Citation
Lappas, Tasso (1993) An adaptive Lagrangian method for computing 1D reacting flows, and, The theory of Riemann invariant manifolds for the compressible Euler equations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/bw0e2789. https://resolver.caltech.edu/CaltechETD:etd09192007130342
Abstract
In the first part of this thesis, a method for computing onedimensional, unsteady compressible flows, with and without chemical reactions, is presented. This work has focused on accurate computation of the discontinuous waves that arise in such flows. The main feature of the method is the use of an adaptive Lagrangian grid. This allows the computation of discontinuous waves and their interactions with the accuracy of fronttracking algorithms. This is done without the use of additional grid points representing shocks, in contrast to conventional, fronttracking schemes. The Lagrangian character of the present scheme also allows contact discontinuities to be captured easily. The algorithm avoids interpolation across discontinuities in a natural and efficient way. The method has been used on a variety of reacting and nonreacting flows in order to test its ability to compute complicated wave interactions accurately and in a robust way. In the second part of this thesis, a new approach is presented for computing multidimensional flows of an inviscid gas. The goal is to use the knowledge of the onedimensional, characteristic problem for gas dynamics to compute genuinely multidimensional flows in a mathematically consistent way. A family of spacetime manifolds is found on which an equivalent 1D problem holds. These manifolds are referred to as Riemann Invariant Manifolds. Their geometry depends on the local, spatial gradients of the flow, and they provide locally a convenient system of coordinate surfaces for spacetime. In the case of zero entropy gradients, functions analogous to the Riemann invariants of 1D gas dynamics can be introduced. These generalized Riemann Invariants are constant on the Riemann Invariant Manifolds. The equations of motion are integrable on these manifolds, and the problem of computing the solution becomes that of determining the geometry of these manifolds locally in spacetime. The geometry of these manifolds is examined, and in particular, their relation to the characteristic surfaces. It turns out that they can be spacelike or timelike, depending on the flow gradients. An important parameter is introduced, which plays the role of a Mach number for the wave fronts that these manifolds represent. Finally, the issue of determining the solution at points in spacetime, using information that propagates along spacelike surfaces is discussed. The question of whether it is possible to use information outside the domain of dependence of a point in spacetime to determine the solution is discussed in relation to the existence and uniqueness theorems, which introduce the concept of domain of dependence. This theory can be viewed as an extension of the method of characteristics to multidimensional, unsteady flows. There are many ways of using the theory to develop practical, numerical schemes. It is shown how it is possible to correct a conventional, secondorder Godunov scheme for multidimensional effects, using this theory. A family of secondorder, conservative Godunov schemes is derived, using the theory of Riemann Invariant Manifolds, for the case of twodimensional flow. The extension to three dimensions is straightforward. One of these schemes is used to compute two standard test cases and a twodimensional, inviscid, shear layer.
Item Type:  Thesis (Dissertation (Ph.D.))  

Degree Grantor:  California Institute of Technology  
Division:  Engineering and Applied Science  
Major Option:  Aeronautics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Group:  GALCIT  
Thesis Committee: 
 
Defense Date:  16 September 1992  
Funders: 
 
Record Number:  CaltechETD:etd09192007130342  
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd09192007130342  
DOI:  10.7907/bw0e2789  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  3644  
Collection:  CaltechTHESIS  
Deposited By:  Imported from ETDdb  
Deposited On:  17 Oct 2007  
Last Modified:  16 Apr 2021 22:32 
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