Outline Outline Reynolds Equation Reynolds Equation Eddy - PowerPoint PPT Presentation

Outline Outline � Reynolds Equation � Reynolds Equation � Eddy Viscosity Models � Eddy Viscosity Models � Mixing Length Model Mixing Length Model � � Near Wall Flows Near Wall Flows � ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi Navier Navier- -Stokes Stokes t ∂ n ⎛ ⎞ ∂ ∂ ∂ ∂ u i = 2 i u u p u o ⎜ ⎟ ρ + = − + µ P i i i 0 u ⎜ ⎟ ∂ d ∂ ∂ ∂ ∂ ∂ j t x x x x x e ⎝ ⎠ x j i j j i i F a Turbulence Turbulence t a Mean y ′ = + Instantaneous t u U u U = ′ u i = i Velocity c u 0 o Velocity i i l e V ′ = + ′ p = p P p P = 0 p t ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi 1

+ t T Time Time 1 Properties ′ Properties u i = ∫ p = = 0 ' 0 u lim u dt i i Averaging Averaging → ∞ T T t 0 ′ ′ ′ ′ ′ ′ ≠ ≠ i ≠ +∞ Ensemble Ensemble u u u 0 u u 0 p ' u 0 ∫ < >= u d u i j k i j u u f ( ) i i Averaging Averaging − ∞ ′ ∂ i = u =< >= Ergodicity Ergodicity ′ ′ = = u u U U u U u 0 0 i i i ∂ i j i j x j ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi Reynolds Equation Reynolds Equation Reynolds Equation Reynolds Equation ⎛ ⎞ ′ ′ ∂ ∂ ∂ ∂ ∂ ⎛ ⎞ ⎡ ⎤ ∂ 2 ∂ ∂ ∂ ∂ u u U U P U U U U U ⎜ ⎟ ⎜ ⎟ ρ + = − + µ − ρ ′ ′ i j ρ + = − δ + µ + − ρ i i i ⎢ j ⎥ i i i U U P ( ) u u ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ j j ij i j ⎢ ⎥ t x x x x t x x x x x ⎝ ⎠ ⎣ ⎦ ⎝ ⎠ j j j i j i j j j ∂ i = Turbulence Turbulence U ⎛ ⎞ ′ ′ ′ ′ ′ 0 − ρ − ρ − ρ 2 u u v u w ∂ ⎜ ⎟ Stress Stress x i ⎜ ⎟ = − ρ ′ ′ − ρ ′ − ρ ′ ′ τ T 2 u v v v w ⎜ ⎟ Turbulence Turbulence Reynolds Reynolds ′ ′ τ = − ρ = τ ⎜ ′ ′ ′ ′ ′ ⎟ T T − ρ − ρ − ρ 2 u u u w v w w ⎝ ⎠ Stress Stress Stress Stress ij i j ji ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi 2

y Prandtl Assumption Prandtl Assumption Boussineq Boussineq Eddy Viscosity Model Eddy Viscosity Model U 1 1 dU ⎛ ⎞ ′ ′ ∂ ′ ′ 2 2 l ρ ∂ 2 2 ( u ) ~ ( v ) ~ U u u U l ⎜ ⎟ dy τ = − δ + µ + j T k k i ⎜ ⎟ ∂ ∂ ij ij T 3 x x ⎝ ⎠ l ′ ′ τ = − ρ j i T u v Ludwig Prandtl Ludwig Prandtl Eddy Eddy Mixing Mixing ∂ ∂ µ = ρν U U τ = ρ T 2 Viscosity Length l Viscosity Length T T ∂ y dy dU von von Eddy ∂ Eddy dy = κ U l ∂ τ = τ = ρν U Karman T T Karman 2 υ = d U Viscosity Viscosity l 2 ∂ 12 T ∂ y T 2 y dy ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi Inertial Inertial Turbulent Stress=Wall Shear Stress Turbulent Stress=Wall Shear Stress Shear Velocity Shear Velocity Sublayer Sublayer τ 2 * ⎛ ∂ ⎞ Inertial Inertial dU u = U * = 0 u τ = ρκ ⎜ ⎟ 2 2 Sublayer Sublayer y ⎜ ⎟ ρ κ ∂ 0 ⎝ ⎠ y dy y Turbulence Turbulence y Scales Scales U 1 = + = + U ln y c κ * τ u = κ l y o B ≈ * 5 u y Wall Wall + = 1 y + = + + U ln y B ν Units Units κ = von Karman von Karman κ 0 . 4 constant constant + < ≤ von von Karman Karman 30 y 300 ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi 3

+ Turbulent stress is negligible < ≤ Turbulent stress is negligible 0 y 5 30 dU τ = µ + U 0 dy 20 + + = + U 2 . 5 ln y 5 . 5 dU 2 = ν * u 10 dy + = y + + = y U + + + dU U = y 12 30 300 1 + dy ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi � Reynolds Equation Reynolds Equation � � Eddy Viscosity Models � Eddy Viscosity Models � Mixing Length Model Mixing Length Model � � Near Wall Flows Near Wall Flows � ME 637-Particle-II G. Ahmadi ME 637-Particle-II G. Ahmadi 4