COMPUTATIONAL NUCLEAR THERMAL HYDRAULICS

COMPUTATIONAL NUCLEAR THERMAL HYDRAULICS

Cho, Hyoung Kyu

Department of Nuclear Engineering

Seoul National University

TWO-FLUID MODEL SOLVER-2: SEMI-IMPLICIT SCHEME

2

Contents

 Introduction

 Governing equations

 Semi-implicit scheme (ICE) for two-phase flow

 Conclusion

Introduction

 Semi-implicit scheme or ICE (Implicit Continuous Eulerian)

 Numerical algorithm for many nuclear thermal-hydraulic analysis codes

 RELAP, MARS, SPACE, COBRA, CUPID

 For the coupled scalar equations

 Mass and energy equations cannot be decoupled even for an incompressible flow

 Due to the phage change terms

 Decoupling between the mass and energy equations – Instability may occur

– Phage terms are very sensitive at the temperature variation

Governing equations

 Two-fluid model

 Total 8 equations for 2D flow

 2 continuity equations

 4 momentum equations

 2 energy equations

g g

g g g

g u

t +∇⋅ =Γ

∂

∂ (

α ρ

) (

α ρ

 ) _{l l l l}u_{l l}

t +∇⋅ = Γ

∂

∂ (

α ρ

) (

α ρ

 )

( )

( )

( ) ( ) (17)

g g g t

g g g g g g g g g g g g

u u u P g M

t

α ρ α ρ α α τ α τ α ρ

∂ + ∇ ⋅ = − ∇ + ∇ ⋅ + ∇ ⋅ + +

∂

    

( )

( )

( ) ( ^t) (18)

l l l

l l l l l l l l l l l l

u u u P g M

t

α ρ α ρ α α τ α τ α ρ

∂ + ∇ ⋅ = − ∇ + ∇ ⋅ + ∇ ⋅ + +

∂

    

P t S

u P

E u

e t e

g g

E g

g D

g g

g g g g

g

g ∂

− ∂ +

⋅

∇

−

=

⋅

∇

∂ +

∂

α

α ρ

α ρ

α

) ( ) ( ) _,

(  

P t S

u P

E u

e t e

l l

E l

l D

l l

l l l l

l

l ∂

− ∂ +

⋅

∇

−

=

⋅

∇

∂ +

∂ (

α ρ

) (

α ρ

 ) (

α

 ) _,

α

Governing equations

 Two-fluid model

 Phage changer terms

g g

g g g

g u

t +∇⋅ =Γ

∂

∂ (

α ρ

) (

α ρ

 ) _{l l l l}u_{l l}

t +∇⋅ = Γ

∂

∂ (

α ρ

) (

α ρ

 )

l li

gi

sat l

il sat

g ig

g h h

T T H T

T

H = −Γ

−

− +

= −

Γ ( ) ( )

) (

)

, ( l

sat il li gi

gi g

sat ig li

gi li g

E H T T

h h

T h T

h H h

S h  −











− −

 −













− −

=

P t S

u P

E u

e t e

g g

E g

g D

g g

g g g g

g

g ∂

− ∂ +

⋅

∇

−

=

⋅

∇

∂ +

∂

α

α ρ

α ρ

α

) ( ) ( ) _,

(  

P t S

u P

E u

e t e

l l

E l

l D

l l

l l l l

l

l ∂

− ∂ +

⋅

∇

−

=

⋅

∇

∂ +

∂ (

α ρ

) (

α ρ

 ) (

α

 ) _,

α

Semi-implicit scheme for two-phase flow

 Numerical Procedure of ICE Scheme

 Same with the two-phase SMAC

Solve momentum eqs. with Pⁿ for Intermediate Velocity u*

Solve pressure correction eqs.

for new time step pressure Pⁿ⁺¹

Update new time step velocity uⁿ⁺¹

Update new time step scalar variables

Time

Advancing

Semi-implicit scheme for two-phase flow

 Solution procedure for the momentum equations

 Simplified virtual mass term

 Implicit velocity for the interfacial drag

g M T

g g g g

g g g g g

g

g u u u P S

t (  )+∇⋅(   ) = − ∇ +∇⋅[ ( + )]+  ^,

∂

∂

α ρ α ρ α α τ τ

g wl g

td g

lift g

vm g

drag gi

g

ig u M M M M M

M   _,  _,  _,  _,  _, +

+ +

+ +

Γ

ig =

g g g

M g M

S  

+

=

α ρ

,

g M g

vm g

drag gi

g g

g g g g g

g

g u u u P u M M S

t (  )+∇⋅(   ) = − ∇ +Γ  +  ^, +  ^, + ' ^,

∂

∂

α ρ α ρ α

[

^{( )}

]

^{( )}

) (

) (

) (

) (

g g g g g

g g g g

g g g

g g g g g

g g

g g g g

g g g g

g g

u u u

t u u

u t u

t u u u

u t u







 





 







ρ α ρ

α ρ

α

ρ α ρ

α ρ

α ρ

α ρ

α

⋅

∇ + Γ +

⋅

∇

−

∂ +

= ∂

⋅

∇

∂ + + ∂

∂

= ∂

⋅

∇

∂ +

∂

g g

g g g

g u

t +∇⋅ = Γ

∂

∂ (

α ρ

) (

α ρ

 )

Semi-implicit scheme for two-phase flow

 Solution procedure for the momentum equations

[ ]

g M g

vm g

drag g

g g gi g g

g g g

g g g g g

g g

g M g

vm g

drag gi

g g

g g g g g

g g g g

g g g

g M g

vm g

drag gi

g g

g g g g g

g g

S M

M P u

u u

u u

t u u

S M

M u

P u

u u

t u u

S M

M u

P u

u t u

, ,

,

, ,

, ,

, ,

' )

( )

(

' )

( )

(

' )

( )

(





 









 

 

 







 





 







+ +

+

∇

− Γ

− Γ

=

⋅

∇

−

⋅

∇

∂ +

∂

+ +

+ Γ

+

∇

−

=

⋅

∇ + Γ +

⋅

∇

−

∂ +

∂

+ +

+ Γ

+

∇

−

=

⋅

∇

∂ +

∂

α ρ

α ρ

α ρ

α

α ρ

α ρ

α ρ

α

α ρ

α ρ

α

(

_{EV CD}

)

_{g EV l EV g}

g CD l

EV g

g gi g CD

EV

g = Γ −Γ Γ u −Γ u = Γ u −Γ u − Γ −Γ u =Γ u −Γ u

Γ

(

l g

)

g drag g vm g M g

EV g

g g g

g g g g g

g

g u u u u u u P M M S

t u

, ,

, '

) (

)

(         



+ +

+

∇

−

− Γ

=

⋅

∇

−

⋅

∇

∂ +

∂

α ρ α ρ α

ρ

α

Semi-implicit scheme for two-phase flow

 Solution procedure for the momentum equations

( )

( )

g M

g l gl m l g VM gl l

g gl

g g

l EV g

g g g

g g g g g

g g

S

t u C u

u u F

P u

u u

u u

t u u

,

,

'

) (

) (

) (





 













 

+

∂

− + ∂

−

−

∇

−

− Γ

=

⋅

∇

−

⋅

∇

∂ +

∂

ρ α α α

ρ α ρ

α ρ

α

(

l g

)

g drag g vm g M g

EV g

g g g

g g g g g

g

g u u u u u u P M M S

t u

, ,

, '

) (

)

(         



+ +

+

∇

−

− Γ

=

⋅

∇

−

⋅

∇

∂ +

∂

α ρ α ρ α

ρ α

) (

) (

| 8 |

1

,g i c D g l g l gl g l

drag A C u u u u F u u

M = −

ρ

 −  − = −  − 

( ) ( )

_



 











 + ⋅∇

∂

− ∂







 + ⋅∇

∂

= _{vm g l m} ∂ ^l _{l l} ^g _{g g}

g

vm u u

t u u

t u C u

M   



 

 _,

α α ρ

Semi-implicit scheme for two-phase flow

 Solution procedure for the momentum equations

( )

t V u dV u

t

u _{g g}

g g g

g

g

ρ α ρ δ

α

^ = ^{ − }

∂

∫

∂ ^*

( )

∑ ( )

∫

∫

∫ ∑

∫

⋅

=

⋅

=

⋅

∇

⋅

=

⋅

=

⋅

∇

f g g g f

g S

g g g g g

g g g

f g g g g f

S

g g g g g

g g g

S u u

S d u u

dV u

u

S u u S

d u u dV

u u

 

 









 

 









ρ α ρ

α ρ

α

ρ α ρ

α ρ

α

) (

) (

Finite Volume Discretization for intermediate velocity

( )

( )

g M

g l gl m l g VM gl l

g gl

g g

l EV g

g g g

g g g g g

g g

S

t u C u

u u F

P u

u u

u u

t u u

,

,

'

) (

) (

) (





 













 

+

∂

− + ∂

−

−

∇

−

− Γ

=

⋅

∇

−

⋅

∇

∂ +

∂

ρ α α α

ρ α ρ

α ρ

α

( )

t V u C u

t V u C u

t dV u C u

V u u F dV u u F

V u dV

u V

u dV

u PV

PdV

g l

gl m l g VM gl g

l gl m l g VM gl g

l gl m l g VM gl

l g gl l

g gl

g Ev g

Ev l

Ev l

Ev g

g

ρ δ α δ α

ρ α α ρ

α α

α α

) (

) (

) (

) (

,

*

* ,

,

*

*

*

*





























− −

= −

∂

−

∂

−

=

−

Γ

= Γ

Γ

= Γ

∇

=

∇

∫

∫

∫

∫

∫

Implicit treatment of phase change and friction terms !

Semi-implicit scheme for two-phase flow

 Solution procedure for the momentum equations

( ) ( ) ( )

V S

t V u C u

t V u C u

V u u F V u V

u PV

S u u

S u u t V

u u

g M g

l gl m l g VM gl g

l gl m l g VM gl

l g gl g

Ev l

Ev g

f g g g f

g

f g g g g f

g g g g

, ,

*

* ,

*

*

*

*

*

) ' (

) (

) (

 















 

 



 



− +

− − +

−

− Γ

− Γ

+

∇

=

⋅

−

⋅

− +

∑

∑

ρ δ α δ α

ρ α α α

ρ α ρ

δ α ρ

α

Finite Volume Discretization for intermediate velocity

( )

( )

g M

g l gl m l g VM gl l

g gl

g g

l EV g

g g g

g g g g g

g g

S

t u C u

u u F

P u

u u

u u

t u u

,

,

'

) (

) (

) (





 













 

+

∂

− + ∂

−

−

∇

−

− Γ

=

⋅

∇

−

⋅

∇

∂ +

∂

ρ α α α

ρ α ρ

α ρ

α

Semi-implicit scheme for two-phase flow

 Solution procedure for the momentum equations

( ) ( ) ( )

V S

t V u C u

t V u C u

V u u F V u V

u PV

S u u

S u u t V

u u

g M g

l gl m l g VM gl g

l gl m l g VM gl

l g gl g

Ev l

Ev g

f g g g f

g

f g g g g f

g g g g

, ,

*

* ,

*

*

*

*

*

) ' (

) (

) (

 















 

 



 



− +

− − +

−

− Γ

− Γ

+

∇

=

⋅

−

⋅

− +

∑

∑

ρ δ α δ α

ρ α α α

ρ α ρ

δ α ρ

α

( ) ( ) ( )

V S

t V u C u

t V u C u

V u u F V u V

u PV

S u u

S u u t V

u u

l M g

l gl m l g VM gl g

l gl m l g VM gl

l g gl g

Cond l

Cond l

f

l f l l l

f

l f l l l l

l l l

, ,

*

* ,

*

*

*

*

*

) ' (

) (

) (

 















 

 



 



− +

− +

−

− +

Γ + Γ

−

∇

=

⋅

−

⋅

− +

∑ ∑

ρ δ α δ α

ρ α α α

ρ α ρ

δ α ρ

α Two linear equations

with two unknowns

Semi-implicit scheme for two-phase flow

 Solution procedure for the momentum equations

( )

[ ]

[

, ¹

]

^{* 2}

* 1

, b

u V V

F t V C

u V V

F t V C

l Cond gl

gd m l g VM gl l l

g Cond gl

gl m l g VM

gl 



 Γ = + +

+ +

Γ

−

−

−

−

−

δ ρ

α α ρ

α

δ ρ

α α

( )

[ ]

[

, ¹

]

^{* 1}

* 1

, b

u V V

F t V C

u V V

F t V C

l Ev gl

gd m l g VM gl

g Ev gl

gl m l g VM gl g

g 



 Γ =

−

−

− +

Γ + +

+

−

−

δ ρ

α α

δ ρ

α α ρ

α

( ) ( ) ( )

V S t V

u C u

t V u C u

V u u F V u V

u PV

S u u

S u u t V

u u

g M g

l gl m l g VM gl g

l gl m l g VM gl

l g gl g

Ev l

Ev g

f g g g f

g

f g g g g f

g g g g

, ,

*

* ,

*

*

*

*

*

) ' (

) (

) (

 















 

 



 



− +

− − +

−

− Γ

− Γ

+

∇

=

⋅

−

⋅

− +

∑

∑

ρ δ α δ α

ρ α α α

ρ α ρ

δ α ρ

α

( ) ( ) ( )

V S t V

u C u

t V u C u

V u u F V u V

u PV

S u u

S u u t V

u u

l M g

l gl m l g VM gl g

l gl m l g VM gl

l g gl g

Cond l

Cond l

f

l f l l l f

l f l l l l

l l l

, ,

*

* ,

*

*

*

*

*

) ' (

) (

) (

 















 

 



 



+

− +

− +

−

− +

Γ + Γ

−

∇

=

⋅

−

⋅

− +

∑

∑

ρ δ α δ α

ρ α α α

ρ α ρ

δ α ρ

α

Two linear equations

with two unknowns

Semi-implicit scheme for two-phase flow

 Solution procedure for the momentum equations

( )

[ ]

[

, ¹

]

^{* 2}

1 *

, b

u V V

F t V C

u V V

F t V C

l Ev gl

gd m l g VM gl l l

g Cond gl

gl m l g VM

gl =

Γ + +

+ +

Γ

−

−

−

−

−

δ ρ

α α ρ

α

δ ρ

α α

( )

[ ]

[

, ¹

]

^{* 1}

1 *

, b

u V V

F t V C

u V V

F t V C

l Ev gl

gd m l g VM gl

g Cond

gl gl

m l g VM gl g

g =

Γ

−

−

− +

Γ + +

+

−

−

δ ρ

α α

δ ρ

α α ρ

α

 

 



= 

 

 



 



 





x x l

g

b b u

u a

a

a a

, 2

, 1

*

*

22 21

12 11

 

 



= 

 

 



 



 





y y l

g

b b v

v a

a

a a

, 2

, 1

*

*

22 21

12 11

 

 



 



 



= 

 

 





⁻

x x l

g

b b a

a

a a

u u

, 2

, 1 1

22 21

12 11

*

*

 

 



 



 



= 

 

 





⁻

y y l

g

b b a

a

a a

v v

, 2

, 1 1

22 21

12 11

*

*

Repeat the same procedure for every cell

for intermediate velocity !

Semi-implicit scheme for two-phase flow

 Solution procedure for the momentum equations Relation between u* and u

ⁿ⁺¹

V a P

a

a a

c c a

a

a a

b b a

a

a a

u

u

_n

l g l

g

 ∇



 



 



 



− 

 

 



 



 



= 

 



 

 



 



= 

 

 





^{− − −}

α α

1

22 21

12 11

2 1 1

22 21

12 11

2 1 1

22 21

12 11

*

*













( ) ^P

A P A

A P A u

u

u u

l n g

n l

g l

n l

g n

g

 ∇ δ



 



= 

−

 ∇



 



− 

 =



 





−

−

₊

+ +

1

* 1

* 1









P A

u u

P A

u u

l l

n l

g g

n g

δ δ

∇

−

=

∇

−

=

+ +

* 1

* 1









V a P

a

a a

c c a

a

a a

b b a

a

a a

u

u

_n

l g n

l n

g 1

1

22 21

12 11

2 1 1

22 21

12 11

2 1 1

22 21

12 11

1

1 +

−

−

− +

+

 ∇



 



 



 



− 

 

 



 



 



= 

 



 

 



 



= 

 

 





α

 α











Semi-implicit scheme for two-phase flow

Solve momentum eqs. with Pn for Intermediate Velocity u*

Solve pressure correction eqs.

for new time step pressure Pⁿ⁺¹

Update new time step velocity uⁿ⁺¹

Update new time step scalar variables

Time Advancing

Semi-implicit scheme for two-phase flow

 Solution procedure for the pressure correction equations

 SMAC: derived from the continuity equations

 ICE: derived from four scalar equations (2 continuity and 2 energy)

 Discretized gas continuity equation

g g

g g g

g u

t +∇⋅ =Γ

∂

∂ (

α ρ

) (

α ρ

 ) _{l l l l}u_{l l}

t +∇⋅ = Γ

∂

∂ (

α ρ

) (

α ρ

 )

l li

gi

sat l

il sat

g ig

g h h

T T H T

T

H = −Γ

−

− +

= −

Γ ( ) ( )

cell g g g

g g

g g

g g

g V

dV t dV t

dV t

t ∆

= +

∂ + ∂

∂

= ∂

∂

∂

∫ ∫

∫

⁽

^{α ρ}

⁾

^{α ρ ρ α α δρ ρ δα}

( ) ( )

∫ ∑

∫

^{∇⋅ + = + ⋅ = + ⋅}

f f

n f g g g S

n g g g n

g g

g u ¹ dV u ¹ dS u ¹ S

) (

)

(

α ρ α ρ α ρ

cell g

gdV = Γ V

∫

Γ

cell li

gi

n sat n

l il n

sat n

g ig

g V

h h

T T

H T

T dV H

−

− +

= − Γ

+ +

+

∫

⁽ +^{1 1) ( 1 1)}

Semi-implicit scheme for two-phase flow

 Solution procedure for the pressure correction equations

 Primary variables

 Unknowns

g g

g g g

g u

t +∇⋅ =Γ

∂

∂ (

α ρ

) (

α ρ

 ) _{l l l l}u_{l l}

t +∇⋅ = Γ

∂

∂ (

α ρ

) (

α ρ

 )

l li

gi

sat l

il sat

g ig

g h h

T T H T

T

H = −Γ

−

− +

= −

Γ ( ) ( )

cell g g g

g g

g g

g g

g V

dV t dV t

dV t

t ∆

= +

∂ + ∂

∂

= ∂

∂

∂

∫ ∫

∫

⁽

^{α ρ}

⁾

^{α ρ ρ α α δρ ρ δα}

( ) ( )

∫ ∑

∫

^{∇⋅ + = + ⋅ = + ⋅}

f f

n f g g g S

n g g g n

g g

g u ¹ dV u ¹ dS u ¹ S

) (

)

(

α ρ α ρ α ρ

cell g

gdV = Γ V

∫

Γ

cell li

gi

n sat n

l il n

sat n

g ig

g V

h h

T T

H T

T dV H

−

− +

= − Γ

+ +

+

∫

⁽ +^{1 1) ( 1 1)}

P e e_{g l}

l

g

α

α

1 1

1

1, ⁺ , ⁺ , ⁺

+ n

sat n

l n g n

g T T T

ρ

Semi-implicit scheme for two-phase flow

 Solution procedure for the pressure correction equations

 Primary variables

 Unknowns

P e e_{g l}

l

g

α

α

1 1

1

1, ⁺ , ⁺ , ⁺

+ n

sat n

l n g n

g T T T

ρ

( ) (

gⁿ

)

n g n

g n g

n n n g

g n

g e e

P e

P P  −











∂ + ∂

 −



 





∂ + ∂

= ^{+ +}

+1

ρ

1

ρ

1

ρ ρ

( ) (

lⁿ

)

n l n

l n l

n n n l

l n

l e e

P e

P P  −

 





∂ + ∂

 −



 





∂ + ∂

= ^{+ +}

+1

ρ ρ

1

ρ

1

ρ

( ) (

gⁿ

)

n g n

g n g

n n n g

g n

g e e

e P T

P P T T

T  −











∂ + ∂

 −



 





∂ + ∂

= ^{+ +}

+1 1 1

( ) (

lⁿ

)

n l n

l n l

n n n l

l n

l e e

e P T

P P T T

T  −



 





∂ + ∂

 −



 





∂ + ∂

= ^{+ +}

+1 1 1

(

^{n n}

)

n n sat

sat n

sat P P

P T T

T  −



 





∂ + ∂

= ⁺

+1 1

Linearization

Replace the secondary

variables in the scalar eqs.