Distribution and number of plumes and initial conditions

5.4.3.1 Position and spacing of plumes

The exact localization of plume generation was investigated. Under the random initial condition introduced in [142], the location is hard to be predicted. Deterministic initial conditions are considered to investigate the formation and location of descending plumes.

Many initial conditions among the set of tested distribution of bacterium settings are able to trigger convective patterns.

All numerical results are computed at Le_τ = 5, Pr_τ = 500, Ra_τ = 2000, S = 10, and H = 4 to ensure the formation of bacterium rich plumes. Firstly, a profile given by the form of wave function cos(2πx/3) is considered to determine the two initial layers of bacteria. Simulation results and initial conditions are shown in figure 5.7. The upper layer has a higher bacterium density than the lower layer. The plumes form at the initial location of the crest of the wave function where there is a larger number of bacteria. By reversing the initial conditions with a denser inferior layer (figure 5.8), the plumes also form at the same location. We can observe that distinct initial conditions can lead to the formation of a very similar pattern (figures 5.7-5.8).

We vary the profile of the curve between the upper and lower layers in the initial condition with different wave functions cos(3πx/2), cos(5πx/2), and cos(2πx). The wavenumber is defined as the spatial frequency of waves per unit distance. The wave-length is defined as the domain horizontal wave-length divided by the wavenumber. Increasing the wavenumber of the initial wave function causes the formation of additional plumes to occur (figures 5.9 and 5.10). In all simulations, the locations of the plumes correspond to sites of initially greater local bacterium density.

When the wavenumber in the initial profile is large enough, the number of plumes is

5.4. NUMERICAL RESULTS AND DISCUSSION

Figure 5.8: Numerical results for n with respect to the deterministic initial conditions.

At the initial time, bacterium density is higher in the upper layer. (a) initial condition is n₀ = [^{0.5, y}≥0.5+0.1 cos(2πx/3)

1, y<0.5+0.1 cos(2πx/3) ; (b) simulation result at t = 1.2 for the initial condition given in (a) ; (c) initial condition is n₀ = [^{0.5, y}≥0.5−0.1 cos(2πx/3)

1, y<0.5−0.1 cos(2πx/3) ; (d) simulation result at t = 1.2 for the initial condition given in (c).

-2 0 2

Figure 5.9: Numerical results for n with respect to the deterministic initial conditions.

At the initial time, bacterium density is higher in the upper layer. (a) initial condition is n₀ = [1, y≥0.5+0.1 cos(5πx/2)

0.5, y<0.5+0.1 cos(5πx/2); (b) simulation result at t = 1.2 for the initial condition given in (a) ; (c) initial condition is n₀ = [^{1, y}≥0.5−0.1 cos(5πx/2)

0.5, y<0.5−0.1 cos(5πx/2); (d) simulation result at t = 1.2 for the initial condition given in (c).

5.4. NUMERICAL RESULTS AND DISCUSSION

Figure 5.10: Numerical results for n with respect to the deterministic initial conditions.

At the initial time, bacterium density is higher in the lower layer. (a) initial condition is n0 = [0.5, y≥0.5+0.1 cos(5πx/2)

1, y<0.5+0.1 cos(5πx/2) ; (b) simulation result at t = 1.2 for the initial condition given in (a) ; (c) initial condition is n₀ = [^{0.5, y}≥0.5−0.1 cos(5πx/2)

1, y<0.5−0.1 cos(5πx/2) ; (d) simulation result at t = 1.2 for the initial condition given in (c).

not equal to the wavenumber fixed by the initial condition. Three plumes, at most, form (figures 5.11 to 5.14). Nonetheless, more than three plumes can form, but they merge later (figure 5.13). Plume merging was previously observed in [142]. The plume merging mechanism is analogous to that of the Rayleigh-Bénard flow. Spacing between plumes seems to be intrinsic to the system and the position can only be predicted in very specific cases such as the examples given above.

The randomly perturbed bacterium density is defined as follows [142]:

n(x) = 0.8 + 0.2ε(x), (5.22)

with ε being a random number with a uniform probability distribution over [0, 1]. Simu-lation results with the randomly perturbed initial condition are given in figure 5.15. The plume-to-plume spacing is not fixed but the number of plumes in the solution remains the same as that with the solution obtained subject to the deterministic initial condition illustrated in figures 5.11 to 5.14.

In figures 5.7–5.15, plumes form at the border of the domain. These border plumes seem to be caused by the no-slip boundary condition on the velocity depending on the arrangement of the convection cells. Bacteria first agglomerate at the surface near the wall. Because the fluid velocity is equal to zero at the wall boundary, a large amount of bacteria remains close to the wall. At a location away from the wall, the velocity is non-zero, thus bacteria descend into the fluid and form a plume at the border.

The location of the plumes may only be predicted in very simple tests for which the wavelengths of initial conditions are lower than the wavelength of the process. Despite the difficulty to predict the site of plume formation, the system presents a dominant wave-length of the fingering instability. The wavewave-length and wavenumber are redefined such as those given in [145]. The wavelength is the length of the domain divided by the num-ber of plumes and the wavenumnum-ber 2π divided by the wavelength. The wavelength and

5.4. NUMERICAL RESULTS AND DISCUSSION

Figure 5.11: Numerical results for n with respect to the deterministic initial conditions.

At the initial time, bacterium density is higher in the upper layer. (a) initial condition is n₀ = [^{1, y}≥0.5+0.1 cos(3πx/2)

0.5, y<0.5+0.1 cos(3πx/2); (b) simulation result at t = 1.2 for the initial condition given in (a) ; (c) initial condition is n₀ = [^{1, y}≥0.5−0.1 cos(3πx/2)

0.5, y<0.5−0.1 cos(3πx/2); (d) simulation result at t = 1.2 for the initial condition given in (c).

-2 0 2

Figure 5.12: Numerical results for n with respect to the deterministic initial conditions.

At the initial time, bacterium density is higher in the lower layer. (a) initial condition is n₀ = [0.5, y≥0.5+0.1 cos(3πx/2)

1, y<0.5+0.1 cos(3πx/2) ; (b) simulation result at t = 1.2 for the initial condition given in (a) ; (c) initial condition is n₀ = [^{0.5, y}≥0.5−0.1 cos(3πx/2)

1, y<0.5−0.1 cos(3πx/2) ; (d) simulation result at t = 1.2 for the initial condition given in (c).

5.4. NUMERICAL RESULTS AND DISCUSSION

Figure 5.13: Numerical results for n with respect to the deterministic initial conditions.

At the initial time, bacterium density is higher in the upper layer. (a) initial condition is n₀ = [^{1, y}≥0.5+0.1 cos(2πx)

0.5, y<0.5+0.1 cos(2πx); (b) simulation result at t = 1.2 for the initial condition given in (a) ; (c) initial condition is n₀ = [^{1, y}≥0.5−0.1 cos(2πx)

0.5, y<0.5−0.1 cos(2πx); (d) simulation result at t = 1.2 for the initial condition given in (c).

-2 0 2

Figure 5.14: Numerical results for n with respect to the deterministic initial conditions.

At the initial time, bacterium density is higher in the lower layer. (a) initial condition is n₀ = [0.5, y≥0.5+0.1 cos(2πx)

1, y<0.5+0.1 cos(2πx) ; (b) simulation result at t = 1.2 for the initial condition given in (a) ; (c) initial condition is n₀ = [^{0.5, y}≥0.5−0.1 cos(2πx)

1, y<0.5−0.1 cos(2πx) ; (d) simulation result at t = 1.2 for the initial condition given in (c).

5.4. NUMERICAL RESULTS AND DISCUSSION

Figure 5.15: Initial random condition given in (5.22) (left) and the corresponding numer-ical results for n at t = 1.2 (right).

number of plumes wavelength wavenumber

2 2 3.14

2 2 3.14

2 2 3.14

2 2 3.14

Table 5.4: The predicted number of plumes, wavenumber and wavelength for ` = 2. Each row corresponds to a different simulation result subject to the randomly perturbed initial condition given in (5.22).

wavenumber are similar for all simulation tests whatever the length of the domain is (ta-bles 5.4 to 5.7).

5.4.3.2 Growth of plumes

We now arbitrarily define the plumes by the isoline n = 1. This choice allows us to track the stack layer and the plumes being formed. The layer where n > 1 represents the layer with a high bacterium density from which plumes form and descend in the fluid.

On the other hand, the layer with n < 1 corresponds to the depletion layer, from which bacteria have migrated to the stack layer.

We define the growth rate of the plume amplitudes by G = (A_t− A_t−1)/∆t. The plume amplitude A_t is computed by measuring the distance from the surface to the tip of the plume at time t (table 5.8) and ∆t is the time increment. Data obtained can be interpolated by the exponential function g(t) = exp{αt + β} as the amplitudes of the descending plumes grow exponentially (figure 5.16). The exponential growth is well un-derstood in the conventional Rayleigh-Taylor convection. Subsequent to the exponential

5.4. NUMERICAL RESULTS AND DISCUSSION

number of plumes wavelength wavenumber

3 2 3.14

3 2 3.14

4 1.5 4.19

3 2 3.14

Table 5.5: The predicted number of plumes, wavenumber and wavelength for ` = 3. Each simulation result is subject to the random initial condition given in (5.22).

number of plumes wavelength wavenumber

5 1.6 3.93

5 1.6 3.93

5 1.6 3.93

5 1.6 3.93

Table 5.6: The predicted number of plumes, wavenumber and wavelength for ` = 4. Each simulation result is subject to the random initial condition given in (5.22).

Table 5.7: The predicted number of plumes, wavenumber and wavelength for ` = 5. Each simulation result is subject to the random initial condition given in (5.22).

number of plumes wavelength wavenumber

5 2 3.14

6 1.67 3.77

5 2 3.14

5 2 3.14

5.4. NUMERICAL RESULTS AND DISCUSSION

Figure 5.16: Growth rate G of plume amplitude (dots) corresponding to the data in table 5.8. The growth rate is interpolated by the exponential function g(t) = e^αt+β (line).

growth phase, the growth rate of amplitude decreases, as plumes get closer to the bottom of the container.