Introduction to CFD -Theory - 2

8/13/2019 Introduction to CFD -Theory - 2

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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


16/58

D

T CFD :

F V (80%).

F E (15%).

O (5%)F ()

S .

B .

V .

L / B.

P :

3 :M ,

;

D ,

;

I ( ), .


17/58

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


56/58

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

. ,

Documents

Introduction to CFD -Theory - 2