Vorticity Transport in Turbulent Flows

Vorticity Transport in Turbulent FlowsVorticity EquationsThe Navier-Stokes and continuity equations are given asu 2⎛ ∂ ⎞ρ ⎜ + u ⋅ ∇u⎟= −∇p+ µ∇ u(1)⎝ ∂t⎠∇ ⋅ u . (2)The Navier-Stokes equation may be restated asu 2∂| u | p+ ? × u + ∇ = −∇ + ν∇2 u∂t2 ρ(3)where the vector identity22| u || u |u ⋅ ∇u= ( ∇ × u)× u + ∇ = ? × u + ∇(4)22is used, and vorticity is defines as? = ∇ × u(5)Taking curl of the Navier-Stokes equation given by (3), we find the vorticitytransport equation. That is∂∂t? 2+ ∇ × (?× u)= ν∇ ?(6)Here the vector identity22∇ ×∇ × u = ∇∇ ⋅ u − ∇ u = −∇ u(7)is used. Noting that∇ × (?× u)= u ⋅∇?− ? ⋅∇u+ ? ( ∇ ⋅ u)− u(∇ ⋅ ? )= u ⋅∇?− ? ⋅ ∇u(8)Equation (6) may be restated asME6371G. Ahmadi

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

Vorticity Energy EquationsMean Flow⎛⎞⎜∂ ∂ 1U ⎟⎛⎞j ⎜ ΩiΩi⎟ =+t x⎝ ∂ ∂j ⎝ 2 ⎠1444424⎠4443ConvectiveTransportof meanVorticity+Ωiω′jd′ij14243( Ω ω′ u′)∂−i i j∂xj1442443transport by turbulencevelocity-vorticity interactionstretchingof vorticityby mean shear flow+∂Ωiu′ω′j i∂xj14243gradient productionof fluctuating vorticity2∂ ⎛ 1 ⎞+ ν ⎜ ΩiΩi ⎟ −∂xj∂xj ⎝ 2 ⎠144424443viscousdiffusion+ΩiΩjDij14243stretchingof vorticityby mean shear flow∂Ωi∂Ωiν∂xj∂xj14243dissipation of mean vorticitywherej( Ω ω′ u′)∂− i i j= transport by turbulence velocity-vorticity interaction∂xu∂Ωi′jω′i= gradient production of fluctuating vorticity∂xjΩ Ω D = stretching of vorticity by mean shear flowijijΩ = stretching of turbulence vorticity by turbulence deformation rateiω′jd′ij2∂ ⎛ 1 ⎞ν ⎜ ΩiΩi ⎟ = viscous transport (diffusion)∂x∂x⎝ 2 ⎠j∂Ωji i− υ = dissipation of mean vorticity∂x ∂xj∂ΩjME6373G. Ahmadi

Fluctuating Flow⎛⎜∂U+⎝ ∂tj∂∂xj⎞⎟⎛1 ⎞⎜ i i ⎟ =ω′ω′⎠⎝2 ⎠Ωjω′id′ij14243( u′ω′ω′)∂Ωi1 ∂− u′ω′j i−j i i+∂xj2 ∂xj14243 1442443gradient production diffusion of turbulenceof fluctuating vorticity vorticity by turbulence+productionby mixedturbulence-mean flowstretchingω′ω′i jd′ij123productionof turbulence vorticityby turbulentstretching2∂ ⎛ 1 ⎞+ ν ⎜ ω′ω′i i ⎟ −∂xj∂xj ⎝ 2 ⎠14 44 24443diffusion by viscosity+∂ω′i∂ω′iν∂xj∂xj14243ω′ω′i jDij14243productionof turbulence vorticityby mean shear flow stretchingdissipation destructionwhere′∂Ω′i− u jω i= gradient production of fluctuating vorticity∂xj1 ∂− 2 ∂xj′ω′i jd′ij( u′ω ′ ω′)jii= diffusion of turbulence vorticity by turbulenceω = production of turbulence vorticity by turbulent stretchingω ′ ′ D = production of turbulence vorticity by mean shear flow stretchingi ω j′j id′ijΩ ω = production by mixed turbulence-mean flow stretching2∂ν∂x∂xj∂ω′ ∂ω′⎛ 1 ⎞⎜ ω′ω′i i ⎟ = diffusion by viscosity⎝ 2 ⎠ji i− ν = dissipation destruction∂x ∂xjjME6374G. Ahmadi