Periodic modulation of the space-filling nature of the turbulent flame leads to spiky heat release oscillations

Abstract

Spiky oscillations are characterized by slow-fast dynamics and are observed in excitable media such as neuronal membranes and cardiac cells. In a turbulent reactive flow system, we observe that the heat release rate exhibits self-sustained periodic, spiky oscillations in synchrony with the sinusoidal periodic acoustic pressure oscillations. These self-sustained oscillations are a consequence of thermoacoustic instability, which arises due to a positive feedback between the acoustic and the heat release rate fields. One of the primary mechanisms for fluctuations in the heat release rate is the modulation in the topology of the flame, a thin interface separating the reactants and products. In this work, we explore the dynamics of the space-filling nature of the flame, quantified by its fractal dimension in relation to the spiky heat release rate oscillations in a turbulent reactive flow system. We discover that the periodic oscillatory dynamics in the space-filling nature of the flame lead to periodic, spiky heat release rate oscillations. Based on this result, we show that the spiky oscillations in the heat release rate can be approximated as $e^{\mathrm{sin}(ωt)}$ during the dynamical state of generalized synchronization between the heat release rate and the acoustic pressure oscillations in a turbulent reactive flow system. In synchronization theory, generalized synchronization is characterized by an emergent functional relationship, $Φ$, between the interacting subsystems. We unravel this emergent functional relation between the heat release rate and the acoustic pressure oscillations in a turbulent reactive flow system. It is intriguing that the dynamics of a far-from-equilibrium complex system can be represented by such simple mathematical relations.

Figures (10)

• Figure 1: Experimental observation of periodic spiky oscillations in heat release rate. (a) Schematic of the turbulent reactive flow system comprising air and fuel inlets, a mixing tube, a combustion chamber, and a decoupler. The acoustic pressure, $p^\prime (t)$, is measured using a piezoelectric transducer mounted on the combustion chamber. A photomultiplier tube (PMT) outfitted with a $\mathrm{CH}^\star$ filter is used to obtain the global heat release rate, $\dot{q}^\prime(t)$. The airflow is seeded with TiO2 particles, and an Nd:YLF laser is used as the light source to illuminate the flow field. The spatially resolved Mie-scattering images of TiO2 particles and the CH$^{\star}$ chemiluminescence images are recorded using two different high-speed cameras. (b) The time series of the acoustic pressure (blue) and the heat release rate (red) oscillations at a Reynolds number value of $3.71 \times 10^4$. The acoustic pressure follows a sinusoidal variation. On the other hand, the heat release rate appears to be spiky, a flat variation followed by a sudden rise in its amplitude. (c) Amplitude spectrum of the acoustic pressure and the heat release rate oscillations. Both $p^\prime$ and $\dot{q}^\prime$ oscillate with the same frequency of 120 Hz.
• Figure 2: Sudden heat release arises due to the emergent large-scale structure. (a) Planner Mie scattering images of TiO2 particles and (b) chemiluminescence images of $\mathrm{CH}^\star$ radical corresponding to the instances c-I to c-VI. The chemiluminescence intensity is a qualitative measure of heat release rate. (c) The acoustic pressure overlaid with the global heat release rate oscillations. Over one acoustic cycle, the sequence follows. (c-I) The flow field is characterized by small-scale structures indicated with circles in a-I. (c-II) Vortex roll-up entraining the fresh reactants into the hot product side, marked with an arrow in a-II. (c-III) The emergent large-scale structure, highlighted with circles in a-III. The localized patches of heat release rate are highlighted with rectangles in b-III. (c-IV, c-V) Sudden and high heat release rate over the spatial domain in b-IV. (c-VI) Eventually, the entrained reactants are burned, and the flow field is characterized by small-scale structures.
• Figure 3: The fractal dimension of the flame contours oscillates periodically during the sustained periodic oscillatory state. (a-I to a-VI) Evolution of flame contours corresponding to the instances b-I to b-VI during the periodic oscillatory state. These flame contours are obtained from the Mie scattering images of TiO2 particles shown in Fig. \ref{['fig: CH6 flameDynamics_physical']}a. (b) The heat release rate oscillations overlaid with the corresponding fractal dimension values of the flame contours. At b-I, the flame contours extend along the ${x}$ direction. (b-II) Onset of vortex roll-up. (b-III) Emergent large-scale structure due to the vortex roll-up and interaction between small-scale vortices. (b-IV, b-V) Flame contours begin to disappear. (b-VI) Eventually, the flame contour extends along the ${x}$ direction, a similar structure to that at the beginning of the cycle. Corresponding to the instance, b-III, multiple scales of wrinkles enhance the space-filling of flame contours. The value of $D_\mathrm{f}$ oscillates periodically raghunathan2020multifractal, suggesting periodic emergence of large coherent structures during self-sustained periodic acoustic pressure oscillations.
• Figure 4: The variation of $\dot{q}=ae^{b~\mathrm{sin}(\omega t+\phi_1)}$ for different values of $b$, by considering $\omega=2\pi \times 120$, $a=1$ and $\phi_1=0$.
• Figure 5: Probability of recurrence ($\mathcal{P}(\tau)$) for the time series of $p^\prime(t)$ and $\dot{q}^\prime(t)$, shown in Fig. \ref{['fig: experimental setup']}b, with time lag $\tau$. The strong correspondence in the recurrence of $p^\prime$ and $\dot{q}^\prime$ is an indication of generalised synchronization between $p^\prime$ and $\dot{q}^\prime$.