Dynamical-systems analysis and unstable periodic orbits in reacting flows behind symmetric bluff bodies

Dynamical systems analysis is performed for reacting flows stabilized behind four symmetric bluff bodies to determine the effects of shape on the nature of flame stability, acoustic coupling, and vortex shedding. The task requires separation of regular, repeatable aspects of the flow from experimental noise and highly irregular, nonrepeatable small-scale structures caused primarily by viscous-mediated energy cascading. The experimental systems are invariant under a reflection, and symmetric vortex shedding is observed throughout the parameter range. As the equivalence ratio—and, hence, acoustic coupling—is reduced, a symmetry-breaking transition to von Karman vortices is initiated. Combining principal-components analysis with a symmetry-based filtering, we construct bifurcation diagrams for the onset and growth of von Karman vortices. We also compute Lyapunov exponents for each flame holder to help quantify the transitions. Furthermore, we outline changes in the phase-space orbits that accompany the onset of von Karman vortex shedding and compute unstable periodic orbits (UPOs) embedded in the complex flows prior to and following the bifurcation. For each flame holder, we find a single UPO in flows without von Karman vortices and a pair of UPOs in flows with von Karman vortices. These periodic orbits organize the dynamics of the flow and can be used to reduce or control flow irregularities. By subtracting them from the overall flow, we are able to deduce the nature of irregular facets of the flows.

Figure 1

Side view of imaging setup with Phantom v7.1 high-speed camera and mirror.

Figure 2

Power spectrum of time series for (chemiluminescence) intensities at a point $\mathbf{x}$ for flow at $\varphi =0.7$. Spectrum beyond 400 Hz does not appear to have a structure. Time filtering of the signal is implemented using the filter function given in Eq. (1).

Figure 3

The 500th panel of original and time-filtered flow, illustrating that the rapidly evolving small-scale motions have been filtered. Size of the images is approximately $100\times 66$ mm.

Figure 4

First nine coherent structures for flow at $\varphi =1.1$, all of which are nearly symmetric about the horizontal midline. Our analysis is predicated on the assumption that asymmetric components of these modes are due to slight experimental nonuniformities and dynamics of the small-scale irregular structures. The scale of individual images, and images in subsequent figures, is approximately $100\times 66$ mm.

Figure 5

(a) Latency and (b) fractional latency of the symmetric (red) and antisymmetric (blue) components of the first 15 coherent structures. Observe that latencies of the antisymmetric components are small (by a factor of $\sim$100) compared to those of the corresponding symmetric components. Furthermore, they remain nearly unchanged as the mode number $n$ increases, supporting our assertion that the antisymmetric components are caused by experimental noise, irregular motions, and/or line-of-sight detection. The fractional latencies of the two components approach each other as $n$ increases, indicating that, for large $n$, the symmetric components are corrupted as well.

Figure 6

First nine coherent structures for flow at $\varphi =0.8$. Modes 5 and 6 (as well as 9 and 12, not pictured) appear to be primarily antisymmetric.

Figure 7

(a) Latency and (b) fractional latency of symmetric and antisymmetric components of coherent structures 0–14 for the flow at $\varphi =0.8$. Note that the dominant components of modes 5, 6, 9, and 12 are antisymmetric, suggesting the presence of von Karman vortex shedding.

Figure 8

Bifurcation diagram for onset of von Karman vortex shedding.

Figure 9

Changes in phase portrait [here the projection to $(a_0\left(t\right),a_2\left(t\right))]$ as $\varphi$ changes from 1.1 to 0.7. Transition to the onset of von Karman vortices is accompanied by the change of the phase portrait from an inverted-$\mathsf{U}$ structure to a filled-loop structure. Similar changes are observed in other projections as well.

Figure 10

Lyapunov exponent of the flow in the range of equivalence ratios studied. Note that the exponent increases with the onset of von Karman vortex shedding.

Figure 11

Schematic examples of Poincaré maps for flows. (a) Periodic orbit of the flow intersects a single point ${\mathcal{P}}_0$ on the Poincaré section repeatedly; ${\mathcal{P}}_0$ is a fixed point of the Poincaré map. (b) Quasiperiodic orbits of the flow intersects the Poincaré section on a curve. Two successive crossings ${\mathcal{P}}_0$ and ${\mathcal{P}}_1$ are shown. The Poincaré map ${\mathcal{P}}_0\to {\mathcal{P}}_1$ is quasiperiodic.

Figure 12

(a) Intersections of the flow at $\varphi =1.1$ with the Poincaré section $a_1=0$. First crossings for close returns, defined by $\varepsilon =0.01$, are shown by red closed circles, while returns to the section are shown by green closed triangles. Dashed lines join pairs of first crossings and returns; i.e., represent the Poincaré map. Open circles are the remaining crossings of the orbit with the Poincaré section, whose returns are farther than $\varepsilon$ from the first crossing. (b) Dynamics of $a_1\left(t\right)$ between the first crossing and return for the seven close returns identified in (a). All seven appear to be organized around a single periodic orbit. The cycle, marked by red circles, is estimated to be the mean of the seven close returns.

Figure 13

Projections of the periodic orbit of the flow at $\varphi =1.1$ to the $(a_1,a_2)$ and $(a_1,a_3)$ planes.

Figure 14

Projections of the periodic orbit of the flow at $\varphi =1.04$ to the $(a_1,a_2)$ and $(a_1,a_3)$ planes. Note that the projections are qualitatively similar to those for the flow at $\varphi =1.1$ (see Fig. 13).

Figure 15

Projections of two periodic orbits of the flow at $\varphi =0.8$ to the $(a_1,a_2)$ and $(a_1,a_3)$ planes. (a) and (b) show projections of one cycle and (c) and (d) those of the other.

Figure 16

Simultaneous snapshots of (a) the original flow at equivalence ratio $\varphi =1.1$ and (b) recurrent flow. Note that the former contains noise and other irregular small-scale structures. The size of the images is approximately $100\times 66$ mm.

Figure 17

Simultaneous snapshots of (a) the flow at equivalence ration $\varphi =0.8$ and (b) recurrent flow. Symmetric and antisymmetric components of the latter are given in (c) and (d), respectively.

Figure 18

Differences in images from equivalent time points of four close returns and corresponding snapshots of periodic flow at $\varphi =1.1$. Images represent noise and other irregular facets of the flow. Note that the images have “similar” statistical features, but the precise structures depend on the close return.

Figure 19

Bifurcation diagrams for onset and growth of von Karman vortex shedding with decreasing $\varphi$ for bluff bodies of several shapes. In each case the bifurcation appears between equivalence ratios $\varphi =0.9$ and $\varphi =1.0$.

Figure 20

Projections to a subspace defined by two eigenvectors illustrate changes of the phase portrait prior to and following the onset of von Karman vortex shedding for four bluff bodies. Images suggest that the flow behind the v-gutter contains the least noise or irregularities.

Figure 21

Growth of Lyapunov exponents for flow behind four bluff bodies as the equivalence ratio is reduced. Expansion rates for flow behind cylindrical, square, and v-gutter bluff bodies are similar. Flow behind the flat plate shows different behavior.

Figure 22

Periodic orbits prior to and subsequent to the onset of von Karman vortex shedding for the flow behind the four bluff bodies.