Linear stability analysis of a two-fluid model for air-water bubble columns

Linear stability analysis is performed for the two-dimensional, two-fluid model for gas-liquid flow applied in our previous computational study of bubble columns [Monahan, S.M., Vitankar, V.S., Fox, R.O., 2005. CFD predictions for flow-regime transitions in bubble columns. A.I.Ch.E. Journal 51, 1897-1923]. The growth rate and the velocity of propagation for a small-amplitude disturbance wave are shown to be highly dependent on the wave number, the direction of propagation, and the two-fluid model parameters. Two types of vertical instabilities are identified: one corresponding to the classical analysis of Jackson [1963. The mechanics of fluidized beds. I: the stability of the state of uniform fluidization. Transactions of the Institution of Chemical Engineers 41, 13-21] for the one-dimensional model, and the other due to a second pair of roots to the characteristic equation of the linearized two-dimensional model. Numerical simulations keeping one type or the other of the roots stable (or unstable) show distinctly different dynamics and suggest that large-scale instabilities seen experimentally may be associated with the second type of instability. The latter leads to instability in the horizontal velocities and is associated with a positive lift coefficient in flows without mean shear in the presence of isotropic bubble-bubble interactions (i.e., “bubble pressure”). This instability is thus different than previously reported instabilities due to negative lift or cooperative/hindered rise.