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Numerical methods in combustion

*One of the activities of the *Scientific
Computing Group* of Eindhoven University
of Technology is the development and analysis of numerical methods
used in computational combustion. The research is focused on the numerical
simulation of laminar flames.*
This work fits into a larger programme of both experimental and numerical
research. The main targets of the program are to obtain insight in the
behaviour of burners and to develop software for simulating burner systems.

*Temperature (left side) and CH4 concentration (right)
in a premixed methane/air flame*
With this software, we can predict the influence of the burner geometry
and variations in the composition of natural gas on the combustion process
in domestic appliances. Also, we may predict the composition of exhaust
gases of small domestic and industrial burners. This second objective is
particularly interesting, due to increasingly strict environmental requirements
imposed by the government.

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Mathematical description

A laminar flame can be considered as the flow of a reacting gas mixture,
and can hence be described by *flow equations*, which model the fluid
flow, and *combustion equations*, which model the chemical reactions
taking place in the mixture. The flow equations describe conservation of
mass and conservation of momentum, (due to problems with screen resolution,
the equations on this page may be incorrectly displayed; right click a
formula and select view image to see it in true size)
The combustion equations describe conservation of energy and conservation
of the mass of each chemical species in the mixture, viz.
The set of conservation laws has to be completed with two *consitutive
relations*: an equation of state and a thermodynamic relation:
The governing equations are nonlinear and strongly coupled. Many complicated
chemical reactions take place, each of which has its own typical time scale.
This leads to large differences in time scales in the problem. Also, the
size of the chemically active layer is small (typically 10^{-4}
m) compared to the size of the computational domain (10^{-1} m),
which leads to large differences in geometry; the combustion variables
(e.g. temperature) vary rapidly in the active layer.

*Measurements on a Bunsen flame with cooled walls*
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Numerical simulation

For the numerical simulation of laminar flames, we first discretize the
conservation laws in space using a finite volume scheme: the computational
domain is covered with *control volumes*, and the conservation laws
are integrated over each control volume. This approach leads to a system
of discrete conservation laws, each of which describes the balance of the
*fluxes*
into and out of a control volume and the production in the control volume.
A finite volume method with an exponential fitting scheme for the flux
computation has been developed for a general convection-diffusion equation;
this scheme is second order accurate for both strongly diffusive and strongly
convective flows.
For instationary calculations, a time integration method has to be chosen.
Due to the large restrictions on the time step caused by the stiff chemical
kinetics, diffusive and convective phenomena, and pressure wave effects,
explicit methods are infeasible. Therefore, implicit methods are needed
for discretizing the equations in time.

Discretization in space and time leads to a system of algebraic equations,
which may be solved using an iterative method such as Gauss-Seidel iteration.
The convergence of the iterative method is accelerated using a multigrid
scheme.

*OH and N2 concentrations in a methane/air flame*
At present, research is carried out on the use of *pressure correction
schemes* to solve the flow equations. Apart from that, the large gradients
of the combustion variables in the chemically active layer call for a small
grid size in this region, whereas far less resolution is needed away from
the chemical reaction zone. Therefore, the use of composite grids in simulations
seems natural, and application of so-called *local defect correction*
methods for computations on non-uniform grids is studied.

*A local defect correction algorithm implicitly yields
a*

*discretization on a composite grid that consists of
a global*

*coarse grid with one or more regions of local uniform*

*refinement by applying discretizations on the uniform*

*grids only*
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Partners in research

The Scientific Computing Group cooperates with the following organizations
in combustion research:
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More information

For more information, please contact one of the following people: