Aspects ThØoriques et NumØriques pour les Fluides ......u+ rp= ˆf in each i(i= 1;2) divu= 0 in (1) where (ur)u= d i=1ui@iuand u= ui, p=piare the velocity and pressure unknown of

Aspects Théoriques et Numériques pour les Fluides IncompressiblesTheoretical and Numerical Issues of Incompressible Fluid Flows

Chapter 4: Two-uids and two-phase ows

Instructors: Pascal Frey, Yannick Privat

Sorbonne Université, CNRS4, place Jussieu, Paris

master ANEDP, 2019-2020

Section 4.1Modelling

Problem statement

The numerical modelling and resolution of biuid problems

• is needed because of small time and length scalesreliable experiments are impossible,

• investigate and understand physical phenomena,• requires the accurate discretization and the tracking ofthe interface separating two immiscible uids.

The major challenges are related to• the evolution of the interface and• the induced changes of its geometry and topology.

Here, we consider the dynamics of interface deformation in low Reynolds number ow.This topic has interest in wide variety of elds: chemical and petroleum engineering,geophysics, biology, . . .

Aspects théoriques et numériques pour les uides incompressibles M2 ANEDP, UPMC, 2019-2020 3/ 41

Simulation of bifluid flows

Several diculties may jeopardize the resolution:

• large jumps of viscosity and density between the uids must be properly taken intoaccount and resolved to satisfy momentum balance,

• mass conservation is especially important in interfacial ows,

• the surface tension force must be considered in the model and accurately evaluated,

• the resolution of the interface must be preserved at all stages, even in the extremecases of folding, merging and breaking.


Related work

Since the seminal work of Harlow and Welch (1965), numerous methods have beenproposed. Comprehensive surveys written by:

1. Anderson D.M., McFadden G.B., Wheeler A.A., Diuse-interface methods in uid mechanics, Annu. Rev. FluidMech., 1998; 30: 139-165.

2. Cuvelier C., Schulkes R.M., Some numerical methods for thecomputation of capillary free boundaries governed by theNavier-Stokes equations, D. Reidel Publishing Company,Dordrecht, 1986.

3. Shyy W., Udaykumar H.S., Rao M.M., Smith R.W., Computa-tional Fluid Dynamics with moving Boundaries, Taylor &Francis, 1996.

4. Smolianski A., Numerical modeling of two uid interfacialows, PhD thesis, Jyväskylä University, 2001.

Downloaded 08 Apr 2009 to 134.157.51.244. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

Algorithms for uid ows are subdivided into two classes, namely Lagrangian andEulerian approaches.


Lagrangian vs. Eulerian approaches

1. Lagrangian methods: interface tracking• follow the interface evolution using a set of markers and deform the grid.• each grid cell contains the same uid part throughout the whole computation.• face diculties in handlingmarkers when the interface becomes highly stretchedor distorded, and when the topology changes.

2. Eulerian techniques: interface capturing• introduce a scalar valued level set function to dene the interface manifold,• xed coordinate system, the uid travels from one grid cell to another.• topology changes easily handled, but mass conservation may be a real concern.• expressions of the interface normal and curvature from the level set function.

1. Dervieux A., Thomasset F., A nite element method for the simulation of a Rayleigh-Taylor in-stablity, in Approximation Methods for Navier-Stokes problems, Lecture Notes in Mathematics,1980; 771, Springer-Verlag, Berlin, 145-158.


Interface capturing

The numerical resolution strategy is

• set in the context of Eulerian and interface-capturing methods.

• involving an important feature: mesh adaptation using unstructured (anisotropic)triangulations.

Our choice is motivated by the following arguments:1. necessity to deal with complex interfacial motionsand topology changes;

2. ow resolution decoupled from the advection part;3. anisotropic mesh adaptation for accurate represen-tation of the interface with a minimal number of un-knowns;

4. biuid resolution allows large viscosity ratios;5. the advection term treated by the method of charac-teristics combined with a Galerkin FE scheme.

Furthermore, uid coalescence and detachment can be eciently treated with the sharp interfacedenition.Aspects théoriques et numériques pour les uides incompressibles M2 ANEDP, UPMC, 2019-2020 7/ 41

Hypothesis and notations

• Suppose Ω is an open bounded computational domain in Rd, the outer boundaryis denoted Σ,

• the subdomains denoted Ω1(t) and Ω2(t), ∂Ωi(t) is the boundary of Ωi(t),

• the interface between the uids by: Γ(t) = ∂Ω1(t) ∩ ∂Ω2(t).

• suppose also that: Ω1(t) ∪Ω2(t) = Ω and Ω1(t) ∩Ω2(t) = ∅.

• the domains Ωi can have several connected components, and the interface Γ(t)

can possibly intersect the outer boundary Σ.

Ω2(t)

Σ

Γ(t)

Ω2(t)

Ω1(t)

CHAPTER 1. THE MODEL EQUATIONS

1n

Σ

Γ(t)

Ω2(t)

Ω1(t)

Figure 1.4: Example of configuration of bifluid flow computational domain.

Notice that there is no temporal derivative in this equation. Physically however, thisdoes not means that the flow is steady. This only reflects that the forces exerted onthe fluid are in a state of dynamic equilibrium as a result of a rapid diffusion of themomentum. Hence, the transient character of the solution is related to the motionof the two fluids and of the interface.

We have assumed that the surface tension effect must be taken into account atthe interface. Therefore, this system is endowed with conditions on the continuity ofthe velocity and on the balance of the normal stress with the surface tension acrossthe interface [GLM06]:

u1 − u2 = 0

(σ1 − σ2) · n1 = −γ κn1(1.2)

where– σ = µ(∇u + (∇u)t) − p I denotes the stress tensor,– n1 is the unit exterior normal vector to Γ(t) of Ω1(t) pointing from Ω1(t) toΩ2(t) (we assume that Γ(t) is sufficiently smooth),

– γ > 0 is the surface tension coefficient assumed to be constant along theinterface,

– κ is the signed mean curvature of the interface, being positive if the interfacecurve/surface bends towards Ω1(t) and negative otherwise.

These equations are completed with some appropriate boundary conditionson the outer boundary Σ. We can set the classical Dirichlet, Neumann or mixed

16


Model equations

• we introduce two scalar functions µ and ρ for the viscosity and density of the uiddened on the whole domain Ω as follows:

µ = χ1µ1 + χ2µ2, ρ = χ1ρ1 + χ2ρ2

where χi is the characteristic function of the domain Ωi, µi and ρi are dynamicviscosities and densities of each uid (i=1,2), respectively.

• we also assume that ρ(x, t) = ρ(x) and µ(x, t) = µ(x).

The ow of the incompressible viscous uid is governed by the Navier-Stokesequations:

ρ

(∂u

∂t+ (u · ∇)u

)− µ∆u+∇p = ρf in eachΩi (i = 1,2)

div u = 0 inΩ(1)

where (u · ∇)u = Σdi=1ui∂iu and

• u = ui, p =pi are the velocity and pressure unknown of the ow in Ωi (i=1,2)

• f is an internal force exerted on the uid (e.g. gravity).


Boundary conditions

To ensure the well-posedness of the problem, Equations (1) need to be complementedby

• boundary conditions: u = uD on ΣD Dirichlet σn = uN on ΣN Newmann u.n = 0, or αu.τ + τ.σn = 0 on ΣS Slip

• assuming Σ = ΣD ∪ΣN ∪ΣS is split into a nite number of components corre-sponding to dierent types of boundary conditions,

• τ, n are unit tangent vector and unit exterior normal vector to Γ and

• σ is the stress tensor dened as:

σ = µ(∇u+ t∇u

)− p Id .


Boundary conditions: numerical issuesNote: in practice,

• the slip condition can be imposed by adding the integral∫

ΣS

α(u.τ).(v.τ)ds

in the left hand side of the variational formulation of problems (see after) and

• the condition u.n|ΣS= 0 can be treated as a Dirichlet condition on u.

• in case of α = 0 (slip without friction), this integral term vanishes and we canimplement the condition u.n|ΣS

= 0 generally by adding∫

ΣS

A ∗ (u.n).(v.n)ds

in the left hand side and multiplying the components on ΣS of right hand side withA where A is so-called penalization number (values about 106).


Initial and interfacial condition

• initial condition: divergence-free velocity eld u0(x) specied over the domain Ω

at time t = 0, i.e.

u(x,0) = u0(x) .

• at the interface Γ, two conditions are imposed :

u1 − u2 = 0 continuity of the velocity(σ1 − σ2) · n1 = −γκn1 normal stress balanced with surface tension

(2)

where n1 is the unit normal vector to Γ(t), exterior to Ω1(t), γ > 0 is the constant surface tension coecient along the interface, κ is the algebraic mean curvature of the interface, being positive if the interfacecurve/surface bends towards Ω1 and negative otherwise.


Section 4.2Evolution of the interface

level set formalism

Auxiliary function

The rst problem we face is to describe the interface Γ in proper way.Following [Set99], we introduce an auxiliary function, level set function, dened as thesigned distance function to the interface Γ, i.e., φ(x) = ±d(x,Γ),

and we have

φ(x) > 0 , if x ∈ Ω1 ,

φ(x) < 0 , if x ∈ Ω2 ,

φ(x) = 0 , if x ∈ Γ .

Signed distance function to the interface Γ.

[Set99] Sethian J.A., Level Set Methods and Fast Marching Methods, Cambridge University Press, 1999.Aspects théoriques et numériques pour les uides incompressibles M2 ANEDP, UPMC, 2019-2020 14/ 41

Level sets and interface capturing

• the evolution of the interface may induce geometry and topology changes.

• fortunately, with the level set formulation, at each time step t, the uid interface Γ

is associated with the zero isocontour of the continuous function φ:

Γ(t) = x ∈ Ω : φ(x, t) = 0 , φ(x, t) = ± miny∈Γ(t)

‖x− y‖ . (3)

• the interface Γ is solution, at each t, of the advection equation:

∂φ

∂t(x, t) + u(x, t) · ∇φ(x, t) = 0, ∀ (x, t) ∈ Ω× R+

φ(x,0) = φ0(x), ∀x ∈ Ω(4)

where φ0(x) is the signed distance function to Γ0 and u is the vector eld denedfrom the velocity eld along the interface (solving the uid equations).


Definition of the velocity field

• the evolution of the interface depends only on the ow eld in its vicinity, and noton the whole domain Ω.

• moreover, in case of complex displacements, sharp velocity variations may causeuncontrolled oscillations and jeopardize the numerical stability of subsequent al-gorithms.

• hence, we extend and regularize the velocity of ows u (taken only along Γ) into avector eld u (dened on Ω) before solving the advection:

−α∆u+ u = 0 in Ω

u = 0 on Σ

u = u on Γ

(5)

where small α > 0 can be interpreted as a regularization lengthscale (balance be-tween keeping u and the level of regularization).It is a more regular inner product than L2(∂Ω) over functions on Ω.


Extension of the velocity field

• let V be a Hilbert space which is composed of functions enjoying the desired regu-larity for u (usually V=H1(Ω)),

• let a(·, ·) be a coercive bilinear form on V which is close to I, so that u is close to u:

∀φψ ∈ V , a(φ, ψ) = α∫

Ω∇φ.∇ψ +

∫

Ωφψ =

∫

Γφψ , (6)

for a small α > 0 which can be interpreted as a regularization lengthscale.

• then, u is searched as the unique solution in V to the variational problem:

∀φ ∈ V , a(u, φ) =∫

Γuφds .

• this problem can easily be solved using a nite element method.


Numerical scheme

The biuid problem can be numerically resolved using the following general scheme :

1. Initialization: level set function φ0, velocity u0, mesh T0h .

2. At each time step tn = n∆t:(a) solve the Navier-Stokes equations for (u, p)n;(b) dene a regularized velocity eld u (close to u, matching u along Γ);(c) solve the advection equation for φ (new location of the interface);(d) generate a conforming mesh Tn+1

h rened in the vicinity of Γ);(e) regularize φ, update density ρ and viscosity µ;(f) interpolate (u, p)n onto Tn+1

h .

3. Resume step 2 until the nal time is reached.

Note: this scheme is masking some complex and necessary numerical routines :solution interpolation, error estimate and mesh adaptation.


Section 4.3Numerical resolution

Numerical methods

• The method of characteristics is known to be very ecient for solving advection-diusion problems, including the Navier-Stokes equations.

• Here, we use this method not only for solving the advection of the interface, butalso for solving the nonlinear convective term in Navier-Stokes equations.

• a Cauchy problem: given an initial function φ0(x) : Ω→ R and given a velocity eld u(x, t) : Ω→ Rd dened on Ω, nd φ(x, t) : Ω× [O, T ]→ R soving:

∂φ

∂t(x, t) + u(x, t)∇φ(x, t) = 0 ∀(x, t) ∈ Ω× (0, T )

φ(x,0) = φ0(x)∀x ∈ Ω .(7)


Method of characteristics

The problem (7) is solved by following backward the characteristic curves of the uidparticules :

• given a particule x ∈ Ω at time s, its curve is described by the following equations:

dX(x, s; t)

dt= u(X(x, s; t), t) ∀t ∈ (0, t)

X(x, s; s) = x(8)

where X(x, s; t) is the position of x at the time t.

• the rst equation of (7) implies that φ(x, t) is constant along the characteristic linesX(x, s; t),

• hence the solution of the Cauchy problem (7) writes:

φ(x, t) = φ0(X(x, t; 0),0) ∀(x, t) ∈ Ω× [0, T ] (9)


Method of characteristics: discretization

• the interval [0, T ] is divided into a nite number of intervals∆t of the form (tn−1, tn)

with tn = n∆t,

• the discretization in time of the equations (8), for all n, reads:

dX(x, tn; t)

dt= u(X(x, tn; t), t) ∀t ∈ (tn−1, tn)

X(x, tn; tn) = x(10)

• we compute only φ(x, tn) for all n, so if denote φ(x, tn) by φn(x), by substitutingthe time interval [tn−1, tn] into (9) we obtain the following result:

φn(x) = φn−1(X(x, tn; tn−1)) ∀x ∈ Ω (11)

whereX(x, tn; tn−1) is the position at the time tn−1 of the characteristic emergingfrom x at the time tn.

• numerically, the expression (11) can be solved using a Lagrange interpolation, i.eφn(x) is computed by takinh into account the values of φn−1(x) at the degree offreedoms of element K which contains X(x, tn; tn−1).


Numerical approximation of the characteristic curves

• only an approximation uh of u is known at the vertices of a triangulation Tnh ,

• compute an approximation of Xh(x, tn; tn−1), solution at the time tn−1 of the ap-proximated characteristic curve:

dXh(x, tn; t)

dt= uh(Xh(x, tn; t), t)

Xh(x, tn; tn) = x(12)

that implies the "formal" expression:

Xh(x, tn; t) = x−∫ tn

tuh(Xh(x, tn; t), t)dt (13)

• the simplest algorithm is obtained directly from (13) as:

Xh(x, tn; tn−1) = x−∆tuh(x) (14)


Numerical approximation of the characteristic curves

• the characteristic curve is considered as a straight line connecting point x and thefoot Xh(x, tn; tn−1). small time step

• in practice, given a substep δt in [tn−1, tn], suppose there exists ∆t = Mδt:(0) Xh(x, tn; tn) = x

(i) Euler’s scheme: (m = 1, ...,M)

Xh(x, tn; tn−mδt) = Xh(x, tn; tn− (m−1)δt)− δtuh(Xh(x, tn; tn−mδt));

(ii) Runge-Kutta 4 scheme:

Xh(x, tn; tn −mδt) = Xh(x, tn; tn − (m− 1)δt)−δt

6(v1 + 2v2 + 2v3 + v4)

with

v1 = uh(Xh(x, tn; tn −mδt))

v2 = uh(Xh(x, tn; tn −mδt)−δt

2v1)

v3 = uh(Xh(x, tn; tn −mδt)−δt

2v2)

v4 = uh(Xh(x, tn; tn −mδt)− δtv3) .


Navier-Stokes: discretization in time

• time discretisation of the Navier-Stokes equations based on method of characteris-tics.

• choice motivated by the fact that this scheme is known to be unconditionally stable,and very ecient on adapted meshes.

• main idea: hide the nonlinear convective part of Navier-Stokes equations in theCauchy problem (8),

• the operator ∂∂t + u.∇ may be turned into a total derivative ddt,

• so the equation (1) can be recast into the following form:

ρdu(X(x, s; t), t)

dt− µ∆u+∇p = ρf (15)



• time-dependent Navier-Stokes problem is now rewritten as follows,with un(x) = u(x, tn), in each Ωi (i = 1,2):

ρun(x)− un−1 Xn−1(x)

∆t− µ∆un(x) +∇pn(x) = ρfn

div un(x) = 0 inΩ(16)

or equivalently, in each Ωi (i = 1,2):ρun(x)

∆t− µ∆un(x) +∇pn(x) = ρfn + ρ

un−1 Xn−1(x)

∆tdiv un(x) = 0 inΩ

(17)

where Xn−1(x) denotes X(x, tn; tn−1) and un−1 Xn−1(x) corresponds to thevelocity at this location at time tn−1.



• To summarize, the resolution consists in performing two consecutive steps:1. approximate the characteristic curves Xn−1(x).2. solve the resulting generalized Stokes system.

• Remarks: the approximation of characteristic curvesXn−1(x) in each time interval [tn−1, tn]

for the Navier-Stokes problem is implemented as before, except that here un isunknown, and thus Xn−1(x) is associated with un−1 (and not un as in the ad-vection equation).

it is worth to forecast some diculties: characteristic curvesmay cross the bound-ary and thus it is needed to retain the last integration point.


Navier-Stokes: variational formulation

• in each time interval [tn−1, tn], we have to solve an unsteady Stokes problem (17).

• suppressing the dependency on n, the equation (17) becomes:αρu− µ∆u+∇p = ρf + αρw in each Ωi (i = 1,2)

div u = 0 in Ω,(18)

where α denotes1

∆tand w represents un−1 Xn−1.

• we set the spaces V = (H10(Ω))d if ΣD ≡ Σ andM = L2

0(Ω) for Dircihlet con-ditions orM = L2(Ω) if Σ\ΣD is not empty.

• without loss of generality, we consider here the variational formulation for Stokesproblem in case of homogenous Dirichlet boundary condition, i.e. u|Γ = 0.



• introducingD(u) = 12(∇u+t∇u), the symmetric gradient of u (the rate of deforma-

tion tensor) and thanks to the incompressibility condition, we have: div(2D(u)) =

∆u.

• using Green’s formula on each domain Ωi yields:

−∫

Ωidiv(2µD(ui))vidx =

∫

Ωi2µD(ui) : ∇vidx−

∫

∂Ωi2µD(ui)ni · vids

=∫

Ωi2µD(ui) : D(vi)dx−

∫

∂Ωi2µD(ui)ni · vids

∫

Ωi∇pi · vidx = −

∫

Ωipi div vidx+

∫

∂Ωipini · vids

where A : B = Σni,j=1AijBij .

• nally, we come to the following equation:

α∫

Ωρu · vdx+

∫

Ω2µD(u) : D(v)dx−

∫

Ωpdiv vdx =

∫

Ωρf · vdx+ α

∫

Ωρw · vdx

+2∑

i=1

∫

∂Ωi2µD(ui)ni · vids−

2∑

i=1

∫

∂Ωipini · vids .

(19)



• using the interface and geometric conditions n2 = −n1 on Γ, v = 0 on Σ we have:2∑

i=1

∫

∂Ωi2µD(ui)ni · vids−

2∑

i=1

∫

∂Ωipini · vids =

2∑

i=1

∫

Γ(2µD(ui)− piI)nivids

=∫

Γ(σ1n1v1 − σ2n1v2)ds

=∫

Γ−γκn1 · vds

• the variational formulation of the homogeneous problem reads: given the functionsf, w, µ, ρ (supposed constant in each time interval) and the constant α; nd u ∈ Vand p ∈M solving:

α∫

Ωρu · vdx+

∫

Ω2µD(u) : D(v)dx−

∫

Ωpdiv vdx =

∫

Ωρf · vdx

+α∫

Ωρw · vdx−

∫

Γγκn1 · vds ∀v ∈ V

∫

Ωq div udx = 0 ∀q ∈M


Navier-Stokes: spatial discretization

• Galerkin nite element approximation leads to the discrete problem:nd (uh, ph) ∈ Vh ×Mh s.t :

a(uh, vh) + b(vh, ph) = l(vh) , ∀vh ∈ Vh,

b(uh, qh) = 0 ∀qh ∈Mh(20)

with Vh ⊂ V and Mh ⊂ M two families of nite dimensional subspaces anda(uh, vh), b(vh, ph), l(vh) are bilinear and linear forms dened on Vh×Vh, Vh×Mh

and Vh respectively as follows:

a(uh, vh) =∑

K∈Thα∫

Kρuh · vhdx+

∑

K∈Th

∫

K2µD(uh) : D(vh)dx

b(vh, ph) =∑

K∈Th

∫

K−ph div vh

l(vh) =∑

K∈Th

∫

Kρfh · vhdx+

∑

K∈Thα∫

Kρwh.vhdx+ lΓh(vh)

(21)

where the term lΓh(vh) = −∫Γhγκn1

h · vh is a discretization of the surface tension−∫Γ γκn

1 · v.


Variational formulation: existence and uniqueness

• the existence and the uniqueness of a solution to the weak formulation of the gen-eralized Stokes problem can be established, see [EG04] or [Qua09].

• proof relies on:i) the ellipticity of the form a(., .) (Poincaré-Friedrichs inequality);ii) the compatibility of the spaces, velocity and pressure (Babuska-Brezzi inf-sup

condition on the form b(., .)), i.e. there existing a positive constant C such that:

infq∈M

supv∈V

b(v, p)

‖v‖1‖q‖0≥ C > 0 (22)

where ‖v‖1 =(Σdi=1‖vi‖1

2)1/2

and ‖.‖1, ‖.‖0 are standard notations of normsin the Sobolev spaces H1(Ω), L2(Ω) respectively.

References

[EG04] Ern A. and Guermond J.L., Theory and Practice of Finite Elements, 159. Springer, (2004).[Qua09] Quarteroni A., Numerical Models for Dierential Problems, 2, Springer, (2009).


Variational formulation: compatibility condition

• The generalized Stokes problem also involves a compatibility condition: the discretespaces for the velocity and the pressure need to be compatible (see section onStokes).

• in practice, mini elements (P1-bubble/P1) or Taylor-Hood elements (P2/P1) areused to solve it.

• the problem (20) leads to solve the sparse symmetric linear system:(A Bt

B 0

)(UP

)=

(F0

)(23)

where A,B and F correspond to the bilinear forms ah, bh and to the right-handside, respectively.

• this linear system is solved using Uzawa’s method or by a penalty method.


Approximation of surface tension term

• Γh is a piecewise ane approximation of Γ included in Th.

• Using a quadrature formula on each edge E ⊂ Γh, we write:∫

Γhγ κ vh · n1 ds =

∑

E⊂Γh

∫

Eγ κ vh · n1 ds =

∑

E⊂Γh

|E|2

∑

xi∈Eγ κ(xi) vh(xi) · n1(xi)

=∑

xi∈Γh

γ κ(xi) vh(xi) · n1(xi)∑

E 3xi

|E|2.

(24)where n1 is the unit normal vector with respect to Ω1.

• how to evaluate n1 and κ at each point of Γh ?

• in principle, with level sets, function φ is used to compute normal and curvaturealong Γ:

n =∇φ|∇φ|

∣∣∣∣∣φ=0

, κ = div n = div

(∇φ|∇φ|

)∣∣∣∣∣φ=0

. (25)

but it is too sensitive to numerical artifacts.


Approximation of surface tension term

We propose another scheme to evaluate the dierential quantities.

• suppose x−1, xi, xi+1 are three successivepoints along Γh

• dene n1(xi) as the unit vector orthogonalto the edge vector ei+1,i−1, considered asthe approximation of the tangent τ(xi).

Ω1

Γ

xi+1

xi

xi−1

n1

Ω2

• the local radius of curvature ρ(xi) is then approximated by:

ρ(xi) =1

4

(〈ei,i−1, ei,i−1〉〈−n1(xi), ei,i−1〉

+〈ei,i+1, ei,i+1〉〈−n1(xi), ei,i+1〉

)

and the local mean curvature is dened as

κ(xi) = 1/ρ(xi) .


Numerical scheme

• time interval [0, T ] is divided into N intervals [tn−1, tn],

• at the iteration n, given Tnh , solve Navier-Stokes and advection equations to obtainthe solutions (un, pn) and φn,

• the scheme is an iterative procedure that involves mesh adaptation, it reads:1. init t = 0: u0 = u0(x), p0 = p0(x), φ0 = φ0(x), T0

h ,2. for n = 1, ..., N do

mesh input outputTn−1h (un−1, pn−1, φn−1)

Navier-Stokes Tn−1h (un−1,pn−1) (un, pn)

velocity extension Tn−1h (un|Γ) un

level set advection Tn−1h (un, φn−1) φn

mesh adaptation Tn−1h (un, φn) Tnh

L2-projection Tn−1h , Tnh (un, pn, φn) Tnh ,(u

n, pn, φn)


Section 4.4Numerical results

2d rising bubble

• study of the rising and the deformation of a single bubble under gravity in a uidconned in a rectangular domain Ω = [0,4]× [0,10].

• initial conguration : circular bubble of radius r = 0.5 centered at [2,1.5]

• initial mesh contains 2,494 nodes

• boundary conditions : no-slip condition (u=0) on the horizontal walls andfree-slip condition (τ · σn = 0 and u · n = 0) on the vertical walls.

• Reynold number and Bond number (or Eotvos number) characterize the simulations:

Re =ρ1√g(2r)3/2

µ1, Bo =

4ρ1gr2

γ(26)

• parameters : constant densities and viscositiesρ1 = 100kg.m−3, µ1 = 0.1kg.m−1.s−1, ρ2 = 1.0kg.m−3, µ2 = 0.01kg.m−1.s−1,gravity g = 9.81 10−3m.s−2.surface tension γ = 6.10−3N.m−1.


2d rising bubble

Rising bubble: 2d domain t = 5s t = 10s


2d rising bubble: surface tension coefficient

γ = 6e− 5 γ = 6e− 3

γ = 2.5e− 2 γ = 9e− 2


3d rising bubble

• Extension to 3d of the rising bubble problem : bubble with diameter 0.5m initializedat [0.75,0.75,1.0] in a domain Ω = [0,1.5]× [0,1.5]× [0,4.5].

• at low Reynold number, the shape of the bubble deforms slowly and becomes dim-pled ellipsoidal and more distorted as time increases.

t = 0 s t = 2.0 s t = 4.0 s t = 8.0 s t = 10.0 s


Documents

Aspects ThØoriques et NumØriques pour les Fluides ......u+ rp= ˆf in each i(i= 1;2) divu= 0 in (1) where (ur)u= d i=1ui@iuand u= ui, p=piare the velocity and pressure unknown of