G equation

 ( Fluid Dynamics ) 

In Combustion, G equation is a scalar $G(\backslash mathbf\{x\},t)$ 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.

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

$\backslash frac\{\backslash partial\; G\}\{\backslash partial\; t\}\; +\; \backslash mathbf\{v\}\backslash cdot\backslash nabla\; G\; =\; S\_T\; |\backslash nabla\; G|$

where

• $\backslash mathbf\{v\}$ is the flow velocity field
• $S\_T$ is the local burning velocity with respect to the unburnt gas

The flame location is given by $G(\backslash mathbf\{x\},t)=G\_o$ which can be defined arbitrarily such that $G(\backslash mathbf\{x\},t)>G\_o$ is the region of burnt gas and

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Local burning velocity According to Matalon–Matkowsky–Clavin–Joulin theory, the burning velocity of the Flame stretch, for small curvature and small strain, is given by

$\backslash frac\{S\_T\}\{S\_L\}\; =\; 1\; +\; \backslash mathcal\{M\}\_c\; \backslash delta\_L\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{n\}\; +\; \backslash mathcal\{M\}\_s\; \backslash tau\_L\; \backslash mathbf\{n\}\backslash mathbf\; n:\; \backslash nabla\backslash mathbf\; v$

where

• $S\_L$ is the burning velocity of unstretched flame with respect to the unburnt gas
• $\backslash mathcal\{M\}\_c$ and $\backslash mathcal\{M\}\_s$ are the two , associated with the curvature term $\backslash nabla\; \backslash cdot\; \backslash mathbf\{n\}$ and the term $\backslash mathbf\{n\}\backslash mathbf\; n:\; \backslash nabla\backslash mathbf\; v$ corresponding to flow strain imposed on the flame
• $\backslash delta\_L$ are the laminar burning speed and thickness of a planar flame
• $\backslash tau\_L=\backslash delta\_L/S\_L$ is the planar flame residence time.

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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 $\backslash mathbf\{v\}=(0,U)$, 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 $\backslash alpha$. 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

$U\backslash frac\{\backslash partial\; G\}\{\backslash partial\; y\}\; =\; S\_L\; \backslash sqrt\{\backslash left(\backslash frac\{\backslash partial\; G\}\{\backslash partial\; x\}\backslash right)^2+\; \backslash left(\backslash frac\{\backslash partial\; G\}\{\backslash partial\; y\}\backslash right)^2\}$

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

$U\; =\; S\_L\backslash sqrt\{1+\; \backslash left(\backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\}\backslash right)^2\},\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\}\; =\; \backslash frac\{\backslash sqrt\{U^2-S\_L^2\}\}\{S\_L\}$

which upon integration gives

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

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

$G(x,y)\; =\; \backslash frac\{\backslash left(U^2-S\_L^2\backslash right)^\{1/2\}\}\{S\_L\}\backslash left(|x|-\; \backslash frac\{b\}\{2\}\backslash right)\; +\; y$

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

$L\; =\; \backslash frac\{b\backslash left(U^2-S\_L^2\backslash right)^\{1/2\}\}\{2S\_L\}$

and the flame angle $\backslash alpha$,

$\backslash tan\; \backslash alpha\; =\; \backslash frac\{b/2\}\{L\}\; =\; \backslash frac\{S\_T\}\{\backslash left(U^2-S\_L^2\backslash right)^\{1/2\}\}$

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

$\backslash sin\backslash alpha\; =\; \backslash frac\{S\_L\}\{U\}\; .$

In fact, the above formula is often used to determine the planar burning speed $S\_L$, by measuring the wedge angle.

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