Effect of viscosity on a well-posed one-dimensional two-fluid model for wavy two-phase flow beyond the Kelvin-Helmholtz instability: limit cycles and chaos

摘要

The one-dimensional (1D) Euler formulation of the Two-Fluid Model (TFM) becomes ill-posed by the Kelvin-Helmholtz (KH) instability which is not present in the 1D Euler equations of single phase flow, though it is present in the multidimensional equations where the single phase flow problem is also ill-posed. It is well known that the 1D TFM may be rendered well posed by including appropriate short wave physics, i.e., by making it more complete. In the case of near horizontal stratified flow, surface tension is the appropriate force which restores the KH force at short wavelengths. A linear stability analysis shows that the model is well-posed, but that material waves grow at a finite rate beyond a critical wavelength. Upon further nonlinear development the wave fronts become steep in a similar fashion to Shallow Water Theory waves and the 1D TFM is bounded by viscosity due to dissipation in shock like structures. A 1D TFM numerical simulation of near horizontal stratified two-phase flow is performed where the TFM, including surface tension and viscous stresses, is simplified to a two-equation model using the fixed flux approximation. As the angle of inclination of the channel is increased, i.e., increasing the body force to drive the flow, the flow becomes KH unstable and waves appear that develop steep fronts. It is shown that these waves grow until they reach a limit cycle due to viscous dissipation. Upon further inclination of the channel, chaos is observed. The appearance of chaos in a 1D TFM implies a nonlinear process equivalent to the Kolmogorov's turbulent cascade that transfers energy intermittently from long wavelengths where energy is produced to short wavelengths where energy is dissipated by viscosity, so that an averaged energy equilibrium in frequency space is attained. Boundedness is a necessary condition for a chaotic TFM, i.e., a nonlinearly well behaved model, but the more restrictive hyperbolic condition of the Euler formulation of the 1D TFM, i.e., zero wave growth at all wavelengths, is not necessary. In other words, it is not necessary to remove the KH instability to have a well behaved TFM.