Velocity of weakly turbulent flames of finite thickness

The velocity increase of a weakly turbulent flame of finite thickness is investigated using analytical theory developed in previous papers. The obtained velocity increase depends on the flow parameters: on the turbulent intensity, on the turbulent spectrum and on the characteristic length scale. It also depends on the thermal and chemical properties of the burning matter: thermal expansion, the Markstein number and the temperature dependence of transport coefficients. It is shown that the influence of the finite flame thickness is especially strong close to the resonance point, when the wavelength of the turbulent harmonic is equal to the cut off wavelength of the Darrieus–Landau instability. The velocity increase is almost independent of the Prandtl number. On the contrary, the Markstein number is one of the most important parameters controlling the velocity increase. The relative role of the external turbulence and the Darrieus–Landau instability for the velocity increase is studied for different parameters of the flow and the burning matter. The velocity increase for turbulent flames in methane and propane fuel mixtures is calculated for different values of the equivalence ratio. The present theoretical results are compared with previous experiments on turbulent flames. In order to perform the comparison, the theoretical results of the present paper are extrapolated to the case of a strongly corrugated flame front using the ideas of self-similar flame dynamics. The obtained theoretical results are in a reasonable agreement with the experimental data, taking into account the uncertainties of both the theory and the experiments. It is shown that in many experiments on turbulent flames the Darrieus–Landau instability is more important for the flame velocity than the external turbulence.     

Large scale effects in weakly turbulent premixed flames

In this study we numerically investigate large scale premixed flames in weakly turbulent flow fields. A large scale flame is classified as such based on a reference hydrodynamic lengthscale being larger than a neutral (cutoff) lengthscale for which the hydrodynamic or Darrieus–Landau (DL) instability is balanced by stabilizing diffusive effects. As a result, DL instability can develop for large scale flames and is inhibited otherwise. Direct numerical simulations of both large scale and small scale three-dimensional, weakly turbulent flames are performed at constant Karlovitz and turbulent Reynolds number, using two paradigmatic configurations, namely a statistically planar flame and a slot Bunsen flame. As expected from linear stability analysis, DL instability induces its characteristic cusp-like corrugation only on large scale flames. We therefore observe significant morphological and topological differences as well as DL-enhanced turbulent flame speeds in large scale flames. Furthermore, we investigate issues related to reaction rate modeling in the context of flame surface density closure. Thicker flame brushes are observed for large scale flames resulting in smaller flame surface densities and overall larger wrinkling factors.     

Hydrodynamic instability of premixed flame propagating in narrow planar channel in the presence of gas flow

The paper analyses the hydrodynamic instability of a flame propagating in the space between two parallel plates in the presence of gas flow. The linear analysis was performed in the framework of a two-dimensional model that describes the averaged gas flow in the space between the plates and the perturbations development of two-dimensional combustion wave. The model includes the parametric dependences of the flame front propagation velocity on its local curvature and on the combustible gas velocity averaged along the height of the channel. It is assumed that the viscous gas flow changes the surface area of the flame front and thereby affects the propagation velocity of the two-dimensional combustion wave. In the absence of the influence of the channel walls on the gas flow, the model transforms into the Darrieus–Landau model of flame hydrodynamic instability. The dependences of the instability growth rate on the wave vector of disturbances, the velocity of the unperturbed gas flow, the viscous friction coefficients and other parameters of the problem are obtained. It is shown that the viscous gas flow in the channel can lead, in some cases, to a significant increase in instability compared with a flame propagating in free space. In particular, the instability increment depends on the direction of the gas flow with respect direction of the flame propagation. In the case when the gas flow moves in the opposite direction to the direction of the flame propagation, the pulsating instability can appear.     

Strain rates,flow patterns and flame surface densities in hydrodynamically unstable,weakly turbulent premixed flames

Recent numerical and experimental studies have unveiled a potentially marked difference between the laminar as well as turbulent propagation of premixed flames exhibiting Darrieus–Landau (DL) (or hydrodynamic) instabilities from flames for which instabilities are inhibited. In this study we utilize two-dimensional numerical simulations of slot burner flames as well as experimental Propane–Air Bunsen flames to analyse differences in turbulent propagation, strain rate and induced flow patterns of hydrodynamically stable and unstable flames. We also investigate the effects of hydrodynamic instability on quantities which are directly related to reaction rate closure models, such as flame surface density and stretch factor. A clear enhancement of turbulent flame speed can be observed for unstable flames, generally mitigated at higher turbulence intensity, which is attributed to a flame area increase induced by the characteristic cusp-like DL-induced corrugation, absent in stable flames, which occurs concurrently and in synergy with turbulent wrinkling. Unstable flames also exhibit, both numerically and experimentally, a different correlation between strain rate and flame curvature and are observed to give rise to a channeling of the induced flow in the fresh mixture. Conditionally averaged flame surface density is also observed to attain smaller values in unstable flames, as a result of the thicker turbulent flame brush, indicating that closure models should incorporate instability-related parameters in addition to turbulence-related parameters.     

Impact of the gravity field on stability of premixed flames propagating between two closely spaced parallel plates

The stability of a planar flame front propagating between two parallel adiabatic plates inclined at an arbitrary angle is investigated in the frame of narrow-channel approximation. It is demonstrated that buoyancy forces can suppress the hydrodynamic (Darrieus–Landau) and cellular (diffusive-thermal) instabilities for sufficiently large value of the gravity parameter for the case of downward-propagating flames. The stability analysis reveals that in the case of oscillatory diffusive-thermal instability, the flame front cannot be stabilized in the similar way. Finally, the stability results are compared satisfactorily with unsteady numerical simulations.     

Analytical Interaction of the Acoustic Wave and Turbulent Flame

A modified resonance model of a weakly turbulent flame in a high-frequency acoustic wave is derived analytically. Under the mechanism of Darrieus-Landau instability, the amplitude of flame wrinkles, which is as functions of the expansion coefficient and the perturbation wave number, increases greatly independent of the stationary＇ turbulence. The high perturbation wave number makes the resonance easier to be triggered but weakened with respect to the extra acoustic wave. In a closed burning chamber with the acoustic wave induced by the flame itself, the high perturbation wave number is to restrain the resonance for a realistic flame.     

Propagation of Darrieus–Landau unstable laminar and turbulent expanding flames

The propagation of laminar and turbulent expanding flames subjected to Darrieus–Landau (DL), hydrodynamic instability was experimentally studied by employing stoichiometric H₂/O₂/N₂ flames under quiescent and turbulent conditions performed in a newly developed medium-scale, fan-stirred combustion chamber. In quiescent environment, DL unstable laminar flame exhibits three-stage propagation, i.e. smooth expansion, transition acceleration, and self-similar acceleration. The self-similar acceleration is characterized by a power-law growth of acceleration exponent, α, with normalized Peclet number, which is different from the usually suggested self-similar propagation with a constant α. The imposed turbulence advances the onset of both transition acceleration and self-similar acceleration stages and promotes the strength of flame acceleration as additional wrinkles are invoked by turbulence eddies. A DL–turbulent interaction regime is confirmed to be the classical corrugated flamelets regime. Furthermore, the DL instability significantly facilitates the propagation of expanding flames in medium and even intense turbulence. The development of DL cells is not suppressed by turbulence eddies, and it needs to be considered in turbulent combustion.     

Turbulent flame and the darrieus–landau instability in a three-dimensional flow

The velocity of a weakly turbulent flame influenced by the Darrieus–Landau (DL) instability in a three-dimensional geometry is investigated on the basis of a model nonlinear equation. The equation takes into account realistically large thermal expansion of burning matter, external turbulence and thermal conduction related to small, but finite flame thickness. An external turbulent flow is imitated by a model obeying the Kolmogorov law. The effects of the DL instability and external turbulence are studied, first separately and then as they influence the flame dynamics together for different values of the turbulent intensity, different thermal expansion of the burning matter and different length scales of the hydrodynamic motion controlled by the width of a hypothetic tube with ideally adiabatic walls. The velocity increase obtained is in a good agreement with experimental results in the case of relatively weak turbulent intensity.     

Accelerating flames in tubes—an analysis

Flame acceleration in tubes is studied. A tube filled with flammable mixture is closed at one end and open to the atmosphere at its second end. When ignition takes place near the closed end, it is well-known from experiments that the flame may accelerate, oscillate and eventually reach considerable speeds. A one-dimensional analysis is presented, based upon the assumption that the flame front propagates at a speed that is small compared to the speed of sound. The analysis leads to a construction of the complete unsteady solution. Results from the analysis and from a numerical simulation are compared. They are similar enough to validate the analysis. The tube acoustics are set in motion by the expansion of the fluid due to ignition at the closed end. Subsequently, both spectrum and amplitude evolve because of the motion of the temperature interface, and because of forcing by the flame front, which the analysis precisely quantifies. Oscillations in the front position are strong enough to result in flow reversal. In addition, the induced periodic acoustic acceleration of the temperature and density interface will periodically make the flame front Rayleigh–Taylor unstable, which should result in the dramatic increase in the propagation speed seen in experiments.     

Phase synchronization and collective interaction of multiple flamelets in a laboratory scale spray combustor

In this paper, we investigate the coupled behvior of the acoustic field in the confinement and the unsteady flame dynamics in a laboratory scale spray combustor. We study this interaction during the intermittency route to thermoacoustic instability when the location of the flame is varied inside the combustor. As the flame location is changed, the synchronization properties of the coupled acoustic pressure and heat release rate signals change from desynchronized aperiodicity (combustion noise) to phase synchronized periodicity (thermoacoustic instability) through intermittent phase synchronization (intermittency). We also characterize the collective interaction between the multiple flamelets anchored at the flame holder and the acoustic field in the system, during different dynamical states observed in the combustor operation. When the signals are desynchronized, we notice that the flamelets exhibit a steady combustion without the exhibition of a prominent feedback with the acoustic field. In a state of intermittent phase synchronization, we observe the existence of a short-term coupling between the heat release rate and the acoustic field. We notice that the onset of collective synchronization in the oscillations of multiple flamelets and the acoustic field leads to the simultaneous emergence of periodicity in the global dynamics of the system. This collective periodicity in both the subsystems causes enhancement of oscillations during epochs of amplitude growth in the intermittency signal. On the contrary, the weakening of the coupling induces suppression of periodic oscillations during epochs of amplitude decay in the intermittency signal. During phase synchronization, we notice a sustained synchronized movement of all flamelets with the periodicity of the acoustic cycle in the system.     

Parametric resonance of truncated conical shells rotating at periodically varying angular speed

Parametric resonance of a truncated conical shell rotating at periodically varying angular speed is studied in this paper. Based upon the Love?s thin shell theory and generalized differential quadrature (GDQ) method, the equations of motion of a rotating conical shell are derived. The time-dependent rotating speed is assumed to be a small and sinusoidal perturbation superimposed upon a constant speed. Considering the periodically rotating speed, the conical shell system is a parametric excited system of the Mathieu–Hill type. The improved Hill?s method is utilized for parametric instability analysis. Both the primary and combination instability regions for various natural modes and boundary conditions are obtained numerically. The effects of relative amplitude and constant part of periodically rotating speed and cone angle on the instability regions are discussed in detail. It is shown that for the natural mode with lower circumferential wavenumber, only the primary instability regions exist. With the increasing circumferential wavenumber, the instability widths are reduced significantly and the combination instability region might appear. The results for different boundary conditions are substantially similar. Increasing the constant rotating speed (or cone angle) all lead to the movements of instability regions and the appearance of combination instability region. The former will cause the instability width increasing, while the latter will reduce the instability width. The variation of length-to-radius ratio only causes the movements of instability regions.     

Flame-acoustic coupling of combustion instability in a non-premixed backward-facing step combustor: the role of acoustic-Reynolds stress

Combustion instability in a laboratory scale backward-facing step combustor is numerically investigated by carrying out an acoustically coupled incompressible large eddy simulation of turbulent reacting flow for various Reynolds numbers with fuel injection at the step. The problem is mathematically formulated as a decomposition of the full compressible Navier–Stokes equations using multi-scale analysis by recognising the small length scale and large time scale of the flow field relative to a longitudinal mode acoustic field for low mean Mach numbers. The equations are decomposed into those for an incompressible flow with temperature-dependent density to zeroth order and linearised Euler equations for acoustics as a first order compressibility correction. Explicit coupling terms between the two equation sets are identified to be the flow dilatation as a source of acoustic energy and the acoustic Reynolds stress (ARS) as a source of flow momentum. The numerical simulations are able to capture the experimentally observed flow–acoustic lock-on that signifies the onset of combustion instability, marked by a shift in the dominant frequency from an acoustic to a hydrodynamic mode and accompanied by a nonlinear variation of pressure amplitude. Attention is devoted to flow conditions at two Reynolds numbers before and after lock-on to show that, after lock-on, the ARS causes large-scale vortical rollup resulting in the evolution of a compact flame. As compared to acoustically uncoupled simulations at these Reynolds numbers that show an elongated flame with no significant roll up and disturbance in the upstream flow field, the ARS is seen to alter the shear layer dynamics by affecting the flow field upstream of the step as well, when acoustically coupled.     

Parametric instabilities of a viscous liquid layer covered by a thin elastic plate under vertical periodic motion

Parametric instabilities of a horizontal liquid layer with a finite depth covered by a thin elastic plate under a vertical periodic motion are investigated with account taken of the viscosity of the liquid layer. The primary regions and the secondary ones of dynamic instabilities are determined by using the equation of a thin elastic plate including the normal component of the viscous stress, but not the tangential component of it. The critical amplitude of the imposed oscillation, beyond which a parametric instability occurs (that is, the neutral stability curves) is found in the space of the frequency and amplitude of the imposed vertical oscillation. These results are confirmed by experimental ones for a liquid layer of glycerine covered with a thin rubber plate.     

Analytical investigation on unstable vibrations of single-mesh gear systems with velocity-modulated time-varying stiffness

Time-varying mesh stiffness parametrically excites gear systems and causes severe vibrations and instabilities. Taking speed fluctuations into account, the time-varying mesh stiffness is frequency modulated, and more complex instabilities might arise. Considering two different speed fluctuation models, parametric instability associated with velocity-modulated time-varying stiffness is analytically investigated using a typical single-mesh gear system model. Closed-form approximations are obtained by perturbation analysis, and verified by numerical analysis. The effects of the amplitude of the mesh stiffness variation, the characteristics of speed fluctuations and damping on parametric instability are systematically examined.     

Detailed simulation of laser-induced ignition,spherical-flame acceleration,and the origins of hydrodynamic instability

Ignition of a lean hydrogen–oxygen premixture by focused-laser-induced breakdown and subsequent three-dimensional expanding-flame instabilities are simulated in high detail. Both diffusive–thermal and hydrodynamic (Darrieus–Landau) instabilities are active and accelerate the flame expansion. The fluid is a partially-ionized gas in local thermodynamic equilibrium with detailed kinetics and transport models, starting from initial conditions from an auxiliary simulation based on a two-temperature local thermodynamic non-equilibrium model. After the decay of the initial laser-induced plasma, the r ～ t^1.5 growth in time of the flame radius matches theory and experimental observations. Based on hydrodynamic theory for spherical-flame propagation, a global Karlovitz number is defined as the ratio of the hydrodynamic to flame-distortion time scales. It initially increases during the diffusive–thermal instability stage, then with the onset of significant baroclinic torque, this trend reverses, with vorticity production becoming the dominant mechanism of instability.     

Effect of rectified and modulated sine forces on chaos in Duffing oscillator

The effect of rectified and modulated sine forces on the onset of horseshoe chaos is studied both analytically and numerically in the Duffing oscillator. With single force analytical threshold condition for the onset of horseshoe chaos is obtained using the Melnikov method. The Melnikov threshold curve is drawn in a parameter space. For the rectified sine wave, onset of cross-well asymptotic chaos is observed just above the Melnikov threshold curve. For the modulus of sine wave long time transient motion followed by a periodic attractor is realized. The possibility of controlling of chaos by the addition of second modulated force is then analyzed. Parametric regimes where suppression of horseshoe chaos occurs are predicted analytically and verified numerically. Interestingly, suppression of chaos is found in the parametric regimes where the Melnikov function does not change sign.     

Darrieus - Landau instability,growing cycloids and expanding flame acceleration

A premixed flame, propagating away from a point ignition source into an unlimited domain displays an increasing flame speed after the flame size has grown beyond a transition radius. Experiments by Gostintsev et al are described by the relation R = R₁ + At^3/2, where t is the time from ignition and, where S_L is the flame burning velocity and is the thermal diffusivity. The non-dimensional function a() is determined from the experimental results to be equal to 0.002², where is the density ratio across the flame.

In the present work, two-dimensional Lagrangian simulations of flame propagation also display a radial growth with a 3/2 power-law behaviour. This is a potential flow model - no vorticity is included. Hence, the Darrieus - Landau hydrodynamic instability by itself can generate flame acceleration. The numerical results are summarized by the relation R = R1+(2/40)L(SLt/L)3/2, where L is a reference length and is the volume production ratio, = - 1. Equating the zone of velocity jump in the numerical scheme with the temperature jump in hydrocarbon flames allows a definition of an effective thermal diffusivity in the numerical work as n = 0.0081SLL. With this relation, the radial growth is given as, in good agreement with the experimental result and the numerical results of Filyand et al.     

Numerical simulation of flames as gas-dynamic discontinuities

The dynamics of thin premixed flames is computationally studied within the context of a hydrodynamic theory. A level-set method is used to track down the flame, which is treated as a free-boundary interface. The flow field is described by the incompressible Navier–Stokes equations, with different densities for the burnt and unburnt gases, supplemented by singular source terms that properly account for thermal expansion effects. The numerical scheme has been tested on several benchmark problems and was shown to be stable and accurate. In particular, the propagation of a planar flame front and the dynamics of hydrodynamically unstable flames were successfully simulated. This includes recovering the planar front in narrow domains, the Darrieus–Landau linear growth rate for long waves of small amplitude, and the nonlinear development of cusp-like structures predicted by the Michelson–Sivashinsky equation for a small density change. The stationary flame of a Bunsen burner with uniform and parabolic outlet flows were also simulated, showing in particular a careful mapping of the flow field. Finally, the evolution of a hydrodynamically unstable flame was studied for finite amplitude disturbances and realistic values of thermal expansion. These results, which constitute one of the main objectives of this study, elucidate the effect of thermal expansion on flame dynamics.     

Exploring a critical diameter for thermo-acoustic instability of downward propagating flames in tubes

Thermo-acoustic oscillations are observed when a flame ignited at open end of a tube propagates towards the closed end due to interaction between unsteady heat release rate fluctuations from flame and acoustic fluctuations. In our past work, it was found that thermo-acoustic instability increases with decreasing diameter from 7.0 cm to 3.0 cm. A recent study in flame propagation in Hele–Shaw cells showed that thermo-acoustic instability is not observed for plate separation less than or equal to 0.4 cm. Thermoacoustic instabilities cannot be observed in very narrow tubes due to excessive damping from the wall. This opens up the possibility of a critical diameter where thermo-acoustic instability would be maximum. In this work we perform flame propagation experiments with diameter of combustion tube in the range 0.5 cm to 3 cm for a fixed length of 70.2 cm. It was found that thermo-acoustic parametric instability begins at lowest laminar burning velocity when the diameter is around 1.0 cm. This diameter is termed as critical diameter. Critical diameter is found to be independent of Lewis number of mixtures. Existence of a critical diameter is thus proved experimentally. Growth rates of primary instability increase with decreasing diameter and show a maximum around the critical diameter and decrease with further decrease in tube diameter. But, growth rates of secondary instability as well as maximum pressure fluctuation amplitude decreases continuously with decreasing diameter. Mechanisms responsible for these observations and existence of a critical diameter are clarified.     

Instabilities and sound speed of a Bose-Einstein condensate in the Kronig-Penney potential

We analyze the full set of Bloch wave stationary solutions for a Bose-Einstein condensate in the Kronig-Penney potential. We investigate the Landau instability and dynamical instability of the Bloch states in the lowest Bloch band, including the loop if it appears. The stability phase diagrams are shown to be similar as in the case of the sinusoidal optical lattice potential. We also compute the speed of sound as a function of the potential strength. The trend is shown to be similar to the sinusoidal case, reflecting our general conclusion that, in any one-dimensional periodic potential, the sound speed always falls monotonically with increasing potential strength, no matter whether the atomic interaction is weak or strong. The Kronig-Penney potential, being analytically tractable and hence more advantageous than the sinusoidal potential, therefore serves as a good model for understanding the phenomena of Bose-Einstein condensation.