G equation
In Combustion, G equation is a scalar field equation which describes the instantaneous flame position, introduced by Forman A. Williams in 1985Williams, F. A. (1985). Turbulent combustion. In The mathematics of combustion (pp. 97-131). Society for Industrial and Applied Mathematics.Kerstein, Alan R., William T. Ashurst, and Forman A. Williams. "Field equation for interface propagation in an unsteady homogeneous flow field." Physical Review A 37.7 (1988): 2728. 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.GH Markstein. (1951). Interaction of flow pulsations and flame propagation. Journal of the Aeronautical Sciences, 18(6), 428-429.Markstein, G. H. (Ed.). (2014). Nonsteady flame propagation: AGARDograph (Vol. 75). Elsevier.Markstein, G. H., & Squire, W. (1955). On the stability of a plane flame front in oscillating flow. The Journal of the Acoustical Society of America, 27(3), 416-424.
Mathematical description
The G equation reads asPeters, Norbert. Turbulent combustion. Cambridge university press, 2000.Williams, Forman A. "Combustion theory." (1985).
:
where
- is the flow velocity field
- is the local burning velocity with respect to the unburnt gas
The flame location is given by which can be defined arbitrarily such that is the region of burnt gas and
=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
S_L is the burning velocity of unstretched flame with respect to the unburnt gas\mathcal{M}_c and\mathcal{M}_s are the two Markstein numbers, associated with the curvature term\nabla \cdot \mathbf{n} and the term\mathbf{n}\mathbf n: \nabla\mathbf v corresponding to flow strain imposed on the flame\delta_L are the laminar burning speed and thickness of a planar flame\tau_L=\delta_L/S_L 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
:
If a separation of the form
:
which upon integration gives
:
Without loss of generality choose the flame location to be at
:
At the flame tip, we have
:
and the flame angle
:
Using the trigonometric identity
:
In fact, the above formula is often used to determine the planar burning speed