G equation


In Combustion, G equation is a scalar field equation which describes the instantaneous flame position, introduced by Forman A. Williams in 1985 in the study of premixed turbulent combustion. The equation is derived based on the Level-set method. The equation was studied by George H. Markstein earlier, in a restrictive form.

Mathematical descriptionPeters, Norbert. Turbulent combustion. Cambridge university press, 2000.Williams, Forman A. "Combustion theory." (1985).

The G equation reads as
where
The flame location is given by which can be defined arbitrarily such that is the region of burnt gas and is the region of unburnt gas. The normal vector to the flame is.

Local burning velocity

The burning velocity of the stretched flame can be derived by subtracting suitable terms from the unstretched flame speed, for small curvature and small strain, as given by
where
The G equation has an exact expression for a simple slot burner. Consider a two-dimensional planar slot burner of slot width with a premixed reactant mixture is fed through the slot with constant velocity, where the coordinate is chosen such that lies at the center of the slot and lies at the location of the mouth of the slot. When the mixture is ignited, a flame develops from the mouth of the slot to certain height with a planar conical shape with cone angle. In the steady case, the G equation reduces to
If a separation of the form is introduced, the equation becomes
which upon integration gives
Without loss of generality choose the flame location to be at. Since the flame is attached to the mouth of the slot, the boundary condition is, which can be used to evaluate the constant. Thus the scalar field is
At the flame tip, we have, the flame height is easily determined as
and the flame angle is given by
Using the trigonometric identity, we have