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I CFD
E: .@..
&
, 15 2011
C F D
CFD ,
, ,
.
+
CFD :
S A ;
N ;
A
H
T CFD . M
H CFD .
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F
CFD :
S : A , O ; H ; D ; F , , ; P, , ; C ; S; C; H ; C
C
C
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R
LAGRANGIAN APPROACH:
P (,,)
SUBSTANTIAL DERIVATIVE ( )
I ()
R
EULERIAN APPROACH:
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M
SURFACE FORCES:
BODY FORCES: , , ,
(
)
M
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E
Conservation equation for internal energy
Conservation equation for mechanical energy
E
F
=
F
= , =
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E
5 EQUATIONS
C (1)
M (3) E (1)
11 UNKNOWNS
2 TD (, , , T
= (, T) = (, T)
= RT = T
V (3)
V (6)
Liquids and gases flowing at low speeds behave as incompressiblefluids: without density variation there is no linkage between the energy
equation and the mass and momentum conservation. The flow fieldcan be considered mass and momentum conservation only
V
I N
F ()
S
F
F
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NS
NS
sMx
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NS
M
E
NS
C
MATHEMATHICALLY CLOSED PROBLEM!
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G
A :
=1,,,, (T,)
D
TRANSPORT EQUATIONFOR PROPERTY DIFFERENTIAL FORM
dV
An
V
S
G
I 3D (CV)
G
INTEGRAL FORM
Rate of change of thetotal amount of fluid
property in the controlvolume
Net rate ofdecrease of fluidproperty of the
fluid element due toconvection
Net rate ofincrease of fluidproperty of the
fluid elementdue to diffusion
Net rate of increaseof fluid property as
result of sources
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G
I
F
S M
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C
:
F () .
DISCRETIZATION METHOD
, .
T /
.
C :
N :
T
(.. , , ) . T
. E , )
A ; D
I , .
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C
CONVERGENCECRITERIA
SOLUTION METHODLinearization and iterative
techniques.
FINITE APPROXIMATIONApproximations
for the derivatives at thegrid points for FD
Approximating surface andvolume integrals for FV
NUMERICAL GRID
Structured,unstructured, etc
REFERENCE SYSTEMCartesian, cylindrical coord.etc.
Fixed or moving, etc
DISCRETIZATIONMETHOD
Differential eq. Algebraic eq.(FD, FE, FV)
MATHEMATICALMODEL
Set of partial differentialor integro-differential
equations and BCs
NUMERICAL SOLVER
T
T (, .).
S () T ( )
( 2D) ( 3D) , .. (, , ).
E 4 2D 6 3D
T ,
. S .
I
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T
B
T
U
FVM FEM
A , / 2D, /
3D.
.
N . T ,
.
S .
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T
H
T
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D
T CFD :
F V (80%).
F E (15%).
O (5%)F ()
S .
B .
V .
L / B.
P :
3 :M ,
;
D ,
;
I ( ), .
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D : F D
M
O (E 18 )
C .
S
A ,
T .
O , .
ADV:
FDM .
I
DIS:
C .
R
D : F VM
C
T (CV), CV. A CV
.
I CV (CV) .
S .
O CV, .
ADV: , .
DIS: 3D. T
FV : ,, .
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F V M
F V M
N
A
A
I
U
L
Q
H
I
A
GENERIC CONSERVATION EQUATIONIN INTEGRAL FORM
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N
T
(CV) .
nodes centered in CVsnodes centered in CVsnodes centered in CVsnodes centered in CVs CV faces centered between nodesCV faces centered between nodesCV faces centered between nodesCV faces centered between nodes
Nodal value mean over CV More accurate CDS approximation
A
=
=
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A
T , S. T , (CV
) . T :
1.
;
2. (CV
) ()
A
A 1:
T INTERPOLATION
( ).
T (CV )
INTERPOLATION ( ).
T , (CV )
INTERPOLATION ( ,
).
2 order accuracy
4 order accuracy
2 order accuracy
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I
A 2: T (CV ).
, , ,
CV .
INTERPOLATION
U I (UDS)
L I (CDS)
Q U I (QUICK)
HO S
O
T
B
,
CV
CV.
T
CV . W ,
A . T
CV;
,
,
.
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S L ES
S
D FDM FVM ,
G
LU
TDMA
G
LU
TDMA
I , , .
I , ( CFD )
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D : G
T A21 0
1 A21/ A11 2 (1 A21/ A11)
E
U
N = 3/3
B = 2
/2,
P
FORWARDFORWARDFORWARDFORWARD
ELIMINATIONELIMINATIONELIMINATIONELIMINATION
BACK SUBSTITUTIONBACK SUBSTITUTIONBACK SUBSTITUTIONBACK SUBSTITUTIONHIGH COSTHIGH COSTHIGH COSTHIGH COST
I
A .
T
I T
M (P )
Correction or updateCorrection or updateCorrection or updateCorrection or update(approximation of the iteration(approximation of the iteration(approximation of the iteration(approximation of the iteration
error)error)error)error)
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C C I E
I .
T
; U,
.
I
norm
S
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U
T
T
: ( ) ,
( )
T , (" )
M
(ODE)
M I V P ODE: T L M
TL M
ODE
.
1= + , 2 = + ,. . .
+ =
+ :
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M I V P
ODE: T L M
EXPLICIT OR FORWARDEXPLICIT OR FORWARDEXPLICIT OR FORWARDEXPLICIT OR FORWARDEULEREULEREULEREULER
IMPLICIT OF BACKWARDIMPLICIT OF BACKWARDIMPLICIT OF BACKWARDIMPLICIT OF BACKWARDEULEREULEREULEREULER
MIDPOINT RULEMIDPOINT RULEMIDPOINT RULEMIDPOINT RULE
TRAPEZOID RULETRAPEZOID RULETRAPEZOID RULETRAPEZOID RULE
implicit methods
M I V P ODE:PC M
.
.
,
.
T E :
*+1,
Second orderaccurate!
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S N S
NS
Steady 2D flowSteady 2D flowSteady 2D flowSteady 2D flow
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S
E (, , T, .)
E
I :
I
Storage locations for v velocities
Storage locations for u velocities
NON ZERO!NON ZERO!NON ZERO!NON ZERO!
BACKWARD STAGGERED GRID
D
T (, J) :
T : (1,J), (+1,J), (,J1),(,J+1)
J (, QUICK, .)
(F) (D) J
E. (1, J)
( )
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D
G ,
I
I
, ,
P : SIMPLE
S I M P L E (SIMPLE)
P S (1972)
M
A *
T
= *+
= *+ = *+
T .
M :
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SIMPLE
T :
NEGLECTED to simplify the equstions:MAIN APPORXIMATION OF SIMPLE!
EQUATION FOR PRESSURE CORRECTION
SIMPLE
EQUATION FOR PRESSURE CORRECTION
I,J
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SIMPLE
*, *, *, *
STEP 1
STEP 2
STEP 3
STEP 4
( )
u*, v*
p
, ,*+,
p, u, v, *
?
* =
* =
* =
* = ,
NO
YES
64
T
Large StructureSmall Structure
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65
T
I () U () (R
)
( ) ( ) ( ) iiiii uUtxutxutxu ',,, ' +=+= ( ) 0,' =txui
Point velocity measurement in a turbulent flow
66
D
T = /2
T
P D F: *
M
H
*)**(**)( dPROBdP +
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67
T
4
13
=
k
2
1
4
1
4
3
L
k
LL
k
L
k
ReReu
uRe
L===
3 :
K :
R (>
,
.
2
1
=
k
( )41
=ku
Scale di Kolmogorov
Length
Time
Velocity
.
68
E
= 2/.
T
T
( =2/).
T
E()
Energy containing rangeInertial subrangeDissipation range
Transfer of energy tosmaller scales
k lDI lEI L
Production PDissipation
=
=3
1
2'
2
1
i
iuk
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69
F :
L (+
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71
D N S (DNS)
,:
4/9
3
L
k
celle ReL
N =
2/1
L
k
Lt ReN =
411Re /tcelle NN timeCPU
Integral scale
Kolmogorov scales
72
D N S (DNS)
ADVANTAGES
H ;
N
P
DISADVANTAGES
O R
A 3D;
R BC
Particle-laden jet,Longmire and
Eaton(JFM, 1992)
Boundariesof a DNSdomain
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73
R N S E
(RANS)
RANSRANSRANSRANS
74
R N S E(RANS)
A 3 3 :
N
T
REYNOLDS STRESSESREYNOLDS STRESSESREYNOLDS STRESSESREYNOLDS STRESSES
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77
RANS:
: R
Navier- Stokes equation
RANS
Closure problem:Reynolds Stresses?
Problem:eddy viscosity?
Reynolds decomposition
EDDY VISCOSITY MODELS(Boussinesq hypothesis)
Mixing length modelk- modelk- model
DIRECT MODELS
Reynolds stressmodels
Anisotropic turbulence
Algebraic stressmodels
78
RANS:
K
S 2D
V
E
PRANDTLS MIXINGPRANDTLS MIXINGPRANDTLS MIXINGPRANDTLS MIXINGLENGTH MODELLENGTH MODELLENGTH MODELLENGTH MODEL
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79
RANS:
T
T ; R,
Velocity scaleVelocity scaleVelocity scaleVelocity scale Length scaleLength scaleLength scaleLength scale
80
RANS:
P P L S (1974)
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81
RANS:
ADVANTAGES
S
W
S
W
DISADVANTAGES
P
P
P
P R
Problems with RANS:-Low Re number flows
Inaccuracy of log law
-Rapidly changing flowsIn 2 eq. Models the Reynoldsstresses are proportional to thedeformation rate but thisholds only when productionand dissipation of k are inbalance
-Stress anisotropy-Adverce pressure gradientsand recirculation regions-Extra strains (curvature,rotation)
82
RANS: M SST
M (1992) : ,
.
H
S
T ( =
I
R =1, ,1=2, ,2=1.17, 2=0.44, 2=0.083, *=0.09
B
C1 C2 , (FC= 1 )
L
( ) ( )kk
ij
j
iijij
t
xx
k
x
UEEgraddivdiv
t
+
+
+=+
2,
2
22
1,
23
22U
Extra term!Extra term!Extra term!Extra term!
( ) 21 1 CFCFC CC +=
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83
RANS: R (RSM)
6 R
P P
D D
D
T R
T
1 .
Lots of constants!Lots of constants!Lots of constants!Lots of constants!
84
RANS: R (RSM)
BC
I: R
O/ :
F : R= 0 = 0
: R L R
ADVANTAGES
P
A R
W , ,
DISADVANTAGES
V (7 PDE)
N
P
C
0= nRij 0= n0= nRij
0= n
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85
L E S (LES)
Turbulent flow
Large energetic eddies are resolved
Small universal eddies are modelled
FILTERFILTERFILTERFILTER to separate small and large eddies
LES
RANS
DNS
86
LES:
S N S
( ) ( ) ( ) 321 ''',, dxdxdxtGt x',x'x,x +
+
+
=
Filtered functionFiltered functionFiltered functionFiltered function Unfiltered functionUnfiltered functionUnfiltered functionUnfiltered function
Filter cutoff widthFilter cutoff widthFilter cutoff widthFilter cutoff width
Spatial filteringSpatial filteringSpatial filteringSpatial filtering
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87
LES:
T
G
S
( )
>
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89
LES: SGS
( )( )
( ) jijijijiji
jijijijijijijjiijijiij
uuuuuuuuuu
uuuuuuuuuuuuuuuuuuuu
''''
''''''
+++=
=+++=++==
Leonard stressesEffects at theresolved scale
cross-stressesInteractions between
SGS ans resolvededdies
LES Reynoldsstresses
Convectivemomentum transferdue to interactoins
between SGS eddiesjijiij uuuuL = jijiij uuuuC '' +=
jiij uuR ''=
THE SGS STRESSES MUST BE MODELLED!THE SGS STRESSES MUST BE MODELLED!THE SGS STRESSES MUST BE MODELLED!THE SGS STRESSES MUST BE MODELLED!
90
LES: SL SGS
S (1963)
, B
T SGS R
P K (2000)
SGS
ijii
i
j
j
iSGSijiiijSGSij R
x
u
x
uRER
3
12
3
12 +
=+=
ijii
i
j
j
iSGSijiiijSGSij
x
u
x
uE
3
12
3
12 +
=+= SGS turbulence modelSGS turbulence modelSGS turbulence modelSGS turbulence model
( ) ( )
===
i
j
j
iijijijSGSSGSSGS
x
u
x
uEEECEC
2
122
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91
L E S
Filtered with width = 8DFiltered with width = 4D
DNS
FLOW OVER A FLAT PLATEFLOW OVER A FLAT PLATEFLOW OVER A FLAT PLATEFLOW OVER A FLAT PLATE
M
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2D
rpc
cvDnf
2
11
==
=
=
c
pp
p
D
18
2
=
N
N
N N
> 5 D 1/3
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P
dispersed flow
.
O , ..
T A , ..
T A , ..
F A
, .
H , ..
H , ..
increasingmassorvolumefraction
M
W ?
W ?
C ?
C
?
A
.
From A. Bakker
/
L D P
A S
E
E G
V F
S
F
S
E
R
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M
C (RANS, LES, DN)
D : E L
2 :
( )
Eulerian/Eulerian
k = 1 conitnuum phase, k = 2 dispersed phase
Eulerian/Lagrangian
Continuum phase
Dispersed phase
L/E
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A
I (E/E E/L) :
I (.. )
F E/E :
F E/L (L) (E)
I (, , )
M
E/E
ADV:
S
L L/E
S
( )DIS
D
L/E
ADV:
B
,
D
( , ), ()
().
CON
H
D
Usually E/L for volume fractions of thedispersed phase
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O : A S M
(ASM) S
.
A .
RESTRICTIONS
A < 0.001 0.01.
O .
N .
O .
N
N .
O : ASM
C
O
O
M
D
M
E
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O : V F (VOF)
A I .
S
T : J
M
S
I :
F ( )
B
V F (VOF)
A (
).
F , :
= 0 ( )
= 1 ( )
0 < < 1
T ()
:
M
C
kS
xu
t i
kj
k
=+
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105
R
106
R
T
Y=
D= (2/)
E
( ) ( ) kkkkk gradYDdivYdivt
Y
&+=+
U
Reaction source:Reaction source:Reaction source:Reaction source:generation or distructiongeneration or distructiongeneration or distructiongeneration or distructionof chemical species kdueof chemical species kdueof chemical species kdueof chemical species kdue
to chemical reactionto chemical reactionto chemical reactionto chemical reaction
( ) ( ) ( ) radk
N
k
k
kkh
St
pYgradh
Schgraddivhdiv
t
h+
+
+=+
=1
11
U
Net rate of increaseof enthalpy due to
diffusion alonggradients of
enthalpy
Net rate of increase ofenthalpy due to mass
diffusion alonggradients of species
concentration
Net rate ofincrease of
enthalpy due topressure work
Net rate ofincrease of
enthalpy due toradiative heat
transfer
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107
K
B A
C :
C
D/
S
G
107
108
F N S (FANS)
W
R
F
i
i
i uu
u
==
ii
uu = ''
iii uuu +=
W :
R T
T
M
R
C
R R
i
k
tk
tik
x
Y
ScuY
=
''''
it
ti
x
huh
=Pr
''''
0=
+
i
i
xu
t
k
i
k
i
i
ik
i
ikkw
x
J
x
uY
x
uY
t
Y&+
=
+
''''
j
i
ij
ji
ij
i
ijjg
xx
p
x
uu
x
uu
t
u
+
+
=
+
''''
( ) radiiijihiii
''
i
''
i
iQFuuJ
xx
uh
x
uh
t
h&+++
=
+
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109
T
T :
C
M
R,
,
,
CHEMISTRY TURBULEN
CE
T
,
,
LrC
T
S
ulDa
/
/ '
==
110
C
()
(Y) (T
), (
)
(CMC ).
S
T
L RRA
()
, ,
, .
R
(EDC, PASR) (
H H
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E D
M H (19)
I 1 F+O(1+)P
111
R = (R = (
112
E D/F R
I 1 F+O(1+)P
R = (R = (
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113
E D C
FINE STRUCTURES
H
P S R
F
T*, *, *
T, ,
m&
m&
RANS CODE VARIABLES
F
T*, *, *
T, ,
0ci m&
P
T :
PROGRESS VARIABLE
=0
=1
MIXTURE FRACTION Z
Z = 1
Z=0
114
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115
T
T
. ,
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