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 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]} The G equation reads ^[6]

${\displaystyle \rho \left({\frac {\partial G}{\partial t}}+\mathbf {v} \cdot \nabla G\right)={\dot {m}}|\nabla G|}$

where ${\displaystyle \rho }$ is the flow density, ${\displaystyle v}$ is the flow velocity and ${\displaystyle {\dot {m}}={\dot {m}}(\mathbf {x} ,t)}$ is the normal mass flux entering any particular level set ${\displaystyle G(\mathbf {x} ,t)=}$constant.

Mathematical description

The G equation reads as ^{[7] [8]}

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

where

• ${\displaystyle \mathbf {v} }$ is the flow velocity field,
• ${\displaystyle S_{T}={\dot {m}}/\rho _{u}}$ is the local burning velocity with respect to the unburnt gas with density ${\displaystyle \rho _{u}}$.

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, pointing towards the burnt gas, is ${\displaystyle \mathbf {n} =\nabla G/|\nabla G|}$.

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

${\displaystyle {\frac {S_{T}}{S_{L}}}=1+{\mathcal {M}}_{c}\delta _{L}\nabla \cdot \mathbf {n} +{\mathcal {M}}_{s}\tau _{L}\mathbf {n} \mathbf {n}$ :\nabla \mathbf {v} }

where

• ${\displaystyle S_{L}}$ is the burning velocity of unstretched flame with respect to the unburnt gas
• ${\displaystyle {\mathcal {M}}_{c}}$ and ${\displaystyle {\mathcal {M}}_{s}}$ are the two Markstein numbers, associated with the curvature term ${\displaystyle \nabla \cdot \mathbf {n} }$ and the term ${\displaystyle \mathbf {n} \mathbf {n}$ :\nabla \mathbf {v} } corresponding to flow strain imposed on the flame
• ${\displaystyle \delta _{L}}$ are the laminar burning speed and thickness of a planar flame
• ${\displaystyle \tau _{L}=\delta _{L}/S_{L}}$ is the planar flame residence time.

A simple example - Slot burner

Slot burner

The G equation has an exact expression for a simple slot burner. Consider a two-dimensional planar slot burner of slot width ${\displaystyle b}$. The premixed reactant mixture is fed through the slot from the bottom with a 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 premixed flame develops from the mouth of the slot to a certain height ${\displaystyle y=L}$ in the form of a two-dimensional wedge shape with a wedge angle ${\displaystyle \alpha }$. For simplicity, let us assume ${\displaystyle S_{T}=S_{L}}$, which is a good approximation except near the wedge corner where curvature effects will becomes important. In the steady case, the G equation reduces to

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

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

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

which upon integration gives

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

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

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

At the flame tip, we have ${\displaystyle x=0,\ y=L,\ G=0}$, which enable us to determine the flame height

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

and the flame angle ${\displaystyle \alpha }$,

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

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

${\displaystyle \sin \alpha ={\frac {S_{L}}{U}}.}$

In fact, the above formula is often used to determine the planar burning speed ${\displaystyle S_{L}}$, by measuring the wedge angle.

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. Rajamanickam, P., & Daou, J. (2024). Hydrodynamic theory of premixed flames under Darcy's law. Physics of Fluids, 36(12).
7. Peters, Norbert. Turbulent combustion. Cambridge university press, 2000.
8. Williams, Forman A. "Combustion theory." (1985).