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

Outline Outline 4 Mean Flow Energy Budget 4 Mean Flow Energy Budget 4 Turbulence Energy Budget 4 Turbulence Energy Budget 4 Simple Shear Flows 4 Simple Shear Flows ME 637-Particle II G. Ahmadi ME 637-Particle II G. Ahmadi u 3 / Λ u 3 / Λ Reynolds Equation Reynolds Equation ⎡ ⎤ ⎡ ⎤ ∂ ∂ ∂ U U P ′ ′ + = − + ⎢ ⎥ i i U ⎢ U U u u ⎥ ⎛ ⎞ ∂ ′ ′ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ρ j i j i j 2 u u ⎢ ⎥ ⎣ ⎦ U U P U t x 2 x ⎣ ⎦ ⎜ ⎟ ρ + = − + µ − ρ i j 1 4 4 4 2 4 4 4 3 j i i i i 1 4 4 4 2 4 4 4 3 U ⎜ ⎟ ∂ ∂ ∂ ∂ ∂ ∂ j t x x x x x ⎝ ⎠ diffusion by turbule nce convection j i j j j ∂ ∂ ∂ U U ∂ 2 U U U i i ′ ′ + ν − ν + i i i u u ∂ ∂ ∂ ∂ ∂ ∂ i j x x 2 x x x i = U 1 4 2 4 j j 4 3 4 1 4 2 4 3 j j j 0 1 4 2 4 3 ∂ x viscous diffusion fluctuatio n energy viscous i production dissipatio n ν u 2 / Λ 2 ν u 2 / Λ 2 u 3 / Λ ME 637-Particle II G. Ahmadi ME 637-Particle II G. Ahmadi 1

u 3 / Λ u 3 / Λ u 3 / Λ Reynolds Equation ⇒ ⇒ Navier- -Stokes Stokes Eq Eq. . – – Reynolds Equation Navier ⎡ ⎤ ′ ′ ′ ′ ′ ∂ ∂ ∂ ∂ u p u u u k k U ′ ′ + = − − + ⎢ j i i j ⎥ Fluctuation Velocity Equation Fluctuation Velocity Equation i U u u ∂ ∂ ∂ ∂ ρ j i j ⎢ ⎥ t x x x 2 ⎣ ⎦ 1 42 43 j 1 4 2 4 3 j j 1 4 4 4 2 4 4 4 3 ′ ′ ′ ′ ∂ ∂ ∂ ∂ ∂ u u U u 1 p ′ ′ + + + = − Convection Pr oduction turbulent Diffusion i i i i U u u ∂ ∂ ∂ ∂ ρ ∂ j j j ∂ ∂ ′ ∂ ′ t x x x x 2 k u u j j j j + ν − ν i i ∂ ∂ ∂ ∂ ′ ′ ∂ x x x x ′ ∂ 2 u u u 1 4 2 j 4 3 j 1 4 2 j 4 3 j + ν + i j i ∂ ∂ ∂ Viscous Dissipatio n x x x Dissipatio n j j j ν u 2 / Λ 2 νυ 2 / η 2 ME 637-Particle II G. Ahmadi ME 637-Particle II G. Ahmadi Turbulence Kinetic Energy Equation ′ ⎡ ′ ′ ′ ⎤ ′ ∂ ∂ ∂ u p u u U p = − ′ + − ′ ′ − ε + j ⎢ i i ⎥ 1 0 u ( ) u u ′ ∂ ∂ ρ ∂ ρ ∂ ′ ′ ′ ∂ ⎡ ′ ′ ′ ⎤ ∂ ′ 2 1 2 ⎣ ⎦ u y 2 y x D u u p u u U p ′ ′ ′ = − + − − ε + j j i i i i i ⎢ u ( ) ⎥ u u ∂ ρ ∂ ρ ∂ j i j ⎣ ⎦ Dt 2 x 2 x x ⎛ ′ ′ ⎞ ′ ′ 1 ′ ∂ ∂ ∂ j j j u u U 1 p u ⎜ ′ ⎟ ′ ′ = − − − ε + 2 1 1 1 1 u 0 u u u ⎜ ⎟ 1 ∂ 2 1 2 ∂ ρ ∂ 2 y ⎝ 2 ⎠ y 3 x = U ( U ( y ), 0 , 0 ) ⎡ ⎤ ∂ ′ ′ ′ ′ ∂ ′ 1 1 ′ p u u 1 p u ′ = − + − ε + ⎢ 2 2 ⎥ 2 2 0 u ( ) u ∂ 2 ρ ρ ∂ 2 y ⎣ 2 ⎦ 3 y 2 ∂ ∂ = = ⋅ ∇ = U 0 ∂ ∂ ⎡ ⎤ 1 ′ ′ ′ ′ ′ x z ∂ ∂ u u 1 p u ′ = − − ε + 2 u ⎢ 3 3 ⎥ 3 0 u ( ) 3 ∂ ρ ∂ 2 2 y ⎣ 2 ⎦ 3 z ME 637-Particle II G. Ahmadi ME 637-Particle II G. Ahmadi 2

Concluding Remarks Concluding Remarks 4 Mean Flow Energy Budget 4 Mean Flow Energy Budget 4 Turbulence Energy Budget 4 Turbulence Energy Budget 4 Simple Shear Flows 4 Simple Shear Flows ME 637-Particle II G. Ahmadi ME 637-Particle II G. Ahmadi 3