Three-dimensional flow driven pore-crack networks in ...

Theoretical and Applied Fracture Mechanics 53 (2010) 9–41

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Three-dimensional flow driven pore-crack networks in porous composites: Boltzmann Lattice method and hybrid hypersingular integrals B.J. Zhu *, Y.L. Shi Key Laboratory of Computational Geodynamics, College of Earth Science, Graduate University of Chinese Academy of Sciences, Beijing 100049, PR China

a r t i c l e

i n f o

Article history: Available online 16 December 2009 Keywords: Hypersingular integral equation method Lattice Boltzmann method Flow driven pore-crack networks Strain energy density function Stress intensity factor Various porous composites

a b s t r a c t This study introduces a hybrid hypersingular integral equation-Lattice Boltzmann method (HHIE-LBM) for analyzing extended 3D flow driven pore-crack networks problem in various porosity composites. First, the extended hybrid electronic–ionic, thermal, magnetic, electric and force coupled fields’ pressure and velocity boundary conditions for HHIE-LBM model are established, and the closed form solutions of extended distribution functions are given. Second, an extended 3D flow driven pore-crack networks problem in various porosity composites is translated into a coupled of HHIE-LBM equations, and the pore-crack networks propagation parameters are analyzed. Third, the extended dynamic stress intensity factors (EDSIFs) are calculated by using the parallel numerical technology and the visualization results are presented. Last, the relationship between the EDSIFs and the differential porosity is discussed, and several rules have been found, which can be utilized to understand the extended fluid flow mechanism in various porosity composites and analyze the extended fluid flow varying mechanism on coseismal slip. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Fluid driven fracture is one of the most important geophysical phenomenon, especially for seismic trigger events at seismogenic zone during inter-seismic period [1,2]. Groundwater has an important role in the whole water resources system, with the increasing demand of the information on groundwater hydrology and hydraulics, the mechanism of fluid and fluid flow driven pore-crack networks in the aquifer need special attention as the main resources of groundwater. As one of the most popular fluid simulation method, Lattice Boltzmann method has been widely applied in studying of fluid problem, and a lot of landmark achievements have been obtained [3–15]. On boundary condition aspect. In [16], a hexagonal Lattice gas model and modeling the 2D Navier–Stokes equation is developed. In [17], a cellular automaton model to simulate the process of seism genesis is presented. A supplementary rule for computing the boundary distribution is proposed, and presented 3D bodycentered-cubic lattices are presented for Poiseuille flow [11]. In [12], a hydrodynamic boundary condition for Lattice Boltzmann simulations is developed. In [10], the pressure and velocity boundary for 2D/3D Lattice Boltzmann BGK model is proposed.

* Corresponding author. Tel.: +86 010 88256475/305 799 9713; fax: +86 010 88256485. E-mail address: [email protected] (B.J. Zhu). 0167-8442/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2009.12.002

Extended Fluid (electronic–ionic, thermal, magnetic, electric and force) flow pore-crack network problem is an interdisciplinary issues, a lot of research results have been obtained in different fields of study. In [18], two dimensional falling drops under gravity for some range of Eotvos and Ohnesorge numbers is simulated. Based on the work in [19], the problem of stress intensity factors statistics in a randomly cracked solid and found that SIF distribution follows the Gnedenko–Gumber asymptotic rule is addressed [20]. The solute transport in a single fracture with the combination of the Lattice Boltzmann method is analyzed, and this study provide a new path of applying the LBM in solute transport simulation in fractures [21]. But there is little research about the extended 3D flow driven pore-crack networks problem in various porosity composites under multiple coupled electronic–ionic, thermal, magnetic, electric and force fields. In this paper, bases on the multiple scale fracture mechanics/ physics theory [22–27], the hybrid hypersingular integral equation-Lattice Boltzmann method (HHIE-LBM) proposed by the authors is defined by combined with extended hypersingular integral equation method [3,4,7,8,28–32] and 3D Lattice Boltzmann method [3,4,7,8,32], and the typical extended 3D flow driven pore-crack networks model for various porosity composites is analyzed by using this method. First, the extended hybrid multiple coupled D3Q27 lattice cubic is created and the extended hybrid electronic–ionic, thermal, electromagnetic (weak and strong coupled) and force coupled fields

10

B.J. Zhu, Y.L. Shi / Theoretical and Applied Fracture Mechanics 53 (2010) 9–41

pressure and velocity boundary conditions for the HHIE-LBM model are established. Then, using the HHIE-LBM method, the extended 3D flow driven pore-crack networks problem in various porosity composites is translated into a set of coupled HHIE-LBM equations, in which the unknown functions are the extended displacement ratio discontinuities, and the extended dynamic stress intensity factors (EDSIFs) and the extended strain energy density function (ESEDF) is derived. Third, the extended dynamic stress intensity factors (EDSIFs) are calculated by using the parallel numerical technology and the visualization results are calculated. The results are presented toward demonstrating the applicability of the proposed method. In addition, the relationship between the EDSIFs and differential porosity are discussed, and several rules have been found, which can be utilized to help understand the extended fluid flow mechanism in various porosity composites and analyze the extended fluid flow varying mechanism on coseismal slip.

1 1 1 where ami n ¼ 7 18. i ¼ 3 di0 þ 18 dim þ 36 din m ¼ 1 6 After we define the similar lattice velocity efi and the distribution function fif (i = 0, 18) at position X(x, y, z) and time t for the fluid flow, the pressure and velocity boundary conditions for fluid flow problem (force field) can be obtained. The thermal flow density, velocity and equilibrium distribution function for incompressible and compressible model can be defined as

2. Basic equation

fieq t ðx; tÞ ¼ ati qtin þ ati qtin

In the present paper, summation from 1 to 3 over repeated lowercase, and of 1 to 7 in uppercase subscripts is assumed, and a subscript comma denotes the partial differentiation with respect to the extended coordinates. The constitutive relationships can be written as

€J ðEIJKL Z_ KL Þ;I þ F_ J ¼ qU

qtin ¼

14 X

@ q d ln q ½ðkc;i Þ;i þ q þ qui q;i ¼ 0 @t dc ðqui Þ;t þ ðqui uj Þ;j ¼ sij;j þ qfib

fieq t ðx; tÞ

ð14Þ

t i

t in

t i

t in ini

¼a q þa q

mw qmw ¼ in ini ua

qmw in ¼

" 2 2 # 3 uta eti uta 9 eti uta þ c2s 2c4s 2c2s

ð15Þ

" # 2 eti uta 9 eti uta 3ðuta Þ2 þ c2s 2c4s 2c2s

ð16Þ

12 X

fimw emw ia

ð17Þ

ð5Þ

Z h i jn jr/a j2 þ naa Wð/a Þ þ f ðnÞ 2 V @na na g ¼ r þ ð1Þa r Krl @t q @g gg g þ gr ¼ r P r @t q q

Fðnaa ; /a Þ ¼

ð6Þ ð7Þ ð8Þ

The electronic–ionic density, velocity and equilibrium distribution functions for incompressible and compressible model can be defined as [33–35]

fimi emi ia

ð9Þ

12 X

fimw

ð18Þ

i¼0

fieq

ð4Þ

The Helmholtz free energy and transport equation can be written as

fieq

mw

mw ðx; tÞ ¼ amw i qin " 2 # mw 2 mw 9 emw 3 umw 3emw a i ua mw mw i ua þ ai qin ini þ c2s 4c4s 2c2s

ð19Þ

mw

mw ðx; tÞ ¼ amw i qin " 2 # mw 2 mw mw 9 emw 3 umw a i ua mw mw ei ua þ ai qin þ c2s 4c4s 2c2s

ð20Þ

where amw ¼ 0di0 þ 18 dim m ¼ 1 12. i After the lattice velocity eei and the distribution function fie (i = 0, 12) at position X(x, y, z) and time t for the electric fluid flow are defined. Obtained are similar pressure and velocity boundary conditions for electric fluid flow field. The extended magnetic density, velocity and equilibrium distribution functions for incompressible and compressible model are defined as

qmin ini uma ¼

i¼0

6 X

fim em ia

ð21Þ

i¼0

fimi

ð10Þ

i¼0

mi mi mi mi fieqincom ðx; tÞ ¼ ami i qin þ ai qin ini

fit

1 where ati ¼ 29 di0 þ 19 dim þ 72 din m ¼ 1 6n ¼ 7 14. The strong couple electromagnetic density, velocity and equilibrium distribution functions for incompressible and compressible model can be defined as

ð3Þ

ðqui Þ;t þ ðqui uj Þ;j ½lðui;j þ uj;i 2eii dij =3Þ pdij ;j qfib ¼ 0

18 X

ð13Þ

i¼0

ð2Þ

The continuity equation, the conservation of momentum and the Navier–Stokes equations can be written as

qmi in ¼

fit etia

i¼0

i¼0

r D ¼ q r E ¼ B_ r B ¼ 0 r H ¼ J þ D_

18 X

14 X

ð1Þ

The Gauss’ law, Faraday’s law and Ampere’s law can be written as

mi qmi in ini ua ¼

qtin ini uta ¼

mi mi 2 mi 2 ! mi ei ua u emi i ua þ a2 c2s 2c4s 2cs

qmin ¼

6 X

fim

m m m fieq m ðx; tÞ ¼ am i qin þ ai qin

ð11Þ 2 !

mi mi 2 mi mi ei ua u emi mi mi mi mi i ua fieqcom ðx; tÞ ¼ ami þ a2 i qin þ ai qin c2s 2c4s 2cs

m m m fieq m ðx; tÞ ¼ am i qin þ ai qin

ð12Þ

ð22Þ

i¼0

" 2 # mi 2 m 9 em 3 um 3em a i ua i ua þ ini c2s 4c4s 2c2s

ð23Þ

" 2 # m 2 m 9 em 3 um 3em a i ua i ua þ c2s 4c4s 2c2s

ð24Þ

1 1 1 where ami i ¼ 6 di0 þ 72 dim þ 36 din m ¼ 1 4n ¼ 5 6.

11


3. Boundary conditions for multiple coupled fields The extended hybrid cubic lattice D3Q27 model is defined by combined D3Q19 (electric–ionic field) model, D3Q19 (force field) model, D3Q15 (thermal field) model, D3Q13 (Maxwell equationstrong electromagnetic coupled field) model, D3Q13 (electric field) model and D3Q7 (magnetic field) model for multiple coupled fields, the extended multiple coupled pressure and velocity condition and the extended distribution functions for all possible case were established, see Fig. 1.

The presentations of the extended lattice velocity ei ðX; tÞ (i = 0 26) at position X(x, y, z) and time t, and the distribution functions for hybrid D3Q27 model are given in Table 1, where clkn (l = 1 27,k = 1 6,n = 1 6) is coupled coefficient matrix (27 6 6). The extended electronic–ionic distribution functions fimi (i = 0 18), flow distribution functions fif (i = 0 18), extended thermal distribution functions fit (i = 0 15), the extended strong couple electromagnetic distribution functions fimw (i = 0 12), the extended electric distribution functions fie (i = 0 12) and the extended magnetic distribution functions fim (i = 0 7) are listed in

Fig. 1. Extended boundary for hybrid cubic Lattice D3Q27 multiple coupled fields.

Table 1 The lattice velocity ei (i=0 26) for the multiple coupled fields. ei

fi

ei1

ei2

ei3

e0

P6

0

0

0

1

0

0

E

1

0

0

W

0

1

0

R

0

1

0

F

0

0

1

N

0

0

1

S

1

1

0

RE

1

1

0

FE

1

1

0

RW

1

1

0

FW

1

0

1

NE

e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15 e16 e17 e18 e19 e20 e21 e22 e23 e24 e25 e26

f 1 ei 1 t 1 mw 1 e 1 m 1 n¼1 ðf0 c 1n þ f0 c 2n þ f0 c 3n þ f0 c 4n þ f0 c 5n þ f0 c 6n Þ P6 f 2 ei 2 t 2 m 2 n¼1 ðf1 c 1n þ f1 c 2n þ f1 c 3n þ f1 c 6n Þ P6 f 3 ei 3 t 3 m 3 n¼1 ðf2 c 1n þ f2 c 2n þ f2 c 3n þ f2 c 6n Þ P6 f 4 ei 4 t 4 m 4 ðf c þ f c þ f c þ f n¼1 3 1n 3 3n 3 c 6n Þ 3 2n P6 f 5 ei 5 t 5 m 5 ðf c þ f c þ f c þ f n¼1 4 1n 4 3n 4 c 6n Þ 4 2n P6 f 6 ei 6 t 6 m 6 n¼1 ðf5 c 1n þ f5 c 2n þ f5 c 3n þ f5 c 6n Þ P6 f 7 ei 7 t 7 m 7 n¼1 ðf6 c 1n þ f6 c 2n þ f6 c 3n þ f6 c 6n Þ P6 f 7 ei 8 mw 7 ðf c þ f c þ f c þ f1e c75n Þ n¼1 7 1n 1 4n 7 2n P6 f 9 ei 9 mw 9 e 9 n¼1 ðf8 c 1n þ f8 c 2n þ f3 c 4n þ f3 c 5n Þ P6 f 10 ei 10 mw 10 e 10 n¼1 ðf9 c 1n þ f9 c 2n þ f4 c 4n þ f4 c 5n Þ P6 f 11 ei 11 mw 11 e 11 n¼1 ðf10 c 1n þ f10 c 2n þ f2 c 4n þ f2 c 5n Þ P6 f 12 ei 12 mw 12 e 12 ðf c þ f c þ f c þ f n¼1 11 1n 5 5 c 5n Þ 4n 11 2n P6 f 13 ei 13 mw 13 e 13 n¼1 ðf12 c 1n þ f12 c 2n þ f8 c 4n þ f8 c 5n Þ P6 f 14 ei 14 mw 14 e 14 n¼1 ðf13 c 1n þ f13 c 2n þ f7 c 4n þ f7 c 5n Þ P6 f 15 ei 15 mw 15 e 15 ðf c þ f c þ f c þ f n¼1 14 1n 6 4n 6 c 5n Þ 14 2n P6 f 16 ei 16 mw 16 e 16 ðf c þ f c þ f c þ f n¼1 15 1n 9 4n 9 c 5n Þ 15 2n P6 f 17 ei 17 mw 17 e 17 n¼1 ðf16 c 1n þ f16 c 2n þ f11 c 4n þ f11 c 5n Þ P6 f 18 ei 18 mw 18 e 18 n¼1 ðf17 c 1n þ f17 c 2n þ f12 c 4n þ f12 c 5n Þ P6 t 20 f c n¼1 7 3n P6 t 20 n¼1 f7 c 3n P6 t 21 n¼1 f8 c 3n P6 t 22 f n¼1 9 c 3n P6 t 23 n¼1 f10 c 3n P6 t 24 f n¼1 11 c 3n P6 t 25 f n¼1 12 c 3n P6 t 26 n¼1 f13 c 3n P6 t 27 f n¼1 14 c 3n

C

1

0

1

NW

1

0

1

SE

1

0

1

SW

0

1

1

NR

0

1

1

SR

0

1

1

NF

0

1

1

SF

1

1

1

RNE

1

1

1

FSW

1

1

1

RSE

1

1

1

FNW

1

1

1

FNE

1

1

1

RSW

1

1

1

FSE

1

1

1

RNW

12


Appendix A. More further information about clkn can be found in Ref. [36].

p ffr:9 ¼

f f mi qmi in ufr:y þ qin ufr:y

þ

6

p ¼ f8 þ ffr:19

p:t p:t p:t qp:t in ðufr:x þ ufr:y þ ufr:z Þ

ð28Þ

12

mw mw m m m 4f 8 þ 4f 3 f1 þ f2 f11 þ f12 f13 þ f14 þ 3qmw in ðufr:x þ ufr:y Þ þ 3qin ðufr:x þ ufr:y Þ

ð29Þ

4

3.1. Front-rear flow

p ffr:3 ¼

f f mi mw mw m m qmi in ufr:y þ qin ufr:y þ qin ufr:y þ qin ufr:y þ 7f 4 f0 f1 f5 f2 f6

3

As shown in Fig. 1a, when the electronic–ionic flow direction is from front to rear, after streaming, the unknown distribution functions are ffr:i (i = 26, 15, 19, 9, 3, 7, 24, 16, 21), on the contrary, after

p ffr:7 ¼

p ffr:15

f f mi qmi in ufr:y þ qin ufr:y

þ

6

¼

p ¼ ffr:16

p:mw p:e p:e p:e 3qp:mw ðup:mw in fr:x þ ufr:y Þ þ 3qin ðufr:x þ ufr:y Þ 4f 10 4f 2 þ f1 f2 þ f11 f12 þ f13 f14

p:mi p:f p:f qp:mi in ufr:y þ qin ufr:y

6 f f mi qmi in ufr:y þ qin ufr:y

6

ð30Þ

þ

4 p:mw p:m p:m þ 4f 18 þ 4f 10 f5 þ f6 f11 f12 þ f13 þ f14 3qp:mw ðup:mw up:m in fr:y þ ufr:z Þ þ 3qin fr:y þ ufr:z

þ

4

p:mw p:e p:e p:e 3qp:mw ðup:mw in fr:y ufr:z Þ þ 3qin ðufr:y ufr:z Þ þ 4f 17 þ 4f 12 f5 þ f6 f11 f12 þ f13 þ f14

4

ð31Þ

ð32Þ

ð33Þ

streaming, the unknown distribution functions are frf:i (i = 22, 17, 23, 10, 4, 8, 20, 18, 25).

For the rear inlet and front outlet case, frfp :i (i = 22, 17, 23, 10, 4, 8, 20, 18, 25) can be defined as

3.1.1. PC (pressure condition) p (i = 26, 15, 19, 9, 3, For the front inlet and rear outlet case, the ffr:i 7, 24, 16, 21) can be defined as

frfp :22 ¼ f9

p ffr:26 ¼ f13

p ffr:21 ¼ f10 þ

p ffr:24 ¼ f11 þ

frfp :17 ¼ frfp :4 ¼

frfp :8 ¼

q

p:t p:t in ðufr:x

up:t fr:y

up:t fr:z Þ

q

p:t t in ðufr:x

þ

up:t fr:y

up:t fr:z Þ

12

q

t t in ðufr:x

þ

utfr:y

utfr:z Þ

6

þ

p:t p:t p:t qp:t in ðurf :x þ urf :y urf :z Þ

12 p:t p:t p:t qp:t in ðurf :x þ urf :y urf :z Þ

12

frfp :20

ð27Þ

12

p:mi p:f p:f qp:mi in urf :y qin urf :y

frfp :23 ¼ f12

ð26Þ

ð34Þ

12

frfp :25 ¼ f14

ð25Þ

12

p:t p:t qtin ðup:t rf :x þ urf :y urf :z Þ

¼ f7

p:t p:t p:t qp:t in urf :x þ urf :y þ urf :z

12

p:mw p:e p:e p:e 3qp:mw ðup:mw in rf :y urf :z Þ þ 3qin urf :y urf :z 3f 11 þ 4f 16 þ f5 f6 þ f12 f13 f14 4

p:mw p:e p:e 4f 9 þ 4f 4 þ f1 f2 þ f11 f12 þ f13 f14 3qp:mw up:mw 3qp:e in in urf :x þ urf :y rf :x þ urf :y 4

4

ð37Þ

ð38Þ

ð39Þ

3

p:mw p:e p:e p:e 4f 15 þ 4f 9 f5 þ f6 f11 f12 þ f13 þ f14 þ 3qp:mw ðup:mw in rf :y þ urf :z Þ 3qin ðurf :y þ urf :z Þ

ð36Þ

p;mi p:f p;f p:t p;t p:m p:m 7f 3 f0 f1 f5 f2 f6 þ qmi in urf :y þ qin urf :y þ qin urf :y þ qin uu:y

frfp :18 ¼

ð35Þ

p:mi p:f p:f qp:mi in urf :y þ qin urf :y

6

p:mi p:f p:f qp:mi in urf :y qin urf :y

6

ð40Þ

ð41Þ

13


3.1.2. VC (velocity condition) v (i = 26, 15, 19, 9, 3, 7, For the front inlet and rear outlet case, ffr:i 24, 16, 21), can be defined as

fv

fr:26

fv

fr:24

¼ f13 þ

¼ f11 þ

v ¼f þ ffr:21 10

fv

fr:19

¼ f8 þ

v ¼ ffr:15

v ¼ ffr:9

v ¼ ffr:3

qvin:t ðutfr:x þ utfr:y þ utfr:z Þ

ð42Þ

12

qvin:t ðutfr:y utfr:x utfr:z Þ

frfv :25 ¼ f14 fv

rf :17

ð43Þ

12 v :t þ uv :t uv :t Þ qvin:t ðufr:x fr:y fr:z

qvin:t ðuvfr:x:t þ uvfr:y:t þ uvfr:z:t Þ

ð45Þ

12

f6 4f 18 4f 10 f5 f11 f12 þ f13 þ f14 2 v :mw v :mw v :mw :e :e :e 3qfr:in ufr:y þ ufr:z uvfr:y þ uvfr:z þ 3qvfr:in þ 4

6

:f :mi v :mi qvfr:in uy þ qvfr:in uvy :f

qv :mi

v :f

qfr:in fr:in þ :f :mi qvfr:in 3 qvfr:in 3

v :mi qfr:in

:f qvfr:in

frfv :4 ¼

!

þ :f :mi 6 qvfr:in 3 qvfr:in 3 4f 8 þ 4f 3 þ f2 f1 f11 þ f12 f13 þ f14 2 v :mw v :mw v :mw :e :e :e 3qfr:in ufr:x þ ufr:y uvfr:x þ uvfr:y þ 3qvfr:in þ 4 þ

ð46Þ

:mi v :mi qvfr:in uy þ qfr:in uy

v :mi qfr:in

qfr:in

ffr:16 ¼

v :f v :f :mi v :mi qvfr:in uy þ qfr:in uy

6

þ

þ

fv

rf :8

¼

ð47Þ

þ :f :mi qvfr:in 3 qvfr:in 3

:mi qvfr:in

:f qvfr:in

qvin:t ðurfv :t:x þ uvrf:t:y uvrf:t:z Þ

ð49Þ

frfv :20 ¼ f7

qvin:t ðuvrf:t:x þ uvrf:t:y þ uvrf:t:z Þ

rf :23

¼ f12

qvin:t uvrf:t:x þ uvrf:t:y uvrf:t:z 12

6 v :mw þ 3qv :e uv :e þ uv :e 3qvrf:mw uvrf:mw :in :x þ urf :y rf :in rf :x rf :y 4

ð56Þ

qvrf:f:in

!

v :f v :f mi qvrf:mi :in uy þ qrf :in uy

6

v :mw 3qvrf:mw uvrf:mw :in :x þ urf :y

þ 3qvrf:e:in urfv :e:x þ urfv :e:y

4

qvrf:mi :in

¼

þ

qvrf:f:in

ð58Þ

!

qvrf:mi qvrf:f:in 3 :in 3

2f 15 þ 2f 9 þ f6 f5 f11 f12 þ f13 þ f14 2 v :mi þ qv :f uv :f qvrf:mi :in uy rf :in y

6 v :mw þ 3qv :e uv :e þ uv :e 3qvrf:mw uvrf:mw :in :y þ urf :z rf :in rf :y rf :z 4

ð59Þ

3.2. South-north flow

ð50Þ

ð52Þ

12

v :e v :e qvin:mi uvrf:mi :y þ qin urf :y

þ v :e qrf :in 3 2f 9 f2 þ 2f 4 þ f1 þ f11 f12 þ f13 f14 2

ð51Þ

12

!

qvrf:mi :in 3

For the rear inlet and front outlet case, frfv :i (i = 22, 17, 23, 10, 4, 8, 20, 18, 25) can be defined as

frfv :22 ¼ f9v

ð55Þ

2f 7 þ f1 þ f2 f11 þ f12 f13 þ f14 2

qvrf:mi :in

rf :18

þ :f :mi qvfr:in 3 qvfr:in 3

4

4

3

fv

!

4f 17 þ 4f 12 þ f6 f5 f11 f12 þ f13 þ f14 2 :mw :mw :mw :e :e :e þ 3qvfr:in 3qvfr:in uvfr:y uvfr:z uvfr:y uvfr:z

qvrf:mi qvrf:f:in 3 :in 3

2f 16 þ f11 þ f6 f5 f12 þ f13 þ f14 2 v :mw v :mw v þ 3qvrf:e:in uvrf:e:y uvrf:e:z 3qrf :in urf :y urf:mw :z

!

6 f2 4f 10 4f 2 f1 f11 þ f12 f13 þ f14 2 v :mw v :mw v :mw :e :e :e 3qfr:in ufr:x þ ufr:y ðuvfr:x þ uvfr:y Þ þ 3qvfr:in þ 4

þ

!

v :mi qv :e uv :e qv :t uv :t qv :m uv :m 11f 3 f0 f1 f5 f2 f6 qvrf:mi in rf :y in :in urf rf :in rf rf :y

3

v :f

qvrf:f:in

ð57Þ

:mi v :mi :e v :e :t :m qvfr:in uy þ qvfr:in uy þ qvin:t uvfr:y þ qvin:m uvfr:y þ 11f 4 þ f1 þ f5 þ f2 þ f6

v :f

qvrf:mi :in

qvrf:mi qvrf:e:in :in þ v :mi 2ðqrf :in 3Þ 2ðqvrf:e:in 3Þ

frfv :10 ¼

!

v :f v :f :mi v :mi qvfr:in uy þ qfr:in uy

v :f

fv

6

ð44Þ

12

ð54Þ

12

v :mi þ qv :f uv :f qvrf:mi :in uy rf :in y

ð48Þ v ¼ ffr:7

¼

qvin:t uvrf:t:x þ uvrf:t:y uvrf:t:z

As shown in Fig. 1b, when the extended fluid flow direction is from south to north, after streaming, the unknown distribution functions are fsn:i (i = 26, 12, 22, 15, 5, 17, 19, 11, 23), on the contrary, the unknown distribution functions are fns:i (i = 20, 14, 24, 18, 6, 16, 25, 13, 21). 3.2.1. PC p (i = 26, 12, 22, 15, 5, For the south inlet and north outlet case, fsn:i 17, 19, 11, 23) can be defined as p t fsn:26 ¼ f13

ð53Þ

p ¼ f8 fsn:19

p:t p:t qtin up:t sn:x þ usn:y þ usn:z

12

p:t p:t p:t qp:t in usn:x usn:y þ usn:z

12

ð60Þ

ð61Þ

14


p fsn:26 ¼ f12

p ¼ f9 fsn:22

p fsn:12 ¼

p ¼ fsn:15

p ¼ fsn:5

p:t

p:t p:t qin up:t sn:x usn:y þ usn:z

ð62Þ

12

p:t p:t p:t qp:t in ðuns:x uns:y þ uns:z Þ

ð63Þ

12

p:mi p:f p:f qp:mi in usn:z þ qin usn:z

6

þ

3

ð66Þ

p:mi p:f p:f qp:mi in usn:z þ qin usn:z

6 p:mi p:f p:f qp:mi in usn:z þ qin usn:z

6

þ

p fns:20 ¼ f7 þ

þ

p ¼ f11 þ fns:24

p ¼ f14 þ fns:25

p ¼ fns:6

p:t

p:t p:t qin up:t ns:x uns:y þ uns:z

ð69Þ

v fsn:22 ¼ f9

p ¼ fns:13

ð79Þ

12 :t :t :t qvin:t ðuvsn:x uvsn:y þ uvsn:z Þ

ð80Þ

12

ð67Þ

p:t p:t p:t qp:t in uns:x uns:y þ uns:z

fv

sn:12

ð70Þ ð71Þ

12 p:t p:t qin up:t ns:x þ uns:y þ uns:z

¼

ð68Þ

:t :t :t qinv :t ðuvsn:x uvsn:y þ uvsn:z Þ

ð81Þ

12

v :f v :f :mi mi qvsn:in uz þ qsn:in uz

:mi qvsn:in

:f qvsn:in

!

þ v :f qsn:in 3 f1 f2 f7 þ f8 f9 f10 2f 13 2 :mw v :mw þ uv :mw þ 3qv :e uv :e þ uv :e 3qvsn:in usn:y sn:z sn:y sn:z sn:in þ 4 6

:mi qvsn:in 3

ð82Þ

ð72Þ

12

p:mw p:e p:e p:f p:f 3qp:mw uns:y þ up:mw þ 3qp:e qp:mi up:mi ns:z ns:z þ qin uns:z in in uns:y þ uns:z f1 þ f2 f7 f8 þ f9 þ f10 þ 2f 11 þ 2f 5 in 4 6

ð73Þ

e:mw p:mw p:f p:f e:mw 2f 15 þ 2f 10 f3 þ f4 f7 þ f8 f9 þ f10 þ 3qp:mw ðup:mw uns:x þ ue:mw qp:mi up:mi ns:x þ uns:z Þ þ 3qin ns:z þ qin uns:z ns:z in in 6 4

ð74Þ

p:t p:m p:m p:mi p:mi p:f p:f qp:t in uns:z þ qin uns:z qin uns:z qin uns:z 11f 5 f0 f4 f3 f1 f2

p ¼ fns:16

ð65Þ

:t :t :t qvin ðuvsn:x uvsn:y þ uvsn:z Þ

v fsn:23 ¼ f12

12

p:t

v ¼ f8 fsn:19

12 p:t

ð64Þ

p:mw p:e p:e mi þ 3qp:e 4f 14 þ 4f 6 f1mi þ f2mi f7mi f8mi þ f9mi þ f10 þ 3qp:mw usn:y þ up:mw sn:z in in usn:y þ usn:z 4

p:t p:t qin up:t ns:x uns:y þ uns:z

p ¼ f10 þ fns:21

p ¼ fns:18

ð78Þ

12

p:mw p:e p:e p:e 4f 16 þ 4f 11 f3 þ f4 f7 þ f8 f9 þ f10 þ 3qp:mw ðup:mw sn:x þ usn:z Þ þ 3qin ðusn:x þ usn:z Þ in 4

p For the north inlet and south outlet case, fns:i (i = 20, 14, 24, 18, 6, 16, 25, 13, 21) can be defined as

p fns:14 ¼

:t :t :t qinv :t ðuvsn:x þ uvsn:y þ uvsn:z Þ

p:mw p:mw p:e p:e p:e mi 4f 18 þ 4f 10 þ f3mi f4mi þ f7mi f8mi þ f9mi f10 þ 3qmw in ðusn:x þ usn:z Þ þ 3qin ðusn:x þ usn:z Þ 4

p:e 3qp:e up:e sn:y þ usn:z þ in 4

p fsn:11 ¼

v fsn:26 ¼ f13

p:mw p:e p:e p:e p:f p:f 3qp:mw ðup:mw qp:mi up:mi sn:y þ usn:z Þ þ 3qin ðusn:y þ usn:z Þ f8 þ 4f 13 þ 3f 7 f1 þ f2 þ f9 þ f10 sn:z þ qin usn:z in þ in 4 6

p:mi p:e p:e p:m p:m qp:mi in usn:z þ qin usn:z þ qin usn:z þ 7f 6 f0 f4 f3 f2 f1

p fsn:17 ¼

3.2.2. VC v (i = 26, 12, 22, 15, 5, For the south inlet and north outlet case, fsn:i 17, 19, 11, 23) are defined as

3 p:mw p:e p:e p:e p:f p:f 2f 17 þ 2f 12 þ f3 f4 þ f7 f8 þ f9 f10 3qp:mw ðup:mw qp:mi up:mi ns:x þ uns:z Þ 3qin ðuns:x þ uns:z Þ sn:z þ qin usn:z in in 4 6

p:mw p:e p:e p:f p:f 3qp:e 4f 12 þ 5f 8 þ f1 f2 þ f7 f9 f10 3qp:mw uns:y þ up:mw qp:mi up:mi ns:z ns:z þ qin uns:z in in uns:y þ uns:z in 4 6

ð75Þ

ð76Þ

ð77Þ

15


v fsn:15 ¼

v ¼ fsn:5

:f :mi mi qvsn:in uz þ qvsn:in ufz

:f :mi qvsn:in qvsn:in þ :mi :f 2ðqvsn:in 3Þ 2ðqvsn:in 3Þ

þ

!

6 f3 f4 þ f7 f8 þ f9 þ f10 þ 2f 18 2 v :mw v :e :mw :mw :e :e 3qvsn:in usn:x þ uvsn:z usn:x þ uvsn:z þ 3qvsn:in þ 4

v ¼ fns:18

:f :mi v :mi qvns:in uz þ qvns:in uvz :f

! f7 2f 15 2f 10 f3 f4 þ f8 þ f9 f10 2 ns:in ns:in v :e v :mw v :mw v :mw v :e v :e 3q ðu þ uns:x Þ þ 3qns:in uns:z þ uns:x þ ns:in ns:z ð92Þ 4

ð83Þ

:f :mi v :mi :t :m qvsn:in uz þ qvsn:in uvz :f qvin:t uvsn:z þ qvin:m uvsn:z þ 11f 6 þ f0 þ f4 þ f3 þ f2 þ f1

6

v ¼ fns:6

3

qv :f

:mi qvns:in

ns:in þ qv :mi 3 qv :f 3

:f :mi v :mi :t m:t f0 þ f4 þ f3 þ f1 þ f2 7f 5 qvns:in uz qvns:in uvz :f qvin:t uvns:z qm:t in uns:z 3

ð93Þ

ð84Þ :f :mi v :mi qvsn:in uz þ qvsn:in uvz :f

:mi qvsn:in

:f qvsn:in

!

fv

¼

v fsn:11

f3 f4 þ f7 f8 þ f9 f10 2f 16 2f 11 2 v :mw v :e v :mw v :mw :e v :e 3qsn:in usn:x þ usn:z þ 3qvsn:in usn:x þ usn:z þ 4 ! :f :f :mi v :mi :mi qvsn:in uz þ qvsn:in uvz :f qvsn:in qvsn:in :f ¼ :mi 6 qvsn:in 3 qvsn:in 3 f1 f2 þ f7 þ f8 f9 f10 2f 14 2f 6 2 :mw :mw :mw :e :e :e 3qvsn;in uvsn:y þ uvsn:z uvsn:y þ uvsn:z þ 3qvsn;in þ 4

sn:17

6

þ

v fns:16 ¼

:mi :f qvsn:in 3 qvsn:in 3

ð85Þ

ns:20

¼ f7 þ

v :t uv :t þ uv :t qvin:t uns:x ns:y ns:z

ns:21

¼ f10 þ

:t :t :t qvin:t uvns:x uvns:y þ uvns:z

ns:24

¼ f11 þ

qvin:t utns:x utns:y þ utns:z

v fns:14 ¼

ð88Þ ð89Þ

12

v fns:25 ¼ f14 þ

ð87Þ

12

fv

qvin:t utns:x þ utns:y þ utns:z

¼ fwe:11

p:mi p;f p:f qp:mi in uwe:x þ qin uwe:x

p ¼ fwe:8

p ¼ fwe:1

6

þ

6 :mw :mw :mw :e v :e uv :e 3qvns:in uvns:z uvns:y uns:z þ 3qvns:in ns:y 4

þ

ð95Þ

As shown in Fig. 1c, when the extended fluid flow direction is from west to east, after streaming, the unknown distribution functions are fwe:i (i = 23, 11, 19, 8, 1, 7, 25, 13, 21), on the contrary, the unknown distribution functions are few:i (i = 22, 12, 26, 10, 2, 9, 20, 14, 24). 3.3.1. PC p (i = 23, 11, 19, 8, 1, 7, For the west inlet and east outlet case, fwe:i 25, 13, 21) as defined as

qv :f

:mi qvns:in

!

p ¼ f10 þ fwe:21

p ¼ f14 þ fwe:25

ð91Þ

p ¼ f8 þ fwe:19

p:t p:t p:t qp:t in uwe:x þ uwe:y uwe:z

p:t p:t p:t qp:t in uwe:x uwe:y uwe:z

3

ð96Þ ð97Þ

12

p:t p:t p:t qp:t in uwe:x uwe:y þ uwe:z

12

p:t p:t p:t qp:t in uwe:x uwe:y uwe:z

ð98Þ

12

p:mw p:e p:e p:e 4f 9 þ 5f 4 f3 f15 f16 þ f17 þ f18 þ 3qp:mw ðup:mw we:x þ uwe:z Þ þ 3qin ðuwe:x þ uwe:z Þ in 4

p:mi p:f p:f p:t p:t p:m p:m qp:mi in uwe:x þ qin uwe:x þ qin uwe:x þ qin uwe:x f0 f4 f3 f5 f6 þ 7f 2

12

p:mw p:e p:e þ 3qp:e 4f 14 þ 3f 6 þ f5 þ f15 f16 þ f17 f18 þ 3qp:mw uwe:x þ up:mw we:y in in uwe:x þ uwe:y þ 4

p:mi p:f p:f qp:mi in uwe:x þ qin uwe:x

6

v :f v :f :mi v :mi qvns:in uz þ qns:in uz

ð90Þ

þ v :f ns:in :mi 6 qvns:in 3 qns:in 3 f7 2f 11 2f 5 þ f1 f2 þ f8 f9 f10 2 v :mw v :mw v :mw :e :e :e 3qns:in uns:y þ uns:z þ 3qvns:in uvns:y þ uvns:z þ 4

f1 f2 þ f7 þ f8 f9 f10 þ 2f 12 þ 2f 8 2

p fwe:23 ¼ f12 þ

12

:f :mi v :mi qvns:in uz þ qvns:in uvz :f

:f qvns:in

:mi qvns:in

3.3. West-east flow

12

fv

ð94Þ

þ :f :mi qvns:in 3 qvns:in 3

v (i = 20, 14, 24, 18, 6, 16, For the north inlet and south outlet case, fns:i 25, 13, 21) are defined as

fv

þ :f ns:in :mi qvns:in 3 qvns:in 3 2f 17 þ 2f 12 þ f3 f4 þ f7 f8 þ f9 f10 2 v :f v :f :mi v :mi qvns:in uz þ qns:in uz 6 v :e :mw :e :e 3qv :mw uv :mw þ uvns:z uns:x þ uvns:z þ 3qvns:in þ ns:in ns:x 4 !

v ¼ fns:13

ð86Þ

qv :f

:mi qvns:in

!

ð99Þ

ð100Þ

ð101Þ

ð102Þ

16

p fwe:7


¼

p fwe:13


6

¼

p:mw p:e p:e þ 3qp:e 4f 10 þ 4f 8 þ f3 f4 þ f15 þ f16 f17 f18 þ 3qp:mw uwe:x up:mw we:y in in uwe:x uwe:y þ 4

ð103Þ

p:mw p:e 4f 12 þ 4f 8 þ f5 f6 þ f15 f16 þ f17 f18 þ 3qmw uwe:x up:mw þ 3qein up:e we:y we:x uwe:y in : 4

ð104Þ


6

p For the east inlet and west outlet case, few:i (i = 22, 12, 26, 10, 2, 9, 20, 14, 24) are defined as,

p few:22

¼

f9p:t

p ¼ f11 þ few:24

p few:20

p few:26

p:t in

q

p:t

qin

up:t ew:x

up:t ew:x

up:t ew:y

12

up:t ew:y

þ 12

up:t ew:z up:t ew:z

¼ f7 þ

p:t p:t p:t qp:t in uew:x þ uew:y uew:z

¼ f13

p few:12 ¼

p ¼ few:2

þ

p:mi p:f p:f qp:mi in uew:x þ qin uew:x

6

ð106Þ

ð119Þ ð107Þ

v fwe:23 ¼ f12 þ

þ

3

ð120Þ ð108Þ

ð109Þ

p:mw p:e p:e p:e 4f 7 þ 4f 1 f3 þ f4 f15 f16 þ f17 þ f18 3qp:mw ðup:mw ew:x uew:z Þ 3qin ðuew:x uew:z Þ in 4

6


6

ð110Þ

¼ f14 þ

v ¼ f10 þ fwe:21

:t :t v :t þ uv :t qvwe:in uvwe:x uwe:y we:z 12

v fwe:19 ¼ f8 þ

:t :t v :t uv :t qvwe:in uvwe:x uwe:y we:z

we:in

þ þ

ð115Þ

ð116Þ

ew:in þ 3 qv :f 3

ew:in f15 f10 f2 þ f3 f4 þ f16 f17 f18 2 :f :f :mi v :mi qvew:in uew:x þ qvew:in uvew:x

6 v :mw :mw :mw v :e uv :e uv :e þ 3qwe:in 3qvwe:in uwe:x uvwe:z we:x we:z þ 4 !

ð117Þ

þ :mi qvew:in 3 qv :f

ew:in

! f14 þ f5 þ f6 þ f15 f16 þ f17 f18 2 3

:mi qvew:in ew:in :f :mi v :mi qvew:in ux þ qvew:in uvx :f

3

f3 þ 2f 12 þ 2f 8 f4 þ f15 þ f16 f17 f18 2 þ

6

:mw :mw :mw :e v :e uv :e qvwe:in uvwe:x uvwe:z uwe:x þ qvwe:in we:z

4

For the east inlet and west outlet case, f v

ew:i

6 :mw :mw :mw v :e uv :e þ uv :e 3qvwe:in uvwe:x þ uvwe:y þ 3qwe:in we:x we:y 4

14, 24) are defined as,

ð118Þ

:f :mi v :mi qvew:in ux þ qvew:in uvx :f

þ

ð121Þ

:f qvew:in

:mi qvew:in

v ¼ few:13

v :f qew:in

þ 3 qv :f

:mi qvew:in

ð113Þ

!

qv :f

:mi qvew:in

v ¼ few:7

þ

12

:mi qvew:in

¼

:t :t v :t uvwe:x uvwe:y uwe:z

ð114Þ

12

qv :t

ð112Þ

p:mw p:e p:e f3 þ f4 f15 f16 þ f17 þ f18 3qp:mw uew:x þ up:mw 3qp:e ew:y in in uew:x þ uew:y 4

12

ð111Þ

p:mw p:e p:e p:e f3 þ f4 f15 f16 þ f17 þ f18 þ 3qp:mw ðup:mw ew:x þ uew:z Þ þ 3qin ðuew:x þ uew:z Þ in 4

:t :t v :t uv :t qvwe:in uvwe:x þ uwe:y we:z

ew:11

:f :mi v :mi :t :m :m :m qvew:in ux þ qvew:in uvx :f þ qvwe:in uvwe:x þ qvwe:in uvwe:x þ 11f 2 þ f0 þ f4 þ f3 þ f5 þ f6

3.3.2. VC v (i = 23, 11, 19, 8, 1, 7, For the west inlet and east outlet case, fwe:i 25, 13, 21), are defined as

fv

v ¼ few:1

p:mw p:e þ 3qein up:e 4f 13 þ 4f 7 þ f5 f6 þ f15 f16 þ f17 f18 þ 3qmw uew:x up:mw ew:y ew:x uew:y in 4


p ¼ f11 þ f5 few:14

we:25

6 v :mw v :e :mw :mw :e :e 3qvse:in uwe:x þ uvwe:z uwe:x þ uvwe:z þ 3qvse:in þ 4

p:mi p:f p:f p:t p:t p:m 9f 1 þ f0 þ f4 þ f3 þ f5 þ f6 qp:mi in uew:x qin uew:x qin uew:x qin 3

p ¼ f8 þ f3 few:9

fv

ew:in

! 2f 9 f3 f4 f15 f16 þ f17 þ f18 2

:f :mi v :mi qvew:in ux þ qvew:in uvx :f

þ

12

6

p: ¼ few:10

ð105Þ

ew:in þ 3 qv :f 3

p:t p:t p:t qp:t in uew:x uew:y þ uew:z


:mi qvew:in

12

qv :f

:mi qvew:in

v :mi ¼ few:8

v few:26 ¼ f13

:t v :t uv :t þ uv :t qvew:in uew:x ew:y ew:z

12

ð122Þ

(i = 22, 12, 26, 10, 2, 9, 20,

ð123Þ

17


v few:22 ¼ f9

:t :t :t v :t qvew:in uvew:x uvew:y uew:z

v ¼ f11 þ few:24

:t :t :t :t qvew:in uvew:x þ uvew:y uvew:z

fv

ew:12

:t :t :t :t qvew:in uvew:x þ uvew:y uvew:z

v ¼ few:2

Fig. 2. Flow driven pore-crack network model.

q

v :e v :e ew:in uew:x uew:y

ew:14

ð127Þ

1

6

:mw :mw :e :e þ 3qvin:e uvew:x 3qvin:mw uvew:x uvew:z uvew:z

4

v :f

qwe:in we:in v :mi 3 þ v :f qwe:in qwe:in 3

Z

þ1

1

Z

> > :

ð128Þ

!

2f 8 þ f3 þ f4 f15 f16 þ f17 þ f18 2

:f :mi v :mi qvwe:in ux þ qvwe:in uvx :f

> :

þ1

1

4

As shown in Fig. 2, one typical extended 3D flow driven porecrack networks model for various porosity composites is constructed through digitized technology by using the slices which were scan from the high resolution X-ray CT facility, which domain size is 50 50 100 mm3(1434 1434 840lu3). 4.1. HIE-LBM equations for the flow driven pore-crack network problem Using the boundary conditions in Eqs. (25)–(130), and the main-part method given in the Refs. [30,36], the flow driven pore-crack network problem in various composites can be translate in to a series HHIE-LBM equations, which the unknown functions is extended discontinue displacement ratio functions. After the complicated mathematical derivation, the closed form formulation of the HHIE-LBM can be expressed as

1 3 9 6 6 P P > 3 2 2 2 2 ~ ~_ 7 ðf > _ b ðfi!0:26 Þ þ 3r 4 r ;a k33 s2i t 1i u r ðc D s ðd q t Þ u Þ Þ þ ðdab 3r ;a r ;b Þ 3r ;a r ;b 0 i!0:26 a b i 44 0 i B C 7 = 0 i¼1 i¼1 B CdS7ds ds ¼ p_ a @ A 5 > Sþ > ~ ; ~_ b ðfi!0:26 Þ _ b ðfi!0:26 Þ þ K ab u þ r 7 K ab1 þ r 5 K ab2 þ r 3 K ab3 u

2 Z 6 6 4

0

2 0 1 3 9 6 6 P 6 6 P P P > Þ Þ ! 2 1 Z 2 3 m t~ 4 2 # m~ = _ _ ~ u u r r A t q ðf þ r q t ðf þ 3r k r v k q ðf Þ u ;a 3 a ;a 6 i!0:26 C i i i a i!0:26 i i n i!0:26 i i i 6 B 7 n¼3 i¼1 i¼1 i¼1 4 @ AdS5ds ds0 ¼ p_ m þ > ~ R S ; ~_ J ðfi!0:26 Þ _ J ðfi!0:26 Þ þ Sþ K mJ u þ r 7 K mJ1 þ r 5 K mJ2 þ r 3 K mJ3 u

0 1 3 9 6 6 P P > ! ! 2 # 2~ 2 4 6~ > _ u u r ðd 3r r Þ A k t ðf Þ þ 3r k r A v k k q ðf Þ Z ab ;a ;b 3 a ;a i 3b i b i!0:26 i i i 33 i 7 i!0:26 C = 7 > B þ1 6 i¼1 i¼1 6 B C 7 dS7ds ds0 ¼ p_ 7 6 B 6 C 6 > 1 4 Sþ @ P 7 A 5 > ~ Þ P > > ~_ J ðfi!0:26 Þ > > _ J ðfi!0:26 þ K 6J u r K 6J1 þ r 5 K 6J2 þ r 3 K 6J3 u þ ; :

8 > > > þ1 > > > < f ðj2 Þ f ðji Þ ¼ > f > ðj3 Þ > : f ðj4 Þ

ji ¼

for nano case

0 6 j1 6 0:001

for micro case

0:001 6 j2 6 0:01

for meso case

0:01 6 j2 6 0:1

for macro case 0:1 6 j2 6 1

ui /i qc2s ðs 0:5ÞDtL2 pðWin Wout Þr2

½K17 ¼ ½ K 1

K2

ð140Þ

K3

K4

Fig. 5a. Velocity in x direction as function of time steps.

ð141Þ K5

K6

K7

ð142Þ

More information about the coefficient matrix [A]7 can be found in reference [43]. The relationship between the SED criterion that can be applied to fatigue and the Richter magnitude scale in earthquake can be further extended by using the extended volume energy density theory [40–42,44,45]. Note that the SED criterion can simultaneously account for the onset of stable fracture initiation and unstable crack propagation, in addition to fatigue. Moreover, Eq. (139) will always be positive definite while other criteria such as energy release rate and the path independent integral may yield negative results. 5. Numerical solution and discussion Fig. 5b. Velocity in y direction as function of time steps.

The detailed description about pressure and velocity condition on the flow driven pore-crack networks model is shown in Fig. 3, the initial pressure is added to the top (inlet) and bottom (outlet), the initial velocity in z direction is 0.0005785. The pressure and velocity parameters, which are added on the simulate model at the initial time, are shown in the Table 2. 5.1. Compliance of boundary condition and convergence of numerical solution From Fig. 4, we can see that the velocity in x, y and z direction at 30,000 time steps possess a stable value in the domain size, which present the numerical method for multiple 3D flow driven porecrack networks is stable and convincing. Fig. 5 shows the velocity and density (pressure) as function of time step in the core domain, it is shown that the value of velocity and density increase with time step increasing, and increasing gradient is decrease with time step increasing, when time step is

Fig. 5c. Velocity in z direction as function of time steps.


21

Fig. 6c. U3 radiation distributions as function of x in oyz plane.

Fig. 5d. Density as function of time steps.

Table 3 Memory calculation and CPU time. Lx (pix)

Ly (pix)

Lz (pix)

Number frame

Frame rate

CPU

CPU time (s)

Memory used (GB)

CPU load

1434 1434

1434 1434

840 840

10 15

3000 3000

84 84

3181 8470

102.26 102.26

0.9 0.9

fluid velocity in x, y and z direction are 0, 0 and 0.0025 cm/s, respectively. The time steps is 80,000, the inlet is minus z direction, the fluid velocities is Ui(x,y,z,t), and the radiation distribution for pressure condition in the core area as a function of x, y and z are shown in Figs. 6–8. Figs. 9, 10 and 11 show that the variation of flow in x, y and z direction with the position on the core area, the numerical solution agrees well with the analytic solution. From the results of Figs. 6–11, we can obtain that the changing rule of velocity and flow as function of pressure, initial velocity conditions.

Fig. 7a. U2 radiation distributions as function of z in oxy plane.

Fig. 7b. U2 radiation distributions as function of y in oxz plane.


Fig. 7c. U2 radiation distributions as function of x in oyz plane.

5.3. Compare with nuclear magnetic resonance method

Fig. 6b. U3 radiation distributions as function of y in oxz plane.

In order to further verify the correctness of our numerical method, we compare our numerical results with nuclear magnetic resonance results. A relationship between HHIE-LBM units and SI units is developed by using non-dimensional Reynolds number as a conversion parameters. Nuclear magnetic resonance method is based

22



Fig. 9c. q3 radiation distributions as function of y in oxz plane.

Fig. 10a. q2 radiation distributions as function of z in oxy plane. Fig. 8b. U1 radiation distributions as function of x in oyz plane.

Fig. 10b. q2 radiation distributions as function of x in oyz plane. Fig. 8c. U1 radiation distributions as function of y in oxz plane.

Fig. 9a. q3 radiation distributions as function of z in oxy plane. Fig. 10c. q2 radiation distributions as function of y in oxz plane.

Fig. 9b. q3 radiation distributions as function of x in oyz plane.

Fig. 11a. q1 radiation distributions as function of z in xoy plane.


Fig. 11b. q1 radiation distributions as function of x in oyz plane.

23

Fig. 11c. q1 radiation distributions as function of y in oxz plane.

Fig. 12. Lattice Boltzmann model (Lx = 100, Ly = 100, Lz = 300).

Fig. 13. Nuclear magnetic model (Lx = 100, Ly = 100, Lz = 107).

on real measurements, this relationship allow us to compare velocity magnitude and direction results with those that can occur under real fields conditions for two cases (low case and high case) under compressible and incompressible assumption, respectively.

Figs. 12 and 13 give the detailed domain size and geometric shapes of LBM model and NMR model, respectively. In order to ensure that the domain size and geometric shapes parameter are identical, we subtracted a new domain 100 100 107 (from 69

24


Fig. 14. The contour of U3 as the function of z in xoy plane.

Fig. 15a. The vector of U3 as the function of z in oxy plane.

to 176 in z direction on oxy section) from the LBM model, the more detail information about the NMR model is listed in Appendix A. Fig. 14 presents the contour of velocity between the LBM and NMR model. Figs. 15, 16 and 17 present the vector of velocity between the LBM and NMR mode. From above results, we can obtain

that the vector of velocity in x, y and z direction through different numerical model (LBM and NMR) has the same result. Fig. 18 presents the direction of velocity in x, y and z direction between the LBM and NMR model for incompressible and compressible condition under differential initial pressure and velocity


Fig. 15b. The contour of U3 as the function of x in oyz plane.

Fig. 15c. The contour of U3 as the function of y in oxz plane.


25

26


Fig. 16b. The vector of U2 as the function of x in oyz plane.

Fig. 16c. The vector of U2 as the function of y in oxz plane.



27

Fig. 17b. The vector of U1 as the function of x in oyz plane.

Fig. 17c. The vector of U1 as the function of y in oxz plane.

value (case I refers to high initial condition and case II refers to low initial condition). Fig. 19 presents the magnitude of velocity in x, y and z direction between the LBM and NMR model for incompressible and compressible condition under differential initial pressure and velocity value. Fig. 20 presents the error analysis between incompressible and compressible condition under different initial pressure and velocity value (case I means high initial condition and case II means low initial condition). From Figs. 18 and 19, we can see that the vector of velocity in x, y and z direction through different numerical model (LBM and NMR) has the same result. It is shown in Fig. 20, when we use incompressible distribution function to simulate the fluid flow driven pore-crack problem for porosity composites, the error is less than ±1.0E-3, the HHIE-LBM numerical method is proved correctness and reliability. 5.3.1. Intrinsic permeability and reynolds Based on the Darcy’s law, the intrinsic permeability in LBM model and physical model can be defined as following

qi qc2s ðs 0:5ÞDtL ui /i qc2s ðs 0:5ÞDtL ¼ DW pðWin Wout Þr2 2 ui /i qc2s ðs 0:5ÞDtL Lphysical physical ¼ pðWin Wout Þr2 LLBM

ji ¼ ji

ð143Þ

The more detailed description is listed in the Appendix A. Take the parameters in Table 4 into Eq. (143), we can obtain the value intrinsic permeability in the core model. Figs. 21 and 22 show the LBM and physical and intrinsic permeability as function of x, y and z coordinate, respectively.

5.4. Extended stress intensity and special area The numerical results in Fig. 23 shows that extended dimensionless extended stress intensity factors varying with x, y and z in the core area, which provided the most dangers position in the whole core area. When the extended stress intensity factor reach the criterion value the area will reach the destroy limit, the porecrack networks propagation begins. The relatively specific danger areas are shown in the Fig. 24.

28


Fig. 18a. The vector of U1 as the function of x in oyz plane for case I.

Fig. 18b. The vector of Ui as the function of x in oyz plane for case II.

The relationship between the DSIFs and various porosity in Figs. 23 and 24 can be obtained. The danger area located at the lowest various porosity areas, when the porosity is fixed, with the extended initial pressure and velocity increasing, the extended pore-crack stresses increase and reach the maximum; when the ex-

tended initial pressure and velocity are fixed, with the porosity decreasing, the extended pore-crack stresses increase and reach the maximum when porosity is 0.45. The results can help explain the experience results of fluid flow varying mechanism on coseismal slip in references [1,2].


29

Fig. 19a. The contour of Ui as the function of x in oyz plane for case I.

Fig. 19b. The contour of Ui as the function of x in oyz plane for case II.

6. Concluding remarks and future work In the present article, a 3D fluid flow driven pore-crack network propagation mechanism in various porosity composites under fully coupled hybrid electronic–ionic, thermal, magnetic, electric and force fields is investigated by the hybrid hypersingular integral equation-Lattice Boltzmann method (HHIE-LBM). This method has been proposed here for the first time, and the following conclusions can be drawn from our results.

The extended hybrid multiple coupled D3Q27 lattice cubic is created and the extended hybrid electronic–ionic, thermal, electromagnetic (weak and strong coupled cases) and force couple fields pressure and velocity boundary conditions for the HHIE-LBM are established. The HHIE-LBM is proposed by the authors, and based on the method; the extended 3D flow driven pore-crack networks problem in various porosity composites is translated into a set of coupled HHIE-LBM equations, in which the unknown functions are the extended displacement ratio discontinuities.

30


Fig. 20. The error analysis of the LBM model for cases I and II.

Table 4 Parameters for intrinsic permeability. cs

s

Dt

L

q

win

wout

r

Lphysical

LLBM

pffiffiffi 1= 3

1

1s

300lu

1mu/lu3

0.2570 mulu1 s2

0.4097 mulu1 s2

90.5 lu

230 mm

280lu

Fig. 21a. Intrinsic permeability in x direction.

Fig. 21b. Intrinsic permeability in y direction.


Fig. 21c. Intrinsic permeability in z direction.

Fig. 22b. Intrinsic physical permeability in y direction.

Fig. 22a. Intrinsic physical permeability in x direction.

Fig. 22c. Intrinsic physical permeability in z direction.

Fig. 23a. Dimensionless model III SIFs radiation distributions.

31

32


Fig. 23b. Dimensionless model I SIFs radiation distributions.

Fig. 23c. Dimensionless model II SIFs radiation distributions.

Fig. 23d. Dimensionless electric SIFs radiation distributions.


Fig. 23e. Dimensionless magnetic SIFs radiation distributions.

Fig. 23f. Dimensionless thermal SIFs radiation distributions.

Fig. 24a. Critical areas according to model III SIFs.

Fig. 24b. Critical area according to model II SIFs.

33

34


Fig. 24c. Critical area according to magnetic SIFs.

Fig. 24f. Critical areas according to thermal SIFs.

Fig. 24d. Critical area according to electric SIFs.

Last, the extended volume energy density function for determining the combine effect of the EDSIFs is derived, and it can be used to describe the pore-crack network propagation mechanism in various porosity composites (different crack scale) under multiple coupled fields (strong and weak case); it establish the relationship between the fatigue criterion in facture theory and the Richter magnitude scale in geophysics theory, which can be utilized to help understand the extended fluid flow mechanism in various porosity composites and analyze the extended fluid flow varying mechanism on coseismal slip. The future work will focus on using the extended volume energy density function to analyze the facture mechanism of the 3D flow driven pore-crack network problem in various porous composites. This can provide a combined evaluation of the different effects that can influence potential fracture of the system. In particular, the SED criterion can be used to analyze the Three Gorges Dam problem. It can be further explored to compare the results with other fracture criteria connected with piezoelectric materials that have led to negative results [46–48].

Acknowledgements The authors would like to thank the editors for their help and thank the anonymous reviewer’s constructive comments, and this work was supported by a grant from the National Natural Science Foundation of China (No. D0408/409740594) and by a grant from China Postdoctoral Science Foundation (No. 20080430073).

Appendix A A.1. BC for the electronic and ionic field A.1.1. Front-rear flow mi (i = 15,9,16,3,7) PC: For the front inlet and rear outlet case, ffr:i can be defined as,

Fig. 24e. Critical areas according to magnetic SIFs.

The EDSIFs are calculated by using parallel numerical method and the visualization results are calculated. The results are presented toward demonstrating the applicability of the proposed method. The relationship between the EDSIFs and differential porosity is discussed, and several rules have been found.

p:mi p:mi ffr:3 ¼ f4mi þ qp:mi fr:in ufr:y =3;

p:mi p:mi mi mi ffr:7 ¼ f10 þ qmi fr:in ufr:y =6 c fr:1 =4

p:mi p:mi p:mi ffr:9 ¼ f8mi þ qp:mi fr:in ufr:y =6 þ c fr:1 =4;

p:mi p:mi p:mi mi ffr:16 ¼ f17 þ qp:mi fr:in ufr:y =6 þ c fr:2 =4

p:mi p:mi p:mi mi ffr:15 ¼ f18 þ qfr:in ufr:y =6 cmi fr:2 =4

35


For the rear inlet and front outlet case, frfp:mi :i (i = 17,10,18,4,8) can be defined as p:mi p;mi mi frfp:mi :4 ¼ f3 qrf :in urf :y =3;

p:mi mi mi mi frfp:mi :10 ¼ f7 qrf :in urf :y =6 þ cfr:1 =4

p:mi p:mi mi mi frfp:mi :8 ¼ f9 qrf :in urf :y =6 cfr:1 =4;

p:mi p:mi mi mi frfp:mi :18 ¼ f15 qrf :in urf :y =6 þ c fr:2 =4

A.1.2. South-north flow PC: For the south inlet and (i = 5,11,12,15,17) can be defined as p:mi p:mi fsn:5 ¼ f6mi þ qp:mi sn:in usn:z =3;

north

outlet

case,

p:mi fsn:i

p:mi mi p:mi mi fsn:11 ¼ f14 þ qp:mi sn:in usn:z =6 c sn:1 =4

p:mi p:mi mi p:mi fsn:12 ¼ f13 þ qp:mi sn:in usn:z =6 c sn:1 =4; p:mi p:mi mi p:mi fsn:15 ¼ f18 þ qp:mi sn:in usn:z =6 þ c sn:2 =4

p:mi p:mi p:mi mi frfp:mi :17 ¼ f16 qrf :in urf :y =6 c fr:2 =4

p:mi p:mi mi p:mi fsn:17 ¼ f16 þ qp:mi sn:in usn:z =6 c sn:2 =4

where

p:mi For the north inlet and south outlet case, fns:i (i = 6,13,14,16,18) can be defined as

mi mi mi mi mi mi cmi fr:a ¼ ðf14 f1 þ f2 f11 þ f12 f13 Þd1a mi mi mi mi mi d2a þ f6 f5mi f11 f12 þ f13 þ f14

up:mi fr:y

p:mi p:mi fns:6 ¼ f5mi qp:mi ns:in uns:z =3;

mi mi mi mi ¼ 1 2ðf4mi þ f8mi þ f10 þ f17 þ f18 Þ þ f0mi þ f1mi þ f11 þ f5mi

p:m mi mi mi mi mi þ f12 þ f2 þ f14 þ f6 þ f13 =qfr:in

mi mi mi mi mi mi mi mi up:mi rf :z ¼ 1 þ 2ðf3 þ f7 þ f9 þ f15 þ f16 Þ þ f0 þ f1 þ f11

p:mi mi mi mi mi mi mi þ f5 þ f12 þ f2 þ f14 þ f6 þ f13 =qrf :in v :mi (i = 15,9,16,3,7), can VC: For the front inlet and rear outlet case, ffr:i

be defined as v :mi ¼ f mi þ ffr:3 4

:mi v :mi qvfr:in ufr:y

3

v :mi ¼ f mi þ ffr:7 10


6

v :mi cmi qfr:in fr:1 :mi 2ðqvfr:in 3Þ

v :mi cmi qfr:in sn:1 v :mi 3Þ ; 6 2ðqfr:in v :mi v :mi v :mi mi v :mi ¼ f mi þ qfr:in ufr:y qfr:in cfr:2 ffr:15 18 v :mi 3Þ 6 2ðqfr:in v :mi ¼ f mi þ ffr:9 8

v :mi ¼ f mi þ ffr:16 17


;


6

v :mi qvrf:mi :in urf :y

3

6

p:mi p:mi mi p:mi ¼ f17 qmi fns:16 in usn:z =6 þ c sn:2 =4

where mi mi mi mi mi mi þ cp:mi sn a ¼ ðf1 f2 þ f7 þ f8 f9 f10 Þd1a mi ðf3mi f4mi þ f7mi f8mi þ f9mi f10 Þd2a

mi mi mi mi mi mi mi mi up:mi sn:z ¼ 1 2ðf6 þ f13 þ f14 þ f16 þ f18 Þ þ f0 þ f4 þ f3

mi þ f2mi þ f9mi þ f8mi þ f1mi þ f7mi =qp:mi þf10 sn:in mi mi mi mi mi mi mi mi mi up:mi ns:z ¼ 1 2ðf5 þ f11 þ f12 þ f15 þ f17 Þ þ f0 þ f4 þ f3 þ f8

p:mi mi mi mi mi mi þ f1 þ f7 þ f10 þ f2 þ f9 =qns:in

þ

v :mi ¼ f mi þ fsn:5 6

v :mi cmi qfr:in fr:2 v :mi 3Þ 2ðqfr:in

;

:mi ¼ f7mi frfv :10

qvin:mi uvrf:mi :y 6

þ

mi qvrf:mi :in c fr 1 2ðqvrf:mi :in 3Þ

v :mi qvrf:mi :in cfr:2 2ðqrfv :mi :in 3Þ


v :mi ¼ f mi fns:6 5 v :mi ¼ f mi fns:13 12

v :mi ¼ f mi fns:18 15

v :mi ¼ uv :mi þ 2ðf mi þ f mi þ f mi þ f mi þ f mi Þ þ f mi þ f mi þ f mi þ f mi qfr:in fr:y 4 8 10 17 18 0 1 11 5

v :mi ¼ f mi fns:16 17

v :mi mi mi mi mi mi mi mi mi mi qrfv :mi :in ¼ ufr:y þ 2ðf3 þ f7 þ f9 þ f15 þ f16 Þ þ f0 þ f1 þ f11 þ f5

þ

mi f12

þ

f2mi

þ

mi f14

þ

f6mi

þ

mi f13

3 :mi mi qvsn:in uz

;


v :mi umi qsn:in z

6

:mi mi qvsn:in csn 1 v :mi 2ðqsn:in 3Þ

:mi mi qvsn:in uz

6

:mi v :mi qvsn:in csn 2 v :mi 2ðqsn:in 3Þ

v :mi (i = 6,13,14,16,18) can For the north inlet and south outlet case, fns:i be defined as

where

mi mi mi þ f12 þ f2mi þ f14 þ f6mi þ f13

:mi mi qvsn:in uz

:mi v :mi qvsn:in csn 1 ; v :mi 6 2ðqsn:in 3Þ v :mi mi v :mi v :mi v :mi ¼ f mi þ qsn:in uz þ qsn:in csn 2 fsn:15 18 :mi 6 2ðqvsn:in 3Þ


mi qrfv :mi :in cfr:1 :mi frfv :8 ¼ f9mi ; 6 2ðqrfv :mi :in 3Þ v :mi mi qvrf:mi qrfv :mi :in urf :y :in cfr:2 :mi mi ¼ f15 þ frfv :18 6 2ðqrfv :mi :in 3Þ

:mi mi ¼ f16 frfv :17

p:mi mi p:mi mi fns:18 ¼ f15 qp:mi ns:in uns:z =6 c sn:2 =4

v :mi (i = 5,11,12,15,17) VC: For the south inlet and north outlet case, fsn:i can be defined as



p:mi p:mi mi p:mi ¼ f12 qp:mi fns:13 ns:in uns:z =6 þ c sn:1 =4;

þ

For the rear inlet and front outlet case, frfv :i:mi (i = 17,10,18,4,8) can be defined as :mi frfv :4 ¼ f3mi

p:mi p:mi mi p:mi fns:14 ¼ f11 qp:mi ns:in uns:z =6 c sn:1 =4

:mi mi qvns:in uz

3 :mi mi qvns:in uz

6 :mi m qvns:in uz

6 :mi mi qvns:in uz

6

v :mi ¼ f mi ; fns:14 11

:mi mi qvns:in uz

þ

:mi v :mi qvns:in csn 1 v :mi 2ðqns:in 3Þ

:mi mi qvns:in csn 2 ; :mi 2ðqvns:in 3Þ

þ

:mi mi qvns:in cns 2 v :mi 2ðqns:in 3Þ

6

:mi v :mi qvns:in csn 1 ; v :mi 2ðqns:in 3Þ

where :mi mi mi mi mi mi mi mi mi mi qvsn:in ¼ umi z þ 2ðf6 þ f13 þ f14 þ f16 þ f18 Þ þ f0 þ f4 þ f3 þ f10

þ f2mi þ f9mi þ f8mi þ f1mi þ f7mi

36


:mi mi mi mi mi mi mi mi mi mi qvns:in ¼ umi z þ 2ðf5 þ f11 þ f12 þ f15 þ f17 Þ þ f0 þ f4 þ f3 þ f8

þ

f1mi

þ

f7mi

mi f10

þ

þ

f2mi

þ

where

f9mi

:mi mi mi mi mi mi mi mi mi qvew:in ¼ umi x þ 2ðf2 þ f9 þ f10 þ f12 þ f14 Þ þ f0 þ f4 þ f3

A.1.3. West-east flow p:mi (i = 1,7,8,11,13) PC: For the west inlet and east outlet case, fwe:i can be defined as p:mi fwe:1 ¼ f2mi þ

p:mi ¼ f9mi þ fwe:8 p:mi fwe:13

q

mi p:mi in uwe:x

3

q

p:mi mi ¼ f10 þ fwe:7

;

mi p:mi in uwe:x

q

mi p:mi in uwe:x

6

cmi we 1

mi þ þ f p:mi ¼ f14 6 4 we:11 qmi up:mi cmi we:x mi ¼ f12 þ in we 2 6 4

4

mi p:mi in uwe:x

q

6

þ

p:mi qmi in uew:x

p:mi few:9 ¼ f8mi

3

p:mi ¼ f7mi few:10

p:mi qmi in uew:x

þ

6

cmi we 4

p:mi qmi in uew:x

6

cmi we 4

4

a

1

p:mi qmi in uew:x

6

cmi we 4

2

v :mi (i = 1,7,8,11,13) can VC: For the west inlet and east outlet case, fwe:i be defined as

v :mi ¼ f mi þ few:8 9

3

v :mi ¼ f mi þ ; few:7 10

:mi mi qvew:in ux

6


:mi mi qvew:in ux

6

:mi mi qvew:in ux

6

:mi mi qvew:in cwe 1 ; :mi 2ðqvew:in 3Þ

qv :mi cmwe 1 ; þ ew:in 2ðqv :mi 3Þ ew:in

6

:mi mi qvew:in cwe 1 :mi 2ðqvew:in 3Þ

:mi mi qvwe:in ux

6

:mi mi qvwe:in cwe 1 ; :mi 6 2ðqvwe:in 3Þ v :mi mi v :mi mi v :mi ¼ f mi qwe:in ux qwe:in cwe 2 fwe:14 11 :mi 6 2ðqvwe:in 3Þ

v :mi ¼ f mi fwe:12 13

:mi mi qvwe:in ux

6

þ

:mi mi qvwe:in cwe 2 :mi 2ðqvwe:in 3Þ

¼

þ

3

p:t ; ffr:7

¼

f8t

þ

p:t p:t p:t qp:t fr:in ufr:x þ ufr:y þ ufr:z

p:t p:t p:t qp:t fr:in ufr:x þ ufr:y ufr:z

:mi mi qvwe:in cwe 1 :mi 2ðqvwe:in 3Þ

f3t

12

;

12 v :mw 3qvrf:mw uvrf:mw :in :y þ urf :z 4

p:t p:t p:t qp:t fr:in ufr:x þ ufr:y þ ufr:z

;

12

p;t qp:t rf :in urf :y

3

frfp:t:10 ¼ f9t

;

frfp:t:8

¼

f7t

p:t p:t p:t qp:t rf :in urf :x þ urf :y þ urf :z

p:t p:t p:t qp:t rf :in urf :x þ urf :y urf :z

frfp:t:13

¼

t f14

;

12 p:t p:t p:t qp:t rf :in urf :x þ urf :y urf :z

t frfp:t:11 ¼ f12

12

12

p:t p:t p:t qp:t rf :in urf :x þ urf :y urf :z

;

12

where

3qp:t fr:in 2

p:mw ¼ f7mw þ ðf1t f2t Þ fsn:8

p:mw p:mw 3qp:mw sn;in usn:z usn:y 4

t t t þ f2t þ f6t þ f1t þ f0t þ f5t =qp:t ¼ 1 2 f4t þ f8t þ f10 þ f11 þ f13 fr:in

up:t rf :x ¼

v :mi mi v :mi mi v :mi ¼ f mi qwe:in ux ; f v :mi ¼ f mi qwe:in ux þ fwe:2 1 we:10 7

3

frfp:t:4

up:t fr:y

v :mi (i = 2, 9, 10, 12, 14) can For the east inlet and west outlet case, few:i be defined as,

v :mi ¼ f mi fwe:9 8

p:t qp:t fr:in ufr:y

p:t t ¼ f13 þ ffr:14

up:t fr:x ¼

qv :mi cmwe 2 þ ew:in :mi 2ðqvew:in 3Þ

v :mi mi v :mi ¼ f mi þ qew:in ux few:13 12

For the rear inlet and front outlet case, frfp:t:i (i = 4, 8, 10, 11, 13) can be defined as

mi mi ¼ 1 2 f1mi þ f7mi þ f8mi þ f11 þ f13 þ f0mi þ f4mi þ f3mi

mi mi mi mi mi mi mi þ f5 þ f17 þ f15 þ f18 þ f6 þ f16 =qin

:mi mi qvew:in ux

¼

f4t

:mw ¼ f9mw frfv :10

þ mi mi mi mi d1a ¼ f3mi þ f4mi f15 f16 þ f17 þ f18 mi mi mi mi mi mi f5 f6 þ f15 f16 þ f17 f18 d2a


p:t ffr:3

p:t t ¼ f10 þ ffr:9

mi mi mi mi mi mi mi mi mi up:mi we:x ¼ 1 2 f2 þ f9 þ f10 þ f12 þ f14 þ f0 þ f4 þ f3 þ f5

mi mi mi mi mi m þ f17 þ f15 þ f18 þ f6 þ f16 =qin up:mi ew:x

mi mi mi mi þ f15 þ f18 þ f6mi þ f16 þ f5mi þ f17

A.2.1. Front-rear flow p:t (i = 3,7,9,12,14) PC: For the front inlet and rear outlet case, ffr:i can be defined as

;

where

cmi we

A.2. BC for the thermal field

cmi we 2

1

p:mw mw p:mi mi f6mw f7mw þ f8mw Þfew:14 ¼ f11 up:mw rf :z ¼ 2qrf :in ðf5

:mi mi mi mi mi mi mi mi mi qvwe:in ¼ umi x þ 2ðf1 þ f7 þ f8 þ f11 þ f13 Þ þ f0 þ f4 þ f3

cmi we 1

p:mi For the east inlet and west outlet case, few:i (i = 2,9,10,12,14) can be defined as,

p:mi few:2 ¼ f1mi

mi mi mi mi þ f15 þ f18 þ f6mi þ f16 þ f5mi þ f17

3qp:t rf :in 2

ðf2t f1t Þ; up:t rf :z ¼

6 t p:t p:t p:t f6 f5t qp:t rf :in urf :x þ qrf :in urf :y 5

t

p:t t t t t t t t t t up:t rf :y ¼ 1 2 f7 þ f9 þ f12 þ f14 þ f3 þ f2 þ f6 þ f1 þ f0 þ f5 =rf :in v :t (i = 3, 7, 9, 12, 14) can VC: For the front inlet and rear outlet case, ffr:i be defined as p:mw fns:6 ¼ f5mw þ

p:mw p:mw 3qp:mw ns;in uns:y þ uns:z

; 4 mw mw ¼ 2q þ f4 f1 þ f2mw ; qvin:t utfr:x þ utfr:y utfr:z v :t ¼ f t þ ffr:9 10 12 qvin:t utfr:x þ utfr:y utfr:z v :t ¼ f t þ ffr:12 11 12 up:mw sn:z

p:mw sn;in

f3mw

37


v :t ¼ f t þ ffr:14 13

qvin:t utfr:x þ utfr:y þ utfr:z

up:t sn:z

12

v :t (i = 17, 10, 18, 4, 8) can be For the rear inlet and front outlet case, ffr:i defined as :t frfv :4 ¼ f3t

qvin:t utrf :y 3

:t ¼ f9t frfv :10

:t ¼ f7t ; frfv :8

qinv :t utrf :x þ utrf :y þ utrf :z

qvin:t utrf :x þ utrf :y utrf :z

:t t frfv :11 ¼ f12

where

12

up:t ns:x ¼

6 t p:t p:t p:t ðf f4t þ qp:t ns:in usn:y qns:in usn:z Þ 5 3

t

p:t t t t t t t t t t up:t ns:z ¼ 1 2 f14 þ f7 þ f5 þ f10 þ f11 þ f1 þ f0 þ f2 þ f3 þ f4 =qns:in

; v :t (i = 5, 7, 10, 11, 14) VC: For the south inlet and north outlet case, fsn:i can be defined as

12

:t t frfv :13 ¼ f14

t

¼ 1 2 f12 þ f9t þ f6t þ f8t þ f3t þ f1t þ f6t þ f2t þ f3t þ f4t =qp:t sn:in

qvin:t utrf :x þ utrf :y utrf :z 12

qvin:t utrf :x þ utrf :y utrf :z

p:t fsn:5 ¼ f6t

p:t qp:t sn:in usn:z

3

;

p:t ¼ f8t fsn:7

q

p:t sn:in

t

6 f5t f6 qvrf:t:in utrf :y þ qvrf:t:in utrf :x

qvrf:t:in 6 t t :t v :t ut 6 f6 f5 qvfr:in utfr:x þ qfr:in fr:y utfr:z ¼ v :t qfr:in 6

p:t fsn:5 ¼ f6t

q

p:t p:t sn:in usn:z

3

p:t ¼ f9t ; fsn:10

p:t sn:in

q

ns:13

up:t sn:x

up:t sn:y

12

þ

up:t sn:z

:t qvns:in utns:x utns:y þ utns:z

t 3qp:t sn:in f2 f1t ; 2 t 3qp:t ns:in ¼ f1 f2t 2

þ

12

:t qvns:in utns:x utns:y þ utns:z

ns:12

¼

t f11

up:t ns:y

;

12 :t qvns:in utns:x utns:y þ utns:z

þ

12

where

t 3 f ft utsn:y ¼ v :t qsn:in 3 1 2

12

:t :t 6 f3t f4t qvns:in utns:y þ qvns:in utns:z v :t qns:in 6 :t :t 6 f3t f4t qvsn:in utsn:z þ qvsn:in utsn:y ¼ v :t 6 qsn:in

utns:x ¼ utsn:x

6 t p:t p:t p:t ðf f5t qp:t sn:in us:y þ qsn:in usn:z Þ; 5 6

;

A.2.3. West–east flow p:t (i = 1, 7, 9, 11, 13) PC: For the west inlet and east outlet case, fwe:i can be defined as

12

up:t sn:x ¼

12

f v :t

:t qvns:in utns:x þ utns:y þ utns:z

p:t fwe:1 ¼ f2t þ

p:t qp:t we:in uwe:x

3

where

up:t sn:y ¼

;

p:t p:t p:t qp:t ns:in uns:x uns:y þ uns:z

p:t p:t p:t qp:t ns:in uns:x uns:y þ uns:z

3

:t t qvsn:in ¼ utz þ 2 f12 þ f9t þ f6t þ f8t þ f3t þ f1t þ f6t þ f2t þ f3t þ f4t ;

p:t p:t qp:t up:t ns:x þ uns:y þ uns:z p:t t fns:13 ¼ f14 þ ns:in ; 12 p:t p:t qp:t up:t ns:x uns:y þ uns:z p:t fns:8 ¼ f7t þ ns:in 12 p:t t fns:12 ¼ f11 þ

:t qvns:in utns:z

t 3 utns:y ¼ v :t f ft qns:in 3 2 1

12

p:t t ¼ f10 þ ; fns:9

12

p:t p:t p:t qp:t sn:in ðusn:x usn:y þ usn:z Þ

3

:t t t t þ f1t þ f0t þ f2t þ f3t þ f4t ; qvns:in ¼ utz þ 2 f14 þ f7t þ f5t þ f10 þ f11

p:t For the north inlet and south outlet case, fns:i (i = 6, 8, 9, 12, 13) can be defined as

p:t fns:6 ¼ f5t þ

¼

t f14

v :t ¼ f t þ fns:8 7

p:t p:t q usn:x þ up:t sn:y þ usn:z p:t t ¼ f13 fsn:14 ; p:t 12p:t p:t p:t q usn:x usn:y þ usn:z p:t ¼ f8t sn:in fsn:7 12

p:t qp:t ns:in uns:z

v :t ¼ f t þ fns:6 5

f v :t

;

v :t (i = 6, 8, 9, 12, 13) can For the north inlet and south outlet case, fns:i be defined as

v :t ¼ f t þ fns:9 10

p:t sn:in

p:t t ¼ f12 fsn:11

;

A.2.2. South-north flow p:t (i = 5, 7, 10, 11, PC: For the south inlet and north outlet case, fsn:i 14) can be defined as

12 p:t up:t sn:y þ usn:z 12

p:t p:t p:t qp:t sn:in usn:x usn:y þ usn:z

p:t t ¼ f12 fsn:11

t t qrfv :t:in ¼ uty þ 2 f7t þ f9t þ f12 þ f14 þ f3t þ f2t þ f6t þ f1t þ f0t þ f5t ; t 3 utfr:x ¼ v :t f ft qfr:in 3 2 1 utrf :z ¼

up:t sn:x

t 3 ¼ v :t f ft qfr:in 3 1 2

12

p:t p:t p:t qp:t sn:in usn:x þ usn:y þ usn:z

v :t ¼ ut þ 2 f t þ f t þ f t þ f t þ f t þ f t þ f t þ f t þ f t þ f t ; qfr:in y 4 8 10 11 13 2 6 1 0 5

utrf :x

p:t p:t p:t qp:t sn:in usn:x usn:y þ usn:z

p:t t ¼ f13 fsn:14

12

p:t ¼ f9t ; fsn:10

p:t p:t p:t qp:t we:in uwe:x uwe:y þ uwe:z p:t we:in

q

up:t we:x

p:t p:t p:t qp:t we:in uwe:x uwe:y uwe:z

p:t t ¼ f14 þ fwe:13 p:t ¼ f8t þ fwe:7

p:t t ¼ f10 þ ; fwe:9

12 p:t up:t we:y uwe:z 12

12 ;

38


p:t fwe:11

¼

t f12

þ

p:t p:t p:t qp:t we:in uwe:x þ uwe:y uwe:z

12

p:t For the east inlet and west outlet case, few:i (i = 2, 8, 10, 12, 14) can be defined as,

p:t few:2 ¼ f1t

q

p:t few:10 ¼ f9t p:t few:14 p:t few:8

¼

¼

p:t few:12

t f13

f7t

¼

3

12

p:t p:t p:t qp:t ew:in uew:x uew:y þ uew:z

12

p:t p:t p:t qp:t ew:in uew:x þ uew:y uew:z

;

A.3.1. Front-rear flow (i = 2,3,10,12) PC: For the rear inlet and front outlet case, frfp:mw :i can be defined as

12

þ

p:t p:t p:t qp:t ew:in uew:x þ uew:y uew:z

:t uvew:y

A.3. BC for the strong coupled electromagnetic field

f4 f3 2 6 t v :t ut f f6t qvew:tv n:in utew:z qew:in ¼ ew:x ; 5 5 6 :t :t ¼ f6t f5t þ qvwe:in utwe:x qvwe:in utwe:z 5

uew:z ¼

:t uvwe:y

;

p:t p:t p:t qp:t ew:in uew:x uew:y uew:z

þ

t f11

p:t p:t ew:in uew:x

mw frfp:mw :12 ¼ f11

12

where

up:t we:x

t t ¼ 1 2 f7t þ f11 þ f1t þ f9t þ f13 þ f3t þ f0t þ f4t þ f5t þ f6t =qp:t we:in ;

up:t ew:z

t 3qp:t ew:in ¼ f4 f3t 2

up:t ew:x

t 2ðf14

¼1

up:t we:z

t 3qp:t we:in ¼ f3 f4t 2

þ

t f10

þ

f2t

þ

t f12

þ

f8t Þ

þ

f3t

þ

f0t

þ

f4t

þ

f5t

þ

f6t

mw frfp:mw :10 ¼ f9 p:t ew:in ;

=q

3

v :t ¼ f t þ fwe:13 14 v :t ¼ f t þ fwe:7 8

f v :t

we:11

¼

t f12

v :t ¼ f t þ ; fwe:9 10

:t qvwe:in utwe:x utwe:y utwe:z

12

;

12 :t qvwe:in utwe:x þ utwe:y utwe:z

3

v :t ¼ f t ; few:10 9

v :t ¼ f t few:14 13 v :t ¼ f t þ few:8 7

ew:12

¼

t f11

þ

v :mw ¼ f mw þ ffr:4 3

;


12

4 p:mw 3qp:mw up:mw rf :in rf :y urf :x

4

:mw :mw :mw 3qvfr:in uvfr:x þ uvfr:y 4 :mw :mw :mw 3qvfr:in uvfr:y þ uvfr:z

;

4 :mw :mw :mw 3qvfr:in uvfr:x þ uvfr:y 4 v :mw v :mw :mw 3qfr:in ufr:y uvfr:z

f3 f4

;

4

12

:t t t qvwe:in ¼ utwe:x þ 2ðf7t þ f11 þ f1t þ f9t þ f13 Þ þ f3t þ f0t þ f4t þ f5t þ f6t ; :t 3qvwe:in v :t t t

2

For the rear inlet and front outlet case, frfv :i:mw (i = 2, 3, 10, 12) can be defined as

where

uwe:z ¼


:t qvew:in utew:x þ utew:y utew:z

:t qvew:in utew:x þ utew:y utew:z

p:mw 3qp:mw up:mw rf :in rf :y þ urf :z

mw mw mw þ f3mw þ f10 þ f12 Þ þ f8mw þ f5mw þ f0mw up:mw fr:y ¼ 1 2ðf2

p:mw mw mw þ f6 þ f7 =qfr:in


12

12

f v :t

4

v :mw (i = 1, 4, 9, 11) can VC: For the front inlet and rear outlet case, ffr:i be defined as

:t qvew:in utew:x utew:y utew:z

:t qvew:in utew:x utew:y þ utew:z

;

mw up:mw ¼ 2qp:mw þ f6mw f7mw þ f8mw Þ; fr:x fr:in ðf5 mw ¼ 2qp:mw þ f6mw þ f7mw f8mw up:mw fr:z fr:in f5

12

:t qvew:in utew:x

þ

up:mw rf :y

mw mw þ f4mw þ f9mw þ f11 Þ þ f8mw þ f5mw þ f0mw up:mw rf :y ¼ 1 2ðf1

þ f6mw þ f7mw =qp:mw rf :in

v :t (i = 2, 8, 10, 12, 14) can be For the east inlet and west outlet case, few:i defined as,

v :t ¼ f t few:2 1

4 up:mw rf :x

p:mw mw up:mw f6mw þ f7mw f8mw ; rf :x ¼ 2qrf :in f5 p:mw mw f6mw f7mw þ f8mw up:mw rf :z ¼ 2qrf :in f5

12

:t qvwe:in utwe:x utwe:y þ utwe:z

:t qvwe:in ðutwe:x utwe:y utwe:z Þ

þ

where

v :t (i = 1, 7, 9, 11, 13) can VC: For the west inlet and east outlet case, fwe:i be defined as :t qvwe:in utwe:x

3q

p:mw rf :in

¼ f4mw frfp:mw :3

6 t p:t p:t p:t f f6t qp:t ew:in uew:z qew:in uew:x ; 5 5 6 p:t p:t p:t ¼ f6t f5t þ qp:t we:in uwe:x qwe:in uwe:z 5

v :t ¼ f t þ fwe:1 2

frfp:mw ¼ f1mw 2

p:mw 3qp:mw up:mw rf :in rf :y urf :z

up:t ew:y ¼ up:t we:y

:t t t t qvew:in ¼ utew:x þ 2ðf14 þ f10 þ f2t þ f12 þ f8t Þ þ f3t þ f0t þ f4t þ f5t þ f6t ; v :t 3qew:in t v :t t

frfv 2:mw ¼ f1mw :mw ¼ f4mw frfv :3

v :mw v :mw 3qvrf:mw :in ðurf :x þ urf :y Þ

; 4 v :mw 3qvrf:mw uvrf:mw :in :x þ urf :y 4

39


:mw frfv :10 ¼ f9mw

:mw mw ¼ f11 frfv :12

v :mw 3qvrf:mw uvrf:mw :in :y þ urf :z 4 v :mw 3qvrf:mw uvrf:mw :in :y urf :z

v :mw (i = 5, 8, 9, 12) can VC: For the south inlet and north outlet case, fsn:i be defined as

;

v :mw ¼ f mw þ fsn:5 6

4


where :mw uvfr:x ¼ :mw uvfr:z ¼

2 v :mw 2 3qfr:in

f5mw þ f6mw f7mw þ f8mw ;

v :mw fsn:12

v :mw ¼ uv :mw þ 2ðf mw þ f mw þ f mw þ f mw Þ þ f mw þ f mw qfr:in fr:y 2 3 10 12 8 5

uvrf:mw :x ¼ uvrf:mw ¼ :z

2 3qrfv :mw :in 2

f5mw

þ

f7mw

f8mw

2

f5mw f6mw f7mw þ f8mw 3qrfv :mw :in 2

v :mw ¼ f mw fns:11 12

A.3.2. South–north flow p:mw (i = 5, 8, 9, 12) PC: For the south inlet and north outlet case, fsn:i can be defined as p:mw fsn:8

¼

f7mw

þ

p:mw mw ¼ f11 þ fsn:12

p:mw fsn:9

mw ¼ f10 þ

p:mw ¼ f6mw þ fsn:5

p:mw p:mw 3qp:mw sn;in usn:z usn:y 4 p:mw p:mw 3qp:mw sn;in ðusn:x þ usn:z Þ

p:mw fns:11

¼

mw f12

p:mw mw ¼ f10 þ fns:10 p:mw fns:6

¼

f5mw

þ

v :mw ¼ f mw þ fns:6 5

4 p:mw p:mw 3qp:mw sn;in usn:y þ usn:z

4 p:mw p:mw 3qp:mw ns;in ðuns:x þ uns:z Þ

3q

:mw uvns:x

¼ 2q

þ

mw 2 f f4mw þ f1mw f2mw ; :mw 3qvns:in 2 3 mw 2 ¼ f f4mw f1mw þ f2mw :mw 3qvns:in 2 3

;

f2mw

f4mw f1mw þ f2mw

p:mw p:mw 3qp:mw we:in ðuwe:x uwe:z Þ ; 4 p:mw p:mw p:mw 3q ðuwe:x þ uwe:z Þ ¼ f4mw þ we:in 4 p:mw 3qp:mw up:mw we:x uwe:y ¼ f8mw þ we:in ; 4 p:mw p:mw p:mw 3q uwe:x þ uwe:y ¼ f6mw þ we:in 4

p:mw fwe:1 ¼ f2mw þ

mw mw mw þ f6mw þ f10 þ f11 Þ þ f2mw þ f3mw þ f0mw up:mw sn:x ¼ 1 2ðf7

p:mw mw mw þ f4 þ f1 =qsn;in

up:mw ns:x ¼ 2q

mw 2 f3 þ f4mw f1mw þ f2mw ; :mw 3qvsn:in 2 mw 2 ¼ f3 þ f4mw þ f1mw f2mw :mw 3qvsn:in 2

A.3.3. West–east flow p:mw (i = 1,3,5,7) can PC: For the west inlet and east outlet case, fwe:i be defined as

p:mw fwe:7

f1mw

4

mw p:mw up:mw þ f4mw f1mw þ f2mw ; sn:z ¼ 2qsn;in f3 mw p:mw up:mw þ f4mw þ f1mw f2mw sn:x ¼ 2qsn;in f3

f4mw

:mw :mw :mw 3qvns:in uvns:y þ uvns:z

:mw :mw mw qvns:in ¼ uvns:x þ 2ðf8mw þ f12 þ f9mw þ f5mw Þ þ f2mw þ f3mw þ f0mw þ f4mw þ f1mw

;

p:mw fwe:3

:mw :mw 3qns:in uvns:x þ uvns:z ; 4

:mw uvns:z ¼

4

p:mw mw ns;in f3 p:mw mw ns;in f3

4 v :mw

;

where

up:mw ns:z

;

p:mw p:mw 3qp:mw ns;in ðuns:x þ uns:z Þ

4 p:mw uns:y þ up:mw ns:z

4

:mw :mw mw mw qvsn:in ¼ uvsn:x þ 2 f7mw þ f6mw þ f10 þ f11 þ f2mw þ f3mw þ f0mw þ f4mw þ f1mw

4

p:mw ns;in

:mw :mw :mw 3qvns:in uvns:x þ uvns:z

where

up:mw sn:x

4

p:mw p:mw 3qp:mw ns;in uns:z uns:y

:mw :mw :mw 3qvns:in uvns:y þ uvns:z

:mw uvsn:z ¼

;

p:mw For the north inlet and south outlet case, fns:i (i = 6, 7, 10, 11) can be defined as

p:mw fns:7 ¼ f8mw

v :mw ¼ f mw þ fns:10 10

4 p:mw p:mw 3qp:mw sn;in ðusn:z usn:x Þ

:mw :mw 3qsn:in uvsn:x þ uvsn:z ; 4 v :mw v :mw v :mw 3q u þ usn:z mw ¼ f11 þ sn:in sn:x 4

v :mw ¼ f mw fns:7 8

;

v :mw mw mw qvrf:mw þ f4mw þ f9mw þ f11 Þ þ f8mw þ f5mw þ f0mw þ f6mw þ f7mw :in ¼ urf :y þ 2ðf1

4 v :mw

v :mw (i = 6, 7, 10, 11) can For the north inlet and south outlet case, fns:i be defined as

f6mw

; 4 :mw :mw :mw 3qvsn:in uvsn:y þ uvsn:z


mw 2 mw mw mw v :mw 2 f5 þ f6 þ f7 f8 3qfr:in

þ f0mw þ f6mw þ f7mw

:mw :mw :mw 3qvsn;in uvsn:y þ uvsn:z

;

mw mw þ f12 þ f9mw þ f5mw Þ þ f2mw þ f3mw up:mw ns:x ¼ 1 2ðf8

mw mw mw þ f0 þ f4 þ f1 =qp:mw ns;in

p:mw fwe:5

p:mw For the east inlet and west outlet case, few:i (i = 2, 4, 6, 8) can be defined as,

p:mw p:mw 3qp:mw ew:in ðuew:x uew:z Þ ; 4 p:mw p:mw p:mw ðuew:x þ uew:z Þ 3q ¼ f3mw þ ew:in 4

p:mw few:2 ¼ f1mw p:mw fwe:4

40

p:mw few:6 p:mw few:8


p:mw 3qp:mw up:mw ew:x þ uew:y ¼ f5mw ew:in ; 4 p:mw p:mw p:mw 3q uew:x uew:y ¼ f7mw þ ew:in 4

:mw :mw mw mw mw qvwe:in ¼ uvwe:x þ 2ðf2mw þ f4mw þ f6mw þ f8mw Þ þ f9mw þ f12 þ f0mw þ f11 þ f10

p:mw mw we:in f12 mw p:mw we:in f12

¼ 2q ¼ 2q

þ þ

mw f11 mw f11

þ

mw f10 mw f10

þ

f9mw ; f9mw

mw mw up:mw þ f4mw þ f6mw þ f8mw Þ þ f9mw þ f12 we:x ¼ 1 2ðf2

p:mw mw mw mw þ f0 þ f11 þ f10 =qwe:in p:mw mw mw mw mw ; up:mw ew:y ¼ 2qew:in f12 f11 f10 f9 p:mw mw mw mw mw up:mw ¼ 2 q f þ f þ f f ew:z 3 4 1 2 ew:in mw up:mw þ f1mw þ f5mw þ f7mw þ f2mw þ f3mw ew:x ¼ 1 2 f3

þ f0mw þ f4mw þ f1mw =qp:mw ew:in v :mw (i = 1,3,5,7) can be VC: For the west inlet and east outlet case, fwe:i defined as

v :mw fwe:1 v :mw fwe:3

v :mw fwe:5 v :mw fwe:7

v :mw 3qv :mw uv :mw uwe:z ¼ f2mw þ we:in we:x ; 4 v :mw v :mw v :mw 3q u þ uwe:z ¼ f4mw þ se:in we:x 4 v :mw uv :mw þ uv :mw 3qwe:in we:x we:y ¼ f6mw þ ; 4 v :mw uv :mw uv :mw 3qwe:in we:x we:y ¼ f8mw þ 4

v :mi (i = 2, 4, 6, 8) can be deFor the east inlet and west outlet case, few:i fined as,

v :mw few:2

f v :mw ew:4

v :mw few:6 v :mw few:8

v :mw 3qv :mw uv :mw uew:z ¼ f1mw ew:in ew:x ; 4 v :mw 3qv :mw uv :mw þ uew:z ¼ f3mw þ ew:in ew:x 4 v :mw uv :mw þ uv :mw 3qew:in ew:x ew:y mw ; ¼ f5 4 v :mw uv :mw uv :mw 3qew:in ew:x ew:y ¼ f7mw þ 4

where

mw 2 mw mw þ f10 f9mw ; f12 þ f11 :mw 3qvwe:in 2 mw 2 mw mw ¼ f10 þ f9mw f12 þ f11 v :mw 3qwe:in 2

:mw uvwe:y ¼

uv :mw we:z

mw 2 mw mw f10 f9mw ; f f11 :mw 3qvew:in 2 12 mw 2 ¼ f3 þ f4mw þ f1mw f2mw :mw 3qvew:in 2

up:mw ew:y ¼

where

up:mw we:y up:mw we:z

up:mw ew:z

:mw p:mw qvew:in ¼ uew:x þ 2ðf3mw þ f1mw þ f5mw þ f7mw Þ þ f2mw þ f3mw þ f0mw þ f4mw þ f1mw

A.4. BC for magnetic field A.4.1. Front-rear flow p:m can be defined as PC: For the front inlet and rear outlet case, ffr:3

p:m m m m m m m ffr:3 ¼ f4m þ qp:m fr:in 2f4 f0 f1 f5 f2 f6 =3

For the rear inlet and front outlet case, frfp:m :4 can be defined as

p:m m m m m m m m frfp:m :4 ¼ f3 qrf :in þ 2f3 þ f0 þ f1 þ f5 þ f2 þ f6 =3

v :m can be defined as VC: For the front inlet and rear outlet case, ffr:3

v :m ¼ f m þ uv :m um þ 2f m þ f m þ f m þ f m þ f m þ f m =3 ffr:3 4 fr:y y 4 0 1 5 2 6

:m For the rear inlet and front outlet case, frfv :4 can be defined as

:m m m m m m m m frfv :4 ¼ f3m uvrf:m :y uy þ 2f3 þ f0 þ f1 þ f5 þ f2 þ f6 =3

A.4.2. South–north flow p:m can be defined PC: For the south inlet and north outlet case, fsn:5 as

p:m m m m m m m fsn:5 ¼ f6m þ qp:m sn:in 2f6 f0 f4 f3 f2 f1 =3

p:m For the north inlet and south outlet case, fns:6 can be defined as

p:m m m m m m m fns:6 ¼ f5m qp:m ns:in 2f5 f0 f4 f3 f1 f2 =3 v :m can be defined as VC: For the south inlet and north outlet case, fsn:5

v :m ¼ f m þ uv :m um þ 2f m þ f m þ f m þ f m þ f m þ f m =3 fsn:5 6 sn:z z 6 0 4 3 2 1 v :m can be defined as For the north inlet and south outlet case, fns:6

v :m ¼ f m uv :m um þ 2f m þ f m þ f m þ f m þ f m þ f m =3 fns:6 5 ns:z z 5 0 4 3 1 2

A.4.3. West–east flow p:m can be defined as PC: For the west inlet and east outlet case, fwe:1

p:m m m m m m m fwe:1 ¼ f2m þ qp:m we;in 2f2 f0 f4 f3 f5 f6 =3

p:m For the east inlet and west outlet case, few:2 can be defined as

p:m m m m m m m few:2 ¼ f1m qp:m ew;in 2f1 f0 f4 f3 f5 f6 =3

v :m can be defined as VC: For the west inlet and east outlet case, fwe:1

Fig. A.1. The more detail description about the NMR model, equipment and experience illustrative diagram.


v :m fwe:1

m :m ¼ f2m þ uvwe:x ux þ 2f2m þ f0m þ f4m þ f3m þ f5m þ f6m =3

v :m can be defined as For the east inlet and west outlet case, few:2

v :m ¼ f m uv :m um þ 2f m þ f m þ f m þ f m þ f m þ f m =3 few:2 1 ew:x x 1 0 4 3 5 6

q¼

k¼ ¼

k

DW

qc2s ðs 0:5ÞDt L

qqc2s ðs 0:5ÞDtL DW

qqc2s ðs 0:5ÞDtL DW 2:75 104 lus

1

3

1ms=lu ð1=6Þ Dts 243lu 1

0:0034mulu s2 3

4

¼

;k ¼

2:75 10 lu 1ms=lu ð1=6Þ Dts 243lu

Rehigh ¼ Relow ¼

1 0:0034mulu s2

U real Lreal

v real U real Lreal

v real

¼ ¼

2:5 104 m=s 5:0 102 m 1:0 106 m2 =s 1:3 104 m=s 5:0 102 m 1:0 106 m2 =s

2

¼ 3:276lu Dt

ms mu

¼ 12:5; ¼ 6:5

Rev LBM 12:5 ð1=6Þ : ¼ ¼ 0:01157; 180 LLBM Rev LBM 12:5 ð1=6Þ ¼ ¼ ¼ 0:0060164 180 LLBM

U LBM jhigh ¼ U LBM jlow

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