Ghosh, Shubhro (1994) The role of various geometrical structures in scalar advection-diffusion. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-10192005-143443
This thesis is divided in two parts: in Part I, using a time-periodic perturbation of a two-dimensional steady separation bubble on a plane no-slip boundary to generate chaotic particle trajectories in a localized region of an unbounded boundary-layer flow, we study the impact of various geometrical structures that arise naturally in chaotic advection fields on the transport of a passive scalar from a local "hot spot" on the no-slip boundary. We consider here the full advection-diffusion problem, though attention is restricted to the case of small scalar diffusion, or large Peclet number. In this regime, a certain one-dimensional unstable manifold is shown to be the dominant organizing structure in the distribution of the passive scalar. In general, it is found that the chaotic structures in the flow strongly influence the scalar distribution while, in contrast, the flux of passive scalar from the localized active no-slip surface is, to dominant order, independent of the overlying chaotic advection. Increasing the intensity of the chaotic advection by perturbing the velocity field further away from integrability results in more non-uniform scalar distributions, unlike the case in bounded flows where the chaotic advection leads to rapid homogenization of diffusive tracer. In the region of chaotic particle motion the scalar distribution attains an asymptotic state which is time-periodic, with the period same as that of the time-dependent advection field. Some of these results are understood by using the shadowing property from dynamical systems theory. The shadowing property allows us to relate the advection-diffusion solution at large Peclet numbers to a fictitious zero-diffusivity or frozen-field solution - the so-called stirring solution - corresponding to infinitely large Peclet number. The zero-diffusivity solution is an unphysical quantity, but it is found to be a powerful heuristic tool in understanding the role of small scalar diffusion. A novel feature in this problem is that the chaotic advection field is adjacent to a no-slip boundary. The interaction between the necessarily non-hyperbolic particle dynamics in a thin near-wall region with the strongly hyperbolic dynamics in the overlying chaotic advection field is found to have important consequences on the scalar distribution: that this is indeed the case is shown using shadowing. Comparisons are made throughout with the flux and the distributions of the passive scalar for the advection-diffusion problem corresponding to the steady, unperturbed, integrable advection field. In Part II, the transport of a passive scalar from a no-slip boundary into a two-dimensional steady boundary-layer flow is studied in the vicinity of a laminar separation point, where the dividing streamline - which is also a one-dimensional unstable manifold - is assumed to be normal to the boundary locally near the separation point. The novelty of the ensuing convection-diffusion process derives from the convective transport normal to the active boundary resulting from convection along the dividing streamline, and because of which the standard thermal boundary-layer approximations become invalid near the separation point. Using only the topology of the laminar, incompressible separated flow, a local solution of the Navier-Stokes equations is constructed in the form of a Taylor-series expansion from the separation point. The representation is universal, without regard to the outer inviscid flow and it is used in obtaining an asymptotically exact solution for the steady scalar distribution near the separation point at large Peclet number, using matched asymptotic expansions. The method demonstrates the application of local solutions of the Navier-Stokes equations in seeking asymptotic solutions to convection-diffusion problems. Verification of the asymptotic result is obtained from numerical computations based on the Wiener bundle solution - which is particularly well-suited to the large-Peclet-number transport problem.
|Item Type:||Thesis (Dissertation (Ph.D.))|
|Degree Grantor:||California Institute of Technology|
|Division:||Chemistry and Chemical Engineering|
|Major Option:||Chemical Engineering|
|Thesis Availability:||Public (worldwide access)|
|Defense Date:||26 May 1994|
|Author Email:||ghoshs (AT) utrc.utc.com|
|Default Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Imported from ETD-db|
|Deposited On:||20 Oct 2005|
|Last Modified:||26 Dec 2012 03:06|
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