G equation

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

${\displaystyle \rho \left({\frac {\partial G}{\partial t}}+\mathbf {v} \cdot \nabla G\right)={\dot {m}}|\nabla G|}$

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