G equation

In Combustion, G equation is a scalar $G(\mathbf {x} ,t)$ field equation which describes the instantaneous flame position, first introduced by Forman A. Williams in 1985 as a model for a premixed turbulent combustion.

Mathematical description^[1]^[2]

The G equation reads as

{\frac {\partial G}{\partial t}}+\mathbf {v} \cdot \nabla G=S_{L}|\nabla G|

where

$\mathbf {v}$ is the flow velocity field
$S_{L}$ is the local burning velocity

The flame location is given by $G(\mathbf {x} ,t)=G_{o}$ which can be defined arbitrarily such that $G(\mathbf {x} ,t)>G_{o}$ is the region of burnt gas and $G(\mathbf {x} ,t)<G_{o}$ is the region of unburnt gas. The normal vector to the flame is $\mathbf {n} =-\nabla G/|\nabla G|$ . The burning velocity is constant for the unstretched flame and for the stretched flame the expression is given by

S_{L}=S_{L}^{o}-S_{L}^{o}{\mathcal {L}}\kappa -{\mathcal {L}}S

where

$S_{L}^{o}$ is the burning velocity of unstretched flame
$S=-\mathbf {n} \cdot \nabla \mathbf {v} \cdot \mathbf {n}$ is the strain rate
${\mathcal {L}}$ is the Markstein length, proportional to the laminar flame thickness $l_{f}$ , the constant of proportionality is Markstein number ${\rm {Ma}}$ (Clavin-Williams equation for one step chemistry with large activation energy)
$\kappa =\nabla \cdot \mathbf {n} =-{\frac {\nabla ^{2}G-\mathbf {n} \cdot \nabla (\mathbf {n} \cdot \nabla G)}{|\nabla G|}}$ is the flame curvature, which is positive if the flame front is convex with respect to the unburnt mixture and vice versa.

A simple example - Slot burner

This example is given by Norbert Peters. The G equation has an exact expression for a simple slot burner. Consider a two-dimensional planar slot burner of slot width $b$ with a premixed reactant mixture is fed through the slot with constant velocity $\mathbf {v} =(0,U)$ , 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 flame develops from the mouth of the slot to certain height $y=L$ with a planar conical shape with cone angle $\alpha$ . In the steady case, the G equation reduces to

U{\frac {\partial G}{\partial y}}=S_{L}{\sqrt {\left({\frac {\partial G}{\partial x}}\right)^{2}+\left({\frac {\partial G}{\partial y}}\right)^{2}}}

If a separation of the form $G(x,y)=y+f(x)$ is introduced, the equation becomes

U=S_{L}{\sqrt {1+\left({\frac {\partial f}{\partial x}}\right)^{2}}},\quad {\text{or}}\quad {\frac {\partial f}{\partial x}}={\frac {\sqrt {U^{2}-S_{L}^{2}}}{S_{L}}}

which upon integration gives

f(x)={\frac {(U^{2}-S_{L}^{2})^{1/2}}{S_{L}}}|x|+C,\quad \Rightarrow \quad G(x,y)={\frac {(U^{2}-S_{L}^{2})^{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 is

G(x,y)={\frac {(U^{2}-S_{L}^{2})^{1/2}}{S_{L}}}\left(|x|-{\frac {b}{2}}\right)+y

At the flame tip, we have $x=0,\ y=L,\ G=0$ , the flame height is easily determined as

L={\frac {b(U^{2}-S_{L}^{2})^{1/2}}{2S_{L}}}

and the flame angle $\alpha$ is given by

\tan \alpha ={\frac {b/2}{L}}={\frac {S_{L}}{(U^{2}-S_{L}^{2})^{2}}}

Using the trigonometric identity $\tan ^{2}\alpha =\sin ^{2}\alpha /(1-\sin ^{2}\alpha )$ , we have

\sin \alpha ={\frac {S_{L}}{U}}

References

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

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G equation

Mathematical description[1][2]

A simple example - Slot burner

References

Mathematical description^[1]^[2]