Fixed-Flux Model Chaos

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


The material wave growth of a well-posed Two-Fluid Model (TFM) beyond the Kelvin–Helmholtz (KH) instability has been approached in the previous chapter. The question that remains is what happens to the nonlinear evolution of the waves after their initial growth. Whitham (Linear and Nonlinear Waves, Wiley, New York, 1974) obtained a set of nonlinear solutions for Shallow Water Theory (SWT) consisting of shocks and expansion waves and identified the kinematic SWT instability. In Chap.  2 it was shown that the TFM may be approximated with the Fixed-Flux Model (FFM), which is equivalent to SWT up to the KH instability. Beyond that the FFM is unique and its nonlinear behavior is now addressed.

The chapter begins with the Kreiss–Yström equations (KY). It has already been shown in Sect.  2.6.3 that the linear stability of the KY equations is remarkably similar to the FFM with artificial viscosity. We apply some of the standard tools of system dynamics and chaos theory to the KY equations to obtain the largest Lyapunov Exponent and the fractal dimension. Fixed point, limit cycles and strange attractors are encountered on the route to chaos. We then proceed to the more difficult case of the FFM without artificial viscosity. A well-posed FFM is validated with a new experiment that is similar to Thorpe’s (Journal of Fluid Mechanics, 39, 25–48, 1969) but focuses on the chaotic behavior of the waves past their initial growth, which Thorpe didn’t report. Then, long-term simulations of the FFM with periodic boundary conditions are performed to obtain the largest positive Lyapunov exponent and the fractal dimension. The Largest Lyapunov exponent turns out one order of magnitude smaller than the linear counterpart and eventually diverging trajectories become bounded by a strange attractor, i.e., Lyapunov stability.

Ultimately, the FFM chaotic behavior newly encountered differs significantly from the well-known linear theory. Therefore, it is important to distinguish between linear stability, which only determines whether the unstable TFM blows up instantaneously (ill-posed) or exponentially (well-posed) and which is valid for a very short interval of time, and nonlinear stability, which determines whether the problem is bounded in the long term.


One-dimensional Two-fluid model Fixed-flux Stratified Two-phase flow Kelvin–Helmholtz Characteristics Dispersion Nonlinear Kreiss–Ystrom Fixed flux Chaos Lyapunov Fractal dimension Convergence 

Supplementary material

Video 1.1 (MP4 5,801 kb)


  1. Abarbanel, H. D. I. (1996). Analysis of observed chaotic data. New York: Springer.CrossRefzbMATHGoogle Scholar
  2. Andritsos, N., & Hanratty, T. J. (1987). Interfacial instabilities for horizontal gas–liquid flows in pipelines. International Journal of Multiphase Flow, 13, 583–603.CrossRefGoogle Scholar
  3. Arai, M. (1980). Characteristics and stability analyses for two-phase flow equation systems with viscous terms. Nuclear Science and Engineering, 74, 77–83.Google Scholar
  4. Barmak, I., Gelfgat, A., Ullmann, A., Brauner, N., & Vitoshkin, H. (2016). Stability of stratified two-phase flows in horizontal channels. Physics of Fluids, 28, 044101.CrossRefGoogle Scholar
  5. Barnea, D., & Taitel, Y. (1994). Interfacial and structural stability of separated flow. International Journal of Multiphase Flow, 20, 387–414.CrossRefzbMATHGoogle Scholar
  6. Lopez de Bertodano, M. A., Fullmer, W.D., & Clausse, A. (2016). One-dimensional two-fluid model for wavy flow beyond the Kelvin-Helmholtz instability: Limit cycles and chaos. Nuclear Engineering and Design. Retrieved from, Scholar
  7. Fullmer, W. D., Lopez de Bertodano, M. A., & Ransom, V. H. (2011). The Kelvin–Helmholtz instability: Comparisons of one- and two-dimensional simulations. In 14th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-14), Toronto.Google Scholar
  8. Fullmer, W. D., Lopez de Bertodano, M. A., & Clausse, A. (2014). Analysis of stability, verification and chaos with the Kreiss–Yström equations. Applied Mathematics and Computation, 248, 28–46.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Grassberger, P., & Procaccia, I. (1983). Characterization of strange attractors. Physical Review Letters, 50(5), 346–349.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Gidaspow, D. (1974) Round Table Discussion (RT-1-2): Modeling of Two-Phase Flow, Proc. 5th Int. Heat Transfer Conf., Tokyo, Japan, September 3–7.Google Scholar
  11. Hyman, L. M., & Nicolaenko, B. (1986). The Kuramoto-Sivashinsky equation: A bridge between PDE’S and dynamical systems. Physica D Nonlinear Phenomena, 18, 113–126.MathSciNetCrossRefzbMATHGoogle Scholar
  12. Keyfitz, B. L., Sever, M., & Zhang, F. (2004). Viscous singular shock structure for a non-hyperbolic Two Fluid model. Nonlinearity, 17, 1731–1747.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Kocamustafaogullari, G. (1985). Two-fluid modeling in analyzing the interfacial stability of liquid film flows. International Journal of Multiphase Flows, 11, 63–89.CrossRefzbMATHGoogle Scholar
  14. Kreiss, H.-O., & Yström, J. (2002). Parabolic problems which are ill-posed in the zero dissipation limit. Mathematical and Computer Modelling, 35, 1271–1295.MathSciNetCrossRefzbMATHGoogle Scholar
  15. Picchi, D., Correra, S., & Poesio, P. (2014). Flow pattern transition, pressure gradient, hold-up predictions in gas/non-Newtonian power-law fluid stratified flow. International Journal of Multiphase Flow, 63, 105–115.MathSciNetCrossRefGoogle Scholar
  16. Picchi, D., & Poesio, P. (2016). A unified model to predict flow pattern transitions in horizontal and slightly inclined two-phase gas/shear-thinning fluid pipe flows. International Journal of Multiphase Flow, 84, 279–291.MathSciNetCrossRefGoogle Scholar
  17. Ramshaw, J. D., Trapp, J. A. (1978). Characteristics, stability and short wavelength phenomena in two-phase flow equation systems. Nuclear Science and Engineering, 66, 93–102.Google Scholar
  18. Richardson, L. F. (1926). Atmospheric Diffusion Shown on a Distance-Neighbour Graph. Proceedings of the Royal Society of London. Series A, 110, 709–737.CrossRefGoogle Scholar
  19. Sprott, J. C. (2003). Chaos and time series analysis. Oxford, UK: Oxford University Press.zbMATHGoogle Scholar
  20. Thorpe, J. A. (1969). Experiments on the instability of stratified shear flow: Immiscible fluids. Journal of Fluid Mechanics, 39, 25–48.CrossRefGoogle Scholar
  21. Vaidheeswaran, A., Fullmer, W. D., Chetty, K., Marino, R. G., & Lopez de Bertodano, M. (2016). Stabiility analysis of chaotic wavy stratified fluid-fluid flow with the 1D fixed-flux two-fluid model. In Proceedings of ASME 2016 HT/FEDSM/ICNMM, Washington, DC, USA, July 10–14.Google Scholar
  22. Whitham, G. B. (1974). Linear and nonlinear waves. New York: Wiley.zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • Martín López de Bertodano
    • 1
  • William Fullmer
    • 1
  • Alejandro Clausse
    • 2
  • Victor H. Ransom
    • 1
  1. 1.School of Nuclear EngineeringPurdue UniversityWest LafayetteUSA
  2. 2.School of Exact SciencesUniversity of Central Buenos Aires & CONICET, National Atomic Energy CommissionBuenos AiresArgentina

Personalised recommendations