Outline Outline Reynolds Equation Reynolds Equation Eddy - - PowerPoint PPT Presentation

SLIDE 1

1

• G. Ahmadi

ME 637-Particle-II

• G. Ahmadi

ME 637-Particle-II

Outline Outline

• Reynolds Equation

Reynolds Equation

• Eddy Viscosity Models

Eddy Viscosity Models

• Mixing Length Model

Mixing Length Model

• Near Wall Flows

Near Wall Flows

• G. Ahmadi

ME 637-Particle-II

Navier Navier-

• Stokes

Stokes

j j i 2 i j i j i

x x u x p x u u t u ∂ ∂ ∂ µ + ∂ ∂ − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ρ

x u

i i =

∂ ∂ Turbulence Turbulence

i i

u U = ui = ′

p P p ′ + =

p P = p = ′

u U u ′ + =

• G. Ahmadi

ME 637-Particle-II

t

Mean Velocity Instantaneous Velocity V e l

• c

i t y a t a F i x e d P

• i

n t

SLIDE 2

2

• G. Ahmadi

ME 637-Particle-II

Time Time Averaging Averaging

∫

+ ∞ →

=

T t t i T i

dt u T 1 lim u

Ensemble Ensemble Averaging Averaging

∫

+∞ ∞ −

>= < u u d ) ( f u u

i i

i i i

U u u >= =<

Ergodicity Ergodicity

• G. Ahmadi

ME 637-Particle-II

Properties Properties

u u

j i

≠ ′ ′

ui = ′

' p =

u ' p

i ≠

′

u u u

k j i

≠ ′ ′ ′ u U u U

j i j i

= ′ = ′

x u

j i =

∂ ′ ∂

• G. Ahmadi

ME 637-Particle-II

Turbulence Turbulence Stress Stress Reynolds Equation Reynolds Equation

j j i j j i 2 i j i j i

x u u x x U x P x U U t U ∂ ′ ′ ∂ ρ − ∂ ∂ ∂ µ + ∂ ∂ − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ρ x U

i i =

∂ ∂

T ji j i T ij

u u τ = ′ ′ ρ − = τ

• G. Ahmadi

ME 637-Particle-II

Turbulence Turbulence Stress Stress Reynolds Reynolds Stress Stress Reynolds Equation Reynolds Equation

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ′ ′ ρ − ∂ ∂ + ∂ ∂ µ + δ − ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ρ

j i i j j i ij j j i j i

u u ) x U x U ( P x x U U t U

⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ′ ρ − ′ ′ ρ − ′ ′ ρ − ′ ′ ρ − ′ ρ − ′ ′ ρ − ′ ′ ρ − ′ ′ ρ − ′ ρ − =

2 2 2 T

w w v w u w v v v u w u v u u τ

SLIDE 3

3

• G. Ahmadi

ME 637-Particle-II

Boussineq Boussineq Eddy Viscosity Model Eddy Viscosity Model ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ µ + δ ′ ′ ρ − = τ

i j j i T ij k k T ij

x U x U 3 u u

T T

ρν = µ

Eddy Eddy Viscosity Viscosity y U

T T T 12

∂ ∂ ρν = τ = τ

• G. Ahmadi

ME 637-Particle-II

l l

U

y

dy dU ~ ) v ( ~ ) u (

2 1 2 2 1 2

l ′ ′

v u

T

′ ′ ρ − = τ

dy U y U

2 T

∂ ∂ ∂ ρ = τ l y U

2 T

∂ ∂ = υ l

Eddy Eddy Viscosity Viscosity

Prandtl Prandtl Assumption Assumption

Mixing Mixing Length Length

2 2

dy U d dy dU κ = l

von von Karman Karman

Ludwig Ludwig Prandtl Prandtl

• G. Ahmadi

ME 637-Particle-II

Inertial Inertial Sublayer Sublayer

y

Inertial Inertial Sublayer Sublayer

• τ

ρ τ =

*

u

Turbulence Turbulence Scales Scales

y κ = l

Shear Velocity Shear Velocity

4 . = κ

von von Karman Karman constant constant

von von Karman Karman

• G. Ahmadi

ME 637-Particle-II

Turbulent Stress=Wall Shear Stress Turbulent Stress=Wall Shear Stress

2 2 2

y U y ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ρκ = τ

y u dy dU

*

κ =

c y ln 1 U u U

*

+ κ = =

+

B y ln 1 U + κ =

+ +

ν =

+

y u y

*

5 B ≈

300 y 30 ≤ <

+

Wall Wall Units Units

SLIDE 4

4

• G. Ahmadi

ME 637-Particle-II

dy dU µ = τ

dy dU u

2 *

ν =

1 dy dU =

+ +

+ + = y

U

5 y ≤ <

+

Turbulent stress is negligible Turbulent stress is negligible

• G. Ahmadi

ME 637-Particle-II

+

y

+

U

5 . 5 y ln 5 . 2 U + =

+ + + + = y

U

30 12 300 10 20 30

• G. Ahmadi

ME 637-Particle-II

• G. Ahmadi

ME 637-Particle-II

• Reynolds Equation

Reynolds Equation

• Eddy Viscosity Models

Eddy Viscosity Models

• Mixing Length Model

Mixing Length Model

• Near Wall Flows

Near Wall Flows