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 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

ρ ( ∂ G ∂ t + v ⋅ ∇ G ) = m ˙ | ∇ G | {\displaystyle \rho \left({\frac {\partial G}{\partial t}}+\mathbf {v} \cdot \nabla G\right)={\dot {m}}|\nabla G|}

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

Mathematical description

The G equation reads as

∂ G ∂ t + v ⋅ ∇ G = S T | ∇ G | {\displaystyle {\frac {\partial G}{\partial t}}+\mathbf {v} \cdot \nabla G=S_{T}|\nabla G|}

where

  • v {\displaystyle \mathbf {v} } is the flow velocity field,
  • S T = m ˙ / ρ u {\displaystyle S_{T}={\dot {m}}/\rho _{u}} is the local burning velocity with respect to the unburnt gas with density ρ u {\displaystyle \rho _{u}}.

The 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 ) > G o {\displaystyle G(\mathbf {x} ,t)>G_{o}} is the region of burnt gas and G ( x , t ) < G o {\displaystyle G(\mathbf {x} ,t)<G_{o}} is the region of unburnt gas. The normal vector to the flame, pointing towards the burnt gas, is n = ∇ G / | ∇ G | {\displaystyle \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

S T = S L + M c δ L ( S L − v ⋅ n ) ∇ ⋅ n − M t δ L ∇ t ⋅ v t {\displaystyle S_{T}=S_{L}+{\mathcal {M}}_{c}\delta _{L}(S_{L}-\mathbf {v} \cdot \mathbf {n} )\nabla \cdot \mathbf {n} -{\mathcal {M}}_{t}\delta _{L}\nabla _{t}\cdot \mathbf {v} _{t}}

where

  • S L {\displaystyle S_{L}} is the burning velocity of unstretched flame with respect to the unburnt gas
  • M c {\displaystyle {\mathcal {M}}_{c}} and M t {\displaystyle {\mathcal {M}}_{t}} are the two Markstein numbers, associated with the curvature and tangential straining; ∇ t ⋅ v t = − n ⊗ n : ∇ v − ( v ⋅ n ) ∇ ⋅ n {\displaystyle \nabla _{t}\cdot \mathbf {v} _{t}=-\mathbf {n} \otimes \mathbf {n} :\nabla \mathbf {v} -(\mathbf {v} \cdot \mathbf {n} )\nabla \cdot \mathbf {n} } is the surface divergence of the tangential velocity v t = ( I − n ⊗ n ) v {\displaystyle \mathbf {v} _{t}=(\mathbf {I} -\mathbf {n} \otimes \mathbf {n} )\mathbf {v} }
  • δ L {\displaystyle \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 {\displaystyle b}. The premixed reactant mixture is fed through the slot from the bottom with a 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 premixed flame develops from the mouth of the slot to a certain height y = L {\displaystyle y=L} in the form of a two-dimensional wedge shape with a wedge angle α {\displaystyle \alpha }. For simplicity, let us assume S T = S L {\displaystyle 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 ∂ G ∂ y = S L ( ∂ G ∂ x ) 2 + ( ∂ G ∂ y ) 2 {\displaystyle 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 ) {\displaystyle G(x,y)=y+f(x)} is introduced, then the equation becomes

U = S L 1 + ( ∂ f ∂ x ) 2 , ⇒ ∂ f ∂ x = U 2 − S L 2 S L {\displaystyle 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 ) = ( U 2 − S L 2 ) 1 / 2 S L | x | + C , ⇒ G ( x , y ) = ( U 2 − S L 2 ) 1 / 2 S L | x | + y + C {\displaystyle 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 {\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 − S L 2 ) 1 / 2 S L ( | x | − b 2 ) + y {\displaystyle 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 {\displaystyle x=0,\ y=L,\ G=0}, which enable us to determine the flame height

L = b ( U 2 − S L 2 ) 1 / 2 2 S L {\displaystyle L={\frac {b\left(U^{2}-S_{L}^{2}\right)^{1/2}}{2S_{L}}}}

and the flame angle α {\displaystyle \alpha },

tan ⁡ α = b / 2 L = S T ( U 2 − S L 2 ) 1 / 2 {\displaystyle \tan \alpha ={\frac {b/2}{L}}={\frac {S_{T}}{\left(U^{2}-S_{L}^{2}\right)^{1/2}}}}

Using the trigonometric identity tan 2 ⁡ α = sin 2 ⁡ α / ( 1 − sin 2 ⁡ α ) {\displaystyle \tan ^{2}\alpha =\sin ^{2}\alpha /\left(1-\sin ^{2}\alpha \right)}, we have

sin ⁡ α = S L U . {\displaystyle \sin \alpha ={\frac {S_{L}}{U}}.}

In fact, the above formula is often used to determine the planar burning speed S L {\displaystyle S_{L}}, by measuring the wedge angle.