Evolution of energy in flow driven by rising bubbles

We investigate by direct numerical simulations the flow that rising bubbles cause in an originally quiescent fluid. We employ the Eulerian-Lagrangian method with two-way coupling and periodic boundary conditions. In order to be able to treat up to 288000 bubbles, the following approximations and simplifications had to be introduced, as done before, e.g., by Climent and Magnaudet, Phys. Rev. Lett. 82, 4827 (1999). (i) The bubbles were treated as point particles, thus (ii) disregarding the near-field interactions among them, and (iii) effective force models for the lift and the drag forces were used. In particular, the lift coefficient was assumed to be 1/2, independent of the bubble Reynolds number and the local flow field. The results suggest that large-scale motions are generated, owing to an inverse energy cascade from the small to the large scales. However, as the Taylor-Reynolds number is only in the range of 1, the corresponding scaling of the energy spectrum with an exponent of $-5/3$ cannot develop over a pronounced range. In the long term, the property of local energy transfer, characteristic of real turbulence, is lost and the input of energy equals the viscous dissipation at all scales. Due to the lack of strong vortices, the bubbles spread rather uniformly in the flow. The mechanism for uniform spreading is as follows. Rising bubbles induce a velocity field behind them that acts on the following bubbles. Owing to the shear, those bubbles experience a lift force, which makes them spread to the left or right, thus preventing the formation of vertical bubble clusters and therefore of efficient forcing. Indeed, when the lift is artificially put to zero in the simulations, the flow is forced much more efficiently and a more pronounced energy that accumulation at large scales (due to the inverse energy cascade) is achieved.

Figure 1

(a) Total fluid energy as a function of time. The straight dashed line indicates the potential flow result $E=\alpha v_T^2/4$. (b) Total fluid energy in simulation with (solid line) and without the lift (dashed line) as functions of time.

Figure 2

Behavior of the unidimensional Taylor-Reynolds number as function of time, in the $x$ (pluses), $y$ (circles), and $z$ (diamonds) directions.

Figure 3

(a) Energy spectra for the simulation that includes lift forces obtained by averaging on four different time intervals: $0<t/{\tau }_b<6\times {10}^2$ (solid line), $12\times {10}^2<t/{\tau }_b<18\times {10}^2$ (dashed line), $24\times {10}^2<t/{\tau }_b<30\times {10}^2$ (dotted line), and $100\times {10}^2<t/{\tau }_b<106\times {10}^2$ (dot-dashed line). The straight line indicates the behavior in homogeneous and isotropic turbulence. The employed time period is a compromise of, on one hand, having sufficient statistics and, on the other hand, being able to follow the time evolution (see Fig. 1). We also tried slightly larger and smaller time periods and get similar results as those shown in this figure. So our results are robust. (b) Energy spectra for the simulation without lift forces obtained by averaging on four different time intervals. For the various symbols, look at the caption of (a).

Figure 4

(a) Time average of the contribution of the bubble forcing to the energy spectrum, as defined in Eq. (8) (solid line), and of the viscous energy dissipation $D\left(k\right)=2\nu k^{2}E\left(k\right)$ (dotted line), in the simulation with the lift force. (b) Time average of the contribution of the bubble forcing to the energy spectrum (solid line) and of the viscous energy dissipation $D\left(k\right)=2\nu k^{2}E\left(k\right)$ (dotted line), in the simulation without the lift force.

Figure 5

Energy spectra, in the statistically stationary state, for case (a) (dashed line) and (d) (solid line) of Table .

Figure 6

Sketch of the action of the lift force on a bubble rising in the wake of another one: the lift tends to expel the bubble from the wake.

Figure 7

Autocorrelation function ${\rho }_{12}\left(\mathbf{R}\right)$ as a function of the distance $\left|\mathbf{R}\right|$, in the horizontal $x\text{−}y$ directions (open symbols) and in the vertical $z$ direction (filled symbols). The results indicate simulations without the lift force (diamonds) and with the lift force (squares).