0 0 . . 09 09 T ~ T ~ Viscosity u u Plane - - PowerPoint PPT Presentation

SLIDE 1

1

• G. Ahmadi

ME 639-Turbulence

• G. Ahmadi

ME 639-Turbulence

Outline

• Introduction to Turbulence Modeling
• Eddy Viscosity Models
• Prandtl Mixing Length Model
• One Equation Model
• Two-Equation Model
• Stress Transport Model
• G. Ahmadi

ME 639-Turbulence

Boussinesq Eddy Viscosity

k 3 2 x U x U u u

ij i j j i T j i

                    k 3 2 x U x U u u

ij i j j i T j i

                   

m mu

u 2 1 k   

m mu

u 2 1 k   

y U v u

T 

      y U v u

T 

     

m T ~

 u 

m T ~

 u 

m

 m 

Thin Shear Layer

Eddy Viscosity Mixing Length

• G. Ahmadi

ME 639-Turbulence

Eddy Viscosity Mixing Length Plane Jet Circular Jet Mixing Layer

y U ~ u

m 

  y U ~ u

m 

  y U

2 m T

     y U

2 m T

    

y U y U v u

2 m

         y U y U v u

2 m

           09 .

m 

  09 .

m 

  075 .

m 

  075 .

m 

  07 .

m 

  07 .

m 

SLIDE 2

2

• G. Ahmadi

ME 639-Turbulence

Mixing Length Boundary Layer

                     

m

y y y                       

m

y y y  41 .   41 .  

09 .

0 

 09 .

0 



1 1

  /

• 

/

m

  / y

• U

/ U

• G. Ahmadi

ME 639-Turbulence

Nikuradse Pipe Flows

4 2 m

r y 1 06 . r y 1 08 . 14 . r                      

4 2 m

r y 1 06 . r y 1 08 . 14 . r                      

 m =0.4 y Nikuradse

• r

/ y

m/ro

1

• G. Ahmadi

ME 639-Turbulence

Mixing Length

T 



T 



y U    y U   

When max T T

| 8 . 0   

max T T

| 8 . 0   

Experiment

T 



T 



U

T



T T

   

Hot

Cold

• G. Ahmadi

ME 639-Turbulence

Reattachment Point

Mixing Length

T 



T 



Experiment Maximum Heat Transfer

SLIDE 3

3

• G. Ahmadi

ME 639-Turbulence

Kolmogorov- Prandtl Turbulence Kinetic Energy



2 1 T

k   

2 1 T

k  

i iu

u 2 1 k   

i iu

u 2 1 k   

• G. Ahmadi

ME 639-Turbulence

Turbulence Kinetic Energy

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

x x k x u x u x U u u ) P 2 u u ( u x k dt d                                  

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

x x k x u x u x U u u ) P 2 u u ( u x k dt d                                  

• G. Ahmadi

ME 639-Turbulence

Thin Shear Layer

                       y U v u ) P 2 u u ( v y dt dk

i i

                       y U v u ) P 2 u u ( v y dt dk

i i

y U v u

T 

      y U v u

T 

     

y k P u u 2 1 v

k T i i

                    y k P u u 2 1 v

k T i i

                   

Turbulence Stress Diffusion

• G. Ahmadi

ME 639-Turbulence

Dissipation



2 3 D

k c   

2 3 D

k c  



2 3 D 2 T k T

k c y U y k y dt dk                             

2 3 D 2 T k T

k c y U y k y dt dk                             Closed k-Equation

SLIDE 4

4

• G. Ahmadi

ME 639-Turbulence

Near a Wall



2 3 D 2 T

k c y U             

2 3 D 2 T

k c y U            



2 3 D T 2

k c ) v u (     

2 3 D T 2

k c ) v u (    

2 2 D

k ) v u ( c   

2 2 D

k ) v u ( c   



2 1 T

k   

2 1 T

k  

3 . ~ 25 . k v u     3 . ~ 25 . k v u    

08 . ~ 07 . cD  08 . ~ 07 . cD 

1

k 

 1

k 



• G. Ahmadi

ME 639-Turbulence

Distribution of Length Scale

k c v u

2 1 D

   k c v u

2 1 D

  

y U c k

2 1 D 2 1

  

 

y U c k

2 1 D 2 1

  

  2 2 2 1 D

y U c v u              

  2 2 2 1 D

y U c v u              

 

y u y U

*

    y u y U

*

   

2 *

u v u    

2 *

u v u    

m 4 1 D 4 1 D

c y c     

m 4 1 D 4 1 D

c y c      08 . cD  08 . cD 

4 .   4 .  

y 2 .   y 2 .  

• G. Ahmadi

ME 639-Turbulence

Achievements of the Model

• Heat transfer in the heat exchanger
• Separated flows

Short Comings

• Transport of the turbulent length scale
• Little advantage over the mixing length
• G. Ahmadi

ME 639-Turbulence