A class of compressible multiphase flow models

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A class of compressible multiphase flow models

Jean-Marc Hérard

To cite this version:

Jean-Marc Hérard. A class of compressible multiphase flow models. Comptes Rendus. Mathématique,

Académie des sciences (Paris), 2016, 354 (9), pp.954-959. �10.1016/j.crma.2016.07.004�. �hal-01348880�

A class of compressible multiphase flow models

Jean-Marc H´erard

1 EDF R&D, 6 quai Watier, F-78400 Chatou

2 I2M, Aix Marseille Universit´e, 39 rue Joliot Curie, F-13453 Marseille

Keywords:Multi-phase flows/ entropy inequality / hyperbolic models / closure laws / relaxation effects.

Abstract:We propose in this note a class of entropy-consistent hyperbolic models for multi-phase barotropic flows. Relevant closure laws are derived and discussed.

Introduction: The accurate modelling of multiphase flows with mixtures involving several components is crucial for several highly unsteady applications for petroleum engineering, but also for nuclear safety applications and more generally for thermal- hydraulics. Many studies in the nuclear framework, for instance those that aim at predicting hydrogen risk, vapor explosion, or similar fast transient situations, require models that comply with some basic specifications, in order to handle strong rarefac- tion waves as well as shock waves. Rather recent proposals have arisen within the last twenty years, at least for two-phase flow models. Some among them ([1, 2, 5, 9, 10, 14]), which rely on the two-fluid approach, enable meaningful unsteady computations.

However, only few multiphase flow models have emerged in the past in order to tackle three-phase flows or even multiphase situations. Some among the latter assume a sys- tem of PDE for mass balance of components, while simplified momentum equations are considered (see for instance [3, 7] for flows in reservoirs). More recently, a couple of contributions, among which we may cite [11, 12, 15, 16], has given focus on the modeling of mass, momentum and energy balances for three-phase flow situations, and even more. The main objective of the present contribution is to give some new insight on this particular topic, while considering three-phase or four-phase models in order to account for unsteady compressible flows. We only consider here barotropic situations for sake of simplicity. We first give emphasis on the modelling of interfacial transfer of momentum for multi-component flows. Next we discuss relevant closure laws for pres- sure and velocity relaxation terms, but also for the interface velocity that governs the evolution of statistical fractions. Finally we give some closure laws for mass transfer.

1 A class of compressible multiphase flow models

We considerNdistinct compressible phases. We also assume that components are -at least slightly- compressible. Thus the starting point is the governing set of equations:







∂_t(α_k) +Vi(Y)∂_x(α_k) =φ_k(Y);

∂t(m_k) +∂x(m_kU_k) =0 ;

∂_t(m_kU_k) +∂_x m_kU_k²+α_kP_k(ρ_k)

+Σ^N_l=1,l6=kΠ_kl(Y)∂_x(α_l) =m_kS_k(Y). (1)

where we notem_k=α_kρ_k, and as usualα_k,ρ_k,U_krepresent the mean statistical fraction, the mean density and the mean velocity in phasek. Mean densities are positive, and the constraint :

Σ^N_k=1α_k=1

holds everywhere, at any time. The interfacial transfer termsφk(Y),S_k(Y)are such that:

Σ^N_k=1φ_k(Y) =0 ; Σ^N_k=1m_kS_k(Y) =0.

Thus the main unknown is:

Y= (α₁, ...,αN−1,ρ1,U₁, ...,ρN,U_N)^t (2)

It lies inR^p, withp=3N−1. The functionsP_k(ρ_k)are classically chosen such that c²_k=P_k⁰(ρ_k)>0. We also defineψ_k(ρ_k)such that:

ψ_k⁰(ρ_k) =P_k(ρ_k) ρ_k²

(3) and the entropy of the mixtureη(Y)is defined as:

η(Y) =1

2Σ^N_k=1m_kU_k²+Σ^N_k=1m_kψ_k(ρ_k). (4) From now on, we will assume that the velocityVi(Y)is a convex combination of phasic velocitiesU_k, so that we may write:

Vi(Y) =Σ^N_k=1a_k(Y)U_k (5) whereΣ^N_k=1a_k(Y) =1, and 0≤a_k(Y).

We define the quantityA(Y,∂x(Y))such that:

A(Y,∂_x(Y)) =Σ^N_k=1 Σl6=k(P_k(Vi(Y)−U_k) +U_kΠkl(Y))∂_x(α_l)

(6) Using this definition, we can obtain the governing equation ofη(Y)for smooth solu- tions of (1), which reads:

∂_t(η(Y)) +∂_x(fη(Y)) =RHSη(Y)−A(Y,∂_x(Y)) (7) setting:

RHS_η(Y) =Σ^N_k=1(m_kS_k(Y)U_k−φk(Y)P_k) (8) f_η(Y) =Σ^N_k=1

U_k²

2 +ψ_k(ρ_k) +P_k ρ_k

m_kU_k. (9)

We wonder now whether there exists a unique set ofN(N−1)functionsΠ_kl(Y) withk6=l that guarantees the minimal entropy dissipation A(Y,∂_x(Y)) =0, when N≤4.

Proposition 1(Closure laws for interfacial pressures)

Smooth solutions of system(1)comply with the constraintA(Y,∂x(Y)) =0, iff:

• N=2:

Π12(Y) =Π21(Y) = (1−a1(Y))P₁+a1(Y)P₂ (10)

• N=3:



















Π12(Y) = (1−a1(Y))P₁+a1(Y)P₂; Π21(Y) =a2(Y)P₁+ (1−a₂(Y))P₂; Π13(Y) = (1−a₁(Y))P₁+a₁(Y)P₃; Π31(Y) =a₃(Y)P₁+ (1−a₃(Y))P₃; Π23(Y) = (1−a₂(Y))P₂+a₂(Y)P₃; Π32(Y) =a₃(Y)P₂+ (1−a₃(Y))P₃;

(11)

• N=4:

Πkl(Y) = (1−a_k(Y))P_k+a_k(Y)P_l (i f: 1≤k<l≤4);

Πkl(Y) =a_l(Y)P_k+ (1−a_l(Y))P_l (i f: 1≤l<k≤4). (12) Sketch of proof: The proof is obtained by construction. It is almost obvious when N=2, but more tedious when N=3 orN=4. First it is necessary to rewrite the scalar quantity A(Y,∂x(Y))in terms of theN−1 independent gradients∂x(α_l)for l =1→N−1 (since∂x(α_N) =−Σ^N−1_l=1∂x(α_l)). All cofactors must be set to zero, which results in a new set of(N−1)scalar equationsLHS_k(Y) =0. For each equation among these, one must again rewrite quantities in terms ofN−1 independent relative velocities(U_N−U_l)forl=1→N−1, and also use the form (5) in order to obtain (U_l−Vi(Y))in terms of the latter relative velocities and of thea_l(Y). Moreover, one needs to take into account the constraint:

Σ^N_k=1

Σ^N_l=1,l6=kΠ_kl(Y)∂_x(α_l)

=0

that arises since these represent interfacial transfer terms inside the mixture. This ends up in a system ofN(N−1)scalar equations, which is linear with respect to theΠ_kl(Y).

It only remains to find theunique N(N−1)solutionsΠ_kl(Y)of the latter system.

Hence, once thea_k(Y)in (5) are given, there exists a unique choice for theΠ_kl(Y).

Note that, unlike for two-phase flows, and for a given couple of phases(k,l), there exists a disequilibrium at the(k,l)interface when three (or four) phases occur, since:

Π_kl(Y)−Π_lk(Y) = (1−a_k(Y)−a_l(Y))(P_k−P_l) for: k<l

is non zero unless a perfect pressure equilibrium holds between the three (or four) phases. This was actually expected, since the quantityΠkl∂x(α_l) +Πlk∂x(α_k)is no longer null, for given(k,l)withk6=l, whenN>2. Moreover, it clearly arises that Πkl(Y)is an average of pressuresP_kandP_l.

Proposition 2 (Entropy inequality for multi-phase flow models)

We consider some fixed phase index k₀∈1, ..,N. We assume that closure laws for interfacial quantitiesφ_k(Y),S_k(Y)comply with the two constraints:

0≤Σ^N_k=1 φ_k(Y)(P_k−P_k0)

;

0≤Σ^N_k=1 m_kS_k(Y)(U_k0−U_k) (13) then smooth solutions of system(1)satisfy the following inequality:

∂_t(η(Y)) +∂_x(fη(Y))≤0 (14)

for the minimal entropy dissipation model associated with:A(Y,∂x(Y)) =0.

The proof is straightforward. We may now give some admissible form for the pres- sure relaxation terms.

Proposition 3 (Pressure-velocity relaxation terms for multi-phase flow models) Assume that closure laws forφk(Y),S_k(Y)take the form:

φ_k(Y) =Σ^N_l=1(d_kl(Y)(P_k−P_l));

m_kS_k(Y) =Σ^N_l=1(e_kl(Y)(U_l−U_k)) (15) with: 0<d_kl(Y) =d_lk(Y), and: 0≤e_kl(Y) =e_lk(Y),then the pressure-velocity relax- ation termsφ_k(Y),S_k(Y)comply with the entropy inequality(14).

Proof: It is classical forN=2. We skip the caseN=3, and we only consider here the caseN=4. We define:x=P₁−P₂,y=P₁−P₃,z=P₁−P₄. The remaining pressure desequilibria may be written as follows: P4−P3=y−z, P4−P2=x−z, P3−P2=x−y. Hence we may compute:

σ4=Σ⁴_k=1(φ_k(Y)(P_k−P₁)) which turns to be:

σ4=d₂₁(Y)x²+d₃₁(Y)y²+d₄₁(Y)z²+d₄₂(Y)(z−x)²+d₃₂(Y)(y−x)²+d₄₃(Y)(y−z)² Thusσ4is strictly positive unlessP₁=P₂=P₃=P₄.

A similar proof holds for velocity relaxation contributions. We emphasize first that the counterpart of properties 2,3 also holds for non isentropic two or three-phase flow models (see [5, 12, 15]). QuantitiesS_k(Y)stand for drag effects between phases; be- sides, pressure relaxation terms φ_k(Y)are already present in all standard two-phase flow models such as those described in [1, 14] for instance. Physically relevant pres- sure relaxation time scales associated with the d_kl were proposed in [8]. One may nonetheless wonder whether these relaxation terms act as expected. Actually, the fol- lowing result clearly provides some assessment of the latter claim. For that purpose, we consider some flow in a box (thus neglecting all convective effects), so that system (1) reduces to:







∂t(α_k) =φk(Y);

∂_t(m_k) =0 ;

∂_t(m_kU_k) =m_kS_k(Y).

(16) Proposition 4 (Pressure relaxation for barotropic three-phase flow models) We set: N=3, and we assume for sake of simplicity that pressure relaxation time scales are equal, so that:φ_k(Y) =d(Y)Σ^N_l=1(P_k−P_l). We also define :

EP(Y) = ((P₁−P₂)²+ (P₁−P₃)²+ (P₂−P₃)²)/2.

Then solutions of (16)comply with:

0≤EP(Y)(t)≤EP(Y)(0)×exp

−6 Z t

0

f_min^P (t)dt

if the frequency0< f_min^P (t)denotes some positive lower bound of(ρ_kc²_kd(Y)/α_k)(t) (for k=1,3).

Proof: We define: y=P₁−P₂ and: x=P₂−P₃, thus: P₁−P₃=y+x, and : EP(Y) =x²+y²+xy. We use the notation:βk=ρkc²_k/α_k. Using the second equation of (16), which gives:∂t(ρ_k) =−ρ_k∂t(α_k)/α_k, and hence:∂t(P_k) =−ρ_kc²_k∂t(α_k)/α_k, it clearly arises that solutions of (16) agree with:

∂_t(x) =−β₂∂_t(α₂) +β₃∂_t(α₃)

∂t(y) =−β₁∂t(α₁) +β2∂t(α₂)

Since ∂t(α_k) =2d(Y)(P_k−P_lm), with: P_lm= (P_l+Pm)/2, fork,l,m non equal in {1,2,3}³, we get at once:

∂_t(EP(Y)) =−d(Y) β₂(x−y)²+β₃(2x+y)²+β₁(x+2y)² which yields:

∂t(EP(Y))≤ −f_min^P (t) (x−y)²+ (2x+y)²+ (x+2y)² or alternatively:

∂_t(EP(Y))≤ −6f_min^P (t)EP(Y)(t) which ends up with the above statement.

This property is still valid for four-phase flow models (see [13]). Considering the same assumption of a flow in a box (16), a similar property may be obtained for velocity relaxation effects in three-phase flow models, considering the counterpart ofEP(Y):

EU(Y) = ((U₁−U₂)²+ (U₁−U₃)²+ (U₂−U₃)²)/2.

It now remains to select admissible closure laws for the interface velocityVi(Y) which governs the statistical fractions evolution. The specifications that are enforced here correspond to the fact that theα_l should be perfectly advected (if all phase pres- sures are in equilibrium), and thus without any thickening; as a consequence, one must enforce that the field associated with the eigenvalueλ=Vi(Y)should be linearly de- generated. The next proposition illustrates that feature, and it is indeed a well-known result for two-phase flow models (see [4] and [5] for instance, for barotropic and non isentropic models respectively, and also [6] for some generalization):

Proposition 5 (Admissible interface velocity in barotropic three-phase flow mod- els) We set: N=3, and we still assume that:Vi(Y) =Σ^N_k=1a_k(Y)U_k, with:Σ^N_k=1a_k(Y) = 1. We set: a_k(Y) =m_k/M where: M =Σ^N_k=1m_k. Then the field associated with:

λ1,2=Vi(Y)is linearly degenerated.

Proof: It is straightforward but cumbersome. We definer₁(Y),r₂(Y)the two right eigenvectors associated with the eigenvalue:λ =Vi(Y), it then only remains to check that:∇Yλ1,2(Y).r₁(Y) =∇Yλ1,2(Y).r₂(Y) =0.

A similar result holds for four-phase flow models. The structure of the contact wave associated withλ =Vi(Y)is examined in detail in [13] forN=3 andN=4. Actu- ally the connection of states through the latter wave is very similar to what happens in

two-phase flow models (see [4, 5]). Eventually, we may give the expected result, that is:

Proposition 6 (Structure of the convective part of system (1))

We assume that: |U_k−Vi(Y)|/c_k6=1. Then system (1) is hyperbolic, since all eigen- values are real, and the set of right eigenvectors spans the whole space of statesR^p.

2 Taking mass transfer into account

We consider now the following system for a mixture ofNphases with possible mass transfer between phases. This reads:







∂_t(α_k) +Vi(Y)∂_x(α_k) =φ_k(Y);

∂_t(m_k) +∂_x(m_kU_k) =Σ^N_l=1,l6=kΓ_kl(Y) =G_k(Y);

∂_t(m_kU_k) +∂_x m_kU_k²+α_kP_k(ρ_k)

+Σ^N_l=1,l6=kΠ_kl(Y)∂_x(α_l) =m_kS_k(Y) +S^G_k(Y). (17) with:

S^G_k(Y) =Σ^N_l=1,l6=kVkl(Y)Γ_kl(Y) and:Γkl(Y) +Γlk(Y) =0. We also enforce the law:

Vkl(Y) =βkl(Y)U_k+ (1−β_kl(Y))U_l,

with:βkl(Y)∈[0,1]. We evenmore assume symmetry, that is:Vkl(Y) =Vlk(Y), which means that:β_kl(Y) +β_lk(Y) =1. The termΓ_kl(Y)simply denotes the interfacial mass transfer between phaseskandl. Of course, we still consider the previous closure laws forφ_k(Y),S_k(Y)andΠ_kl(Y).

The time evolution of the entropyηis now governed by:

∂t(η(Y)) +∂x(fη(Y)) =RHS^G_η(Y) (18) but the source term on the right handside becomes:

RHS^G_η(Y) =Σ^N_k=1(m_kS_k(Y)U_k−φ_k(Y)P_k) +Σ^N_k=1

U_k(Σ^N_l=1,l6=kVkl(Y)Γ_kl(Y)−^U₂^kG_k(Y)) +G_k(Y)(^Pk

ρ_k+ψk(ρ_k)) (19) Thus we get :

Proposition 7 (An entropy-consistent closure law for the interfacial mass transfer) Assume that: βkl(Y) =1/2, and also that f_kl(Y) = f_lk(Y)>0. Then the following closure law for the mass transfer:

Γkl(Y) =f_kl(Y)

(P_l

ρ_l+ψl(ρ_l))−(P_k

ρ_k+ψ_k(ρ_k))

(20) complies with the entropy inequality for smooth solutions Y of (17):

∂t(η(Y)) +∂x(f_η(Y))≤0 (21)

The proof is simple and left to the reader, who is refered to [13]. The latter reference also provides more details and a thorough analysis of the statistical fraction LD wave λ =Vi(Y). Details on (unique) jump conditions can also be found therein. The exten- sion to the framework of non-isentropic multiphase multi-component flows is currently investigated for 4≤N.

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