G equation

In Combustion, G equation is a scalar

field equation which describes the instantaneous flame position, introduced by Forman A. Williams in 1985[1][2] in the study of premixed turbulent combustion.

The equation is derived based on the Level-set method.

The equation was first studied by George H. Markstein, in a restrictive form for the burning velocity and not as a level set of a field.

[3][4][5] The G equation reads as[6][7] where The flame location is given by

is the region of burnt gas and

is the region of unburnt gas.

The normal vector to the flame, pointing towards the burnt gas, is

According to Matalon–Matkowsky–Clavin–Joulin theory, the burning velocity of the stretched flame, for small curvature and small strain, is 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

The premixed reactant mixture is fed through the slot from the bottom with a constant velocity

lies at the center of the slot and

lies at the location of the mouth of the slot.

When the mixture is ignited, a premixed flame develops from the mouth of the slot to a certain height

in the form of a two-dimensional wedge shape with a wedge angle

, which is a good approximation except near the wedge corner where curvature effects will becomes important.

In the steady case, the G equation reduces to If a separation of the form

is introduced, then 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

Thus the scalar field is At the flame tip, we have

, which enable us to determine the flame height and the flame angle

, Using the trigonometric identity

, we have In fact, the above formula is often used to determine the planar burning speed

, by measuring the wedge angle.