICF) are successful. A study of 1D and 2D simulations are presented for investigating RT instability growth in deceleration stage of imploding geometries. Two cases of laser-driven implosion geometry, Cartesian and cylindrical, are used to study late stage deceleration-phase RT instability development on the interior surface of imploding targets. FLASH's hydrodynamic (HD) and magnetohydrodynamic (MHD) modeling capabilities are used for different laser and target parameters in order to study the RT instability and the impact of externally applied magnetic fields on their evolution. Several simulation regimes have been identified that provide novel insight into the impact that a seeded magnetic field can have on RT instability growth and the conditions under which magnetic field stabilization of the RT instability is observable. Finally, future work and recommendations are made. / Master of Science / The direction for the future of renewable energy is uncertain at this time; however, it is known that the future of human energy consumption must be green in order to be sustainable. Fusion energy presents an opportunity for an unlimited clean renewable energy source that has yet to be realized. Fusion is achieved only by overcoming the earthly limitations presented by trying to replicate conditions at the interior of stellar structures. The pressures, temperature, and densities seen in the interior of stars are not easily reproduced, and thus human technology must be developed to reach these difficult stellar conditions in order to harvest fusion energy. There are two main branches of developmental technology geared towards achieving the difficult conditions controlled nuclear fusion presents, magnetic confinement fusion (MCF) and inertial confinement fusion (ICF)[17]. Yet in both approaches barriers exist which have thwarted the efforts toward reaching fusion ignition which must be addressed through scientific discovery. Successfully reaching ignition is only the first step in the ultimate pursuit of a self sustaining fusion reactor. This work will focus on the experimental ICF configuration, and on one such inhibitor toward achieving ignition, the Rayleigh-Taylor (RT) instability. The RT instability develops on the surfaces of the fusion fuel capsules, targets, and causes nonuniform compression of the target. This nonuniform compression of the target leads to lower pressures and densities through the material mixing of fusion fuel and the capsule shell, which ultimately leads to challenges with reaching fusion ignition. The work presented here was performed utilizing the University of Chicago's FLASH code, which is a state-of-the-art open source radiation magneto-hydrodynamic (MHD) code used for plasma and astrophysics computational modeling [11]. Simulations of the RT instability are performed using FLASH in planar and cylindrical geometries to explore fundamental Rayleigh-Taylor instability evolution for these two different geometries. These geometries provide easier access for experimental diagnostics to probe RT dynamics. Additionally, the impact of externally applied magnetic fields are explored in an effort to examine if and how the detrimental instability can be controlled.

Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro / Em suma, esta tese propõe uma metodologia de acompanhamento de interfaces móveis que baseia-se no método dos conjuntos de nível aqui chamado de método das curvas de nível, uma denominação baseada nas aplicações em que as interfaces são representadas por curvas acoplado a uma implementação semi-Lagrangeana, para problemas em meios porosos. Embora esta técnica possa, em princípio, ser aplicada a qualquer problema físico que apresente uma interface móvel, nesta tese são focados escoamentos em meios porosos consolidados e saturados por um ou dois fluidos imiscíveis e incompressíveis. Adicionalmente, um método iterativo paralelizável para a resolução de sistemas de equações lineares definidos em redes, que podem ser reduzidos à forma das equações fundamentais de equilíbrio, é empregado na determinação dos campos de velocidade associados aos escoamentos em meios porosos. O cenário semi-Lagrangeano acoplado ao método das curvas de nível é comparado com a implementação utilizando o bem conhecido esquema up-wind. Um exaustivo estudo realizado revela a superioridade da metodologia proposta frente à concorrente utilizando o up-wind. Finalmente, o método das curvas de nível com implementação semi-Lagrangeana (método semi-Lagrangeano das curvas de nível), e o método iterativo para a determinação do campo de velocidades são aplicados no estudo de problemas transientes em meios porosos que apresentam instabilidades dos tipos Saffman-Taylor e Rayleigh-Taylor. Este estudo envolve uma análise de estabilidade linear, a introdução de diversas perturbações trigonométricas na interface e a sua evolução não-linear. / Briefy, this thesis proposes a method for capturing moving interfaces based on the level set method coupled to a Semi-Lagrangian implementation for problems in porous media. Although this method could, in principle, be applied to any physical problem with moving interfaces, we foccus, in this thesis, on flows inside a consolidated porous media saturated by one or two imiscible and incompressible fluids. Besides, a parallelizable iterative method for solving linear systems defined on a network that can be reduced to the fundamental equilibrium equations, is employed to determine the velocity field associated with the flow in a porous medium. The semi-Lagrangian scheme coupled with the level set method is compared with the well-known implementation with the up-wind scheme. An exhaustive study is performed and reveals the superiority of the proposed scheme in relation to the competing one using the up-wind method. Finally, the level set method with semi-Lagrangian implementation and the iterative method for determining the velocity field are applied to the study of transient problems in porous media which present Saffman-Taylor and Rayleigh-Taylor instabilities. This study involves the application of a linear stability analysis, the introduction of several trigonometric perturbations to the interface and its non-linear evolution.

Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro / Em suma, esta tese propõe uma metodologia de acompanhamento de interfaces móveis que baseia-se no método dos conjuntos de nível aqui chamado de método das curvas de nível, uma denominação baseada nas aplicações em que as interfaces são representadas por curvas acoplado a uma implementação semi-Lagrangeana, para problemas em meios porosos. Embora esta técnica possa, em princípio, ser aplicada a qualquer problema físico que apresente uma interface móvel, nesta tese são focados escoamentos em meios porosos consolidados e saturados por um ou dois fluidos imiscíveis e incompressíveis. Adicionalmente, um método iterativo paralelizável para a resolução de sistemas de equações lineares definidos em redes, que podem ser reduzidos à forma das equações fundamentais de equilíbrio, é empregado na determinação dos campos de velocidade associados aos escoamentos em meios porosos. O cenário semi-Lagrangeano acoplado ao método das curvas de nível é comparado com a implementação utilizando o bem conhecido esquema up-wind. Um exaustivo estudo realizado revela a superioridade da metodologia proposta frente à concorrente utilizando o up-wind. Finalmente, o método das curvas de nível com implementação semi-Lagrangeana (método semi-Lagrangeano das curvas de nível), e o método iterativo para a determinação do campo de velocidades são aplicados no estudo de problemas transientes em meios porosos que apresentam instabilidades dos tipos Saffman-Taylor e Rayleigh-Taylor. Este estudo envolve uma análise de estabilidade linear, a introdução de diversas perturbações trigonométricas na interface e a sua evolução não-linear. / Briefy, this thesis proposes a method for capturing moving interfaces based on the level set method coupled to a Semi-Lagrangian implementation for problems in porous media. Although this method could, in principle, be applied to any physical problem with moving interfaces, we foccus, in this thesis, on flows inside a consolidated porous media saturated by one or two imiscible and incompressible fluids. Besides, a parallelizable iterative method for solving linear systems defined on a network that can be reduced to the fundamental equilibrium equations, is employed to determine the velocity field associated with the flow in a porous medium. The semi-Lagrangian scheme coupled with the level set method is compared with the well-known implementation with the up-wind scheme. An exhaustive study is performed and reveals the superiority of the proposed scheme in relation to the competing one using the up-wind method. Finally, the level set method with semi-Lagrangian implementation and the iterative method for determining the velocity field are applied to the study of transient problems in porous media which present Saffman-Taylor and Rayleigh-Taylor instabilities. This study involves the application of a linear stability analysis, the introduction of several trigonometric perturbations to the interface and its non-linear evolution.