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|} . The G equation has the form of the Hamilton-Jacobi equation, an equation in analytical mechanics used to model particle dynamics as propagation of waves.
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
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