Detonation initiation by compressible turbulence thermodynamic fluctuations

Abstract

Theory and computations have established that thermodynamic gradients created by hot spots in reactive gas mixtures can lead to spontaneous detonation initiation. However, the current laminar theory of the temperature-gradient mechanism for detonation initiation is restricted to idealized physical configurations. Thus, it only predicts conditions for the onset of detonations in quiescent gases, where an isolated hot spot is formed on a timescale shorter than the chemical and acoustic timescales of the gas. In this work, we extend the laminar temperature-gradient mechanism into a statistical model for predicting the detonability of an autoignitive gas experiencing compressible isotropic turbulence fluctuations. Compressible turbulence forms non-monotonic temperature fields with tightly-spaced local minima and maxima that evolve over a range of timescales, including those much larger than chemical and acoustic timescales. We examine the utility of the adapted statistical model through direct numerical simulations of compressible isotropic turbulence in premixed hydrogen-air reactants for a range of conditions. We find strong, but not conclusive, evidence that the model can predict the degree of detonability in an autoignitive gas due to turbulence-induced thermodynamic gradients.

Introduction

Turbulence compressibility, autoignitive combustion, nonlinear interactions between turbulence and chemistry, and transitions to detonation are all fundamental aspects of many different high-speed reacting flows, from human-scale engineered flows such as supersonic combustion ramjet engines (scramjets) [1] to astrophysical flows such as Type 1a supernovae [2]. It is now well-established that thermodynamic gradients within localized hot spots (also termed exothermic centers) can lead to the direct initiation of detonations in auto-igniting gas mixtures [3], [4], [5], [6]. It is also established that nonlinear turbulence-chemistry interactions alone can induce deflagration-to-detonation transition (DDT) in high-speed turbulent flames. This can occur via the same localized mechanism of hot-spot autoignition [7], [8], where the necessary thermodynamic gradient is created by the turbulent flame rather than an external source. Alternatively, DDT can be induced through a self-reinforcing amplification of an initial large-scale pressure wave generated during a self-acceleration event of an unsteady turbulent flame propagating with a speed above that of a Chapman-Jouguet (CJ) deflagration [9], [10]. All three mechanisms — autoignitive hot-spot initiation, local hot-spot DDT, and turbulence-driven DDT in super-CJ flames — rely, at least in part, on a thermomechanical feedback loop between chemical heat release and pressure in order to amplify an acoustic wave into a detonation wave. This unsteady feedback loop is described in detail for the particular case of sub-detonable shock waves, termed shock wave amplification by coherent energy release, or SWACER, in Lee et al. [4]. Qualitatively, the critical difference between the detonation mechanisms is the source that initiates and drives the feedback mechanism: autoignition or turbulent deflagration.

In autoignitive gaseous flows, spatial gradients in temperature, pressure, and species mixture fractions produce spatial gradients in the chemical ignition delay time, denoted t_ign. As long as advective and diffusive processes operating on the temperature and species gradients are slow compared to the range of ignition delay times present [7], a monotonic gradient will form a spontaneous reaction front that propagates with a speed equal to the inverse of the local ignition delay time gradient magnitude, namely $u_{\mathrm{sp}}={\left|\nabla t_{\mathrm{ign}}\right|}^{-1}$.

Neglecting any thermomechanical response of the gas to initial conditions or subsequent heat release, an isolated hot spot with a constant u_sp as the initial condition will form either a subsonic or supersonic spontaneous ignition wave that will transition to a conventional deflagration or detonation wave within specific regimes of the spontaneous wave-speed. These four propagation regimes are delineated by the magnitude of u_sp with respect to the local CJ detonation speed, D_CJ, the upstream speed of sound, a, and the laminar deflagration speed, S_L [5]. Only in the case where a < u_sp < D_CJ will the spontaneous ignition wave formed from initial conditions transition to a detonation wave.

When the thermomechanical response of the gas to both the initial conditions and heat release are taken into account, a far larger range of u_sp can transition to a detonation wave than would be expected from an analysis of the initial conditions alone [4], [7], [11]. Most notably, when S_L < u_sp < a, a spontaneous subsonic ignition wave would be predicted to form and propagate through the hot spot without transitioning to another regime [5]. However, the thermomechanical response of the surrounding gas to the initial creation of the hot spot will form an acoustic wave, independent of the ignition wave [4], [11]. Due to the monotonically-decreasing speed of sound, the downstream pressure increase due to heat release from the subsonic ignition wave will propagate upstream faster than the leading acoustic wave, allowing the trailing subsonic ignition wave to amplify the leading acoustic wave. In turn, the ignition-delay-time gradient between the acoustic and ignition waves will be smaller than in the initial conditions, and therefore the ignition wave will accelerate towards the acoustic wave. Consequently, the heat release rate will increase within the reaction zone of the wave. This process is self-reinforcing, in the same manner as the SWACER feedback mechanism [4], and will eventually accelerate the ignition wave to supersonic speeds. At this point, the ignition wave will overtake and coalesce with the leading acoustic wave and transition to a detonation wave. This broader temperature-gradient-based mechanism for the direct initiation of detonation waves has been extensively validated from both an analytical perspective, through asymptotic analysis and reduced order models, and from a numerical perspective, through one- and two-dimensional (1D and 2D, respectively) numerical simulations of simplified hot spot configurations. The reader is directed to Refs. [12], [13] for comprehensive reviews of the foundational literature.

Both Khokhlov [7] and Bradley [6] numerically studied the thermomechanical response of 1D hot spots to chemical heat release in order to discern the critical range of hot-spot radii and temperature-gradient magnitudes that lead to detonation initiation. In addition to the ratio a/u_sp, these studies determined that detonation initiation also depends on the ratio of the characteristic time scale of exothermic heat release, t_exo, within the hot spot to the acoustic residence time of the hot spot, r₀/a, where r₀ is the initial radius prior to thermomechanical relaxation. The importance of this acoustic-exothermicity coupling was expanded upon in subsequent studies [14], [15], [16], [17], [18], establishing that the detonability of an isolated hot spot can be characterized by two non-dimensional parameters, given as

$$\xi =\frac{a}{u_{\mathrm{sp}}}=a|\nabla t_{\mathrm{ign}}|\approx a{t}_{\mathrm{ign}}\frac{T_{\mathrm{a}}}{T^2}{\left|\nabla T\right|}_{\mathrm{hs}},$$

$$\zeta =\frac{r_0}{a{t}_{\mathrm{exo}}},$$

where T is the temperature, $T_{\mathrm{a}}=\mathrm{d}\left(\ln t_{\mathrm{ign}}\right)/\mathrm{d}(1/T)$ is the effective activation temperature of the chemistry, |∇T|_hs is the average or constant temperature gradient of the hot spot, and t_ign, t_exo, T, and T_a are each evaluated from the initial conditions at the radial midpoint, r₀/2. It should be noted that no assumptions have been made about the explosive or autoignition limits of the reactant mixture in deriving ξ and ζ. The only assumptions are that the two chemical timescales, t_ign and t_exo, are quantifiable, or alternatively that |∇t_ign| is well approximated by t_ignT_a|∇T|_hs/T².

As described by Gu et al. [14], the acoustic-exothermicity coupling, ζ, predicts the detonability of the hot spot according to the following criteria:

• If ζ ≫ 1, then the spontaneous ignition wave will form much faster than an acoustic wave can exit the hot spot, and the two waves will coalesce and transition into a fully-developed detonation wave before both waves exit the region of negative speed-of-sound gradient in the hot spot that enables the self-reinforcing thermomechanical coupling between the waves.

• If ζ ≈ 1, then the ignition wave will form in the same time that the acoustic wave will have crossed the initial hot spot radius, and whether a detonation wave forms will be strongly dependent on the value of ξ, because the continuous thermomechanical relaxation of the hot spot may continue to decelerate the acoustic wave long enough to allow for the self-reinforcing thermomechanical coupling to bring the two waves together.

• If ζ ≪ 1, then the ignition wave will not form before the acoustic wave has left the hot spot and ceased to decelerate, and none of the heat release within the ignition wave will be capable of amplifying the acoustic wave in a self-reinforcing manner.

In accordance with these criteria, the acoustic-induction coupling, ξ, has upper and lower detonability limits that correspond to a/S_L and a/D_CJ in the asymptotic limit ζ ≫ 1, whereas at ζ ≈ 1, the upper and lower detonability limits contract to a single point, typically in the range 2 < ξ < 7, forming a “detonation peninsula” within the $\zeta -\xi$ parameter space [18].

The temperature-gradient mechanism of detonation initiation is, however, limited to the prediction of detonations in localized and isolated hot spots that are formed on timescales shorter than, or comparable to, chemical and acoustic timescales. In the case of highly turbulent autoignitive flows, by contrast, turbulence compressibility can generate non-monotonic temperature fields with tightly-spaced minima and maxima that vary over a wide range of length and time scales, including those much larger than chemical and acoustic length and time scales. As such, there is currently no predictive model for the initiation of detonations by compressible turbulence fluctuations in an autoignitive flow. Therefore, in this work, we adapt these governing non-dimensional parameters of the laminar temperature-gradient mechanism into a statistical model for the a priori prediction of spontaneous detonation initiation by thermodynamic gradients formed from flow-field fluctuations in compressible homogeneous isotropic turbulence (HIT). We then use three-dimensional (3D) direct numerical simulations (DNS) of compressible HIT in premixed hydrogen-air reactants over a small — but critical — range of conditions to examine the utility of the adapted statistical model for predicting compressible turbulence detonability.

The present focus on autoignitive HIT is motivated by prior studies of velocity and thermodynamic fluctuations in both non-reacting and reacting turbulent flows. For non-reacting equilibrium turbulent flows at very high Reynolds numbers, it is widely hypothesized that the smallest scales behave in a universal manner independent of any particular large-scale flow geometry, and, as a consequence, the turbulent fluctuations must be statistically homogeneous and isotropic [19], [20], [21]. In this context, the interactions of HIT with chemistry can be viewed as a model problem that approximates, on a statistical basis, the dynamics of high-pass-filtered turbulent fluctuations in a material volume of reacting fluid advecting with the bulk flow of any sufficiently high Reynolds number combustion process that has the same initial reactant and final product states [22], [23], [24], [25]. Similarly, the effect of compressible HIT thermodynamic fluctuations on autoignition can be isolated from the effect of turbulent fluctuations in reactant mixture fractions by analyzing fully premixed reactants. In this case, the assumptions of small-scale homogeneity and premixed reactants allow for highly detailed analyses of turbulence-chemistry interactions, including multi-scale and cross-scale interactions [10], [26], [27], [28], [29], [30].

The present work draws upon previous 2D DNS studies of superimposed temperature and turbulent kinetic energy fluctuations at conditions relevant to homogeneous charge compression ignition engines to study the importance of turbulent mixing and diffusive timescales on the development of reaction fronts [31], [32], [33], [34], [35], and the dynamical impact of chain-branching-regulated chemical kinetics, including negative-temperature-coefficient behavior, compared to the thermally-regulated one-step Arrhenius chemistry models [36], [37]. Additionally, Yu and Bai [38] compared saddle-shaped and spherically-curved reaction fronts, as well as the effects of 3D turbulence on autoignition, in what may be the only computational study to date that utilizes 3D DNS with detailed chemistry to focus exclusively on premixed autoignition waves. Furthermore, the present work draws upon previous studies of 3D compressible HIT, particularly 3D DNS studies of compressible turbulence thermodynamics [39], [40], [41]. However, no previous study has performed DNS of turbulent premixed autoignition where the controlling temperature gradients were a direct result of compressible turbulence fluctuations. This intrinsic, turbulence-induced mechanism of detonation initiation is the primary focus of the present study.

This paper is organized as follows. First, we extend the dimensional analysis for detonation initiation in isolated hot spots to turbulence-induced thermodynamic fluctuations in compressible mixtures. Next, we describe details of the numerical simulation algorithms and procedures. This is followed by a detailed analysis of the results, with conclusions presented at the end.