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.

Mathematical description
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The G equation reads as


 * $$\frac{\partial G}{\partial t} + \mathbf{v}\cdot\nabla G = U_L |\nabla G| $$

where
 * $$\mathbf{v}$$ is the flow velocity field
 * $$U_L$$ is the local burning velocity

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 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


 * $$U_L = S_L - S_L \mathcal{L} \kappa - \mathcal{L} S$$

where
 * $$S_L$$ is the burning velocity of unstretched flame
 * $$S=-\mathbf{n}\cdot\nabla\mathbf{v}\cdot\mathbf{n} $$ is the term corresponding to the imposed strain rate on the flame due to the flow field
 * $$\mathcal{L}$$ is the Markstein length, proportional to the laminar flame thickness $$\delta_L$$, the constant of proportionality is Markstein number $$\mathcal{M}$$
 * $\kappa = \nabla\cdot\mathbf{n} = -\frac{\nabla^2 G - \mathbf{n}\cdot\nabla(\mathbf{n}\cdot\nabla G)}{|\nabla G|}$ 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 $$b$$ with a premixed reactant mixture is fed through the slot with 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 flame develops from the mouth of the slot to certain height $$y=L$$ with a planar conical shape with cone angle $$\alpha$$. In the steady case, the G equation reduces to


 * $$U\frac{\partial G}{\partial y} = U_L \sqrt{\left(\frac{\partial G}{\partial x}\right)^2+ \left(\frac{\partial G}{\partial y}\right)^2} $$

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


 * $$U = U_L\sqrt{1+ \left(\frac{\partial f}{\partial x}\right)^2}, \quad \text{or} \quad \frac{\partial f}{\partial x} = \frac{\sqrt{U^2-U_L^2}}{U_L}$$

which upon integration gives


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

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


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

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


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

and the flame angle $$\alpha$$ is given by


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

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


 * $$\sin\alpha = \frac{U_L}{U}$$