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.