* Corresponding author, E-mail: [email protected]
김 종 태,*1 박 익 규,1 조 형 규,2 김 경 두,2 정 재 준1
NUMERICAL METHOD FOR TWO-PHASE FLOW ANALYSIS USING SIMPLE-ALGORITHM ON AN UNSTRUCTURED MESH
Jongtae Kim,*1 Ik-Kyu Park,1 Hyung-Kyu Cho,2 Kyung Doo Kim2 and Jae Jun Jeong1
For analyses of multi-phase flows in a water-cooled nuclear power plant, a three-dimensional SIMPLE-algorithm based hydrodynamic solver CUPID-S has been developed. As governing equations, it adopts a two-fluid three-field model for the two-phase flows. The three fields represent a continuous liquid, a dispersed droplets, and a vapour field. The governing equations are discretized by a finite volume method on an unstructured grid to handle the geometrical complexity of the nuclear reactors. The phasic momentum equations are coupled and solved with a sparse block Gauss-Seidel matrix solver to increase a numerical stability. The pressure correction equation derived by summing the phasic volume fraction equations is applied on the unstructured mesh in the context of a cell-centered co-located scheme. This paper presents the numerical method and the preliminary results of the calculations.
Key Words : Two-Phase Flow SIMPLE SIMPLE Algorithm Unstructured Mesh
1.
(two-phase flow)
,
.
, (
) , (volume
fraction) (continuous field) (dispersed
field) .
.
CFD adaptive mesh, VOF,
level-set [1-3]
(capturing) (tracking) .
(short-term) ,
(mid or long-term)
. Euler-Euler
Eulerian
Navier-Stokes .
.
. CFD
.
CFX (dispersed phase) ( )
MUSIC .
2-field
. Kunz [4]
4-field ,
Antal [5]
multi-field .
, SIMPLE
CUPID-S (Component Unstructured Program for Interfacial Dynamics-based on SIMPLE algorithm) .
2 (two-fluid) 3 (three-field) [6]
, SIMPLE
MCBA(Mass Conservation-Based Algorithm)[7]
.
(hybrid mesh, mixed element mesh)
.
. SIMPLE
(phase separation)
2.
, 2 (continuous
liquid phase), (droplets phase), (vapor
phase) 3 .
.
2.1 2 3
v, l, d k
.
∇ ∙
(1)
, ( - )
.
, . .
∇⋅ ∇
∇⋅
(2)
CUPID-S
, ,
(drag force) ,
(virtual mass force) .
, (thermal equilibrium)
.
∇⋅ ∇⋅
∇
⋅∇ ″ ′ (3)
(system) (closure) .
(4)
2.2
, ,
.
∇
(5)Green-Gauss
. (5)
(Finite Volume Metohd, FVM) (cell)
(discretization) .
SIMPLE (implicit method)
, Euler
.
,
(under-relaxation) . ,
2
(implicit) (explicit)
(deferred correction method) . (delta form) (dual-time integration) [8].
∇
(6)(6) (5) (pseudo-time) .
,
,
. SIMPLE
.
(time-lagging)
. 1
(upwind scheme) 1 , 2
TVD . 2
(reconstruction) (gradient)
. (primary)
(cross)
. (skewness)
. (6)
[8] .
(6) (7)
.
(7),
. (iteration step)
.
.
(7) (matrix) .
2.3
.
.
.
, . 0
(singularity)
. (minimum) 10-8
.
.
. -
SIMPLE
.
Euler (8) .
(8)
.
(9)
. (9) (8) (10)
.
(10)
.
(11)
x- (12)
.
∇ ∇⋅
(12)
x- .
(13) .
∇⋅
(13)
,
.
(13) (12) (14)
.
∇⋅ ∇⋅
∇⋅
(14)
(stability) .
,
. ,
.
. 3
(phasic coupling)
(spatial coupling) . 3 3
.
(15)
ncell
× block matrix
. ×
, (diagonal)
(off-diagonal) .
block matrix 0
.
block matrix Gauss-Seidel .
2.4
m
* .
′
(16)
(15) 2
(17) .
(17)m
.
(18)(17) (18)
.
′
′
(19) ′ .
(19)
,
SIMPLE ′≈
. SIMPLE
.
′
′
′
′
′
(20)SIMPLEC ′≈ ′
, (19) SIMPLEC
.
′
′
′
′
′
(21)
2.5
.
(mass conservation-based algorithm MCBA) (geometric conservation- based algorithm GCBA)[9]
MCBA .
MCBA . ,
. (normalize) .
⋅
(22)
(22)
. [9] Moukalled, Darwish MCBA
.
0 ,
(22) 0
. CUPID-S
(22)
. (22)
(known)
.
′ (23)
′ (24)
′
(25)
′ ′
≈ ′ ′ (26)
(25), (26) (22) .
′
′
′
(27)
(27) n+1
.
′ chain-rule
(20), (21) .
′
′ ′ (28)
′
′ ′∇′ (29)
′′∇′ (30)
(28), (29), (30) (27) (31) .
′
′
∇′
(31)
(31) .
, , ,
. .
′
′
′
(32)
.
∇′
′ ′⋅
∇′ ⋅∇′ ⋅
⋅
(33)
(skewness)
(33)
.
SIMPLE 0
(34) .
∇′≈
∙
′ ′ (34)(31)
.
′
′ (35)
2.6
,
- .
,
. SIMPLE
, ,
(coupled algorithm)
.
Fig. 1 Progress of phasic volume fractions and velocity fileds along time for a phase separation in a square chamber, left: liquid phase, right: vapor phase
.
, [9]
(steam table) .
2.7
, SIMPLE
MCBA . SIMPLE
.
.
, , .
.
.
.
.
3.
SIMPLE
(phase separation) , .
3.1
.
2
0
. CUPID-S 2
2 m .
× 2
400
, 0
1 (slip)
.
. Fig. 1
. ,
, 10
. CUPID-S
.
2
[10].
.
2m 1 ,
500 ,
2 .
Elevation [m]
Vaporvolumefraction
0 0.5 1 1.5 2
0 0.2 0.4 0.6 0.8 1
t=2s
t=2s
t=4s t=6s
t=4s
t=8s
t=6s t=8s t=10s t=10s Numerical solution Analytical solution
Fig. 2 Phase separation in a vertical pipe, comparison of the vapor-liquid interface propagation between numerical and analytical solutions
Fig. 3 Phase separation in a vertical pipe, vapor volume fractions at 2s and 6s
0.1 m
2.0 m 0.1 m
2.0 m
(a)
(b)
Fig. 4 Calculation of the flashing in a 2-D channel (a) 2-D mesh, (b) Pressure distribution
Fig. 2
.
.
. Fig. 3 2 6
,
.
3.2 2 (flashing)
(flashing) . CUPID-S
2
. 0.1m 2m
Fig. 4(a) .
250 , 4.0 m/s
, 450K, 1MPa . 10
1MPa 0.854MPa . 1MPa 0.854MPa 453.04K 446.27K ,
2 .
10 2
- 2
, 13 . Fig. 4(b) 13
. Fig. 5
.
0.6 .
(interfacial)
1
x[m]
temperature[K]
0 0.5 1 1.5 2
445 446 447 448 449 450
vapor
vaportemperature liquidtemperature saturationtemperature liquid
Fig. 6 Flashing in a 2-D channel, distributions of the water and vapor temperatures along the channel
x[m]
streamwisevelocity[m/s]
0 0.5 1 1.5 2
4 6 8 10 12
liquid vapor
Fig. 7 Flashing in a 2-D channel, distributions of the water and vapor velocities along the channel
x[m]
Phasicvolumefraction
0 0.5 1 1.5 2
0 0.2 0.4 0.6 0.8 1
liquidphase vaporphase
Fig. 5 Flashing in a 2-D channel, distributions of the water and vapor volume fractions along the channel
.
,
.
Fig. 6 ,
.
, .
, Fig. 5
. Fig. 7
. CUPID-S , ,
.
4.
2-
3- SIMPLE
, ,
. SIMPLE
,
,
.
, ,
.
.
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,”
, pp.31-35.
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,” KAERI
/TR-2314/2002.
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