Vorticity Transport in Turbulent Flows

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