The G-equation is a premixed flame-front tracking model. The transport equation governing the unsteady evolution of a propagating flame interface is (derivation can be found in [514]),
 | (8–72) |
where 
is the fluid density, 
is the
fluid velocity vector, 
is the laminar flame speed (interface normal propagation), 
is the diffusivity and 
is the flame curvature.
For turbulent flames, Equation 8–72 can be Favre
Reynolds-averaged or spatially-filtered to provide transport equations
for the flame mean position,
 | (8–73) |
and the variance of the flame position,
 | (8–74) |
| where | |
 is the turbulent flame speed | |
 is a diffusivity
term defined as 
| |
 is the turbulent velocity scale | |
 is a modeling constant
taken from the  turbulence model (default=0.09) | |
 is a modeling constant = 2.0 | |
 is the turbulent
Schmidt number | |
 is the turbulent length scale | |
 is the flame curvature, defined
as  , where 
|
In Equation 8–74, 
indicates
that the diffusion term is only applied parallel to the flame front [514], the normal component being accounted for
in the turbulent burning velocity. In practice this makes only a minor
difference to results and can lead to convergence problems, and is
therefore disabled by default.
Ansys Fluent also offers the option of using an algebraic expression
to calculate the flame position variance, 
, instead of Equation 8–74:
 | (8–75) |
where 
is
the effective turbulent viscosity.
For more information, see the following section:
Equation 8–72 and Equation 8–73 do not contain a diffusion term and sharp interfaces remain sharp
at all times. Special numerical techniques such as the Volume-of-Fluid
(VOF) and Marker-and-Cell (MAC) have been devised to solve such equations.
In Ansys Fluent, the mean G-equation Equation 8–73 is
solved using a Level-set method. Here, 
represents the
signed mean distance to the flame front, and hence the flame front
is the 
isosurface. Since 
is the distance
to the flame front, 
is constrained to be unity everywhere in the flow
field. The standard Ansys Fluent transport equation machinery is used
to solve for 
(Equation 8–73) over a time step. However, at the end of the time step, the 
field is typically is not exactly equal to the mean
flame distance (and 
is not identically equal to 1), and this condition
is enforced by a process called re-initialization. In Ansys Fluent this
is done by constructing a faceted representation of the flame front
from the 
field. Then, in every cell, 
is set to the geometric distance to the nearest
flame front facet. 
is positive in the burnt region
downstream of the flame front and negative in the unburnt region upstream
of the flame front.
Given the mean flame front position, the mean progress variable is calculated according to a Gaussian distribution depending on proximity to the flame front and the G-equation variance:
 | (8–76) |
Mean properties, such as the mean density, temperature and species mass fractions, are calculated from the mean reaction progress variable, as in the C-equation model.