1.2 Physics of Premixed Combustion
This section presents a brief introduction to the modelling of the premixed turbulent combustion
and its governing equations.
In premixed combustion, fuel and oxidizer are mixed at the molecular level prior to ignition. Combustion
occurs as a flame front propagates into the unburnt reactants. Examples of premixed
combustion include aspirated internal combustion engines, lean-premixed gas turbine combustors,
and gas-leak explosions.
The effect of turbulence is that it wrinkles and stretches the propagating laminar flame sheet, increasing
the sheet area and, in turn, the effective flame speed. The large turbulent eddies tend to
wrinkle and corrugate the flame sheet, while the small turbulent eddies, if they are smaller than the
laminar flame thickness, may penetrate the flame sheet and modify the laminar flame structure.
As the premixed flame is a reaction wave propagating from burned to fresh gases, the basic parameter
is known to be the progress variable. In the fresh gas, the progress variable is conventionally
put to zero. In the burned gas, it equals unity. Across the flame, the intermediate values describe
the progress of the reaction. A progress variable can be set with the help of any quantity, like
temperature, reactant mass fraction, provided it is bounded by a single value in the burned gas and
another one in the fresh gas. The progress variable is usually named c, in usual notations[1].
c =
T ? Tf
Tb ? Tf
(1.1)
Where b stands for burned gas, and f stands for fressh gas. It is seen that c is a normalization of a
scalar quantity.
In OpenFOAM, the flame front propagation is modelled by solving a transport equation for the
density-weighted mean reaction regress variable denoted by b(eq 1.7), where:
b = 1 ? c (1.2)
?
?t(?b) + ?.(?~ub) ? ?.(
µt
Sct
?b) = ??Sc (1.3)
Sct =
µ
?D : turbulent Schmidt number.
Sc : reaction regress source term (the dimesnion is [T
?1
] ), and is modeled as equation 1.4:
?Sc = ?uSu?|?b| (1.4)
By substituting equation 1.4 to equation 1.7 we would have:
?
?t(?b) + ?.(?~ub) ? ?.(
µt
Sct
?b) = ??uSu?|?b| (1.5)
where:
b : mean reaction regress variable
Su : laminar flame speed [m/s]
D : diffusion coefficient [m2/s]
? : Turbulent flame velocity and laminar flame velocity ratio
?u : density of unburnt mixture [kg/m3
]
Based on this definition:
b=1 : unburnt mixture
b=0 : burnt mixture
The value of b is defined as a boundary condition at all flow inlets. It is usually specified as
either 0 (unburnt) or 1 (burnt). OpenFOAM has a premixed turbulent combustion model based
on the reaction regress variable(b=1-c) approach. Information about this model is provided in the
bEqn.H file of the XiFoam solver.