In Combustion, G equation is a scalar ${\displaystyle G(\mathbf {x} ,t)}$ 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.^[3][4][5]

Mathematical description

Sources:^[6][7]

The G equation reads as

${\displaystyle {\frac {\partial G}{\partial t}}+\mathbf {v} \cdot \nabla G=U_{L}|\nabla G|}$

where

• ${\displaystyle \mathbf {v} }$ is the flow velocity field
• ${\displaystyle U_{L}}$ is the local burning velocity

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

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 ${\displaystyle b}$ with a premixed reactant mixture is fed through the slot with constant velocity ${\displaystyle \mathbf {v} =(0,U)}$, where the coordinate ${\displaystyle (x,y)}$ is chosen such that ${\displaystyle x=0}$ lies at the center of the slot and ${\displaystyle y=0}$ 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 ${\displaystyle y=L}$ with a planar conical shape with cone angle ${\displaystyle \alpha }$. In the steady case, the G equation reduces to

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

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

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

which upon integration gives

${\displaystyle f(x)={\frac {\left(U^{2}-U_{L}^{2}\right)^{1/2}}{U_{L}}}|x|+C,\quad \Rightarrow \quad G(x,y)={\frac {\left(U^{2}-U_{L}^{2}\right)^{1/2}}{U_{L}}}|x|+y+C}$

Without loss of generality choose the flame location to be at ${\displaystyle G(x,y)=G_{o}=0}$. Since the flame is attached to the mouth of the slot ${\displaystyle |x|=b/2,\ y=0}$, the boundary condition is ${\displaystyle G(b/2,0)=0}$, which can be used to evaluate the constant ${\displaystyle C}$. Thus the scalar field is

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

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

${\displaystyle L={\frac {b\left(U^{2}-U_{L}^{2}\right)^{1/2}}{2U_{L}}}}$

and the flame angle ${\displaystyle \alpha }$ is given by

${\displaystyle \tan \alpha ={\frac {b/2}{L}}={\frac {U_{L}}{\left(U^{2}-U_{L}^{2}\right)^{1/2}}}}$

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

${\displaystyle \sin \alpha ={\frac {U_{L}}{U}}}$

References