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 first studied by George H. Markstein, in a restrictive form for the burning velocity and not as a level set of a field. The G equation reads
where is the flow density, is the flow velocity and is the normal mass flux entering any particular level set constant.

Mathematical description

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, pointing towards the burnt gas, is.

Local burning velocity

According to Matalon–Matkowsky–Clavin–Joulin theory, the burning velocity of the stretched flame, for small curvature and small strain, is given by
where
  • is the burning velocity of unstretched flame with respect to the unburnt gas
  • and are the two Markstein numbers, associated with the curvature term and the term corresponding to flow strain imposed on the flame
  • are the laminar burning speed and thickness of a planar flame
  • is the planar flame residence time.

    A simple example - Slot burner

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, 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 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. For simplicity, let us assume, 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, the boundary condition is, which can be used to evaluate the constant. 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.