Premixed flame response to equivalence ratio perturbations

This paper studies the heat-release oscillation response of premixed flames to oscillations in reactant stream fuel/air ratio. Prior analyses have studied this problem in the linear regime and have shown that heat release dynamics are controlled by the superposition of three processes: flame speed, heat of reaction, and flame surface area oscillations. Each contribution has somewhat different dynamics, leading to complex frequency and mean fuel/air ratio dependencies. The present work extends these analyses to include stretch and non quasi-steady effects on the linear flame dynamics, as well as analysis of nonlinearities in flame response characteristics. Because the flame response is controlled by a superposition of multiple processes, each with a highly nonlinear dependence upon fuel/air ratio, the results are quite rich and the key nonlinearity mechanism varies with mean fuel/air ratio, frequency, and amplitude of excitation. In the quasi-steady framework, two key mechanisms leading to heat-release saturation have been identified. The first of these is the flame-kinematic mechanism, previously studied in the context of premixed flame response to flow oscillations and recently highlighted by Birbaud et al. (Combustion and Flame 154 (2008), 356–367). This mechanism arises due to fluctuations in flame position associated with the oscillations in flame speed. The second mechanism is due to the intrinsically nonlinear dependence of flame speed and mixture heat of reaction upon fuel/air ratio oscillations. This second mechanism is particularly dominant at perturbation amplitudes that cause the instantaneous stoichiometry to oscillate between lean and rich values, thereby causing non-monotonic variation of local flame speed and heat of reaction with equivalence ratio.     

Nonlinear kinematic response of premixed flames to harmonic velocity disturbances

This paper analyzes the nonlinear dynamics of premixed flames responding to harmonic velocity disturbances. These nonlinear dynamics were studied by solving a constant flame speed front tracking equation for the flame’s response to harmonically oscillating velocity disturbances. The solution to these equations is used to quantify the transfer function relating the ratio of the normalized flame area to velocity fluctuations, G = (A′/A_o)/(u′/u_o), upon the amplitude of velocity oscillations, ε = u′/u_o. Due to nonlinearities, the amplitude of this transfer function relative to its linear value decreases with increasing amplitude of velocity oscillation, u′/u_o. In contrast, the transfer function phase exhibits almost no amplitude dependence. The velocity amplitude where transfer function nonlinearities become significant depends strongly upon three parameters: a Strouhal number, St = ωL_f/u_o (where L_f is the flame length), the ratio of the flame length to width, β = L_f/R, and the flame shape in the absence of perturbations (i.e., conical, inverted wedge, etc.). In the linear case, the transfer function, G, depends only upon an algebraic combination of the first two parameters, given by St₂ = St (1 + β²)/β². In general, however, G exhibits a distinct dependence upon both parameters St and β. In particular, we show that the nonlinear response of G is an intrinsically dynamic phenomenon; i.e., its quasi-steady response (St 1) is purely linear. As such, nonlinearity is enhanced with increasing Strouhal numbers. In contrast, nonlinearity is suppressed at large β values; as such, the response of a long flame remains quite similar to its linear value, even at large ε values where the flame front exhibits substantial corrugation and cusping. Finally, we show that the response of conical flames remains much more linear at comparable disturbance amplitudes than for “V” or wedge-shaped flames. These predictions are shown to be consistent with available experimental data.     

Temporal evolution of flame stretch due to turbulence and the hydrodynamic instability

The temporal evolution of the strain rate on a turbulent premixed flame was measured experimentally using cinema-stereoscopic particle image velocimetry. Turbulence strains a flame due to velocity gradients associated both directly with the turbulence and those caused by the hydrodynamic instability, which are initiated by the turbulence. The development of flame wrinkles caused by both of these mechanisms was observed. Wrinkles generated by the turbulence formed around vortical structures, which passed through the flame and were attenuated. After the turbulent structures had passed, the hydrodynamic instability flow pattern developed and caused additional strain. The hydrodynamic instability also caused the growth of small flame front perturbations into large wrinkles. In the moderately turbulent flame investigated, it was found that the evolution of the strain rate caused by turbulence–flame interactions followed a common pattern involving three temporal regimes. In the first, the turbulence exerted extensive (positive) strain on the flame, creating a wrinkle that had negative curvature (concave towards the reactants). This was followed by a transition period, leading into the third regime in which the flow pattern and strain rate were dominated by the hydrodynamic instability mechanism. It was also found that the magnitudes of the strain rate in the first and third regimes were similar. Hence, the hydrodynamic instability mechanism caused significant strain on a flame and should be included in turbulent combustion models.     

Mechanism of combustion dynamics in a backward-facing step stabilized premixed flame

Combustion dynamics leading to thermoacoustic instability in a rearward-facing step stabilized premixed flame is experimentally examined with the objective of investigating the fluid dynamic mechanism that drives heat release rate fluctuations, and how it couples with the acoustic field. The field is probed visually, using linear photodiode arrays that capture the spatiotemporal distribution of CH* and OH*; an equivalence ratio monitor; and a number of pressure sensors. Results show resonance between the acoustic quarter wave mode of the combustion tunnel and a fluid dynamic mode of the wake. Under unstable conditions, the flame is convoluted around a large vortex that extends several step heights downstream. During a typical cycle, while the velocity is decreasing, the vortex grows, and the flame extends downstream around its outer edge. As the velocity reaches its minimum, becoming mostly negative, the vortex reaches its maximum size, and the flame collides with the upper wall; its leading edge folds, trapping reactants pockets, and its trailing edge propagates far upstream of the step. In the next phase, while the velocity is increasing, the heat release grows rapidly as trapped reactant’ pockets are consumed by flames converging towards their centers, and the upstream flame is dislodged back downstream. The heat release rate reaches its maximum halfway into the velocity rise period, leading the maximum velocity by about 90°. In this quarter-wave mode, the pressure leads the velocity by 90° as well, that is, it is in phase with the heat release rate. Numerical modeling results support this mechanism. Equivalence ratio contribution to the instability mechanism is shown to be minor, i.e., heat release dynamics are governed by the cyclical formation of the wake vortex and its interaction with the flame.     

Subgrid combustion modeling of 3-D premixed flames in the thin-reaction-zone regime

Large eddy simulation (LES) is used to investigate three-dimensional (3D) lean premixed turbulent methane–air flames in the thin-reaction-zone regime. In this regime, the Kolmogorov scale is smaller than the preheat zone thickness, but larger than the reaction zone thickness. Past numerical studies of similar flames were primarily direct numerical simulation either in two-dimensions or using the artificially thickened flame approach in 3D. For an LES the effect of small (unresolved) scales on the scalar field must be, modeled accurately to capture the correct flame structure. A subgrid combustion model based on the linear-eddy-mixing (LEM) model is used within an LES framework (called LEM–LES hereafter) to capture the 3D flame-structure of the highly stretched premixed flames. A finite-rate, one-step methane–air chemistry with a non-unity Lewis number formulation is used in this study. The simulated flame structure resembles flames experimentally studied in the thin-reaction-zone regime. Even though the preheat zone is broadened by the penetration of small eddies, the chemical reaction zone remains thin and localized. This feature is captured properly in the current LEM–LES approach. The flame structure and other statistics such as the flame area evolution, curvature, and strain-rate statistics computed using the LEM–LES are also in good agreement with the past DNS studies.     

Ethanol turbulent spray flame response to gas velocity modulation

A numerical investigation of the interaction between a spray flame and an acoustic forcing of the velocity field is presented in this paper. In combustion systems, a thermoacoustic instability is the result of a process of coupling between oscillations in heat released and acoustic waves. When liquid fuels are used, the atomisation and the evaporation process also undergo the effects of such instabilities, and the computational fluid dynamics of these complex phenomena becomes a challenging task. In this paper, an acoustic perturbation is applied to the mass flow of the gas phase at the inlet and its effect on the evaporating fuel spray and on the flame front is investigated with unsteady Reynolds averaged Navier-Stokes numerical simulations. Two flames are simulated: a partially premixed ethanol/air spray flame and a premixed pre-vaporised ethanol/air flame, with and without acoustic forcing. The frequencies used to perturb the flames are 200 and 2500 Hz, which are representative for two different regimes. Those regimes are classified based on the Strouhal number St = (D/U)f_f: at 200 Hz, St = 0.07, and at 2500 Hz, St = 0.8. The exposure of the flame to a 200 Hz signal results in a stretching of the flame which causes gas field fluctuations, a delay of the evaporation and an increase of the reaction rate. The coupling between the flame and the flow excitation is such that the flame breaks up periodically. At 2500 Hz, the evaporation rate increases but the response of the gas field is weak and the flame is more stable. The presence of droplets does not play a crucial role at 2500 Hz, as shown by a comparison of the discrete flame function in the case of spray and pre-vaporised flame. At low Strouhal number, the forced response of the pre-vaporised flame is much higher compared to that of the spray flame.     

Intensity distributions near focus of strongly converging spherical waves diffracted at a circular aperture

Based on the Rayleigh–Sommerfeld diffraction integral, the diffraction of converging spherical waves at a circular aperture is studied in a general case. The expression for the intensity near focus of strongly converging spherical diffracted waves is derived, which reduces to the well known result expressed in terms of Lommel functions for the case of weakly converging spherical diffracted waves. The intensity distributions at the geometrical focal plane and along the axis are given. Numerical comparative examples are presented to illustrate the more general applicability of our results.     

Effects of pressure on flame propagation in a premixture containing fine fuel droplets

Combustion experiments on fuel droplet–vapor–air mixtures have been performed with a rapid expansion apparatus which generates monodispersed droplet clouds with narrow diameter distribution using the condensation method. The effects of fine fuel droplets on flame propagation were investigated for ethanol droplet–vapor–air mixtures at various pressures from 0.2 to 1.0 MPa. A stagnant fuel droplet–vapor–air mixture, generated in a rapid expansion chamber, was ignited at the center of the chamber using an ignition wire. Spherical flame propagation under constant-pressure conditions was observed with a high-speed video camera and flame speed was measured. Total equivalence ratio, and the ratio of liquid fuel mass to total fuel mass, was varied from 0.6 to 1.4 and from zero to 56%, respectively. The mean droplet diameter of fuel droplet–vapor–air mixtures was set at 8.5 and 11 μm. It was found that the flame speed of droplet–vapor–air mixtures less than 0.9 in the total equivalence ratio exceeds that of premixed gases of the same total equivalence ratio at all pressures. The flame speed of fuel droplet–vapor–air mixtures decreases as the pressure increases in all total equivalence ratios. At large ratios of liquid fuel mass to total fuel mass, the normalized flame speed (the flame speed of droplet–vapor–air mixtures divided by the flame speed of the premixed gas with the same total equivalence ratio), increases with the increase in pressure for fuel-lean mixtures, and it decreases for fuel-rich mixtures. The outcome is reversed at small ratios of liquid fuel mass to total fuel mass; the normalized flame speed decreases with the increase in pressure for fuel-lean mixtures, and increases for fuel-rich mixtures. The results suggest that the increase in pressure promotes droplet evaporation in the preheat zone.     

Unsteady response of chain-branching premixed-flames to pressure waves

Motivated by some recent experimental results that are incompatible with the standard one-step model, we present an analytical study of the unsteady response of flames to acoustic waves in the framework of a simple two-step chemistry model, using conditions that are appropriate to approximately describe lean methane–air flames. The calculated response functions are qualitatively different from those obtained with the standard one-step model. The results are sufficiently encouraging to suggest that the analysis should be extended in the near future to more detailed kinetics schemes.     

Solutions to the Boltzmann Equation in the Boussinesq Regime

We consider a gas in a horizontal slab in which the top and bottom walls are kept at different temperatures. The system is described by the Boltzmann equation (BE) with Maxwellian boundary conditions specifying the wall temperatures. We study the behavior of the system when the Knudsen number is small and the temperature difference between the walls as well as the velocity field is of order , while the gravitational force is of order ². We prove that there exists a solution to the BE for which is near a global Maxwellian, and whose moments are close, up to order ², to the density, velocity and temperature obtained from the smooth solution of the Oberbeck–Boussinesq equations assumed to exist for .     

A new approach to treating pressure oscillations in combustion instability phenomena

Developing exact models of combustion instabilities is not an easy task to carry out and requires a great deal of time prior to obtaining success. The present study proposes a low-order model for pressure oscillations that does not require any knowledge of the systems, any new physical findings nor intricate details regarding its operating condition. This new approach is obtained using a Modified Van der Pol’s equation (MVDP) which is tuned by use of a Dual Extended Kalman Filter (DKEF) as a recursive estimator with perspectives in control by computer. This phenomenological model is used to predict the pressure signal from a variety of different combustors. Input data were taken from experimental cases such as a Rijke tube, a gas turbine and a liquid-fuel aero-engine combustor. Furthermore, a simulation considering high frequency oscillations to show the capability of the new approach is presented. In all cases, the results demonstrated the feasibility of applying the tractable model MVDP and DKEF running together to investigate pressure oscillations in practical cases.     

Unstable combustion induced by oblique shock waves at the non-attaching condition of the oblique detonation wave

The instability of oblique shock wave (OSW) induced combustion is examined for a wedge with a flow turning angle greater than the maximum attach angle of the oblique detonation wave (ODW), where archival results rarely exist for this case in previous literatures. Numerical simulations were carried out for wedges of different length scales to account for the ratio of the chemical and fluid dynamic time scales. The results reveal three different regimes of combustion. (1) No ignition or decoupled combustion was observed if a fluid dynamic time is shorter than a chemical time behind an OSW. (2) Oscillatory combustion was observed behind an OSW if a fluid dynamic time is longer than a chemical time behind an OSW and the fluid dynamic time is shorter than the chemical time behind a normal shock wave (NSW) at the same Mach number. (3) Detached bow shock-induced combustion (or detached overdriven detonation wave) was observed if a fluid dynamic time is longer than a chemical time behind a NSW. Since no ignition or decoupled combustion occurs as a very slow reaction and the detached wave occurs as an infinitely fast reaction, the finite rate chemistry is considered to be the key for the oscillating combustion induced by an OSW over a wedge of a finite length with a flow turning angle greater than the maximum attach angle for an ODW. Since this case has not been previously reported, grid independency was tested intensively to account for the interaction between the shock and reaction waves and to determine the critical time scale where the oscillating combustion can be observed.     

A moving grid finite element method applied to a model biological pattern generator

Many problems in biology involve growth. In numerical simulations it can therefore be very convenient to employ a moving computational grid on a continuously deforming domain. In this paper we present a novel application of the moving grid finite element method to compute solutions of reaction–diffusion systems in two-dimensional continuously deforming Euclidean domains. A numerical software package has been developed as a result of this research that is capable of solving generalised Turing models for morphogenesis.