= u lim u dt Averaging i i Averaging i = - 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 639-Turbulence ME 639-Turbulence G. Ahmadi G. Ahmadi Navier Navier- -Stokes Stokes + t T Time Time 1 ∫ = u lim u dt ∂ Averaging i i ⎛ ⎞ Averaging → ∞ ∂ ∂ ∂ ∂ i = T u T 2 u u p u ⎜ ⎟ ρ + = − + µ t 0 i i i u 0 ⎜ ⎟ ∂ ∂ ∂ ∂ ∂ ∂ j x ⎝ t x ⎠ x x x j i j j i +∞ Ensemble Ensemble ∫ < >= u d u Turbulence u u f ( ) Turbulence Averaging i i Averaging − ∞ ′ = + u U u ′ U = u i = u 0 i i =< >= Ergodicity Ergodicity u u U ′ = + ′ p = i i i P = p P p 0 p ME 639-Turbulence ME 639-Turbulence G. Ahmadi G. Ahmadi 1

Reynolds Equation Reynolds Equation Properties ′ Properties u i = p = 0 ' 0 ⎛ ⎞ ∂ ′ ′ ∂ ∂ ∂ ∂ 2 u u U U P U ⎜ ⎟ ρ + = − + µ − ρ i j i i i U ⎜ ⎟ ∂ ∂ ∂ ∂ ∂ ∂ j t x x x x x ⎝ ⎠ ′ ′ ′ ′ ′ ′ ≠ ≠ i ≠ j i j j j u u u 0 u u 0 p ' u 0 i j k i j ∂ i = U 0 ∂ x i ′ ∂ i = u Turbulence ′ ′ Turbulence = = ′ ′ τ = − ρ = τ T T U u U u 0 0 u u ∂ i j i j Stress Stress ij i j ji x j ME 639-Turbulence ME 639-Turbulence G. Ahmadi G. Ahmadi Reynolds Equation Boussineq Eddy Viscosity Model Eddy Viscosity Model Reynolds Equation Boussineq ⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ∂ ∂ ρ ′ ′ ∂ ∂ ∂ ∂ ∂ U U U U U u u U ⎜ ⎟ ′ ′ ⎜ ⎟ ρ + = − δ + µ + − ρ ⎢ j ⎥ τ = − δ + µ + i i i U P ( ) u u T j k k i ⎜ ⎟ ∂ ∂ ∂ ∂ ∂ ⎜ ⎟ j ij i j ⎢ ⎥ t x x x x ⎝ ⎠ ⎣ ⎦ ∂ ∂ ij ij T 3 x x j j j i ⎝ ⎠ j i Turbulence Turbulence ⎛ ⎞ Eddy Eddy ′ ′ ′ ′ ′ − ρ − ρ − ρ 2 u u v u w ⎜ ⎟ µ = ρν Stress Stress Viscosity Viscosity ⎜ ⎟ T T = − ρ ′ ′ − ρ ′ − ρ ′ ′ τ T 2 u v v v w ⎜ ⎟ Reynolds Reynolds ⎜ ′ ′ ′ ′ ′ ⎟ − ρ − ρ − ρ ∂ 2 u w v w w U ⎝ ⎠ τ = τ = ρν Stress Stress T T ∂ 12 T y ME 639-Turbulence ME 639-Turbulence G. Ahmadi G. Ahmadi 2

y Inertial Prandtl Assumption Prandtl Assumption Inertial Shear Velocity Shear Velocity Sublayer Sublayer U 1 1 τ dU ′ ′ Inertial Inertial 2 2 l 2 2 = ( u ) ~ ( v ) ~ * 0 l u dy Sublayer ρ Sublayer Turbulence Turbulence l y ′ ′ τ = − ρ T u v Scales Scales Ludwig Prandtl Ludwig Prandtl τ Mixing Mixing ∂ ∂ = κ U U l τ = ρ y T 2 Length l o Length ∂ y dy dU von von Eddy Eddy dy κ = von Karman Karman von = κ l ∂ 0 . 4 U Karman Karman 2 υ = d U Viscosity Viscosity l 2 constant constant ∂ T 2 von Karman von Karman y dy ME 639-Turbulence ME 639-Turbulence G. Ahmadi G. Ahmadi Turbulent Stress=Wall Shear Stress Turbulent Stress=Wall Shear Stress < + ≤ Turbulent stress is negligible Turbulent stress is negligible 0 y 5 2 * ⎛ ∂ ⎞ dU u dU U = τ = µ τ = ρκ ⎜ ⎟ 2 2 y ⎜ ⎟ κ ∂ 0 0 ⎝ ⎠ dy y dy y dU U 1 2 = ν = + = + * u U ln y c κ * dy u B ≈ + = y * 5 u y Wall Wall + = 1 + + y + = + + dU U ln y B ν U = Units Units 1 κ + dy + < ≤ 30 y 300 ME 639-Turbulence ME 639-Turbulence G. Ahmadi G. Ahmadi 3

30 + U 20 + + = + U 2 . 5 ln y 5 . 5 10 + = y + U + y 12 30 300 ME 639-Turbulence ME 639-Turbulence G. Ahmadi G. Ahmadi Concluding Remarks Concluding Remarks � Reynolds Equation Reynolds Equation � � Eddy Viscosity Models � Eddy Viscosity Models � Mixing Length Model Mixing Length Model � � Near Wall Flows Near Wall Flows � ME 639-Turbulence G. Ahmadi 4