Outline Outline 4 Mean Flow Energy Budget 4 Mean Flow Energy - - PowerPoint PPT Presentation

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• G. Ahmadi

ME 637-Particle II

• G. Ahmadi

ME 637-Particle II

Outline Outline

4 4Mean Flow Energy Budget Mean Flow Energy Budget 4 4Turbulence Energy Budget Turbulence Energy Budget 4 4Simple Shear Flows Simple Shear Flows

• G. Ahmadi

ME 637-Particle II

Reynolds Equation Reynolds Equation

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

x u u x x U x P x U U t U ∂ ′ ′ ∂ ρ − ∂ ∂ ∂ µ + ∂ ∂ − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ρ

x U

i i =

∂ ∂

• G. Ahmadi

ME 637-Particle II

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

production energy n fluctuatio j i j i n dissipatio viscous diffusion viscous i i j j 2 nce by turbule diffusion j i j i i convection i i j j

x U u u j x i U j x i U 2 U U x x u u U P U x 2 U U x U t ∂ ∂ ′ ′ + ∂ ∂ ∂ ∂ ν − ∂ ∂ ∂ ν + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ′ ′ + ρ ∂ ∂ − = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ ∂ + ∂ ∂

u3/Λ u3/Λ u3/Λ νu2/Λ2 νu2/Λ2

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• G. Ahmadi

ME 637-Particle II

Fluctuation Velocity Equation Fluctuation Velocity Equation

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

x u u x x u x p 1 x u u x U u x u U t u ∂ ′ ′ ∂ + ∂ ∂ ′ ∂ ν + ∂ ′ ∂ ρ − = ∂ ′ ∂ ′ + ∂ ∂ ′ + ∂ ′ ∂ + ∂ ′ ∂

Navier Navier-

• Stokes

Stokes Eq Eq. . – – Reynolds Equation Reynolds Equation⇒ ⇒

• G. Ahmadi

ME 637-Particle II

4 3 4 2 1 4 3 4 2 1 4 4 4 3 4 4 4 2 1 4 3 4 2 1 43 42 1

n Dissipatio j i j i n Dissipatio Viscous j j 2 Diffusion turbulent j i i j j

• duction

Pr j i j i Convection j j

x u x u x x k 2 u u u p u x x U u u x k U t k ∂ ′ ∂ ∂ ′ ∂ ν − ∂ ∂ ∂ ν + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ′ ′ ′ + ρ ′ ′ ∂ ∂ − ∂ ∂ ′ ′ − = ∂ ∂ + ∂ ∂

u3/Λ u3/Λ u3/Λ νu2/Λ2 νυ2/η2

• G. Ahmadi

ME 637-Particle II j j j i j i i i j j i i

x u p x U u u ) 2 u u p ( u x 2 u u Dt D ∂ ′ ∂ ρ ′ + ε − ∂ ∂ ′ ′ − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ′ ′ + ρ ′ ′ ∂ ∂ − = ′ ′

Turbulence Kinetic Energy Equation

) , ), y ( U (

1

= U

z x = ∇ ⋅ = ∂ ∂ = ∂ ∂ U

• G. Ahmadi

ME 637-Particle II

j j 1 2 1 i i 2

x u p y U u u ) 2 u u p ( u y ′ ∂ ′ ∂ ρ ′ + ε − ∂ ∂ ′ ′ − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ′ ′ + ρ ′ ′ ∂ ∂ − = x u p 3 1 y U u u 2 u u u y

1 1 2 1 1 1 2

∂ ′ ∂ ρ ′ + ε − ∂ ∂ ′ ′ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′ ′ ′ ∂ ∂ − = y u p 3 1 ) 2 u u p ( u y

2 2 2 2

∂ ′ ∂ ρ ′ + ε − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ′ ′ + ρ ′ ′ ∂ ∂ − = z u p 3 1 ) 2 u u ( u y

3 3 3 2

∂ ′ ∂ ρ ′ + ε − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ′ ′ ′ ∂ ∂ − =

2 1

u 2 1 ′

2 3

u 2 1 ′

2 2

u 2 1 ′

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• G. Ahmadi

ME 637-Particle II

Concluding Remarks Concluding Remarks

4 4Mean Flow Energy Budget Mean Flow Energy Budget 4 4Turbulence Energy Budget Turbulence Energy Budget 4 4Simple Shear Flows Simple Shear Flows

• G. Ahmadi

ME 637-Particle II