A preconditioned Roe-Turkel method for a two-phase
compressible flow model at low Mach number
Marica Pelanti
ENSTA ParisTech
Institute of Mechanical Sciences and Industrial Applications (IMSIA)
UMR 9219 ENSTA PT - EDF - CNRS - CEA
Collaborator: Keh-Ming Shyue, National Taiwan University
Séminaire GT LRC Manon
LJLL, UPMC, April 5, 2016
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
1 / 29
Outline
1 Simulation of liquid-vapor flows
2 Two-phase flow model
6-equation model with pressure relaxation
3 Numerical solution
Wave Propagation Schemes
4 Difficulties at low Mach number
Roe-Turkel method for the two-phase model
5 Conclusions and Perspectives
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
2 / 29
Simulation of liquid-vapor flows
Simulation of liquid-vapor flows
Cavitating flows: formation and dynamic evolution of vapor cavities in a
liquid due to pressure variations.
May involve complex thermo-hydrodynamic processes:
liquid/vapor transition, interfaces, shocks, wave interaction
• Important to account for compressibility of both liquid and gas
phases to correctly capture wave dynamics.
? Compressible multiphase flow models (Temperature non-equilibrium)
? Diffuse-interface approach (interface = artificial mixing zone)
? Finite-Volume Godunov-type schemes (Riemann Solvers)
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
3 / 29
Two-phase flow model
6-equation model with pressure relaxation
1-velocity 6-equation two-phase model with stiff mechanical relaxation
Variant of 6-equation model of Saurel–Petitpas–Berry, JCP 2009.
∂t α1 + ~u · ∇α1 = µ(p1 − p2 ),
∂t (α1 ρ1 ) + ∇ · (α1 ρ1 ~u) = 0,
∂t (α2 ρ2 ) + ∇ · (α2 ρ2 ~u) = 0,
∂t (ρ~u) + ∇ · (ρ~u ⊗ ~u) + ∇ (α1 p1 + α2 p2 ) = 0,
∂t (α1 E1 ) + ∇ · (α1 (E1 + p1 )~u) +Υq,∇q = −pI µ(p1 − p2 ),
∂t (α2 E2 ) + ∇ · (α2 (E2 + p2 )~u) −Υq,∇q = pI µ(p1 − p2 ),
Υq,∇q = ~
u · (Y1 ∇(α2 p2 )−Y2 ∇(α1 p1 )) .
q = [α1 , αk ρk , ρ~
u, αk Ek ]T ; αk = volume fraction; ρk = phasic density; ρ =
P
α k ρ k ; Yk =
αk ρ k
.
ρ
Ek = Ek + 21 ρk ~
u·~
u = phasic total energy; Ek = ρk εk = phasic internal energy; ~
u = velocity.
µ = pressure relaxation parameter; pI =
Z2 p1 +Z1 p2
Z1 +Z2
= interface pressure; pk = pk (Ek , ρk ).
Instantaneous mechanical equilibrium: µ → +∞
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
4 / 29
Two-phase flow model
6-equation model with pressure relaxation
1-velocity 6-equation two-phase model with stiff mechanical relaxation
Variant of 6-equation model of Saurel–Petitpas–Berry, JCP 2009. + thermo-chemical relaxation
∂t α1 + ~u · ∇α1 = µ(p1 − p2 ),
∂t (α1 ρ1 ) + ∇ · (α1 ρ1 ~u) = M,
∂t (α2 ρ2 ) + ∇ · (α2 ρ2 ~u) = −M,
∂t (ρ~u) + ∇ · (ρ~u ⊗ ~u) + ∇ (α1 p1 + α2 p2 ) = 0,
∂t (α1 E1 ) + ∇ · (α1 (E1 + p1 )~u) +Υq,∇q = −pI µ(p1 − p2 ) + Q + eI M,
∂t (α2 E2 ) + ∇ · (α2 (E2 + p2 )~u) −Υq,∇q = pI µ(p1 − p2 ) − Q − eI M,
Υq,∇q = ~
u · (Y1 ∇(α2 p2 )−Y2 ∇(α1 p1 )) .
P = µ(p1 − p2 ) = volume transfer
Q = heat transfer, M = mass transfer for modelling liquid-vapor transition
Q = θ(T2 − T1 ), M = ν(g2 − g1 )
[Pelanti–Shyue JCP 2014]
Here Hp: Q = M = 0. (mechanical cavitation only)
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
4 / 29
Two-phase flow model
Limit pressure equilibrium model
Homogeneous non-equilibrium 6-equation model: Hyperbolic system
p
(frozen) sound speed = cf = Y1 c21 + Y2 c22 , ck = phasic sound speed, Yk =
αk ρ k
ρ
For µ → ∞ 6-eq. model reduces to (hyperbolic) 5-eq. pressure equilibrium model
∂t α1 + ~
u · ∇α1 = α1 α2
ρ2 c2 −ρ1 c1
α2 ρ1 c1 +α1 ρ2 c2
∇·~
u,
[Kapila et al. 2001]
∂t (α1 ρ1 ) + ∇ · (α1 ρ1 ~
u) = 0,
∂t (α2 ρ2 ) + ∇ · (α2 ρ2 ~
u) = 0,
∂t (ρ~
u) + ∇ · (ρ~
u⊗~
u) + ∇p = 0,
∂t E + ∇ · ((E + p)~
u) = 0.
Wood’s (equilibrium) sound speed cW
r cW =
1
ρ
α1
ρ1 c 2
1
+
α2
ρ2 c2
2
−1
cW ≤ cf sub-characteristic condition
Note: 6-eq. model numerically advantageous and allows incorporating thermo-chemical
relaxation in phasic energy eqs.
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
5 / 29
Numerical solution
Fractional step method
Numerical solution
Hyperbolic (non-conservative) system with source term:
∂t q + ∇ · F + ς(q, ∇q) = ψµ (q).
q = [α1 , α1 ρ1 , α2 ρ2 , ρ~
u, α1 E1 , α2 E2 ]T , ς(q, ∇q) = [~
u · ∇α1 , 0, 0, 0, Υq,∇q , −Υq,∇q ]T ,
F(q) = [0, α1 ρ1 ~
u, α2 ρ2 ~
u, ρ~
u⊗~
u + (α1 p1 + α2 p2 )I, α1 (E1 + p1 )~
u, α2 (E2 + p2 )~
u]T ,
ψµ (q) = [P, 0, 0, 0, −pI P, pI P]T , P = µ(p1 − p2 )
Fractional step algorithm:
1. Homogeneous hyperbolic system. Solve via Finite Volume Wave
Propagation schemes ∂t q + ∇ · F(q) + ς(q, ∇q) = 0.
2. Pressure relaxation (mechanical equilibrium).
Solve in the limit µ → +∞ system of ODEs ∂t q = ψµ (q).
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
6 / 29
Numerical solution
Wave Propagation FV Schemes
Wave Propagation Schemes
CLAWPACK software
[LeVeque 1997]
(Homogeneous) Hyperbolic System (1D): ∂t q + A(q)∂x q = 0.
Godunov-type method: Use solution of Riemann problems at cell interfaces xi+1/2
System + I.C.
(
Qn
if x ≤ xi+1/2 ,
i
q(x, tn ) =
Qn
i+1 if x > xi+1/2 .
Qn
i−1
Qn
i x
i+ 1
2
Qn
i+1
l
Approximate Riemann Solver → {Wi+1/2
, sli+1/2 } = Riemann pb. solution at xi+1/2
W l = waves propagating at speeds sl
= Qn
Qn+1
i −
i
1D:
A± ∆Qi+1/2 =
P
∆t
(A+ ∆Qi−1/2
∆x
l
±
l
l (si+1/2 ) Wi+1/2
+ A− ∆Qi+1/2 )
= left/right-going fluctuations at xi+1/2
Note: For conservation laws ∂t q + ∂x f (q) = 0, A(q) = f 0 (q):
Qn+1
= Qn
i −
i
n
n
∆t
(Fi+1/2
−Fi−1/2
),
∆x
Fi+1/2 = f (Qi , Qi+1 ) = numerical flux at xi+1/2
A+ ∆Qi+1/2 = Fi+1/2 − f (Qi ),
Marica Pelanti (ENSTA ParisTech)
A− ∆Qi+1/2 = f (Qi+1 ) − Fi+1/2
Séminaire GT LRC Manon
05/04/2016
7 / 29
Numerical solution
Roe-type solver
Roe-type solver
2-phase system: ∂t q + ∂x f (q) + ς(q, ∂x q) = 0, q = [α1 , α1 ρ1 , α2 ρ2 , ρu, α1 E1 , α2 E2 ]T .
Quasi-linear form: ∂t q + A(q)∂x q = 0.
Solve exactly R. P. with data q` , qr for a linearized system: ∂t q + Â(q` , qr )∂x q = 0.
 = Â(q` , qr ) = Roe matrix,
 = A(q̂),
g1 , Y
\
\
q̂ = (û, Ŷ1 , uY
1 H1 , Y
2 H2 ),
defined to ensure conservation of α1 ρ1 , α2 ρ2 , ρu, E = α1 E1 + α2 E2 :
P6
j=1
Âξj ∆q (j) = ∆f (ξ) ,
∆(·) = (·)r − (·)` ,
û =
ξ = 2, 3, 4,
and
P6
j=1 (Â5j
+ Â6j )∆q (j) = ∆fE ,
fE (q) = u(E + α1 p1 + α2 p2 ) = flux function of mixture total energy E.
√
√
u` ρ` + ur ρr
,
√
√
ρ` + ρr
Ŷk =
√
√
Y k ` ρ` + Y k r ρr
,
√
√
ρ` + ρr
√
√
k )r ρr
dk = (uYk )` √ρ` + (uY
uY
,
√
ρ` + ρr
Y\
k Hk =
gk = 1 ûŶ + uY
dk ,
uY
2
√
√
(Yk Hk )` ρ` + (Yk Hk )r ρr
,
√
√
ρ` + ρr
k = 1, 2.
g1 + uY
g2 = û.
Note Ŷ1 + Ŷ2 = 1 and uY
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
8 / 29
Numerical solution
sl = λ̂l ,
Roe solver speeds and waves:
Roe-type solver
W l = ζ̂l r̂l ,
l = 1, . . . 6.
λ̂1 = û − ĉ , λ̂2 = λ̂3 = λ̂4 = λ̂5 = û , λ̂6 = û + ĉ ,
q
2
2
d2
[
\ û2
ĉ = Yd
Y
1 c1 + Y2 c2 ,
k ck = κk Yk Hk − 2 Ŷk + χk Ŷk .
Roe eigenvalues:
Roe right eigenvectors:

0

Ŷ1


Ŷ2

R̂ = 
û − ĉ

 \
g1 ĉ
 Y1 H1 − uY
g2 ĉ
\
Y2 H2 − uY
(SG EOS pk = κk Ek + χk ρk − Πk )
0
0
0
0
2
−κ
κ
1
2
−χ
κ
1
1
0
0
1
û
κ2
+κ
1
0
û2
2
0
1
0
û
1
−χ
+
κ
1
1
0
0
0
û2
2
Π1 −Π2
κ1
0
Ŷ1
Ŷ2
û + ĉ
g
\
Y
1 H1 + uY1 ĉ
0
g
\
Y
2 H2 + uY2 ĉ
0





.



ζ̂ = R̂−1 ∆q = wave strengths:
ζ̂1,6 =
ζ̂3 =
ζ̂2 =
∆pm ∓ĉρ̂∆u
,
2ĉ2
pm = α1 p1 + α2 p2 ,
m
∆(α2 ρ2 ) − Ŷ2 ∆p
, ζ̂4 = ∆(α1 ρ1 )
ĉ2
∆pm \
g2 ∆u.
∆(α2 E2 ) − ĉ2 Y2 H2 − ρ̂ uY
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
ρ̂ =
−
√
ρ` ρr ,
m
Ŷ1 ∆p
,
ĉ2
ζ̂5 = ∆α1 ,
05/04/2016
9 / 29
Numerical solution
Stiff mechanical relaxation
Stiff mechanical relaxation
Solve in the limit µ → +∞ system of ODEs, q = [α1 , α1 ρ1 , α2 ρ2 , ρ~
u, α1 E1 , α2 E2 ]T ,
ψµ (q) = [P, 0, 0, 0, −pI P, pI P]T ,
∂t q = ψµ (q),
P = µ(p1 − p2 ).
(·)0 = quantities at initial time coming from solution homogeneous system
(·)∗ = quantities at final time: mechanical equilibrium p∗1 = p∗2 = p∗I = p∗
Clearly: (αk ρk )∗ = (αk ρk )0 , k = 1, 2, (ρ~
u)∗ = (ρ~
u)0 , E ∗ = (α1 E1 )∗ + (α2 E2 )∗ = E 0 .
Now approximate pI as convex average: p̄I = (1 − ϑ)p0I + ϑp∗ , ϑ ∈ [0, 1], then
(αk Ek )∗ − (αk Ek )0 = (αk Ek )∗ − (αk Ek )0 = (−1)k p̄2I (α1∗ − α10 ),
E.g. ϑ = 1/2 ⇒ p̄I =
∗
p0
I +p
2
k = 1, 2.
(equiv. Hp. pI linear with α1 ), ϑ = 1 ⇒ p̄I = p∗ .
⇒ Two equations for equilibrium values α1∗ and p∗ . (SG EOS: quadratic eq. for p∗ )
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
10 / 29
Numerical solution
Stiff mechanical relaxation
Stiff mechanical relaxation
Solve in the limit µ → +∞ system of ODEs, q = [α1 , α1 ρ1 , α2 ρ2 , ρ~
u, α1 E1 , α2 E2 ]T ,
ψµ (q) = [P, 0, 0, 0, −pI P, pI P]T ,
∂t q = ψµ (q),
P = µ(p1 − p2 ).
(·)0 = quantities at initial time coming from solution homogeneous system
(·)∗ = quantities at final time: mechanical equilibrium p∗1 = p∗2 = p∗I = p∗
Clearly: (αk ρk )∗ = (αk ρk )0 , k = 1, 2, (ρ~
u)∗ = (ρ~
u)0 , E ∗ = (α1 E1 )∗ + (α2 E2 )∗ = E 0 .
Now approximate pI as convex average: p̄I = (1 − ϑ)p0I + ϑp∗ , ϑ ∈ [0, 1], then
(αk Ek )∗ − (αk Ek )0 = (αk Ek )∗ − (αk Ek )0 = (−1)k p̄2I (α1∗ − α10 ),
E.g. ϑ = 1/2 ⇒ p̄I =
∗
p0
I +p
2
k = 1, 2.
(equiv. Hp. pI linear with α1 ), ϑ = 1 ⇒ p̄I = p∗ .
⇒ Two equations for equilibrium values α1∗ and p∗ . (SG EOS: quadratic eq. for p∗ )
Note: Mixture-energy-consistent scheme thanks to total-energy-based formulation
? Consistency with conservation law of mixture total energy E at the discrete level
? Consistency by construction with mixture EOS E 0 = α1∗ E1 (p∗ , ρ∗1 ) + α2∗ E2 (p∗ , ρ∗2 )
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
10 / 29
Numerical experiments
Two-phase flow model with heat and mass transfer
Two-phase model with heat and mass transfer - Numerical tests
Cavitation wake generated by a high-speed underwater projectile (u = 600 m/s)
600 × 200 cells, CFL = 0.5, 2nd order.
High-pressure fuel (dodecane) injector
400 × 160 cells, CFL = 0.5, 2nd order
[Pelanti–Shyue JCP 2014]
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
11 / 29
Difficulties at low Mach number
Difficulties of compressible flow solvers at low Mach number
Upwind schemes: highly efficient for transonic and supersonic flow regimes
Well-known difficulties at low Mach number regimes:
• Severe loss of accuracy and efficiency as M =
|~
u|
c
→0
• Time step for stability of time-explicit schemes ∆t = O(M )
(CFL constraint)
→ Critical for liquid-gas flows due to large & rapid variation of acoustic impedance.
Mach number may range from
M 1 in nearly incompressible
liquid to large values M > 1 in
liquid-gas mixture.
Wood’s sound speed cW
(pressure equilibrium)
1
ρc2
W
=
α1
ρ1 c2
1
+
α2
ρ2 c2
2
From Fundamentals of Multiphase Flows, C. E. Brennen
Here: Focus on loss of accuracy problem as M → 0
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
12 / 29
Difficulties at low Mach number
Literature
Extensive literature on the asymptotic behavior of Euler’s and compressible Navier–Stokes eqs.
in the low Mach limit (continuous and discrete problem) and numerical strategies
Work on (genuine) multiphase compressible flow models still limited
A. J. Chorin, Bull. Amer. Math. Soc. (1967)
S. Klainerman and A. Majda, Comm. Pure Appl. Math. (1981)
B. van Leer, W. Lee, and P. Roe, AIAA Technical Papers (1991)
E. Turkel, J. Comput. Phys. (1987), Appl. Numer. Math. (1993)
Y.-H. Choi and C. Merkle, J. Comput. Phys. (1993)
G. Volpe, AIAA J. (1993)
R. Klein, J. Comput. Phys. (1995)
B. Müller, J. Eng. Math. (1998)
H. Bijl and P. Wesseling, J. Comput. Phys. (1998)
H. Guillard and C. Viozat, Computers. & Fluids (1999)
S. Jin, SIAM J. Sci. Comput. (1999)
A. Meister, SIAM J. Appl. Math. (2000)
G. Metivier and S. Schochet, Arch. Ration. Mech. Anal. (2001)
C.-D. Munz, S. Roller, R. Klein, and K. J. Geratz, Computers. & Fluids (2003)
H. Guillard and A. Murrone, Computers & Fluids (2004)
P. Birken and A. Meister, Proc. Appl. Math. Mech. (2005)
T. Alazard, Arch. Ration. Mech. Anal. (2006)
D. Vidovic, A. Segal, and P. Wesseling, J. Comput. Phys. (2006)
M. Bilanceri, F. Beux, and M. V. Salvetti, Computers & Fluids (2006)
X.-S. Li and C.-W. Gu, J. Comput. Phys. (2008)
A. Murrone and H. Guillard, Computers and Fluids (2008)
H. Guillard, Computers & Fluids (2009)
X.-S. Li, C.-W. Gu, and J.-Z. Xu, Computers & Fluids (2009)
B. Braconnier and B. Nkonga, J. Comput. Phys. (2009)
F. Rieper and G. Bader, J. Comput. Phys. (2009)
S. Dellacherie J. Comput. Phys. (2010)
S. Dellacherie, P. Omnes, and F. Rieper, J. Comput. Phys. (2010)
P. Degond and M. Tang, Communications in Computational Physics (2011)
F. Rieper, J. Comput. Phys. (2010), J. Comput. Phys. (2011)
S. Le Martelot, B. Nkonga, and R. Saurel, J. Comput. Phys. (2013)
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
13 / 29
Difficulties at low Mach number
Two-phase model in the low Mach number limit
Two-phase flow model in the low Mach number limit
Asymptotic behavior of solutions for M∗ → 0: non-dimensionalization of equations
+ expansion in powers of M∗ , (·) = (·)[0] + (·)[1] M∗ + (·)[2] M∗2 + . . . + . . .
Physical flow model approximated by 6-equation model with µ → ∞:
5-equation pressure equilibrium model (equilibrium sound speed cW )
Asymptotic limit 5-equation model for MW∗ =
∇p[0] = ∇p[1] = 0
⇒
|~
u∗ |
cW∗
→0
[Murrone–Guillard 2008]
2
p(~x, t) = p[0] (t) + p[2] (~
x, t)MW∗
+ ....
Order 1 system (Hp. p[0] (t) = p0 = const.)
[0]
[0]
ρk + ~
u[0] ∇ · ρk = 0, k = 1, 2,
∂t ~
u[0] + ∇ · (~
u[0] ⊗ ~
u[0] ) +
∇ · u~[0] = 0,
[0]
1
∇p[2]
ρ[0]
= 0,
[0] [0]
[0] [0]
ρ[0] = α1 ρ1 + α2 ρ2 ,
[0]
∂t α1 + ~
u[0] · ∇α1 = 0,
Limit model for both 5-eq. and 6-eq. model with µ → ∞ as Mf ≤ MW → 0.
Hp. Well-prepared initial data: p(~
x, 0) = p0 +O(M∗2 ), ~
u(~
x, 0) = ~
u0 (~
x) + O(M∗ ), ∇·~
u0 = 0.
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
14 / 29
Difficulties at low Mach number
Two-phase model in the low Mach number limit
Continuous vs. discrete 2-phase model in the low Mach number limit
As for Euler’s equations, in the limit MW ∗ → 0 [Guillard–Viozat 99, Murrone–Guillard 08]
Discrete solutions support pressure perturbations of wrong order of magnitude in MW∗
• Continuous physical flow model:
• Upwind space discretizations:
2
p(~
x, t) = p[0] (t) + p[2] (~
x, t)MW∗
+ ....
p(~
x, t) = p[0] + p[1] (~
x, t)MW∗ + . . .
For both discretizations of the 5-eq. model and of the 6-eq. model with µ → ∞
Failure to approximate incompressible limit linked to this inconsistency.
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
15 / 29
Difficulties at low Mach number
Two-phase model in the low Mach number limit
Continuous vs. discrete 2-phase model in the low Mach number limit
As for Euler’s equations, in the limit MW ∗ → 0 [Guillard–Viozat 99, Murrone–Guillard 08]
Discrete solutions support pressure perturbations of wrong order of magnitude in MW∗
• Continuous physical flow model:
• Upwind space discretizations:
2
p(~
x, t) = p[0] (t) + p[2] (~
x, t)MW∗
+ ....
p(~
x, t) = p[0] + p[1] (~
x, t)MW∗ + . . .
For both discretizations of the 5-eq. model and of the 6-eq. model with µ → ∞
Failure to approximate incompressible limit linked to this inconsistency.
Scheme for the 6-equation model: low Mach number difficulty arises in the Step 1
→ approximation of the non-equilibrium homogeneous system
Useful to analyze asymptotic behaviour of continuous homogeneous 6-eq. model:
pm = α1 p1 + α2 p2 = effective pressure
pm (~
x, t) =
[0]
pm (t)
Note: Mf =
|~
u|
cf
+
≤
[2]
2
pm (~x, t)Mf∗
|~
u|
cW
+ ...
[0]
[1]
∇pm = 0, ∇pm = 0
⇒
as Mf∗ ≤ MW∗ → 0
= MW (sub-characteristic condition cW ≤ cf )
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
15 / 29
Difficulties at low Mach number
Two-phase model in the low Mach number limit
Roe-type method semi-discrete 2D equations
2D wave propagation scheme in semi-discrete form
1
dqij
1
+ ∆x
A+ ∆Qi−1/2,j +A− ∆Qi+1/2,j + ∆y
B+ ∆Qi,j−1/2 + B− ∆Qi,j+1/2 = 0,
dt
A± ∆Q, B± ∆Q = fluctuations in x and y directions (plane wave Riemann problems).
Hp: ∆x = ∆y. Reference cell : J = (i, j), ν(J) = {(i − 1, j), (i + 1, j), (i, j − 1), (i, j + 1)},
∆JK (·) ≡ (·)J − (·)K , ~
nJK = unit normal vector to interface JK from J to K, U = ~
u·~
n.
Non-dimensionalization + asymptotic expansions in powers of Mf∗
→ analogous results as for Roe’s scheme for Euler eqs. with pm playing the role of p
[0]
Hp. p1 = p2 = pm = p0 on boundaries ⇒ pm,J = const. ∀J.
−1
Then momentum equation at order Mf∗
gives
P
[1]
[0] [0]
nJK + ρ̂JK ĉJK ~nJK ∆JK U [0] = 0.
K∈ν(J) pm,K ~
[1]
→ admits non-constant solutions for order 1 pressure pm .
⇒ solutions of discrete equations contain pressure perturbations O(Mf∗ )
2
while continuous model supports perturbations O(Mf∗
).
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
16 / 29
Low Mach number preconditioning
Low Mach number preconditioning
Classical low Mach strategy: preconditioning of the numerical dissipation term V∆Q
associated to the spatial discretization of the convective portion of the system
For (conservative) Euler’s equations ∂t q + ∂x f (q) = 0, q = [ρ, ρu, E]T :
Semi-discrete scheme:
dqi
dt
Numerical flux: Fi+1/2 =
+
1
(Fi+1/2
∆x
1
(f (qi )
2
− Fi−1/2 ) = 0
+ f (qi+1 )) −
1
2
V∆Qi+1/2
V∆Qi+1/2 = Θi+1/2 (qi+1 − qi ), Θ = numerical dissipation matrix
Preconditioning: Correct dissipation term via preconditioning matrix P .
Note: scheme retains conservation property and time-consistency
Roe-Turkel scheme for Euler equations [Guillard–Viozat 1999]
Roe’scheme: Θ = |Â| replaced by ΘP = P −1 |P Â|.
Turkel’s preconditioner: P ϕ = diag(β 2 , 1, 1), β = O(M ) if M ≤ 1, β = 1 otherwise,
for entropic variables ϕ = [p, u, s]T → P (q) =
∂q
P ϕ ∂ϕ
∂ϕ
∂q
Discrete eqs. with preconditioned dissipation recover correct scaling of pressure
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
17 / 29
Low Mach number preconditioning
Roe-Turkel method for the two-phase model
Wave propagation schemes in viscous form
Hyperbolic system (1D): ∂t q + A(q)∂x q = 0.
Semi-discrete WP scheme:
A± ∆Qi+1/2 =
P
dqi
dt
1
(A+ ∆Qi−1/2
∆x
+
l
±
l
l (si+1/2 ) Wi+1/2
,
W l , sl = waves and speeds
Viscous form: A± ∆Qi+1/2 = 21 ∆fˆi+1/2 ±
∆fˆi+1/2 =
P
l
l
sli+1/2 Wi+1/2
,
+ A− ∆Qi+1/2 ) = 0.
1
2
V∆Qi+1/2 =
V∆Qi+1/2 .
P
l
l
|sli+1/2 |Wi+1/2
= dissipation term.
Note: ∆fˆ(ξ) = ∆f (ξ) ∀ conserved quantity ξ
Roe-type scheme for 2-phase model: V∆Qi+1/2 = |Âi+1/2 |(qi+1 − qi )
|Â| = dissipation matrix
Marica Pelanti (ENSTA ParisTech)
(Â = Roe matrix )
Séminaire GT LRC Manon
05/04/2016
18 / 29
Low Mach number preconditioning
Roe-Turkel method for the two-phase model
Wave propagation schemes in viscous form
Hyperbolic system (1D): ∂t q + A(q)∂x q = 0.
Semi-discrete WP scheme:
A± ∆Qi+1/2 =
P
dqi
dt
1
(A+ ∆Qi−1/2
∆x
+
l
±
l
l (si+1/2 ) Wi+1/2
,
W l , sl = waves and speeds
Viscous form: A± ∆Qi+1/2 = 21 ∆fˆi+1/2 ±
∆fˆi+1/2 =
P
l
l
sli+1/2 Wi+1/2
,
+ A− ∆Qi+1/2 ) = 0.
1
2
V∆Qi+1/2 =
V∆Qi+1/2 .
P
l
l
|sli+1/2 |Wi+1/2
= dissipation term.
Note: ∆fˆ(ξ) = ∆f (ξ) ∀ conserved quantity ξ
Roe-type scheme for 2-phase model: V∆Qi+1/2 = |Âi+1/2 |(qi+1 − qi )
|Â| = dissipation matrix
(Â = Roe matrix )
Preconditioning: correct V∆Q at low Mach number → V P∆Q
P
lP
V P∆Qi+1/2 = l |slP
| Wi+1/2
, W lP, slP = preconditioned waves and speeds
i+1/2
Preconditioned Roe-type scheme: replace |Â| by P −1 |P Â|.
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
18 / 29
Low Mach number preconditioning
Roe-Turkel method for the two-phase model
Turkel-type preconditioner for the two-phase model
Conservative variables: q = [α1 , α1 ρ1 , α2 ρ2 , ρu, α1 E1 , α2 E2 ]T .
Choice of entropic variables: ϕ = [pm , u, s1 , s2 , Y1 , α1 ]T
pm = α1 p1 + α2 p2 = effective pressure
u = velocity, sk = phasic entropy, Y1 =
α1 ρ1
=
ρ
mass fraction, α1 = volume fraction.
Turkel-type preconditioner: P ϕ= diag(β 2 , 1, 1, 1, 1, 1)
→ P (q) =
∂q
P φ ∂φ
∂ϕ
∂q
β = O(M∗ ) for M∗ ≤ 1, β = 1 otherwise.
Note: for Mf ≤ MW 1 can set M∗ = Mf but need M∗ = MW if transonic regions
Eqs. for ϕ:
∂t pm + u ∂x pm + ρ c2f ∂x u = 0,
∂t u + u ∂ x u +
1
∂ p
ρ x m
c2f = Y1 c21 + Y2 c22 ,
c2k =
∂pk ∂ρk s
k
= 0,
∂t s1 + u ∂x s1 = 0,
∂t s2 + u ∂x s2 = 0,
∂t Y1 + u ∂x Y1 = 0,
∂t α1 + u ∂x α1 = 0.
Only acoustic waves are altered by preconditioning
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
19 / 29
Low Mach number preconditioning
Roe-Turkel method for the two-phase model
Roe-Turkel scheme - Preconditioned waves and speeds
A±P ∆Q = 12 ∆fˆi+1/2 ±
V P∆Qi+1/2
P
V P∆Qi+1/2 , ∆fˆi+1/2 = 6l=1 sl W l , sl = λ̂l , W l = ζ̂l r̂l
P
P6
−1
lP
lP
˜P
= Pi+1/2
|Pi+1/2 Âi+1/2 |(qi+1 − qi ) = 6l=1 ζ̂lP |λ̂P
l |r̂l =
l=1 |s |W
1
2
lP
= ζ̂lP r̂˜lP, r̂˜lP = P −1 r̂lP, ζ̂ P = R̂P −1∆q
slP = λ̂P
l, W
P
{λ̂P
l , r̂l } = eigenstr. of P Â
Only acoustic waves 1,6 are corrected at low Mach number, interface waves unaltered
ζ̂1,6 =
1
2ĉ
λ̂1,6 = û ∓ ĉ
∆pm
∓ ρ̂∆u
ĉ
0

Ŷ
1


Ŷ2

=
û ∓ ĉ

 \
g1 ĉ
 Y1 H1 ∓ uY
g
\
Y
2 H2 ∓ uY2 ĉ

r̂1,6
Xβ = ((1 −
β 2 )û)2
+
(2βĉ)2
→
λ̂P
1,6 =
→
P
ζ̂1,6
= √1
(1 + β 2 )û ∓
Xβ









1
2
p Xβ
∆pm
2
∓(λ̂P
1,6 −ûβ )
∓ ρ̂∆u
0

Ŷ1


Ŷ2

2
=
û + (λ̂P
1,6 − ûβ )

 \
P
g
 Y1 H1 + uY1 (λ̂1,6 − ûβ 2 )
g P − ûβ 2 )
\
Y
2 H2 + uY2 (λ̂

→
P
r̂˜1,6









1,6
β = min(max(M̃ , ), 1), β ≥ 1 ⇒ Roe
Note. Recovers Roe-Turkel scheme for Euler eqs. with pm playing the role of p.
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
20 / 29
Low Mach number preconditioning
Roe-Turkel method for the two-phase model
Asymptotic analysis of Roe-Turkel discrete equations
Similar to Roe-Turkel method for Euler eqs., effect of preconditioning: terms of order
−2
−1
Mf∗
appear in phasic mass and energy equations and all terms Mf∗
disappear.
[0]
X
K∈ν(J)
[0]
Ŷk,JK
[0]
r
, ∆JK pm = 0 ,
[0]
X̂
X
K∈ν(J)
β̃,JK
X
[0]
pm,K ~
nJK +
[0]
2~
ûJK
K∈ν(J)
k = 1, 2, β̃ =
β
,
Mf∗
β̃,JK
[0]
ÛJK ~
nJK
+
r
[0]
X̂
Y\
[0]
k Hk JK
r
∆JK pm = 0,
[0]
X̂
[0]
β̃,JK
[0]
∆JK pm = 0, X̂
2
= (Û [0] )2 + 2β̃ĉ[0] .
β̃,JK
[0]
[1]
Note: eqs. of same form with pm replaced by pm .
[0]
[1]
Under suitable Hp. on boundaries: pm,J , pmJ = const. ∀J
[2]
2
⇒ pm (~
x, t) = p0 + pm (~x, t)Mf∗
+ . . . as for continuous 6-eq. homogeneous model
Under suitable Hp. on boundary and initial conditions perturbations of p1 , p2 , α1
2
coming from solution homogeneous system are O(Mf∗
).
→ equilibrium pressure computed from relaxation step support perturbations of
2
O(MW∗
), MW∗ ≥ Mf∗ , consistent with continuous relaxed model.
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
21 / 29
Low Mach number preconditioning
Numerical experiments
Two-phase liquid-gas channel flow
Liquid water with initial small amount of gas, αg0 = 10−3 , p0 = 106 Pa.
cW0 = 1000.17 m/s, cf0 = 1625.51 m/s
u0 = (2, 5, 10, 20) m/s → M0 =
M0 = 0.01999
u0
cW0
= (0.00199, 0.0049, 0.0099, 0.0199)
Roe’s scheme
Roe-Turkel
100 × 25 cells.
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
22 / 29
Low Mach number preconditioning
Numerical experiments
Two-phase liquid-gas channel flow
u0 = 2 m/s, M0 = 0.00199
100 × 25 cells.
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
23 / 29
Low Mach number preconditioning
Numerical experiments
Two-phase liquid-gas channel flow
Transitional interval toward stationary conditions, u0 = 2 m/s, M0 = 0.00199
Bottom pressure fluctuations
Marica Pelanti (ENSTA ParisTech)
Bottom volume fraction fluctuations
Séminaire GT LRC Manon
05/04/2016
24 / 29
Low Mach number preconditioning
Numerical experiments
Two-phase liquid-gas channel flow
δpmax =
pmax −pmin
p0
= maximum pressure fluctuations
δαg max = αg max − αg min = maximum volume fraction fluctuations
u0 [m/s]
M0
δpmax (Roe-Turkel)
δpmax (Roe)
δαg max (Roe-Turkel)
δαg max (Roe)
20
0.01999
0.24553
0.59704
2.9309 × 10−4
6.1179 × 10−4
10
0.00999
0.06135
0.27800
6.3731 × 10−5
2.6857 × 10−4
5
0.00499
0.01535
0.13519
1.5473 × 10−5
1.3122 × 10−4
2
0.00199
0.00247
0.05373
2.4742 × 10−6
5.2907 × 10−5
Roe-Turkel: δpmax , δαg max = O(M02 ) → consistent with asymptotic behaviour continuous model
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
25 / 29
Low Mach number preconditioning
Numerical experiments
Transonic liquid-gas channel flow
Min = 0.0738. Shock formation in the divergent.
pin = 1.3 × 107 Pa, uin = 100 m/s, pout = 9 × 106 Pa, αg0 = 10−3
Note: To handle all regimes
need to set β = O(MW )
201 × 25 cells
—– Sonic line MW = 1
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
26 / 29
Low Mach number preconditioning
Numerical experiments
Nozzle flow with liquid-vapor transition
Heat transfer Q and mass transfer M included in the 6-equation model.
Q = θ(T2 − T1 ), M = ν(g2 − g1 ), θ, ν → +∞ at metastable interfaces Tliq > Tsat
Nozzle flow with abrupt outlet pressure drop → growth of vapor cavities
t = 0 : subsonic steady flow, Min = 0.0230, pin = pout = 106 Pa, Yv0 = 5.482 × 10−6 .
t > 0 : outlet pressure abruptly decreased to pout = 0.7 × 106 Pa.
100 × 24 cells
—– Sonic line Meq = 1
Mach number ranges from Mmin = 0.0195 to Mmax = 6.4595. Note: ceq ≤ cW ≤ cf .
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
27 / 29
Conclusions and Perspectives
Conclusions and Perspectives
New preconditioned method for compressible two-phase flow model with relaxation
? Roe-type scheme with Turkel-type preconditioning for low Mach number regimes
Extension of Roe-Turkel method of Guillard–Viozat 1999
Discrete eqs. support pressure fluctuations O(M 2 ) as continuous relaxed model.
Accuracy at low Mach number, preserves performance for higher Mach number.
• Preconditioning of the numerical dissipation matrix cures the accuracy problem.
However: severe stability constraint with time-explicit schemes: ∆t = O(M 2 )
(Preconditioned dissipation matrix has eigenvalue growing as M −2 [Birken–Meister])
Planned work: development of efficient implicit time integration techniques
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
28 / 29
Merci beaucoup de votre attention !!
ENSTA ParisTech cavitation tunnel
Marica Pelanti (ENSTA ParisTech)
Séminaire GT LRC Manon
05/04/2016
29 / 29