Onset of chaotic advection in open flows

In this paper we investigate the transition to chaos in the motion of particles advected by open flows with obstacles. By means of a topological argument, we show that the separation points on the surface of the obstacle imply the existence of a saddle point downstream from the obstacle, with an associated heteroclinic orbit. We argue that as soon as the flow becomes time periodic, these orbits give rise to heteroclinic tangles, causing passively advected particles to experience transient chaos. The transition to chaos thus coincides with the onset of time dependence in open flows with stagnant points, in contrast with flows with no stagnant points. We also show that the nonhyperbolic nature of the dynamics near the walls causes anomalous scalings in the vicinity of the transition. These results are confirmed by numerical simulations of the two-dimensional flow around a cylinder.

Figure 1

Part of a numerically calculated streakline for $w=2$ in the wake of the obstacle. Ten thousand particles were injected per period at position $(x,y)=(-3,0)$ into the flow, and their subsequent positions were plotted every period, at times given by $t\mathrm{mod}T=0$, until they leave the time-dependent region and travel downstream. The solid black area on the left is the cylindrical obstacle.

Figure 2

Ratio of the number $N_v$ of particles entering the wake to all $N$ particles injected into the flow at time $t=0$, as a function of the bifurcation parameter, with random initial positions distributed uniformly in the region $-16.0<x<-1.1$, $-0.05<y<0.05$. The function log(.) denotes the natural logarithm. In (a) the original model with bifurcation parameter $w$ is investigated by injecting $N=1\times {10}^6$ particles; the fitted curve corresponds to $N_v∕N\sim \exp \left(k{w}^{-\eta }\right)$, where $k=-15.44$ and $\eta =1.046$. In (b) the alternative model with bifurcation parameter $\beta$ is investigated; the fitted curve corresponds to $N_v∕N\sim \exp \left(k{\beta }^{-\eta }\right)$, where $k=-2.807$ and $\eta =0.101$. In these computations, $N=1\times {10}^6$ is used for parameters $\beta >2\times {10}^{-6}$, whereas $N=5\times {10}^6$ is used for smaller $\beta$. We verified that, even though the particular value of the ratio $N_v∕N$ depends on where the particles are introduced in the flow, the scaling coefficient $\eta$ is independent of this.

Figure 3

Schematic picture of the manifolds escaping from the separation points in the flow around an obstacle. In (a), the manifolds are shown for an autonomous system, such that they form separatrices. In (b), the manifolds for a time-periodic flow are depicted, showing a heteroclinic tangle. Images (c) and (d) show the manifolds and streamlines in the case of persistent vortices.

Figure 4

Visualization of the heteroclinic tangle for $w=2$. Unstable manifolds are visualized by injecting $100\times 100$ particles close to the separation points and plotting their subsequent positions stroboscopically for 18 periods. The stable manifolds are visualized by injecting $800\times 800$ particles uniformly in the region $0.9<x<1.6$, $-0.5<y<0.5$ and checking if they leave downstream (reach $x=+10$) or pass through the region $1.25<x<1.35$, $-0.15<y<0.1$; the boundary between these two outcomes marks the stable manifold of the saddle point.

Figure 5

Results for the modified model. (a) Streamlines of the modified model with $\beta =0$, representing a time-independent flow with two stationary vortices. (b) Heteroclinic tangle for $\beta =0.001$. The stable manifolds of the saddle point and the unstable manifolds of the separation points on the surface of the cylinder were computed in the same way as in Fig. 4.

Figure 6

Escape time for the model of Eq. (8), with initial conditions taken at the $x=0$ segment, as a function of the initial $y$ coordinate. Escape was defined to occur when the particles reach $x=20$, after which they are washed downstream. The parameters are $V=2$, $\sigma =2$, $L=5$, $u=1$, $k=1$. (a) uses $\mu =2$ and shows regular (nonchaotic) advection, while (b) uses $\mu =10$ and shows chaotic advection.