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

$\frac{\partial G}{\partial t}+\mathbf{v}\cdot \mathrm{\nabla }G=S_T|\mathrm{\nabla }G|$

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

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

$\frac{S_T}{S_L}=1+{{ℳ}}_c{\delta }_L\mathrm{\nabla }\cdot \mathbf{n}+{{ℳ}}_s{\tau }_L\mathbf{n}\mathbf{n}:\mathrm{\nabla }\mathbf{v}$

where

• $S_L$ is the burning velocity of unstretched flame with respect to the unburnt gas
• ${{ℳ}}_c$ and ${{ℳ}}_s$ are the two Markstein numbers, associated with the curvature term $\mathrm{\nabla }\cdot \mathbf{n}$ and the term $\mathbf{n}\mathbf{n}:\mathrm{\nabla }\mathbf{v}$ 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

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

which upon integration gives

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

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 $$\begin{gathered} |x|=b/2, \\ y=0 \end{gathered}$$, the boundary condition is $G(b/2,0)=0$, which can be used to evaluate the constant $C$. Thus the scalar field is

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

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

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

and the flame angle $\alpha$,

$\tan \alpha =\frac{b/2}{L}=\frac{S_T}{{(U^2-S_L^2)}^{1/2}}$

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