Numerical Simulation of Complex and Multiphase Flows.fayssal/CONFS/Exp_Quivy.pdf · Numerical Simulation of Complex and Multiphase Flows 1 Introduction : – Presentation of VF scheme

Numerical Simulation of Complex and Multiphase Flows

Numerical Simulation of Complex and Multiphase Flows.

IGESA. Porquerolles. 18th - 22nd April 2005

Non Homogenous Riemann Solver to simulate two-phaseflows.

K. Mohamed(1), L. Quivy(1),(2), F. Benkhaldoun(1),(2)

(1) LAGA, Université Paris XIII, FRANCE(2) CMLA, ENS de Cachan, FRANCE

Porquerolles 2005 1


1 Introduction :

– Presentation of VF scheme for non hongeneous systems, using flux

values instead of eigenvectors.

– Scheme analysis.

– Numerical results in 1D and 2D.

Porquerolles 2005 2


2 Equations : balance laws

∂U(x, t)

∂t+

d∑

j=1

∂Fj(U(x, t))

∂xj= Q(x, t, U), (1)

x = (x1, x2, ..., xd) ∈ D ⊂ Rd, t > 0,

U : D × R+ −→ Ω : physical values (p components),

Ω open bounded in Rp,

Fj (1 ≤ j ≤ d) : flux functions.

Non homogeneous term Q(x, t, U) (source terms or non conservative

terms).

U(x, 0) = U0(x) : initial condition + boundary conditions.

⇒Modelisation of shallow water or multiphase flows.

Porquerolles 2005 3


1D uniform case :∂U

∂t+∂F (U)

∂x= Q(x,U), 0 < t < T,

U(x, 0) = U0(x), ∀x ∈ Ω ⊂ R.

SRNHR schemeUnj+ 1

2=

1

2(Unj + Unj+1)−

αnj+ 1

2

2Snj+ 1

2

[F (Unj+1)− F (Unj )

]+αnj+ 1

2

2

∆x

Snj+ 1

2

Qnj+ 12

Un+1j = Unj − rn

[F (Unj+ 1

2)− F (Unj− 1

2)]

+ τnQnj ,

where Snj+ 1

2

= maxp=1,...,m

(∣∣λnp,i∣∣ ,∣∣λnp,i+1

∣∣) : local Rusanov velocity,

αnj+ 1

2

real and positive parameter, rn = τn

∆x .

Remark : αnj+ 1

2

is a parameter which aims to control numerical diffusion of

scheme.For example, for linear scalar equation on uniform mesh, αn

j+ 12

= 1

corresponds to Lax-Wendroff scheme.

Porquerolles 2005 4


3 Scheme analysis :Hypothesis : (scalar case)H1) f ′ and f” have a constant sign and does not vanish.H2) u0 has a constant sign and does not vanish. Suppose that0 < um ≤ u0(x) ≤ uM .

Proposition 3.1 Suppose that αnj+ 1

2

= (αnj+ 1

2

)1 =Snj+ 1

2

snj+ 1

2

, ∀j, ∀n, with

Snj+ 12

= max(|f ′(unj )|, |f ′(unj+1)|

)and

snj+ 12

= min(|f ′(unj )|, |f ′(unj+1

)|).

Denoting by αn = sup(j∈Z)

Snj+ 1

2

snj+ 1

2

, αmax =fpMfpm

,

fpm = min(|f ′(um)|, |f ′(uM )|), fpM = max(|f ′(um)|, |f ′(uM )|) and

M = sup | f ′(w) |, w ∈ w inR/ |w| ≤ αmax ‖ u0 ‖∞.Then, under condition αnMrn ≤ 1, SRNHR scheme is TV D, satisfy

maximum principle and then is L∞ stable.

Porquerolles 2005 5


Proposition 3.2 Suppose now that αnj+ 1

2

= (αnj+ 1

2

)2 = rnSnj+ 1

2

. Then

scheme SRNHR writes

unj+ 1

2

= 12 (unj + unj+1)− rn

2

[f(unj+1)− f(unj )

]

un+1j = unj − rn

[f(un

j+ 12

)− f(unj− 1

2

)].

which is secund order Richtmeyer scheme.

Then, the use of limiter theory :

αnj+ 12

= Φj(αnj+ 1

2)1 + (1− Φj) (αnj+ 1

2)2 (2)

where Φj is a limiter function (Superbee, Van-Leer,...).

Porquerolles 2005 6


4 Stationnary states preserving :

For Saint-Venant equations :

∂h

∂t(x, t) +

∂hu

∂x(x, t) = 0

∂hu

∂t(x, t) +

∂(hu2 + gh2

2 )

∂x(x, t) = −gh(x, t)

dz

dx(x)

h0(x), u0(x), z(x) given

Proposition 4.1 Under condition that source term is discretized as

Qnj = − 1

8∆xg(unj−1 + 2unj + unj+1)(zj+1 − zj−1)

SRNHR scheme satisfy exact C-property.

Riemann invariants for Saint-Venant equations are Wk = u+ (−1)k2c.

Porquerolles 2005 7


At step n, for each cell, local Rusanov velo-

city : Snj+ 1

2= max

p

(max

(|λnp,j |, |λnp,j+1|

))

If (Wkj+1 = Wkj ) then

θWk = 0

elseVelocity at interface :

cj+ 12

=√ghj+hj+1

2,

Vj+ 12

=(uj+uj+1)

2,

λkj+ 1

2

= Vj+ 12

+ (−1)k cj+ 12

If (λkj+ 1

2

> 0) then

θWk =Wkj

−Wkj−1

Wkj+1−Wkj

elseθWk =

Wkj+2−Wkj+1

Wkj+1−Wkj

end Ifend If

sj+ 12

= min(|λ1j+ 1

2

|, |λ2j+ 1

2

|)If (sj+ 1

2< ε ) then

sj+ 12

= ε

end IfIf (θW1 < 0 or θW2 < 0) then

φ = 0

else

φ = limiter functionend Ifαnj+ 1

2=

Snj+ 1

2sj+ 1

2

(1− φ) + φrnSnj+ 1

2.

Porquerolles 2005 8


5 1D homogeneous Saint-Venant equations :

Consider a dam break with solution containing a shock wave and a

rarefaction wave.

∂h

∂t(x, t) +

∂(hu)

∂x(x, t) = 0

∂(hu)

∂t(x, t) +

∂(hu2 + gh2

2

)

∂x(x, t) = 0

(3)

with initial conditions : h0(x) =

6 if x ≤ 6

2 if x > 6., and u0(x) = 0, ∀x.

Mesh contains 100 points and results are given at t = 0.4.

Porquerolles 2005 9


0 2 4 6 8 10 122

2.5

3

3.5

4

4.5

5

5.5

6Hauteur de l’eau, SRNHR, t=0.4, np=100

solution exacteSRNHR−alpha limiteROE

Homogeneous Shallow Water,

water level, SRNHR and Roe

schemes.

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5solution exacteSRNHR−alpha limiteROE


water velocity, SRNHR and Roe

schemes.

Porquerolles 2005 10


0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5alpha−limite


variations of αnj+ 1

2.

0 2 4 6 8 10 12−20

−15

−10

−5

0

5

10

15

20Invariant de Riemann 1Invariant de Riemann 2


Riemann invariants.



−7 −6.5 −6 −5.5 −5 −4.5 −4 −3.5 −3 −2.5 −2−6

−5

−4

−3

−2

−1

0

log(dx)

log

(L1

−e

rre

ur)

SRNHR−alpha−cstPENTE=0.887ROEPENTE=0.83RUSANOV PENTE=0.77

np=100

np=200

np=400

np=800

np=1600

np=3200

np=6400

np=12800

FIG. 1 – Homogeneous Shallow Water, water level, error curve.



6 1D homogeneous Euler equations :

Shock tube with initial conditions

ρ0(x, y) =

1 kg/m3 si x ≤ 0

0.01 kg/m3 si x > 0,

u0(x, y) = 0m/s, ∀x ∈ [−10; 10],

P0(x, y) =

105 Pa si x ≤ 0

103 Pa si x > 0,

Mesh : 800 points ; cfl = 0.95 ; t = 0.01.

Comparaison between SRNHR scheme and VFRoe scheme.

First characteristic field is a rarefaction wave which contains a sonic point.

Roe scheme needs to add entropic correction.



−10 −8 −6 −4 −2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Densité du fluide, CFL=0.75, t=0.01, Maill=400

SRNHR ROE

Fluid density ; SRNHR and Roe

schemes.

−10 −8 −6 −4 −2 0 2 4 6 8 100

100

200

300

400

500

600

700Vitesse du fluide, CFL=0.75,T=0.01, MAILLAGE=400

SRNHR ROE

Fluid velocity ; SRNHR and Roe

schemes.



7 1D non homogeneous Saint-Venant

equations :Consider a dam break over a step. Source term describes bottom geometry.

∂h

∂t(x, t) +

∂(hu)

∂x(x, t) = 0

∂(hu)

∂t(x, t) +

∂(hu2 + gh2

2 )

∂x(x, t) = −gh(x, t)

dz

dx(x)

z(x) =

0 if x ≤ 6

1 if x > 6

h0(x) =

5 if x ≤ 6

1 if x > 6.

u0(x) = 0.

(4)



Comparaison between SRNHR and Vazquez (equilibrium) schemes.

Mesh : 100 points ; results are given at t = 0.5.

0 2 4 6 8 10 121

1.5

2

2.5

3

3.5

4

4.5

5Hauteur de l’eau, t=0.5, np= 100

solution exacteSRNHR−alpha limiteVAZQUEZ

Non homogeneous Shallow

Water, water level, SRNHR and

Vazquez schemes.

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5Vitesse de l’eau, t=0.5, np=100

solution exacteSRNHR−alpha limiteVAZQUEZ

Non homogeneous Shallow

Water, water velocity, SRNHR

and Vazquez schemes.



−5 −4.5 −4 −3.5 −3 −2.5 −2−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

log(dx)

log

(L1

−E

rre

ur)

SRNHR−alpha limitéPente=0.82SRNHR−alpha=1.1Pente=0.734VazquezPente=0.51

np=100

np=200

np=400

np=800

np=1600

FIG. 2 – Non homogeneous Shallow Water, error curve, (water level), SRNHR and

Vazquez schemes.



8 2D SRNHR schemeNon homogeneous Saint-Venant equations :

∀(x, y) ∈ Ω ⊂ R2, t ∈ R+,

∂h

∂t(x, y, t) +

∂(hu)

∂x(x, y, t) +

∂hv

∂y(x, y, t) = 0

∂(hu)

∂t(x, y, t) +

∂(hu2 + 12gh2)

∂x(x, y, t) +

∂hv

∂y(x, y, t) = −gh(x, y, t)

dz

dx(x, y)

∂(hv)

∂t(x, y, t) +

∂hv

∂x+∂(hv2 + 1

2gh2)

∂y(x, y, t) = −gh(x, y, t)

dz

dy(x, y)

(5)

with initial conditions

h(x, y, 0) = h0(x, y)

(hu)(x, y, 0) = (hu)0(x, y)

(hv)(x, y, 0) = (hv)0(x, y)

z(x, y) given.



Using projection on normal direction at cell interface (R. Abgrall & al.,

2003) :

Denoting by ~V = (u v)t, ~η = (nx ny)t,

U = ~V · ~η = unx + vny and V = ~V · ~η⊥ = −uny + vnx, where U is the

projection of ~V on ~η and V , the projection of ~V on ~η⊥.

⇒

∂h

∂t+∂hU

∂η= 0

∂hU

∂t+∂(hU2 + 1

2gh2)

∂η+ gh

dz

dη= 0

∂hV

∂t+∂hUV

∂η= 0.



Numerical results :

Initial conditions : h0(x, y) =

6 if x ≤ 6, ∀y ∈ [0; 1]

2 if x > 6, ∀y ∈ [0; 1]

u0(x, y) = v0(x, y) = 0, ∀x ∈ [0; 12], ∀y ∈ [0; 1].

PSfrag replacements

12 m

zl = 0 m

zr = 1 mhr = 2 m

hl = 6 m

(u, v)l = (0, 0) m/s (u, v)r = (0, 0) m/s



Results with unstructured mesh 100× 10.

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

2

2.5

3

3.5

4

4.5

5

5.5

6

Hauteur de l’eau 2D, SRNHR

2

2.5

3

3.5

4

4.5

5

5.5

6

Non homogeneous 2D St-

Venant, water level ; SRNHR

scheme.

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.3

2.6

2.9

3.1

3.4

3.74

4.3

4.6

4.9

5.1

5.4

5.7

x (m)

y (

m)

Isovaleurs de la hauteur d’eau, SRNHR, alpha=1.2, t=0.4

Isolines ; water level ; SRNHR

scheme.



0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

−0.5

0

0.5

1

1.5

2

2.5

3Vitesse de l’eau 2D, SRNHR

0

0.5

1

1.5

2

2.5


Venant, water velocity ; SRNHR

scheme.

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.2

0.39

0.59

0.79

0.98

1.2

1.4

1.6

1.8

2

2.2

2.4 2.6

x (m)

y (

m)

Isovaleurs de la vitesse de l’eau , SRNHR, alpha=1.2, t=0.4

Isolines ; water velocity ;

SRNHR scheme.



0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

−2

0

2

4

6

8

10

Débit 2D, SRNHR

0

1

2

3

4

5

6

7

8

9


Venant, water flow ; SRNHR

scheme.

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 0.67

1.3

2

2.7

3.3

4

4.7

5.3

6

6.7

7.3 8

8.7

x (m)

y (

m)

Isovaleur du débit de l’eau, SRNHR, alpha=1.2, t=0.4

Isolines ; water flow ; SRNHR

scheme.



9 Two phase flow

1D case :

∂U

∂t+∂f(U)

∂x= Q1(x,U) +Q2(x,U)

U(x, 0) = U0(x)

(6)

with

U = (µvρv µvρvuv µlρl µlρlul)t,

f(U) = (µvρvuv µvρvu2v µlρlul µlρlu

2l )t,

Q1(x,U) = (0 − µv∂P

∂x0 − µl

∂P

∂x)t,

Q2(x,U) = (0 µvρvg 0 µlρlg)t.

Non hyperbolic and non conservative problem.



1D Ransom ProblemInitial condition :

∀x ∈ [x0, xl], µv(t = 0) = 0.2,

ul(t = 0) = 10,

uv(t = 0) = 0,

p(t = 0) = 105,

ρv(t = 0) = 1,

ρl(t = 0) = 988, 0638.

Boundary conditions :

– inlet (x0 = 0) :

µv(0, t) = 0.2,

ul(0, t) = 10, uv(0, t) = 0.

– outlet (xl = 12) :

p (12, t) = 105.

g

PSfrag replacements

inlet

outlet

g

12 m

ul = 10 m/svv = 0 m/s

µv = 0.2 m/s

P = 105Pa

gaz

liquid



0 2 4 6 8 10 120.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5analytic solutionSRNHR48pSRNHR120pSRNHR150pRUSANOV3000p

FIG. 3 – Void fraction.



10 Discretization of source termSRNHR scheme :

Unj+ 12

= (νUnj + (1−ν)Unj+1)−αnj+ 1

2

2snj+ 1

2

[f(Unj+1)−f(Unj )

]−αnj+ 1

2

2

∆x

snj+ 1

2

Qnj+ 12

Un+1j = Unj − r

[f(Unj+ 1

2

)− f

(Unj− 1

2

))]

+ ∆tnQnj ,

Qnj =1

2∆x

(µnj)

(Pj+1 − Pj−1) (second step).

Qnj+ 12

=µnj+ 1

2

2

(Pj+1 − Pj)h

(first step).

µnj+ 1

2

: intermediate value between µnj and µnj+1.

To preserve stationnary states, 2 choices :

µnj+ 1

2

= 12 (µnj + µnj+1) : results given before.

µnj+ 1

2

= µnj+ 1

2

obtained with Unj+ 1

2

computed at same step→ better in this

case.



0 2 4 6 8 10 120.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Taux de vide,1D,t=0.6

solution exacteSRNHR−48pSRNHR−100pSRNHR−200pSRNHR−400p

Void fraction ; µnj+ 1

2= µn

j+ 12

.

−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

log(dx)lo

g(L

1−

Err

eu

r)

Courbe d’erreur, Taux de vide,SRNHR, cfl=0.5, t=0.6

SRNHR− choix 1PENTE=0.533

np=24

np=48

np=96

np=200

np=400

Error curve ; Void fraction ;

µnj+ 1

2= µn

j+ 12

.



11 Model with interfacial pressure

Source term writes now :

Q1(x,W ) = (0 −µv∂P

∂x−(P−Pi)∂µv∂x 0 −µl

∂P

∂x−(P−Pi)

∂µl∂x

)t,

Q2(x,W ) = (0 µvρvg 0 µlρlg)t,

with P − Pi = ρv(uv − ul)2 : interfacial pressure.

→ Hyperbolic problem



0 2 4 6 8 10 120.2

0.25

0.3

0.35

0.4

0.45

0.5Taux de vide, 1D,cfl=0.5, t=0.6


Void Fraction (with interfacial

pressure) ; µnj+ 1

2= µn

j+ 12

.

−5.5 −5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

log(dx)

log

(L1

−E

rre

ur)

Courbe d’erreur,Taux de vide avec la pression interfaciale, cfl=0.5, t=0.6

SRNHR −Choix 1PENTE=0.4 np=24

np=48

np=96

np=200

np=400

np=800

np=1600 np=2000

Error curve ; Void Fraction

(with interfacial pressure) ;

µnj+ 1

2= µn

j+ 12

.



0 2 4 6 8 10 120.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Taux de vide,1D, t=0.6


Void Fraction (with

interfacial pressure) ;

µnj+ 1

2= 1

2(µnj + µnj+1).

−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

log(dx)

log

(L1

−E

rre

ur)

Courbe d’erreur, Taux de vide avec la pression interfaciale, cfl=0.5, t=0.6

SRNHR−Choix 2PENTE=0.46 np=24

np=48

np=96

np=200

np=400

np=500

Error curve ; Void Fraction

(with interfacial pressure) ;

µnj+ 1

2= 1

2(µnj + µnj+1).



µnj+ 1

2

= 12 (µnj + µnj+1) µn

j+ 12

= µnj+ 1

2

Classical Refinement until Refinement until

model 150 points 400 points

Interfacial Refinement until Refinement until

pressure 500 points 2000 points



2D case :

∂W

∂t+∂F (W )

∂x+∂G(W )

∂y= Q1(x, , y,W ) +Q2(x, y,W )

W (x, y, 0) = W0(x, y),(7)

with W (x, y, t) = (µlρl µlρlul µlρlvl µvρv µvρvuv µvρvvv)t,

F (W (x, y, t)) = (µlρlul µlρlu2l µlρlulvl µvρvuv µvρvu

2v µvρvuvvv)

t,

G(W (x, y, t)) = (µlρlvl µlρlulvl µlρlv2l µvρvvv µvρvuvvv µvρvv

2v)t,

Q1(x, y,W ) = (0 − µl∂P

∂x− µl

∂P

∂y0 − µv

∂P

∂x− µv

∂P

∂y)t,

Q2(x, y,W ) = (0 µlρlg 0 0 µvρvg 0)t.

ρk, µk, uk, vk : density, void fraction, velocities.P : commun pressure.P = Avρ

γv , ρl = KlP

a, Av, γ, Kl, a : constantes.



µv = 0.6. UK Drag mesh 48× 10.

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

Taux de vide, SRNHR

0.6

0.62

0.64

0.66

0.68

0.7

0.72

Void fraction ; SRNHR scheme

(α = 2).

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

4Isovaleur de taux de vide

Void fraction ; Isolines ; SRNHR

scheme (α = 2).



0 2 4 6 8 10 12

0

0.2

0.4

0.6

0.8

1

10

11

12

13

14

15

16

Vitesse du liquide, SRNHR

10.5

11

11.5

12

12.5

13

13.5

14

14.5

15

15.5

Liquid velocity ; SRNHR scheme

(α = 2).

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

4Isovaleur de la vitesse du liquide

Liquid velocity ; Isolines ;

SRNHR scheme (α = 2).



Conclusion :

– Robust and efficient scheme for non homogeneous systems.

– Do not need calculus of jacobien fields decomposition.

– Accurate results obtained with few mesh points.


Documents

Numerical Simulation of Complex and Multiphase Flows.fayssal/CONFS/Exp_Quivy.pdf · Numerical Simulation of Complex and Multiphase Flows 1 Introduction : – Presentation of VF scheme