2012_computer&fluids_longoetal

A finite element Galerkin/least-squares method for computation of multicomponent
compressible–incompressible flows
Antonella Longo a,⇑
, Michele Barsanti b
, Andrea Cassioli c
, Paolo Papale a
a
Istituto Nazionale di Geofisica e Vulcanologia, Sede di Pisa, I-56126 Pisa, Italy
b
Dipartimento di Matematica Applicata, University of Pisa, Pisa, Italy
c
Dipartimento di Sistemi e Informatica, University of Florence, Florence, Italy
a r t i c l e i n f o
Article history:
Received 21 February 2012
Received in revised form 30 May 2012
Accepted 3 July 2012
Available online 21 July 2012
Keywords:
Stabilized finite element method
Navier–Stokes equations
Compressible–incompressible flow
Multicomponent flow
a b s t r a c t
The space–time Galerkin/least-squares finite element method with discontinuity capturing (ST-GLSDC),
developed by Hughes and collaborators [Shakib et al. A new finite element formulation for computational
fluid dynamics: X. The compressible Euler and Navier–Stokes equations. Comput Methods Appl Mech Eng
1991;89:141–219], allows to study both compressible and incompressible single-fluid one-component
flows. It is effective in the stabilization of the numerical solution without introducing excessive overdif-
fusion. In this work the development by Hauke and Hughes [A comparative study of different sets of
variables for solving compressible and incompressible flows. Comput Methods Appl Mech Eng
1998;153:1–44] to pressure primitive variables is extended to single-fluid multicomponent compressible
and incompressible flows of gas–liquid mixtures at local mechanical and chemical equilibrium. The sta-
bilized algorithm is implemented in a parallel C++ library, which is tested on several benchmarks. The
solution of the system of equations for the conservation of mass of each component, and of momentum
and energy of the global mixture, requires the introduction of mass fractions as primitive variables to
describe mixture composition. The weak formulation, the stabilization parameters, and the time-
marching algorithm are rewritten in terms of the expanded set of variables, keeping similarity with
the formulation in pressure variables.
Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction
Flows of compressible–incompressible, multicomponent multi-
phase fluid mixtures are of common interest in theoretical, geo-
physical, and industrial CFD [1,2].
The computation of incompressible and compressible fluid
dynamics generally requires completely different numerical
approaches; only a few stabilized finite element formulations for
compressible–incompressible flows have been developed. The com-
pressible formulation in augmented conservation variables [3],
derived from the streamline-upwind/Petrov–Galerkin techniques
[4–6], properly computes the incompressible limit. The characteris-
tic-based split procedure with Taylor-Galerkin/pressure-correction
scheme, suitable for both compressible and incompressible regime
[7,8], combines the methods [9–11] for compressible flows.
The Galerkin/least-squares formulation adds the least-squares
term as a weighted residual. It was first introduced for compress-
ible single-fluid flows [12], subsequently recast in entropy vari-
ables, supplemented with a shock-capturing operator and with a
full space–time discretization to obtain the space–time Galerkin/
least-squares formulation with discontinuity-capturing operator
(ST-GLSDC) [13]. A compressible entropy formulation for multi-
component mixtures of ideal and perfect gases was also developed
[14,15].
The single-fluid ST-GLSDC method and other slight variants
were successfully applied to incompressible flows [16,17] and are
well behaved in the incompressible limit in the context of primi-
tive or entropy variables [18,19]. The unified approach by [18,19]
is the development of ST-GLSDC to handle the whole spectrum of
compressible–incompressible regimes, employing the same set of
variables.
The stability, accuracy, and convergence of ST-GLSDC are well
established, so that this method is a reliable basis for the present
formulation. The ST-GLSDC also allows accessory techniques such
as domain decomposition, local time-stepping and linear solution
algorithms, and can be efficiently solved with a GMRES developed
for the non-symmetric linear systems arising from the discretiza-
tion [20]. The feasibility of computational improvements in accu-
racy and speed is a practical requirement for the simulation of
multicomponent flows where the number of unknowns increases
drastically as components are added.
0045-7930/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.compfluid.2012.07.008
⇑ Corresponding author. Tel.: +39 0508911939; fax: +39 0508911942.
E-mail address: longo@pi.ingv.it (A. Longo).
URL: http://www.pi.ingv.it/~longo (A. Longo).
Computers & Fluids 67 (2012) 57–71
Contents lists available at SciVerse ScienceDirect
Computers & Fluids
journal homepage: www.elsevier.com/locate/compfluid

The formulation presented in this paper extends from single-
fluid flows to multiphase multicomponent homogeneous gas–
liquid mixture flows at mechanical and chemical local equilibrium
the ST-GLSDC method [19] employing primitive variables, and gen-
eralizes to primitive variables and fluids with general equations of
state the multicomponent entropy method [14]. The weak formu-
lation, the stabilization parameters, and the algorithmic imple-
mentation at elemental level are written in order to solve the
conservation equations for the mass of each component rather
than for the fluid mass, along with the conservation equations
for momentum and energy of the mixture as a whole. The addi-
tional terms due to increased degrees of freedom in the formula-
tion are derived within the weak formulation. The expressions
for the stabilization parameters of the original ST-GLSDC, namely
the s parameter and the discontinuity capturing operator, are re-
placed with analogous ones accounting for the mixture compo-
nents, keeping dimensional consistency and simplicity. The
criterion followed in this extension is to keep the robustness of
the original method maintaining a reasonable computing effort.
Several finite element methods have been developed for multi-
component and/or multiphase flows. Some of them deal with par-
ticular subjects such as diffusion or sintering of phases, others are
restricted to incompressible flows with the SUPG formulation
[21–25]. The method proposed in this work allows the solution
of both compressible and incompressible large number of prob-
lems such as shock wave interaction with contact discontinuities,
evolution of internal interfaces, bubbly flows with evaporation or
gas dissolution.
2. Physical model
The model considers a compressible–incompressible multicom-
ponent multiphase mixture at mechanical, thermal and chemical
local equilibrium. Components can be in gaseous or liquid state,
and undergo instantaneous phase change. Chemical reactions
resulting in component production or consumption are not al-
lowed to occur. Gas–liquid mixtures may contain a continuous
and a dispersed phase, or phases separated by internal interfaces.
The governing equations are mass conservation of each compo-
nent, and momentum and energy conservation of the mixture as a
whole:
ðqykÞ;t þ ðquiykÞ;i ¼ ÀJk
i;i for k ¼ 1; . . . ; n ð1Þ
ðqujÞ;t þ ðquiuj þ pdijÞ;i ¼ ðsjiÞ;i þ qbj for j ¼ 1; . . . ; d ð2Þ
ðqetÞ;t þ ðquiet þ puiÞ;i ¼ sijuj À qi À
Xn
k¼1
Jk
i hk
!
;i
þ qðbiui þ rÞ ð3Þ
where all symbols are defined in the Nomenclature or the Appen-
dixes. Indexes after an inferior comma represent variables with re-
spect to which partial differentiation is computed; the summation
convention on repeated indexes is applied throughout. The mixture
Nomenclature
b body force vector per unit mass
c mixture sound velocity
d = 1, 2 or 3 number of spatial dimensions
e specific internal energy of mixture
et specific total energy of mixture
F source vector
Fadv
advective flux vector
Fdiff
diffusive flux vector
g partial specific Gibbs free energy of mixture
gk partial specific Gibbs free energy of component k
gp
k partial specific Gibbs free energy of component k in
phase p
h specific enthalpy of mixture
Jk
i mass diffusion flux of component k in ith direction
k specific kinetic energy of mixture
Kij diffusivity matrices
M Mach number
M molar mass of mixture
Mk molar mass of component k
n number of components
p pressure
Pr ¼ l
qj Prandtl number
qi diffusive heat flux in ith direction
r heat source per unit mass
Rk specific gas constant of component k
Re ¼ quL
l Reynolds number
S source matrix
Sc ¼ l
qD Schmidt number
t time
T temperature
u velocity vector
U conservative variables vector
V entropy variables vector
xk mixture molar fraction of component k
xp
k mixture molar fraction of component k in phase p
yk mixture mass fraction of component k
yp
k mixture mass fraction of component k in phase p
y = (y1,. . .,yn) vector of mass fraction of components
Y primitive variables vector
dij kronecker delta
gp
k mass fraction of component k in phase p with respect to
component k
j thermal conductivity
l first viscosity coefficient of mixture
mh
discontinuity capturing operator
np
k molar fraction of component k in phase p with respect
to phase p
q mixture density
s viscous stress tensor
sU intrinsic time-scale matrix for conservation formulation
sV intrinsic time-scale matrix for entropy formulation
sY intrinsic time-scale matrix for primitive formulation
()k kth component
()i ith spatial direction
()p
phase p
ðÞqyk
index for the qyk conservative variable entry
()qy indexes for the qy1, . . ., qyn conservative variable entries
ðÞqui
index for the qui conservative variable entry
()qu indexes for the qu conservative variable entries
ðÞqet
index for the qet conservative variable entry
ðÞgÀk
T
indexes for the g1Àk
T ; . . . ; gnÀk
T entropy variable entries
ðÞu
T
indexes for the u
T entropy variable entries
ðÞÀ1
T
index for the 1
T entropy variable entry
(),y partial derivatives with respect to y1, . . ., ynÀ1 primitive
variables
(),p partial derivative with respect to p primitive variable
(),u partial derivatives with respect to u1, u2, u3 primitive
variables
(),T partial derivative with respect to T primitive variable
(),i partial derivative with respect to the ith spatial direc-
tion
(),t partial derivative with respect to time
58 A. Longo et al. / Computers & Fluids 67 (2012) 57–71

is assumed to follow the ideal solution model, which can be used for
both liquid and gaseous mixtures [26]. The constitutive relations for
thermodynamic potentials, physical and chemical properties of the
mixture and components in their phases, and chemical equilibria
complete the system of equations (see Appendix A).
The composition of the mixture is described by n À 1 indepen-
dent mass fraction of components yk and by the partition coeffi-
cients of components in their phases, which depend on pressure
and temperature (see Appendix A). Chemical reactions are ne-
glected, and mixtures of pure components are approximated as
mixtures of nearly pure components as discussed in Section 4,
therefore mass fractions yk never vanish. Different components
are miscible, i.e. interdiffusion is allowed. The diffusive fluxes of
mass of components, Jk
i , are expressed by the generalized Fick’s
law for multicomponent fluids (A.18) [27,28]. The stress tensor sij
(A.16) assumes Newtonian rheology. The energy flux (first term
at the RHS of energy Eq. (3)) is due to viscous dissipation sijuj, heat
conduction qi expressed by Fourier’s law (A.17), and interdiffusion
of components carrying a specific enthalpy hk. The heat source per
unit mass r includes the latent heat due to phase change. Mass dif-
fusion due to pressure gradients, Soret and Dufour effects, and sur-
face tension, are neglected.
3. Numerical formulation
The system of Eqs. (1)–(3) is written in terms of the vector of
conservation variables U using compact notation [19]:
AU;t þ Fadv
i;i ¼ Fdiff
i;i þ F ð4Þ
U ¼ q
y1
..
.
yn
u1
u2
u3
et
0
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
A
ð5Þ
with the advective and diffusive flux vectors in the ith direction, Fadv
i
and Fdiff
i , and the source vector F reported in (B.2)–(B.4)
In order to solve the incompressible limit, the conservation vari-
ables U are replaced by the pressure primitive variables Y which
are employed as unknowns:
Y ¼
y1
..
.
ynÀ1
p
u1
u2
u3
T
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
ð6Þ
The n À 1 independent mass fractions of components yk, pressure p
and temperature T represent the n + 1 thermodynamic quantities
necessary to completely describe the compressible mixture of n
components at chemical equilibrium. The pressure primitive vari-
ables (p,u,T) for simple fluid compressible–incompressible flows
proved to be a reliable choice in terms of accuracy, convergence,
computational costs and robustness of the related numerical algo-
rithm [19]. Extention to the Y vector (6) involves inclusion of the
mass fraction of components, and makes the calculation of physical
properties and boundary conditions easier.
The system of Eq. (4) is rewritten in terms of primitive variables
in quasi-linear form [19]:
U;Y Y ;t þ Fadv
i;Y Y ;i ¼ ðKijY ;jÞ;i À SY ð7Þ
The matrices U,Y, Fi, Y
adv
of partial derivatives of U and Fadv
i with re-
spect to Y are reported in (B.5), (B.18). The linearization requires
that the diffusive flux and source vector are expressed as
Fdiff
i ¼ KijY ;j where Kij are the diffusivity matrices (B.32), and
F = ÀSY where S is the source matrix (B.36) [13].
A similar multicomponent formulation has been introduced in
[14], where a vibrational energy term, neglected here, is consid-
ered. The analysis in [14] takes into account only entropy variables
and compressible mixtures of ideal or perfect gases. The multicom-
ponent formulation in this work, based on pressure primitive vari-
ables, allows the solution of compressible as well as
incompressible flows of gas, liquid, and gas–liquid mixtures with
general EOS. The tight similarity between the method of this work
and those in [19,14] allows to develop all the necessary stabiliza-
tion terms, while keeping the advantages of the numerical method
for single-fluid flow.
4. Stabilizing method
The remarkable robustness of the ST-GLSDC method originates
from the full space–time discretization, the least-squares and the
discontinuity capturing operators [13].
Given the space–time domain Q = X Â I, the total time interval I
is partitioned into sub-intervals In ¼ tþ
n ; tÀ
nþ1
Â Ã
to obtain a sequence
of time slabs Qn = X Â In. In each slab Qn the space domain X is
decomposed into (nel)n elements Xe
n, obtaining ST finite elements
Qe
n ¼ Xe
n Â In. Within each Qe
n, the trial solution and weighting
functions are approximated by k-th order interpolation polynomi-
als, continuous over Qn, but discontinuous across successive time
intervals In. The ST-GLSDC weighted residual formulation for the
solution of (7) in primitive variables is [19]:
Z
Qn
ðÀW;t Á UðYÞ À W;i Á Fadv
i ðYÞ þ W;i Á KijY ;j À W Á FÞdQ
þ
Z
X
W tÀ
nþ1
À Á
Á U Y tÀ
nþ1
À ÁÀ Á
À W tþ
n
À Á
Á U Y tÀ
n
À ÁÀ ÁÀ Á
dX
þ
XðnelÞn
e¼1
Z
Xe
n
ðLT
WÞ Á sY ðLY À FÞdQ þ
XðnelÞn
e¼1
Z
Xe
n
mh
gij
W;i Á U;Y Y ;jdQ
¼
Z
Pn
WðÀFadv
i ðYÞ þ Fdiff
i ðYÞÞnidP ð8Þ
where L ¼ U;Y @;t þ Fadv
i;Y @;i À ð@;iÞðKij@;jÞ is the differential operator
associated with the -linear form of the conservation Eq. (7), gij
= [-
ninj] is the controvariant metric tensor, and Pn is the boundary of
Qn with outward unit normal in i-th direction ni. The first and last
integrals are the Galerkin terms, and the second is the jump term.
The third integral is the least-squares term, in which the weighted
residual is multiplied by the matrix of intrinsic time-scales sY
appropriate for the equations in primitive variables. The fourth inte-
gral is the discontinuity capturing term.
The definition of sY used in this work is an extension of the non-
diagonal snd [29], which is relatively easy to implement and does
not require huge CPU resources, while being robust and efficient.
The tau matrix for conservation variables sU is the basis for the
construction of the non-diagonal sY:
sY ¼ Y ;U sU ð9Þ
The generalization of sU to multicomponent flows is obtained by
replacing the diagonal entry relative to the mass conservation equa-
tion with n diagonal terms sqyk
for k = 1, . . ., n, relative to mass con-
servation of components. Due to the diagonal form of sU the
A. Longo et al. / Computers & Fluids 67 (2012) 57–71 59

definition of the new terms by dimensional consistency is
straightforward.
Numerical test cases performed in this work showed an im-
proved stability of results in the incompressible regime, when
the intrinsic time scale of advection, 1/ke
[29], is corrected by add-
ing the quantity h
e
diff =2juj, following the same strategy as for the
diagonal sd [29]. The resulting expression for sU is:
sU ¼ diagðsqy1
; . . . ; sqyn
; squ1
; squ2
; squ3
; sqet
Þ ð10Þ
where
sqyk
¼ min
Dt
2
;
1
ke þ
h
e
diff
2juj
;
me
h
e
diff
À Á2
4Dk
!
for k ¼ 1; . . . ; n ð11Þ
squi
¼ min
Dt
2
;
1
ke þ
h
e
diff
2juj
;
qme
h
e
diff
À Á2
4l
!
for i ¼ 1; . . . ; d ð12Þ
sqet
¼ min
Dt
2
;
1
ke þ
h
e
diff
2juj
;
qcvme
h
e
diff
À Á2
4j
!
ð13Þ
The artificial diffusivity coefficient, mh
, in the discontinuity-cap-
turing term in (8) has the Hughes and Mallet form as in [19]:
mh
¼max 0;
ðLY ÀFÞÁV;U ðLY ÀFÞ
gijY ;i ÁVT
;Y U;Y Y ;j
" #1=2
À
ðLY ÀFÞÁsV ðLY ÀFÞ
gijY ;i ÁVT
;Y U;Y Y ;j
" #0
@
1
A
ð14Þ
where the vector V of entropy variables, and the matrices of its
derivatives with respect to U and Y, V,U and V,Y, are defined below
and include the multicomponent terms added in this work.
The vector of entropy variables is [14]:
V ¼
1
T
g1 À juj2
2
..
.
gn À juj2
2
u1
u2
u3
À1
0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
A
ð15Þ
The calculation of V,Y depends on the partial specific Gibbs free
energy of each component in the gas–liquid mixture gk (A.1). The
expression corresponding to ideal solution mixture is reported in
(B.38). The symmetric and positive definite matrix V,U is (B.45). It
is analogous to (C4) in [14], which was obtained under the more
restrictive assumption of ideal gas mixture.
The sV matrix for entropy variables in (14) is obtained from sU
(10) through the chain rule:
sV ¼ V;UsU ð16Þ
Internal interfaces between pure immiscible fluids represent a
well-known difficulty in dealing with two-phase or multi-fluid
flows, requiring appropriate techniques of front tracking with
adaptive meshes, or interface capturing with auxiliary functions
defined on fixed grids [30]. In the single-fluid multicomponent
multiphase method of this work pure immiscible components are
modeled as nearly pure components with very small coefficients
of diffusion [31,32]. This approximation prevents divergence of
partial derivatives, while keeping physical properties close to those
of pure components. The time–space evolution of the internal dis-
continuities is thus computed within the ST-GLSDC formulation by
solving conservation equations with the steep gradients of compo-
sition across the interface.
5. Solution technique
The ST-GLSDC weak formulation (8) generates two non-linear
systems of discretized equations using the linear-in-time approxi-
mation by [13] with piecewise functions linear in time and space.
These systems are linearized in each time slab by means of a third-
order predictor multi-corrector algorithm [13] and solved with the
block-diagonal pre-preconditioned GMRES [20,33].
Convergence is controlled at each iteration step i P 2 for each
variable on the global vector of nodal unknown upgrades Dy
ðiÞ
1 ;
. . . ; Dy
ðiÞ
nÀ1; DpðiÞ
; Du
ðiÞ
1 ; Du
ðiÞ
2 ; Du
ðiÞ
3 ; DTðiÞ
. The algorithm is said to
converge if the highest L1
norm of the updates is less than a given
small number :
max
u¼y1;ÁÁÁ;ynÀ1;p;u1;u2;u3;T
ðkDuðiÞ
k1Þ 6 for i P 2 ð17Þ
where
kDuðiÞ
k1 ¼
max
j¼1;ÁÁÁ;nnp
DuðiÞ
j

max
j¼1;ÁÁÁ;nnp
Duð1Þ
j
ð18Þ
The kÁk1norm is chosen because it corresponds to the maximum er-
ror occurring in the solution.
The convergence criterion (17) is applied with = 0.1, and it is
typically satisfied after 2–5 corrections of the time-marching algo-
rithm. The value represents the upgrade within the multi-correc-
tor loop with respect to the first correction, and is sufficient to
obtain accurate solutions.
The block-diagonal pre-preconditioned GMRES [20] is used to
solve the linearized systems, employing the implicit form of the
tangent matrices [13], selected for their stabilizing properties,
and recalculated at each multicorrector step.
The tolerance of the GMRES solver is set to 10À9
, generally
requiring 2 cycles out of an allowed maximum of 4, each with a
maximum allowed number of 200 iterations, using a 40-dimen-
sional Krylov space [34].
The finite element discretization involves bilinear quadrilateral
elements in space, and linear in time. Gauss integration is used
with 2 Â 2 points in space and two points in time. The spatial dis-
cretization is uniform unless differently required by the features of
the flow or the shape of the domain. The time step is selected to
satisfy the CFL condition, although the ST-GLSDC method does
not require strict accordance to the CFL condition [13].
The algorithm has been implemented as a C++ object oriented
library, with an extensive use of design patterns and following
the template meta-programming paradigm. Taking advantage of
the intrinsic parallel nature of finite element computations, the
code has been designed to be an MPI-based parallel one.
A modular organization has been deployed, reflecting the finite
element computation structure:
linear algebra support (vectors,matrices, linear system solvers,
. . .);
finite element numerics (elements, nodes,degree of freedom,
. . .);
discretization of the fluid-dynamics equations (element inte-
gration, assembly, . . .);
initial and boundary conditions;
Part of the parallel computations rely on the widely used
Trilinos package (developed by the Sandia Laboratories, [33]).
The latter provides high-level representation for distributed matri-
ces and vectors, as well as efficient tools for parallel computation
(matrix multiplication, GMRES linear system iterative solver, etc.).
60 A. Longo et al. / Computers Fluids 67 (2012) 57–71

Mesh design and partitioning among processors is performed
by METIS ([35]).
Specific tests confirm that the implementation performance
scales satisfactory with the number of processors and problem
dimension, both in terms of memory usage and cpu time.
6. Numerical cases
The ST-GLSDC multicomponent formulation developed in this
work is verified and validated with laboratory experiments, exact
solutions, and numerical results from the literature. The consid-
ered cases span the compressible–incompressible regimes, and in-
volve low to high viscosity fluids, with Re 2 [0–106
], Pr 2 [1–7],
Sc 2 [0–106
] and M 2 [0–3]. CFL numbers are in the range 10À2
–
104
.
The complete list of test cases is reported below; details and re-
sults can be found at www.pi.ingv.it/$longo.
Incompressible regime:
1. 1D transport of compositional interfaces.
2. Poiseuille flow [19] without (A) or with (B) sharp composi-
tional discontinuities of passive tracers.
3. Driven cavity flow [8,19] with or without sharp composi-
tional discontinuities of passive tracers.
4. Kelvin–Helmholtz instability without gravity [36].
5. Two-layers gravitational instability due to thermal or com-
positional gradients.
6. Broken dam [36,32].
Compressible regime:
7. 1D shock with constant composition [19].
8. Oblique shock with constant composition [19], and with
compositional front of passive tracer crossing the shock
region at different angles.
9. Flow over a flat plate with constant composition [19].
10. Acoustic wave with constant composition [13].
11. Steady wave in a resonant pipe.
12. Strong 1D shock and shock-interface interaction [37].
13. 2D shock-interface interaction [38].
14. 1D equilibrium bubbly flow: (A) half-pressurized tube, (B)
constant inflow chamber.
Table 1
Fluid physical properties, initial and boundary conditions for driven cavity (case 3B).
Physical properties
q = q0(1 + b(p À p0)) at T0 = 298.15 K
q0 = 998.2 kg/m3
, b = 0.45 GPaÀ1
, p0 = 105
Pa
l = 103
Pa s
D ¼ 10À6
m2
=s
Initial conditions Boundary conditions
u1 = u1 = 0 Top: u1 = 1 m/s, u2 = 0
p = 105
Pa Sides and bottom: u1 = u2 = 0
Composition: see Fig. 1a Middle bottom: p = 105
Pa
Table 2
Physical properties, initial and boundary conditions for broken dam (case 6).
Physical properties
Water ([36]) Water ([32]) Air ([36,32])
q = q0(1 + b(p À p0)) at T0 = 298.15 K qwater ¼ pþcp
ðcÀ1Þe
qair = p/RT
q0 = 1000 kg/m3
c = 4.4 R = 288.29 J/ kg K
b = 0.45 GPaÀ1
e = 776.5 kJ/kg T = 298.15 K
p0 = 105
Pa p = 6 Á 108
Pa
lwater = 1.01À3
Pa s lwater = 1.01À3
Pa s lair = 1.81À5
Pa s
D ¼ 10À10
m2
=s D ¼ 10À10
m2
=s D ¼ 10À10
m2
=s
Initial conditions Boundary conditions
u1 = u2 = 0 Everywhere: u1 = u2 = 0
Hydrostatic p top: p = 105
Pa
Everywhere: T = 298.15 K
(a) (b)
(c) (d)
Fig. 1. Driven cavity: (a–c) velocity field (reported only on 1/4 of the grid nodes to increase visibility of vectors) and mass fraction of one of the components in the binary
mixture (colorbar); contours correspond to 0.5 mass fraction. (a) Initial conditions; (b and c) transient phases of mixing before final homogenization; (d) steady state velocity
(t P 8 s): ux along the center vertical line and uy along the center horizontal line (dashed line: [19], solid line: this work).
A. Longo et al. / Computers Fluids 67 (2012) 57–71 61

The method outlined in this work is first validated on bench-
marks for simple fluid flows with passive uniform or non-uniform
compositional tracers (cases 1, 2, 3, 7, 8, 9, 10, 11). The calculations
show that coupling between (p,u,T) and yk variables does not de-
grade the accuracy of solutions, or the capability of preventing
oscillations and overdiffusion.
Cases (4, 5, 6, 12, 13, 14) considers different components that
have different physical and chemical properties, either constant
or dependent on the primitive variables Y. The incompressible
cases (4, 5, 6) show tests on flows instability. Case (4) is high Re
with forced convection and shear stresses as destabilizing forces;
cases (5, 6) are low Re with natural convection due to gravity
destabilization. The initial hydrostatic pressure profile in cases (5,
6) is calculated using a fourth-order Runge–Kutta integration of
the density field.
Cases (12, 13) consist in shock-interface interactions, with the
formation and propagation of shock waves and transport of com-
positional fronts.
The purpose of tests (4, 5, 6, 12 and 13) is to check the robust-
ness of the algorithm in tracking sharp compositional interfaces.
The results show that numerical oscillations of the solution for
mass fraction at the moving contact discontinuities do not exceed
10À3
relative error.
Cases (14) involve mixtures where components are present
both in the gas and liquid state and gas dissolution/exsolution
occurs.
In the following, selected representative cases (3, 6, 12, 13, 14)
are reported and briefly discussed.
6.1. Driven cavity with sharp compositional discontinuities
The driven cavity (case 3B) is a standard example of viscous,
incompressible flow. The top boundary of the cavity slides at con-
stant velocity in the direction of increasing x causing recirculation
inside (Fig. 1). Details of boundary and initial conditions are the
same as in [19] and reported in Table 1. The multicomponent fluid
consists of a mixture of two liquids with identical physical proper-
ties, labeled with two different ‘‘colors’’. The resulting flow is char-
acterized by Re $ 1 and Sc $ 106
. The initial condition for
composition consists of stripes parallel to the cavity sides
(Fig. 1a), and the walls are impermeable. The 1 Â 1 m domain is
discretized into 20 Â 20 square elements, the time step is 10À3
s
at the beginning of simulation, increasing to 103
s in the steady
state.
Recirculation deeply deforms the stripes (Fig. 1b and c), and
mixing proceeds until uniform composition is reached. The steady
t = 0.281 s
t = 0.222 s
t = 0.164 st = 0 s
t = 0.109 s
t = 0.066 s
x (m) x (m)
y(m)y(m)y(m)
a
this work
Murrone and Guillard (2005)
Fig. 2. Broken dam: initial conditions, and contours corresponding to 0.5 volume fraction at different times (this work: solid line; [32]: dashed line).
Experiment (Martin and Moyce, 1952)
Murrone and Guillard (2005)
Present calculation (MG05)
Nakayama and Shibata (1998)
Present calculation (NS98)
Dimensionlessfrontpositionx/a
2
3
1
4
6
7
8
5
0.4
1.0
0.8
0.6
0.2
Dimensionlessheighty/(2a)
Dimensionless time t(g/a)0.5 Dimensionless time t(2g/a)0.5
(a) (b)
0 1 2 3 4 5 60 1 2 3 4
Fig. 3. (a) Dimensionless height of the column, (b) dimensionless front position. Comparison between experimental data (crosses), numerical results from [32] (dotted line,
MG05), from [36] (dashed line, NS98), and present calculations. Cases MG05 and NS98 correspond to conditions similar to [32] (dash-dotted line) and [36] (solid line),
respectively.

state velocity profiles (t P 8 s) are coincident with those from [19]
(Fig. 1d), showing that the solution of composition conservation
equations does not generate spurious numerical effects. The solu-
tion for mass fractions does not present oscillations. The final uni-
form composition has a value coincident with the initial weighted
average of the three stripes.
Stability of the numerical code is demonstrated by obtaining
the same results of [19] with the coarsest grid size of 20 Â 20 ele-
ments used.
6.2. Broken dam
The broken dam (case 6) is a test on body force contribution and
advection of internal interfaces, with fluids being water and air,
treated as nearly pure components, with a negligible coefficient
of diffusion [31,32]. The two examples in [36,32] are reproduced:
the finite element method [36] tracks the free internal surface solv-
ing the advection equation for the fractional volume of liquid,
whereas [32] is a five equation reduced Eulerian diffuse interface
model.
Physical properties, boundary and initial conditions adopted in
this paper are reported in Table 2. In both cases [36,32] the pres-
sure at the top boundary of the domain is fixed at 105
Pa and the
hydrostatic pressure profile is assumed as initial condition for
water.
The constant densities used in [36] are modeled in the present
work with the perfect gas law for air and the linearized equation of
state for water at STP already adopted in the driven cavity case (see
Table 1).
The stiffened gas and water equations of state (87.1) and (87.2)
in [32] are used for the second example. The stiffened gas equation
of state (87.1) corresponds to the perfect gas law for air. The Eule-
rian model in [32] assumes inviscid fluids, while the present work
employes constants viscosities and mixing laws as in [36].
The same grid spacing 25 Á 10À4
m and 25 Á 10À3
m, respectively,
is used in each case [36,32]. The time step is in both cases 10À4
s as
in [36].
Fig. 2 reports the contour line of 0.5 volume fraction calculated
with the present model, along with the same contour lines from
[32]. According to the color map from Fig. 16, p. 695, of [32], the
interface is diffused over about 4 grid elements for a total of
4 Â 25 Á 10À4
m. According to the contour lines of / = 0.1, 0.5, and
0.9 volume fractions form Fig. 7, p. 199, of [36], the interface is dif-
fused over about 4 grid elements, for a total of 4 Â 25 Á 10À3
m. In
the present work the same interface diffusion is found:
4 Â 25 Á 10À4
m for the test case from [32], and 4 Â 25 Á 10À3
m
for the test case from [36]. Other authors have simulated the bro-
ken dam test benchmark by using the ghost-fluid method [39,40].
It is noteworth that interface diffusion in such cases (11 Á 10À4
–
37.5 Á 10À2
m) embraces the range found here and in [36,32].
t = 0.001 s
x (m)
y(m) Contours of:
1.012 10
5
Pa
1.013 10
5
Pa
1.014 10
5
Pa
1.015 10
5
Pa
1.016 10
5
Pa
1.017 10
5
Pa
0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
t = 0.002 s
x (m)
y(m)
0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
t = 0.003 s
x (m)
y(m)
0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
t = 0.004 s
x (m)
y(m)
0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
Fig. 4. Propagation of the sound wave from the falling wall of water. Contours of pressure at different times from present calculations, corresponding to case NS98.
Table 3
Physical properties, initial and boundary conditions for 1D shock-interface interaction
(case 12).
Physical properties
Gas 1 Gas 2
q2 = p/R1T q2 = p/R2T
R1 = 1621.4 J/kg K R2 = 292.8 J/kg K
c1 = 1.67 c2 = 1.4
cv1
¼ 2420 J=kg K cv2
¼ 732 J=kg K
j1 = 10À6
m2
/s j2 = 10À6
m2
/s
D12 ¼ 10À9
m2
=s D21 ¼ 10À9
m2
=s
Initial conditions
Gas 1 in x 6 0.5 m Gas 2 in x 0.5 m
u1 = u2 = 0 m/s u1 = u2 = 0 m/s
q = 14.55 kg/m3
q = 1.16 kg/m3
p = 194.3 Á 105
Pa p = 1.0 Á 105
Pa
T = 823.66 K T = 293.52 K
Boundary conditions: tube closed at both ends

Fig. 3 shows the comparison between the present two simula-
tion results and those from [36,32], along with the experimental
data from [41]. There is a better agreement with the results from
[36] than with those from [32]. For each pair of simulations (pres-
ent and [36], present and [32]) the discrepancy with the experi-
mental results from [41] is of the same magnitude.
The formulation in this work allows the solution of the propaga-
tion of the sound wave from the falling wall of water into the adja-
cent air (Fig. 4). The broken dam case involves within the same
computational domain the incompressible behavior of the falling
water and the subsonic compressible behavior of the air, repre-
senting a test for the capability of simultaneous computation of
compressible–incompressible multicomponent flows.
6.3. Strong 1D shock and shock-interface interaction
Test A of [37] (case 12) is a shock tube filled in the first and sec-
ond half with two different gases initially divided by a diaphragm,
and the left half is at higher pressure and temperature than the right
one. Initial conditions and physical properties of gases are the same
as Test A in [37] and reported in Table 3. When the diaphragm is re-
moved, a 1D strong shock developes at the contact interface be-
tween the two gases, propagates rightwards (Fig. 5) at M = 1.5 and
displaces the interface. The fully conservative model [37] is devoted
to the simulation of compressible multicomponent flows with
shock-interface interaction for inviscid perfect gases. This model
generates oscillation-free solutions across material interfaces,
(d)
(b)
0
2
4
6
8
10
12
14
16
Density(kg/m3)
(a)
1.4
1.45
1.5
1.55
1.6
1.65
1.7
RatioofSpecificHeats
(e)
1.4
1.45
1.5
1.55
1.6
1.65
1.7
RatioofSpecificHeats
(f)
0
200
400
600
800
1000
1200
1400
1600(c)
Velocity(m/s)
Exact
Other models
(Wang et al., 2004)
Wang at al. (2004)
Present model
Exact
Other models
(Wang et al., 2004)
Wang at al. (2004)
Present model
Exact
Other models
(Wang et al., 2004)
Wang at al. (2004)
Present model
Exact
Other models
(Wang et al., 2004)
Wang at al. (2004)
Present model
Exact
Other models
(Wang et al., 2004)
Wang at al. (2004)
Present model
Exact
Other models
(Wang et al., 2004)
Wang at al. (2004)
Present model
2.0
1.5
Pressure(107Pa)
1.0
0.5
0.0
Internalenergy(107J/kg)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
x (m)x (m)
0.7 0.75 0.8 0.85 0.90 0.2 0.4 0.6 0.8 1.0
x (m)
0 0.2 0.4 0.6 0.8 1.0
x (m)
0 0.2 0.4 0.6 0.8 1.0
x (m)
0 0.2 0.4 0.6 0.8 1.0
x (m)
0 0.2 0.4 0.6 0.8 1.0
Fig. 5. Test A of [37] at 2 Á 10À4
s. (a) Density, (b) pressure, (c) velocity, (d) specific internal energy, (e and f) ratio of specific heats.

employing a finite difference numerical scheme with a fast Riemann
solver, plus a MUSCL scheme to calculate fluxes with second-order
accuracy. The 1 m-length tube is discretized into 400 Â 1 elements
which ensure an accuracy in the resolution of the contact disconti-
nuity comparable to that in [37]. The time step is 10À6
s. Fig. 5 shows
the comparison between the present model results, those in [37]
and other conservative models reported in [37].
6.4. 2D shock-interface interaction
The 2D shock-interface interaction test (case 13) consists in the
reflection and refraction of a Mach 2 planar shock over an oblique
contact discontinuity and the displacement of the interface [38].
The finite volume method in [38] was developed for inviscid
perfect gases, with an explicit conservative discretization, calcula-
tion of the fluxes with Roe’s approximate Riemann solver for multi-
ple species, and second order accuracy in space with the MUSCL
technique. A correction of the total energy was introduced in order
to avoid oscillations through contact discontinuities.
The physical properties of gases, initial and boundary conditions
and time step in [38] are used (see Fig. 6 and Table 4). The compu-
tational domain is 0.5 Â 0.6 m, discretized into 100 Â 100 elements;
the time step is 2.5 Á 10À6
s. The calculated density distribution in
Fig. 6 shows that the model is able to correctly predict the positions
of the reflected shock and of the compositional discontinuity. On the
contrary, the determination of the reflected shock is less accurate,
approximating the uncorrected solution of [38].
6.5. Gas–liquid equilibrium bubbly mixtures
Mixtures (cases 14A,B) where components are present both in
the gas and dissolved liquid phase involve the solubility of a gas
Fig. 6. 2D interaction of a shock with a contact discontinuity from [38]. Top panel:
initial conditions (see also Table 4). Bottom panel: density colormap and contours
of 3, 4.5 and 15 kg/m3
resulting from this work (black line) and exact solution
(dashed line) at t = 0.00125 s, see also exact solution reported in Table 4.
Table 4
Physical properties, initial and boundary conditions for 2D shock-interface interaction (case 13).
Fluid properties, EOS: q = p/R T
Gas 1 Gas 2
c1 = 1.1 c2 = 1.4
R1 = 214.28 J/kg K R2 = 214.28 J/kg K
cv1 = 2142.9 J/kg K cv2 = 535.7 J/kg K
j1 = 10À6
W/m K j2 = 10À6
W/m K
D12 ¼ 10À9
m2
=s D21 ¼ 10À9
m2
=s
Initial conditions and exact solution (see also [38] and Fig. 6)
Region (1) Region (2) Region (3) Region (4) Region (5)
Gas 2 Gas 1 Gas 2 Gas 1 Gas 2
ux = 0 ux = 0 ux = 375 m/s ux = 262.33 m/s ux = 326.19 m/s
uy = 0 uy = 0 uy = 0 uy = À85.08 m/s uy = 66.24 m/s
q = 1.56 kg/m3
q = 5.19 kg/m3
q = 4.15 kg/m3
q = 24.43 kg/m3
q = 5.10 kg/m3
p = 1.0 Á 105
Pa p = 1.0 Á 105
Pa p = 4.5 Á 105
Pa p = 6.01 Á 105
Pa p = 6.01 Á 105
Pa
T = 300.0 K T = 89.96 K T = 506.25 K T = 114.79 K T = 550.25 K
Boundary conditions
Left side: as region (1), right side: as region (2)
Top: u2 = 0, bottom: u2,2 = 0
Table 5
Physical properties, initial and boundary conditions for gas-equilibrium bubbly
mixture cases (case 14A,B).
Fluid properties
Water and dissolved CO2 Exsolved CO2
q = q0(1 + b(p À p0)) q = p/RT
q0 = 998.2 kg/m3
R = 188.9116 J/kg K
b = 0.45 GPaÀ1
p0 = 105
Pa
T0 = 298.15 K
l = 10À3
Pa s l = 10À5
Pa s
D ¼ 10À6
m2
=s D ¼ 10À6
m2
=s

in the liquid. The ideal behavior of infinitely diluted solutions is
assumed, so that the solute follows the Henry’s laws p = Hx, where
x is the mole fraction of the solute in the liquid, p is its equilibrium
partial pressure, and H is the Henry’s law constant. In all examples
the liquid solvent is water, the volatile component is carbon diox-
ide (H = 1.447 Á 108
Pa at 298.15 K), and the possible presence of
small amounts of solvent in the gas phase is neglected. The phys-
ical properties, initial and boundary conditions are reported in
Table 5. Mixture viscosity is calculated as in (A.20). The two cases
reported below do not include a comparison with previous calcu-
lation or experiments, since they have been specifically designed
to test the multicomponent compressible–incompressible features
of the present numerical model.
Case A. 1D half-pressurized tube. Case A consists of a 1D tube of
1-m length, closed at both ends, filled with a uniform mixture of
water and carbon dioxide with mass fractions yH2O ¼ 0:998 and
yCO2
¼ 0:002 respectively. A diaphragm separating two regions
with different pressure is initially located in the middle of the
tube. The initial pressure in the two halves of the tube is
1.12 Á 105
Pa on the right and 1.2 Á 105
Pa on the left (Fig. 7,
see pressure in first row), so that gas is present only in the
low pressure part of the tube according to Henry’s law (Fig. 7,
see gas volume fraction in first row). The fluid in the two halves
of the tube is initially at rest (Fig. 7, see velocity in first row).
The tube is discretized in 200 Â 1 elements, and the time step
is 10À6
s. At time t = 0 the diaphragm is suddenly removed.
The results in Fig. 7 show that the flow is characterized by an
oscillating behavior, where the mixture accelerates alterna-
tively towards the right (t = [0–9] Á 10À3
s, [2.8–4.4] Á 10À2
s,
[6.4–8.1] Á 10À2
s) and left ([1.0–2.7] Á 10À2
s, [4.5–6.3] Á 10À2
s)
ends of the tube. The initial rightward flow is due to the initial
pressure gradient. This flow produces compression of the mix-
ture ahead of the moving fluid and rarefaction behind it, result-
ing in a decrease of the gas volume fraction ahead and an
increase behind. The presence of the closed end downstream
produces a significant increase of pressure at the right end of
the tube, so that at a certain time the pressure gradient is
reversed and a leftward flow starts. These oscillations repeat
many times, slowly damping out. Although the maximum
observed velocities are very small, the use of a compressible
model is required to account for density changes which are
explicitly dependent on time and which govern the dynamics
of the liquid–gas mixture in the tube.
Case B. Constant inflow chamber. Case B consists of a 1D tube 1 m
long filled with a gas–liquid mixture of water and carbon diox-
ide with 0.98 and 0.02 mass fractions, respectively, at a pressure
of 1.2 Á 105
Pa, resulting in a gas volume fraction of 0.85. The
1.12
1.2
Pressure (× 105 Pa)
0
0.06
Gas volume fraction
−0.05
0.05
0 s
Velocity (m/s)
1.12
1.2
0
0.06
−0.05
0.05
0.001 s
1.12
1.2
0
0.06
−0.05
0.05
0.003 s
1.12
1.2
0
0.06
−0.05
0.05
0.009 s
1.12
1.2
0
0.06
−0.05
0.05
0.011 s
1.12
1.2
0
0.06
−0.05
0.05
0.015 s
1.12
1.2
0
0.06
−0.05
0.05
0.019 s
1.12
1.2
0
0.06
−0.05
0.05
0.023 s
1.12
1.2
0
0.06
−0.05
0.05
0.027 s
1.12
1.2
0
0.06
−0.05
0.05
0.032 s
1.12
1.2
0
0.06
−0.05
0.05
0.037 s
1.12
1.2
0
0.06
−0.05
0.05
0.042 s
0 0.2 0.4 0.6 0.8 1
1.12
1.2
x (m)
0 0.2 0.4 0.6 0.8 1
0
0.06
x (m)
0 0.2 0.4 0.6 0.8 1
−0.05
0.05
0.047 s
x (m)
Fig. 7. Pressure (left), gas volume fraction (center) and velocity (right) along the initially half-pressurized tube between 0 and 0.05 s.

right end of the tube is closed, while a water-carbon dioxide
mixture with 0.998 and 0.002 mass fractions, respectively,
enters the left end of the tube with a constant velocity of 1 m/
s. The entering mixture has a zero gas volume fraction, accord-
ing to Henry’s law. The tube is discretized in 200 Â 1 elements,
and the time step is 10À4
s. The injected liquid progressively
occupies the tube (Fig. 8a), compressing the two-phase mixture
inside the tube and causing an increase of the dissolved carbon
dioxide (Fig. 8b), a decrease to zero of the gas volume fraction
(Fig. 8c), and an increase both in density and pressure
(Fig. 8d). Once the gas is completely dissolved after 0.5 s, a fur-
ther inflow produces a much steeper pressure increase (Fig. 8d).
7. Conclusions
A space–time Galerkin/least-squares with discontinuity captur-
ing finite element method for compressible and incompressible
flows has been developed extending [19,14] to single-fluid, multi-
component, locally homogeneous gas–liquid mixtures, overcoming
the limitation of perfect gas mixtures. Our method allows the com-
putation of flows involving mixtures with the possibility of choosing
arbitrary EOS and including exsolution of gases. The s matrix and
discontinuity capturing operators have been extended to account
for chemical components and allow tracking of internal interfaces.
The numerical code, developed as a library of C++ classes, has
been tested and validated on many benchmark cases both in the
compressible and incompressible regimes. Two additional cases
(gas–liquid equilibrium bubbly mixture) describing time evolution
of gas–liquid partition with or without transport of compositional
discontinuities have been introduced. The new numerical model
presented here can be successfully employed in the study of a vari-
ety of cases involving fundamental as well as applicative (indus-
trial, geophysical, etc.) areas.
At present the code has been developed and tested in 2D carte-
sian coordinates; extension to 2D cylindrical and 3D cartesian
coordinates, as well as addition of other physico-chemical pro-
cesses (chemical reaction kinetics, crystallization, non-ideal behav-
ior of components,. . .) is in progress.
Acknowledgements
This work has been supported by GNV Projects 2004-06/V3_2
and 2007-09/V1.
Appendix A. Mixture model
The physical model considers an ideal mixture of n components,
that may be either in the liquid or gaseous state. The partial spe-
cific (per unit mass) Gibbs free energy of component k in phase
p, component k, and mixture are [42]:
gp
k ¼ gp;0
k ðp; TÞ þ RkT ln np
k ðA:1Þ
gk ¼
X
p
gp
k gp
k ðA:2Þ
g ¼
X
k
ykgk ¼
X
k;p
yp
k gp
k ðA:3Þ
where gp;0
k is the specific Gibbs free energy in the standard state, np
k
is mole fraction of component k in phase p with respect to phase p,
and gp
k ¼ yp
k =yk is the weight fraction of component k in phase p
with respect to component k. In the assumption of ideal mixture,
np
k and gp
k depend only on pressure and temperature. Either np
k or
gp
k are given by the relations for chemical equilibrium, and can be
derived the ones from the others through a linear relationship. Eq.
0 0.2 0.4 0.6 0.8 1
0.01
0.02
x (m)
TotalCO2massfraction
t=0
t=0.5 s
0 0.2 0.4 0.6 0.8 1
0.01
0.02
x (m)
DissolvedCO2massfractions
t=0
t=0.5 s
0 0.2 0.4 0.6 0.8 1
0
0.4
0.8
x (m)
Gasvolumefraction
t=0
t=0.5 s
0 0.1 0.2 0.3 0.4 0.5
2
3
4
5
6
7
t (s)
log10p,log10ρ
p
ρ
(a) (b)
(c) (d)
Fig. 8. Liquid solution in equilibrium with a gas: profiles from t = 0–0.5 s each 0.05 s, of (a) mass fraction of total carbon dioxide, (b) mass fraction of dissolved carbon dioxide
(solid line) and correspondent total carbon dioxide (dashed line), (c) gas volume fraction, (d) pressure and density at the closed end of the tube as a function of time.

(A.1) is used for both liquid and gaseous mixtures. The partial deriv-
atives entering the V,Y matrix (14), (B.38) can be derived from (A.1):
@gp
k
@p

T;n
¼
1
qp
k
ðA:4Þ
@ gp
k =T
À Á
@T

p;n
¼ À
h
p
k
T2
ðA:5Þ
@gp
k
@np
h

p;T;np
Á ½h;nŠ
¼
RkT
np
k
dkh ðA:6Þ
where notation np
Á ½h; nŠ means that all ns but np
h and np
n are constant.
Specific internal energy and enthalpy of component k in phase p,
component k and mixture are:
ep
k ¼ cp
vkT ðA:7Þ
h
p
k ¼ cp
pkT ¼ ep
k þ
p
qp
k
ðA:8Þ
ek ¼
X
p
gp
k ep
k ðA:9Þ
hk ¼
X
p
gp
k h
p
k ðA:10Þ
e ¼
X
k
ykek ¼
X
k;p
yp
k ep
k ðA:11Þ
h ¼
X
k
ykhk ¼
X
k;p
yp
k h
p
k ¼ e þ
p
q
ðA:12Þ
where qp
k ; cp
vk; cp
pk are density and specific heat coefficients at con-
stant volume and pressure of component k in phase p. The equation
of state qp
k ¼ qp
k ðp; TÞ may have a general form.
The ideal solution mixture implies the following thermody-
namic relations [26]:
1
q
¼
X
k;p
yp
k
qp
k
¼
X
k
yk
qk
ðA:13Þ
@ð1=qÞ
@yk

p;T;yÁ½k;nŠ
¼
1
qk
À
1
qn
ðA:14Þ
@e
@yk

p;T;yÁ½k;nŠ
¼ ek À en ðA:15Þ
where notation yÁ[k,n] means that all ys but yk and yn are constant.
The total specific energy of the mixture in (3) is et ¼ e þ juj2
2
.
Diffusion is modeled with the linear fluxes of momentum, heat
and mass [43]:
sij ¼ lðui;j þ uj;iÞ þ kuk;kdij ðA:16Þ
qi ¼ ÀjT;i ðA:17Þ
Jk
i ¼ Àq
XnÀ1
h¼1
Dkhyh;i for k ¼ 1; . . . ; n À 1; and Jn
i ¼ 1 À
XnÀ1
h¼1
Jk
i ðA:18Þ
with k ¼ lb À 2
3
l, the viscosity coefficients, and neglecting the bulk
viscosity lb under the Stokes’ assumption [15]. The viscosity l is
calculated with the standard rules of mixing [44] for one phase
mixtures and with a semi-empirical relation [45] for bubbly
mixtures:
l ¼ expð
X
k
xk ln lkÞ for one phase mixture ðA:19Þ
l ¼ lc
1 À
ad
adm
À2:5adm ld
þ 0:4lc
ld þ lc
lp ¼ exp
X
k
np
k ln lp
k
! for bubbly mixture
ðA:20Þ
where lc
and ld
are viscosities of the continuous and dispersed
phases, ad
is the volume fractions of the dispersed phase and
adm
= 0.75 is the volume fractions of the dispersed phase at the
maximum package. The thermal diffusion coefficient j is calculated
as:
j ¼
X
k;p
xp
k jp
k ðA:21Þ
where jp
k is the thermal diffusion coefficient of component k in
phase p, and xp
k is the molar fraction of component k in phase p
in the mixture. The mass fluxes Jk
i s are written with the general-
ized Fick’s law derived from the Maxwell–Stefan equations, with
Dkh the matrix of generalized Dkh Fick’s diffusion coefficients
[27,28].
The mixture molar mass M is:
1
M
¼
X
k
yk
Mk
¼
X
k;p
yp
k
Mk
ðA:22Þ
The sound velocity in the mixture is calculated as [46]:
c ¼
X
k;p
ap
k
qp
k cp
k
q
!À1
2
ðA:23Þ
where ap
k and cp
k are volume fraction and sound velocity of compo-
nent k in phase p in the mixture.
Appendix B. Numerical method
The generalization of ST-GLSDC formulation to multicompo-
nent, compressible–incompressible flows makes use of the conser-
vation U, entropy V, and primitive Y variables vectors:
U ¼ q
y1
..
.
ynÀ1
yn
u1
u2
u3
et
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
V ¼
1
T
g1 À k
..
.
gnÀ1 À k
gn À k
u1
u2
u3
À1
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
Y ¼
y1
..
.
ynÀ1
p
u1
u2
u3
T
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
ðB:1Þ
The flux and source vectors in the system of conservation Eq. (4)
are:
Fadv
i ¼ qui
y1
..
.
yn
u1
u2
u3
et
0
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
A
þ p
0
..
.
0
d1i
d2i
d3i
ui
0
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
A
ðB:2Þ
Fdiff
i ¼
0
..
.
0
s1i
s2i
s3i
sijuj
0
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
A
þ
ÀJ1
i
..
.
ÀJn
i
0
0
0
Àqi À
Xn
k¼1
Jk
i hk
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
ðB:3Þ

F ¼ q
0
..
.
0
b1
b2
b3
biui þ r
0
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
A
ðB:4Þ
The quasi-linear form (7) of the system of equations and the
discontinuity capturing operator (14) involve the matrices of par-
tial derivatives of conservation variables and advective fluxes with
respect to primitive variables, U,Y and Fi, Y
adv
, the diffusivity matri-
ces Kij, the source matrix S, the matrices of partial derivatives of
entropy variables with respect to primitive and conservation vari-
ables V,Y and V,U. Matrices are decomposed in submatrices and re-
ported in compact form below.
The U,Y derivatives are:
ðB:5Þ
where the first index indicates conservation variables, derived with
respect to the second index indicating primitive variables. Subma-
trices in (B.5) are:
Uqy; y ¼
y1r1 þ q y1r2 . . . y1rnÀ1
y2r1 y2r2 þ q . . . y2rnÀ1
..
. ..
. ..
. ..
.
ynÀ1r1 ynÀ1r2 . . . ynÀ1rnÀ1 þ q
ynr1 À q ynr2 À q . . . ynrnÀ1 À q
2
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
5
ðB:6Þ
Uqy;p ¼ qbyT
ðB:7Þ
Uqy; u ¼ 0nÂ3 ðB:8Þ
Uqy;T ¼ ÀqayT
ðB:9Þ
Uqu; y ¼
u1r1 . . . u1rnÀ1
u2r1
..
.
u2rnÀ1
u3r1 . . . u3rnÀ1
2
6
6
4
3
7
7
5 ðB:10Þ
Uqu;p ¼ qbuT
ðB:11Þ
Uqu; u ¼ qI3Â3 ðB:12Þ
Uqu;T ¼ ÀqauT
ðB:13Þ
Uqet;y ¼ ðetr1 þ qey
1; . . . ; etrnÀ1 þ qey
nÀ1Þ ðB:14Þ
Uqet;p ¼ ep
1 ðB:15Þ
Uqet;u ¼ qu ðB:16Þ
Uqet;T ¼ ep
4 ðB:17Þ
where
rk ¼ Àq2 1
qk
À 1
qn

ey
k ¼ ek À en
ep
1 ¼ qbðh þ kÞ À aT ep
4 ¼ Àqaðh þ kÞ þ qcp
where a and b are the coefficients of isobaric expansion and isother-
mal compressibility, rk and ey
k
are defined in the present work,
whereas ep
1 and ep
4 have been introduced in (A.19, A.20) of [19].
The partial derivatives of advective fluxes with respect to prim-
itive variables are:
ðB:18Þ
where
Fadv
1

qy; u
¼
qy1 0 0
..
. ..
. ..
.
qyn 0 0
2
6
6
4
3
7
7
5 ðB:19Þ
Fadv
2

qy; u
¼
0 qy1 0
..
. ..
. ..
.
0 qyn 0
2
6
6
4
3
7
7
5 ðB:20Þ
Fadv
3

qy; u
¼
0 0 qy1
..
. ..
. ..
.
0 0 qyn
2
6
6
4
3
7
7
5 ðB:21Þ
Fadv
1

qu;p
¼
qbu2
1
À Á
þ 1
qbu1u2
qbu1u3
0
B
@
1
C
A ðB:22Þ
Fadv
2

qu;p
¼
qbu1u2
qbu2
2
À Á
þ 1
qbu2u3
0
B
@
1
C
A ðB:23Þ
Fadv
3

qu;p
¼
qbu1u3
qbu2u3
qbu2
3
À Á
þ 1
0
B
@
1
C
A ðB:24Þ
Fadv
1

qu; u
¼
2qu1 0 0
qu2 qu1 0
qu3 0 qu1
2
6
4
3
7
5 ðB:25Þ
Fadv
2

qu; u
¼
qu2 qu1 0
0 2qu2 0
0 qu3 qu2
2
6
4
3
7
5 ðB:26Þ
Fadv
3

qu; u
¼
qu3 0 qu1
0 qu3 qu2
0 0 2qu3
2
6
4
3
7
5 ðB:27Þ
Fadv
i

qet;p
¼ uiep
2 ðB:28Þ
Fadv
1

qet;u
¼ ep
3 þ qu2
1; qu1u2; qu1u3
À Á
ðB:29Þ
Fadv
2

qet;u
¼ qu1u2; ep
3 þ qu2
2; qu2u3
À Á
ðB:30Þ
Fadv
3

qet;u
¼ qu1u3; qu2u3; ep
3 þ qu2
3
À Á
ðB:31Þ
and ep
2 ¼ ep
1 þ 1, and ep
3 ¼ qet þ p have been defined in (A.19)-(A.20)
of [19]. The diffusivity matrices are:

ðB:32Þ
The Dij submatrices are of dimensions n Â (n À 1) and form the
mass diffusive fluxes (A.18) written in terms of the first n À 1 inde-
pendent yk as needed by the quasi-linear form (7):
Dij ¼ 0 for i – j ðB:33Þ
Dii ¼ ÀqD ðB:34Þ
e:
D ¼
D11 . . . D1 nÀ1
..
. ..
. ..
.
Dn 1 . . . Dn nÀ1
2
6
6
4
3
7
7
5 ðB:35Þ
The null matrix (0nÂ1j0nÂ3j0nÂ1) in (B.34) corresponds to neglecting
the Soret effect and mass diffusion caused by pressure gradients.
The energy flux due to interdiffusion of components corresponds
to the 1 Â (n À 1) vector hT
Dij, where hT
= (h1, . . ., hn) is the vector
of specific enthalpies of components. The kij submatrices corre-
spond to the diffusivity matrices reported in (A.51)-(A.59) in [19]
where the first line has been eliminated according to the present
extension of the formulation to multicomponent fluids.
The source matrix is:
ðB:36Þ
where the first index refers to conservation variables and the sec-
ond to primitive variables, and subvectors are:
Squ;T ¼ À
q
T
b
T
; Sqet ;u ¼ Àqb; Sqet;T ¼ À
qr
T
ðB:37Þ
The matrix of partial derivatives V,Y is:
ðB:38Þ
where the first index indicates entropy variables, derived with re-
spect to the second index that indicates the primitive variables.
The detailed expressions of all submatrices are:
VgÀk
T
;y ¼
R1 x11 þ 1
y1

R1x12 ... R1x1ðnÀ1Þ
R2x21 R2 x22 þ 1
y2

... R2x2ðnÀ1Þ
..
. ..
. ..
. ..
.
RnÀ1xðnÀ1Þ1 RnÀ1xðnÀ1Þ2 ... RnÀ1 xðnÀ1ÞðnÀ1Þ þ 1
ynÀ1

Rn xn 1 À 1
yn

Rn xn 2 À 1
yn

... Rn xnðnÀ1Þ À 1
yn

2
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
5
ðB:39Þ
VgÀk
T
;p
¼
1
T
1
q1
..
.
1
qn
0
B
B
B
@
1
C
C
C
A
ðB:40Þ
VgÀk
T
;u
¼ À
1
T
u1 u2 u3
..
. ..
. ..
.
u1 u2 u3
2
6
6
4
3
7
7
5 ðB:41Þ
VgÀk
T
;T
¼
1
T
k À h1
..
.
k À hn
0
B
B
@
1
C
C
A ðB:42Þ
Vu
T
;T ¼ À
1
T
u1
u2
u3
0
B
@
1
C
A ðB:43Þ
VÀ1
T
;T ¼
1
T
ðB:44Þ
where xij ¼
P
pgp
i
np
n
yn
À
np
j
yj

. The matrix of partial derivatives V,U is:
V;U ¼ V;Y Y ;U ¼ V;Y UÀ1
;Y ðB:45Þ
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