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Fixed-Flux Model

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Two-Fluid Model Stability, Simulation and Chaos

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Abstract

In this chapter, the compressible, one-dimensional Two-Fluid Model (TFM) for stratified two-phase flow is introduced first. For conditions of practical interest, a characteristic analysis demonstrates that the acoustic roots are always real and that the origin of ill-posedness (or well-posedness) of the model lies in the material roots. Therefore, compressibility is neglected and an incompressible model is used to derive a reduced two-equation model by applying the fixed-flux approximation, which is the key to study the local material waves in isolation, and is referred to as the Fixed-Flux Model (FFM). It is demonstrated that the FFM reduces to the well-known Shallow Water Theory (SWT) under some conditions. While it is not possible to cover all the possible local instabilities for stratified flow with the FFM, we are interested in two significant cases: SWT and Kelvin–Helmholtz (KH) instabilities, otherwise known as kinematic and dynamic instabilities.

Furthermore, the local linear material stability analysis allows us to address the question of the ill-posed TFM, which is caused by the KH instability which also precisely differentiates the FFM from SWT. The dispersion analysis of the FFM shows the well known results that the hydrostatic force makes the TFM stable up to the KH instability and surface tension makes the unstable model well-posed beyond it (Ramshaw and Trapp, Nuclear Science and Engineering, 66, 93–102, 1978). However, the well-posed FFM is still Lyapunov unstable and a bounding nonlinear viscous mechanism will be analyzed in Chap. 4, in terms of SWT material shocks (Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974). Finally, the numerical stability and convergence of a few finite-difference schemes typically used to simulate TFM problems is also addressed with the von Neumann stability analysis of the FFM.

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Authors and Affiliations

  1. School of Nuclear Engineering, Purdue University, West Lafayette, IN, USA

    Martín López de Bertodano, William Fullmer & Victor H. Ransom

  2. School of Exact Sciences, University of Central Buenos Aires & CONICET, National Atomic Energy Commission, Buenos Aires, Argentina

    Alejandro Clausse

Authors
  1. Martín López de Bertodano
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  2. William Fullmer
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  3. Alejandro Clausse
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  4. Victor H. Ransom
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de Bertodano, M.L., Fullmer, W., Clausse, A., Ransom, V.H. (2017). Fixed-Flux Model. In: Two-Fluid Model Stability, Simulation and Chaos. Springer, Cham. https://doi.org/10.1007/978-3-319-44968-5_2

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  • DOI: https://doi.org/10.1007/978-3-319-44968-5_2

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