# 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.

## 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

## 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

## Partners in research

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