Physics: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. The G equation reads

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

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

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,
• $S_T=\dot{m}/{\rho }_u$ is the local burning velocity with respect to the unburnt gas with density ${\rho }_u$.

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

The G equation has the form of the Hamilton-Jacobi equation, an equation in analytical mechanics used to model particle dynamics as propagation of waves.

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

$S_T=S_L+{{ℳ}}_c{\delta }_L(S_L-\mathbf{𝐯}\cdot \mathbf{𝐧})\mathrm{\nabla }\cdot \mathbf{𝐧}-{{ℳ}}_t{\delta }_L{\mathrm{\nabla }}_t\cdot {\mathbf{𝐯}}_t$

where

• $S_L$ is the burning velocity of unstretched flame with respect to the unburnt gas
• ${{ℳ}}_c$ and ${{ℳ}}_t$ are the two Markstein numbers, associated with the curvature and tangential straining; ${\mathrm{\nabla }}_t\cdot {\mathbf{𝐯}}_t=-\mathbf{𝐧}\otimes \mathbf{𝐧}:\mathrm{\nabla }\mathbf{𝐯}-(\mathbf{𝐯}\cdot \mathbf{𝐧})\mathrm{\nabla }\cdot \mathbf{𝐧}$ is the surface divergence of the tangential velocity ${\mathbf{𝐯}}_t=(\mathbf{𝐈}-\mathbf{𝐧}\otimes \mathbf{𝐧})\mathbf{𝐯}$
• ${\delta }_L$ is the laminar flame thickness.

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

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

At the flame tip, we have $x=0,y=L,G=0$, 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.