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Numerical modeling of laminar flames
M. Graziadei
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Introduction
The equations that describe laminar flames are characterized by the
presence of high activity regions, where most of the activity is concentrated.
There, gradients are quite large compared to those in the rest of the domain.
When solving such problems nume- rically, this solution behaviour requires
a much finer grid in the high activity region, or flame front, than in
the zones where the solution is fairly smooth.
Laminar reacting flow equations
Laminar recting flows can be described by the following equations
Continuity equation
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Bunsen flame: graphical representation of the temperature.
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Momentum equation
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Energy equation
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Species equation
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- Discretization: finite difference or finite volume method.
- A coarse grid covers the entire domain.
- In the high activity region a finer grid is required to get
the desired accuracy.
Local Defect Correction (LDC) method
The LDC method uses the solution computed on the fine grid to improve
the coarse grid solution. It works as follows:
- Compute the coarse grid solution.
- Define a fine grid BVP by interpolating the BCs on the interface
between the fine and the coarse grid from the coarse grid solution.
- Compute the solution of the fine grid problem
- The fine grid solution is then used to estimate the defect
of discretization on the coarse grid.
- Finally, to correct the coarse grid problem, add the defect
of discretization to its right-hand side.
The method is used in an iterative way. It converges very fast: tipically
one iteration suffices.
Test problem
LDC technique has been implemented with several fine grid types.
The following test problem is described by the two-dimensional convection-diffusion-reaction
equation
where f is such that the exact solution is given by
- s(x) describes the flame front shape;
 determines the steepness of u(x) in the vicinity
of the flame front.
It can be solved either with a slanting or with a curvilinear fine
grid.
Slanting grid
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Curvilinear grid
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Slanting grid
- Advantage: the system of equations maintains the same level
of complexity; it needs only to be rotated.
- Drawback: The number of fine grid points is redundant.
Curvilinear orthogonal grid
They are made such that a set of coordinate lines follows the level
curves.
- Advantage: the number of fine grid points can be reduced. In
fact, the step size along the coordinate lines tracing the level curves
can be bigger than the step size in the orthogonal direction. Furthermore,
the lines can be more dense just close to the high activity region and
less dense far away from it.
- Drawback: the system of equations becomes more complex. In
fact, we want to solve our system in a rectangular domain, which has
thus to be mapped into the curvilinear grid.
Future developments
Application of LDC in combination with slanting and curvilinear fine
grids to real flame problems.
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