Advisor: Prof. P. K. Yeung

**Committee Members:**

Prof. P. K. Yeung (Committee Chair, AE)

Prof. Edmund Chow (CSE)

Prof. Suresh Menon (AE)

Prof. Stephen Ruffin (AE)

Prof. Donald Webster (CEE)

**LAGRANGIAN INVESTIGATIONS OF TURBULENT DISPERSION**

**AND MIXING USING PETASCALE COMPUTING**

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**Abstract:**

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In many fields of science and engineering important to society, such as study of air/water quality, pollutant dispersion, cloud physics, design of improved combustion devices, etc., the ability of turbulent flow to provide efficient transport of entities such as pollutants, vapor droplets, fuel/oxidizer, etc. is of critical importance. To understand and hence develop proper predictive tools for such transported entities, it is necessary to understand turbulence from a Lagrangian perspective (of an observer moving with the flow), including the interaction between turbulent transport and molecular diffusion. Usually, in both direct numerical simulations (DNS) and experiments, a population of fluid particles is tracked forward in time (forward tracking) from specified initial conditions to understand how a cloud of material spreads in a turbulent flow. However the process of turbulent mixing occurs when material located at different regions at previous times is brought together at a later time. In such a scenario, it is more important to track the particles backward in time (backward tracking). Backward tracking is also important from a modeling perspective, which would help address questions about the dynamical origins of a patch of contaminant material, or a highly convoluted multi-particle cluster. Furthermore, it can also be shown that the n^{th} moment of a passive scalar field can be directly related to the backward in time statistics of an n-particle cluster. Although conceptually simple, backward tracking is very difficult to accomplish due to time irreversibility of Navier-Stokes equations, and thus not very well understood in literature.

In this work, we use DNS of stationary isotropic turbulence to investigate the process of backward and forward dispersion using state of the art computing facilities. A new massively parallel computational framework has been developed to enable particle tracking in DNS at Petascale problem sizes, performing up to 40X faster than the previous implementation. We have also implemented an efficient and statistically robust approach to extract backward and forward statistics via the post-processing of trajectory data stored in DNS of fluid particles and diffusing molecules (that undergo Brownian motion relative to the fluid). Detailed results are first obtained for pairs of fluid particles. An important consequence of applying Kolmogorov’s similarity hypotheses to Lagrangian statistics of particle pairs is the universal t^{3} scaling (Richardson’s scaling) at intermediate times. Backward dispersion is found to be faster at intermediate times resulting in a higher Richardson constant, though the scaling is not as robust as in forward dispersion. Extensions to higher-order moments of the separation are also addressed. Statistics of the trajectories of molecules taken singly and in pairs are investigated. The separation statistics of molecular pairs exhibit more robust Richardson scaling compared to fluid particles. An important innovation in this work is to demonstrate explicitly the practical utility of a Lagrangian description of turbulent mixing, where molecular displacements and separations in the limit of small backward initial separations can be used to calculate the evolution of scalar fluctuations resulting from a known source function in space. Lagrangian calculations of production and dissipation rates of the scalar fluctuations are shown to agree very well with Eulerian results for the case of passive scalars driven by a uniform mean-gradient. The well-known scalar dissipation anomaly is accordingly addressed in a Lagrangian context. Extensions to three- and four-particle clusters of fluid particles and molecules are also addressed.