Vorticity Transport in Turbulent Flows

or∂ ? + u ⋅ ∇?= ? ⋅∇u+ ν∇2 ?∂t∂ ? + u ⋅ ∇?= d ⋅ ? + ν∇2 ?∂t(9)(10)where d is the deformation rate tensor.During a turbulent motion u and ? become random functions of space and time,and they may be decomposed into a mean and fluctuating parts. i.e.,u = U + u′, Ui= ui, u′ i= 0 , (11)? = O + ? ′ , O = ? , ? ′ = 0 , (12)where U and O are the mean quantities and u ' and ?' are the fluctuating parts. As wasnoted before, a bar on the top of the letter stands for the (time) averaged quantity.Substituting (11) and (12) into Equation (10) after averaging we find the meanvorticity transport equation. That is∂Ω∂ti+ Uj∂Ω∂xji∂ω′iu′j= −∂xj∂ω′ju′i+∂xj+ Ωj∂U∂xij2∂ Ωi+ ν∂x∂xjj(13)Subtracting (13) from (10) leads to the equation for the vorticity fluctuation field. That is∂ω′i∂t+ Uj∂ω′i= −u′j∂xj∂Ω∂xij− u′j∂ω′∂xij+ Ω d′jij+ ω′ Djij+ ω′ d′jij2∂ ω′i+ ν∂x∂xjj∂ω′iu′j+∂xj− ω′ d′jij(14)Equation (14) may be used to derive the mean square vorticity or dissipation ratetransport equations.ME6372G. Ahmadi