Breaking the symmetry of a wavy channel alters the route to chaotic flow

We numerically explore the two-dimensional, incompressible, isothermal flow through a wavy channel, with a focus on how the channel geometry affects the routes to chaos at Reynolds numbers between 150 and 1000. We find that (i) the period-doubling route arises in a symmetric channel, (ii) the Ruelle-Takens-Newhouse route arises in an asymmetric channel, and (iii) the type-II intermittency route arises in both asymmetric and semiwavy channels. We also find that the flow through the semiwavy channel evolves from a quasiperiodic torus to an unstable invariant set (chaotic saddle), before eventually settling on a period-1 limit-cycle attractor. This study reveals that laminar channel flow at elevated Reynolds numbers can exhibit a variety of nonlinear dynamics. Specifically, it highlights how breaking the symmetry of a wavy channel can not only influence the critical Reynolds number at which chaos emerges, but also diversify the types of bifurcation encountered en route to chaos itself.

Figure 1

Computational domain for the (a) symmetric, (b) asymmetric and (c) semiwavy channels used in this study. The geometrical parameters are $H/a=12/7$ and $L/a=8$ [9, 25, 26]. The flow is two-dimensional, incompressible, isothermal, and fully developed. The flow enters the channel from the left inlet and exits at the right outlet.

Figure 2

Period-doubling route to chaos in the flow through a symmetric wavy channel. The bifurcation diagram is shown in panel (a). Five different states are highlighted: (b)–(f), black, a period-1 limit cycle, (g)–(k), yellow, $T^2$ quasiperiodicity, (l)–(p), green, a period-2 limit cycle, (q)–(u), purple, a period-4 limit cycle, and (v)–(z), blue, low-dimensional chaos. Skipping (a), the columns show, from left to right, the time series of $v$, the phase portrait, the one-sided Poincaré map, the correlation sum, and the slope of the correlation sum.

Figure 3

Ruelle-Takens-Newhouse and intermittency routes to chaos in the flow through an asymmetric wavy channel. The bifurcation diagram is shown in (a). Five different states are highlighted: (b)–(f) black, a period-1 limit cycle, (g)–(k), yellow, $T^2$ quasiperiodicity, (l)–(p), blue, low-dimensional chaos, (q)–(s), red intermittency, and (t)–(x), black, a period-5 limit cycle. Skipping (a), the columns show, from left to right, the time series of $v$, the phase portrait, the one-sided Poincaré map, the correlation sum, and the slope of the correlation sum.

Figure 4

Type-II intermittency in the flow through an asymmetric wavy channel at $\mathrm{Re}=490$: (a) the time trace of $v$ showing chaotic bursts amidst a periodic background, (b) the probability distribution of the interchaos time, (c) the recurrence plot featuring kitelike structures, (d) the PSD where the periodic epochs are dominated by narrowband components at $f_1$ and $f_2$ ($f_2\simeq 2f_1$) and the chaotic bursts are dominated by broader peaks, (e) the Poincaré section featuring spiraling patterns, and (f) the average interchaos time versus the normalized bifurcation parameter.

Figure 5

Intermittency route to chaos in the flow through a semi-wavy channel. The bifurcation diagram is shown in panel (a). Three different states are highlighted: (b)–(f), black, a period-1 limit cycle, (g)–(i), red, intermittency, and (j)–(n), blue, low-dimensional chaos. Skipping (a), the columns show, from left to right, the time series of $v$, the phase portrait, the one-sided Poincaré map, the correlation sum, and the slope of the correlation sum.

Figure 6

Type-II intermittency in the flow through a semiwavy channel at $\mathrm{Re}=860$: (a) the time trace of $v$ showing chaotic bursts amidst a periodic background, (b) the probability distribution of the interchaos time, (c) the recurrence plot featuring kitelike structures, (d) the PSD where the periodic epochs are dominated by narrowband components at $f_1, f_2$ and $f_3$ ($f_2\simeq 2f_1$ and $f_3\simeq 3f_1$) and the chaotic bursts are dominated by broader peaks, (e) the Poincaré map featuring spiraling patterns, and (f) the average interchaos time versus the normalized bifurcation parameter.

Figure 7

Evidence of an unstable invariant set (chaotic saddle) in the flow through a semiwavy channel at two different $\mathrm{Re}$ values: (a)–(c) $\mathrm{Re}=835$ and (d)–(f) $\mathrm{Re}=840$. From left to right, they show the time trace of $v$, the phase portrait, and the Poincaré map.

Figure 8

Instantaneous and time-averaged streamlines and streamwise velocity contours for the three channel geometries. The blue, green, and red regions denote reversed flow ($u/u_0<0$), decelerated flow ($0<u/u_0<1$), and accelerated flow ($u/u_0>1$), respectively. $P_k$, period-$k$ limit cycle; $QP$, quasiperiodicity; $C$, chaos; $I$, intermittency.

Figure 9

Grid independence check using the dimensionless streamwise velocity profile $u/u_0$ for the (a) symmetric, (b) asymmetric, and (c) semiwavy channels.

Figure 10

Computational domain for the (a) symmetric, (b) asymmetric, and (c) semiwavy channels.

Figure 11

Validation of our numerical framework via streamlines: (a), (c) our simulations and (b), (d) those by Harikrishnan et al. [9].

Figure 12

Validation of our numerical framework via nondimensional $z$-vorticity contours: (a), (c) our simulations and (b), (d) those by Harikrishnan et al. [9].