Aspects Théoriques et Numériques pour les Fluides .... Frey, chap4.pdfAspects Théoriques et Numériques pour les Fluides Incompressibles TheoreticalandNumericalIssuesofIncompressibleFluidFlows

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

Chapter 4: Two-fluids and two-phase flows

Instructors: Pascal Frey, Yannick Privat

Sorbonne Université, CNRS4, place Jussieu, Paris

master ANEDP, 2018

Section 4.1

Modelling

Problem statement

The numerical modelling and resolution of bifluid 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 of the in-terface separating two immiscible fluids.

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 flow.

This topic has interest in wide variety of fields: chemical and petroleum engineering, geophysics,biology, . . .

Aspects théoriques et numériques pour les fluides incompressibles M2 ANEDP, UPMC, 2018 3/ 41

Simulation of bifluid flows

Several difficulties may jeopardize the resolution:

• large jumps of viscosity and density between the fluids must be properly taken into accountand resolved to satisfy momentum balance,

• mass conservation is especially important in interfacial flows,

• 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 extreme cases offolding, merging and breaking.


Related work

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

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

2. Cuvelier C., Schulkes R.M., Some numerical methods forthe computation of capillary free boundaries governed by theNavier-Stokes equations, D. Reidel Publishing Company, Dor-drecht, 1986.

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

4. Smolianski A., Numerical modeling of two fluid interfacialflows, 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 fluid flows are subdivided into two classes, namely Lagrangian and Eulerianapproaches.


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 fluid part throughout the whole computation.

• face difficulties in handling markers when the interface becomes highly stretched or dis-torded, and when the topology changes.

2. Eulerian techniques: interface capturing

• introduce a scalar valued level set function to define the interface manifold,

• fixed coordinate system, the fluid 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 finite element method for the simulation of a Rayleigh-Taylorinstablity, 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) triangula-tions.

Our choice is motivated by the following arguments:

1. necessity to deal with complex interfacial motions andtopology changes;

2. flow resolution decoupled from the advection part;

3. anisotropic mesh adaptation for accurate representation ofthe interface with a minimal number of unknowns;

4. bifluid resolution allows large viscosity ratios;

5. the advection term treated by the method of characteristicscombined with a Galerkin FE scheme.

Furthermore, fluid coalescence and detachment can be efficiently treated with the sharp interface definition.


Hypothesis and notations

• Suppose Ω is an open bounded computational domain in Rd , the outer boundary is denotedΣ,

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

• the interface between the fluids 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 possiblyintersect 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 fluid definedon 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 dynamic viscositiesand densities of each fluid (i=1,2), respectively.

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

The flow of the incompressible viscous fluid is governed by the Navier-Stokes equations: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 flow in Ωi (i=1,2)

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


Boundary conditions

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

• 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 finite number of components corresponding todifferent types of boundary conditions,

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

• σ is the stress tensor defined as:

σ = µ(∇u + t∇u

)− p Id .


Boundary conditions: numerical issues

Note: in practice,

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

ΣSα(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 can implementthe condition u.n|ΣS = 0 generally by adding

∫

ΣSA ∗ (u.n).(v .n)ds

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


Initial and interfacial condition

• initial condition: divergence-free velocity field u0(x) specified over the domain Ω at timet = 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 = −γκ n1 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 coefficient along the interface,

κ is the algebraic mean curvature of the interface, being positive if the interface curve/surfacebends towards Ω1 and negative otherwise.


Section 4.2

Evolution of the interface

level set formalism

Auxiliary function

The first problem we face is to describe the interface Γ in proper way.Following [Set99], we introduce an auxiliary function, level set function, defined as the signeddistance 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.


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 fluid interface Γ isassociated 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 field defined from thevelocity field along the interface (solving the fluid equations).


Definition of the velocity field

• the evolution of the interface depends only on the flow field in its vicinity, and not on thewhole domain Ω.

• moreover, in case of complex displacements, sharp velocity variations may cause uncontrolledoscillations and jeopardize the numerical stability of subsequent algorithms.

• hence, we extend and regularize the velocity of flows u (taken only along Γ) into a vectorfield u (defined 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 between keepingu 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 regularity 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) =

∫

Γu φds .

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


Numerical scheme

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

1. Initialization: level set function φ0, velocity u0, mesh T 0h .

2. At each time step tn = n∆t:

(a) solve the Navier-Stokes equations for (u, p)n;

(b) define a regularized velocity field 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 T n+1h refined in the vicinity of Γ);

(e) regularize φ, update density ρ and viscosity µ;

(f) interpolate (u, p)n onto T n+1h .

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

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


Section 4.3

Numerical resolution

Numerical methods

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

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

• a Cauchy problem:

given an initial function φ0(x) : Ω→ R and

given a velocity field u(x, t) : Ω→ Rd defined on Ω,

find 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 fluid particules :

• 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 first 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 finite number of intervals ∆t of the form (tn−1, tn) withtn = 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 substituting the timeinterval [tn−1, tn] into (9) we obtain the following result:

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

where X(x, tn; tn−1) is the position at the time tn−1 of the characteristic emerging from x

at the time tn.

• numerically, the expression (11) can be solved using a Lagrange interpolation, i.e φn(x) iscomputed by takinh into account the values of φn−1(x) at the degree of freedoms of elementK 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 T nh ,

• compute an approximation of Xh(x, tn; tn−1), solution at the time tn−1 of the approximatedcharacteristic 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 the footXh(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 characteristics.

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

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

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

dt ,

• 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) = ρf n

div un(x) = 0 inΩ

(16)

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

∆t− µ∆un(x) +∇pn(x) = ρf n + 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 the velocity atthis 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 curves Xn−1(x) in each time interval [tn−1, tn] forthe Navier-Stokes problem is implemented as before, except that here un is unknown,and thus Xn−1(x) is associated with un−1 (and not un as in the advection equation).

it is worth to forecast some difficulties: characteristic curves may cross the boundary andthus 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 and M = L2

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

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



• introducing D(u) = 12(∇u + t∇u), the symmetric gradient of u (the rate of deformation

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))v idx =

∫

Ωi2µD(ui) : ∇v idx −

∫

∂Ωi2µD(ui)ni · v ids

=

∫

Ωi2µD(ui) : D(v i)dx −

∫

∂Ωi2µD(ui)ni · v ids

∫

Ωi∇pi · v idx = −

∫

Ωipi div v idx +

∫

∂Ωipini · v ids

where A : B = Σni ,j=1Ai jBi j .

• finally, we come to the following equation:

α

∫

Ωρu · vdx +

∫

Ω2µD(u) : D(v)dx −

∫

Ωp div vdx =

∫

Ωρf · vdx + α

∫

Ωρw · vdx

+2∑

i=1

∫

∂Ωi2µD(ui)ni · v ids −

2∑

i=1

∫

∂Ωipini · v ids .

(19)



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

2∑

i=1

∫

∂Ωi2µD(ui)ni · v ids −

2∑

i=1

∫

∂Ωipini · v ids =

2∑

i=1

∫

Γ(2µD(ui)− pi I)niv ids

=

∫

Γ(σ1n1v1 − σ2n1v2)ds

=

∫

Γ−γκn1 · vds

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

α

∫

Ωρu · vdx +

∫

Ω2µD(u) : D(v)dx −

∫

Ωp div vdx =

∫

Ωρf · vdx

+α

∫

Ωρw · vdx −

∫

Γγκn1 · vds ∀v ∈ V

∫

Ωq div udx = 0 ∀q ∈ M


Navier-Stokes: spatial discretization

• Galerkin finite element approximation leads to the discrete problem:find (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 finite dimensional subspaces and a(uh, vh),b(vh, ph), l(vh) are bilinear and linear forms defined on Vh×Vh, Vh×Mh and Vh respectivelyas 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 γκn1·

v .


Variational formulation: existence and uniqueness

• the existence and the uniqueness of a solution to the weak formulation of the generalizedStokes 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 conditionon 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‖12)1/2

and ‖.‖1, ‖.‖0 are standard notations of norms in 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 Differential Problems, 2, Springer, (2009).


Variational formulation: compatibility condition

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

• in practice, mini elements (P1-bubble/P1) or Taylor-Hood elements (P2/P1) are used tosolve 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-hand side,respectively.

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


Approximation of surface tension term

• Γh is a piecewise affine 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 3 xi

|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 curvature along 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 differential quantities.

• suppose x−1, xi , xi+1 are three successive pointsalong Γh

• define n1(xi) as the unit vector orthogonal to theedge vector ei+1,i−1, considered as the approxi-mation 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 defined 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 T nh , solve Navier-Stokes and advection equations to obtain thesolutions (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), T 0h ,

2. for n = 1, ..., N domesh input outputT n−1h (un−1, pn−1, φn−1)

Navier-Stokes T n−1h (un−1,pn−1) (un, pn)

velocity extension T n−1h (un|Γ) un

level set advection T n−1h (un, φn−1) φn

mesh adaptation T n−1h (un, φn) T nh

L2-projection T n−1h , T nh (un, pn, φn) T nh ,(u

n, pn, φn)


Section 4.4

Numerical results

2d rising bubble

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

• initial configuration : 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, gravityg = 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 0.5 m initialized at[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 dimpled ellip-soidal 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


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Aspects Théoriques et Numériques pour les Fluides .... Frey, chap4.pdfAspects Théoriques et Numériques pour les Fluides Incompressibles TheoreticalandNumericalIssuesofIncompressibleFluidFlows