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

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• G. Ahmadi

ME 637-Particle II

• G. Ahmadi

ME 637-Particle II

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

• G. Ahmadi

ME 637-Particle II

Navier Navier-

• Stokes Equation

Stokes Equation

u u u u

2

p t ∇ µ + −∇ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∇ ⋅ + ∂ ∂ ρ

u ⋅ ∇

u u u ω u

2 2

p 2 | | t ∇ ν + ρ −∇ = ∇ + × + ∂ ∂

2 | | 2 | | ) (

2 2

u u ω u u u u u ∇ + × = ∇ + × × ∇ = ∇ ⋅

Vector Identity Vector Identity

• G. Ahmadi

ME 637-Particle II

Vorticity Vorticity

Vector Identities Vector Identities

u ω × ∇ =

ω u ω ω

2

) ( t ∇ ν = × × ∇ + ∂ ∂

Vorticity Vorticity Transport Transport Eq Eq. . u u u u

2 2

−∇ = ∇ − ⋅ ∇∇ = × ∇ × ∇

2

• G. Ahmadi

ME 637-Particle II

Vector Identities Vector Identities

Vorticity Vorticity Transport Transport Eq Eq. .

ω u ω ω u ω

2

t ∇ ν + ∇ ⋅ = ∇ ⋅ + ∂ ∂

ω ω d ω u ω

2

t ∇ ν + ⋅ = ∇ ⋅ + ∂ ∂

u ω ω u ω u u ω u ω ω u u ω ∇ ⋅ − ∇ ⋅ = ⋅ ∇ − ⋅ ∇ + ∇ ⋅ − ∇ ⋅ = × × ∇ ) ( ) ( ) (

• G. Ahmadi

ME 637-Particle II

Turbulent Flows Turbulent Flows

Mean Vorticity Transport Equation Mean Vorticity Transport Equation

u U u ′ + =

i i

u U = ui = ′

ω Ω ω ′ + =

ω Ω = = ′ ω

j j i 2 j i j j i j j j i j i j i

x x x U x u x u x U t ∂ ∂ Ω ∂ ν + ∂ ∂ Ω + ∂ ′ ω′ ∂ + ∂ ′ ω′ ∂ − = ∂ Ω ∂ + ∂ Ω ∂

• G. Ahmadi

ME 637-Particle II

Instantaneous Vorticity Transport Equation Instantaneous Vorticity Transport Equation

ij j j j i j j i 2 ij j ij j ij j j i j j i j j i j i

d x u x x d D d x u x u x U t ′ ω′ − ∂ ′ ω′ ∂ + ∂ ∂ ω′ ∂ ν + ′ ω′ + ω′ + ′ Ω + ∂ ω′ ∂ ′ − ∂ Ω ∂ ′ − = ∂ ω′ ∂ + ∂ ω′ ∂

• G. Ahmadi

ME 637-Particle II

( )

4 3 4 2 1 4 4 4 3 4 4 4 2 1 3 2 1 4 3 4 2 1 4 3 4 2 1 4 4 3 4 4 2 1 4 4 4 4 3 4 4 4 4 2 1

city mean vorti

• f

n dissipatio j i j i diffusion viscous i i j j 2 flow shear mean by vorticity

• f

stretching ij j i flow shear mean by vorticity

• f

stretching ij j i vorticity g fluctuatin

• f

production gradient j i i j n interactio vorticity

• velocity

nce by turbule transport j i i j ity meanVortic

• f

Transport Convective i i j j

x x 2 1 x x d D x u u x 2 1 x U t ∂ Ω ∂ ∂ Ω ∂ ν − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Ω Ω ∂ ∂ ∂ ν + ′ ω′ Ω + Ω Ω + ∂ Ω ∂ ω′ ′ + ′ ω′ Ω ∂ ∂ − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Ω Ω ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ u3/Λ3 u3/Λ3 u3/Λ3 u3/Λ3 u3/Λ3 νu2/Λ4 = u3/Λ3 RΛ

3

• G. Ahmadi

ME 637-Particle II

( )

4 3 4 2 1 4 4 4 3 4 4 4 2 1 3 2 1 3 2 1 3 2 1 4 4 3 4 4 2 1 4 3 4 2 1

n destructio n dissipatio j i j i ty by viscosi diffusion i i j j 2 stretching flow mean

• turbulence

mixed by production ij i j stretching flow shear mean by vorticity e turbulenc

• f

production ij j i stretching nt by turbule vorticity e turbulenc

• f

production ij j i nce by turbule vorticity e turbulenc

• f

diffusion i i j j vorticity g fluctuatin

• f

production gradient j i i j i i j j

x x 2 1 x x d D d u x 2 1 x u 2 1 x U t ∂ ω′ ∂ ∂ ω′ ∂ ν − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ω′ ω′ ∂ ∂ ∂ ν + ′ ω′ Ω + ω′ ω′ + ′ ω′ ω′ + ω′ ω′ ′ ∂ ∂ − ∂ Ω ∂ ω′ ′ − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ω′ ω′ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂

u3/Λλ2 u3/Λ3 u3/Λ2λ u3/λ3 u3/Λλ2 u3/Λ3 νu2/λ2Λ2 = u3/Λ3 νu2/λ2η2

• G. Ahmadi

ME 637-Particle II

2 2 i i

u ~ λ ω′ ω′ Λ ω′ ′

2 i i

u ~ u

2 2 ij j

u ~ d Λ ′ ω′ ...) b a ( u ~

ij ij 2 2 j i

+ Λ λ + δ λ ω′ ω′

2 3 2 2 ij j i j i j i

u u u ~ D x U Λ λ = Λ Λ λ λ ω′ ω′ = ∂ ∂ ω′ ω′

• G. Ahmadi

ME 637-Particle II

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

• G. Ahmadi

ME 637-Particle II