Two - Particle Dispersion Models for Turbulent Flows
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Date
2014-08-18
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Johns Hopkins University
Abstract
This dissertation focuses on the behavior of $\langle |\br|^2\rangle$ in a turbulent flow, where $\br=\bx_1-\bx_2$ is the separation distance between particle pairs. There are three main contributions of this thesis:
\begin{itemize}
\item We developed a new formalism for the study of backward dispersion for both deterministic and stochastic tracers. We performed a systematic numerical study of deterministic tracers and investigated small and long time scaling laws, revealing a small time $t^4$ scaling and verifying the $t^3$ behavior at long time. This $t^4$ is a higher order term expansion of the dispersion after the ballistic $t^2$ and a short time $t^3$ term. We have also shown analytically how the Batchelor range persists for all times. For the stochastic tracers, we analytically computed an exact $t^3$ term in the separation using Ito calculus. We have shown numerically that this term is dominant at long times and that it seems to correspond to the asymptotic behavior of the deterministic tracers. These tracers are the mass-less inertia free particles following the velocity field lines.
\item We solved the "inverse problem" for fluid particle pair statistics by showing that the time evolution of the probability distribution of pair separations is the exact solution of a diffusion equation with suitable diffusivity. The "inverse problem" refers to this general framework that is used to convert observed measurements into information about a physical object or system. In contrast to common assumptions, we have shown that short time correlation is not a necessary condition for the system to be described by a diffusion equation. We have shown the assumptions necessary to arrive to the Kraichnan and Lundgren diffusion model and have studied numerically what these assumptions imply.
\item We developed an analytical model for Gaussian random fields in an effort to compare it with kinematic simulation models. We have shown that our model has a very rich physics and agrees with much of the kinematic simulation physics at low Reynolds number. Unlike kinematic simulations, within our model, we can drive the system to very high Reynolds numbers and observe the asymptotic behaviors. The insights provided by our model and its asymptotic behavior can yield further arguments in the debate on the actual behavior of the kinematic simulations.
\end{itemize}
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Keywords
Turbulence Theory