Stability and convergence of computational eulerian two-fluid model for a bubble plume

Abstract

The Eulerian two-fluid model (TFM) of Ishii (1975) is used to analyze the dynamics of an air-water bubble plume. The focus is on the effect of the linear stability, in particular the ill-posed condition, on the nonlinear stability of the TFM. It is well-known that the TFM for bubbly flows is ill-posed as an initial value problem in the absence of short wavelength physics for non-zero slip velocities. It is also known that the 1-D TFM can be made conditionally well-posed (for void fraction <26%) by adding momentum transfer due to interfacial pressure difference and virtual mass. However, there is still the possibility of a TFM being ill-posed in regions of higher void fractions (void fraction >26%). Physically, as the void fraction increases, the bubbles tend to undergo collisions, and the momentum transfer due to this mechanism may become significant. In the current study a bubble collision model adapted from the work of Alajbegovic et al. (1999) is used for CFD TFM calculations for bubbly flows using an LES approach. It is shown by linear stability analysis and non-linear simulations that the collision term makes the TFM unconditionally well-posed and stable in a non-linear sense.Secondly, computational grid convergence tests are performed with the well-posed CFD TFM. It is observed that the coarse grid solution exhibits an unphysical limit cycle behavior that is inconsistent with turbulence, but as the mesh is refined the solution becomes chaotic. CFD TFM simulations commonly employ a grid restriction to avoid ill-posed behavior. However, this is unnecessary with a well-posed Eulerian TFM derived from first principles using the continuum assumption. Once the restriction is removed by adding appropriate short-wavelength physics, i.e., interfacial pressure difference and collision mechanism, convergence may be approached in a statistical sense consistent with a turbulent CFD model.