Download - Miet 2394 Cfd Lecture 7(1)
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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
+