G equation

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 studied by George H. Markstein earlier, in a restrictive form.[3][4]

Mathematical description[5][6]

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 ${\displaystyle \mathbf {n} =-\nabla G/|\nabla G|}$.

Local burning velocity

The burning velocity of the stretched flame can be derived by subtracting suitable terms from the unstretched flame speed, for small curvature and small strain, as given by

${\displaystyle U_{L}=S_{L}-S_{L}{\mathcal {L}}\kappa -{\mathcal {L}}S}$

where

• ${\displaystyle S_{L}}$ is the burning velocity of unstretched flame
• ${\displaystyle S=-\mathbf {n} \cdot \nabla \mathbf {v} \cdot \mathbf {n} }$ is the term corresponding to the imposed strain rate on the flame due to the flow field
• ${\displaystyle {\mathcal {L}}}$ is the Markstein length, proportional to the laminar flame thickness ${\displaystyle \delta _{L}}$, the constant of proportionality is Markstein number ${\displaystyle {\mathcal {M}}}$
• ${\displaystyle \kappa =\nabla \cdot \mathbf {n} =-{\frac {\nabla ^{2}G-\mathbf {n} \cdot \nabla (\mathbf {n} \cdot \nabla G)}{|\nabla G|}}}$ 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

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

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

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

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

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

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

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

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

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

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

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