UMD Theses and Dissertations
Permanent URI for this collectionhttp://hdl.handle.net/1903/3
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Item Mathematical Problems Arising When Connecting Kinetic to Fluid Regimes(2008-07-31) Sun, Weiran; Levermore, Charles D.; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In this dissertation we study two problems that are related to the question of how to obtain appropriate macroscopic descriptions of a gas from its microscopic formulation. Mathematically, fluid equations formulate the macroscopic dynamics of a gas while kinetic equations are used to study the microscopic world. One can derive fluid equations from kinetic equations through formal asymptotic expansions like those of Hilbert or Chapman-Enskog. The first problem we study relates to the justification of the steps in those formal expansions, while the second relates to the well-posedness of a resulting fluid system. The first problem we study is that of establishing a Fredholm alternative for the linearized Boltzmann collision operator. The Fredholm alternative is used in both the formal asymptotic derivations and the rigorous justifications of fluid approximations to the Boltzmann equation. Results of this type have been obtained for collision kernels satisfying the Grad angular cutoff assumption. However, because DiPerna-Lions' renormalized solutions for the Boltzmann equation are established for more general collision kernels, it is interesting to extend the Fredholm property of the linearized Boltzmann operator to these collision kernels. We show that under a weak cutoff assumption, the linearized Boltzamnn operator does satisfy the Fredholm alternative. The second problem we study is the well-posedness of a dispersive fluid system that is formally obtained via an asymptotic expansion of the Boltzmann equation as a first correction to the compressible Navier-Stokes system. This system is degenerate in both dissipation and dispersion. Therefore the theory for strictly dispersive systems does not apply directly. To prove the well-posedness of this degenerate system, we need to study different smoothing effects for different components of the solution. We show that using the regularization effects including dispersion and dissipation, this system has a unique smooth solution for a finite time.Item On The Fluid Dynamics of Virtual Impaction and The Design of a Slit Aerosol Sampler(2006-09-18) Charrouf, Marwan; Calabrese, Richard V.; Chemical Engineering; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)It has been long established that Reynolds number effects can lead to flow instabilities and/or transition from laminar to turbulent flow regimes. The nature of free shear jets is well understood and heavily covered in the fluid mechanics literature. On the other hand, the study of confined nozzles presents some challenges and is still a developing area of research. In this work, we focus on quasi-impinging jets, such as the ones feeding into a virtual impactor. Virtual impactors are popular, inexpensive aerosol collection devices capable of separating airborne solid particles. Recently they found increased application in areas that require concentration of dilute aerosols, such as biological-laden flows. In essence, this research is motivated by the need to fundamentally understand the fluid-particle interaction mechanisms entailed during virtual impaction. To this end, we rely on theoretical insight gained by numerical analysis of the classical equations within a one-way coupled Lagrangian framework. In the first part of this investigation we perform a direct transient simulation of the two-dimensional incompressible Navier-Stokes equations for air as the carrier phase. The momentum and continuity equations are solved by FLUENT. The solutions of three separate computations with jet Reynolds numbers equal to 350, 2100, and 3500 are analyzed. The 2-D time-mean results established the nature of the jet potential core and clarifications about the role of the Reynolds number were proposed. Transient analysis deciphered the characteristics of the mirrored Kelvin-Helmholtz instability, along with particle-eddy interaction mechanisms. In the second part we perform a large eddy simulation (LES) on a domain of a real-life sampler. The Lagrangian dynamic residual stress model is implemented and validated for two canonical turbulent flows. The newly contrived code is then applied to the study of a prototype device. A three-dimensional growth mechanism is proposed for the jet mixing layers. The Lagrangian dynamic model LES exhibited significant regions of high subgrid turbulent viscosity, compared to the dynamic Lilly-model simulation, and we were able to identify the origin, and learn the dynamics of five key coherent structures dominant during transition. Comparison with preliminary experimental data for the aerosol separation efficiency showed fairly good agreement.