G equation

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

Mathematical description

The G equation reads as

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

where

• $\mathbf {v}$ is the flow velocity field
• $S_{T}$ is the local burning velocity with respect to the unburnt gas

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

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

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

where

• $S_{L}$ is the burning velocity of unstretched flame with respect to the unburnt gas
• ${\mathcal {M}}_{c}$ and ${\mathcal {M}}_{s}$ are the two Markstein numbers, associated with the curvature term $\nabla \cdot \mathbf {n}$ and the term $\mathbf {n} \mathbf {n} :\nabla \mathbf {v}$ 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 $\mathbf {v} =(0,U)$, 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 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 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, then the equation becomes

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

which upon integration gives

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

Without loss of generality choose the flame location to be at $G(x,y)=G_{o}=0$. 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

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

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

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

and the flame angle $\alpha$,

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

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

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

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