Objectives Objectives Turbulent Jet Flows Turbulent Jet Flows - PowerPoint PPT Presentation

Objectives Objectives � Turbulent Jet Flows � Turbulent Jet Flows � Momentum Integral � Momentum Integral � Similarity Variable � Similarity Variable � Eddy Viscosity Model � Eddy Viscosity Model � Similarity Solution � Similarity Solution � Thermal Plume � Thermal Plume G. Ahmadi G. Ahmadi ME 639-Turbulence ME 639-Turbulence Equation of Motion Equation of Motion y y ∂ ∂ ∂ U U + + ′ ′ = U V u v 0 ∂ ∂ ∂ x y y R +∞ x x d ∫ = 2 Momentum Integral U dy 0 Momentum Integral U s U s dx − ∞ +∞ ρ ∫ = = U 2 dy M Jet Momentum − ∞ G. Ahmadi G. Ahmadi ME 639-Turbulence ME 639-Turbulence 1

Similarity Variables Continuity Continuity ∂ ψ = U = = U m n l ∂ ∂ cx Dx ∂ U V y + = s Stream Stream 0 ∂ ∂ x y Function Function ∂ ψ n = Momentum Momentum = − 1 − = − V 2 m 1 2 m n Equation Equation ∂ x y ξ = Momentum Momentum + = 1 Self Similar Solutions = − Self Similar Solutions 2 m n 0 l m Integral Integral ( ) 2 = ξ ( ) U U f ′ ′ − = ξ 2 u v U g 1 − = s Then Then s = l Dx U Cx 2 s G. Ahmadi G. Ahmadi ME 639-Turbulence ME 639-Turbulence ∂ ξ Turbulence Shear Stress Turbulence Shear Stress y = ∂ ξ ξ Momentum Equation Momentum Equation 1 1 y y = = = − = − ξ = ∂ ∂ ∂ 2 l y Dx x Dx x U l Dx 2 ′ ′ − = ν ′ ′ ′ ′ + = − T ∂ F 2 u v F F g Mean U- -Velocity Velocity Mean U y Stream Function Stream Function D 1 − F 1 ( ) ( ) Eddy viscosity Eddy viscosity ′ = ξ ψ = ξ 2 CDx 2 U Cx F ν 1 1 g ′ ′ = = = T g F Mean V- -Velocity Velocity Mean V ′ ′ Turbulent Shear Stress Turbulent Shear Stress l U R F R ∂ ψ ( ) 1 ⎛ ⎞ s T − 1 T − g − ′ ′ = ξ ′ = − = − − ξ ⎜ ⎟ 2 1 V CDx 2 F F u v C x ∂ ⎝ ⎠ x 2 G. Ahmadi G. Ahmadi ME 639-Turbulence ME 639-Turbulence 2

ξ ′ 1 ( ) ′ ′ ′ + = ′ = = = 2 ′ ′ ′ ′ Mean Mean f F sec h + = F F F c ξ F F F 0 2 2 1 Velocity Velocity cosh 2 Boundary Conditions Boundary Conditions Similarity Variables Similarity Variables ( ) ( ) ( ) ′ ∞ = ′ = = 1 F 0 F 0 0 F 0 1 ⎛ ⎞ = U d 2 = l ⎜ ⎟ 0 . 078 x s 2 . 7 ⎝ ⎠ U x j ξ = +∞ F 2 tanh Mean Velocity Momentum Mean Velocity Momentum ∫ = 2 2 U dy U d 2 Integral Integral j − ∞ G. Ahmadi G. Ahmadi ME 639-Turbulence ME 639-Turbulence Nondimensional Nondimensional Mean Velocity Turbulence Shear Stress 1 0.03 0.02 0.8 0.01 0.6 f( ξ) g( ξ ) 0 0.4 -0.01 0.2 -0.02 0 -0.03 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 ξ ξ G. Ahmadi G. Ahmadi ME 639-Turbulence ME 639-Turbulence 3

U s ∂ ∂ ∂ g T W W ′ ′ + + = y 0 U W u w ∂ ∂ ∂ x z x T 0 ∂ ∂ ∂ R T T + + ′ ′ = R U W T u 0 ∂ ∂ ∂ g x z x U s U j ∂ ∂ U W z + = 0 ∂ ∂ x z x +∞ q ∫ = = T Wdx const ρ Transient Turbulent Jet Transient Turbulent Jet c Heat Source Heat Source X/d − ∞ p 5 10 G. Ahmadi G. Ahmadi ME 639-Turbulence ME 639-Turbulence Nondimensional Mean Velocity Concluding Remarks Concluding Remarks and Mean Temperature 1 � Turbulent Jet Flows � Turbulent Jet Flows 0.8 � Momentum Integral � Momentum Integral 0.6 f, θ � Similarity Variable � Similarity Variable 0.4 � Eddy Viscosity Model � Eddy Viscosity Model 0.2 � Similarity Solution � Similarity Solution 0 -4 -3 -2 -1 0 1 2 3 4 � Thermal Plume � Thermal Plume ξ G. Ahmadi G. Ahmadi ME 639-Turbulence ME 639-Turbulence 4