Mode-locked rotating detonation waves: Experiments and a model equation

Direct observation of a rotating detonation engine combustion chamber has enabled the extraction of the kinematics of its detonation waves. These records exhibit a rich set of instabilities and bifurcations arising from the interaction of coherent wave fronts and global gain dynamics. We develop a model of the observed dynamics by recasting the Majda detonation analog as an autowave process. The solution fronts become attractors of the engine, i.e., mode-locked rotating detonation waves. We find that denotative energy release competes with dissipation and gain recovery to produce the observed dynamics and a bifurcation structure common to other driven-dissipative systems, such as mode-locked lasers.

Figure 1

Section view of the rotating detonation engine (RDE) used for this study. The engine geometry is such that gaseous methane and oxygen are directed into a narrow annular gap through a set of propellant injectors. A spark plug ignites the mixture, which rapidly transitions to a number of circumferentially traveling detonation waves.

Figure 2

(a) A high-speed camera frame from an experiment shows the location of rotating detonation waves in the annulus of a RDE. Overlaid is a rendering of the propellant injection scheme. (b) Tracking the detonations through time yields a spatial-temporal view of their kinematics. Line slopes correspond to speed. The vertical cut in (b) is synchronized with the states of (a) and (c). (c) The phase difference $\mathrm{\Psi }$ for the waves seen in (a) is not $\pi$, though eventually the phase difference approaches this stable value.

Figure 3

Representative wave nucleation process in a startup transient (a) in an experiment and (b) in a simulation of the proposed model, displayed as pseudocolor plots of amplitude (arb. units). As seen in the wave reference frame of (a) and (b), the oscillatory phase difference between the two waves immediately after nucleation decays through time as the two waves become mode locked. (b) $s=3.5$. The instantaneous speeds of the waves along $\mathrm{\Psi }=0$ in (a) and (b) are given in (c).

Figure 4

Representative destruction of a wave (a) in an experiment and (b) in a simulation of the model, shown in the wave-attached reference frame as pseudocolor plots of amplitude (arb. units). Oscillations in $\mathrm{\Psi }$ grow exponentially until one wave overruns the other. For a given injection function $\beta$ and loss $\epsilon$, the oscillation period and phase difference growth rate are parameterized by the change in $s$ and $u_p$. (b) $s=2$ with a $-20\%$ change in $s$ applied to the mode-locked state. The instantaneous speeds of the waves along $\mathrm{\Psi }=0$ in (a) and (b) are given in (c).

Figure 5

Space-time history of mode-locked modulation of wave speeds (a) in an experiment and (b) in a simulation, in the wave reference frame. The instantaneous speeds of the waves along $\mathrm{\Psi }=0$ in (a) and (b) are shown in (c). The accompanying spectra show clear sidebands symmetric about the carrier frequency. The frequency shown in the spectra is scaled by the average transit time of the mode-locked wave, $L/D_{\mathrm{wave}}$. The abscissa magnitude corresponds to a count of the waves in the domain. As shown, the dominant frequency is three waves with sidebands near two and four waves.

Figure 6

Space-time history of plane-wave pulsation mode of operation (a),(b) in an experiment and (c),(d) in a simulation. Simulation parameters are those listed in Table 1 with $q_0=6$ and $\epsilon =1.0$. The deactivation and reactivation of the injectors give rise to a resonance between the combustion and propellant injection.

Figure 7

The influence of the state of the domain, $\eta$, on the gain recovery function $\beta$ following the functional form of Eq. (5). The solid line is the activation function used for simulations in Sec. 4.

Figure 8

Nucleation and mode locking of detonations from a single pulse initial condition ($s=3.5$). Vertical lines in the $\theta -t$ diagram correspond to the shown simulation snapshots. The initial sech-pulse rapidly transitions to a CJ detonation. In regions where $\eta$ is low, the injectors behave steadily. However, as the wave reaches its tail, $\eta$ is everywhere elevated and the injection is severely curtailed. A second wave forms from the self-steepening of parasitic deflagration. After wave nucleation, the two waves behave dispersively and their phase differences approach $\pi$.

Figure 9

Number of waves and wave speed through a sweep of the bifurcation parameter $s$ for linear and nonlinear loss terms. The final states have been approached from “above,” i.e., via a relaxation of three waves to two or one wave. In the transition from one to two waves for the nonlinear loss case, a series of period-halving bifurcations increase order in the system to eventually form two mode-locked waves with zero oscillation in the phase difference. The qualitatively similar experimental bifurcation structure is published in Bykovskii et al. [11]. The energy balance dynamics in laser cavities produce a similar cascaded bifurcation structure, including chaotic interpulse regimes [9, 34].

Figure 10

Integrated pixel intensity displayed through $\theta$ for an experiment through which a mode transition from (a) one to (b) two waves occurs. A similar induced bifurcation in a simulation is displayed in (c) and (d). Of note is (i) the decrease of wave amplitude between one and two waves, (ii) the increase in background luminosity (or, in the simulation, base state of $\eta$), and (iii) the local increase of the magnitude of the state preceding the waves. The wave speeds decrease about 10% through the bifurcation, though this decrease is attributable to both a reduction in wave amplitude and an increase in parasitic deflagration. Once the parasitic deflagration can self-steepen to form a shock (see Fig. 8), a bifurcation of the number of waves occurs.

Figure 11

Increasing the magnitude of the loss coefficient $\epsilon$ from 0.11 to 0.3 increases the traveling wave speed to 117% (up from 74%) of the CJ value referenced to an ambient state of ${\eta }_0=0$. The simulation is otherwise identical to that of Fig. 8.

Figure 12

Ring fiber laser configuration with localized gain (pump) and loss (output coupler where a fraction of the energy is directed out of the system). The collection of soliton pulses within the cavity are formed by the intensity discrimination (nonlinear loss) provided by the saturable absorber. The solitons seek maximal and symmetric phase differences between one another in an analogous process to that of rotating detonation waves.

Figure 13

2D control volume for model derivation.