Applications of the lattice Boltzmann multiphase models The application of the LBM multiphase method goes hand in hand with

model development. Successful examples include flow in porous media, Rayleigh–Taylor instability simulation and mixing, boiling dynamics, ternary flow (Chen et al.,2000), and spinodal decomposition (Alexander, Chen, and Grunau, 1993; Osborn et al., 1995).

6.3.2.1 Multiphase flow in porous media

Lattice Boltzmann multiphase fluid flow models have been extensively used to simulate multicomponent flow in porous media to understand the fun- damental physics associated with enhanced oil recovery, including perme- abilities. The LBM is particularly useful for this problem because of its capability of handling complex geometrical boundaries and varying physi- cal parameters, including viscosity and wettability. In the work by Buckles

et al. (1994), Martys and Chen (1996), and Ferreol and Rothman (1995), re-

alistic sandstone geometries from oil fields were used. Very complicated flow patterns were observed. The numerical values of the relative permeability as a function of percent saturation of wetting fluid agree qualitatively with experimental data.

Gunstensen and Rothman (1993) studied the linear and nonlinear multicomponent flow regimes corresponding to large and small flow rates,

Lattice Boltzmann Methods 173 respectively. They found, for the first time, that the traditional Darcy law must be modified in the nonlinear regime because of the capillary effect. Zhang, Zhang, and Chen (2000) studied flow in porous media at the pore scale with lattice Boltzmann simulations on pore geometries reconstructed from computed microtomographic images. Pore scale results were analyzed to give quantities such as permeability, porosity, and specific surface area at various scales and at various locations. With this, some fundamental issues such as scale dependency and medium variability were assessed quantita- tively. More specifically, they quantified the existence and size of the well- known concept of representative elementary volume (REV). They suggested that, for heterogeneous media, a better measure may be the so-called “sta- tistical REV,” which has weaker requirements than does the deterministic REV. Kang et al. (2002a) further incorporated surface chemical reactions to study corrosion in porous media and developed a unified lattice Boltzmann method for flow in multiscale porous media (Kang et al.,2002b).

6.3.2.2 Rayleigh–Taylor instability

The quantitative accuracy of the lattice Boltzmann multiphase model was best demonstrated in studies by He et al. (1999a,b) on the Rayleigh–Taylor instability. He et al. used a phase-function model and the simulation results agree well with those reported in the existing literature.

Figure 6.2 shows the growth of the Rayleigh–Taylor instability from a small single-mode perturbation in a computational domain of width W . The Atwood number A = (ρh−ρl)/(ρh+ρl), where ρhand ρlare densities of heavy

and light fluids, respectively, is 0.5. The Reynolds number Re =√W gW/ν

is 2048. The simulation was carried out on a 256×1024 grid. The gravity was chosen so that √W g = 0.04. The interface was represented by 19 equally

spaced density contours. During the early stages (t < 1.0), the growth of the fluid interface remains symmetric up and down. Later, the heavy fluid falls down as a spike and the light fluid rises to form bubbles. Starting from

t = 2.0, the heavy fluid begins to roll up into two counter-rotating vortices.

This phenomenon was first computed by Daly (1967) and studied further by other authors (Youngs,1984; Tryggvason,1988; Glimm et al.,1986; Mulder

et al.,1992). At a later time (t = 3.0), these two vortices become unstable

and a pair of secondary vortices appear at the tails of the roll-ups. The roll- ups and vortices in the heavy fluid spike are due to the Kelvin–Helmholtz instability (Daly, 1967). The shapes of the fluid interface in the current study compare well with previous studies (Daly,1967; Tryggvason, 1988).

In the later stage of the Rayleigh–Taylor instability (t = 5.0), the heavy fluid falling down gradually forms one central spike and two side spikes. It is

Fig. 6.2. Evolution of the fluid interface from a single-mode perturbation in the Rayleigh–Taylor instability. The Atwood number is 0.5 and the Reynolds number is 2048. A total of 19 density contours are plotted. The time is measured in units ofW/g.

interesting to note that only the side spikes of the heavy fluid experience the Kelvin–Helmholtz instability and the interface along these two spikes is stretched and folded into very complicated shapes. The mixing of heavy and light fluid is significant. On the other hand, the interface along the central spike, as well as the fronts of both bubble and spike, remain relatively smooth.

A more interesting study is the simulations of multiple-mode Rayleigh– Taylor instability. The simulation was carried out on a 512× 512 grid. The Reynolds number was chosen as 4096 and the Atwood number was fixed at 0.5. Gravity was chosen so that √W g = 0.08. The initial perturbation of

Lattice Boltzmann Methods 175 the fluid interface was given by:

h = n

ancos(knx) + bnsin(knx), (6.62)

where kn = 2nπ/W is the wavenumber. The amplitudes, an and bn, were randomly chosen from a Gaussian distribution. A total of 10 modes (n ∈ [21, 30]) was used.

The evolution of this multiple-mode Rayleigh–Taylor instability is shown in Fig. 6.3 at six time instants. The early stage (t < 1.0) is characterized by the growth of structures with small wavenumbers. The heavy fluid falls down as slender spikes while the light fluid rises up as small bubbles. The amplitudes of the perturbation have grown much larger than the initial wave- length at t = 1.0. The interaction among small structures becomes obvious at t = 2.0 and continues throughout the simulation. Three features can be observed during this stage. First, the larger bubbles rise faster than the smaller ones. This is because the bubble at this stage moves proportionally to its size. Second, small bubbles continue to merge into larger ones. At the time of t = 4.0, the initial perturbation has totally disappeared and

t = 1.0 t = 2.0 t =3.0

t = 4.0 t = 5.0 t = 6.0

Fig. 6.3. Rayleigh–Taylor instability from a multiple mode perturbation. The Atwood number is 0.5 and the Reynolds number is 4096. The time is measured in units ofW/g.

the dominant wavelength becomes W/2. Notice that this long wavelength does not exist in the initial perturbation. Third, the interaction among bub- bles leads to a turbulent mixing layer. The thickness of this mixing layer increases with time.

The same LBM multiphase flow model was also used to study the three- dimensional Rayleigh–Taylor instability – a task challenging many conven- tional multiphase schemes. He et al. (1999b) carried out a three-dimensional Rayleigh–Taylor instability simulation in a rectangular box with a square horizontal cross-section. The height–width aspect ratio was fixed at 4:1. Gravity pointed downward and surface tension was neglected. For simplic- ity, the kinematic viscosities were chosen to be the same for both the heavy and light fluids. Periodic boundary conditions were applied at the four sides, while no-slip boundary conditions were applied at the top and bottom walls. The instability develops from the imposed single-mode initial perturbation:

h(x, y) W = 0.05  cos  2πx W  + cos  2πy W  , (6.63)

where h is the height of the interface and W is the box width. The origin of the coordinates is at the lower left bottom corner of the box.

The results are presented in dimensionless form. The box width, W , is taken as the length scale and T = W/g is taken as the time scale. As

before, the relevant dimensionless parameters are the Reynolds number and the Atwood number.

The typical evolution of the fluid interface in the three-dimensional Rayleigh–Taylor instability is shown in Fig.6.4. The Atwood and Reynolds numbers are 0.5 and 1024, respectively. The simulation is carried out on a

128× 128 × 512 grid. Presented in the figure are views of the interface from

both the heavy fluid side (left panels) and the light fluid side (right panels). As expected, the heavy and light fluids penetrate into each other as time increases. The light fluid rises to form a bubble and the heavy fluid falls to generate a spike. Furthermore, there is an additional landmark feature that distinguishes this interface from that observed in a two-dimensional Rayleigh–Taylor instability, namely the four saddle points at the middle of the four sides of the computational box shown in Fig.6.4(a). The evolution of the interface around these saddle points is one of the unique features of the three-dimensional Rayleigh–Taylor instability.

As shown in Fig. 6.4, the interface remains rather simple during the early stages (Fig.6.4a) but becomes more complicated as time increases. For this typical case, the Kelvin–Helmholtz instability does not develop until t = 2.0 when the first roll-up of the heavy fluid appears in the neighborhood of the

Lattice Boltzmann Methods 177

(a) (b)

(c) (d)

saddle point

Fig. 6.4. Evolution of the fluid interface from a single-mode perturbation at (a) t = 1.0, (b) t = 2.0, (c) t = 3.0, and (d) t = 4.0. The Atwood number is 0.5 and the Reynolds number 1024. Time is measured in units of W/g. The

interface is viewed from the heavy fluid side (left panel) and from the light fluid side (right panel). The interfaces in the left panel are shifted W/2 in both x- and

y-directions for a better view of the bubble.

saddle points (Fig. 6.4b). The roll-up at the edge of the spike starts at a later time (Fig. 6.4c). At t = 4.0, these roll-ups have been stretched into two extra layers of heavy fluid folded upwards: one forms a skirt around the spike and the other forms a girdle inside the bubble (Fig. 6.4d). Notice also the small curls inside the skirt. These structures are formed by the Kelvin–Helmholtz instability. From the outside, both the bubble and the spike look like mushrooms. A similar evolution of the interface was observed in other numerical simulations (Tryggvason and Unverdi,1990; Li, Jin, and

Glimm, 1996) although those calculations were not carried out to this late stage. This two-layer roll-up phenomenon is another unique feature of the three-dimensional Rayleigh–Taylor instability.

The complicated structures of the interface in the later stages can be seen clearly in horizontal cross-sections. As an example, in Fig. 6.4(d) we show the results at t = 4.0 from the same simulation. We found it helpful to present the interfaces above the saddle points separately from those below. Figure 6.5 shows the density plots at a number of horizontal planes above the saddle point. The planes are labelled by the grid index k ∈ [1, 512] and the plane altitude can be found using z = k/128. At the saddle point level (k = 213), the light fluid exhibits the shape of a “rosette” with the

Fig. 6.5. Density plots at t = 4.0 in horizontal planes above the saddle points. Black represents the heavy fluid and white the light fluid. The plots are shifted

W/2 in both x- and y-directions for a better view of the bubble. The Atwood

number is 0.5 and the Reynolds number 1024. The plane altitude can be calculated using z = k/128.

Lattice Boltzmann Methods 179

Fig. 6.6. Density plots at t = 4.0 in horizontal planes below the saddle points. Black represents the heavy fluid and white the light fluid. The Atwood number is 0.5 and the Reynolds number 1024. The plane altitude can be calculated using

z = k/128.

petals pointing towards the saddle points. Moving upwards, the “rosette” quickly contracts its petals and becomes embedded in a box of heavy fluid (k = 220). At a slightly higher level (k = 230), the surrounding heavy fluid is divided into two parts by a ring of light fluid. This ring of light fluid comes from the outer layer of the mushroom of the bubble, while the heavy fluid between the “rosette” and the light fluid ring comes from the roll-up girdle of the heavy fluid around the saddle points. A similar structure persists all the way up to k = 300. Above that level, the “rosette” of light fluid in the middle gradually transforms into a “diamond” shape (k = 320–360). In the meantime, the girdle of the heavy fluid shrinks and finally disappears at k = 370. Above k = 370, only a bubble of light fluid is visible.

Figure6.6shows density plots for a number of horizontal planes below the saddle points. Across the bottom of the spike, the blob of heavy fluid has a

roughly rectangular shape at k = 20. At slightly higher levels (k = 30–40), this simple geometry is replaced by a configuration with a core of heavy fluid in the middle surrounded by a frame of heavy fluid. The heavy-fluid core forms a cross with its bars oriented diagonally. The frame is a skirt of heavy fluid which has rolled up around the spike. Between the heavy-fluid core and the frame is a gap of light fluid. Moving upwards, the heavy-fluid frame gradually transforms from its original square shape to a roughly circular shape until it finally disappears near k = 120. For k between 70 and 110, we can see another layer of heavy fluid which represents the folding back of the heavy-fluid skirt. The wavy interface indicates that an instability may start to develop in these horizontal planes.

Regarding the instability in horizontal planes, it is interesting to notice what happens at the tips of the cross of the heavy-fluid core. Starting from k = 60, these tips begin to widen and the heavy fluid begins to roll inward. At k = 90, these tips have transformed into an “anchor” shape. The same configuration persists all the way up to the saddle level, although the roll-ups begin to shrink at k = 170. This roll-up phenomenon in the horizontal planes is very similar to that observed in the Richtmyer–Meshkov instability (Zhang and Graham, 1998). However, the phenomena are not exactly the same because in our study we do not have an explicit outward force.