LaMSID - EDF R&D, AMA LATP, University of Provence
Computing water-hammer flows with a two-fluidmodel
AMIS 2012
Yujie LIU
21 Juin 2012
PhD supervisors : Fabien Crouzet, EDF R&D, AMAFrederic Daude, EDF R&D, AMAPascal Galon, CEA SaclayPhilippe Helluy , IRMA StrasbourgJean-Marc Herard, EDF R&D, MFEE
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 1 / 32
Outline
1 Introduction
2 The two-fluid model & water hammer
3 Computation of the two-fluid modelEvolution stepRelaxation step
4 Verification of the two-fluid model
5 Conclusions and perspectives
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 2 / 32
Introduction
Water-hammer transient flow in piping systems of a nuclear power plant
Water-hammer (hydraulic shock) :sudden change of direction or velocity ! large changes in pressureRisk : damage to pipes, supports, and valve ; power plant availability
Figure: Water-hammer ‘classical’ Figure: Are line Civaux (2011)
Objective : Water-hammer simulation(SITAR Project )
Modeling water-hammer flows (Assessing/comparing the di↵erent models)
Computing the model with fluid-structure interaction (FSI) (Europlexus)
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 3 / 32
Introduction
Modeling water-hammer flows
Complex phenomenon : column separation/cavitation...
Phenomenon characteristics :
compressible two-phase flows(water/steam)unsteady flowsshock wavesfast transient
�!�!�!�!
two-phase flow modelinghyperbolic systemunique jump conditionsexplicit scheme
Di↵erent two-phase flow models :
two-fluid approach :the two-fluid model (7 equations)the two-fluid single-pressure model (6 equations)
homogeneous approach :the five-equation models (5 equations)homogeneous relaxation model (4 equations)homogeneous equilibrium model (3 equations)
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 4 / 32
The two-fluid model & water hammer
Outline
1 Introduction
2 The two-fluid model & water hammer
3 Computation of the two-fluid modelEvolution stepRelaxation step
4 Verification of the two-fluid model
5 Conclusions and perspectives
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 5 / 32
The two-fluid model & water hammer
Governing equations of the two-fluid model
7 PDE+ closure (P
I
, VI
)+ EOS(equation of state) : "'
(⇢'
, p'
) + source terms
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
@t
↵v
+VI
@x
↵v
= �v
@t
(↵'
⇢'
) +@x
(↵'
⇢'
u'
) = �'
, ' = l, v
@t
(↵'
⇢'
u'
) +@x
�
↵'
⇢'
u2'
+ ↵'
p'
�
� PI
@x
↵'
= �'
Uint
+D'
@t
(↵'
⇢'
e'
) +@x
[(↵'
⇢'
e'
+ ↵'
p'
)u'
] + PI
@t
↵'
= �'
Hint
+D'
Uint
+Q'
(1)with e
'
= "'
(⇢'
, p'
) + u2'
/2, Uint
= (ul
+ uv
)/2 and Hint
= ul
⇤ uv
/2,
�v
�'
D'
Q'
mass transfermomentum transferenergy transfer
pressure relaxationchemical potential relaxationvelocity relaxationtemperature relaxation
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 6 / 32
The two-fluid model & water hammer
Choice of closure laws for water-hammer flows
Interfacial pressure/velocity :
PI
= pl
, VI
= uv
(Baer-Nunziato)
EOS :
Liquid : Sti↵ened Gas (SG) "l
(⇢l
, pl
) =pl
+ �l
(pl
)1⇢l
(�l
� 1)
Vapor : Perfect Gas (GP) "v
(⇢v
, pv
) =pv
⇢v
(�v
� 1)
Source terms (Entropy-consistent interfacial closure) :
pressure relaxation : �v
= (⌧p
)�1 ↵l
↵v
|Pl
|+|Pv
|(P
v
� Pl
),X
'=l,v
�'
= 0
velocity relaxation : Dv
= (⌧u
)�1 ml
mv
ml
+mv
(ul
� uv
),X
'=l,v
D'
= 0,
with ⌧k
, k =p,u, time scales for pressure/velocity relaxations,m
'
= ↵'
⇢'
, ' = l, v, partial masses.
Remark :
liquid/vapor : PI
= pl
, (pl
)1 >> (pv
)1gas/particles : P
I
= pg
, (pg
)1 << (pp
)1, (GHHN, M2AN 2010)[1]
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 7 / 32
The two-fluid model & water hammer
Main properties of the two-fluid model
Property 1 Hyperbolicity and structure of wavesThe system (1) is hyperbolic unless |u
l
� uv
|= cl
. It admits 7 realeigenvalues :
�1,2 = uv
,�3 = uv
� cv
,�4 = uv
+ cv
,�5 = u
l
,�6 = ul
� cl
,�7 = ul
+ cl
Fields associated with �1,2,5 are linearly degenerate (LD).Other fields are genuinely nonlinear (GNL).
Property 2 Jump conditionsUnique jump conditions hold within each field associated with �
k
.↵l
is uniform apart from the field associated with �1,2 = uv
.Jump conditions in other fields correspond to single phase jump relations.
Property 3 Entropy inequalityDefine the entropy ⌘(W ) = m
l
sl
+mv
sv
and the entropy fluxf⌘
(W ) = ml
sl
ul
+mv
sv
uv
; then smooth solutions W of (1) are such that :
0 @t
(⌘(W )) + @x
(f⌘
(W ))
with the constraint c2'
@p' (s') + @
⇢' (s') = 0 for entropies s'
.
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 8 / 32
Computation of the two-fluid model
Outline
1 Introduction
2 The two-fluid model & water hammer
3 Computation of the two-fluid modelEvolution stepRelaxation step
4 Verification of the two-fluid model
5 Conclusions and perspectives
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 9 / 32
Computation of the two-fluid model
A fractional step method complying with the entropy inequality :
At time interval [tn, tn +�t], given initial values Wn
Evolution step : Wn
convective e↵ects����������������������!hyperbolic homogeneous system
W
8
>
>
<
>
>
:
@t
↵v
+VI
@x
↵v
= 0@t
(↵'
⇢'
) +@x
(↵'
⇢'
u'
) = 0@t
(↵'
⇢'
u'
) +@x
�
↵'
⇢'
u2'
+ ↵'
p'
�
� PI
@x
↵'
= 0@t
(↵'
⇢'
e'
) +@x
[(↵'
⇢'
e'
+ ↵'
p'
)u'
] + PI
@t
↵'
= 0
(2)
Relaxation step : Wsource terms���������! Wn+1
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
@t
↵l
= �l
@t
(↵l
⇢l
) = 0@t
(↵l
⇢l
ul
) = Dl
@t
(↵l
⇢l
el
) + PI
@t
(↵l
) = Dl
Uint
@t
(↵v
⇢v
+ ↵l
⇢l
) = 0@t
(↵v
⇢v
uv
+ ↵l
⇢l
ul
) = 0@t
(↵v
⇢v
ev
+ ↵l
⇢l
el
) = 0
(3)
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 10 / 32
Computation of the two-fluid model Evolution step
Evolution step : an extension of Rusanov scheme
Homogeneous system :
@t
W + @x
(f(W )) + h(W )@x
↵v
= 0 (4)
Extension of Rusanov scheme (SR1) :
hi
(Wn+1i
�Wn
i
) +�tn(Fn
i+1/2 � Fn
i�1/2) +�tn⇣
(↵1)n
i+ 12� (↵1)
n
i� 12
⌘
�n
i
= 0
(5)Flux :
Fn
i+1/2 = 12
�
f(Wn
i
) + f(Wn
i+1)� ri+1/2(W
n
i+1 �Wn
i
)�
ri+1/2 = max(r
i
, ri+1), r spectral radius
(6)
Non-conservative terms :
(↵l
)n
i+ 12= 1
2
⇣
(↵l
)ni
+ (↵l
)ni+1
⌘
, �n
i
= h(Wn
i
) (7)
.Property 4The scheme (5) preserves positive values of partial masses m
'
and void fractions↵'
, if
�tn (ri+1/2 + r
i�1/2) 2CFLhi
, CFL 2]0, 1[
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 11 / 32
Computation of the two-fluid model Evolution step
The second-order Rusanov scheme (SR1-ORDER2 )
Time scheme : second-order Runge-Kutta methodSpace discretisation : minmod reconstruction (Y = (↵
v
, uv
, pv
, sv
, ul
, pl
, sl
)t)
If (�n
i+1 � �n
i
)(�n
i
� �n
i�1) > 0,
(�n
i
)+ = �n
i
+ �ni
hi2, (�n
i
)� = �n
i
� �ni
hi2,
�ni
= sign(�n
i+1 � �n
i
)min
⇢
2
?
?
?
?
�ni+1��n
ihi+1+hi
?
?
?
?
, 2
?
?
?
?
�ni ��n
i�1hi+hi�1
?
?
?
?
�
Else, (�n
i
)+ = (�n
i
)� = �n
i
(8)
Flux :
Fn
i+1/2 =1
2
n
f(W ((Y n
i
)+)) + f(W ((Y n
i+1)�))� r
i+1/2
n
W ((Y n
i+1)�)�W ((Y n
i
)+)oo
(9)Non-conservative terms :
h(W )@x
↵v
�!
⇣
(↵1)n
i+ 12� (↵1)
n
i� 12
⌘
�n
i
hi
(10)
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 12 / 32
Computation of the two-fluid model Evolution step
Verification of the schemes : test cases
Test case 1 (TT, JCP, 2010 ; SWK, JCP, 2006) [6, 4]2 contact discontinuities, 1 rarefaction, 3 shocksEOS : GP(vapor), SG (liquid)
Test case 2 (constructed),A moving void fraction waveEOS : GP(vapor), SG (liquid)
VI
Figure: Structure of fields of case 2
Test case 3 (constructed),A moving void fraction waveEOS : GP(vapor), SG (liquid)Large variation of ↵
v
: 0.05 ! 0.95, equilibrium of other variables
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 13 / 32
Computation of the two-fluid model Evolution step
Verification of the first/second-order Rusanov schemes : test case 1
!0.5 0 0.5
!0.4
!0.2
0
0.2
0.4
0.6
0.8
x
!1
Exact solution
SR1!200!cells
SR1!500!cells
SR1!ORDER2!200!cells
SR1!ORDER2!500!cells
!0.5 0 0.5
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x
!1
Exact solution
SR1!200!cells
SR1!500!cells
SR1!ORDER2!200!cells
SR1!ORDER2!500!cells
!0.5 0 0.5
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
0.8
x
u1
Exact solution
SR1!200!cells
SR1!500!cells
SR1!ORDER2!200!cells
SR1!ORDER2!500!cells
!0.5 0 0.5
!0.5
0
0.5
1
1.5
2
2.5
3
x
p 1
Exact solutionSR1!200!cellsSR1!500!cellsSR1!ORDER2!200!cellsSR1!ORDER2!500!cells
!0.5 0 0.5
1650
1700
1750
1800
1850
1900
1950
2000
2050
x
!2
Exact solution
SR1!200!cells
SR1!500!cells
SR1!ORDER2!200!cells
SR1!ORDER2!500!cells
!0.5 0 0.5
!0.3
!0.25
!0.2
!0.15
!0.1
!0.05
0
x
u2
Exact solution
SR1!200!cells
SR1!500!cells
SR1!ORDER2!200!cells
SR1!ORDER2!500!cells
!0.5 0 0.5
!600
!400
!200
0
200
400
600
800
1000
x
p 2
Exact solutionSR1!200!cellsSR1!500!cellsSR1!ORDER2!200!cellsSR1!ORDER2!500!cells
Figure: SR1, SR1-ORDER2 ; with 200 & 500 cells, CFL = 0.49, t = 0.15.
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 14 / 32
Computation of the two-fluid model Evolution step
Verification of the first/second order Rusanov scheme : test case 1
Rate of convergence observed :
SR1 :1
2, SR1-ORDER2 :
2
3
!5 !4.5 !4 !3.5 !3 !2.5 !2!4.5
!4
!3.5
!3
!2.5
!2
!1.5
!1
!0.5
ln(h)
ln(e
rror
)
!
1!SR1
"1
u1
p1
"2
u2
p2
!1!SR1!ORDER2
"1
u1
p1
"2
u2
p2
C0 h 1/2
C h 2/3
Figure: L-1 norm of the error for varibles ↵v
, ⇢'
, u'
, p'
,' = l, v traced in logarithmicscale, SR1, SR1-ORDER2, CFL = 0.49, t = 0.15
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 15 / 32
Computation of the two-fluid model Evolution step
Verification of the first/second order Rusanov scheme : test case 2
0 200 400 600 800 1000
!0.3
!0.2
!0.1
0
0.1
0.2
0.3
0.4
0.5
x
!1
Exact solution
SR1!200!cells
SR1!500!cells
SR1!ORDER2!200!cells
SR1!ORDER2!500!cells
0 200 400 600 800 1000
8.5
9
9.5
10
10.5
11
11.5
12
x
!1
Exact solution
SR1!200!cells
SR1!500!cells
SR1!ORDER2!200!cells
SR1!ORDER2!500!cells
0 200 400 600 800 1000
9.75
9.8
9.85
9.9
9.95
10
10.05
x 105
x
p 1
Exact solutionSR1!200!cellsSR1!500!cellsSR1!ORDER2!200!cellsSR1!ORDER2!500!cells
0 200 400 600 800 1000
12
12.5
13
13.5
14
14.5
15
15.5
16
16.5
17
x
u1
Exact solution
SR1!200!cells
SR1!500!cells
SR1!ORDER2!200!cells
SR1!ORDER2!500!cells
0 200 400 600 800 1000
999.86
999.88
999.9
999.92
999.94
999.96
999.98
1000
1000.02
x
!2
Exact solution
SR1!200!cells
SR1!500!cells
SR1!ORDER2!200!cells
SR1!ORDER2!500!cells
0 200 400 600 800 1000
2
3
4
5
6
7
8
9
10
x
u2
Exact solution
SR1!200!cells
SR1!500!cells
SR1!ORDER2!200!cells
SR1!ORDER2!500!cells
0 200 400 600 800 1000
6.5
7
7.5
8
8.5
9
9.5
10
10.5
11
x 105
x
p 2
Exact solutionSR1!200!cellsSR1!500!cellsSR1!ORDER2!200!cellsSR1!ORDER2!500!cells
Figure: SR1, SR1-ORDER2 ; 200 & 500 cells, CFL = 0.49, t = 0.25
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 16 / 32
Computation of the two-fluid model Evolution step
Verification of the first/second order Rusanov scheme : test case 2
Rate of convergence observed :
SR1 :1
2, SR1-ORDER2 :
2
3
!5 !4.5 !4 !3.5 !3 !2.5 !2!8
!7
!6
!5
!4
!3
!2
!1
ln(h)
ln(e
rror)
!
1!SR1
"1
u1
p1
"2
u2
p2
!1!SR1!ORDER2
"1
u1
p1
"2
u2
p2
C0 h 1/2
C h 2/3
Figure: SR1, SR1-ORDER2, CFL = 0.49, t = 0.25
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 17 / 32
Computation of the two-fluid model Evolution step
Evolution step : a fractional step method
8
>
>
<
>
>
:
@t
↵v
+ uv
@x
↵v
= 0@t
(↵'
⇢'
) = 0, ' = l, v@t
(↵'
⇢'
u'
) = 0, ' = l, v@t
(↵'
E'
)� pl
uv
@x
↵'
= 0, ' = l, v
(11)
8
>
>
<
>
>
:
@t
↵v
= 0@t
(↵'
⇢'
) + @x
(↵'
⇢'
u'
) = 0, ' = l, v
@t
(↵'
⇢'
u'
) + @x
�
↵'
⇢'
u2'
�
+ @x
(↵'
p'
)� pl
@x
↵'
= 0, ' = l, v@t
(↵'
E'
) + @x
[(↵'
E'
+ ↵'
p'
)u'
] = 0, ' = l, v
(12)
PropertySystem (11) is hyperbolic. It admits seven real eigenvalues. All fields are LD.
�1 = uv
,�2�7 = 0
System (12) is hyperbolic unless : |ul
| = cl
, or |uv
| = cv
. It admits seven realeigenvalues. The 1, 2, 5-fields are LD and other fields are GNL.
�1 = 0,�2 = u
v
, �3 = uv
� cv
, �4 = uv
+ cv
,�5 = u
l
, �6 = ul
� cl
, �7 = ul
+ cl
.
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 18 / 32
Computation of the two-fluid model Evolution step
Evolution step : a fractional step method
First step : Wn ! W ⇤
8
>
>
<
>
>
:
@t
↵v
+ uv
@x
↵v
= 0@t
(↵'
⇢'
) = 0, ' = l, v@t
(↵'
⇢'
u'
) = 0, ' = l, v@t
(↵'
E'
)� pl
uv
@x
↵'
= 0, ' = l, v
(13)
↵v
: Rusanov flux formulation(↵
v
Ev
+ ↵l
El
)⇤i
= (↵v
Ev
+ ↵l
El
)ni
(sl
)⇤i
= (sl
)ni
(14)
Second step :W ⇤ ! Wn+1 : generic Rusanov formulation
PropertyThe scheme preserves positive values of partial masses m
'
and void fractions ↵'
ifthe CFL like condition holds on. The scheme preserves the mean total energy.
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 19 / 32
Computation of the two-fluid model Evolution step
Verification of the fractional step method : test case 1
Rate of convergence observed :
PFRAC3 :1
2,
!5 !4.5 !4 !3.5 !3 !2.5 !2
!3.5
!3
!2.5
!2
!1.5
!1
!0.5
ln(h)
ln(e
rror
)
!
1!SR1
"1
u1
p1
"2
u2
p2
!1!PFRAC3
"1
u1
p1
"2
u2
p2
C0 h1/2
Figure: SR1, PFRAC3, CFL = 0.49, t = 0.15.
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 20 / 32
Computation of the two-fluid model Evolution step
Verification of the fractional step method : test case 2
Rate of convergence observed :
PFRAC3 :1
2,
!5 !4.5 !4 !3.5 !3 !2.5!8
!7
!6
!5
!4
!3
!2
!1
0
ln(h)
ln(e
rror)
!
1!SR1
"1
u1
p1
"2
u2
p2
!1!PFRAC3
"1
u1
p1
"2
u2
p2
C0 h1/2
Figure: SR1, PFRAC3 CFL = 0.49, t = 0.25 .
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 21 / 32
Computation of the two-fluid model Evolution step
Verification of the fractional step method : test case 3
Rate of convergence observed :
PFRAC3 :1
2,
!5.5 !5 !4.5 !4 !3.5 !3 !2.5!5
!4
!3
!2
!1
0
1
ln(h)
ln(e
rror)
!
1
"1
u1
p1
"2
u2
p2
C0 h1/2
Figure: PFRAC3, CFL = 0.49, t = 0.25 .
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 22 / 32
Computation of the two-fluid model Relaxation step
Relaxation step
Wvelocity relaxation�������������! W ⇤ pressure relaxation�������������! Wn+1
i
(15)
Computing the ordinary di↵erential equations :
Velocity relaxation : drag terms
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
@t
↵l
= 0@t
(↵l
⇢l
) = 0@t
(↵l
⇢l
ul
) = Dl
@t
(↵l
⇢l
el
) = Dl
Uint
@t
(↵v
⇢v
+ ↵l
⇢l
) = 0@t
(↵v
⇢v
uv
+ ↵l
⇢l
ul
) = 0@t
(↵v
⇢v
ev
+ ↵l
⇢l
el
) = 0(16)
Proposed in [2](HH, CaF, 2012).
Pressure relaxation :
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
@t
↵v
= �v
@t
(↵v
⇢v
) = 0@t
(↵v
⇢v
uv
) = 0@t
(↵v
⇢v
ev
) + pl
@t
↵v
= 0,@t
(↵l
⇢l
) = 0@t
(↵l
⇢l
ul
) = 0@t
(↵l
⇢l
el
) + pl
@t
↵l
= 0,(17)
Proposed in [1](GHHN, M2AN, 2010)+ Extension to new EOS
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 23 / 32
Computation of the two-fluid model Relaxation step
Pressure relaxation : extension to EOS GP(vapor)+SG(liquid)
The ordinary di↵erential equation (17) is :
8
>
>
>
<
>
>
>
:
@t
↵v
= �v
@t
(mv
ev
) + pl
@t
↵v
= 0@t
(ml
el
)� pl
@t
↵v
= 0@t
(↵'
⇢'
) = 0@t
(↵'
⇢'
u'
) = 0
(18)
An implicit scheme calculates (pn+1l
, pn+1v
,↵n+1v
), solution of
8
<
:
↵n+1v
� ↵⇤v
+�t(✓)�1↵n+1l
↵n+1v
(pn+1l
� pn+1v
) = 0(m
l
el
)n+1 � (ml
el
)⇤ � Pn+1I
(↵n+1v
� ↵⇤v
) = 0(m
v
ev
)n+1 � (mv
ev
)⇤ + Pn+1I
(↵n+1v
� ↵⇤v
) = 0(19)
Property 6Assume that the EOS of the gas phase is GP, and that of the liquid phase is SG,then the scheme (19) admits a unique relevant solution (pn+1
l
, pn+1v
,↵n+1v
) such
that pv
> 0, pl
+ (pl
)1 > 0 and ↵n+1v
lies in [↵M
,↵m
] 2 [0, 1], with
↵M =�v
� 1
�v
↵0v
, ↵m =1 + (�
l
� 1)↵0v
�l
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 24 / 32
Computation of the two-fluid model Relaxation step
Verification of the pressure relaxation scheme
Slope of curves : 0.56Rate of convergence expected : 1‘Exact solution’ : Matlab ODE45 approach (AbsTol : 1e-14)
!5.5 !5 !4.5 !4 !3.5 !3 !2.5 !2!3.5
!3
!2.5
!2
!1.5
!1
!0.5
0
ln(h)
ln(e
rro
r)
!
l
pl
pv
Figure: Curves of convergence of the pressure relaxation scheme.
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 25 / 32
Verification of the two-fluid model
Outline
1 Introduction
2 The two-fluid model & water hammer
3 Computation of the two-fluid modelEvolution stepRelaxation step
4 Verification of the two-fluid model
5 Conclusions and perspectives
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 26 / 32
Verification of the two-fluid model
Simpson Experiment (in process)S A.R., PhDThesis, 1986 [5]
Reservoir
Tuyau 36 m
u
Figure: Simpson ExperimentFigure: Pressure at the valve ofpipe
Combination of many di↵erent phenomena :
Reflection of waves, vaporous cavitation
Regime transition : almost single-phase flow ! mixture of water/steam
Fluid structure interaction
Preliminary questions : Is the two-fluid model able to
retreive the single-phase phenomena at the beginning of experiment ?
capture the speed of sound in the liquid/vapor mixture ?
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 27 / 32
Verification of the two-fluid model
Single-phase phenomena of Simpson experiment
0.9999999990.999999990.99999990.9999990.999990.99990.9990.990.94
5
6
7
8
9
10x 10
5
!l
pres
sion
(p)
Pression!formule de JoukowskyResu numerique 100 maillesResu numerique 1000 maillesResu numerique 10000 mailles
Figure: Pressure magnitude of the first shock wave : p = ↵l
pl
+ ↵v
pv
according todi↵erent values of ↵
l
Joukowsky formula : incompressible single-phase estimation
�p = ⇢l
cl
�u (20)
Numeric : ⌧p
= 10�10s, ⌧u
= 10�10s,
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 28 / 32
Verification of the two-fluid model
Speed of sound in the liquid/vapor mixture
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
500
1000
1500
!l
c
Formule de WallisExperience 250 HZExperience 500 HZResu numérique avec 1000!maillesResu numérique avec 10000!mailles
0.3 0.4 0.5 0.6 0.7 0.8 0.9 120
30
40
50
60
70
80
90
100
!l
c
Formule de WallisExperience 250 HZExperience 500 HZResu numerique avec 1000!maillesResu numérique avec 10000!mailles
Figure: Numerics, Wallis, experiment for di↵erent values of ↵l
Wallis formula :1
mcWallis
2=
↵l
⇢l
cl
2+
↵v
⇢v
cv
2(21)
Experiment : [3] (Karplus, H. B.-1958)
Numerics : ⌧p
= 10�10s, ⌧u
= 10�10s,
Remark : There is no definition of speed of sound for the mixture for the
two-fluid model , here cnum
= dispulse
/time
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 29 / 32
Conclusions and perspectives
Outline
1 Introduction
2 The two-fluid model & water hammer
3 Computation of the two-fluid modelEvolution stepRelaxation step
4 Verification of the two-fluid model
5 Conclusions and perspectives
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 30 / 32
Conclusions and perspectives
Conclusions and perspectives
. Two-fluid model :
stable convection schemes + relaxation schemes (p, u)rather expensive (3D computation ! ! !)
good approximation of speed of sound in the mixturesingle-phase phenomena retrieved
masse transfer relaxation scheme (cavitation) to be testedmore complete realistic calculations to be leaded
. The five-equation models
. Comparison of models + FSI
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 31 / 32
Conclusions and perspectives
Thank you for your attention !
Any questions ?
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 32 / 32
Conclusions and perspectives
T. Gallouet, P. Helluy, J.-M. Herard, and J. Nussbaum.
Hyperbolic relaxation models for granular flows.
Math. Model. and Numer. Anal., 44(2) :371–400, 2010.
J.-M. Herard and O. Hurisse.
A fractional step method to compute a class of compressible gas-liquide flows.
Computers & Fluids, 55 :57–69, 2012.
H. B. Karplus.
The velocity of sound in a liquide containing gas bubbles.
Armour Research Foundation of Illinois Institute of Technology, C00-248, 1958.
D. Schwendeman, C. Wahle, and A. Kapila.
The Riemann problem and a high-resolution Godunov method for a model ofcompressible two-phase flow.
Journal of Computational Physics, 212(2) :490–526, 2006.
A. R. Simpson.
Large water hammer pressures due to column separation in sloping pipes
(transient cavitation).
PhD thesis, University of Michigan, 1986.Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 32 / 32
Conclusions and perspectives
S. Tokareva and E. Toro.
HLLC-type Riemann solver for the BaerNunziato equations of compressibletwo-phase flow.
Journal of Computational Physics, 229 :3573–3604, 2010.
Yujie LIU (EDF R&D, AMA) Water-hammer/Two-fluid model AMIS 2012 32 / 32