An adaptive finite element method for reactive flow problems. (English) Zbl 0896.76034
Heidelberg: Univ. Heidelberg, Naturwissenschaftlich-Mathematische Gesamtfakultät, 109 p. (1998).
The main subject of this work is the development of an adaptive method for the simulation of stationary combustion processes at a low Mach number.
In the first chapter, we present the equations for reactive flow problems in the low-Mach-number limit. The differences to the fully compressible formulation are explained and the underlying models for diffusion are detailed. We discuss the form of the chemical source terms and the character of the non-constant coefficients. Furthermore, we explain briefly the flame-sheet model, since it is used to compare our approach of adaptivity with more classical methods.
An important topic is the discretization of the equations, which is described in the second chapter. We discuss the pressure stabilization and the streamline diffusion for stationary compressible flow at a low Mach number. Since this formulation is non-standard, we derive an error estimate. We apply this discretization to low-Mach-number combustion. Special emphasis is also given to the non-stationary case, since the resulting discretization should be stable for large and small time steps.
In the third chapter, we present a fast and efficient solver for the incompressible Navier-Stokes equations and for reactive flow problems. We present a matrix structure which is especially adapted to reactive flow problems. Since the linear equations are solved by the use of multigrid techniques, we present the smoothing operator. Our approach to adaptivity is presented in the fourth chapter. Here, we explain the concept of error estimation for functionals of the solution. Strategies for mesh adaptation are explained in order to produce “good” meshes. Numerical tests show the reliability and efficiency of the error estimate.
In the final fifth chapter, we apply the developed algorithm to three cobustion problems with different degrees of complexity: the flame-shet problem, an ozone decomposition flame with three species and six reactions, and methane flame in a complex geometry. The thesis is completed with conclusions and ideas for further work.
In the first chapter, we present the equations for reactive flow problems in the low-Mach-number limit. The differences to the fully compressible formulation are explained and the underlying models for diffusion are detailed. We discuss the form of the chemical source terms and the character of the non-constant coefficients. Furthermore, we explain briefly the flame-sheet model, since it is used to compare our approach of adaptivity with more classical methods.
An important topic is the discretization of the equations, which is described in the second chapter. We discuss the pressure stabilization and the streamline diffusion for stationary compressible flow at a low Mach number. Since this formulation is non-standard, we derive an error estimate. We apply this discretization to low-Mach-number combustion. Special emphasis is also given to the non-stationary case, since the resulting discretization should be stable for large and small time steps.
In the third chapter, we present a fast and efficient solver for the incompressible Navier-Stokes equations and for reactive flow problems. We present a matrix structure which is especially adapted to reactive flow problems. Since the linear equations are solved by the use of multigrid techniques, we present the smoothing operator. Our approach to adaptivity is presented in the fourth chapter. Here, we explain the concept of error estimation for functionals of the solution. Strategies for mesh adaptation are explained in order to produce “good” meshes. Numerical tests show the reliability and efficiency of the error estimate.
In the final fifth chapter, we apply the developed algorithm to three cobustion problems with different degrees of complexity: the flame-shet problem, an ozone decomposition flame with three species and six reactions, and methane flame in a complex geometry. The thesis is completed with conclusions and ideas for further work.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76V05 | Reaction effects in flows |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 80A25 | Combustion |
| 76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |
