Well-posedness and convergence of cfd two-fluid model for bubbly flows

The current research is focused on developing a well-posed multidimensional CFD two-fluid model (TFM) for bubbly flows. Two-phase flows exhibit a wide range of local flow instabilities such as Kelvin-Helmholtz, Rayleigh-Taylor, plume and jet instabilities. They arise due to the density difference and/or the relative velocity between the two phases. A physically correct TFM is essential to model these instabilities. However, this is not the case with the TFMs in numerical codes, which can be shown to have complex eigenvalues due to incompleteness and hence are ill-posed as initial value problems. A common approach to regularize an incomplete TFM is to add artificial physics or numerically by using a coarse grid or first order methods. However, it eliminates the local physical instabilities along with the undesired high frequency oscillations resulting from the ill-posedness. Thus, the TFM loses the capability to predict the inherent local dynamics of the two-phase flow. The alternative approach followed in the current study is to introduce appropriate physical mechanisms that make the TFM well-posed. ^ First a well-posed 1-D TFM for vertical bubbly flows is analyzed with characteristics, and dispersion analysis. When an incomplete TFM is used, it results in high frequency oscillations in the solution. It is demonstrated through the travelling void wave problem that, by adding the missing short wavelength physics to the numerical TFM, this can be removed by making the model well-posed. To extend the limit of well-posedness beyond the well-known TFM of Pauchon and Banerjee [1], the mechanism of collision is considered, and it is shown by characteristics analysis that the TFM then becomes well-posed for all void fractions of practical interest. The aforementioned ideas are then extended to CFD TFM. The travelling void wave problem is again used to demonstrate that by adding appropriate physics, the problem of ill-posedness is resolved. ^ Furthermore, issues pertaining to the presence of the wall boundaries need to be addressed in a CFD TFM. A near-wall modeling technique is proposed which takes into account the turbulent boundary conditions and void fraction distribution in the vicinity of the wall. An important consequence of using the proposed technique is that the need of wall force model, which is questionable when applied to air-water turbulent bubbly flows, is eliminated. Also the bubbly TFM near the wall becomes convergent. ^ Finally, the well-posed CFD TFM developed in the present study is checked for grid convergence. Previous researchers have advocated the idea of fixing the minimum grid size based on bubble diameter. This has restricted a thorough verification exercise in the past. It is shown that the grid size criterion can be removed if the model is made well-posed, which also makes sense because a continuum model should be independent of grid size. It is observed that the solution from the coarse grid simulations is a limit cycle whereas upon grid refinement, the solution becomes chaotic which is characteristic of turbulent bubbly two-phase flows. Therefore the grid size restriction may have an unwanted consequence. FFT spectra and time averaged void fraction profiles are used to assess grid convergence since the solutions are chaotic. The energy spectra indicate the Kolmogorov -5/3 scaling commonly used to describe turbulent flows