Shocks and Wave Interactions for Compressible Flow
Open Access
- Author:
- Endres, Erik Edward
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- June 19, 2008
- Committee Members:
- Eric M Mockensturm, Committee Member
Helge Kristian Jenssen, Committee Chair/Co-Chair
Alexi Novikov, Committee Member
Yuxi Zheng, Committee Member - Keywords:
- Euler system
shocks
rarefactions
interactions - Abstract:
- We study the compressible Euler system in one (1-D) or several (multi-D) space dimensions. For the 1-D system we focus on Riemann problems and interactions while for multi-D we consider boundary value problems with spherical or cylindrical symmetry (which transforms the system into 1-D). In 1-D we consider the full system, which includes the energy equation, and for multi-D we also consider isentropic flow. In Chapter 1 we consider 1-D flow of ideal polytropic gases. We resolve all but two possible interactions of elementary waves (shocks, contact discontinuities, and centered rarefactions). This has been done by several authors to varying degrees of completeness. In addition to solving for the outgoing waves we obtain various relations between strengths of incoming waves and outgoing waves. As an application of resolving the interactions, we create a specific interaction pattern by carefully choosing the initial data which is done in Chapter 2. This particular pattern is motivated by examples by Young and Jenssen for certain non-physical systems for which the solution may blow up in sup-norm and/or total variation (for carefully chosen data). In the case of the 1-D Euler system, the chosen data generates two contact discontinuities and repeated reflections of shocks between them. We also impose absorbing boundaries outside the central region in order to make the analysis tractable. By using the relations of the incoming waves and outgoing waves strengths, we are able to show that there will be only a finite number of interactions in any finite time interval and that all conserved quantities are uniformly bounded. This gives that these special solutions of the full Euler system are defined for all time and are uniformly bounded. Further, we comment briefly on a particular scaling of the dependent variables in the Euler system. This provides a large data result for certain types of scaled up data. Chapter 3 and Chapter 4 are devoted to the construction of stationary solutions of both the barotropic and full, compressible Euler system in several space dimensions with spherical or cylindrical symmetry, for a general equation of state. For given Dirichlet data on a sphere or a cylinder we first construct smooth and radially symmetric solutions to the Euler equations in the exterior domain. On the other hand, stationary smooth solutions in the interior domain necessarily become sonic and cannot be continued beyond a critical inner radius. We then use these solutions to construct entropy-satisfying shocks for the Euler equations in the region between two concentric spheres (or cylinders). In addition, for given data on the two concentric spheres (or cylinders) we investigate when a shock solution exists, and how the shock location is related to the boundary data. These shock solutions provide the first step in the construction of smooth solutions to the Navier-Stokes system which are shown to converge to the shock solutions of the Euler system in the small viscosity limit. Only the analysis of the shock solutions for the Euler system will be discussed here.