Dynamical flow characterization of transitional and chaotic regimes in converging–diverging channels

Abstract

Numerical investigation of laminar, transitional and chaotic flows in converging–diverging channels are performed by direct numerical simulations in the Reynolds number range 10 < Re < 850. The temporal flow evolution and the onset of turbulence are investigated by combining classical fluid dynamics representations with dynamical system flow characterizations. Modern dynamical system techniques such as timedelay reconstructions of pseudophase spaces, autocorrelation functions, fractal dimensions and Eulerian Lyapunov exponents are used for the dynamical flow characterization of laminar, transitional and chaotic flow regimes. As a consequence of these flow characterizations, it is verified that the transitional flow evolves through intermediate states of periodicity, two-frequency quasi-periodicity, frequency-locking periodicity, and multiple-frequency quasi-periodicity before reaching a non-periodic unpredictable behaviour corresponding to low-dimensional deterministic chaos.Qualitative and quantitative differences in Eulerian dynamical flow parameters are identified to determine the predictability of transitional flows and to characterize chaotic, weak turbulent flows in converging–diverging channels. Autocorrelation functions, pseudophase space representations and Poincaré maps are used for the qualitative identification of chaotic flows, assertion of their unpredictable nature, and recognition of the topological structure of the attractors for different flow regimes. The predictability of transitional flows is determined by analysing the autocorrelation functions and by representing their attractors in the reconstructed pseudophase spaces. The transitional flow behaviour is examined by the geometric visualization of the evolution of the attractors and Poincaré maps until the appearance of a strange attractor at the onset of chaos. Eulerian Lyapunov exponents and fractal dimensions are quantitative parameters to establish the onset of chaos, the persistence of chaotic flow behaviour, and the long-term persistent unpredictability of chaotic Eulerian flow regimes. Lastly, three-dimensional simulations for converging–diverging channel flow are performed to determine the effect of the spanwise direction on the route of transition to chaos.