= u Vorticity Vorticity u + = + - PowerPoint PPT Presentation

Outline Outline � Vorticity Transport in Viscous Flows Vorticity Transport in Viscous Flows � � Mean Flow Vorticity Energy Budget Mean Flow Vorticity Energy Budget � � Turbulence Mean Square Turbulence Mean Square Vorticity Vorticity � � Order of Magnitude Analysis Order of Magnitude Analysis � ME 637-Particle II G. Ahmadi ME 637-Particle II G. Ahmadi Navier Navier- -Stokes Equation Stokes Equation = ∇ × ω u Vorticity Vorticity ∂ ⎛ u ⎞ ρ + ⋅ ∇ = −∇ + µ ∇ ⎜ u u ⎟ 2 u ∇ ⋅ p u ∂ ⎝ ⎠ t Vector Identities Vector Identities ∇ × ∇ × = ∇∇ ⋅ − ∇ = −∇ u u 2 u 2 u u 2 u 2 | | | | Vector Identity Vector Identity ⋅ ∇ = ∇ × × + ∇ = × + ∇ u u u u ω u ( ) 2 2 ∂ ω Vorticity Vorticity ∂ + ∇ × × = ν ∇ u u 2 ω u 2 ω | | p ( ) + × + ∇ = −∇ + ν ∇ ω u 2 u Transport Eq Eq. . Transport ∂ t ∂ ρ t 2 ME 637-Particle II G. Ahmadi ME 637-Particle II G. Ahmadi 1

Vector Identities Vector Identities Turbulent Flows Turbulent Flows ∇ × × = ⋅ ∇ − ⋅ ∇ + ∇ ⋅ − ∇ ⋅ ω u u ω ω u ω u u ω ( ) ( ) ( ) ′ = + u U u u i = ′ U = u 0 = ⋅ ∇ − ⋅ ∇ u ω ω u i i ′ = + ′ = ω Ω ω Ω = ω ω ∂ 0 ω + ⋅ ∇ = ⋅ ∇ + ν ∇ u ω ω u 2 ω ∂ t Mean Vorticity Transport Equation Mean Vorticity Transport Equation Vorticity Vorticity Transport Eq Eq. . Transport ω′ ′ ω′ ′ ∂ ∂ ∂ Ω ∂ Ω ∂ ∂ Ω ∂ u u 2 ω U + = − + + Ω + ν i j j i i i i i + ⋅ ∇ = ⋅ + ν ∇ U u ω d ω 2 ω ∂ ∂ ∂ ∂ ∂ ∂ ∂ j j t x x x x x x ∂ t j j j j j j ME 637-Particle II G. Ahmadi ME 637-Particle II G. Ahmadi Instantaneous Vorticity Transport Equation Instantaneous Vorticity Transport Equation u 3 / Λ 3 u 3 / Λ 3 u 3 / Λ 3 ⎛ ⎞ ( ) ∂ ∂ ∂ ∂ Ω ⎛ ⎞ 1 ⎜ ⎟ ω′ ′ ′ ω′ + Ω Ω = − Ω + ⎜ ⎟ i U u u ⎜ ⎟ ∂ ∂ ∂ ∂ j i i i i j j i ⎝ ⎠ t x 2 x x ∂ ω′ ∂ ω′ ∂ Ω ∂ ω′ ⎝ ⎠ j j j 1 4 2 4 4 3 4 1 4 2 4 3 1 4 4 4 4 2 4 4 4 4 3 ′ ′ ′ + = − − + Ω i i i i U u u d transport by turbule nce gradient production Convective Transport ∂ ∂ ∂ ∂ j j j j ij velocity - vorticity interactio n of fluctuatin g vorticity of meanVortic ity t x x x u 3 / Λ 3 u 3 / Λ 3 j j j + Ω Ω + Ω ω′ ′ D d i j ij i j ij 1 4 2 4 3 1 2 3 ∂ ω′ ′ ω′ ∂ 2 u stretching of vorticity stretching of vorticity ω′ ω′ ′ ω′ ′ + + + ν + − i j by mean shear flow by mean shear flow i ν u 2 / Λ 4 = u 3 / Λ 3 R Λ D d d ∂ ∂ ∂ j ij j ij j ij ∂ ∂ Ω ∂ Ω 2 ⎛ ⎞ x x x 1 + ν Ω Ω − ν ⎜ ⎟ i i j j j ∂ ∂ ∂ ∂ ⎝ i i ⎠ x x 2 x x j j j j 1 4 4 4 2 4 4 4 3 1 4 2 4 3 viscous diffusion dissipatio n of mean vorti city ME 637-Particle II G. Ahmadi ME 637-Particle II G. Ahmadi 2

u 3 / Λλ 2 u 3 / Λ 3 u 3 / Λλ 2 2 2 2 u u u ⎛ ⎞ ( ) ∂ ∂ ∂ Ω ∂ ⎛ ⎞ ω′ ω′ ′ ω′ ω′ ′ ~ λ 1 1 ⎜ ⎟ ω′ ω′ ′ ω′ ′ ω′ ω′ u ~ d ~ + = − − ⎜ ⎟ U u i u ⎜ ⎟ Λ Λ i i i i j ij ∂ j ∂ i i j i ∂ ∂ j i i 2 2 ⎝ ⎠ t x 2 x 2 x ⎝ ⎠ j j j 1 4 2 4 3 1 4 2 4 4 3 4 gradient production diffusion of turbulenc e of fluctuatin g vorticity vorticity by turbule nce λ u 3 / λ 3 2 u ω′ ω′ + ω′ ω′ ′ + ω′ ω′ u 3 / Λ 2 λ δ + + d D ~ ( a b ...) 1 i 2 j 3 ij 1 i 2 j 3 ij λ Λ i j ij ij 2 production of turbulenc e vorticity production of turbulenc e vorticity ν u 2 / λ 2 η 2 by turbule nt stretching by mean shear flow stretching ν u 2 / λ 2 Λ 2 = u 3 / Λ 3 u 3 / Λ 3 ω′ ω′ ∂ ∂ ∂ 2 ⎛ ⎞ 1 + Ω ω′ ′ + ν ω′ ω′ − ν ⎜ ⎟ i i ∂ λ d 2 3 U u u u ∂ ∂ ∂ ∂ 1 j 2 i 3 ij ⎝ i i ⎠ ω′ ω′ ω′ ω′ x x 2 x x = = i D ~ 1 4 j 4 4 2 j 4 4 4 3 1 4 2 j 4 3 j production by mixed i j ∂ i j ij λ Λ Λ λ Λ 2 2 turbulence - mean flow stretching x diffusion by viscosi ty dissipatio n destructio n j ME 637-Particle II G. Ahmadi ME 637-Particle II G. Ahmadi Concluding Remarks Concluding Remarks � Vorticity Transport in Viscous Flows Vorticity Transport in Viscous Flows � � Mean Flow Vorticity Energy Budget � Mean Flow Vorticity Energy Budget � Turbulence Mean Square � Turbulence Mean Square Vorticity Vorticity � Order of Magnitude Analysis � Order of Magnitude Analysis ME 637-Particle II G. Ahmadi ME 637-Particle II G. Ahmadi 3