computing water-hammer flows with a two-fluid model · the two-ﬂuid model & water hammer main...

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, AMAFrédéric Daude, EDF R&D, AMAPascal Galon, CEA SaclayPhilippe Helluy , IRMA StrasbourgJean-Marc Hérard, 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


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)


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)


The two-fluid model & water hammer

Outline

1 Introduction







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



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

)�1m

l

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


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' .


Computation of the two-fluid model

Outline

1 Introduction






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 : Wnconvective e↵ects��!

hyperbolic homogeneous systemW̃

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 : W̃source 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)


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

�Wni

) +�tn(Fni+1/2 � F

n

i�1/2) +�tn

⇣

(↵1)n

i+ 12� (↵1)

n

i� 12

⌘

�ni

= 0

(5)Flux :

Fni+1/2 =

12

�

f(Wni

) + f(Wni+1)� ri+1/2(Wni+1 �Wni )

�

ri+1/2 = max(ri, ri+1), r spectral radius

(6)

Non-conservative terms :

(↵l

)n

i+ 12= 12

⇣

(↵l

)ni

+ (↵l

)ni+1

⌘

, �ni

= h(Wni

) (7)

.Property 4The scheme (5) preserves positive values of partial masses m

'

and void fractions↵'

, if

�tn (ri+1/2 + ri�1/2) 2CFLhi, CFL 2]0, 1[



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 (�ni+1 � �ni )(�ni � �ni�1) > 0,

(�ni

)+ = �ni

+ �ni

hi2 , (�

n

i

)� = �ni

� �ni

hi2 ,

�ni

= sign(�ni+1 � �ni )min

⇢

2

?

?

?

?

�ni+1��ni

hi+1+hi

?

?

?

?

, 2

?

?

?

?

�ni ��ni�1

hi+hi�1

?

?

?

?

�

Else, (�ni

)+ = (�ni

)� = �ni

(8)

Flux :

Fni+1/2 =

1

2

n

f(W ((Y ni

)+)) + f(W ((Y ni+1)

�))� ri+1/2

n

W ((Y ni+1)

�)�W ((Y ni

)+)oo

(9)Non-conservative terms :

h(W )@x

↵v

�!

⇣

(↵1)n

i+ 12� (↵1)

n

i� 12

⌘

�ni

hi

(10)



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



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

α1

Exact solution

SR1−200−cells

SR1−500−cells

SR1−ORDER2−200−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

ρ1

Exact solution

SR1−200−cells

SR1−500−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



−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

ρ2

Exact solution

SR1−200−cells

SR1−500−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



−0.5 0 0.5

−600

−400

−200

0

200

400

600

800

1000

x

p 2


Figure: SR1, SR1-ORDER2 ; with 200 & 500 cells, CFL = 0.49, t = 0.15.



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

)

α

1−SR1

ρ1

u1

p1

ρ2

u2

p2

α1−SR1−ORDER2

ρ1

u1

p1

ρ2

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




0 200 400 600 800 1000

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

x

α1

Exact solution

SR1−200−cells

SR1−500−cells



0 200 400 600 800 1000

8.5

9

9.5

10

10.5

11

11.5

12

x

ρ1

Exact solution

SR1−200−cells

SR1−500−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


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



0 200 400 600 800 1000

999.86

999.88

999.9

999.92

999.94

999.96

999.98

1000

1000.02

x

ρ2

Exact solution

SR1−200−cells

SR1−500−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



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


Figure: SR1, SR1-ORDER2 ; 200 & 500 cells, CFL = 0.49, t = 0.25





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)

α

1−SR1

ρ1

u1

p1

ρ2

u2

p2

α1−SR1−ORDER2

ρ1

u1

p1

ρ2

u2

p2

C0 h 1/2

C h 2/3

Figure: SR1, SR1-ORDER2, CFL = 0.49, t = 0.25



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 = uv , �3 = uv � cv , �4 = uv + cv ,�5 = u

l

, �6 = ul

� cl

, �7 = ul

+ cl

.



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.



Verification of the fractional step method : test case 1


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

)

α

1−SR1

ρ1

u1

p1

ρ2

u2

p2

α1−PFRAC3

ρ1

u1

p1

ρ2

u2

p2

C0 h1/2

Figure: SR1, PFRAC3, CFL = 0.49, t = 0.15.





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)

α

1−SR1

ρ1

u1

p1

ρ2

u2

p2

α1−PFRAC3

ρ1

u1

p1

ρ2

u2

p2

C0 h1/2

Figure: SR1, PFRAC3 CFL = 0.49, t = 0.25 .





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)

α

1

ρ1

u1

p1

ρ2

u2

p2

C0 h1/2

Figure: PFRAC3, CFL = 0.49, t = 0.25 .


Computation of the two-fluid model Relaxation step

Relaxation step

W̃velocity 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



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



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)

α

l

pl

pv

Figure: Curves of convergence of the pressure relaxation scheme.


Verification of the two-fluid model

Outline

1 Introduction







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/steamFluid 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 ?



Single-phase phenomena of Simpson experiment

0.9999999990.999999990.99999990.9999990.999990.99990.9990.990.94

5

6

7

8

9

10x 10

5

αl

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,



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

αl

c

Formule de WallisExperience 250 HZExperience 500 HZResu numÃ©rique avec 1000−maillesResu numÃ©rique 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

αl

c

Formule de WallisExperience 250 HZExperience 500 HZResu numerique avec 1000−maillesResu numÃ©rique 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


Conclusions and perspectives

Outline

1 Introduction








. 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



Thank you for your attention !

Any questions ?



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IntroductionThe two-fluid model & water hammer Computation of the two-fluid modelEvolution step Relaxation step

Verification of the two-fluid model Conclusions and perspectives

computing water-hammer flows with a two-fluid model · the two-ﬂuid model & water hammer main...

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