By Martín López de Bertodano, William Fullmer, Alejandro Clausse, Victor H. Ransom

This ebook addresses the linear and nonlinear two-phase balance of the one-dimensional Two-Fluid version (TFM) fabric waves and the numerical tools used to unravel it. The TFM fluid dynamic balance is an issue that continues to be open considering that its inception greater than 40 years in the past. the trouble is bold since it consists of the mixed demanding situations of two-phase topological constitution and turbulence, either nonlinear phenomena. the only dimensional strategy allows the separation of the previous from the latter.The authors first examine the kinematic and Kelvin-Helmholtz instabilities with the simplified one-dimensional Fixed-Flux version (FFM). They then study the density wave instability with the well known Drift-Flux version. They show that the Fixed-Flux and Drift-Flux assumptions are complementary TFM simplifications that handle two-phase neighborhood and international linear instabilities individually. moreover, they show with a well-posed FFM and a DFM situations of nonlinear two-phase habit which are chaotic and Lyapunov sturdy. at the sensible part, additionally they verify the regularization of an ill-posed one-dimensional TFM business code. in addition, the one-dimensional balance analyses are utilized to procure well-posed CFD TFMs which are both solid (RANS) or Lyapunov solid (URANS), with the point of interest on numerical convergence.

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3). Of course the analysis of the compressible 1D TFM is much more extensive and only the simplest possible case is considered in this section. For a thorough up-todate analysis of the gas dynamics aspects of the TFM the reader is referred to Stadtke (2006). 2 Characteristics The characteristics analysis follows those of Gidaspow (1974) and Stadtke (2006). Expanding the derivatives in Eqs. 5), the original system of partial differential equations can be cast into vector equation form as: d d A ϕ þ B ϕ ¼ 0; dt dx ð2:6Þ where ϕ ¼ ½α1 ; u1 ; u2 ; pŠT is the vector of the independent variables.

However, the last case is ill-posed elliptic, which leads to difficulties (Barnea and Taitel 1993) and will be analyzed next with a dispersion analysis. 2. 1 is reproduced here, only now F ¼ 0, but the effects of viscosity and surface tension are included. The first step is to linearize the two-equation system using ϕ ¼ ϕ0 þ ϕ0 and keeping only first-order terms with respect to ϕ0 . Then a Fourier solution is applied ϕ0 ¼ ϕ^0 eiðkxÀωtÞ to the linearized equations, where k and ω are the wavenumber and the angular frequency.

The next and more important question is whether a well-posed but unstable TFM is sufficient. Well-posedness, as defined in this section, is a linear stability property. Drew and Passman (1999) argued that a well-posed viscous TFM is still practically ill-posed because the wave growth rate remains high and the model has exponential blow up. Indeed the TFM would be unacceptable if the surface waves kept on growing unboundedly. But linear stability analysis turns out to be insufficient because unstable surface waves in their very nature either break or peak, and both these phenomena are nonlinear.

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