Citation
Ghosh, Shubhro (1994) The role of various geometrical structures in scalar advectiondiffusion. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd10192005143443
Abstract
This thesis is divided in two parts: in Part I, using a timeperiodic perturbation of a twodimensional steady separation bubble on a plane noslip boundary to generate chaotic particle trajectories in a localized region of an unbounded boundarylayer 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 noslip boundary. We consider here the full advectiondiffusion problem, though attention is restricted to the case of small scalar diffusion, or large Peclet number. In this regime, a certain onedimensional 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 noslip 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 nonuniform 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 timeperiodic, with the period same as that of the timedependent 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 advectiondiffusion solution at large Peclet numbers to a fictitious zerodiffusivity or frozenfield solution  the socalled stirring solution  corresponding to infinitely large Peclet number. The zerodiffusivity 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 noslip boundary. The interaction between the necessarily nonhyperbolic particle dynamics in a thin nearwall 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 advectiondiffusion problem corresponding to the steady, unperturbed, integrable advection field. In Part II, the transport of a passive scalar from a noslip boundary into a twodimensional steady boundarylayer flow is studied in the vicinity of a laminar separation point, where the dividing streamline  which is also a onedimensional unstable manifold  is assumed to be normal to the boundary locally near the separation point. The novelty of the ensuing convectiondiffusion 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 boundarylayer approximations become invalid near the separation point. Using only the topology of the laminar, incompressible separated flow, a local solution of the NavierStokes equations is constructed in the form of a Taylorseries 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 NavierStokes equations in seeking asymptotic solutions to convectiondiffusion problems. Verification of the asymptotic result is obtained from numerical computations based on the Wiener bundle solution  which is particularly wellsuited to the largePecletnumber 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) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  26 May 1994 
NonCaltech Author Email:  ghoshs (AT) utrc.utc.com 
Record Number:  CaltechETD:etd10192005143443 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd10192005143443 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  4176 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  20 Oct 2005 
Last Modified:  28 Aug 2015 19:24 
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