Outline Viscous Flow Turbulence Mixing Length Models One-Equation - - PowerPoint PPT Presentation

▶
outline
SMART_READER_LITE
LIVE PREVIEW

Outline Viscous Flow Turbulence Mixing Length Models One-Equation - - PowerPoint PPT Presentation

Feb 08, 2024 441 likes •570 views

ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi Outline Viscous Flow Turbulence Mixing Length Models One-Equation Models Two-Equation Models Stress Transport Models Rate-Dependent Models ME 639-Turbulence G.

slide-1
SLIDE 1

1

  • G. Ahmadi

ME 639-Turbulence

  • G. Ahmadi

ME 639-Turbulence

  • G. Ahmadi

ME 639-Turbulence

  • G. Ahmadi

ME 639-Turbulence

Outline

  • Viscous Flow
  • Turbulence
  • Mixing Length Models
  • One-Equation Models
  • Two-Equation Models
  • Stress Transport Models
  • Rate-Dependent Models
slide-2
SLIDE 2

2

  • G. Ahmadi

ME 639-Turbulence

) ( t         u ) ( t         u

τ f u       dt d τ f u       dt d

τ τ 

T

τ τ 

T

h e        q u : τ  h e        q u : τ 

T h ) T (         q  T h ) T (         q 

Mass Momentum Energy Entropy

  • G. Ahmadi

ME 639-Turbulence

Viscous Fluids Material Frame- Indifference

) u ( G p

j , i kl kl kl

     ) u ( G p

j , i kl kl kl

    

ij ij j , i

d u   

ij ij j , i

d u   

) u u ( 2 1 d

k , l l , k kl

  ) u u ( 2 1 d

k , l l , k kl

 

) u u ( 2 1

k , l l , k kl

   ) u u ( 2 1

k , l l , k kl

  

) d ( F p

ij kl kl kl

     ) d ( F p

ij kl kl kl

    

  • G. Ahmadi

ME 639-Turbulence

Navier- Stokes

kl kl i , i kl

d 2 ) u p (        

kl kl i , i kl

d 2 ) u p (        

2 3     2 3    

   

j j i 2 i j i j i

x x u x p ) x u u t u (               

j j i 2 i j i j i

x x u x p ) x u u t u (               

x u

i i 

  x u

i i 

 

  • G. Ahmadi

ME 639-Turbulence

Reynolds Equation

i i i

u U u   

i i i

u U u   

p P p    p P p   

i i

U u 

i i

U u  ui   ui  

P p  P p  p   p  

j j i j j i 2 . i j i j i

x u u x x U x P 1 x U U t U                   

j j i j j i 2 . i j i j i

x u u x x U x P 1 x U U t U                   

j i T ij

u u      

j i T ij

u u      

Turbulent Stress

slide-3
SLIDE 3

3

  • G. Ahmadi

ME 639-Turbulence

Eddy Viscosity Mixing Length

dy dU v u

T T 21

        dy dU v u

T T 21

       

ij k k i j j i T j i T ij

u u 3 1 ) x U x U ( u u                 

ij k k i j j i T j i T ij

u u 3 1 ) x U x U ( u u                 

y U | y U | l2

m T 21

       y U | y U | l2

m T 21

      

| y U | l2

m T

    | y U | l2

m T

   

y T v T

T T

        y T v T

T T

       

  • G. Ahmadi

ME 639-Turbulence

Eddy Viscosity Free Shear Flows

 u c

T 

  u c

T 

Scale Length   Scale Length   Scale Velocity u  Scale Velocity u 

   c   c

Kinematic Viscosity

Path Free Mean   Path Free Mean   Sound

  • f

Speed c  Sound

  • f

Speed c 

m

c ~  

m

c ~  

y

m

   y

m

  

Near Wall Flows

  • G. Ahmadi

ME 639-Turbulence

Local Equilibrium Production = Dissipation Mixing length Hypothesis

Short Comings of Mixing Length

  • Eddy viscosity vanishes when velocity

gradient is zero

  • Lack of transport of turbulence scales
  • Estimating the mixing length
  • G. Ahmadi

ME 639-Turbulence

Reattachment Point

Mixing Length

T 

T 

Experiment Maximum Heat Transfer

slide-4
SLIDE 4

4

  • G. Ahmadi

ME 639-Turbulence

Exact k-equation Eddy Viscosity

2 / 1 T

k c   

2 / 1 T

k c  

                          

Diffusion Viscous i i j j 2 n Dissipatio j i j i

  • duction

Pr j i j i Diffusion Turbulence i i k k Transport Convective i i

2 u u x x x u x u x U u u ) P 2 u u ( u x 2 u u dt d                                                          

Diffusion Viscous i i j j 2 n Dissipatio j i j i

  • duction

Pr j i j i Diffusion Turbulence i i k k Transport Convective i i

2 u u x x x u x u x U u u ) P 2 u u ( u x 2 u u dt d                               

  • G. Ahmadi

ME 639-Turbulence

Modeled k-equation

                 

n Dissipatio 2 / 3 D

  • duction

Pr j i i j j i T Diffusion j k T j

k c x U ) x U x U ( ) x k ( x dt dk                                   

n Dissipatio 2 / 3 D

  • duction

Pr j i i j j i T Diffusion j k T j

k c x U ) x U x U ( ) x k ( x dt dk                 

  • G. Ahmadi

ME 639-Turbulence

K-equation

             

n Dissipatio 2 / 3 D

  • duction

Pr Diffusion Max

k c y U ak ) Bk ( y dt dk                       

n Dissipatio 2 / 3 D

  • duction

Pr Diffusion Max

k c y U ak ) Bk ( y dt dk         

Short Comings of One-Equation Models

  • Lack of transport of turbulence length scale
  • Estimating the length scale
  • G. Ahmadi

ME 639-Turbulence

Source Secondary z n Dissipatio T 2

  • duction

Pr 2 T 1 Diffusion z T

S ] k c ) y U ( k c [ z ) y z ( y dt dz                          

Source Secondary z n Dissipatio T 2

  • duction

Pr 2 T 1 Diffusion z T

S ] k c ) y U ( k c [ z ) y z ( y dt dz                         

slide-5
SLIDE 5

5

  • G. Ahmadi

ME 639-Turbulence

Scale Time k /

2

  Scale Time k /

2

  Scale Frequency / k

2 

 Scale Frequency / k

2 

Scale Vorticity / k

2 

 Scale Vorticity / k

2 

Rate n Dissipatio x u x u

j i j i

          Rate n Dissipatio x u x u

j i j i

         

 k z   k z 

  • G. Ahmadi

ME 639-Turbulence

                    

n distructio Viscous l k i 2 l k i 2 stretching vortex by Generation l k l i k i Diffusion j j

x x u x x u 2 x u x u x u 2 ) ' u ( x dt d                                                 

n distructio Viscous l k i 2 l k i 2 stretching vortex by Generation l k l i k i Diffusion j j

x x u x x u 2 x u x u x u 2 ) ' u ( x dt d                            

  • G. Ahmadi

ME 639-Turbulence

k E(k)

ε

ε Universal Equilibrium Inertia Subrang e

Kolmogorov

  • G. Ahmadi

ME 639-Turbulence

  

 2 T

k c   

 2 T

k c

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

k 3 2 ) x U x U ( u u            

Momentum Mass

j i j i i

u u x x P 1 dt dU          

j i j i i

u u x x P 1 dt dU          

x U

i i 

  x U

i i 

 

slide-6
SLIDE 6

6

  • G. Ahmadi

ME 639-Turbulence

k-equation -equation

                  

n dissipatio

  • duction

Pr j i i j j i T Diffusion j k T j

x U ) x U x U ( ) x k ( x dt dk                                     

n dissipatio

  • duction

Pr j i i j j i T Diffusion j k T j

x U ) x U x U ( ) x k ( x dt dk                  

                  

n Distructio 2 2 Generation j i i j j i T 1 Diffusion j T j

k c x U ) x U x U ( k c ) x ( x dt d                     

  

                  

n Distructio 2 2 Generation j i i j j i T 1 Diffusion j T j

k c x U ) x U x U ( k c ) x ( x dt d                     

  

  • G. Ahmadi

ME 639-Turbulence

  • G. Ahmadi

ME 639-Turbulence

k- Model

  • G. Ahmadi

ME 639-Turbulence

k- Model

slide-7
SLIDE 7

7

  • G. Ahmadi

ME 639-Turbulence

k- Model

  • G. Ahmadi

ME 639-Turbulence

Algebraic Stress Model

  • G. Ahmadi

ME 639-Turbulence

Algebraic Stress Model

  • G. Ahmadi

ME 639-Turbulence

slide-8
SLIDE 8

8

  • G. Ahmadi

ME 639-Turbulence

  • Eddy viscosity assumption
  • Isotropic eddy viscosity
  • Negligible convection and diffusion
  • f turbulent shear stress
  • Absence of normal stress effects

k ~ u u

j i 

  • G. Ahmadi

ME 639-Turbulence

k i k k i k k i k k k i 2 i k i k i

x U u ) u u ( x u u x x x u x p 1 x u U t u                                 

k i k k i k k i k k k i 2 i k i k i

x U u ) u u ( x u u x x x u x p 1 x u U t u                                 

Fluctuation Velocity

  • G. Ahmadi

ME 639-Turbulence

                                                

Diffusion j i k ik j jk i k j i k strain essure Pr i j j i n Dissipatio k j k i

  • duction

Pr k i k j k j k i Convection j i k k

] u u x ) u u ( ' p u u u [ x ) x u x u ( ' p x u x u 2 ] x U u u x U u u [ u u ) x U t (                                                      

                                                

Diffusion j i k ik j jk i k j i k strain essure Pr i j j i n Dissipatio k j k i

  • duction

Pr k i k j k j k i Convection j i k k

] u u x ) u u ( ' p u u u [ x ) x u x u ( ' p x u x u 2 ] x U u u x U u u [ u u ) x U t (                                                      

  • G. Ahmadi

ME 639-Turbulence

Diffusion ) x u u u u x u u u u x u u u u ( k c u u u

l j i l k l i k l j l k i l i s k j i

                          ) x u u u u x u u u u x u u u u ( k c u u u

l j i l k l i k l j l k i l i s k j i

                          Dissipation

         

ij k j k i

3 2 x u x u 2          

ij k j k i

3 2 x u x u 2

slide-9
SLIDE 9

9

  • G. Ahmadi

ME 639-Turbulence

 

                           

Term Rapid i j j i 1 l m 1 m l Isotropy to turn Re i j j i 1 l m m l 1 1 i j j i

) 2 ( ji ) 2 ( ij 1 ) 1 ( ij

) x u x u ( ) x u ( ) x U ( 2 ) x u x u ( ) x u x u ( ) ( G d ) x u x u ( ' p

     

                                   

x

x x, x

 

                           

Term Rapid i j j i 1 l m 1 m l Isotropy to turn Re i j j i 1 l m m l 1 1 i j j i

) 2 ( ji ) 2 ( ij 1 ) 1 ( ij

) x u x u ( ) x u ( ) x U ( 2 ) x u x u ( ) x u x u ( ) ( G d ) x u x u ( ' p

     

                                   

x

x x, x

Pressure-Strain

  • G. Ahmadi

ME 639-Turbulence

) k 3 2 u u )( k ( c

ij j i 1 ) 1 ( ij

        ) k 3 2 u u )( k ( c

ij j i 1 ) 1 ( ij

       

) P 3 2 P (

ij ij ) 2 ( ji ) 2 ( ij

        ) P 3 2 P (

ij ij ) 2 ( ji ) 2 ( ij

       

) x U u u x U u u ( P

k i k i k j k i ij

           ) x U u u x U u u ( P

k i k i k j k i ij

          

Return to Isotropy Rapid Term

Production

  • G. Ahmadi

ME 639-Turbulence

                                                         

Diffusion l j i l k l i k l j l k j l i k s effects Wall ) w ( ji ) w ( ij strain essure Pr ) 2 ( ji ) 2 ( ij ij j i 1 n Dissipatio ij

  • duction

Pr k j k i k i k j Convection j i k k

]} x u u u u x u u u u x u u u u [ k { x c ) ( ) ( ) k 3 2 u u ( k c 3 2 ] x U u u x U u u [ u u ) x U t (                                                           

                                                         

Diffusion l j i l k l i k l j l k j l i k s effects Wall ) w ( ji ) w ( ij strain essure Pr ) 2 ( ji ) 2 ( ij ij j i 1 n Dissipatio ij

  • duction

Pr k j k i k i k j Convection j i k k

]} x u u u u x u u u u x u u u u [ k { x c ) ( ) ( ) k 3 2 u u ( k c 3 2 ] x U u u x U u u [ u u ) x U t (                                                           

  • G. Ahmadi

ME 639-Turbulence

                       

n Destructio 2 2 Generation k i k i 1 Diffusion i i k k Convection k k

k c x U u u k c ) x u u k ( x c ) x U t (                       

  

                       

n Destructio 2 2 Generation k i k i 1 Diffusion i i k k Convection k k

k c x U u u k c ) x u u k ( x c ) x U t (                       

  

Dissipation

slide-10
SLIDE 10

10

  • G. Ahmadi

ME 639-Turbulence j i j i i j j

u u x x P 1 U ) x U t (               

j i j i i j j

u u x x P 1 U ) x U t (                  

i i

x U   

i i

x U

   , P , u u , U

j i i

   , P , u u , U

j i i

Mass Reynolds 11 Unknowns for 11 Equations

  • G. Ahmadi

ME 639-Turbulence

Gibson and Rodi (1981)

  • G. Ahmadi

ME 639-Turbulence

Mean Velocity and Turbulence Shear Stress

  • G. Ahmadi

ME 639-Turbulence

Turbulence Intensity

slide-11
SLIDE 11

11

  • G. Ahmadi

ME 639-Turbulence

Stress Transport Model k-Equation                                      

n Dissipatio ij Strain essure Pr ij ij ij j i 1

  • duction

Pr k i k j k j k i Diffusion m m k k s j i

3 2 ) P 3 2 P ( ) k 3 2 u u ( k c x U u u x U u u ) j u i u x u u k ( x c u u dt d                                   

                                     

n Dissipatio ij Strain essure Pr ij ij ij j i 1

  • duction

Pr k i k j k j k i Diffusion m m k k s j i

3 2 ) P 3 2 P ( ) k 3 2 u u ( k c x U u u x U u u ) j u i u x u u k ( x c u u dt d                                   

n Dissipatio

  • duction

Pr m k m k Diffusion m m k k s

x U u u ) x k u u k ( x c dt dk                             

n Dissipatio

  • duction

Pr m k m k Diffusion m m k k s

x U u u ) x k u u k ( x c dt dk                             

  • G. Ahmadi

ME 639-Turbulence

) P ( k u u ) D dt dk ( k u u D u u dt d

j i j i ij j i

            ) P ( k u u ) D dt dk ( k u u D u u dt d

j i j i ij j i

           

Rodi’s Assumption

) u u x u u k ( x D

j i l l k k ij

          ) u u x u u k ( x D

j i l l k k ij

         

k i k j k j k i ij

x U u u x U u u P          

k i k j k j k i ij

x U u u x U u u P           ) x k u u k ( x D

l l k k

        ) x k u u k ( x D

l l k k

       

l k l k

x U u u P     

l k l k

x U u u P     

  • G. Ahmadi

ME 639-Turbulence

                          ) 1 P ( c 1 1 P 3 2 P c 1 3 2 k u u

1 ij ij 1 ij j i

                          ) 1 P ( c 1 1 P 3 2 P c 1 3 2 k u u

1 ij ij 1 ij j i

  

 2 T

k c   

 2 T

k c

2 1 1 1

)] 1 P ( c 1 1 [ )] P 1 ( c 1 1 [ c ) 1 ( 3 2 c          

 2 1 1 1

)] 1 P ( c 1 1 [ )] P 1 ( c 1 1 [ c ) 1 ( 3 2 c          

Simple Shear Flow

  • G. Ahmadi

ME 639-Turbulence

  • Available models can predict the mean

flow properties with reasonable accuracy.

  • First-order modeling is reasonable when

turbulence has a single length and velocity scale.

  • The k- model gives reasonable results

when a scalar eddy viscosity is sufficient.

  • The stress transport models have the

potential to be most accurate.

slide-12
SLIDE 12

12

  • G. Ahmadi

ME 639-Turbulence

  • Adjustments of coefficients are needed.
  • The derivation of the models are arbitrary.
  • There is no systematic method for

improving a model when it loses its accuracy.

  • Models for complicated turbulent flows are

not available.

  • Realizability and other fundamental

principles are sometimes violated.

  • G. Ahmadi

ME 639-Turbulence