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Computational FluidDynamics – Lecture 7 

Prof. Jiyuan Tu

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 Why we need turbulenceprediction? 

Determining:

※Frictional drag

※Flow Separation

※Transition from laminar to turbulent flow

※Thickness of boundary layers

※Flow miing rate in reaction ! combustion

※"tent of secondary flows

※Spreading of #ets and wakes$%%

Almost every flow problem in industry is turbulent!

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 Velocity Fluctuation

u

u

t 

( )t ' u

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 What is turbulence?（ I）

u

u

t 

( )t ' u

 Velocity fluctuating in a turbulent flow

0

0

1( , , , )

t T 

t u u x y z t dt  

T 

+

= ∫ 

'( )u u u t  = +

'u u u= −

0 0 0

0 0 0

'2   1 1( ) ( )

1( ) 0

t T t T t T  

t t t u u u dt udt u dt  

T T 

Tu uT  T 

+ + += − = −

= − =

∫ ∫ ∫  (ormal shear stress

0

0

'2 ' 21 ( ) 0t T 

t u u dt  

T 

+= >∫ 

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 What is turbulence?（ II）

At large angles of

attack ,flow may

separate completely

from the top surface of

an airfoil , reducing

lift drastically and

causing the airfoil to

stall

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 What is turbulence?（ III）

ow and high !eynolds number "orte# shedding

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 What is turbulence?（ V ）

 ,rithmetic average of velocity fluctuations and the Root Mean

S-uare of the fluctuations

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 What is turbulence?（ VI）

/a01aminar flow shear stress caused by random motion of molecules

/b0Turbulent flow as series of random three3dimensional eddies

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Drag &Lift Forces （ I）

Flat plate

  Drag Friction Force=

,  0

 D pressureC    =

, D D friction f  C C C = =

2 21 1

2 2

 D f D f   F F C Av C Av= = =

Cylinder 

, , D D pressure D frictionC C C = +

0 LC    =

P1>P2

Symmetry

P1 P2

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Drag &Lift Forces （ II）

Frictional velocity 

 LC  f  C 

 f  C 

  f  C 

  f  C 

 LC  Airfoil

,   D pressureC    ⇓ ,   L pressureC    ⇑

$ % f w wC uτ τ ρ ⇒ =

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Laminar V!urbulence

 aminar and turbulent "elocity profiles

 in the fully de"eloped region

 aminar 

&urbulence

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 Assumptions and"omple#ity 

' '

ij i j R u u ρ = −  ' '1, 2,i j u v= = →

Diret numerial simulation of governing euations is only

possible for simple low"#e flows

$nstead% we solve #eynolds Averaged &avier"Sto'es (#A&S)

euations:

2iji i

k 

k i j j j

 RU U  pU 

 x x x x x ρ µ 

∂∂ ∂∂= − + +

∂ ∂ ∂ ∂ ∂

Steady

$nompressible flow

*it+out body fores

Reynolds stresses

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!urbulent hear tress

 y

uvu t turb

∂

∂=′′−=   µ  ρ τ 

' '

w turb   u vτ τ ρ = = −#eynolds Stress

Assumption

Turbulence viscosity

,odelk    ε −

-+e logarit+mi law : $

$ 2ln 0

u yu

u v= +

0 00 y+<

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$oussines% ypothesis .oussines observed

Turbulent eddies transport momentum similar to molecule

Recall, viscous stresses proportional to mean velocitygradients

By analogy, turbulent stress also proportional to meanvelocity gradients

.oussines proposed eddy visosity onept

  is referred to as t+e eddy visosity or turbulent

visosityt u

' '   2

( )

 ji

ij i j t ij

 j i

uu

 R u u  x x ρ µ δ 

∂∂= − = + −∂ ∂

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k-  !wo'(%uation !urbulence

)odel（ I）

-urbulent /ineti 0nergy '2 '2 '21 ( )2

  u v w= + +

-urbulene Dissipation #ate' '

( )( )i it  j j

u u

 x xε υ    ∂ ∂=

∂ ∂

"rrors

Models

9all treatment

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k-  !wo'(%uation !urbulence

)odel（ II）

 D ! "      −=

0*+0= µ C 

k 

k 

T 

k 

T  "  y

k 

 y x

k 

 x y

k 

 x

k u

t 

k +  

 

  

 ∂∂

∂∂

+   

  

 ∂∂

∂∂

=∂∂

+∂∂

+∂∂

σ 

ν 

σ 

ν v

ε 

ε ε 

ε 

σ 

ν ε 

σ 

ν ε ε ε " 

 y y x x y x

u

t 

T T  +  

 

 

 

 

∂

∂

∂

∂+  

 

 

 

 

∂

∂

∂

∂=

∂

∂+

∂

∂+

∂

∂v

222

2   

 

 

 

 

∂

∂+

∂

∂+

  

 

 

 

 

∂

∂+ 

 

 

 

 

∂

∂=

 x y

u

 y x

u !  T T 

vv

ν ν The production terms

10k σ    = 1ε σ    = 1 1++C ε    =   2   12C ε    =

 D   ε =The destruction terms

ε  µ 

2k C vt  =

 D ! "      −=

)( 21   DC  ! C k 

"  ε ε ε ε 

−=

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k'  !wo'(%uation !urbulence

)odel（ III）   model popular  because it is

!obust

-fficient

.imple to use !easonably successful

-#perience has shown that model is inade/uate for

flows with

.trong cur"ature .trong buoyancy effects

.trong swirl

k    ε −

k    ε −

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!he *+, k-  )odel（ I）

"tend the turbulence model

Renormali:ation ;roup theory basis /R(;0

 ,ble to replace wall function with a fine grid

Turbulence kinetic energy consist of a<

7onvection generation diffusion and dissipation term

The transport e-uation for dissipation consist of a<

7onvection generation diffusion destruction and

additional term related to mean strain and turbulence

-uantities% It has similar structure to standard model

k    ε −

k    ε −

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!he *+, k-  )odel（ II）

( )   R DC  ! C k 

"    −−=   21   ε ε ε ε 

( )

k 

#

 $%

%%%C  R

  o &2

)

)

1

1

+

−

=

01*k σ    = 01*ε σ    = 1 1+2C ε    =   2 1*C ε    =

Some of the coefficients can vary with the solution

0*+0= µ C 

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!he k-ω )odel（ I）

Formulate based on turbulent fre-uencyω

1ower re-uirement for near wall resolution for low3

Reynolds number flows<

Same ;overning e-uation for turbulent energy k 

2

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!he k-ω )odel（ II）

00=′β 

;overning e-uation for turbulent fre-uencyω

The model assumes that<

Model constants

( )    

  

 ∇+⋅∇=⋅∇+

∂∂

ω σ 

 µ  µ ω 

ω 

ω 

)(

t  u ()

t 

 (  

2

2

1

1)1(2   βω 

ω α 

ω 

ω σ  ρα  ω  ( ! k  x x

k  F  k 

 j j−+∂

∂∂∂

−−

ω  ρ  µ 

  k 

t  =

=α  00=β    02=k σ    02=ω σ 

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 !he *eali-able k- )odel（ I） =roposed by (,S, 1ewis modeling group /Shih et

al%544'0 (ote that standard model gives

Reali:able model ensures ?reali:ability@ i%e% =ositivity of normal stresses

k    ε −

'2 0( 1, 2, )uα    α ≥ =' '

'2 '2

1 1( 1,2,3 1,2,)u u

u u

α β 

α β 

α β − ≤ ≤ = =

ε  µ 

2

k C vt  =

ε 

 µ k U 

 A A

C 

 s

$

0

1

+=

02

2   22

2

≤′⇒∂∂

−=′   u xU k 

C k u ε  µ 

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 !he *eali-able k-  )odel（ II）

( )   

 

 

 

 

∂

∂+

∂∂

=+

−=i

 j

 j

iij

T 

 * 

ij x

u

 x

u" 

#+k 

# (C #"  (C " 

2

1,2

2

2

212

1ε 

+= ma#1 %%

C    ( )

  212

2 ij" 

k 

% ε =

 

( )   2

1   %,%,cos

1,cos,0++ ij

ki jk ij

 so   " " " 

" " " , ,  A A   =====   −ϕ ϕ 

  10k σ    = 12ε σ    =2   1C    =

-+e Soures -erms

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!he Wall Function（ I）"perimental 8ata

1ogarithmic

Region

The turbulent boundary layer: respective dimensionlessvelocity profile as a function of the wall distance in

comparison to experimental data

)ln(1   ++ =   -yU κ 

++=  yU 

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!he Wall Function（ II）

Viscous Sublayer :   $

$

 yuu

u v=

2

$   w w y y yuuv v

τ τ 

 ρ ρµ = = =   4w w

u u u

 y y yτ µ µ µ  

∆ ∂= = =

∆ ∂

Normalized variables:

$

u

u u

+

=  $

 yu

 y v

+

=

Normalized law of the wall   u y+ +=

y

5

u

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!he Wall Function（ III）

1ln( ) p

t 

U u -y

u   κ  

+ += =

2

u

u

C 

τ  

κ    =

u

 y

τ  

ε  

κ  

=

,

,,

( )( )

, p   T ' 

T t , T t 

T T C uT u ! 

.

τ   ρ    σ  

σ  σ  

+ + −

= − = +

( )( ) 6( ) 7 2t  t ij ij

k 

div kU div gradk - -  t 

 µ  ρκ   ρ µ µ ρε 

σ 

∂+ = + + × −

∂

2

1 1 2 2

( )( ) 6( ) 7 2t  t ij ijdiv U div grad C f - - C f    

t   ε ε 

ε 

 µ  ρε ε ε  ρε µ ε µ ρ 

σ κ κ  

∂+ = + + × −

∂

High Reynolds Number

Low Reynolds Number

9all38amping functionε 

 ρ  µ   µ  µ 

2k  f  C t  =

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!he Wall Function（ V ）

' 0v   =' 0u   =   0k  →   0ε  →

2

$ y

u

u  

C =

$

0+1 y

u

 yε    =

This is the standard wall function at high

Reynolds number 

$

2ln 0 yu  y

u

+= + ,t low Reynolds number 

 y +$u   wτ    f  C 

Boundary Condition at Solid Walls:

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"omparison of!urbulence（ I）

Turbulence model performance: pressure surface boundary layernormalied mean velocity profile at !" # $ %Uref  & " m's(

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"omparison of!urbulence（ II）

 Turbulence model performance: pressure surface boundarylayer normalied mean velocity profile at !" # $

%Uref  & ) m's(

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 "omparison of!urbulence（ III）

36%&

36%2)

36%2

36%5)

36%5

36%6)

6

35 6 5 2 & ' ) * + . 4 56

8istance from step /AB0

   (  o  r   m  a l i  :  e  d   m  a   i   m  u   m  n  e  g  a t i  v  e  v  e l  o  c i t  y

"perimental data R(; k3ε Reali:able k3ε Standard k3ε

*aximum measured and predicted negative velocity profiles ofthe flow in the recirculation one

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.sing !urbulence )odels（ I） 8alculate !eynolds number and if necessary the swirl number

of the flow

.elect appropriate turbulence model 9or e#ample, use the   model for simple flows with no significant strain rates

(ie pipe flows, channel flows)

!:; model for separated flows, flows with large stream

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.sing !urbulence )odel（ II） Successful turbulence modeling re-uires

engineering #udgment of< Flow physics

7omputer resources available =ro#ect re-uirements

 ,ccuracy

Turn around time

Turbulence models ! near3wall treatments thatavailable

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.sing !urbulence )odel（ III） Modeling =rocedure

7alculate characteristic and determine if

Turbulence needs modeling

"stimate wall3ad#acent cell centroid first beforegenerating mesh

Cegin with SD" /standard 0and change to

R(; RD"SDE or SST if needed

Use RSM for highly swirling flows Use wall functions unless low3Re flow andAor

comple near3wall physics are present

k    ε −

 Re

 y

+