Nonlinear dynamics and intermittency in a turbulent reacting wake with density ratio as bifurcation parameter

The flame or flow behavior of a turbulent reacting wake is known to be fundamentally different at high and low values of flame density ratio $\left({\rho }_u/{\rho }_b\right)$, as the flow transitions from globally stable to unstable. This paper analyzes the nonlinear dynamics present in a bluff-body stabilized flame, and identifies the transition characteristics in the wake as ${\rho }_u/{\rho }_b$ is varied over a Reynolds number (based on the bluff-body lip velocity) range of 1000–3300. Recurrence quantification analysis (RQA) of the experimentally obtained time series of the flame edge fluctuations reveals that the time series is highly aperiodic at high values of ${\rho }_u/{\rho }_b$ and transitions to increasingly correlated or nearly periodic behavior at low values. From the RQA of the transverse velocity time series, we observe that periodicity in the flame oscillations are related to periodicity in the flow. Therefore, we hypothesize that this transition from aperiodic to nearly periodic behavior in the flame edge time series is a manifestation of the transition in the flow from globally stable, convective instability to global instability as ${\rho }_u/{\rho }_b$ decreases. The recurrence analysis further reveals that the transition in periodicity is not a sudden shift; rather it occurs through an intermittent regime present at low and intermediate ${\rho }_u/{\rho }_b$. During intermittency, the flow behavior switches between aperiodic oscillations, reminiscent of a globally stable, convective instability, and periodic oscillations, reminiscent of a global instability. Analysis of the distribution of the lengths of the periodic regions in the intermittent time series and the first return map indicate the presence of type-II intermittency.

Figure 1

Schematic and photograph of the atmospheric pressure, vitiated, bluff-body rig.

Figure 2

Schematic of test section with ballistic bluff body installed.

Figure 3

Schematics of bluff bodies: (a) ballistic; (b) v gutter.

Figure 4

Time series of the transverse position of the flame top edge at $x/D=5$ for (a) ${\rho }_u/{\rho }_b=3.1$, (b) ${\rho }_u/{\rho }_b=2.4$, (c) ${\rho }_u/{\rho }_b=2.0$, and (d) ${\rho }_u/{\rho }_b=1.7$. The lighter grey portions in the time series correspond to aperiodic behavior, while the black portions correspond to regular or nearly periodic behavior.

Figure 5

Bifurcation diagram of the rms amplitude of the time series corresponding to the flame top edge transverse position vs the flame density ratio $\left({\rho }_u/{\rho }_b\right)$ at $x/D=5$ and $x/D=3.95$. The error bars correspond to 95% confidence interval.

Figure 6

Comparison of the top flame edge time series at $x/D=5$ and ${\rho }_u/{\rho }_b=2.0$ with the corresponding flame chemiluminescence images.

Figure 7

A (a) periodic and an (b) aperiodic part of the top flame edge time series at $x/D=5$ and ${\rho }_u/{\rho }_b=1.7$. Two-dimensional phase space plot of the (c) periodic and the (d) aperiodic portion of the time series.

Figure 8

Recurrence plot of (a) periodic part, (b) aperiodic part, and (c) full portion of the flame edge time series at $x/D=5$ and ${\rho }_u/{\rho }_b=1.7$. The threshold for constructing the RPs was kept constant for all the time series and was chosen as the mean phase space diameter.

Figure 9

Recurrence plot of (a) periodic part and (b) aperiodic part of the flame edge time series at $x/D=5$ and ${\rho }_u/{\rho }_b=1.7$. The thresholds for constructing the RP's were chosen such that the recurrence rate (RR) was fixed to be 0.2.

Figure 10

Variation of (a) recurrence rate (RR) and (b) laminarity (LAM) with ${\rho }_u/{\rho }_b$. The threshold was set as the mean phase space diameter, which is of the order of the size of the low-amplitude noisy attractor in the embedded phase space.

Figure 11

Variation of (a) determinism (DET) and (b) entropy (ENTR) with ${\rho }_u/{\rho }_b$. The thresholds were chosen such that the recurrence rate (RR) was kept fixed at 0.2.

Figure 12

Variation of (a) recurrence rate (RR) and (b) laminarity (LAM) with axial position $x/D$ for fixed ${\rho }_u/{\rho }_b$. The threshold is set as the mean phase space diameter, which is of the order of the size of the low-amplitude noisy attractor in the embedded phase space.

Figure 13

Spatial variation of the recurrence rate (RR) for (a) ${\rho }_u/{\rho }_b=1.7$, (b) ${\rho }_u/{\rho }_b=2.0$, (c) ${\rho }_u/{\rho }_b=2.2$, and (d) ${\rho }_u/{\rho }_b=2.4$.

Figure 14

Histogram of periodic length distribution for the intermittent flame edge time series at $x/D=5$ and (a) ${\rho }_u/{\rho }_b=1.7$ and (b) ${\rho }_u/{\rho }_b=2.0$.

Figure 15

Log-log plot of the average periodic region length $〈L〉$ vs the normalized density ratio $E$. The power law $〈L〉\sim E^{-0.97}$ is obtained from the plot with 95% confidence.

Figure 16

The first return map or the Poincaré plot constructed by taking the successive extrema of a periodic portion (black trajectory) followed by an aperiodic portion (light gray trajectory) in the flame edge time series at $x/D=5$ and ${\rho }_u/{\rho }_b=1.7$.

Figure 17

Multifractal spectrum of the flame edge time series at (a) $x/D=5$ and (b) $x/D=3.95$ for varying ${\rho }_u/{\rho }_b$.

Figure 18

Variation of the multifractal width with ${\rho }_u/{\rho }_b$ at fixed $x/D$.

Figure 19

(a) Multifractal spectrum of the flame edge time series at ${\rho }_u/{\rho }_b=1.7$. (b) Variation of multifractal width with $x/D$ at fixed values of ${\rho }_u/{\rho }_b$.

Figure 20

(a) Variation of the Hurst exponent $\left(H\right)$ with ${\rho }_u/{\rho }_b$ for constant values of $x/D$. (b) Variation of Hurst exponent with $x/D$ for constant values of ${\rho }_u/{\rho }_b$.

Figure 21

Flame edge time series at $x/D=5$ and at (a) ${\rho }_u/{\rho }_b=2.9$ and (b) ${\rho }_u/{\rho }_b=1.7$. At ${\rho }_u/{\rho }_b=2.9$, we can observe short term persistence ($y$ being constant for quite some time) in the time series.