Citation
Bhat, Harish Subrahmanya (2005) Lagrangian averaging, nonlinear waves, and shock capturing. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd05262005100534
Abstract
In this thesis, we explore various models for the flow of a compressible fluid as well as model equations for shock formation, one of the main features of compressible fluid flows.
We begin by reviewing the variational structure of compressible fluid mechanics. We derive the barotropic compressible Euler equations from a variational principle in both material and spatial frames. Writing the resulting equations of motion requires certain Liealgebraic calculations that we carry out in detail for expository purposes.
Next, we extend the derivation of the Lagrangian averaged Euler (LAEalpha) equations to the case of barotropic compressible flows. The derivation in this thesis involves averaging over a tube of trajectories centered around a given Lagrangian flow. With this tube framework, the LAEalpha equations are derived by following a simple procedure: start with a given action, expand via Taylor series in terms of smallscale fluid fluctuations, truncate, average, and then model those terms that are nonlinear functions of the fluctuations.
We then analyze a onedimensional subcase of the general models derived above. We prove the existence of a large family of traveling wave solutions. Computing the dispersion relation for this model, we find it is nonlinear, implying that the equation is dispersive. We carry out numerical experiments that show that the model possesses smooth, bounded solutions that display interesting pattern formation.
Finally, we examine a Hamiltonian partial differential equation (PDE) that regularizes the inviscid Burgers equation without the addition of standard viscosity. Here alpha is a small parameter that controls a nonlinear smoothing term that we have added to the inviscid Burgers equation. We show the existence of a large family of traveling front solutions. We analyze the initialvalue problem and prove wellposedness for a certain class of initial data. We prove that in the zeroalpha limit, without any standard viscosity, solutions of the PDE converge strongly to weak solutions of the inviscid Burgers equation. We provide numerical evidence that this limit satisfies an entropy inequality for the inviscid Burgers equation. We demonstrate a Hamiltonian structure for the PDE.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Burgers equation; Hamiltonian partial differential equations; Lagrangian averaging; nonlinear waves; shock waves 
Degree Grantor:  California Institute of Technology 
Division:  Engineering and Applied Science 
Major Option:  Control and Dynamical Systems 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  6 May 2005 
Author Email:  harish.bhat (AT) gmail.com 
Record Number:  CaltechETD:etd05262005100534 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd05262005100534 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  2089 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  26 May 2005 
Last Modified:  26 Dec 2012 02:46 
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