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Two-Fluid Model

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Two-Fluid Model Stability, Simulation and Chaos

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Abstract

The stability analysis of the four-equation incompressible TFM for horizontal stratified flow is presented. Both the characteristic equation and the dispersion relation are obtained. A von Neumann analysis of the semi-implicit First-Order Upwind (FOU) scheme, currently used in industrial reactor safety codes, is performed and compared to the dispersion analysis. A second-order finite difference scheme is then implemented and verified with the modified water faucet problem of Ransom (Course A—Numerical modeling of two-phase flows. Technical Report EGG-EAST-8546, EG&G Idaho, 1989). The difference with the previous chapter is that the flux is not fixed as a spatiotemporal constant. Removing the fixed-flux condition of the previous chapter brings in the implicit pressure Poisson technique which makes the algorithm significantly more elaborate.

This chapter also presents the nonlinear simulations of the four-equation TFM for the experiment of Thorpe (Journal of Fluid Mechanics, 39, 25–48, 1969) that is used to validate the model. The major outcome of this chapter is the implementation of surface tension and of the dissipative Reynolds stresses together into a validated well-posed TFM that has the capability to simulate local instabilities. These mechanisms are neglected in industrial 1D TFM codes but they have a strong impact on the numerical simulations of KH unstable flows for two reasons: they provide physical mechanisms to stabilize the TFM both linearly and nonlinearly and they allow the numerical model to converge in a statistical sense.

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Notes

  1. 1.

    Lax’s Equivalence Theorem : Given a properly posed initial value problem and a finite-difference approximation to it that satisfies the consistency condition, stability is the necessary and sufficient condition for convergence.

  2. 2.

    While this is a rapid ramping up of the axial force, it is a smooth transition. Previous simulations that did not include this tilting essentially impose a step change in the gravity vector at time zero.

  3. 3.

    It should be pointed out that the reported time of onset of instability (Thorpe 1969) had much smaller error bars. However, it should also include “half the time taken to tilt the tube” which was reported to be “usually about 0.25 s.” Therefore, this value is used as the uncertainty.

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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). Two-Fluid Model. In: Two-Fluid Model Stability, Simulation and Chaos. Springer, Cham. https://doi.org/10.1007/978-3-319-44968-5_3

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  • DOI: https://doi.org/10.1007/978-3-319-44968-5_3

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