Effect of Collision Force on Well-Posedness and Stability of the Two-Fluid Model for Vertical Bubbly Flows

It is well-known that an incomplete two-fluid model (TFM) leads to imaginary roots of the characteristic polynomial, thus rendering the model ill-posed. A common approach to fix this problem has been to introduce sufficient numerical/artificial diffusion or nonphysical hyperbolizing terms to stabilize the model. The disadvantage of this approach is that the physical instabilities that can be accurately predicted by the TFM either get severely dampened or disappear entirely. The preferred alternative is to introduce appropriate physics that may stabilize the TFM at short wavelengths while preserving the physical long-wavelength instabilities. For instance, in near-horizontal stratified flows, the appropriate physical mechanism is surface tension. However, it is not apparent what such a mechanism should be in dispersed bubbly flows. Researchers in the past have demonstrated that the inclusion of the momentum transfer due to interfacial pressure along with virtual mass force makes the model conditionally well-posed up to a gas volume fraction of 26%. However, in practice, one may observe bubbly flows having gas concentrations beyond this theoretical limit. Hence, it is important to make the behavior of the TFM well posed for the entire range of gas volume fractions that is physically permissible. In this paper, the often-neglected phenomenon of bubble collisions is considered. The colliding bubbles generate a dispersed-phase pressure that is resistive to increased compaction. The inclusion of bubble pressure in the TFM renders the model well-posed up to the maximum packing limit. Furthermore, it is also shown that the collision force is necessary to predict the wave propagation velocities for bubbly flows over the entire range of void fractions observed in reality. Comparisons are made with the data, and a reasonable agreement is seen. Finally, it is demonstrated with computational fluid dynamics calculations that the addition of appropriate physical mechanisms (i.e., interfacial pressure and collision) makes the multidimensional TFM well-posed.