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

SLIDE 1

1

• G. Ahmadi

ME 639-Turbulence

• G. Ahmadi

ME 639-Turbulence

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

ME 639-Turbulence

R x x y y Us Us

• G. Ahmadi

ME 639-Turbulence

Equation of Motion Equation of Motion Momentum Integral Momentum Integral v u y y U V x U U = ′ ′ ∂ ∂ + ∂ ∂ + ∂ ∂

dy U dx d

2

=

∫

+∞ ∞ −

Momentum Jet M dy U2 = = ρ∫

+∞ ∞ −

SLIDE 2

2

• G. Ahmadi

ME 639-Turbulence

Continuity Continuity Self Similar Solutions Self Similar Solutions

y V x U = ∂ ∂ + ∂ ∂

y U ∂ ψ ∂ =

x V ∂ ψ ∂ − =

Stream Stream Function Function

( )

ξ = f U U

s

( )

ξ = ′ ′ − g U v u

2 s

l y = ξ

• G. Ahmadi

ME 639-Turbulence

Similarity Variables Then Then

Momentum Momentum Integral Integral

m s

cx U =

n

Dx = l

n m 2 1 m 2 − = −

1 n =

n m 2 = +

2 1 m − =

2 1 s

Cx U

−

=

Dx = l

Momentum Momentum Equation Equation

• G. Ahmadi

ME 639-Turbulence

Mean U Mean U-

• Velocity

Velocity

Dx y y = = ξ l

x Dx y x

2

ξ − = − = ∂ ξ ∂ Dx 1 1 y = = ∂ ξ ∂ l

( )

ξ = ψ F CDx 2

1

( )

ξ = ′ ′ −

− g

x C v u

1 2

( )

ξ ′ =

− F

Cx U

2 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ′ ξ − − = ∂ ψ ∂ − =

−

F F 2 1 CDx x V

2 1

Stream Function Stream Function Mean V Mean V-

• Velocity

Velocity Turbulent Shear Stress Turbulent Shear Stress

• G. Ahmadi

ME 639-Turbulence

y U v u

T ∂

∂ ν = ′ ′ −

F g R 1 U

T s T

′ ′ = = ν l

F R 1 g

T

′ ′ =

g D 2 F F F 2 ′ − = ′ ′ + ′

Momentum Equation Momentum Equation Turbulence Shear Stress Turbulence Shear Stress Eddy viscosity Eddy viscosity

SLIDE 3

3

• G. Ahmadi

ME 639-Turbulence

Mean Velocity Mean Velocity

( )

F F F = ′ ′ ′ + ′ ′

1

c F F F = ′ ′ + ′

( )

1 F = ′

( )

F = ∞ ′

( )

F =

2 tanh 2 F ξ =

Boundary Conditions Boundary Conditions

• G. Ahmadi

ME 639-Turbulence

Mean Mean Velocity Velocity

2 h sec 2 cosh 1 F f

2 2

ξ = ξ = ′ =

2 1 j s

x d 7 . 2 U U ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =

x 078 . = l

d U dy U

2 j 2

=

∫

+∞ ∞ −

Similarity Variables Similarity Variables Momentum Momentum Integral Integral

• G. Ahmadi

ME 639-Turbulence 0.2 0.4 0.6 0.8 1

• 4
• 3
• 2
• 1

1 2 3 4 ξ f(ξ) Nondimensional Mean Velocity

• G. Ahmadi

ME 639-Turbulence

• 0.03
• 0.02
• 0.01

0.01 0.02 0.03

• 4
• 3
• 2
• 1

1 2 3 4 ξ g(ξ ) Nondimensional Turbulence Shear Stress

SLIDE 4

4

• G. Ahmadi

ME 639-Turbulence

R Us y X/d

5 10

Transient Transient Turbulent Jet

Turbulent Jet

Uj

• G. Ahmadi

ME 639-Turbulence

Heat Source Heat Source Us x z R g T T g w u x z W W x W U = ′ ′ ∂ ∂ + ∂ ∂ + ∂ ∂ u T x z T W x T U = ′ ′ ∂ ∂ + ∂ ∂ + ∂ ∂ z W x U = ∂ ∂ + ∂ ∂

p

c q const Wdx T ρ = =

∫

+∞ ∞ −

• G. Ahmadi

ME 639-Turbulence 0.2 0.4 0.6 0.8 1

• 4
• 3
• 2
• 1

1 2 3 4 ξ f, θ Nondimensional Mean Velocity and Mean Temperature

• G. Ahmadi

ME 639-Turbulence

Concluding Remarks Concluding Remarks

• 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