# Stability And Objectivity Of A Bubbly And Slug Flow Two-Fluid Model With Wake Entrainment

The current study is aimed at developing a well-posed and objective, i.e., frame invariant, Eulerian one-dimensional (1D) Two-Fluid Model (TFM) to predict flow regime transition from dispersed to clustered bubbly and slug flow for vertical adiabatic two-phase flows. Two-phase flows in general are characterized by local material wave or void fraction wave instabilities and flow regime transitions are one of the important consequences of these instabilities. The physical mechanism of wake entrainment for clustering of dispersed bubbles is proposed, leading to formation of bubble clusters and Taylor bubbles. The focus of the work is on simulation of the local interfacial structures for bubble clusters and Taylor bubbles, using a well-posed, unstable and non-linearly bounded 1D Shallow Water TFM.

The first part of the current study investigates the dynamic behavior of the well posed 1D mechanistic TFM obtained from the averaging approach of Ishii [1], due to wake entrainment instability. For this, a 1D Shallow Water TFM derived from the 1D mechanistic TFM is used, which retains the same dynamic behavior as that of the latter at short wavelengths and the required wake entrainment force is derived mechanistically. Three stability approaches are followed to study the dynamic behavior of the 1D Shallow Water TFM: characteristics, dispersion analysis, and nonlinear numerical simulations. An in-house code is used for the 1D numerical simulations of the growth of void fraction waves due to wake entrainment. The simulation results are validated with the experimental data of Cheng and Azzopardi [2] and Song et al. [3] To conclude the first part, the 1D results of the two-equation Shallow Water TFM are carried over to the complete four-equation TFM for quasi 1D simulations using the commercial CFD code of ANSYS Fluent.

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of this procedure is that the derived equations of motion are objective (Geurst [4]), in particular the added mass terms.

Thus, the second part of the current study focuses on deriving an objective, well-posed and unstable 1D TFM as well as developing a constitutive model for the wake entrainment effect using the variational method. Additional momentum transfer terms present in both the liquid phase and gas phase momentum equations, which render the variational TFM objective, are discussed. The variational method is then used to derive the 1D Shallow Water TFM using the fixed flux assumption. The conservative interfacial momentum transfer terms require formulation of the inertial coupling between the phases. Potential flow theory is first used to derive the inertial coupling coefficient for a single bubble and then for a pair of bubbles to consider interaction between the two bubbles. Then, a lumped parameter model is used to derive the inertial coupling coefficient for the wake entrainment effect. A local drag coefficient is obtained for the non-conservative interfacial drag force from the experimental data using kinematic approximation, i.e., force balance between drag and gravity. The linear and non-linear stability analyses are used to address the stability of the 1D variational Shallow Water TFM. The presence of appropriate short-wave physics makes the 1D Shallow Water TFM hyperbolic well-posed and kinematically unstable. Finally, numerical simulations are performed to demonstrate the development of void fraction waves due wake entrainment. The growth of void fraction waves is non-linearly bounded, i.e., Lyapunov stable. The simulation results are compared with the experimental data to validate the propagation properties of void fraction waves for bubble clusters and Taylor bubbles. This work illustrates the short-wave two-phase flow simulation capability of the TFM for the bubbly to slug flow regime transition.