Юн А.А. Крылов Б.А. Расчет и Моделирование Турбулентных Течений с Теплообменом, Смешением

_________

( ) __________________________________________________

. . , . .

, ,

FASTEST-3D

,

2007

27.5.4 49

49 .. , .. . , , Fastest-3D: . - .: - , 2007. 116 .: .

ISBN 978-5-7035-1854-0

FASTEST-3D.

, , .

: Dr.-Ing. . ; Dr.-Ing. . ISBN 978-5-7035-1854-0 ( ) . ., . ., 2007

. 2007, .

, ,

Fastest-3D

. . 29.08.07

. . 6084 1/16 . . . . 6,74. .-. . 7,25. 250 .

. 3736/2239. .629.

, ., . 4, , -80, -3 125993.

, ., . 4, , -80, -3 125993.

3

5 7 1. 13 1.1. 13 1.2. 16 1.3. 17 1.4. 26 2. 28 2.1. (DNS) 28 2.2. (LES) 29 2.3.

(RANS) 30

2.3.1. 34 2.3.2. 35 2.3.3.

37

2.3.4.

38

2.3.5.

42

2.3.6.

48

2.3.7.

49

2.3.8. 51 2.3.9. 56 2.3.10. 58 3. 66 3.1. 66 3.2. 68 3.3. 72 3.4. 73 3.5. 74 3.6. 75 3.7. 76 3.8. 78 3.9. 79

4

3.10. 82 4. FASTEST-3D 87 4.1. FASTEST-3D 88 4.2. FASTEST-3D 89 4.3. ,

93

4.4. , 97 4.5. 102 4.6. 104 106 111

5

30

. , , (Computational Fluid Dynamics CFD). CFD , . , - , , . .

FASTEST-3D (Flow Analysis Solving Transport Equation Simulating Turbulence 3 Dimensional), (TU Darmstadt). FASTEST-3D , , , , , , . FASTEST-3D

6

SFB, , Siemens, Rolls-Roys, Alstom. . , , Fluent, CFX, ANSYS , .

FASTEST-3D , Linux, . Pentium 100 (32Mb) .

Dr.-Ing. ., Dr.-Ing. . (TU Darmstadt) Prof. Dr.-Ing. M. Schafer, Prof. Dr.-Ing. J. Janicka, Prof. Dr.-Ing. A. Sadiki, Dr.-Ing. D. Sternel.

7

, , , ,I II III IV V , ..

ija 2

2

iC , ic jiC

pc ( ) pd

D

1 2,F F

iFG

f , if

ig 2

k 22

k+ , kk U

+ = l

p kP

Pr Prt R ( )

Re

8

Re p Ret Sc ijS 1

c -

T t , iu u

,

u

, /w w u+ i ju u 2 2

ju V 3 ix y y+

, /u y

, , ,i i i i

, ij 2

2

0 22

9

+ ,

4U + =

i k ( ) ij , t ( )

, t 2

3

2

k

ij 2

2

ij 1c

k , i 1c

10

( ) ( )j ( )div ( )grad

( )b ( )F ( )O ( )P ( )u

AS , BML , , CDS - (Central

Difference Scheme) CFD

(Computational Fluid Dynamics) CLS , , CV (Control Volume)

11

DES (Detached Eddy Simulation)

DH , DNS

(Direct Numerical Simulation) EARSM

(Explicit Algebraic Reynolds Stress Model)

EASFM (Explicit Algebraic Scalar Flux Model)

EDC (Eddy-Dissipation Concept)

EVM (Eddy-Viscosity Model)

GGDH - (General Gradient-Diffusion Hypothesis)

GL , GS , IARSM

(Implicit Algebraic Reynolds Stress Model)

KM , LES (Large

Eddy Simulation) LRR , , LU - (Lower-

Upper decomposition) MPI Message Passing Interface PDF RANS

(Reynolds Averaging based Numerical Simulations)

RSM (Reynolds Stress Model)

RSFM (Reynolds Scalar Flux Model)

12

RST (Reynolds Stress Tensor)

SJ , SGS , , SIMPLE

(Semi Implicit Method for Pressure Linked Equations)

SIP (Strongly Implicit Procedure)

SMC (Second Moment Closure)

SSG , , SST (Shear

Stress Transport) UDS (Upwind

Difference Scheme) URANS

(Unsteady RANS)

WJ , WWJ , , ZFK --

13

1.

: , . 1.1.

[4]

N N

0ii

I II

ut x

+ = , (1.1)

, .

0ii

ux = . (1.2)

14

N N N

Ni j iji

ij i j V

I IIIII IV

u uu p gt x x x

+ = + +

, (1.3)

, - , - ( ), - (). [54]

23

i i ij ij

i j

u u ux x x

= + , (1.4)

ij - ( 1, 0ij iji j i j = = = ).

23

i ji i i ij i

j i i j j

u uu u u up gt x x x x xx

+ = + + + . (1.5)

(1.2) (1.5)

[1]. [4]

21i i i

j ij i j j

u u upu gt x x x x

+ = + + . (1.6)

15

N N

Np i p ij iij i j V

I III IVII

c T u c T uq St x x x

+ = + +

, (1.7)

, - , -

ii

Tqx

= , - , - , ..

, (, ),

ii i i

u D St x x x

+ = + . (1.8) D

DSc= . (1.9)

,

, , . , : ,

p RT= . (1.10)

, , .

16

. , , (, ) . ( ), . , .

1.2. . , , - , ().

: (EulerLagrange) (EulerEuler). . .. , . () (), . . 1.1 . (), ().

17

. 1.1. (), ()

, , , , .. [12], [14], [49].

1.3. , , (). . [63]. [6]: , , . , . : (), .

18

, (, ).

F O PF O P + = , (1.11) : F - ; O - ; P - . . N M :

1 1

N N

kj k kj kk k

= =

= , 1,j M= , (1.12)

k - k kj , kj - k j . (1.12)

1

0N

kj kk

W=

= , 1,j M= , (1.13) kj kj kj = kW k . kj

i k j

:

kj j k kjr W =i

, (1.14)

19

jr - j . ki

k M :

1 1

M m

k kj k j kjj j

W r = =

= = i i . (1.15) (1.15) (1.13)

1 1 1

N M Nk j k kj

j j kr W

= = =

= i

. (1.16)

jr : , ,j f j b jr r r= , (1.17)

, , 1kj

N kf j f j k

k

Yr KW

= =

; , , 1kj

N kb j b j k

k

Yr KW

= =

.

,f jK , ,b jK

, ,, , expf jn f j

f j f jE

K A TRT

= . (1.18)

(1.18) :

,f jA , ,f jnT

,f jE . ,b jK

20

11

0 0

, , exp

N

kjk j ja

b j f jS HpK K

RT R RT

=

= , (1.19)

ap ;

0jS , 0jH -

. (1.12), (1.18). [6]. , . . , , . , . , . . , , . , . , . , . , , . . ,

21

, xNO . , , xNO . (mixture fraction), :

,, ,

O

F O

z zz z

= , (1.20) z . , (1.19), :

ii i i

ut x x x

+ = . (1.21) , , [6]. .

22

. 1.2. . 1.2

. , : ( )k kY Y = , ( )T T = , ( ) = , ( ), it x = . [6], [63]. [6]. , , , . . 1.3 . - [6].

23

T x 0T () maxT . , , : , , , , , ; .

. 1.3. , , . , , - (ZFK) : - maxT ;

24

- l fl ; - s

G.

(1.18) maxT T= [6]

2

max~ exp

p

Esc RT

G

. (1.22)

, ; - maxT . , , . . [40] : , , . .

25

. 1.4.

. 1.4 . , . . , . () . , . , : , , .

, , , . , .

26

1.4. , , . : , . , , :

0 0Re l u= . (1.23)

0l - ; 0u - ; - . . , . . , , . , . . , , , . . ; , , , , . [33] . , . . 1.5 .

27

. 1.5. , : - , - , - , - , - . [40]. (, , ..) , : DNS, LES, RANS URANS.

28

2.

2.1. (DNS)

(DNS)

. , , .

DNS . . :

13 4

k =

, (2.1)

:

12

k = . (2.2)

, . 2.1,

[38].

29

. 2.1. DNS

.

Re 5103 5104 5105 5106 5108 200 Mflop/s

68 . 444 . 610 1 Tflops/s 13 . 88 . 122 , DNS . , , DNS, , .

2.2. (LES) , . , . , , , . LES , , , , DNS. , LES , DNS. LES . , LES RANS. . LES . , ; , LES, , DNS. C

30

, , . LES RANS LES-DES. DES RANS, LES. , , , .. . .

2.3. (RANS) , . : ( ) ( ) ( ),,i i ix t x t x t = + . (2.3) :

( ) ( )0

1lim ,t

i itx t dt

t = . (2.4)

. 2.1

.

31

. 2.1. : () ; ()

, , , () [62].

:

0ii

ux

= ; (2.5)

1i j i ji i ij i j i j

u u u uuu p gt x x x x x

+ = + +

; (2.6)

j jj j j

u D u St x x x + = +

, (2.7)

i ju u - , ju - .

; , (). , . .

32

(2.5) - (2.7) . . (2.5) - (2.7) .

. , , , . , , 1969 . [62], :

ff = . (2.8)

i iu u u= + ; (2.9) i i = + . (2.10)

:

0ii

ut x

+ =

; (2.11)

kj ii i i ijj i j j

u uu up u u gt x x x x

+ = + +

; (2.12)

kj jj j j

uD u

t x x x + = + . (2.13)

33

(2.11) - (2.13), (turbulence- chemistry interaction). (2.13) . , (1.14) - (1.19) ( 1), : ( ),T Y , (2.14) , ( ),T Y . (2.15) , , , (PDF),

( ) ( )P d +

= , (2.16)

, , ; ( )P . ( ) ( )P , (2.16). . [35], , . , - . . ,

34

[35]. , , [6], [40], [63].

2.3.1.

, , . 1877 . [13] , .

23

jii j t i

j i

uuu u kx x

= + . (2.17)

, . t . , t . t , . , . , t tu l , (2.18) tu - ; l - . tu l .

35

. , , , .. :

23

i jij ij

u ua

k = . (2.19)

,

:

2 jt iijj i

uuak x x = +

. (2.20)

, , , . , , , xy , . ija ,

ijS ij , , 2.3.5. 2.3.2.

, t , ,

36

. t 1925 . [62].

2 xt mvlx

= , (2.21)

ml - , . , ml y= , (2.22)

0.39 - ; y - . : [48], [11], [27] .. [62] : , . , ; , . , , , . , . , .

37

2.3.3.

, , ( ) tu l . ,

tu k , k - . (2.18), : *t C k L = , (2.23) C - . ,

32j jt i

j t Dj j k j i j i

u uuk k k ku Ct x x x x x x L

+ = + + +

, (2.24)

C , DC , k - . L .

, , (, , [15]), (, [39]) [62].

, , , , , . , .

38

, , , .. .

2.3.4.

. (1942) [33]. k .

k -

k - , [62]:

( )* *ij ij tj j j j

k k u ku kt x x x x

+ = + + ; (2.25)

( )2ij ij tj j j j

uut x k x x x + = + +

; (2.26)

tk = ; (2.27)

23

jiij i j t ij

j i

u uu u kx x

= = . (2.28)

:

* *9 /100; 3 / 40; 5 / 9; 1/ 2; 1/ 2 = = = = = .

k -

k - , (1945) [17]

39

(1972) [26].

i tj ijj j j k j

k k u kut x x x x

+ = + +

; (2.29)

2

1 2i tj ij

j j j j

uu c ct x k x k x x

+ = + +

; (2.30)

2

tkC = ; (2.31)

23

jiij i j t ij

j i

u uu u kx x

= = . (2.32)

:

1 21.44; 1.92; 0.09; 1.0; 1.3kc c C = = = = = .

k , k .

, (1993) [37], k k . k k , 1F , k k . , k k , 1F ( )11 F . 1F , . , k , t . (SST),

40

[15],

0.31i ju u k = . , SST (shear stress transport), , [16].

:

( )*ij ij k tj j j j

k k u ku kt x x x x

+ = + + ; (2.33)

( )2

1 212(1 ) ;

ij ij t

j j j j

j j

uut x k x x x

kFx x

+ = + + + +

(2.34)

tk = ; (2.35)

23

jiij i j t ij

j i

u uu u kx x

= = + . (2.36)

1 k 1 2 k ,

( )1 1 1 21F F = + . (2.37) . 1. k ():

*

1 1 10.09; 0.075; 0.5; 0.5;k = = = = * 2 *

1 1 10.41; / / = = .

2. k :

*2 2 20.09; 0.0828; 1; 0.856;k = = = =

* 2 *2 2 20.41; / / = = .

41

:

21 * 2 24500arg min max ; ;

k

kky y CD y

= ; (2.38)

2021max 2 ,10k

j j

kCDx x

=

; (2.39)

41 1tanh(arg )F = . (2.40)

,

1a k = (2.41) 1a [15]. , :

tH = , (2.42)

uy

= . , :

1ta k = . (2.43)

, , , SST- . 2F :

42

( )11 1 2maxta ka w F

= , (2.44)

2F (2.40).

( )22 2tanh argF = ; (2.45) 2 2

500arg max 2 / 0,09 ;k yy

=

. (2.46)

:

* 1 1 10.09; 0.075; 0.85; 0.5;k = = = = * 2 *

1 1 1 10.41; / / ; 0.31a = = = .

.

, . , , .

2.3.5. , , . [41], [53]. ija . ,

12

jiij

j i

uuSx x

= + 1

2ji

ijj i

uux x

= .

, , (CLS) [18].

43

( )( )

2 21 2 3

2 2 2 24 52 2

6 72 2

1 13 3

23

. (2.47)

t t t ts

t t

t tij jk ki ij jk ki

a S c S II I c S S c II Ikk kc S S c S S IVI

k kc S S S c S

= + + + +

+ + + + +

c

:

( )( ) ( )( )1.50.3 0.361 exp

exp 0.75max ,1 0.35 max ,c

SS

= + . (2.48)

:

2 2 21 2 3 4 5 6 70.1, 0.1, 0.26, 10 , 0, 5 , 5c c c c c c c c c c = = = = = = = .

, (GL) [23], .

:

( )2 2 21 2 3 13t t ska c S c S S c S II I

= + ,(2.49)

( )

( )2

12 2 2

3 1

3 6 1c

+= + + + ; (2.50)

( ) ( )222 22 3/ 2, /8S = = . (2.51)

44

:

1 2 32 / 21, 1/ 7, 2 / 7 = = = . , , , , . , CLS 7 , . . .

(EARSM) , , .

N

( )1

2

i j k i j i ii k j k

k k kI II III

i j jii j k j jk j ik

k k j i

VIV

ji

k ki

u u u u u u uu u u ut x x x

u u uupu u u p u p ux x x x

uux x

+ =

+ + + +

. (2.52)

VI

; - ; -

45

; - ; - ; - . :

1

,

ij i j l i j i jl

l

ij ij ijij

Da u u u u u u ukuk Pt x k x k

PC

= +

+ + (2.53)

; - ; -

( ' ' ' 'i iij i k j kk k

u uP u u u ux x = , / 2ijP P= ); -

; - ; . , , . . ARSM.

0ij i j l i j ll

Da u u u u u kukt x k x

= . (2.54)

, ARSM t , , i ju u , . ARSM , .

46

1i j ij ij ij iju u PP C

k

= + + . (2.55)

ijP , / 2ijP P= ijC

, . ij ij . . , :

( )21

2

7 18115 11

5 9 2 ,11 3

ij ij ik kj ik kj

ik kj ik kj ij ik ki

CP a S a a

C a S S a a S

+ + = + +

(2.56)

( )3 4 1 2 2 .3ij ij ik kj ik kj ik kj ik kj ij ik kiPA A a A S a a A a S S a a S + = + +

( )( )12

1 2 3 42 2 2 2

11 15 988 11, , ,11 7 1 7 1 7 1 7 1

CCA A A AC C C C

= = = =+ + + + - . . 2.2.

. 2.2. EARSM

1A 2A 3A 4A WJ, [60] 1.20 0 1.80 2.25 LRR, [34] 1.54 0.37 1.45 2.89 SSG, [53] 1.22 0.47 0.88 2.37 GS, [23] 1.22 0.47 5.36 1.68

47

, . ija ijS ij , . [52], [60], [65]. [60] :

( )( )

( )( ) ( )

2 21 2 3 4

2 2 2 25 6

2 2 2 2 2 2 27 8

2 2 2 2 2 29 10

1 13 3

23

23

.

sa S S II I II I S S

S S S S IVI

S S VI S S S S

S S S S

= + + + +

+ + + + + + +

+ +

(2.57)

2

jiij

j i

uuSx x

= + ,

2uu ji

ij x xj i

=

k = : ija a= , ijS S= , ij = . i - S :

2 2

3 2

2 2

, ,

, ,

.

s ij ij ij ji

ij jk ki ij jk ki

ij jk kj jk

II S S S II

III S S S S IV S S

V S S S

= = = = = = = = = =

(2.58)

48

(2.56) (2.57) i . .

2.3.6.

(2.52) (2.56). .

i j ij ij ij ij iju u

C P Dt

+ = + . (2.59)

ijC ; ijP - ; ijD - ; ij - ; ij - . .

, , . .

ijD , , (DH) [19]:

i ji j k s k ll

u uku u u C u ux = , (2.60)

sC 0.22.

49

: .

:

( )

1 sij

ijC a = , (2.61)

1C - .

, . (, , (LRR)) [34]:

( )( )

2

* *2

9 64 25 11 3

7 10 ,11

rij

ij ik kj ik kj km km ij

ik kj ik kj

CS a S S a a S

C a a

+ = + + +

+ (2.62)

2C - . LRR : 1 1.5C = 2 0.582C = [34].

, , k .

, , , , [64] (. .2), , .

2.3.7.

1,

, .

50

:

tjj

ux

= . (2.63)

, , (DH) [19]:

1i t t i jj

u C u ux = . (2.64)

, (KM) [31],

:

1i k k j

i t tj

u u u uu C

k x =

. (2.65)

, , , (AS) [9],

:

1 2 2i j i k k j

i t t tj

u u u u u uu k C C

k xk = +

. (2.66)

:

1 20.22, 0.22t tC C= = .

, :

51

( ) ( )i j i i i i ij

u u u P Dt x + = + + . (2.67)

[25].

, EARSM, , , , (WWJ) [61]:

( )41i ij j kk

ku c B u ux = , (2.68)

B

( ) ( )22 13

1 2

12

1 12 2

s sG Q I G c S c c S cB

G GQ Q

+ + + =

+;

2 31sc c c = , 2 31c c c = + ; 1

1 12 12

PG cr

= ; ,

: 2 21 s sQ c II c II = + , 3 22 2 23 s s sQ c III c c IV= + . :

1 2 3 44.51, 0.47, 0.02, 0.08c c c c = = = = .

2.3.8.

, :

p

kSt

= , (2.69)

52

p - , , k - . . . , .. . , . p Re p ,

Re ppW d

= , (2.70) W - ; pd - .

. : , . :

pVV

CV

= ; (2.71)

pmm

Cm

= , (2.72)

p ; . .

( 1.2)

53

(2.5-2.7). , . , . , . , ( ). , . ( ) , ( ) [49], , .

.

p pd x

udt

=JJG G

. (2.73)

, ,

...p A g vm Sa M Ppdu

m F F F F F Fdt

= + + + + + +JJG JG JG JG JG JG JG

, (2.74)

AF

JG- ; gF

JG-

; vmFJG

- ; SaFJG

- ; MFJG

- ; PFJG

- . , , .

54

pp id

I Tdt =JJJG G

, (2.75)

pI - ; iT G - , .

, (2.74).

, . :

23 Re4

fA p d p rel

p pF m C U

d

=

G. (2.76)

dC

. Re 1p

55

fp

= :

( )vm p vm f pdF m C u udt= G . (2.79) vmC . vmC 0.5.

. . [44]

0.5

21.615AS p reluF d Uy

= G

. (2.80)

.

( )28 relM f p Ma rel relrelU

F d c U = GG G GG , (2.81)

rel

G, relUG

- .

f f f fP p pp p

Du DuF m m g

Dt Dt =

G GG G . (2.82)

,

. (2.5) (2.7), (2.73) (2.75), ,

56

. .

2.3.9. 1.3,

(mixture fraction). :

ki ji i i

u D ut x x x

+ =

. (2.83)

,

2.3.7. , , ( )k kY Y = ( )k kT T = . 2.3. kY PDF

( ) ( )10

k kY Y P d = . (2.84) PDF ( )P - .

, 2 . (2.13),

57

k k22 2

2 2

2 .

jj j

j j j j

j j

uD u u

t x x x x

Sc x x

+ =

(2.85)

(2.85) ,

(2.83). k

ju k2ju

j jx x .

, , ( 2.3.7.).

k tji

ux

=

. (2.86)

:

k2

2 tj

iu

x

=

. (2.87)

[35]:

22 2j jSc x x k

= . (2.88)

[35], [47], [59].

- , . FASTEST-3D : BML Gequation. BML ,

58

(progress variable), , . Gequation . [6], [40], [63]. - .

BML [35]:

( ) k

k,

i ti i j

i j j k j

iu i

uk k ku k u ut x x x x

pu cx

+ = + +

(2.89)

( ) k

k

2

1 2

3 .

i ti i j

i j j j

ij

uu c u u ct x k x k x x

pc u ck x

+ = + +

(2.90)

(2.88) (2.89) BML . 3c 1.44 . (BML), . [35], [47], [59].

2.3.10.

. Ret , .

59

, : , , .

. 2.2.

: 1) ,

: u y+ += , uuu

+ = , u yy + = ,

wu = - , 2) ,

. ,

5 ln 3.05u y+ += + . ,

60

. 0 11.63y+ (. 2.2),

3) 11.63y+ , :

( )1 lnu Ey B+ + = + , (2.91) 0.41 - ; - , ( 8.8E = ); 5.0B - .

, 20% 80% . , , . ( ), . , , . [45].

(, [29]) :

1) 11y+ : t Pr y+ + = , (2.92)

- ; t - ; + - ,

61

2) 0 11.63y+ : t , (2.93)

3) 11.63y+ :

t lnt

k y Dk

+ + = + . (2.94) + :

( )Prt fU + + = + , (2.95)

f [29]

2 / 312.5Pr 2.12ln Pr 1.5f = + + . (2.96)

:

effyT

++= . (2.97)

, . . , , , . , , k (2.29), (2.30)

62

Ret , , s , . , 2 / k (2.30) . 0 = + , , ; 0 - ( - ) .

(Low- Re) .

k [16]

i tj ijj j j k j

k k u kut x x x x

+ = + +

; (2.98)

2

1 1 2 2

;

ij ij

j j

t

j j

uu c f c f Et x k x k

x x

+ = + + + +

(2.99)

0 = + , 0 22ky

= ; (2.100)

/ 222yE e

y += ; (2.101)

2

tkc f = . (2.102)

0.01151 yf e

+= ; 1 1f = ; 2(Re / 6)

2 1 0.22 tf e=

. :

1 21.35; 1.80; 0.09; 1.0; 1.3kc c c = = = = = .

63

, k , .

WJ CLS.

( & ) [60]

( )( ) ( )

( )

2 2 2 221 2 1 3

2 2 21 4 1

2 2 2 2 21 6 1 9

3 4 1 113 3max( , )

12max( , )

2 . (2.103)3

exij seq

s s

eqs s

Ba f S II I f II III II

Bf f S SII II

f S S IVI f S S

= + +

+ + + +

-

1 1 expyfA

++

= , (2.104)

26A+ = .

- (2.86). sII , - eqsII :

21

1

405 5.74216 160

eqs

cIIc

= , (2.105)

1 1.8c = .

64

(Craft, Launder & Suga) [18]:

( )( )

21 2

2 2 23 4 2

2 25 2

6 72 2

13

13

23

;

t t ts

t t

t

t tij jk ki ij jk ki

a S c S II I c S Sk

kc II I c S S

kc S S IVI

k kc S S S c S

= + + + + + + + +

+ +

(2.106)

2

tkc f = ; (2.107)

( )( ) ( )( )1.50.3 0.361 exp

exp 0.75max ,1 0.35 max ,c

SS

= + . (2.108)

( ) ( )1/ 2 21 exp Re /90 Re / 400t tf = . (2.109)

2Re /t k = . (2.110)

:

2 2 21 2 3 4 5 6 70.1, 0.1, 0.26, 10 , 0, 5 , 5c c c c c c c c c c = = = = = = = .

. , - (LB) [8]

0.255Re 2 41(1 ) (1 )Re

tT

tf e = + . (2.111)

65

, ; , , [64] (. .2). ; , , , .

66

3.

, , , . . : , , , . FASTEST3D (Flow Analysis Solving Transport Equations Simulating Turbulence 3 Dimensional) [20]:

- , ;

- ; - ; -

; - (Strongly Implicit Procedure)

; - ,

(domain decomposition). 3.1.

N N

i

i i i IVI II III

u S

t x x x

+ =

, (3.1)

67

; - ; - ; .

. , , :

( ) ( )ii i iV V V V

dV u dV dV S dVt x x x + = , (3.2)

- (, ); - (/ , . .); S - , , ( , ..).

, :

V

div dV ndF S dV = = v . (3.3)

( ) i iiV V

dV u n d S dVt x + = , (3.4)

, V in , n . .

FASTEST-3D (. 3.1).

68

. 3.1.

. . (3.4) . , .

3.2. . . 3.2 . . 1 2 3( , , ) , 1 2 3( , , )x x x . [21] .

69

. 3.2. [21]

1 1 1

1 2 3

2 2 2

1 2 3

3 3 3

1 2 3

x x x

x x xT

x x x

=

. (3.5)

ji j ix x

= . (3.6)

1T ( ) :

70

1 1 1

1 2 3

1 2 2 2

1 2 3

3 3 3

1 2 3

x x x

Tx x x

x x x

=

, (3.7)

( )1 TadT TJ= , (3.8) detJ T= - ( )TadT - , T .

,

1 1j i iji i

xadjx J J

= = (3.9)

1 iji jx J

= , (3.10)

ij - ,

3 3 3 32 2 1 1 1 2 1 2

2 3 3 2 3 2 2 3 2 3 3 2

3 3 3 32 2 1 1 1 2 1 2

3 1 1 3 1 3 3 1 3 1 1 3

3 3 3 32 2 1 1

1 2 2 1 2 1 1

x x x xx x x x x x x x

x x x xx x x x x x x x

x x x xx x x x

=

1 2 1 1

2 1 2 2 2

x x x x

.

(3.11)

71

, (3.10) (3.2),

( ) i i ik ikV V

dV u n d n d S dVt J

+ = . (3.12)

(3.12), [21]

( )j j li n i l kb kiV V

dV u F b u b S dVt V x

+ = , (3.13)

jnbF -

; jkb - . , 1j = East ( E) West ( W); 2j = North ( N) South ( S); 3j = Top ( T) Bottom ( B) (. 3.3).

. 3.3.

72

3.3. East, . 3.4. . P E; , , te, ne, se be.

. 3.4. East East [21]:

( ) ( ) ( )2 2 21 1 1 1 11' 2 3 12

1 1 1 2 1 1 3 1 3 1 331 2 2 2 3 2 1 1 2 2 3 3 3

{

,

ie k i k

F b b b b b

v v

b b b b b b b b b b b b

v v

= + + +

+ + + + + +

(3.14)

i :

73

1 E P = , 2 ne se = , (3.15) 3 te be = .

, , E P , (3.14). ne , se , te , be P, E, N, S, T, B, ne, se, te, be. 3.4. East

( ) ( )1 1 1 1 11 1 2 2 3 3{ } ee k e e eeF b u b u b u b u m = + + = i , (3.16)

emi

East ,

( )1 1 11 1 2 2 3 3em b u b u b u= + +i . (3.17)

0e t w s bnm m m m m m+ + =i i i i i i

. (3.18) FASTEST-3D ; , .

74

3.5.

( - ) (3.1) .

:

( )P

PPV

dV Vt t . (3.19)

. , :

( ) ( ) 1 1 1 1 1 1n nP P n n n n nNb Nb P P PNb

V VA S A

t = + . (3.20)

t , n 1n . (3.20) PA , pS . : ( ) ( )

( )

11 1 1 1 1

1 1 1 1 11 .(3.21)

n nP P n n n n n

P P Nb NbNb

n n n n nP P Nb Nb

Nb

V Vf A A S

t

f A A S

= + + + + + +

0f = , ; 0.5f = , . , , . f ,

75

.

3.6.

3.3-3.5 (3.1). , , . (, - ..) (.3.5).

. 3.5. ,

, ,

76

1

1 1 1 ,

P

P k Pi i iPV

j jj P ji i

j k l

dV V b Vx x J

b V bJ J

= =

(3.22)

j k l PV = .

, ix ( ) ( ) ( )1 2 3

i

pe w i n s i t b iuS p p b p p b p p b= . (3.23)

ig :

i

giuS g V = . (3.24)

(3.22)

i

kuS i

duS :

i ii i

p g d ki u uu uS S S S S= + + + . (3.25)

, :

i ii

g d ki u uuS S S S= + + . (3.26)

3.7.

. . 3.6 .

77

. 3.6. (Upwind Difference

Scheme (UDS)) East P E (. 3.7):

, 0, 0

P eUDSe

E e

f mf

f m>=

78

(P E) (. 3.7) :

( )1CDSe E e P ef f f = + , (3.28)

e PeE P

= . (3.29)

, , , .

(UDS/CDS), FASTEST-3D :

N ( )UDS CDS UDSe e e e

f f f f= + , (3.30)

- .

0 = UDS , 1 = UDS . 0 1, . , UDS CDS UDS .

3.8.

. : Np p Nb Nb P

Nb

A A S =

, (3.31)

79

pA - p P ; NbA - .

(3.31) : (1) (1) (2) (2)

(1) (2)p p Nb Nb Nb Nb P

Nb NbA A A S = , (3.32)

Nb(1)=E, W, N, S, T, B Nb(2)=EN, WN, ES, WS, NT, NB, ST, SB, ET, EB, WT, WB, ENT, ENB, WNT, WNB, EST, ESB, WST, WSB (. 3.3).

, ( )( )(1) (1)

(1)2p p Nb Nb P

NbA A S Nb = . (3.33)

1n +

1n+ ( )( )1 1(1) (1)

(1)2n n np p Nb Nb P

NbA A S Nb+ + = . (3.34)

, . 3.9. . , . - . FASTEST-3D SIMPLE, , , , .

80

[20] [21]. m 1m . (3.34) iu

1 1 1 1, , ( )

m mu m u m m mp i P Nb i Nb P i p

NbA u A u Q p = . (3.35)

pS

1( )mi pp 1mPQ . * 1mp p = *u (3.35) *,i Pu

( )11* * 1 *, ,1 mm u mi P Nb i Nb P iu PNbPu A u Q pA

= + . (3.36)

*iu

, . , * * * * * * *p e w n s t bm F F F F F F = + + , (3.37) *iu

* * 1, ,e e i e i eF u b= . (3.38)

. , ( *,i eu ,

*,i nu , ..) .

81

**iu **p ,

. :

** * cori i iu u u= + , ** * cori i ip p p= + . (3.39) **iu

**p , :

( )11** * 1 **, ,1 mm u mi P Nb i Nb P iu PNbPu A u Q pA

= + . (3.40)

(3.36) (3.40) : ( ),cor cori P i Pu p= . (3.41) e ( ),cor cori e i eu p= . (3.42) 1, ,

cor core e i e i eF u b= (3.43)

* cor cor cor cor cor corp e w n s t bm F F F F F F = + + , (3.44) *Pm , *iu .

(3.42) (3.43) , (3.44),

82

*corcorp p Nb p

Nb NbA p A p m = , (3.45)

, (Strongly Implicit Procedure (SIP)). , , FASTEST-3D :

(3.36) 1m *iu . , *u

* 1mp p = *Pm (3.37).

*Pm (3.45), corp . :

** * corpp p p= + , 0 1p< , (3.46)

P - , . ** mp p= 1m + . corp *iu

**iu ,

**,i eF .., .

3.10. (3.31)

83

A S = , (3.47) A- , (3.31) CV; - , ; S - , , (3.31). (3.47) . A ( FASTEST-3D ). , , , , , , . . . n (3.37) n , . nr ( (3.31)) n nA S r = . (3.48) . : 1n nM N B+ = + . (3.49) nM , (3.49) ( ) ( )1n n nM B M N+ = (3.50)

84

n nM r = . (3.51) 1n n n += , . nN , (3.49) . (3.49) M A. (SIP), [55] [21], FASTEST-3D. M LU ILU: M LU A N= = + . (3.52)

ILU LU, A L U . , L U A (W, E, S, N, B, T, P), LU (SE, NW ..). [55] , , N , LU. N , 0N M A. , (SE, NW, etc.) N (W, E, S, N, B, T, P). . [55] , A. A, N. M, A N, . L U A N.

85

, L U . (3.51) n nLU r = (3.53) 1n n nU L r R = = . (3.54) LU, nR (3.54) . n , : 1n n n+ = + . (3.55) , nr .

. .

, (3.31) , , (3.31) ( SIP).

, , , . :

( )1 1m m new m = + , (3.56)

86

m 1m - m m (m-1) ; new - (3.31) 0 1< . , , . .

87

4. FASTEST-3D

FASTEST-3D

, , , . , , , (, ..) . . , , .

FASTEST-3D Fortran C [20]:

- (); - ; - - ; - , ;

- ;

- ;

- , MPI (Message Passing Interface).

, . ,

88

, .

4.1. FASTEST-3D

, FASTEST-3D Linux fastest3d.tgz. : CD ( cp /media/ /fastest3d.tgz fastest3d.tgz); ( tar xvfz fastest3d.tgz). . 4.1 FASTEST-3D. :

1. ( 4.2). 2. ( 4.2). 3. , ( 4.3).

4. ( 4.4). 5. ( 4.5).

89

. 4.1. FASTEST-3D.

4.2. FASTEST-3D

ICEM CFD.

: , , . grid01.grd grid01.tbc fastest3d/project/grid , test.grd test.tbc. *.grd , *.tbc ( , ..). *.tbc, :

INL- , OUT- , SOL- , MIR- , CON- ;

INL- , OUT- , SOL- , , FLX- , , HTC- ,

90

, MIR- ;

SOL- , , FLX- , .

, fastests3d/project/map/test.md. test.md.

( (1)): ### number of processors 1 ;number of processors

( ): ### processor info 1 1 1 0 ;proc. in i, j, k dir., new processor (block 1) 1 1 1 0 ;proc. in i, j, k dir., new processor (block 2) 1 1 1 0 ;proc. in i, j, k dir., new processor (block 3)

( (1) ):

### number of flow regions 1 ( ): ### flow region description 1 ;flow region for block #1 1 ;flow region for block #2 1 ;flow region for block #3 ( (0)): ### number of rotating regions 0

91

(): ### rotating region description 0 ;block #1 0 ;block #2 0 ;block #3

,

(): ### volume check 1 ;1 = on, 0 = off

( (1),

(2,2,2): ### monitoring point 1 2 2 2 ;block, i, j, k

( (0)):

### turbulence statistics 0 ;input number for turbulence statistics (memory allocation) ; 0 = no memory for turbulence statistics is allocated ; 1 = memory for mean values and reynolds stresses is allocated ; 2 = additional memory for skewness and flatness

,

( (0)): ### number of chemical species 0

( (0)): ### number of chemical reactions 0

92

( (0)): ### number of comb. prog. variables 0

( (isotherm), (turbulent), - (earsm), (steady), (prem) ( ), (radiation) ( )): ### allocation isotherm turbulent earsm steady ;keywords are: ; isotherm or temperature ; steady or unsteady ; laminar or turbulent ; if turbulent____ keps ; | ; |_ earsm ; | ; |_ restr ; | ; |_ les ; ; prem or nothing ; prem is for single reactive scalar ; for Bray-Moss-Libby model ; (premixed combustion) ; ;When "temperature" is given, memory

; for arrays concerning temperature ;calculations and temperature topology ;will created. ; ; radiation or nothing ;(Gas radiation model.Only with

; temperature possible, otherwise

93

; memory for radiation is not allocated) ; ;If no entry point is given, memory will ;be allocated for isotherm, turbulent and ;unsteady calculation.

map3d map fastest3d/project.

4.3. ,

, ( ): fastest3d/project/id/test.id. . ( (channel)): ### title channel (created by Alex Yun) ( (0), (1): ### read restart 0

( (10) , ASCI (0), (10) ): ### write output 10 0 10 ;restart ascii visual (0 = no output) (tecplot6): ### file formats for visual output tecplot6 ;[cfview37|cfview37time] ;[tecplot6|tecplot6temp]

94

;[explorer] (not supported ???) ;[plot3d|plot3dtemp]

( (vel), (pres), (tke), (edis)): ### visual output variables vel pres tke edis ;[vel][pres][tke][edis][temp][conc] ;[den][vis][vism][hcap][hcon][hconm][diff][td] ;[vort][avg][tau][yplus][dist][mix][mixdiff] ;[tauxx][tauyy][tauzz][tauxy][tauyz][tauxz] ;[mixvar][drhodt][srs][usrs][vsrs][wsrs] ( (t), (f), 0.6): ### compressible f 0.6 ;[t/f] [density realxation]

( (vel),

(turb)): ### lcalc vel turb ;[vel][turb|rsm|visles][temp][conc][time][mix] ;[mixvar][tab][srs][scfl][raf][sp1][sp3] ;turb: use a two-equation turbulence model ;rsm: use a reynolds stress turbulence model ;visles: perform a Large Eddy Simulation ;mix: perform combustion progress variables ;mixvar: perform combustion progress variables ; variances equation ;tab: perform laminar/flamelet/ildm chemistry ;srs: solve sing.react.scal. transp.eq.(BML model) ;scfl: solve scal.flux transp.equ. (+BML model) ;raf: solve rad.flux equation ;sp1/sp3 different approximations for the heat flux equation

96

( (1.e-3), (1.e+10)): ### residuum limits 1.e-3 1.e+10 (- (cds)): ### interpolation method cds ;[cds|tbi] : ### underrelaxation 0.3 0.3 0.3 0.9 0.5 0.5 0.7 0.9 0.9 0.9 0.5 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 ;u v w p k eps vis den t c mix restr mixvar srs scfl ildm ildmv rf cst vofr

, , ( (fofi), (10000) , (1.e-3): ### time discretization fofi ;discretization method [fofi|sofi|crni|ruk3|ruk5] 10000 1.e-3 ;no of timesteps,size of timestep

( (none)): ### bouyancy by temperature gradient none ;[none|boussinesq|dengrad] ( (0)): ### prog. variables for combustion 0 ;no of progress variables: mix. frac, mass. fracts

97

( (0,0,0): ### pressure gradient 0. 0. 0. ;pressure gradient in x,y,z direction

4.4. ,

: fastest3d/project/funcbcs.f, fastest3d/project/funcprp.f.

. funcbcs.f. :

function fu(x,y,z,iflr,igb,ifa,totime) function fv(x,y,z,iflr,igb,ifa,totime) function fw(x,y,z,iflr,igb,ifa,totime)

: fu=1.36d0 fv=0.0d0 fw=0.0d0

: function fte(x,y,z,iflr,igb,ifa,totime)

fte=0.3d0 :

function frscxx(x,y,z,iflr,igb,ifa,totime) function frscyy(x,y,z,iflr,igb,ifa,totime) function frsczz(x,y,z,iflr,igb,ifa,totime) function frscxy(x,y,z,iflr,igb,ifa,totime) function frscyz(x,y,z,iflr,igb,ifa,totime) function frsczx(x,y,z,iflr,igb,ifa,totime)

98

: frscxx=0.04d0

frscyy=0.04d0 frsczz=0.04d0 frscxy=0.d0 frscyz=0.d0

frsczx=0.d0 :

function fed(x,y,z,iflr,igb,ifa,totime)

rlength=0.127d0 fed=sqrt(fte(x,y,z,iflr,igb,ifa,totime)**3)/rlength

function ft(x,y,z,iflr,igb,ifa,totime)

:

ft=293.15d0

:

function fuw(x,y,z,iflr,igb,ifa,totime) function fvw(x,y,z,iflr,igb,ifa,totime) function fww(x,y,z,iflr,igb,ifa,totime)

:

fuw=0.d0 fvw=0.d0 fww=0.d0

function ftw(x,y,z,iflr,igb,ifa,totime)

99

:

ftw=293.15d0

function fqw(x,y,z,iflr,igb,ifa,totime)

fqw=0.d0 :

function fmixt(x,y,z,iflr,igb,ifa,totime)

:

if (y.gt.0.5) then fmixt=0.0 else fmixt=1.0

endif

function fmixtvar(x,y,z,iflr,igb,ifa,imphi,totime)

:

if(imphi.eq.1) then c---- mixture fraction variance fmixtvar=0.0d0 elseif(imphi.eq.2) then c---- reaction progress variable 1 variance (ildm) fmixtvar=0.0d0 elseif(imphi.eq.3) then c---- reaction progress variable 2 variance (ildm) fmixtvar=0.0d0 endif

100

function fsrs(x,y,z,iflr,igb,ifa,totime) function fsrscorx(x,y,z,iflr,igb,ifa,totime) function fsrscory(x,y,z,iflr,igb,ifa,totime) function fsrscorz(x,y,z,iflr,igb,ifa,totime)

:

fsrs=0.d0 fsrscorx=0.d0 fsrscory=0.d0 fsrscorz=0.d0

funcprp.f :

function fden(press,tmpr,iflr,sumcm) c---- Air fden=0.96737341d0-0.00028655237d0*tmpr-1463.8211d0/tmpr+ 46843.588d0/t**1.5-347974.74d0/tmpr**2

function fvis(tmpr,iflr) c---- Air fvis=-9.2014393d-6+4.4397544d-9*tmpr-1.4046401d-14*tmpr**2.5+ 3.2787496d-16*tmpr**3+1.5276332d-6*sqrt(tmpr) c---- Luft R=287.1 J/(Kg K) if (nfr.eq.1) rgas=287.1d0

101

function flam(tmpr,iflr) c---- Air flam=0.042491107d0+1.4835066d-6*tmpr**1.5-8.5222871d-12* tmpr**3-0.41142788d0/sqrt(tmpr) function fcp(tmpr,iflr) c---- Air fcp=-0.014860411d0+8.3859987d-10*tmpr**2.5- 1.7304405d-11*tmpr**3+0.002323523d0*sqrt(tmpr) function fbeta(tmpr,iflr) c---- Air fbeta=3.421d-3 function frks(x,y,z,iflr,igb,ifa) c---- Alle Waende sind glatt: frks = 0.d0 c c---- Alle Waende haben die gleiche Rauhigkeit: c frks = 0.001d0 function fmixdiff(t,iflr,imphi)

if (imphi.eq.1) fmixdiff=0.0003d0 if (imphi.eq.2) fmixdiff=0.0003d0 function fschmidtt()

fschmidtt=0.9d0 ( ) function frhoun(f)

frhoun=1.0d0

102

( ) function ftun(f)

ftun=300.0d0

( ) function fulamburn(n,f)

fulamburn=0.5d0 (T_burn - T_unburn) / T_unburn ( ) function fexprat(n,f)

fexprat=6.0d0 fastest3d. 4.5.

CFView Tecplot. . , Linux GNUplot. Linux .

:

- subroutine usrini; - (

) subroutine usrtim; - (

) subroutine util. subroutine util:

c === channel flow === c === Re=12300 === integer npoint parameter (npoint=30) integer icell,ipoint

103

real*8 minabstand,abstand real*8 xst,xend,yst,yend,zst,zend,ddl real*8 xline(1:npoint),yline(1:npoint),zline(1:npoint) real*8 ypoint,upoint,vpoint,wpoint c --------- Profile at x/h=95 --------------- c start and end coordinates xst=12.0d0 xend=12.0d0 ddl=(xend-xst)/dfloat(npoint-1) xline(1)=xst do ipoint=2 , npoint xline(ipoint)=xline(ipoint-1)+ddl enddo yst= 0.0d0 yend=0.127d0 ddl=(yend-yst)/dfloat(npoint-1) yline(1)=yst do ipoint=2 , npoint yline(ipoint)=yline(ipoint-1)+ddl enddo zst=0.025d0 zend=0.025d0 ddl=(zend-zst)/dfloat(npoint-1) zline(1)=zst do ipoint=2 ,npoint zline(ipoint)=zline(ipoint-1)+ddl enddo open(1,file='res/channel.dat') write(1,*)'#Profile at axial position x=12m' write(1,*)'#h, u, uu, uv, vv, ww, te ed' do ipoint = 1 , npoint minabstand=1d+0 c looking for the nearest point do icell=1,nxyza c check the distance abstand=sqrt((xc(icell)-xline(ipoint))**2+ & (yc(icell)-yline(ipoint))**2+

104

& (zc(icell)-zline(ipoint))**2) c update min. distance if necessary if (abstand.lt.minabstand & ) then minabstand=abstand c --- y ypoint=yc(icell) c --- u upoint=u(icell) endif enddo write(1,'(10(e14.7,1x))') & ypoint , upoint enddo close(1)

4.6.

. , , Fastest-3D.

FASTEST-3D 4.3-4.5.

, ; , (. 4.2).

. 4.2.

105

100h , h - . . Re 13700= . 6000CV = . [31]. exp_channel.dat.

. 4.3 , WJ , . AMD Athlon 2000+ 2 . 20 .

. 4.3.

, WJ .

106

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2001. 6. - . .

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111

. .1. .

. -

. .

- 0

i

i

ux

=

-

1i ji iij i

j i j j

u u uu p gt x x x x

+ = + +

-

1j j

j j ju D u S

t x x x

+ = +

( ) . - p

pd x

udt

=JJG G

. - ...p A g vm Sa M Pp

dum F F F F F F

dt= + + + + + +

JJG JG JG JG JG JG JG

. - pp i

dI T

dt =JJJG G

-

0ii

ut x

+ =

- kj ii ii ij

j i j j

u uu up u u gt x x x x

+ = + +

- kjj

j j j

uD u

t x x x + = +

.

i tj ij

j j j k j

k k u kut x x x x

+ = + +

112

. .1.

.

21 2

i tj ijj j j j

uu c ct x k x k x x

+ = + +

. ( )* *ij ij t

j j j j

k k u ku kt x x x x

+ = + +

. .

( )2ij ij tj j j j

uut x k x x x + = + +

. ( )*ij ij k t

j j j j

k k u ku kt x x x x

+ = + +

. .

SST

( )2

1 212(1 )

ij ij t

j j j j

j j

uut x k x x x

kFx x

+ = + + + +

. RST

CLS

( )( )

21 2

2 2 23 4 2

2 25 62 2

7 2

13

13

23

t t ts

t t

t tij jk ki

tij jk ki

a S c S II I c S Sk

kc II I c S S

k kc S S IVI c S S S

kc S

= + + + + + + + + +

+

.

RST

GL ( )21 2

22

313

t

ts

ka c S c S S

c S II I

= + +

113

. .1.

EARSM

.

RST

( ) ( )

( ) ( )( )

2 21 2 3

2 24 5

2 2 2 2 2 26 7

2 2 2 2 28 9

2 2 2 210

1 13 3

2 23 3

sa S S II I II I

S S S S

S S IVI S S VI

S S S S S S

S S

= + + +

+ + + + + + +

+ + ++

- .

RST

WJ, LRR, SSG, GS

( )3 4 12

23

ij ij ik kj ik kj


PA A a A S a a

A a S S a a S

+ = + +

-

RST

( )1

2

i j k i j i ii k j k

k k k

i ji j k i jk j ik

k k

j ji i

j i k k

u u u u u u uu u u ut x x x

u uu u u p u p u

x x

u uu upx x x x

+ = + + +

+ +

.

i tj ijj j j k j

k k u kut x x x x

+ = + +

.

21 1 2 2

ij ij

j j

t

j j

uu c f c f Et x k x k

x x

+ = + + + +

114

. .1.

. RST

CLS

( )( )

1 22

2 2 23 4 2

2 25 62 2

7 2

13

13

23

t t ts

t t

t tij jk ki

tij jk ki

f c f c fa S S II I S S

kkc f II I c f S S

k kc f S S IVI c f S S S

kc f S

= + + + + + + + + +

+

EARSM

.

RST

( )

( ) ( )( )

2 221 2

2 21 3

2 2 21 4 1

2 2 2 2 21 6 1 9

3 4 113max( , )

13

12max( , )

23

exij seq

s s

eqs s

Ba f S II III II

f II I

Bf f S SII II

f S S IVI f S S

= + + +

+ + + +

- .

RST

WJ

( )3 4 12

23

ij ij ik kj ik kj


PA A a A S a a

A a S S a a S

+ = + +

-

tj

ju

x

= GGDH

DH 1i t t i j

ju C u u

x =

115

. .1.

KM 1

i k k ji t t

j

u u u uu C

k x =

AS 1 2 2

i j i k k ji t t t

j

u u u u u uu k C C

k xk = +

EASFM

WWJ ( )41i ij j kk

ku c B u ux =

(BML) . ( ) k

k

ii i j

i j

ti

j k j u i

uk k u k u ut x x

k pu cx x x

+ = + + +

.

( ) kk

2

1 2

3

i ti i j

i j j j

ij

uu c u u ct x k x k x x

pc u ck x

+ = + +

RST

k k( ) k

k k k k

2k i ji j i jS

k k k

j jRii k j k ij j ik

k k u i j

u u uu u u ukCt x x x

u pu pu u u u u c u cx x x x

+ = + + +

BML

k k( ) kk k j j k

2

22

12

k jj jS

k k k

j jS ci j k cj j

k k u i

u u cu c u ckCt x x x

u pcu c u u c c u cx x x c

+ = + + +

, . . 1 .

116

. .2. RANS [64] (

). 10-15

3-72-6

1 1-1,11,1-1,3

3-5

0

5

10

15

k-eps EARSM AdvancedEARSM

RSM Low-Rekeps

Low-ReEARSM

Low-ReRSM

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