G equation explained

In Combustion, G equation is a scalar

G( x ,t) 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]}

Mathematical description

The G equation reads as^{[6] [7]}

(\partial G)/(\partial t) + v ⋅ \nabla G = S _{T} |\nabla G|

where

v is the flow velocity fieldS _{T} is the local burning velocity with respect to the unburnt gas

The flame location is given by

G( x ,t) = G _{o} which can be defined arbitrarily such that G( x ,t)>G _{o} is the region of burnt gas and G( x ,t)<G _{o} is the region of unburnt gas. The normal vector to the flame, pointing towards the burnt gas, is n = \nabla G /|\nabla G| .

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

(S _{T})/(S _{L}) = 1 + l{M} _{c} \delta _{L} \nabla ⋅ n + l{M} _{s} \tau _{L} n n: \nabla v

where

S _{L} is the burning velocity of unstretched flame with respect to the unburnt gasl{M} _{c} and l{M} _{s} are the two Markstein numbers, associated with the curvature term \nabla ⋅ n and the term n n: \nabla 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

b . The premixed reactant mixture is fed through the slot from the bottom with a constant velocity v = ( 0 ,U) , where the coordinate (x,y) is chosen such that x =0 lies at the center of the slot and y =0 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 y = L in the form of a two-dimensional wedge shape with a wedge angle \alpha . For simplicity, let us assume S _{T=S} _{L} , which is a good approximation except near the wedge corner where curvature effects will becomes important. In the steady case, the G equation reduces toU (\partial G)/(\partial y) = S _{L} \sqrt{\left( (\partial G)/(\partial x) \right) ^{2+} \left( (\partial G)/(\partial y) \right) ^{2}}

If a separation of the form

G(x,y) = y + f(x) is introduced, then the equation becomesU = S _{L\sqrt{1+} \left( (\partial f)/(\partial x) \right) ^{2},} ⇒ (\partial f)/(\partial x) = (\sqrt{U ^{2} _{L})/() }

which upon integration gives

f(x) = (\left(U ^{2 \right)} _{L} ^{1/2})/(S _{L}) |x| + C, ⇒ G(x,y) = (\left(U ^{2 \right)} _{L} ^{1/2})/(S _{L}) |x| + y + C

Without loss of generality choose the flame location to be at

G(x,y) = G _{o=0} . Since the flame is attached to the mouth of the slot |x| = b/ 2 , y =0 , the boundary condition is G(b/ 2 , 0 ) =0 , which can be used to evaluate the constant C . Thus the scalar field isG(x,y) = (\left(U ^{2 \right)} _{L} ^{1/2})/(S _{L}) \left(|x| - (b)/(2) \right) + y

At the flame tip, we have

x =0 , y = L, G =0 , which enable us to determine the flame heightL = (b\left(U ^{2 \right)} _{L} ^{1/2})/(2 S _{L})

and the flame angle

\alpha ,\tan \alpha = (b/ 2)/(L) = (S _{T})/(\left(U ^{2 \right)} _{L} ^{1/2}) \tan ^{2\alpha} = \sin ^{2\alpha/\left(1-\sin} ^{2\alpha\right)} , we have\sin\alpha = (S _{L})/(U) .

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

S _{L} , by measuring the wedge angle.

Notes and References

1. Williams, F. A. (1985). Turbulent combustion. In The mathematics of combustion (pp. 97-131). Society for Industrial and Applied Mathematics.
2. 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.
3. GH Markstein. (1951). Interaction of flow pulsations and flame propagation. Journal of the Aeronautical Sciences, 18(6), 428-429.
4. Markstein, G. H. (Ed.). (2014). Nonsteady flame propagation: AGARDograph (Vol. 75). Elsevier.
5. 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.
6. Peters, Norbert. Turbulent combustion. Cambridge university press, 2000.
7. Williams, Forman A. "Combustion theory." (1985).