Immersed Boundary Methods: Current Status and Future Research Directions Amsterdam, 15-17 June 2009
NUMERICAL STUDY ON THE HYDRODYNAMICAL FORCES
ACTING IN A SWARM OF BUOYANCY DRIVEN GAS BUBBLES
J. J. J. Gillissen & H. E. A. van den Akker
Department of Multi-Scale Physics
J.M. Burgers Centre for Fluid Mechanics, Delft University of Technology,
Prins Bernhardlaan 6, 2628 BW Delft, Netherlands; e-mail: j.j.j.gillissen@tudelft.nl
Abstract.
Bubble columns are encountered in a large variety of industrial processes, especially as chemical reactors. Detailed process design requires a good understanding of the hydrodynamics which can be obtained using numerical simulation. As in almost all industrial relevant flows, the simulation of the full Navier Stokes equations is a very demanding, if not impossible task. Therefore, in bubble column engineering, the usual practice is to solve the Reynolds Averaged Navier Stokes equations, in which only mean quantities are explicitly computed, while the effect of the fluctuations are modelled using ad-hoc modelling assumptions.
One important modelling aspect is the formulation of the drag force, which accounts for the momentum transfer between the continuous liquid-phase and the dispersed gas-phase. At small gas fractions (α < 1%) the drag force is almost unaffected by bubble interactions and can accurately be predicted based on theory and experiment. At large α however, neighbouring bubbles effect each other's motion by imposing geometrical restrictions and by modifying the bubble wake structure.
How these physics effect the Reynolds averaged drag force is unclear since the important parameters, such as bubble arrangement and fluid velocity field, are difficult to measure experimentally at large α .
Therefore we numerically investigate an idealized swarms of bubbles, which rise due to gravity in a periodic domain. To reduce the complexity, we consider the limits of (i) negligible bubble coalesence and deformation (spherical bubble shape). With these assumptions the problem is fully characterized by the gas fraction α and the Reynolds number Re=UD/ν, based on the kinematic viscosity of the liquid phase ν, the bubble diameter D and the averaged bubble rise velocity U, which results from the balance between drag force and buoyancy force. In the present work we consider α = 0.02, 0.08, and 0.32 and Re = 100, which for air bubbles in water reasonably conforms to the undeformability assumption [1].
We use a one-field description. The incompressible and isothermal Navier Stokes equations are discretized using a Lattice Boltzmann method [2] on a cubic grid, consisting of 2563
grid-points. Bubble diameter is spanned by 28 grid-points. The no-penetration condition on the bubble surface is enforced using an immersed boundary method. Bubble mass is neglected. Bubble acceleration is computed such that the sum of the corresponding immersed boundary forces exactly balances the buoyancy force. Coalesence is excluded by imposing repulsion forces, based on lubrication theory.
Besides, focussing on the modification of the drag force due to wake destruction, we also outline the details of the method and analyse the accuracy for the case of the rise of a single bubble. Furthermore we explain some of the advantages of the Lattice Boltzmann scheme for immersed boundary problems in fluid mechanics.
Fig. 1. Gray scale shows the dissipation of the kinetic energy in the continuous phase in a plane parallel to the gravitational acceleration at Re = 100 and α =0.32. This computation constitutes 512 equally-sized spherical bubbles. The white, circular areas represent bubbles crossing the plane.
