A Fully Adaptive Front Tracking Method for the Simulation of Two Phase Flows

A Fully Adaptive Front Tracking Method for the Simulation of Two Phase Flows

M.R. Pivello^a,∗, A.S. Neto^a, M.M. Villar^a, A.M. Roma^b, R. Serfaty^c

aUniversidade Federal de Uberlˆandia, Av. Jo˜ao Naves de ´Avila, 2121, Sta. Mˆonica, Uberlˆandia, MG, Brazil

bIME-USP - S˜ao Paulo - SP, Brazil

cCENPES - PETROBRAS - RIO DE JANEIRO - RJ - BRAZIL

Abstract

This work presents a computational framework for the simulation of three-dimensional two- phase flows based on adaptive strategies for space and time discretization. The method is based on the front-tracking method [49], and the discretization of the Eulerian domain employs the Structured Adaptive Mesh Refinement (SAMR) strategy [3]. The time inte- gration algorithm is based on the IMEX scheme, and the time step is calculated based on CFL criteria. The memoryless polygonal simplification algorithm [22] provided by GTS [29]

yields a volume- and shape- preserving remeshing algorithm for theLagrangian framework.

Also, to overcome the non-conservative nature of the velocity interpolation process, a vol- ume recovery step is added to the interface advection, along with a sub-grid undulation removal algorithm [42]. Therefore, additional tools for volume recovery and conservative surface smoothing were implemented. The methodology was applied to a series of rising bubble simulations and validated against experimental results [4]. Finally, the algorithm was applied to the simulation of two cases of bubbles rising in the wobbling regime. The use of adaptive mesh refinement strategies led to physically insightful results, which otherwise would not be possible in a serial code with a uniform mesh.

Keywords: Front Tracking, Adaptive Mesh Refinement, volume preserving, multiphase flow

∗Corresponding Author

Email addresses: [email protected] (M.R. Pivello),[email protected](A.S.

Neto),[email protected](M.M. Villar), [email protected](A.M. Roma), [email protected](R. Serfaty)

1. Background

A body rising or falling under gravity reaches a terminal velocity when the forces acting on it (drag, buoyancy and weight) are in equilibrium. The drag force on rigid bodies depends on the body shape, on the terminal velocity and on the physical properties of the flow.

When the flow has more than on fluid component, the situation becomes more complex.

Fluid bodies, such as bubbles or drops, can deform under the action of the flow and the transfer of momentum across the interface may induce vortices inside the bubble. Therefore, the bubble shape will depend not only on the viscous and interfacial forces, but also on the forces from the surrounding flow [10].

Rising bubble flows can be described in terms of three non-dimensional numbers: the Eotvos number, the Morton number and the Reynolds number:

Eo= g∆ρφ²

σ , (1a)

M = g∆ρµ⁴_c

ρ²_cσ³ , (1b)

Re= ρ_cU φ µc

, (1c)

where g is the gravity acceleration, ρ_c is the density of the continuous phase, φ is the equivalent diameter of the bubble, µ_c is the viscosity of the continuous phase and σ is the interface tension of between the fluid-fluid interface.

Bubbles tend to deform when subject to external flow fields until normal and shear stresses balance at the fluid-fluid interface, and their shape under the action of gravity in an initially quiescent liquid can be grouped into three large categories: spherical, ellipsoidal and spherical-cap or ellipsoidal-cap. If the interfacial tension and/or viscous forces are much more significant than inertia forces, bubbles are termed spherical. Clift et al. [9]

classify a rising bubble as spherical if its aspect ratio (height to width ratio) lies within 10%

of unity. Ellipsoidal bubbles are oblate with a convex shape when viewed from inside and may present axi-symmetry. As inertia forces become more important, ellipsoidal bubbles may undergo periodic dilatation or random wobbling motion, making shape characterization rather difficult [4]. Large bubbles usually have flat or indented bases, without fore-and-aft

symmetry. Their fore-shape may resemble segments cut from spheres or oblate spheroids of low eccentricity, which originated the namesspherical-caporellipsoidal cap. Bubbles in this regime may also develop thin envelopes of dispersed fluid at their bases, usually referred to as skirts. [5]

Since the Morton Number depends only on physical properties, it is widely used for creating experimental correlations. Its values vary over a wide range, from 10⁵ to 10⁻¹⁴, influenced by the viscosity of the continuous phase and on the interface tension. Bubbles immersed in low Morton fluids quickly accelerate to their maximum velocity and then oscil- late before achieving the steady state. Regarding its shape, if the bubble is initially spherical, it becomes increasingly oblate until it reaches the point of maximum velocity. Then, large bubbles assume a spherical shape with a fluctuating base. The path, initially linear, may change to zigzag or spiral after reaching the maximum velocity. Bubbles immersed in high Morton number fluids, on the other hand, have a rectilinear path resulting from a smoother velocity profile, in which the bubble slowly accelerates towards the maximum velocity with- out any overshooting. Also, the spherical cap is achieved without any instability [4].

In the present work, the motion of a single bubble rising in an otherwise quiescent liquid is simulated under various flow regimes, ranging from low-Reynolds, spherical bubbles, to high-Reynolds, low-Morton wobbling bubbles. The simulations are carried on using a Front- Tracking method coupled with a Structured Adaptive Mesh Refinement for solving the Navier-Stokes equations.

1.1. One-Fluid Formulation

The oldest and most successful approach for solving multiphase flows is the so called one- fluid formulation. A single set of Navier-Stokes equations is written for the entire domain, and the phase interfaces are taken into account by adding a singular force term [46].

The Navier-Stokes equations can be solved by any method suitable for constant physical properties, but generally a projection method is employed [47]. Although similar to what is done for single phase flows, projection methods for multiphase flows have some complicating factors: the pressure equation, the advection of material fields and the computation of the

interface tension force [47]. In the pressure equation, the density field is not constant and therefore cannot be separated from the pressure, which may lead to numerical stability issues, especially for large density ratios [46], [17].

Density and viscosity fields are not directly advected. Instead, a marker function H that identifies the location of the different fluids is used. This function is a scalar field so that H=1 identifies the location of one fluid and H=0 identifies the other one, resembling a Heaviside function. However, the jump is smoothed between a few computational cells, so that the gradients of physical properties at the interface are also smoothed.

There are basically two family of methods for representing the interface, according to the reference frame used in the definition of the interface:

• Eulerianmethods: rely on a single, Eulerian, framework to simulate the flow field and the presence of the interface. Level set and volume of fluid are typical examples of this family.

• Lagrangian methods: solve the flow field in the usual Eulerian framework and uses a Lagrangian framework to represent the fluid interface explicitly. Examples: marker particles, Lattice-Boltzmann and front-tracking methods.

Level set methods define the interface as the zero level set of a distance function from the interface, which is advected with the local fluid velocity in an Eulerian manner [45, 44].

Although being conceptually simple and relatively simple to implement, these methods lack volume conservation when the flow becomes too complex, that is, when there is high local deformation on the interface and the vorticity field is moderate to high [1].

Volume of fluid methods (VOF) [16] use an indicator function which yields the volume fraction at a position (x,y,z) at time t and the interface orientation is determined using its gradient [36]. Similarly to level set methods, the interface is advected in an Eulerian manner.

Although in one dimension it is possible to compute exactly the flux of the volume fraction from one cell to the next, the extension to two- and three- dimensions has, however, proven to be challenging [47].

With regards to fragmentation and coalescence, VOF methods share a feature that can be an advantage or a drawback. Since each phase has a specific scalar value attached to it, if two interfaces of the same phase occupy the same computational cell, they will always coalesce. This could be a problem if coalescence is known to not happen at that situation or, on the other hand, an advantage if the merging occurs, because no additional test would be necessary. This is the opposite to what happens with Front Tracking methods, which always demand specific algorithms for the merging and/or the breakup of the interface.

Marker particle methods [32, 50] track each phase with a set of meshlessLagrangianpar- ticles, spread through all the volume occupied by the respective phase. These particles also carry the physical properties of their fluid phases, and a mapping between the Lagrangian and Eulerian domain is used to retrieve the relevant physical properties to solve the Navier- Stokes equations. These methods are extremely accurate and robust and can predict the topology of an interface subjected to considerable shear and vorticity in the fluids sharing the interface. However, it may be necessary to add new marker particles, increasing the com- putational cost considerably. Also merging and breakup of interfaces constitute a problem [1].

The lattice Boltzmann method (LBM) can be viewed as a special, particle-based dis- cretization method to solve the Boltzmann equation [18, 19]. This method is particularly attractive in case multiple moving objects, since no dynamic remeshing is necessary [1].

However, similar to VOF methods, artificial coalescence and fragmentation may occur.

Front-tracking methods [49, 46] rely on an unstructured mesh to track the interface and an Eulerian grid is used to solve the Navier-Stokes equations. This method may be very accurate for representing the interface, but it also requires dynamic remeshing of the Lagrangian interface mesh and a mapping of the Lagrangian data onto the Eulerian grid. Multiple interfaces interacting each other as in coalescence or fragmentation require additional models in order to decide whether the merge/breakup will occur. That is, the use of a separate mesh for tracking the interface enables the implementation of additional criteria for preventing the coalescence of two bubbles/droplets just because they approached each other. This property is advantageous in cases in which swarm effects in dispersed flows need to be

studied. Although in the present work only single bubbles will be studied, this technique will be adopted.

1.2. Front Tracking Methods

Front Tracking Methods were introduced originally in the 1960’s by Richtmyer and Mor- ton [31] and have continually evolved since then. The Navier-Stokes equations are solved in an Eulerian framework and the interface between the phases is tracked by an independent set of marker points which store representative coordinates of the interface. Although stor- ing the markers coordinates is enough to fully determine the interface position, storing the interface as a mesh is useful for computing geometrical properties such like normals, area and the interfacial tension force, not to mention the checking against collisions or merge/breakup phenomena. Since the interface is explicitly represented by its coordinates, the set of mark- ers is said to use a Lagrangianframework, and the mesh which characterizes the interface is termedLagrangian mesh.

There are two possible strategies for mesh communication. Glim and his co-workers [13]

discretized the Navier-Stokes equations using a finite difference method which modifies the stencil in the vicinity of the mesh overlap, so that the vertices of the two meshes match each other. However, for multiphase flows the most widespread methodology is based on the Immersed Boundary Method by Peskin [26], [27], which represents the interface by imposing a force field that is spread on the Eulerian mesh. The Navier Stokes equations are solved based on one-fluid models.

Among the various methodologies developed after Peskin, one of the most widespread in the context of multiphase flows is the work of Tryggvason [49, 46]. Similarly to IBM, the interface between the phases is represented by the interfacial tension force field.

The front-tracking method of Tryggvason [49, 46] is based on the one-fluid formulation and on the immersed boundary method of Peskin [26, 27]. Therefore, it relies on a mesh described in a Lagrangian framework on which the interface tension force is calculated and then spread on the Eulerian grid, at the vicinity of theLagrangianpoints. Since the interface position is known, the interface tension force can be calculated by integrating the interface

tension on a surface element ∆S, according to Eq. (2):

δFσ = Z

∆S

σκnds, (2)

whereσis the surface tension coefficient,κis twice the mean curvature for three-dimensional domains and n is the local normal to the surface. By replacing the geometrical relation κ×n = (n× ∇)×n [48] on Eq. (2) and using the Stokes theorem, the force on a surface element can be computed without explicitly calculating the surface curvature, via Eq. (3):

δF_σ = I

δΓ

σt×ndΓ, (3)

wheredΓ is the boundary of the integration element,t is the unit tangent andn is the unit normal, both computed at the element boundary.

Equation (3) is the basis for computing the surface tension force in front-tracking methods and may be solved in various ways. In the most common approach, the mesh elements are directly used as the integration elements. Tryggvason et al. [46] compute the tangent vectors from the end points of the element edges, but perform a local surface fit in order to calculate the normal vectors. Deen et al. [11] compute the tangent vectors in a similar manner, but use the normals at the adjacent elements as depicted in Fig. (1) on the top left.

Therefore, the resulting force vector lies on the plane defined by the neighbouring element and is perpendicular to the tangent vector. Singh and Shyy [41], on the other hand, use the resultant of t×n computed on both elements sharing an edge (see Fig. (1) on the top right). The resultant of the forces acting on the three edges is used as the force acting on the element. Recently, Tryggvason et al. [48], suggested performing the integration around the mesh vertexes, as depicted in Fig. (1) on the bottom. In this case, the integration element is defined by the centroid of the elements sharing the vertex and the midpoint of the respective edges. This choice changes the tangent and normal definitions. In the two first cases, the tangent vectors are defined by the edges of the mesh elements. In the last case, the tangent is defined by the segment linking the midpoint of an edge to the centroid of the element.

The normal vector, on the other hand, is defined by the mesh element normals in all cases.

Figure 1: Three different ways to calculate the interface tension force on a surface element.

On the top, integration is performed using the mesh element itself (left: Deen et al. [11], right, Singh and Shyy [41]). At the bottom, building the integration elements around the mesh vertexes, based on the centroid of the mesh elements sharing the vertex, proposed by Tryggvason et al. [48].

1.3. Indicator Function

One-fluid formulations rely on a single set of equations for solving the entire flow domain, modelling it as a single fluid with variable physical properties. If the properties are constant inside a given phase, their distribution over the domain are modelled as a Heaviside function.

In this context, the objective of a given indicator function is to supply a scalar field which may be used as a Heaviside function to compute the physical properties which are relevant to the flow. Front tracking methodologies which follow the Tryggvason school usually solve a Poisson equation for the entire domain which yields the scalar field [49], [1], [41]. However,

solving such equation is one of the most expensive parts of a projection-based Navier-Stokes solver.

Alternatively, Mauch [25] has developed a methodology called Closest Point Transform, or CPT, which yields an implicit representation of a surface mesh by its distance field.

Unlike the Poisson equation, which needs to be solved over the entire Eulerian domain, the distance field must be calculated only in the regions around the interface. Ceniceros and Roma [6] implemented a two-dimensional version of CPT as an indicator function in a front-tracking method and applied it to the study of a surface tension-mediated Kelvin Helmholtz instability based on a uniform Eulerian grid and Ceniceros et al. [8] extended the formulation for a two-dimension block-structured adaptive mesh refinement formulation.

2. Mathematical Model

2.1. Overview

Solving a multiphase flow problem with a one-fluid formulation consists in using a sin- gle set of Navier-Stokes equations using variable physical properties, while the fluid-fluid interface is modelled using a source term for the interface tension force, as shown in Eqs.

4-5.

ρ[∂u

∂t + (u· ∇)u] =∇ ·[µ(∇u+∇u^T)]

− ∇p+ρg+fσ,

(4)

∇ ·u= 0. (5)

Here, u is the fluid velocity field, ρ(x, t) is the fluid density, µ(x, t) is the dynamic viscosity, p is the pressure, g is the gravity and f_σ is the interface tension force. Since the distribution of the density and viscosity fields along the domain is determined by the presence of the interface, an indicator function is used for associating each cell in the Eulerian domain to the respective fluid phase.

There are three moments that require communication between the Lagrangian and Eu- lerian frameworks:

• The advection of the interface: it requires the velocity field, which is calculated in the Eulerian framework by solving the Navier-Stokes equations, to be interpolated to the Lagrangianframework.

• The solution of the Navier-Stokes equations: it requires the interface tension force, which is calculated in theLagrangiandomain and distributed over the Eulerian frame- work.

• The transport of the viscosity and density fields: they are performed via an indicator function, rather than solving an advection equation.

Although this algorithm is very simplified and no assumption is made about space or time discretization, it describes the main steps of the front-tracking method. In the remaining of this section, these steps are detailed.

Interpolation and distribution processes are performed as described in [46], using the following equation as the Dirac kernel:

W(r) =







1

4(1 +cos(^π₂r)), r <2,

0, r≥2,

(6) and

r = x−X

h_x ,y−Y

h_y ,z−Z

h_z . (7)

The indicator function consists in assigning a scalar field which covers the entire Eulerian domain, assigning a unique, constant value for each flow phase. The most common approach consists in solving a Poisson equation for the entire domain, so that the source term identifies the density jump at the interface [46].

To overcome this burden, Ceniceros and Roma [6] proposed an alternative approach for the indicator function. Firstly, the Closest Point Transform (CPT) [25] is used to obtain an implicit representation of theLagrangianinterface. Then, a Heaviside function is applied to the indicator function field in order to map it onto the interval [0,1]. Following the idea of a finite thickness interface, a smooth version of the Heaviside function was used ([37, 51]):

H(ϕ) =











1, ϕ > γ

0.5(1 + ^ϕ_γ +_π¹sin(^πϕ_γ )), kϕk ≤γ

0, ϕ < −γ.

(8)

After being mapped, the indicator function can be used for calculating the distribution of the density (ρ) and viscosity (µ) fields are calculated as follows:

ρ(ϕ) =H(ϕ)ρ1+ (1−H(ϕ))ρ2,

µ(ϕ) = H(ϕ)µ1+ (1−H(ϕ))µ2.

(9)

Since CPT is here used as an indicator function, it does not have to be computed over the entire domain. Instead, it is limited to an interval [−γ,+γ], where γ is the absolute value of the largest distance from the interface to a given point in the Eulerian framework.

The remaining of the domain is assigned with a constant value (e.g.: +γ inside the dispersed phase and−γ outside it), while the Poisson equation should be solved over the entire domain.

2.2. Interfacial tension force

In the present work, the approach of Singh and Shyy [41] will be followed. Using the notation shown in Fig. (1) for identifying tangent and normal vectors, Eq. (2) is discretized as follows. Notice that the indexes are not related to Einstein notation.

F_m =X

j

σ(t_m,j ×n_m,j), (10)

whereF_m is the vector force acting on element m, n_m,j is the outer unit-normal associated to edgek and t_m,j is the non-normalized tangent vector at edge j.

2.3. The complete algorithm

The procedure detailed in this section can be put in an algorithmic form as follows:

1 Set up boundary conditions and initial conditions.

2 Set-up physical properties.

3 Calculate the interface force.

4 Solve the Navier-Stokes equations.

5 Update the interface position.

6 Remesh if necessary.

7 Loop over steps 2 through 6 until the end of the simulation.

3. Temporal Discretization

The time discretization scheme presented here is similar to the scheme presented in [8]

with a few changes, such that various semi-implicit schemes can be chosen by setting up values for some parameters. The second-order semi-implicit scheme is given by:

ρⁿ⁺¹(φ)

∆t (α₂uⁿ⁺¹+α₁uⁿ+α₀uⁿ⁻¹) = β₁f(uⁿ) +β₀f(uⁿ⁻¹)+ (11a) λh

θ₂∇²uⁿ⁺¹+θ₁∇²uⁿ+θ₀∇²uⁿ⁻¹

− ∇pⁿ+ρⁿ⁺¹g,

∇ ·uⁿ⁺¹= 0, (11b)

whereλ =kµk∞ and f(u) depends on the diffusive, advective and forcing terms:

f(u) =−λ∇²u+∇ ·h

µ(∇u+∇u^T)i

−u· ∇u+f_σ, (12) and the values ofαi,βi andθi are: α0 = ^(2γ−1)ω_1+ω ²,α1 = (1−2γ)ω−1,α2 = ^1+2γω_1+ω ,β0 = 1 +γ, β₁ =−γω, θ₀ = c

2, θ₁ = 1−γ−

1 + _ω¹

c

2, θ₂ =γ + _2ω^c , where ω = ∆t_n+1/∆t_n is the ratio between two consecutive time steps. A family of numerical schemes involving parametersγ and ccan be derived as shown by Ascher et al. [2]:

• Crank-Nicholson Adams-Bashforth (CNAB): (γ, c) = (0.5,0.0);

• Modified Crank-Nicholson Adams-Bashforth (MCNAB): (γ, c) = (0.5,0.125);

• Crank-Nicholson Leap Frog (CNLF): (γ, c) = (0.0,1.0);

• Semi-Backward Difference (SBDF): (γ, c) = (1.0,0.0).

In the present work, the above SBDF temporal time discretization was chosen along with a variable-size time step.

Applying a fractional step approach to Eqs. (11a) and (11b) yields:

ρⁿ⁺¹

∆t (α₂u^∗+α₁uⁿ+α₀uⁿ⁻¹) =β₁f(uⁿ) +β₀f(uⁿ⁻¹)+ (13) λh

θ₂∇²u^∗+θ₁∇²uⁿ+θ₀∇²uⁿ⁻¹

− ∇pⁿ+ρⁿ⁺¹g, uⁿ⁺¹ =u^∗− ∆t

α₂

∇q

ρⁿ⁺¹, (14)

∇ ·uⁿ⁺¹ = 0. (15)

Once the provisional velocity u^∗ is computed from (13) imposing u^∗ .

=uⁿ⁺¹ on the bound- aries, it is projected onto the space of divergence-free vector fields. This is accomplished by solving the Poisson equation for q defined by (14) and (15) along with homogeneous Neumann boundary conditions∂q/∂n= 0, where n is the outer normal of the domain.

Since a fractional time step is used, linear systems must be solved both for the provisional vector field,u^∗ in (13), and for the pressure incrementq, which is given by:

∇ ·[ 1

ρⁿ⁺¹∇q] = α₂

∆t∇ ·u^∗. (16)

The time step is calculated following stability criteria for explicit schemes, as detailed in [8].

4. Spatial discretization

4.1. Adaptive mesh refinement

Adaptive mesh refinement (AMR) encompasses a set of methodologies which provide the means for locally refining the mesh in regions in which the error in the solution is above some threshold previously specified. These regions are usually identified by some error measure or high gradients of some variable, such as vorticity or, in the case of multiphase flows, the density field, for example. AMR methods can be applied either to structured or unstructured meshes. An example of unstructured adaptive mesh refinement for solving the Navier-Stokes equations is the Moving Mesh Interface Tracking (MMIT) Method of Quan and Schmidt

[30], which is based on the one fluid formulation. However, unlike the front-tracking method of Unverdi and Tryggvason [49], it uses an unstructured mesh to discretize the Eulerian domain, allowing the mesh points in theLagrangianinterface to be also part of the Eulerian mesh.

Regarding specifically Cartesian grids, there are two kinds of AMR strategies widely used in the context of CFD: octree [35] and SAMR [3]. Both methods are based on splitting Cartesian cells in order to obtain a locally refined mesh at some regions of interest. However, in the SAMR approach, cells located at a given refinement level are clustered into cartesian blocks, often calledpatches. The SAMR concept consists in creating a hierarchy of cartesian grids with different levels of refinement which cover the entire domain, concentrating the finer grids on the regions which require special attention. A level of refinement comprises one or more cartesian grid blocks which do not intersect each other and share the same grid spacing.

Figure 2 shows an example of a block-structured adaptive mesh during the simulation of a bubble rising in a quiescent liquid, where the grid was refined based on the gradient of the vorticity and on the presence of the interface.

When the remeshing criteria are based on the magnitude of some variable, ranges of this variable are defined so that each refinement level covers a different range of values of that variable. In this work, the following approach was followed:

Consider a variable Φ, ranging from Φ_minto Φ_maxon the entire Eulerian domain. Thresh- olds defined as fractions of Φ_max are used for flagging bad points on each refinement level.

In the case of the vorticity magnitude, which was used in this work, it was heuristically defined that the range [Φ/10...Φ] would be covered by the finest level. The coverage of the remaining levels would be equally divided in the interval [Φ/10...Φ].

The use of front-tracking methods with block-structured AMR may be traced back to the work of Roma [34], where the IB Method of Peskin [26, 27] was implemented in a SAMR framework and applied it to the solution of the two-dimensional incompressible Eu- ler equations. Griffith and Peskin [15], using the SAMRAI computational framework [14]

employed SAMR in the the simulation of the blood-muscle-valve mechanics of the human

Figure 2: An example of block structured adaptive mesh refinement. The vortex shedding behind a rising bubble is refined based on the gradient of the vorticity. In the simulation, the bubble actually moves in the vertical direction (present work).

heart. Ceniceros et al. [8] extended the work of Roma [34] to two-dimensional, transient two-phase flows and Ceniceros et al. [7] simulated 3D multiphase flows with a phase field model.

5. The Lagrangian framework

5.1. Interface advection

The interface advection is performed in a Lagrangian manner. After the velocity field is found by solving the Navier-Stokes equations, this velocity field is interpolated on the Lagrangian interface. The interface position is then updated by solving the ODE given by

dX(t)

dt =U(X(t), t), (17)

where X(t) is the position of the Lagrangian interface and U(X(t), t) is the velocity of the interface. In the first time step, Eq. (17) is solved using the first order Euler method, given

by:

Xⁿ⁺¹ =Xⁿ+ ∆t·Uⁿ(Xⁿ). (18) After solving the first time step, the Navier-Stokes solver enters the temporal loop. In this situation, a second order time integration scheme may be used, since the temporal history for both, velocity and position of the interface, can be stored. Therefore, the following integration method is used:

Xⁿ⁺¹ =Xⁿ+∆t

2 · Uⁿ(Xⁿ) +Uⁿ⁻¹(Xⁿ⁻¹)

. (19)

Notice that the term Uⁿ⁻¹ can always be obtained, even after a remeshing operation since, in such situation, it can be restored via interpolation of uⁿ⁻¹ on the corresponding position of the Lagrangian interface.

There are two issues that arise as result of the interface advection: the growth of small undulations in some regions where the interface area tend to shrink [42] and the volume change inside the interface. These issues will be addressed next.

5.2. Volume and shape preserving after advecting the interface

A mass conservation issue in Front Tracking arise after the advection of the interface. The most common reason pointed out for this problem is the fact that the interpolation procedure for calculating the velocity field at the interface is not conservative. Another reason, which will be shown in the presentation of the results, is the use of low order schemes for advecting the interface. Regardless of the origin of this issue, the volume change at a given time step is small (O(10⁻⁴)%). However, on the overall simulation it has a cumulative behaviour and, after running tens of thousands of time steps, the volume change can increase significantly.

This problem can be solved by applying a correction step after updating the interface position. Since the volume change in a given time step is small, all change can be assumed to happen in the normal direction. Assuming that the expansion or shrink of the interface is constant along the interface and always in the normal direction, a simple geometrical correction is applied as described in [28].

5.2.1. Interface undulation

Another common problem of the Front Tracking Method in this methodology is the development of small undulations or wrinkles that occurs at the rear of rising bubbles.

According to Sousa et al. (2004), these wrinkles result of variations in the velocity field and may be amplified where the surface area is shrinking. Figure 3 shows an example of this phenomenon, depicting the rear of a bubble in the spherical cap regime (Eo=40, M =0.056, Re=20.6).

Figure 3: Wrinkles at the rear of a rising bubble. Spherical cap regime,Eo = 40,M = 0.056.

Sousa et al. [42] described an algorithm named TSUR-3D (TSUR: Trapezoidal Sub-grid Undulation Removal) which eliminates such undulations by requiring that the normal of an edge, in the plane defined by the normals at the vertices of the edge, be parallel to the resultant of the normals of these vertices. This is done by computing new coordinates for the vertices of the edge while taking into account local volume preservation.

Figure (4) shows the result of the algorithm for a given edge eand its vertices, namedv₁ and v₂. If n₁ and n₂ are the normal vectors atv₁ and v₂, an approximation of the normal vector fore can be given by

n= (n₁+n₂)

k(n1+n2)k. (20)

Letmbe the midpoint of e. The deviation between nand the correct normal at mmay

Figure 4: Undulation Removal procedure: repositioning a generic edge e (Adapted from [42]).

be computed by h₁ and h₂ as follows:

h₁ = (v₁−m)·n, (21a)

h₂ = (v₂−m)·n. (21b)

Next, auxiliary points p1 and p2 are calculated as

p₁ =v₁−h₁n, (22a)

p₂ =v₂−h₂n, (22b)

defining two polyhedra: P₁(v₁,x₁₀, ...,x_1n,p₁) with volume V₁ and P₂(v₂,x₂₀, ...,x_2n,p₂) with volumeV₂.

Usingp₁andp₂ as the new coordinates ofv₁ andv₂ fulfils the objective of matching the normals. However, the local volume conservation has not been taken into account yet. To accomplish that, an average correction will be applied to the coordinates of both vertices, following the direction ofn.

Assume the volumes V₁ and V₂ are computed based on two non-intersecting plane poly- gons: S₁, with areaA₁ and S₂, with areaA₂, such that

V₁ =A₁h₁ and V₂ =A₂h₂. (23)

Let P =P₁∪P₂ be a polyhedron with volume

V =Ah, (24)

wereA=A₁+A₂ andV =V₁+V₂. Using these relations and Eqs. (23) and (24), one may write:

V h = V₁

h₁ +V₂

h₂, (25)

which leads to the following expression forh:

h= V₁+V₂

V1

h1 +^V_h²

2

. (26)

Finally, the new positions of v₁ and v₂ are given by:

v₁ =p₁+hn, (27a)

v₂ =p₂+hn. (27b)

5.3. Surface remeshing

Surface remeshing comprises refining and coarsening procedures. Mesh refinement is a relatively simple task, in which flagged elements or their edges are split until the required characteristic length is achieved. It causes no change in the geometry shape but may lead to some decrease in the mesh quality, depending on the splitting algorithm. Mesh coarsening, on the other hand, replaces an edge or an element by a vertex, and therefore has the potential ability to change the geometry shape. Also, the volume and/or surface area of the geometry must be preserved. In the present work, the Memoryless Simplification Algorithm (MSA) of Lindstrom and Turk [23] was used. The main feature of this method is that it preserves volume, shape and element quality while no time history of the mesh is necessary. This is an interesting feature, especially when large meshes are used. The method is based on edge removal rather than vertex or element removal. The advantage of this approach is that it removes always the same number of geometric entities: one vertex, two faces and three edges [24].

The two most important parameters in this algorithm are the position of the new vertex and the order in which the edges are removed. Vertex positioning is treated as an opti- mization problem and the element shape, the local volume and area are constraints limiting the space for calculating the new coordinates. MSA uses the geometry volume and area to make such decisions. The new vertex position is calculated so that the local volume is not affected by the collapse. If the new vertex is close to a geometric border, the local area is also preserved. The volume or area which are modified by the neighbouring triangles are also taken into account as well as the element quality (Lindstrom; Turk, 1998; Lindstrom; Turk, 1999). In the present work, the GTS computational library (GNU Triangulated Library) provided the underlying implementation of the aforementioned algorithm.

GTS provides a complete data structure for representing a surface geometry with non- structured, triangulated meshes. The library contains support functions for iterating over the geometric entities (vertices, edges and faces), as well as remeshing functionalities, boolean operations and object location with kd-trees and binary trees. The implementation is made in C and the data structure resembles an object oriented framework. Its design is based on the GTK+ architecture and Glib provides all supporting containers which are not strictly geometry-related, such as linked lists, hash tables etc. In order to keep the library as generic as possible, no special attribute is included. However, the inheritance structure implemented allows the user to create new data types to meet specific needs and which are still fully integrated with GTS [29]. Regarding the kind of functionality provided, GTS functions may be divided into four groups:

1. Geometric functions: area, volume, centre of mass, distance.

2. Location functions: point is inside a surface, point belongs to a plane, which is the closest point to some triangle.

3. Mesh-related functions: mesh refinement and coarsening, identification of boundary edges and vertices.

4. Iterator-like functions: provide access to the mesh components (vertices, edges and faces).

6. Results

Results presented in this section will be divided into two groups. Subsection 6.1 shows the verification tests, an important step in which analytical solutions are used in order to verify the convergence order of the code and validation tests through some simulations of bubble flows. Subsection 7 shows the results of the simulation of bubble flows.

6.1. Verification

The verification procedure will be applied separately to the Navier-Stokes solver and the front-tracking tools, but both are based on the Method of Manufactured Solutions (MMS) [33]. This method consists in using analytical functions to set up initial and boundary conditions, as well as source terms when necessary, so that the numerical solution can be compared to analytical functions and the convergence order of the solution can be verified.

One of the few requirements on these functions is that they must satisfy the physical con- straints of the problem that will be simulated by the computational code under test. In the case of a Navier-Stokes solver, the velocity field must be solenoidal and, for multiphase flows, density and viscosity fields must not have a constant value.

6.1.1. Verification of the Navier-Stokes equations

Let r(x, y, z, t) = π(w1x+w2y+w3z) +w4t, where wi are constants. The analytical expressions for the velocity, pressure, density and viscosity fields are given by [39]:

u_e =sin²(r), (28)

v_e =−cos²(r), (29)

w_e = w₁+w₂

w₃ cos²(r), (30)

pe = cos²(r), (31)

ρe = 1 + 0.1 sin²(r), (32)

µe = 1 + 0.2 cos²(r). (33)

where the subscripte stands for the manufactured exact solutions of the primary variables.

Notice that the velocity profile given by eqs. 28 - 30 is solenoidal. Parameters w_i are

used in order to choose which term will be active in the expression for r(x, y, z, t). Here, w1 =w2 =w3 = 2, w4 = 1. The expression for the source term is obtained by isolating the external force term in the momentum equation.

The computational domain is a unitary cube and the time step is calculated dynamically, based on the stability criteria described in [8]. Figure 5 shows an example of the mesh employed in the simulations, displaying the boundaries of each refinement level. Although it is locally refined, it will not dynamically change in time (static AMR). For a mesh with grid spacing ∆x, ∆y and ∆z at the x-, y- and z-directions and characteristic length h = min(∆x,∆y,∆z), the refinement ratioris defined as the quotient between the characteristic length of the current level and the characteristic length of the subsequent refined level. In this work, a constant ratior = 2 was used in all simulations.

Figure 5: Static grid with one refinement level used on the convergence rate analysis.

The error between the numerical and exact solutions is quantified using the L₂-norm:

Φ^h(φ) =L^h₂(φ) = v u u t

1 V_Ω

X

i∈Ω^h_u

(φ_i−φ^e_i)²w_i, (34) whereV_Ω is the volume of the Eulerian domain and the weightsw_i are given by:

w

=



 

 

 

 

 

 

∆x∆y∆z, on the coarse grid,

∆x∆y∆z

r

, on the fine grid, r + 1

2

∆x∆y∆z r

at the coarse-fine grid interface.

(35)

An analysis of spacial convergence rate (= log₂(Φ^2h/Φ^h)) was performed. The coarsest mesh has a grid spacing h₁ = 1/16, and the subsequent meshes are refined with a constant refinement ratio r= 2, leaving the sizes of the patches unchanged. Three sets of cases were run, based on the kind of boundary conditions applied: (i) Dirichlet for the velocity and Neumann for the pressure, (ii) Neumann for the velocity and Dirichlet for the pressure and (iii) Periodicity for both variables.

Table (1) shows the errors and the convergence rate obtained for case 1. As expected, a second order rate was achieved for the velocity, while the pressure yielded a rate between 1 and 2 [CITAR MINION].

Table 1: Verification tests. Dirichlet boundary conditions for velocity and Neumann for the pressure.

convergence rate test: L₂-norm of error and convergence rate () L₂(h₁) ₁ L₂(h₂) ₂ L₂(h₃) ₃ L₂(h₄) u 1.69e−2 1.98 4.28e−3 1.98 1.09e−3 1.99 2.74e−4 v 1.69e−2 1.98 4.27e−3 1.98 1.08e−3 1.99 2.74e−4 w 3.23e−2 1.99 8.15e−3 1.99 2.06e−3 2.00 5.16e−4 p 7.99e−1 1.77 2.35e−1 1.79 6.81e−2 1.78 1.99e−2

A comparison between this case and tables 2 and 3 shows that the velocity field is less sensitive to the grid refinement when Dirichlet boundary conditions are applied. On the other hand, Neumann and Periodic boundary conditions yield slightly higher convergence rates when finer meshes are employed. Similar analysis applies to the pressure field, since case 1 led to higher convergence rates.

Table 2: Verification tests. Neumann boundary conditions for velocity and Dirichlet for the pressure.

convergence rate test: L₂-norm of error and convergence rate () L₂(h₁) ₁ L₂(h₂) ₂ L₂(h₃) ₃ L₂(h₄) u 7.85e−2 1.83 2.19e−2 2.02 5.42e−3 2.00 1.35e−3 v 8.38e−2 1.91 2.23e−2 2.02 5.49e−3 1.99 1.38e−3 w 1.70e−1 1.94 4.44e−2 2.03 1.09e−2 2.03 2.70e−3 p 2.73e−1 1.69 8.49e−2 1.62 2.76e−2 1.60 9.12e−3

Table 3: Verification tests. Navier-Stokes solver run with Periodic boundary conditions for both, velocity and pressure.

convergence rate test: L₂-norm of error and convergence rate () L₂(h₁) ₁ L₂(h₂) ₂ L₂(h₃) ₃ L₂(h₄) u 8.19e−2 1.86 2.25e−2 2.02 5.54e−3 2.00 1.38e−3 v 8.64e−2 1.92 2.28e−2 2.03 5.59e−3 2.00 1.40e−3 w 1.74e−1 1.94 4.52e−2 2.03 1.11e−2 2.01 2.75e−3 p 2.27e−1 1.59 7.53e−2 1.54 2.59e−2 1.55 8.84e−3

6.2. Analytic Shear Flow

This test consists in subjecting a surface, usually spherical, to a solenoidal, periodic, analytic velocity field and assessing its volume change as the flow evolves. Assuming a velocity field with a period T, the surface is stretched until t = T /2 and then returns to its original position and shape. The velocity field may be calculated either directly on the Lagrangian vertices or interpolated from an auxiliary grid, allowing to assess the influence of the interpolation functions on the volume change during the simulation. Regarding the interface advection, two time integration schemes were assessed: first order Euler method and a second order method based on the trapezoidal rule, respectively Eqs. 18 and 19.

The velocity profile is given by equation 36 [41] and the interpolation of the velocity field

was performed using Eqs. (6) and (7) [27].

u(x, y, z, t) = cos(πt/T )sin

(πx)(sin(2πz) − sin(2πy)), v(x, y, z, t) = cos(πt/T )sin

(πy)(sin(2πx) − sin(2πz)), w(x, y, z, t) = cos(πt/T )sin

(πz)(sin(2πy) − sin(2πx)).

(36)

The sphere diameter is set to φ = 0.3m and its centre is C(0.5,0.75,0.5)m. In all cases simulated, the characteristic length of the surface mesh, defined as the mean edge length, was set toξ_L=φ/32.

Figure 6 shows snapshots of an initially spherical shape during a typical simulation at t = 0, t = T /4, t = T /2, t = 3T /4 and t =T. The period is T = 4s, the same which was used in the simulations presented here. The surface is stretched until T = T /2 and then ideally would symmetrically return to its initial position, so that the shape and position should be the same not only when comparing t = 0 and t = T, but also when comparing t=T /4 and t= 3T /4.

Figure 6: Deformation of a spherical surface when subjected to the analytical velocity field given by Eq. 36. Snapshots taken at t= 0, t= ^T₄,t = ^T₂, t= ^3T₄ et =T

Surface stretching requires remeshing, which may also interfere on the volume conser- vation process. Regardless its volume conservation properties, any remeshing algorithm destroys the Lagrangian velocity history, requiring a new interpolation of the velocity field or the use of a first order scheme in the time step immediately after the remeshing oper- ation. Since interpolation schemes are usually non-conservative, a conservative remeshing algorithm could still lead to a non-conserving volume outcome because of the need for ve- locity interpolation. In order to assess the influence of such variables on the volume of the Lagrangian interface, a series of tests was performed, in which three main parameters were

analysed: (i) the velocity field interpolation, (ii) the surface remeshing and (iii) the time integration scheme. The volumetric change was computed as a relative error, based on the initial sphere volume: =

1−^V_V^f

i

×100, whereV_i is the initial volume of the interface and Vf is the volume of the interface after finishing the shear flow. Table (4) shows the eight cases simulated.

In order to assess the ability of the volume recovery algorithm on preserving the original interface volume, the cases described above were simulated again, but now with a volume- recovery step applied at the end of the advection of the surface, at each time step. The objective was to preserve the original volume of the interface, within a tolerance of±0.01%.

Table (4) summarizes the case setup described above. In each of these cases, the simulation was performed with ∆t= 0.001s and with ∆t= 0.0005s.

Table 4: Shear Flow. Case description for each test performed.

Case Description

Volume Velocity Remeshing Recovery Interpolation

1 no no no

2 no no yes

3 no yes no

4 no yes yes

5 yes no no

6 yes no yes

7 yes yes no

8 yes yes yes

Figure (7) shows the volume change as a function of time for cases 1 to 4. The effect of surface remeshing can be assessed by comparing plots in the top of the figure with their corresponding ones on the bottom, while the effect of velocity interpolation appears when the plots on the left are compared with the plots on the right. Comparing cases 1 and 3 clearly shows the influence of the remeshing operation on the volume change, especially on

the second order temporal scheme. Also, notice that although the error between the initial and final volume are negligible for the 2nd order scheme, it reaches a maximum value around 1% when t = T /2. The 1st order scheme, on the other hand, shows a cumulative trend in all cases. This cumulative behaviour also appears in the 2nd order scheme when surface remeshing is enabled, although with much smaller intensity.

(a) case 1 (b) case 3

(c) case 2 (d) case 4

Figure 7: The influence of velocity interpolation and surface remeshing on the preservation of the initial interface volume.

Regarding the volume recovery algorithm, Fig. (8) shows that the volume change was

successfully kept within the specified range of±0.01% of the original volume.

(a) case 5 (b) case 7

(c) case 6 (d) case 8

Figure 8: The influence of the volume recovery algorithm on the preservation of the initial interface volume.

6.3. Spurious currents

Spurious currents or parasitic currents are non-null velocity fields generated in static bubble flow simulations due to numerical errors caused by unbalance that arise during the surface force calculation [41]. Tests for quantifying such currents consist in simulating a bubble or a droplet immersed in a flow with zero gravity and zero initial velocity field. In

such situation the buoyancy force would be null and the bubble/droplet should stand still during the whole simulation. The flow is characterized by two non-dimensional parameters:

the Laplace number and the Capillarity number, respectively Eqs. (37a) and (37b):

La= σρφ

µ² , (37a)

Ca= kuk∞µ

σ . (37b)

wherekuk∞ is the maximum velocity magnitude found in the velocity field. [38].

In order to assess the magnitude of the spurious currents in the present computer code, five cases were simulated, in which the Laplace number ranged from 1.2 to 12000. The re- sults, presented in Table (5), show good agreement with [40] and [42], who report Capillarity numbersCa=O(10⁻⁴). Three levels of refinement were employed, and the mesh resolution at the finest level was set to φ/32, where φ is the bubble diameter.

Table 5: Capillarity as a function of the Laplace number.

La Ca

1.2 9.84e−4 12 1.17e−3 120 1.08e−3 1200 1.09e−3 12000 8.76e−4

7. Rising Bubbles

7.1. Introduction

After the verification tests presented in the previous section, the implementation of the numerical method is thoroughly tested for the simulation of a single rising bubble ascending in various regimes. Initially, a sensitivity analysis was performed, with two objectives: to assess (i) the role of some geometric parameters and (ii) the influence of the density ratio on

the outcome of the flow. The results obtained in this analysis were applied in the setup of the forthcoming cases, which were mainly based on [4] and [17]. In the first group of cases, the ability of the method for simulating different bubble regimes was studied. The second group simulated a series of cases in which the bubbles have the same Eotvos number (Eo), in order to study the influence of the Morton number (M) on the Reynolds (Re) profile.

A third group was simulated aiming at comparing the streamline pattern obtained in the present work against the aforementioned references. Finally, some wobbling simulations were performed, based on the works of [43]. In all cases, the following boundary conditions were applied:

• Velocity: free-slip at the side boundaries, homogeneous Dirichlet at the bottom bound- ary and homogeneous Neumann at the top boundary.

• Pressure: homogeneous Neumann at the side and bottom boundaries and homogeneous Dirichlet at the top boundary.

Two parameters were used for flagging the regions of the Eulerian domain for grid re- finement: the position of the Lagrangian interface and the vorticity magnitude. In all cases simulated, the grid spacing at the finest level was set to φ/32. In the cases of sections 7.2 and 7.3, five levels of refinement were used. The dimensions of the Eulerian domain were set to (8φ×8φ×Lz), and different values were used for the z-dimension: Lz= 16φ, Lz = 24φ and L = 80φ. Notice that an uniform grid with the same grid spacing would have approx- imately 3.355×10⁷ cells for the first case and 1.678×10⁸ on the last case. On the other hand, adaptive grids required, on average, 100 times less. Details about the grid sizes will be presented during the description of each case.

7.2. Sensitivity Analysis

Several geometry-related parameters may influence the outcome of the simulation. In this section these parameters will be studied, based on the simulation of a bubble rising in a quiescent liquid in the spherical cap regime, as suggested by Lemonnier et al. [20]: Eo= 40, M = 0.056, ρ_C = 1000kg/m³, µ_C = 0.273556P a ·s, λ_ρ = 100, λ_µ = 100. The bubble

diameter is φ = 0.02m and the terminal Reynolds is Re= 20.6. ρ_C and µ_C are the density and dynamic viscosity of the continuous phase and λρ and λµ are the density ratio and viscosity ratio between the continuous and dispersed phase, respectively. On all cases run in this sensitivity analysis, the difference between the present work and the aforementioned reference work is quantified by the relative error based on the terminal Reynolds number.

Four cases were run in order to assess the role of the geometric correction tools on the flow simulation. In case A1 the standard Front Tracking Method was used, without any further intervention. In case A2 a volume correction step was applied at the end of the interface advection, while in case A3 only TSUR3D was used. Case A4 applies both, volume recovery and TSUR3D algorithms.

Table 6: A-series: Cases description and error on the terminal Re number. Reference Re= 20.6.

Case Description Re Relative Error (%) A1 Standard FT 23.5 12.3 A2 A1 + VRA 23.3 11.6 A3 A1 + TSUR3D 19.9 3.5 A4 A1 + A2 + A3 19.8 4.0

The effect of the geometric tools on the Re number and terminal velocity is shown in Fig. (9), which depicts the time evolution of the Reynolds number. Notice that the volume recovery algorithm has little influence at the beginning of the flow. In fact, only after reaching the terminal regime a small decrease in the Re number can be seen. Surface wrinkle removal, on the other hand, affects the early stages of the flow by smoothing the transient region and decreasing the Re maximum at the overshoot region. Since the volume is conserved along the simulation, the changes in Reynolds number comes from the bubble ascending velocity, which is clearly shown in Fig. (9).

Figure 10 shows details of the bubble shape at the end of the flow for cases A1 and A3.

Figure 9: A-series: Terminal Reynolds number depending on the geometric intervention applied. Reference Reynolds number: Re= 20.6.

7.2.1. Domain Size

In order to assess the influence of the domain size on the bubble shape and position, as well as on the terminal Reynolds number, four cases were simulated. Assuming gravity acting on the -z direction and a square cross-section in the xy plane, table 7 shows the domain sizes used in the simulation, the terminal Re obtained and the associated relative errors. Volume recovery and wrinkle removal algorithms were applied to all cases.

Figure 11 shows that a domain with cross-sectionL= 4φstill has some influence on the bubble rising velocity. Table 7 shows an error ε = 8.25% for this case, whereas the other 3 cases yielded errors around 4%. Since volume recovery and wrinkle removal algorithms were applied on all cases, no sensible difference in shape or volume was found between the cases. Case A7 was chosen as a commitment solution between the low cost of domain A6 and the larger domain A8, which yields more details about the flow at the cost of time and computer efforts.

Figure 10: Bubble shape at the end of the simulation. Top: no geometric intervention.

Bottom: volume recovery and wrinkle removal.

Table 7: Influence of the domain size on the terminal Reynolds number. All cases were run with Lz = 16φ. Reference Re= 20.6.

Case L Centre Re Error (%)

A5 L= 4φ (2φ,2φ,2φ) 18.9 8.25 A6 L= 6φ (3φ,3φ,2φ) 19.6 4.85 A7 L= 8φ (4φ,4φ,2φ) 19.8 4.0 A8 L= 10φ (5φ,5φ,2φ) 19.9 3.5

7.2.2. Resolution of the Eulerian Mesh

In order to assess the sensitivity of the flow solution to the Eulerian grid resolution, case A4 was simulated using three different grid resolutions: φ/16, φ/32 and φ/64. Since no significant change in the error was detected and the computational cost (in terms of wall time) for the third case was about five times higher than the second case, the grid resolution

Figure 11: A-series. Influence of the domain size on the bubble Re profile. Reference Reynolds number Re= 20.6.

φ/32 was adopted as the standard grid refinement for all cases, along with a domain size

Lx=Ly= 8φ, Lz = 24φ.

7.2.3. Resolution of the Lagrangian Mesh

The influence of the characteristic length of the Lagrangian mesh was studied by simu- lating the test case in three different configurations. The same number of refinement levels and grid spacing was used on all simulations. The bubble interface was discretized with 3 successively refined meshes and all other simulation parameters were kept constant. The length ratio is defined asλ_ξ = ^ξ_ξ^E

L, where ξ_E is the characteristic length of the Eulerian grid, defined at the level containing the Lagrangian mesh, andξL is the mean edge length of the Lagrangian mesh. The value of the error based on the terminal Re is presented in table 8, showing small influence of the Lagrangian mesh resolution on the outcome of the simulation.

Table 8: Influence of the Lagrangian mesh resolution on the terminal bubble Reynolds number. ReferenceRe= 20.6.

Case λ_ξ Re Error (%) A9 1 19.707 4.333 A10 2 19.826 3.757 A11 4 19.803 3.869

7.2.4. Density Ratio

The influence of the density ratio on the bubble ascending velocity was studied for three different bubble regimes. The first case simulated the rising of an oblate ellipsoidal bubble defined by Eo = 243, M = 266 and Re = 7.77 [4], and a density ratio λ_ρ = 1370. Three simulations were performed, for density ratios λ_ρ = 10,100,1000. Figure 12 shows the Reynolds profile for each case simulated, while table 9 shows the terminal shape and Re number. As expected, the error on the terminal Re number decreases as the density ratio approaches the experimental value. However, the average time cost for solving one time step in λ_ρ = 100 increases 50% when compared to λ_ρ = 10, and so does λ_ρ = 1000 when compared toλ_ρ = 100.

7.2.5. Sensitivity Analysis - Conclusions

This section has presented a sensitivity study in order to determine the influence of a set of simulation parameters on the outcome of the flow. Volume recovery and non-physical undulation removal algorithms have shown good results, by maintaining the bubble shape and conserving its volume.

When studying the influence of the Eulerian domain size on the bubble shape and Re number, a domain with 8φ between lateral boundaries has shown to be a commitment solution between computational cost and flow details.

Regarding domain resolution, an Eulerian grid spacing h=φ/32 was chosen along with a Lagrangian mesh with mean edge length as small as half the Eulerian grid spacing at its finest level, since no significant gains were found with further refinement.

Figure 12: Rising bubble. Influence of the density ratio on the ascending velocity.

Therefore, further simulations will adopt the following configuration: Given a bubble with diameter φ, set to rise in the +z-direction, the dimensions of the Eulerian domain will be (8φ×8φ×Lz), and the initial position of the bubble will be (4φ×4φ×2φ). The dimensionLz will depend on the case being simulated, but a minimum lengthLz = 24φwill be adopted. Since the ratio between the Eulerian grid spacing and the mean edge length in the Lagrangian mesh was set to λξ = 2, the Lagrangian mean edge length will be set toξ_L =φ/64. In all cases, the volume recovery algorithm and the non-physical undulation removal will be used. Also, a density ratio λρ= 100 will be adopted.

7.3. Terminal bubble shape and Reynolds number for low Reynolds flows

The ability of the current algorithm for predicting the terminal bubble shape and Reynolds number was tested against the experimental work of Bhaga and Weber [4] for a series of low Re flows in different regimes. Also, two numerical works were used for comparison: Hua and Lou [17] developed an axisymmetric algorithm with two volume-recovery steps for the

Table 9: Influence of the density ratio on the terminal Reynolds number and bubble shape.

Comparison against the experimental work of Bhaga and Weber [4]. Eo = 243, M = 266, Re= 7.77.

Test Case Shape Re Error (ε[%])

Experiment 7.77 -

λ_ρ= 10 7.27 6.44

λ_ρ= 100 7.61 2.06

λ_ρ= 1000 7.67 1.28

Lagrangian interface per time step: after remeshing the interface and after the interface advection. Stene [43] developed a 3D algorithm based on Berger’s SAMR methodology for the discretization of the Eulerian domain. The Lagrangian framework, however, followed the approach of Hua and Lou [17], requiring two volume-recovery steps per time step.

Table 10 shows various bubble regimes, ranging from low Re - low Eo spherical bubbles to moderate Re - moderate Eo skirted bubbles. A comparison with the experimental results shows that the present algorithm was able to predict the bubble shape properly. Regarding the terminal Re number, the maximum error occurred at the creeping flow regime, an so did the work of Hua and Lou [17]. Stene [43] reports, for the same case, a relative errorε= 21%

and suggests that the distance between the bubble and the domain boundaries could have influenced the results at this regime. However, in all cases in this work, the domain size was calculated based on the bubble initial diameter, as explained in section 7.2.5, and similar

approach was also reported in [43].

Table 10: Comparison of terminal shapes and Reynolds number reported by the reference works [4, 17] and the present work.

Bhaga Hua & Lou Present Work Eo = 17.7

M=711 Re=0.232

Re=0.211 Re=0.212 ε= 9.05% ε= 8.62%

Eo = 32.2 M=8.2e-4

Re=55.3 Re = 52.96 Re = 53.2

ε= 4.23% ε= 3.80%

Eo = 243 M=266

Re=7.77 Re = 8.40 Re = 7.61

ε= 8.07% ε= 2.06%

Eo = 115 M=4.63e-3

Re=94.0 Re = 88.70 Re = 90.50

ε= 5.64% ε= 3.73%

Eo = 339 M=43.1

Re=18.3 Re = 17.91 Re = 17.06

ε= 2.13% ε= 6.78%

Eo = 641 M=43.1

Re=30.3 Re = 28.54 Re = 31.47

ε= 1.22% ε= 5.81%

In the remaining cases, the largest errors occurred for bubbles in the skirted regime, probably due to the decrease in the bubble thickness in the skirt regions. Possible remedies could be increasing the Eulerian grid resolution and/or the implementation of an algorithm for dealing with fragmentation mechanisms.

The influence of the Morton number on low Reynolds flows was also studied. Table (11) shows five cases of dimpled spherical cap bubbles atEo = 116 and Morton number ranging fromM = 848 toM = 1.31. As in the previous case, the results were checked against Bhaga and Weber [4] and Hua and Lou [17]. The terminal bubble shape again was well predicted and the highest relative error on the prediction of the terminal Re number occurred for the case with the lowest terminal Reynolds.

Regarding the flow pattern around the bubbles, Bhaga and Weber [4] observed the existence of a closed toroidal wake through the flow visualization using hydrogen tracers, which appear as bright lines in the photographs shown in Table (12). Results obtained in the present work and also reported on Hua and Lou [17] show a secondary wake recirculation just behind the bubble rim. This could be the reason for the bright spots observed in the lower outside part of the bubble rim in the first case. Notice that this recirculation becomes bigger in the second case, which explains the bright region in the photograph of the experimental case. The relative error in the first case was ε= 2.85% and ε= 20.56% in the second case.

7.3.1. Influence of the initial bubble shape

Studying the experimental setup designed by Bhaga and Weber [4] shows that the device used for releasing the bubbles may have influence on the initial bubble shape. The bubble release mechanism is based on a dump cup which temporally holds the bubble and then releases it into an inverted funnel through which the bubble is then released for ascending through the test column. Since this mechanism may lead to different bubble shapes, a study was performed in order to assess the influence of the initial bubble shape on the outcome of the bubble flow [17].