Citation
Landman, Michael J. (1987) New Solutions of an Amplitude Equation Describing Transition in a Laminar Shear Flow. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/CYPPD856. https://resolver.caltech.edu/CaltechETD:etd03032008130724
Abstract
In order to better understand the process of laminarturbulent transition in parallel shear flows, the study of the stability of viscous flow between parallel plates, known as plane Poiseuille flow, is found to be a good prototype. For Reynolds number near the critical value at which a linear instability first appears, Stewartson and Stuart (1971) developed a weakly nonlinear theory for which an amplitude equation is derived describing the evolution of a disturbance in plane Poiseuille flow in two space dimensions. This nonlinear partial differential equation is now commonly known in the literature as the GinzburgLandau equation, and is of the form
(∂A)/(∂t) = (a_{r} + ia_{i})[(∂^{2}A)/(∂x^{2})] + (Re  Re_{c})A + (d_{r} + id_{i})AA^{2}.
This dissertation concentrates on analyzing quasisteady solutions of the GinzburgLandau equation, where
A = e^{iΩt}Φ(x  ct).
These solutions describe modulations to the wave of primary instability, with amplitude which is steady in an appropriate moving coordinate system. The ordinary differential equation describing the spatial structure of quasisteady solutions is viewed as a lowdimensional dynamical system. Using numerical continuation and perturbation techniques, new spatially periodic and quasiperiodic solutions are found which bifurcate from the laminar state and undergo a complex series of bifurcations. Moreover, solitary waves and other solutions suggestive of laminar transition are found numerically for Reynolds number on either side of Re_{c}, connecting the laminar state to finite amplitude states, some of the latter corresponding to known solutions of the full fluid equations. The existence of these new spatially quasiperiodic and transition solutions suggests the existence of a similar class of solutions in the Navier Stokes equations, describing pulses and fronts of instability, as observed experimentally in parallel shear flows.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  Applied Mathematics  
Degree Grantor:  California Institute of Technology  
Division:  Engineering and Applied Science  
Major Option:  Applied Mathematics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  14 April 1987  
Funders: 
 
Record Number:  CaltechETD:etd03032008130724  
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd03032008130724  
DOI:  10.7907/CYPPD856  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  853  
Collection:  CaltechTHESIS  
Deposited By:  Imported from ETDdb  
Deposited On:  03 Mar 2008  
Last Modified:  21 Dec 2019 02:10 
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