Outline Outline Turbulent Wake Flows Turbulent Wake Flows - PowerPoint PPT Presentation

Outline Outline � Turbulent Wake Flows � Turbulent Wake Flows � Momentum Integral � Momentum Integral � Similarity Variable � Similarity Variable � Eddy Viscosity Model � Eddy Viscosity Model � Similarity Solution � Similarity Solution ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi U o Simplified Reynolds Equation Simplified Reynolds Equation ∂ ∂ U ′ ′ + = U 0 u v 0 y ∂ ∂ x y R x Momentum Integral Momentum Integral Momentum Thickness Momentum Thickness U s +∞ +∞ ⎛ − ⎞ ( ) U ∫ ρ ∫ ⎜ ⎟ − = = − ρ θ θ = 2 1 dy U U U dy M U ⎜ ⎟ 0 0 0 ⎝ U ⎠ − ∞ − ∞ 0 ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 1

Drag Coefficient Similarity Variables Drag Coefficient Similarity Variables ρ θ θ U = = 2 m n D M U 2 l Ax Bx = = − = = 0 C s d 1 1 1 d ρ ρ ρ 2 2 2 U d U d U d ∂ 0 0 0 ′ ′ ∂ U U 2 2 2 2 u v U − Then − Then m 1 ~ s ~ x s 2 m n ~ ~ x ∂ ∂ l x x y d c d = θ ≈ × 3 5 1 Re ~ 10 to 3 10 d 2 Momentum and Momentum and 1 − = − ⎧ − = ⎫ = m 1 2 m n n m 1 n Momentum Momentum ⎨ ⎬ Self Similar Solutions Self Similar Solutions 2 + = = − ⎩ m n ⎭ Integral Integral m n 0 1 = − m − ⎛ ⎞ ⎛ ⎞ U U y y ′ ′ 2 = − = ⎜ ⎟ ⎜ ⎟ 2 0 u v U g 1 f 1 − s = ⎝ ⎠ ⎝ ⎠ l = l 2 U U Ax l Bx 2 s 0 ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi y = ∂ ξ ∂ ξ ξ ∂ y ′ ′ − 1 y 2 1 ξ = U u v U g g y = = − = − ν = − = − = − ′ ′ − = ν s l U T ∂ ∂ u v ∂ 1 ∂ 3 ′ l 1 T s x 2 − 2 x U 1 f − ′ 2 Bx 2 U f Bx Bx 2 y ∂ s l y Turbulent Reynolds Number Turbulent Reynolds Number 1 − f Mean Velocity ( ) ( ) Mean Velocity − = ξ = ξ ν U U U f Ax 2 g 1 l U 0 s = − = T = s R Turbulence Turbulence ′ ( ) ( ) T ν − g l ′ ′ U f R = − ξ = − ξ 2 2 1 u v U g A x T s T s Shear Stress Shear Stress 1 ν T ≈ U 0 B ( ) ′ = − Cons tan t Momentum Momentum + ξ ′ = ′ g f f f g R Equation Equation 2 A T ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 2

′ ( ) Nondimensional Mean ′ ξ + = = ′ ′ ξ + = f f const 0 Velocity f f 0 1 0.8 Mean Velocity Mean Velocity Turbulence Shear Stress Turbulence Shear Stress 0.6 f ξ 2 − ξ ξ 2 ′ ′ − U U − 0.4 u v = = = − = − 0 2 f e g e 2 2 0.2 R U U T s s 0 +∞ ⎛ − ⎞ Momentum Momentum -4 -3 -2 -1 0 1 2 3 4 θ ∫ U U = ⎜ ⎟ = π s l 1 dy 2 ⎜ ⎟ ξ Thickness Thickness ⎝ ⎠ U U − ∞ 0 0 ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi Similarity Variables Similarity Variables Nondimensional Turbulence Shear Stress 0.05 ⎧ ⎫ 1 1 1 ⎛ θ ⎛ θ 0.04 ⎞ ⎛ ⎞ ⎞ ⎪ U R ⎪ 2 2 2 = = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ s T 0.03 1 . 58 R T ≈ ⎪ ⎪ 12 . 5 ⎝ ⎠ π ⎝ ⎠ ⎪ ⎝ ⎠ ⎪ U x x 2 2 0.02 0 ⎨ ⎬ 0.01 1 1 1 ⎪ ⎪ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 l 2 g x 2 x 2 2 ⎜ ⎟ = = ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ 0 . 252 ⎜ ⎟ -0.01 θ θ π θ ⎪ ⎝ ⎠ ⎝ ⎠ ⎪ ⎝ ⎠ ⎩ 2 R ⎭ -0.02 T -0.03 -0.04 θ U U l -0.05 = = θ l 0 ν = = θ U 0 . 4 U s 0 . 0319 U -4 -3 -2 -1 0 1 2 3 4 π s 0 T 0 2 12 . 5 ξ R T ≈ 12 . 5 ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 3