Fixed-Flux Model Chaos
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.
KeywordsOne-dimensional Two-fluid model Fixed-flux Stratified Two-phase flow Kelvin–Helmholtz Characteristics Dispersion Nonlinear Kreiss–Ystrom Fixed flux Chaos Lyapunov Fractal dimension Convergence
Video 1.1 (MP4 5,801 kb)
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