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 1985Williams, F. A. (1985). Turbulent combustion. In The mathematics of combustion (pp. 97-131). Society for Industrial and Applied Mathematics.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. 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.GH Markstein. (1951). Interaction of flow pulsations and flame propagation. Journal of the Aeronautical Sciences, 18(6), 428-429.Markstein, G. H. (Ed.). (2014). Nonsteady flame propagation: AGARDograph (Vol. 75). Elsevier.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.

Mathematical description

The G equation reads asPeters, Norbert. Turbulent combustion. Cambridge university press, 2000.Williams, Forman A. "Combustion theory." (1985).

:\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) 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

File:Burner 1D.jpg

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.

References