A mass-averaged velocity model for mixture flows of high
density ratios: Numerical simulations and comparison with
experiments
Jocelyn ÉTIENNE
Laboratoire de Modélisation et Calcul, IMAG, BP 53, 38041 Grenoble, France
Emil J. HOPFINGER
Laboratoire des Écoulements Géophysiques et Industriels, HMG, BP 53, 38041 Grenoble,
France
Pierre SARAMITO
Laboratoire de Modélisation et Calcul, IMAG, BP 53, 38041 Grenoble, France
Binary mixtures of two fluids with a large density difference are present in many environment
and industrial flows, but the numerical simulations of such flows found in literature [14, 4, 12] are
limited to relatively small density variations. The models used are usually based on a mixture
velocity v which is a volume average of the local velocities of each of the constituents. This
choice is convenient because it preserves the continuity equation of homogeneous flows, that is
∇ · v = 0. On the other hand, the momentum equation and boundary conditions are given
in terms of the mass-averaged velocity u, and it is unclear how these boundary conditions can
be applied on v. In Section 1, we give a simple derivation of the model in terms of velocity
u, based on conservation arguments only. We discuss briefly the well-posedness issues for the
model, which has been studied with various restrictions by Kazhikhov and Smagulov [16], Beirao
da Veiga et al. [7], Majda [18], Embid [8], Lions [17] and Bresch et al. [5]. We refer to this
last article for a review of these issues. In Section 2, we recall how the same model and the
model in terms of v can be obtained by means of ensemble averaging, as done for instance by
Joseph and Renardy [15] or Rajagopal and Tao [21]. This leads to define two velocity fields
which describe the motion of each of the constituents, and gives the corresponding conservation
equations. Conservation equations for the mixture (written either in terms of u or v) can then
be deduced, and the velocity difference w between the constituents can be modelled.
Then we introduce in Section 3 a numerical scheme for the model in u and present simultation
results with density ratios up to 100 in the case of the lock-exchange problem. This problem has
the advantage of having a very simple set-up, shown in Figure 1: a channel is divided into two
parts by a vertical splitter-plate and both chambers are filled with gases of different densities.
When one removes the splitter-plate, a lock-exchange flow occurs, that is a dense current of the
heavier gas spreads on the ground of the lighter gas chamber, and a light one on the ceiling
of the other chamber. This flow has been extensively studied in fluid mechanics literature,
both theoretically [2] and experimentally [11], and quantitative results are available to which we
compare our numerical results in Sections 4 and 5.
1
Governing equations
Let us consider an isothermal flow of local density % and velocity u in a domain Ω over a time
span [0, T ]. For a perfect mixture of two incompressible fluids, of density % d (the heavier one)
and of density %` (the lighter one), the local density is % = % d Φd + %` Φ` where Φd , Φ` are the
volumic fraction of the constituents, Φ d + Φ` = 1 and both, %d and %` , are constants. The
`
> 0.
characteristic density ratio is α = %d%−%
`
Our main concern in this section is to take into account the mutual diffusion of the fluids in
the non-homogeneous, incompressible Navier-Stokes equations. We will see that, as in [15], this
yields a non-solenoidal velocity.
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1
b
Yt = 0
p=0
Ω− ,
%`
%d
ey
ex
Yt = 0
a
Lock-gate
2H
%d
%`
ey
ex
20H
10H
Figure 1: The exchange flow: boundary conditions in the initial configuration and experimental
set-up used by Gröbelbauer et al.[11]
1.1
Mass and constituent conservation equations
The mass conservation of constituent i across the surface S of a fixed volume V can be written
as:
Z
Z
Z
∂
%i q i · n dS
%i Φi ui · n dS +
%i Φi dV =
−
∂t V
S
S
where q i is the part of the mass flux which is due to diffusion. Thus Φ i , i = d, ` obey the
equations:
DΦd
+ Φd ∇ · u = −∇ · qd
Dt
(1)
Fick’s law governs the diffusive fluxes of one fluid into the other such that % d qd + %` q` = 0 [15]
1
with %i q i = − ReSc
F√
(Φd )∇Φi , where ReSc = UDH is the product of the Reynolds and Schmidt
numbers, with U = αgH the terminal velocity of a dense fluid parcel in the light fluid, and H
the half-height of the flow domain (see Fig. 1). Since Φ ` = 1 − Φd , we use only Φ = Φd in the
1
[20, 9]. The present
sequel. From kinetic gas theory we know that F (Φ) ought to vary as 1+αΦ
calculations do not include this feature because it was found to destabilize strongly the numerical
schemes. We, therefore, focused on solving accurately the system assuming F (Φ) ≡ 1. The
results show that for the lock-exchange flows this assumption has only a second order influence.
We obtain the continuity and constituents equations:
α DΦ
1 + αΦ Dt
1
DΦ
+ Φ∇ · u =
∇ · (F (Φ)∇Φ)
Dt
ReSc
∇·u=−
(2)
(3)
The equation ∇ · u 6= 0 is unusual. It arises in spite of the incompressibility of each constituent
(i.e., %d and %` are constants) because of the diffusion between the two species. It is readily seen
from equations (2, 3) that when Sc tends to infinity, ∇ · u goes to zero. Otherwise, diffusion
will result in equal and opposite mass fluxes of constituents d and ` across the boundary of
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2
any small volume V(t) entrained by the flow velocity. As a result, since both constituents are
incompressible and of different densities, the volume V(t) will vary; giving ∇ · u 6= 0. Note that
diffusion effects are obviously negligible for Boussinesq conditions, α 1.
1.2
Momentum equation
We can assume that the mixture behaves like a Newtonian fluid, with a dynamic viscosity µ that
may depend on the local composition of the mixture Φ. Therefore, we write µ(Φ) = ηλ(Φ), where
η is a constant reference dynamic viscosity, and λ a non-dimensional function of Φ. Denoting
Du = 21 (∇u + ∇uT ), the momentum equation is:
1
2
1 + αΦ
Du
= −∇p +
∇ · λ(Φ) 2Du − ∇ · u I −
ey
(4)
(1 + αΦ)
Dt
Re
3
α
and Re = %` Uη H . For lock-exchange flows and most gravity-driven flows, the boundary condition
t
for u is either u |∂Ω = 0 (no inflow, no slip condition) or u · n = 0 and ∂u
∂n = 0 where ut =
(u − (u · n)n) is the tangential velocity (no inflow, slip condition). Then, for both mechanical
∂Φ
= 0.
and mathematical reasons, the boundary condition for Φ will be ∂n
The choice of λ(Φ) is subject to a debate. On the one hand, for all gases including the ones used
in the experiments by [11], the dynamic viscosity is much the same and so λ(Φ) ≡ 1. However,
α
< 1.
in this case, existence of a global weak solution [1] is subject to the condition that 2Sc
There is no physical reason for the Schmidt number to behave this way when α varies; indeed,
its value remains of order 1 for common gases. Thus in most cases the above condition is not
met, and there exists a finite time after which the equations have no regular solution anymore.
In practice, this appeared as a blow-up of the numerical solution in the relevant time-range for
lock-exchange flows for α ≥ 60.
α
(1 + αΦ)F (Φ)∇Φ holds,
Bresch et al. [5], on the contrary, show that if the relation ∇λ(Φ) = 2Sc
then the unconditional existence of global weak solutions can be proved. This condition is never
satisfied if we choose λ(Φ) = 1. If we take a constant kinematic viscosity ν = µ% , that is, if
1
. Though we were
λ(Φ) = 1 + αΦ, then the relation is matched for Sc = 12 , and F (Φ) = 1+αΦ
1
not able yet to incorporate F (Φ) = 1+αΦ in our numerical simulations (thus the relation is not
satisfied), the solutions for constant kinematic viscosity (λ(Φ) = 1 + αΦ) and F (Φ) = 1 were
stable in all cases (α up to 100 was tested).
2
Derivation of the model from two-fluid equations
In order to explicit the relation between the model used here and other models used in literature,
we present another derivation based on the works of Joseph and Renardy [15] or Rajagopal and
Tao [21]. Note that this derivation is only valid if each constituent in the mixture is present at
every point of the domain, that is 0 < Φ(x) < 1. If this is the case, the velocity fields u d and
u` can be defined by ensemble averaging the velocities of molecules of each constituent locally
in space.
This gives conservation laws of the form, for i = d, ` :
∂(%i Φi )
+ ∇ · (%i Φi ui ) = 0
∂t
1
%i Φi
∂(%i Φi ui )
+ ∇ · (%i Φi ui ⊗ ui ) = −∇(Φi p) +
∇ · (Φi λ(Φ)σi ) ± ξ(Φ, w) +
g
∂t
Re
α
(5)
(6)
where σi need to be modelled, w = ud − u` and ±ξ is a volumic interaction force between the
two constituents.
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3
Following Rajagopal and Tao [21, chap. 7], we can write mixture equations in terms of the
` Φ` u`
mass-averaged velocity u = %d Φd ud +%
. In this case, the continuity equation is obtained by
%
summing equation (5) for i = d, ` :
∂%
%d Φd ud + % ` Φ` u`
+ ∇ · (%
)=0
∂t
%
in which we recognize equation (2). Similarly, we sum equation (6) for i = d, ` :
∂%u
Φd Φ`
%
+ ∇ · (%u ⊗ u) = −∇p + ∇ · λ(Φ)(Φd σd + Φ` σ` ) − %d %`
w⊗w + g
∂t
%
α
Rajagopal and Tao then propose to assume that the stress tensor in the right hand side is equal to
the Cauchy stress tensor of a Newtonian fluid and deduce the form of σ i . This assumption leads
to the same equation likeequation (4).
Finally, we remark that (1 + αΦ)(u − v) = αΦ(1 − Φ)w,
αΦ(1−Φ)
which yields ∇ · u = ∇ ·
1+αΦ w , and this yields a third equation (3) when combined with
equation (2).
On the other hand, one can also define a volume-averaged velocity v = Φ d ud +Φ` u` . Joseph and
Renardy introduce this field, and summing equation (5) divided by % d and %` respectively for
i = d, ` obtain that ∇ · v = 0. This is an interesting feature for numerical analysists, and among
others, Boyer [4] derivates the momentum equation in terms of v. This is more complex than for
u, and transport terms of the form ∇ · [Φ(1 − Φ)(v ⊗ w + w ⊗ v)] appear. Boyer eliminates them
thanks to order of magnitude arguments. A third equation ∂Φ
∂t + v · ∇Φ + ∇ · (Φ(1 − Φ)w) = 0
completes his model.
Both models need to be closed with an equation prescribing w. In our case of an inert gas
mixture, Fick’s law governs the velocity difference between the two constituents, and can be
1
G(Φ)∇Φ. It can be shown [9] that G(Φ) =
written in its general form as Φ(1 − Φ)w = − ReSc
(1 + αΦ)F (Φ) where F was introduced in Section 1.
3
Numerical approach
The flows of interest, with large density variations, are composed of intrusion fronts, where
density and velocity gradients are locally steep, and of large areas away from these fronts which
have a uniform density and small velocity gradients away from the walls.
This calls for a method capable of automatic and unconstrained mesh adaptation, since the
location of the interface between dense and light parts of the flow is unknown. However, refining
the mesh in areas of steep density gradients makes it difficult to control a numerical stability
condition (CFL). Thus we use the method of characteristics for the time-discretisation of the
convective part of the equations, which is not subject to such a condition [19].
3.1
Discretisation in time
∂
+ u · ∇ by a
The method of characteristics consists in approximating the total derivative ∂t
finite difference in time along the pathlines of the flow. First we define the pathlines with a
mapping X(x, t; t + τ ) between the fluid particles located at x in Ω at time t and the position
these reach when advected by the fluid velocity u over a time-span τ :
Z t+τ
X(x, t; t + τ ) = x +
u(X(x, t; s), s)ds
(7)
t
Then it is easily shown that
∂
+ u · ∇ f (x, t + ∆t) =
∂t
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f (x, t + ∆t) − f (X(x, t + ∆t; t), t)
+ O(∆t).
∆t
4
(8)
Thus, if we can calculate X n an approximation of X(·, tn +∆t; t), then we can define an implicit
Euler scheme between tn and tn+1 = tn + ∆t using this equality.
We cannot apply directly (7) since we have used the unknown velocity u(x, t + τ ) for τ ∈ [0, ∆t],
R t+∆t
while we only know u(x, t), but X n = x − t
u(X(x, t; s), t)ds = X(x, t + ∆t; t) + O(∆t 2 )
can be calculated. This does not affect the order of approximation in (8). Using this, equation
(3) yields a Poisson-like, classical problem, and equations (4, 2) a Stokes-like problem.
3.2
Semi-discrete algorithm
∂Φ
We will restrict ourselves here to the case of closed boundaries (such that u| ∂Ω = 0 and ∂n
= 0),
which is not very stringent since many variable-density problems occur in such configurations.
A slip condition would be a straightforward extension of this scheme, but would make the
notations superfluously complicated. Thus
search
the solution
(Φ, u, p) in V × V 0 d × Q,
we will
R
1
1
2
with V = H (Ω), V0 = H0 (Ω) and Q = q ∈ L (Ω), Ω qdx = 0 . χ is an intermediate variable
in V which stands for −∇ · u.
The variational formulation is written in terms of the multilinear forms:
1
1
a(Φ, u, v) = ∆t
(u, Φv) + Re
2 (Du, λ(Φ)Dv) − 32 (∇ · u, λ(Φ)∇ · v)
b(v, q) = − (q, ∇ · v)
1
1
c(Φ, ψ) = ∆t
(Φ, ψ) + ReSc
(∇Φ, ∇ψ)
Now we discretize the problem by choosing finite element spaces V h and Qh for the approximation
of V and Q. A classical choice for solving the Stokes problem is obtained with the Taylor-Hood
finite element [25], which is a piecewise quadratic approximation of the velocity and a piecewise
linear one for the pressure. The volume fraction Φ is also discretised in a piecewise quadratic
functional space.
Algorithm
Initialization: n = 0. Choose Φ0h some arbitrary function in Vh , with Φ0h (x) ∈ [0, 1], a.e. x ∈ Ω
d .
and ∇Φ0h · n∂Ω = 0, a.e. x ∈ ∂Ω, and u0h in V0,h
Loop: n ≥ 0, assuming (Φn , un ) are given.
• Step 1: Calculate X n (·) with:
X n (x) = x − ∆t unh (x −
∆t n
u (x))
2 h
(9)
• Step 2: Find Φn+1
in Vh such that, for all ψh ∈ Vh ,
h
1 n
n+1
n
n n
Φ ◦ X , ψh .
c(Φh , ψh ) =
Φh χ +
∆t h
(10)
• Step 3: Calculate Γn+1
∈ Vh , such that, for all ψh ∈ Vh ,
h
Γn+1
h , ψh
=
• Step 4: Calculate χn+1
= Γn+1
−
h
h
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Φn+1
− Φnh ◦ X n
α
h
, ψh
1 + αΦnh
∆t
R
n+1
Ω Γh
5
dx .
!
(11)
1.2
1.2
32
1
1
8
2
0.25
−0.25
−2
−8
−32
0
1.5
2
3
0
1.5
3.3
2
3
3.3
Figure 2: Local zoom in Ω showing the non-dimensional vorticity (left) and the mesh used for
its calculation (right) around a lock-exchange dense intruding front with α = 3 at t = 6.
d
• Step 5: Find (un+1
h , ph ) in V0,h × Qh such that,
n+1
a(Φn+1
h , uh , v h )
1
(un ◦ X n , v h ) −
+ b(v h , ph ) =
∆t h
1 + αΦn+1
h
ez , v
α
!
d
∀v ∈ V0,h
(12a)
n+1
b(un+1
h , q h ) = χh , q h
∀qh ∈ Qh
(12b)
Step 1 of the algorithm is more complicated than it appears if one considers that we use unstructured meshes with strong local refinements. This means that the knowledge of the coordinates
of X n (x) does not give directly the element K of the mesh it belongs to, and an efficient search
algorithm is necessary to determine it. Indeed, if N denotes the number of elements in our mesh,
the search algorithm will be used for each degree of freedom in the mesh, that is O(N ) times
per time-step. We propose an algorithm which allows to keep the overall cost of a time-step in
O(N ln N ), and consists for a given mesh in sorting its elements in a localization tree of depth
ln N , which allows a O(ln N ) localization for each degree of freedom.
Step 2 is then a classical elliptic equation to solve, a multifrontal LU -type factorization is used.
Step 3 and 4 are a specific technique that we need to introduce in order for the discrete Stokes
problem (12) to have a solution. Indeed, the subset of V 0d such that (12b) holds is non-empty if
and only if χn+1
∈ Q [6]. Since no discretisation is known which preserves this condition and give
h
optimal error estimates in L2 –norm, we overcome this difficulty by calculating first an optimal approximation Γn+1
of χ(·, tn+1 ) in Step 3 and project it Ronto Q in Step 4. This projection preserves
h
Γn+1 − χ(·, tn+1 ) .
the error estimate because from Schwarz inequality, Ω Γn+1
dx
≤
C
h
h
0,Ω
In Step 5 remains a Stokes-like problem, with the difference that the right hand side of equation
(12b) is not zero. We use an augmented Lagrangian technique with a Uzawa iterative algorithm
for problem (12) as done in [22].
In [9] we prove that this scheme yields optimal error bounds ku−u h kV +kΦ−ΦhkV ≤ C(h2 +∆t)
and that for any ε ≥ 0, for a sufficiently fine mesh and time step we have −ε ≤ Φ nh (x) ≤ 1 + ε
for any x ∈ Ω and tn ∈ [0, T ]. We also explain the difficulty of alternatives to the projection
step 4.
3.3
Mesh adaptation
Mesh adaptation aims at reducing the projection error of the solution onto the finite element
space in which we approximate it. A first guess of the solution at time t n+1 is calculated on a
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6
position y
2
t=5
2
t = 10
2
1.5
1.5
1.5
1
1
1
0.5
0.5
0.5
0
-0.02-0.01 0 0.01 0.02
velocity ux
0
-0.02 -0.01 0 0.01 0.02
velocity ux
0
t = 15
-0.02 -0.01 0 0.01 0.02
velocity ux
Figure 3: Velocity profile at x = 0 and different instants of time in asymptotic theory --- and
in numerical simulations —.
uniform coarse mesh, and is used to generate a new mesh on which a better approximation of the
solution can be calculated. When iterated, this procedure reaches a fixed point corresponding
to the best approximation space of a given dimension for the solution [3]. This process is
handled by the mesh generator bamg [13] for both Φ and u, using refinement ratios of order 10 3
between the coarsest triangle size and the finest one. Figure 2 shows the mesh refined around
the vorticity-sheets of a dense intruding front. The whole of the finite elements resolution is
embedded in the open-source C++ environment rheolef [23].
4
Asymptotic behaviour at the release
Following Stoker[24] who obtained an asymptotic solution for the breaking of a dam, we have
conducted an analytical study of the onset of the lock-exchange flow in the case when α tends to
infinity, noticing that, away from the walls, viscous effects are neglectible in the limit t → 0. The
boundary conditions are shown in Figure 1, and in addition we suppose that the left boundary
is at the infinity. Also, we suppose that the side walls of the channel allow a perfect slip and
thus that the solution is spanwise-invariant (in z direction). Note that since we neglect % ` , only
the left part of the domain Ω− is considered in the calculation. Because α is then
√ infinity, we
do not use the same non-dimensional form as in Section 1, but we use U 0 = gH. Thus in
Lagrange representation with (a, b) the coordinates corresponding to the initial positions of the
particles, if we denote X(a, b; t) and Y (a, b; t) the displacement of the particles and p(a, b; t) the
pressure, the Euler equations can be rearranged such that:
Xtt Xa + (Ytt + 1)Ya + p̃a = 0
(13a)
Xtt Xb + (Ytt + 1)Yb + p̃b = 0
(13b)
Xa Yb − Xb Ya = 1,
(13c)
where p̃ = % H
0 2 p.
dU
The initial conditions correspond to the gate in Figure 1 with the fluid at rest, so that a
Taylor expansion of the displacements around t = 0 gives X(a, b; t) = a + γt 2 + o(t2 ) and
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7
Y (a, b; t) = b + δt2 + o(t2 ), and keeping the O(t2 ) terms in (13):
2γ(1 + γa t2 ) + (2δ + 1)δa t2 + βpa = 0
2
2
(14a)
2γγb t + (2δ + 1)(1 + δb t ) + βpb = 0
(14b)
γa + δ b = 0
(14c)
Substracting equations (14a) and (14b), derivated with respect to b and a respectively, yields
γb − δa = 0.
(15)
We recognize in equations (14c,15) the Cauchy-Riemann conditions, thus, it is necessary and
sufficient that the complex function δ + iγ be an analytic function of a + ib in its domain so that
γ, δ are solutions of the problem.
Now we make use of the boundary conditions. Obviously, δ vanishes for b = 0 and b = 2. For
a = 0, using the free surface condition p = 0, the first order term in (14b) gives δ = − 12 , and for
a → −∞ we have δ → 0. From equations (14c,15), we infer that ∇γ · n ∂Ω− = 0.
Since the system (14c,15) implies that ∆γ = ∆δ = 0 in Ω − , there cannot be more than one
solution for δ, and γ is unique up toR a constant. This constant is easy to determine, since there
2
must be no influx from infinity, so 0 γ(a, b)db tends to 0 when a tends to infinity.
Using the mapping w̄ = cosh π(−a+ib)
, Stoker exhibits an analytic function which enforces the
2
w̄−1
i
; and finally we obtain the initial acceleration:
boundary conditions: δ + iγ = − 2π log w̄+1
!
2 πa
cos2 πb
1
4 + sinh 4
(16)
ln
2γ(a, b) =
2 πa
π
sin2 πb
4 + sinh 4
!
sin πb
2
2
2δ(a, b) = − arctan
(17)
π
sinh πa
2
02
The acceleration is independent of % d , but depends only on UH = g. There is a singularity in
the acceleration at the junction points between the free surface and the walls. This of course
would be damped by viscous forces, nevertheless we can expect a strong boundary layer at these
points. Moreover, since the viscous effects propagate as νt, we can compare the velocity profile
of the solution of a viscous model with the analytical results outside the boundary-layer.
In Figure 3 we have plotted the velocity obtained from asymptotic theory of the a = 0 particles
at time t. For comparison, we have included the velocity of the particles at x = 0 at the same
instant1 obtained from a numerical simulation with α = 79. Figure 3 shows that the numerical
error is small.
5
Results and comparison with experiments
Gröbelbauer et al.[11] have measured the passage time of both the dense and the light fronts
of an exchange flow at fixed positions on the horizontal walls of a channel. They have used
different gas combinations with density ratios α + 1 up to 21.6 (Freon R22 and Helium), with
the set-up shown in Figure 1.
Numerical simulations were carried reproducing strictly the conditions of the experiments, in
2D since we are in laminar conditions [10]. A stick condition was used for all boundaries, which
is known to be the right condition
q for gas-solid interfaces. We present the results in Figure 4 in
∗
`
terms of the parameter % = %%dd −%
+%` .
Therefore, because the Eulerian coordinates of particles at a = 0 are (x, y)T = t2 (γ(0, b), δ(0, b))T , the
comparison is only valid up to the first order in t.
1
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8
u
The main interest of the experiments was to provide a law for the Froude numbers Fr = √front
of
gH
both the light and dense fronts versus the density ratio. In addition
to experimental data, the
√
limit-cases of α = 0 (Fr = 0) and α → ∞ (Fr ` = √12 and Frd = 2 2) are classical results [2, 24].
The light front seems to obey to a roughly linear law in experiments and for the λ(Φ) = 1 in
Figure 4, but concerning
dense front experimental and numerical results with λ(Φ) = % give
√ the√
4
a power law Frd = 2 2(1 − 1 − %∗ ) (see Figure 5).
It is surprising that a different model for each front provides good results. This is investigated
in [10]. Nevertheless, the discrepancy of one model each time finds obviously its origin in the
modelling of the flow, and the good fit of the other, in addition to the results of the previous
section, indicates that the numerical simulations were successful in approximating the model
solution.
6
Conclusions
The direct numerical simulations presented in this paper are, to our knowledge, the first simulations of exchange flows of miscible fluids of large density ratios. The reasons of the difficulty
of the numerical simulation of this problem are exposed and an appropriate numerical scheme
is designed. A finite elements discretization is used, allowing a dynamic mesh adaptation which
is an essential feature in the simulations of this type of flow, and simulations with density ratios
up to 100 are performed. We calculate an asymptotic solution for this flow which compares well
with the numerical solution. Finally, the comparison with the experiments by Gröbelbauer et
al. [11] raise important modelling questions on the binary mixtures flows
Références
[1] S. N. Antonsev, A. V. Kazhikhov, and V. N. Monakov. Boundary value problems in mechanics of nonhomogeneous fluids. North-Holland, 1990.
[2] T. B. Benjamin. Gravity currents and related phenomena. J. Fluid Mech., 31(2), 1968.
[3] H. Borouchaki, P. L. George, F. Hecht, P. Laug, and E. Saltel. Delaunay mesh generation
governed by metric specifications. Part I: Algorithms. Finite Elem. Anal. Des., 25:61–83,
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Fr`
0.8
0.4
Experimental results
Fr` for λ(Φ) = 1
Fr` for λ(Φ) = 1 + αΦ
00.5
0.6
0.7
2
%∗
0.8
0.9
1
0.8
0.9
1
Frd
1.5
1
0.5
Experimental results
Frd for λ(Φ) = 1
Frd for λ(Φ) = 1 + αΦ
0
0.5
0.6
0.7
%∗
Figure 4: Froude numbers Fr` and Frd of the light and dense fronts in experiments and numerical
simulations for both viscosity models. The dotted line joins the theoretical limits for % ∗ = 0 and
%∗ = 1.
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Experimental results
Frd for λ(Φ) = 1
Fr√d for
√ λ(Φ) = 1 + αΦ
2 2 4 1 − %∗
√
2 2 − Frd
2.5
1
0.01
0.1
1 − %∗
1
Figure 5: Correlation law between Froude number Fr d of the dense front and 1 − %∗ .
Figure 6: Non-dimensional vorticity maps for α = 39 at different stages in the flow for the
constant dynamic viscosity model λ(Φ) = 1.
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