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

SLIDE 1

1

• G. Ahmadi

ME 639-Turbulence

• G. Ahmadi

ME 639-Turbulence

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

• G. Ahmadi

ME 639-Turbulence

y x R Us Uo

• G. Ahmadi

ME 639-Turbulence

Simplified Reynolds Equation Simplified Reynolds Equation

v u y x U U0 = ′ ′ ∂ ∂ + ∂ ∂ ( )

θ ρ − = = − ρ∫

+∞ ∞ − 2

U M dy U U U

Momentum Integral Momentum Integral

∫

+∞ ∞ −

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = θ dy U U 1

Momentum Thickness Momentum Thickness

SLIDE 2

2

• G. Ahmadi

ME 639-Turbulence

Drag Coefficient Drag Coefficient

Self Similar Solutions Self Similar Solutions

d 2 d U 2 1 U d U 2 1 M d U 2 1 D C

2 2 2 2 d

θ = ρ θ ρ = ρ − = ρ =

5 3 d

10 3 to 10 ~ Re ×

1 cd =

2 d ≈ θ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = − l y f U U U

s

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ′ ′ − l y g U v u

2 s

• G. Ahmadi

ME 639-Turbulence

Similarity Variables Similarity Variables

m s

Ax U =

n

Bx = l

1 m s

x ~ x U ~ x U

−

∂ ∂

n m 2 2 s

x ~ U ~ y v u

−

∂ ′ ′ ∂ l

n m n m 2 1 m = + − = −

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − = = − n m 1 m n

2 1 m 2 1 n − = =

2 1

Ax U

−

=

2 1

Bx = l

Then Then

Momentum and Momentum and Momentum Momentum Integral Integral

• G. Ahmadi

ME 639-Turbulence 2 1

Bx y y = = ξ l

2 1

Bx 1 y = ∂ ξ ∂

x 2 Bx y 2 1 x

2 3

ξ − = − = ∂ ξ ∂

−

( ) ( )

ξ = ξ = −

− f

Ax f U U U

2 1 s

( ) ( )

ξ − = ξ − = ′ ′

− g

x A g U v u

1 2 2 s

( )

g f f A 2 B U0 ′ = ′ ξ +

Mean Velocity Mean Velocity Turbulence Turbulence Shear Stress Shear Stress Momentum Momentum Equation Equation

• G. Ahmadi

ME 639-Turbulence

y U v u

T ∂

∂ ν = ′ ′ −

f g U 1 f U g U y U v u

s s 2 s T

′ − = ′ − − − = ∂ ∂ ′ ′ − = ν l l

T s T

R 1 f g U = ′ − = ν l

T s T

U R ν = l

t tan Cons

T ≈

ν

f R 1 g

T

′ − =

Turbulent Reynolds Number Turbulent Reynolds Number

SLIDE 3

3

• G. Ahmadi

ME 639-Turbulence

( )

f f = ′ ′ + ′ ξ

const f f = = ′ + ξ

2 s

2

e f U U U

ξ −

= = −

2 s 2 T

U v u e R g

2

′ ′ − = ξ − =

ξ − π = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = θ ∫

+∞ ∞ −

2 U U dy U U 1

s

l

Mean Velocity Mean Velocity Turbulence Shear Stress Turbulence Shear Stress

Momentum Momentum Thickness Thickness

• G. Ahmadi

ME 639-Turbulence 0.2 0.4 0.6 0.8 1

• 4
• 3
• 2
• 1

1 2 3 4 ξ f Nondimensional Mean Velocity

• G. Ahmadi

ME 639-Turbulence

• 0.05
• 0.04
• 0.03
• 0.02
• 0.01

0.01 0.02 0.03 0.04 0.05

• 4
• 3
• 2
• 1

1 2 3 4 ξ g Nondimensional Turbulence Shear Stress

5 . 12 R T ≈

• G. Ahmadi

ME 639-Turbulence

⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ θ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ π ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ θ = θ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ θ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ θ =

2 1 2 1 T 2 1 2 1 2 1 T 2 1 s

x 252 . R 2 2 x x 58 . 1 2 2 R x U U l

5 . 12 R T ≈

s

U 4 . 2 U U θ = π θ = l

θ = = ν

s T

U 0319 . 5 . 12 U l

Similarity Variables Similarity Variables