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

Feb 12, 2023 •411 likes •454 views

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

  1. 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

  2. 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

  3. 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

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