In Combustion, G equation is a scalar field equation which describes the instantaneous flame position, introduced by Forman A. Williams in 1985[1][2] in the study of premixed turbulent combustion. The equation is derived based on the Level-set method. The equation was studied by George H. Markstein earlier, in a restrictive form.[3][4]
The flame location is given by which can be defined arbitrarily such that is the region of burnt gas and is the region of unburnt gas. The normal vector to the flame is .
Local burning velocity
The burning velocity of the stretched flame can be derived by subtracting suitable terms from the unstretched flame speed, for small curvature and small strain, as given by
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 with a premixed reactant mixture is fed through the slot with constant velocity , where the coordinate is chosen such that lies at the center of the slot and 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 with a planar conical shape with cone angle . In the steady case, the G equation reduces to
If a separation of the form is introduced, the equation becomes
which upon integration gives
Without loss of generality choose the flame location to be at . Since the flame is attached to the mouth of the slot , the boundary condition is , which can be used to evaluate the constant . Thus the scalar field is
At the flame tip, we have , the flame height is easily determined as
^Williams, 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.
^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.
^Peters, Norbert. Turbulent combustion. Cambridge university press, 2000.
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In Combustion G equation is a scalar G x t displaystyle G mathbf x t field equation which describes the instantaneous flame position introduced by Forman A Williams in 1985 1 2 in the study of premixed turbulent combustion The equation is derived based on the Level set method The equation was studied by George H Markstein earlier in a restrictive form 3 4 Contents 1 Mathematical description 5 6 1 1 Local burning velocity 2 A simple example Slot burner 3 ReferencesMathematical description 5 6 EditThe G equation reads as G t v G U L G displaystyle frac partial G partial t mathbf v cdot nabla G U L nabla G where v displaystyle mathbf v is the flow velocity field U L displaystyle U L is the local burning velocityThe flame location is given by G x t G o displaystyle G mathbf x t G o which can be defined arbitrarily such that G x t gt G o displaystyle G mathbf x t gt G o is the region of burnt gas and G x t lt G o displaystyle G mathbf x t lt G o is the region of unburnt gas The normal vector to the flame is n G G displaystyle mathbf n nabla G nabla G Local burning velocity Edit The burning velocity of the stretched flame can be derived by subtracting suitable terms from the unstretched flame speed for small curvature and small strain as given by U L S L S L L k L S displaystyle U L S L S L mathcal L kappa mathcal L S where S L displaystyle S L is the burning velocity of unstretched flame S n v n displaystyle 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 L displaystyle mathcal L is the Markstein length proportional to the laminar flame thickness d L displaystyle delta L the constant of proportionality is Markstein number M displaystyle mathcal M k n 2 G n n G G textstyle 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 EditThe G equation has an exact expression for a simple slot burner Consider a two dimensional planar slot burner of slot width b displaystyle b with a premixed reactant mixture is fed through the slot with constant velocity v 0 U displaystyle mathbf v 0 U where the coordinate x y displaystyle x y is chosen such that x 0 displaystyle x 0 lies at the center of the slot and y 0 displaystyle 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 displaystyle y L with a planar conical shape with cone angle a displaystyle alpha In the steady case the G equation reduces to U G y U L G x 2 G y 2 displaystyle 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 displaystyle G x y y f x is introduced the equation becomes U U L 1 f x 2 or f x U 2 U L 2 U L displaystyle 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 U 2 U L 2 1 2 U L x C G x y U 2 U L 2 1 2 U L x y C displaystyle 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 displaystyle G x y G o 0 Since the flame is attached to the mouth of the slot x b 2 y 0 displaystyle x b 2 y 0 the boundary condition is G b 2 0 0 displaystyle G b 2 0 0 which can be used to evaluate the constant C displaystyle C Thus the scalar field is G x y U 2 U L 2 1 2 U L x b 2 y displaystyle 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 displaystyle x 0 y L G 0 the flame height is easily determined as L b U 2 U L 2 1 2 2 U L displaystyle L frac b left U 2 U L 2 right 1 2 2U L and the flame angle a displaystyle alpha is given by tan a b 2 L U L U 2 U L 2 1 2 displaystyle tan alpha frac b 2 L frac U L left U 2 U L 2 right 1 2 Using the trigonometric identity tan 2 a sin 2 a 1 sin 2 a displaystyle tan 2 alpha sin 2 alpha left 1 sin 2 alpha right we have sin a U L U displaystyle sin alpha frac U L U References Edit Williams 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 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 Peters Norbert Turbulent combustion Cambridge university press 2000 Williams Forman A Combustion theory 1985 Retrieved from https en wikipedia org w index php title G equation amp oldid 1030121584, wikipedia, wiki, book, books, library,