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A comparison of different approaches to compute surface tension contribution in incompressible two-phase flows

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We perform a quantitative assessment of different strategies to compute the contribution due to surface tension in incompressible two-phase flows using a conservative level set (CLS) method. More specifically, we compare classical approaches, such as the direct computation of the curvature from the level set or the Laplace-Beltrami operator, with an evolution equation for the mean curvature recently proposed in literature. We consider the test case of a static bubble, for which an exact solution for the pressure jump across the interface is available, and the test case of an oscillating bubble, showing pros and cons of the different approaches.
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A comparison of different approaches to compute surface
tension contribution in incompressible two-phase flows
Giuseppe Orlando(1)
Paolo Francesco Barbante(2), Luca Bonaventura(2)
(1) CMAP, CNRS, ´
Ecole polytechnique, Institute Polytechnique de Paris
Route de Saclay, 91120 Palaiseau, France
giuseppe.orlando@polytechnique.edu
(2) MOX - Dipartimento di Matematica, Politecnico di Milano
Piazza Leonardo da Vinci 32, 20133 Milano, Italy
paolo.barbante@polimi.it, luca.bonaventura@polimi.it
Keywords: Navier-Stokes equations, Incompressible flows, Two-phase flows, Level set, Sur-
face tension, Curvature.
1
arXiv:2402.04670v3 [physics.flu-dyn] 12 Nov 2024
Abstract
We perform a quantitative assessment of different strategies to compute the con-
tribution due to surface tension in incompressible two-phase flows using a conservative
level set (CLS) method. More specifically, we compare classical approaches, such as
the direct computation of the curvature from the level set or the Laplace-Beltrami
operator, with an evolution equation for the mean curvature recently proposed in lit-
erature. We consider the test case of a static bubble, for which an exact solution for
the pressure jump across the interface is available, and the test case of an oscillating
bubble, showing pros and cons of the different approaches.
2
1 Introduction
Interfacial flows with surface tension play an important role in several industrial
and engineering applications [15, 17]. Many modelling approaches have been pro-
posed to capture the motion of the interface. We consider here the conservative level
set (CLS) method, originally proposed in [20], [21], to which we refer for a detailed
description of the scheme. In this framework, the normal to the interface and the cur-
vature are implicitly determined from the level set function. We compare here different
approaches to compute the force due to surface tension. More specifically, we consider
three possible strategies: the use of the Laplace-Beltrami operator, the estimation of
the total curvature directly from the level set function, and the use of an evolution
equation for the mean curvature recently proposed in [23].
The paper is structured as follows: in Section 2, we briefly recall the different
formulations chosen to model the surface tension force. In Section 3, we briefly outline
the numerical method employed for the analysis. Section 4 is devoted to a quantitative
assessment of relations introduced in Section 2 for the test case of a static bubble and
for the test case of an oscillating bubble. Finally, some conclusions and insights for
future work are presented in Section 5.
2 Mathematical model
Let Rd,2d3 be a connected open bounded set with a sufficiently smooth
boundary and denote by xthe spatial coordinates and by tthe temporal coordinate.
The two fluids in are considered immiscible and they are contained in the subdomains
1(t) and 2(t), respectively, so that 1(t)2(t) = Ω. The interface between the two
fluids is denoted by Γ(t), defined as Γ(t) = 1(t)2(t). We consider the classical
unsteady, isothermal, incompressible Navier-Stokes equations without gravity, which
read as follows [22]:
(ρ(x, t)u)
∂t +∇· (ρ(x, t)uu) = −∇p+∇·[2µ(x, t)D(u)] + fσ
∇·u= 0,(1)
for xΩ, t(0, Tf], supplied with suitable initial and boundary conditions. Here,
Tfis the final time, uis the fluid velocity, pis the pressure, ρis the fluid density and
µis the dynamic viscosity. We assume that both density and viscosity are defined as
ρ(x, t) = (ρ1in 1(t)
ρ2in 2(t)and µ(x, t) = (µ1in 1(t)
µ2in 2(t)(2)
with ρ1, ρ2, µ1,and µ2constant values. Moreover, D(u) denotes the symmetric part of
the gradient. Finally, fσrepresents a volumetric force which takes into account surface
tension, defined as [14]
fσ=σκnΓδ(Γ(t)),(3)
where σis the constant surface tension coefficient, nΓis the outward unit normal
to Γ, κ=∇·nΓis the total curvature, and δ(Γ(t)) is the Dirac delta distribution
supported on the interface. In the following, for the sake of simplicity in the notation,
3
we omit the explicit dependence on space and time for the different quantities. As
discussed in [16], we can rewrite the volumetric force as
fσ=∇·[σ(InΓnΓ)δ(Γ)] ,(4)
which corresponds to the application of the Laplace-Beltrami operator. The CLS
method [20, 21] describes implicitly the interface in terms of a regularized Heaviside
function ϕand its evolution equation reads as follows:
∂ϕ
∂t +u· ϕ= 0.(5)
For the sake of completeness, we report the definition of ϕin terms of the signed
distance function φ, employed in the classical level set method [25]. The following
relation is assumed:
ϕ=1
1 + eφ/ε ,(6)
where εhelps smoothing the transition of the discontinuous physical properties between
the two subdomains and it is also known as interface thickness. From definition (6),
it follows that
Γ = x : ϕ=1
2.(7)
The Continuum Surface Force (CSF) approach, introduced in [6], is employed to treat
density, viscosity, and surface tension. More specifically, we set
ρρ2+ (ρ1ρ2)ϕ(8)
µµ2+ (µ1µ2)ϕ. (9)
For the surface tension term, we recall that [9]
δ(Γ) = δϕ1
2|∇ϕ|,(10)
where δϕ1
2is the Dirac delta distribution with support equal to the interface
implicitly described by ϕ=1
2. Hence, we consider the following approximations:
fσ ∇·σ|∇ϕ| ϕ ϕ
|∇ϕ| (11)
fσ σ∇·ϕ
|∇ϕ|ϕ=σϕ·Hϕϕ |∇ϕ|2ϕ
|∇ϕ|3(12)
where we exploit the relation nΓ=ϕ
|∇ϕ|. Notice that, since ϕrepresents a regularized
Heaviside function, we approximate
δϕ1
2= ϕ1
2
θ(φ)1
2
= (φ)
= 1 (13)
with θdenoting the Heaviside function. The second relation in (12) is based on the
so-called Bonnet’s formula [11] for the curvature and Hϕdenotes the Hessian matrix
4
of ϕ. Finally, we propose here another strategy to evaluate fσ. We compute κfrom an
evolution equation for the mean curvature H=κ
2recently proposed in [23]:
∂H
∂t +u· H=H(u)nΓ·nΓ+1
2nΓ: (u)T
1
2(u)TnΓ·(nΓ)nΓ
1
2(nΓnΓI) : hh(u)TnΓiiT,(14)
or, equivalently,
∂H
∂t +u· H=1
2nΓ: (u)T+1
2∇·h(InΓnΓ) (u)TnΓi,(15)
so that
fσ2σHϕ. (16)
The discretization of incompressible Navier-Stokes equations poses several major com-
putational issues. In particular, the velocity uand the pressure pare coupled by the
incompressibility constraint ∇·u= 0. We adopt here the so-called artificial compress-
ibility formulation, originally introduced in [7] and employed in [3], [4], [18], [22], [24]
among many others. The incompressibility constraint is relaxed and a time evolution
equation for the pressure is introduced. Hence, the final form of the system reads as
follows:
(ρu)
∂t +∇· (ρuu) = −∇p+∇·[2µD(u)] + fσ
1
ρ0c2
∂p
∂t +∇· u= 0 (17)
∂ϕ
∂t +u· ϕ= 0,
with cbeing the artificial speed of sound and ρ0being a reference density. A dimen-
sional analysis can be carried out (we refer to [22] for all the details), so as to obtain
the following system of equations:
(ρu)
∂t +∇· (ρuu) = −∇p+1
Re ∇·[2µD(u)] + 1
W e fσ
M2∂p
∂t +∇· u= 0 (18)
∂ϕ
∂t +u· ϕ= 0,
where, with a slight abuse of notation, we employ the same symbols to mark non-
dimensional quantities. Here Re, W e and Mare the Reynolds, Weber, and Mach
number, respectively, defined as
Re =ρref Uref Lref
µref
W e =ρr ef U2
ref Lref
σM=Uref
c,(19)
with Uref being the reference velocity, Lref denoting the reference length, ρref being
the reference density and µref denoting the reference viscosity.
5
3 Numerical method
In this Section, we briefly outline the numerical method employed for the discretiza-
tion of system (18). We refer to [22] for a detailed description of the numerical scheme.
We consider a decomposition of the domain into a family of hexahedra (quadrilat-
erals in the two-dimensional case) Thand denote each element by K. The skeleton E
denotes the set of all element faces and E=EIEB, where EIis the subset of interior
faces and EBis the subset of boundary faces. Classical jump and average operators are
then defined as customary for finite element discretizations. A face e EIshares two
elements that we denote by K+with outward unit normal n+and Kwith outward
unit normal n, whereas for a face e EBwe denote by nthe outward unit normal.
For a scalar function Ψ the jump is defined as
[[Ψ]] = Ψ+n++ Ψnif e EI[[Ψ]] = Ψnif e EB.(20)
The average is defined as
{{Ψ}} =1
2Ψ++ Ψif e EI{{Ψ}} = Ψ if e EB.(21)
Similar definitions apply for a vector function Ψ:
[[Ψ]] = Ψ+·n++Ψ·nif e EI[[Ψ]] = Ψ·nif e EB(22)
{{Ψ}} =1
2Ψ++Ψif e EI{{Ψ}} =Ψif e EB.(23)
We now introduce the following finite element spaces:
Qk=vL2(Ω) : v|KQkK Th(24)
and
Qk= [Qk]d,(25)
where Qkis the space of polynomials of degree kin each coordinate direction. The
finite element spaces that will be used for the discretization of velocity and pressure
are Vh=Qkand Wh=Qk1L2
0(Ω), respectively, where k2. For what concerns
the level set function and the curvature, we consider instead Xh=Qrwith r2,
so that its gradient is at least a piecewise linear polynomial. In the present work, we
consider k=r= 2.
We briefly recall for the convenience of the reader the formulation of the TR-BDF2.
Let t=Tf/N be a discrete time step and tn=nt, n = 0, . . . , N , be discrete time
levels for a generic time dependent problem u=N(u). The incremental form of the
TR-BDF2 scheme can be described in terms of two stages, the first one from tnto
tn+γ=tn+γt, and the second one from tn+γto tn+1, as follows:
un+γun
γt=1
2Nun+γ+1
2N(un) (26)
un+1 un+γ
(1 γ) t=1
2γNun+1+1γ
2 (2 γ)Nun+γ+1γ
2 (2 γ)N(un).(27)
Here, undenotes the approximation at time n= 0, . . . , N. Notice that, in order to
guarantee L-stability, one has to choose γ= 2 2 [12].
6
A reinitialization procedure is adopted for the level set function. We employ the
following PDE [20, 21]:
∂ϕ
∂τ +∇· (ucϕ(1 ϕ)nΓ) = ∇·(βεuc(ϕ·nΓ)nΓ),(28)
where τis an artificial pseudo-time variable, ucis an artificial compression velocity,
and βis a constant. It is important to notice that nΓdoes not change during the
reinitialization procedure, but it is computed using the initial value of the level set
function. The supplementary equation for the mean curvature (14)-(15) has to be
discretized using (16) to evaluate the surface tension contribution. For the sake of
completeness, we report the first stage of the TR-BDF2 scheme for (14)-(15):
Hn+γHn
γt+1
2un+γ
2· Hn+γ+1
2un+γ
2· Hn=
1
21
2nn+γ
Γ:un+γ
2+1
21
2nn
Γ:un(29)
+1
21
2∇·Inn+γ
Γnn+γ
Γun+γ
2Tnn+γ
Γ
+1
21
2∇·h(Inn
Γnn
Γ) (un)Tnn
Γi,
where un+γ
2=1 + γ
2(1γ)unγ
2(1γ)un1is defined by extrapolation. Hence, the
7
weak formulation for (29) reads as follows:
X
K∈ThZK
Hn+γ
γtwd + 1
2X
K∈ThZK
un+γ
2· Hn+γwd
+1
2X
e∈E ZennHn+γun+γ
2oo·[[w]] dΣ1
2X
e∈E Zennun+γ
2oo·Hn+γwdΣ
+1
2X
e∈E Ze
λn+γ
2
2Hn+γ·[[w]] dΣ (30)
=X
K∈ThZK
Hn
γtwd + 1
2X
K∈ThZK
un+γ
2· Hnwd
1
2X
e∈E ZennHnun+γ
2oo·[[w]] dΣ1
2X
e∈E Zennun+γ
2oo·[[Hnw]] dΣ
1
2X
e∈E Ze
λn+γ
2
2[[Hn]] ·[[w]] dΣ
+1
2X
K∈ThZK
1
2nn+γ
Γ:un+γ
2wd + 1
2X
K∈ThZK
1
2nn
Γ:unwd
+1
2X
e∈E Zennnn+γ
Γun+γ
2oo·[[w]] dΣ + 1
2X
e∈E Ze{{∇nn
Γun}} · [[w]] dΣ
1
2X
e∈E Zennnn+γ
Γun+γ
2oo:DDun+γ
2wEEdΣ1
2X
e∈E Ze{{∇nn
Γ}} :⟨⟨wun⟩⟩dΣ
1
2X
K∈ThZK
1
2Inn+γ
Γnn+γ
Γun+γ
2Tnn+γ
Γ· wd
+1
2X
e∈E ZeInn+γ
Γnn+γ
Γun+γ
2Tnn+γ
Γ·[[w]]
1
2X
K∈ThZK
1
2(Inn
Γnn
Γ) (un)Tnn
Γ· wd
+1
2X
e∈E Zenn(Inn
Γnn
Γ) (un)Tnn
Γoo·[[w]] wXh,
with
λn+γ
2= max un+γ
2+·n+
e
,un+γ
2·n
e.(31)
The numerical approximation of non-conservative terms is based on the approach pro-
posed in [5]. We recast the non-conservative term into the sum of two contributions:
the first one takes into account the elementwise gradient, whereas the second one con-
siders its jumps across the element faces. The second TR-BDF2 stage can be described
in a similar manner. Finally, we consider the following spatial discretization for fσ:
fσ2HϕX
K∈ThZK
2Hϕ·φdΩ+X
e∈E Ze
2{{Hϕ}}[[φ]] dΣX
e∈E Ze
2{{H}}[[ϕφ]] dΣ,
(32)
8
which is again based on the approach presented in [5]. Here, φis a basis function of
the finite element space chosen to discretize the momentum equation, i.e. Vh.
4 Numerical results
In this Section, we present the results from simulations comparing the different ap-
proaches presented in Section 2 to evaluate fσ. The numerical method outlined in
Section 3 has been implemented in the framework of the deal .II library [1, 2].
4.1 Static bubble
We consider the 2D stationary bubble in a zero force field described e.g. in [8, 13, 26].
According to the Laplace–Young law [10], the pressure jump across the interface of two
immiscible fluids is directly related to surface tension. Indeed, the following relation
holds:
p=pin pout =σ
R,(33)
where pin and pout are the pressure inside and outside the bubble, respectively, whereas
Ris the radius of the bubble. A bubble with R= 0.25 m centered in (x0, y0) = (0.5,0.5)
in = (0,1)2is considered. The fluid properties are ρ1=ρ2= 104kg m3,µ1=µ2=
1 kg m1s1, and σ= 1 N m1, so that p= 4 N m2. Finally, we set Tf= 25 s and a
fixed time step t= 0.2 s. The artificial speed of sound is set to c1428 m s1, which
is of the same order of magnitude of the speed of sound in water. The reference length
is equal to Lref = 2R= 0.5 m and the reference velocity Uref is chosen in such a way
that the reference pressure pref =ρref U2
ref is unitary. Hence, we get Uref = 0.01 m s1,
so as to obtain Re = 50, W e = 0.5, and M= 7 ×106. Periodic boundary conditions
are considered. Following [8], we consider three metrics to assess the pressure jump
computation:
1. ptotal =pin pout, where the subscripts in denotes quantities inside the bubble
(averaged over cells with r=q(xx0)2+ (yy0)2R) and out quantities
outside the drop (averaged over cells with r > R,
2. ppartial =pin pout, where the subscripts in denotes quantities inside the bubble
(averaged over cells with rR
2and out quantities outside the drop (averaged
over cells with r > 3
2R), so as to avoid the interface region,
3. pmax =pmax pmin, where pmax and pmin are the maximum and minimum
pressure on the whole domain, respectively.
We also monitor the degree of circularity, defined in [14] as
χ=2pπ|2|
Pb
,(34)
with 2being the subdomain occupied by the bubble, |2|being the area of the bubble
and Pbbeing its perimeter. The degree of circularity is the ratio between the perimeter
of a circle with the same area of the bubble and the current perimeter of the bubble
itself. Since, for a perfectly circular bubble, the degree of circularity is unitary, we
expect values equal to one or very close to it.
9
4.1.1 Use of Laplace-Beltrami operator
In this Section, we consider relation (11) to evaluate fσand we report the results in
Table 1. Notice that we add a small number η= 1010 to the denominator |∇ϕ|so
as to avoid division by zero. The same workaround is adopted to evaluate the unit
normal vector in (12). The approximation is very robust and no oscillations for the
pressure jump arise during the time evolution (see Figure 3). However, one can easily
notice that we achieve convergence towards a value which is different with respect
to the analytical one. This is in agreement with what happens using the Volume
of Fluid method, as reported in [8], and we will further discuss this issue in Section
4.1.3. Finally, the degree of circularity is always around 1 for all the configurations,
meaning that the circular shape is preserved. The relative error for χwith Nel is
around 2 ×105. Finally, the generation of spurious currents, which typically appear
in the form of vortices around the interface, is strongly reduced, as evident from Figure
1. This further confirms the robustness of this approach.
Nel ptotal ppartial pmax χ
40 3.37 4.15 4.16 1.00
80 3.37 4.13 4.16 1.00
160 3.37 4.13 4.16 1.00
320 3.37 4.13 4.17 1.00
Table 1: Static bubble test case, pressure jump across the interface and degree of circularity
at final time t=Tfwith fσcomputed using the Laplace-Beltrami operator (11). Nel denotes
the number of elements along each direction.
4.1.2 Computation of the total curvature from the level set
In this Section, we consider relations (12) for fσ. Table 2 shows the results obtained
with Bonnet’s formula. One can easily notice that the results are much less accurate
with respect to those obtained with the Laplace-Beltrami operator. Moreover, the
values of pressure jump are strongly oscillating, as one can notice from Figure 3. The
two relations in (12), namely the computation of fσwith Bonnet’s formula and with the
divergence of the unit normal are equivalent for continuous functions. However, they
yield different values once projected onto discrete functional spaces. Notice also that,
to apply the first relation in (12), we need to project the unit normal vector ϕ
|∇ϕ|onto a
suitable space, so as to compute the divergence. In spite of this discrepancy, analogous
results are achieved as evident from Table 3 and Figure 3. The presence of spurious
oscillations in the curvature field computed from the level set formulation is well known
in the literature and different filtering approaches have been proposed to mitigate this
issue, see e.g. [11] for a comparison different filtering strategies. For the degree of
circularity, we notice a small degradation in the description of the interface, especially
at coarse resolutions. However, good results are obtained overall. The relative error
for χwith Nel = 320 is around 4 ×104for both relations (12).
10
Figure 1: Static bubble test case, velocity magnitude at t=Tfusing (11) with Nel = 320.
Nel ptotal ppartial pmax χ
40 2.81 3.40 10.26 0.98
80 3.00 3.59 3.64 1.00
160 2.95 3.54 3.59 1.00
320 2.94 3.53 3.58 1.00
Table 2: Static bubble test case, pressure jump across the interface and degree of circularity
at final time t=Tfwith fσcomputed using the Bonnet’s formula (12). Nel denotes the
number of elements along each direction.
Nel ptotal ppartial pmax χ
40 2.26 2.81 9.62 0.98
80 2.98 3.57 3.62 1.00
160 2.95 3.54 3.59 1.00
320 2.94 3.53 3.58 1.00
Table 3: Static bubble test case, pressure jump across the interface and degree of circularity
at final time t=Tfwith fσcomputed using the divergence of unit normal vector (12). Nel
denotes the number of elements along each direction.
11
4.1.3 Use of an evolution equation for the curvature
The use of the Laplace-Beltrami operator is a very robust technique, but, as already no-
ticed, it yields slightly inaccurate results. The reason is that there are functions, such as
the unit normal vector or the curvature, whose definitions are properly meaningful only
for the points on the surface Γ [19, 23]. The capillarity flux tensor (InΓnΓ)|∇ϕ|
introduces instead spurious contributions far from the interface. Analogous consider-
ations hold for the total curvature computed from the level set, for which spurious or
even singular values arise if |∇ϕ| 0. In order to overcome this issue, we employ an
evolution equation for the mean curvature, so as to evaluate fσwith (16). This strat-
egy is conceptually similar to the one proposed in [27], where a curvature-augmented
approach to the level set method has been proposed. However, the evolution equation
(14)-(15) is more general and valid for any sufficiently regular moving closed surface.
A key point is the choice of the initial value for the mean curvature H0, so as to avoid
spurious contributions far from the interface. We set therefore
H0=
1
2
1
(xx0)2+(yy0)2if ϕ0>β
h
0 otherwise. (35)
Here, β= 5 ×104,ϕ0is the initial value of the level set function, and h=L
Nelris the
space step size, with L= 1 m being the domain length, Nel denoting the number of
elements along each direction and rbeing the polynomial degree of the finite element
space chosen for the discretization of the mean curvature. Recall that, in the present
work, we take r= 2. The resulting initial datum for the curvature with Nel = 320 is
reported in Figure 2.
Relation (35) allows us to consider contributions of the curvature in fσonly at
the interface and close to it, namely when |∇ϕ|is above a certain threshold. Table
4 shows the obtained results and one can easily notice that we achieve more accurate
results with respect to those obtained with the Laplace-Beltrami operator. Moreover,
ppartial and pmax are closer as the resolution is increased, meaning that the presence
of spurious contributions far from the interface is progressively reduced. No oscillations
arise for the pressure jump, as evident from Figure 3, analogously to what is observed
for the Laplace-Beltrami operator. Finally, for what concerns the degree of circularity,
the relative error with Nel = 320 is around 1 ×105, which is analogous to the value
obtained with the Laplace-Beltrami operator and one order of magnitude lower than
what obtained with (12).
Nel ptotal ppartial pmax χ
40 3.53 4.36 4.40 0.99
80 3.38 4.13 4.15 1.00
160 3.36 4.10 4.11 1.00
320 3.35 4.05 4.06 1.00
Table 4: Static bubble test case, pressure jump across the interface and degree of circularity
at final time t=Tfwith fσcomputed using (16). Nel denotes the number of elements along
each direction.
12
Figure 2: Static bubble test case, initialization of Husing (35) with Nel = 320.
For longer times, the numerical solution of His corrupted. As discussed in [27],
a reinitialization procedure is necessary, analogously to what is done for the level set
and, more generally, for transport equations of interfacial quantities. This issue starts
arising with the formation of spurious currents in the form of vortices around the
interface, as one can notice from Figure 4. Nevertheless, the magnitude of spurious
currents is significantly reduced with respect to (12), meaning that, in this test case,
the present approach is more robust and accurate with respect to the one presented
in Section 4.1.2. The development of suitable reinitialization techniques for (14)-(15)
will be matter of future analysis.
4.2 Oscillating bubble
We now consider a more dynamic example. Starting from the configuration described
in Section 4.1, we modify the initial shape from a circle to an ellipse by scaling the
semi-axes by a factor 1.25 in the x-direction and 0.75 in the y-direction, respectively.
The final time is now Tf= 20 s. Indeed, as already discussed in Section 4.1, for
longer simulation times the approach based on the evolution equation (14)-(15) is
corrupted. We consider two computational grids, composed by Nel = 40 and Nel = 80
elements, along each direction, respectively. We compare the use of the Laplace-
Beltrami operator (11), the use of the Bonnet’s formula (12) and the computation of
surface tension contribution (16) using the evolution equation for the mean curvature
(14)-(15). Figure 5 shows the interface ϕ=1
2at the final time. One can easily notice
13
Figure 3: Static bubble test case, evolution of ppartial with Nel = 320. The black line
reports the results obtained with the Laplace-Beltrami operator (11), the red line shows
the results obtained with the Bonnet’s formula in (12), whereas the blue dots represent
the results achieved with the divergence of the unit normal in (12). Finally, the green line
shows the results obtained with the evolution equation (14)-(15) for the mean curvature and
surface tension computed with (16).
a) b)
Figure 4: Static bubble test case, velocity magnitude at t=Tfwith Nel = 320, a) curvature
computed from divergence of unit normal as in (12), b) curvature computed from evolution
equation (14)-(15) and surface tension evaluated with (16).
14
that oscillations around the configuration captured by (11) are present using Bonnet’s
formula (12). These oscillations are stronger employing the evolution equation for the
mean curvature (14)-(15). Analogous considerations hold using the 80 ×80 grid, even
though the oscillations are reduced as long as the resolution increases, as one can notice
from Figure 6.
Figure 5: Oscillating bubble test case, isoline ϕ=1
2at t=Tfwith Nel = 40. The black line
shows the results with the use of the Laplace-Beltrami operator (11), the red line reports the
results obtained with the use of the Bonnet’s formula (12), whereas the blue line represents
the results achieved using the evolution equation for the mean curvature (14)-(15) to compute
(16). Nel denotes the number of elements along each direction.
5 Conclusions
We have performed a quantitative assessment of different strategies to compute the
contribution due to surface tension in immiscible incompressible flows. The conserva-
tive level set method developed in [22] has been employed for this comparison. The
results show that the use of an evolution equation for the curvature can be considered
15
Figure 6: Oscillating bubble test case, isoline ϕ=1
2at t=Tfwith Nel = 80. The black line
shows the results with the use of the Laplace-Beltrami operator (11), the red line reports the
results obtained with the use of the Bonnet’s formula (12), whereas the blue line represents
the results achieved using the evolution equation for the mean curvature (14)-(15) to compute
(16). Nel denotes the number of elements along each direction.
as a valid alternative to investigate in order to compute this quantity. The most accu-
rate results presented in Section 4.1 are those obtained using the evolution equation for
the mean curvature (14)-(15) to compute the surface tension contribution (16). Issues
are instead present for longer times and for dynamic configurations, as discussed in
Section 4.2, for which this approach, at the current stage, yields less accurate results
and requires therefore further investigation. In future work, we aim to develop suitable
reinitialization techniques for the evolution of the mean curvature, so as to improve the
accuracy for longer simulation times and to incorporate this relation and the analo-
gous one for the interfacial area density described in [23] into classical models for both
incompressible and compressible two-phase flows, so as to improve the computation of
interfacial source term in the case of a not well resolved interface.
16
Acknowledgments
This work has been partially supported by the ESCAPE-2 project, European Union’s
Horizon 2020 Research and Innovation Programme (Grant Agreement No. 800897).
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