arXiv subjects

Joel Daou

Publications and source records attributed to Joel Daou.

13 recordsLinked to original sources

arXiv2026

Premixed flames in a stagnation point flow under Darcy's law

Premixed flames in stagnation point flows are traditionally described using Navier--Stokes equations where inertia and density variations play an important part in determining the flame structure. However, in porous media or Hele-Shaw configurations, Darcy's law replaces the momentum balance, shifting the governing physics to a balance between pressure and viscous forces. This study investigates non-adiabatic strained premixed flames under Darcy's law, pertinent in particular to confined flames in Hele-Shaw burners, accounting for non-unity Lewis numbers and volumetric heat losses. The flame is established in a planar counterflow formed by impinging a cold unburnt gas and a hot burnt gas maintained at the adiabatic flame temperature. We show that the jump in the strain rate across the flame is associated with a jump in viscosity, rather than, as in the classical Navier--Stokes case, a jump in density. Furthermore, the ratio of viscosity to the density-permeability product $\mu/\rho \kappa$, i.e., kinematic viscous resistance, is identified as a key coordinate stretching factor in the mathematical description of the flame structure. This ratio increases significantly across the flame. As a result: (1) the burnt gas acts as a strong viscous barrier, (2) for an increasing strain rate, flame migration towards the burnt gas is hindered, (3) for a decreasing strain rate, migration towards the unburnt gas is promoted, and (4) streamline refraction is augmented. By analysing the burning rate across varying strain rates and heat-loss parameters, we identify distinct extinction and ignition regimes that fundamentally differ from classical combustion theory, thereby providing new insights into flame stabilisation in friction-dominated environments and under confinement.

physics.flu-dyn↗

arXiv2026

Hydrodynamic theory of premixed flames under Darcy's law: Interfacial conditions and effects of nonunity Lewis number and heat loss

Premixed flames propagating in porous media or Hele-Shaw channels are governed by Darcy's law, which accounts for the strong frictional forces imposed by the solid matrix or confining walls. Prior theoretical studies of such flames have typically employed phenomenological Markstein-type corrections and have assumed unity Lewis numbers and adiabatic conditions. In this work, we develop a rigorous hydrodynamic theory for premixed flames under Darcy's law that incorporates nonunity Lewis numbers and heat losses. Using large activation-energy asymptotics and a systematic multiple-scale analysis, we derive the interfacial jump conditions across the flame from first principles. The conventional continuity requirements of mass flux and pressure at an interface under Darcy's law acquire corrections to the finite thickness of the flame. The adiabatic burning rate is shown to involve three distinct Markstein numbers, corresponding to curvature, tangential flow strain, and gravity-induced strain. The gravity term is unique to Darcy's law and has no counterpart in classical Navier--Stokes formulations. Moreover, the curvature Markstein number and the tangential strain Markstein number are found to be unequal, in contrast to the classical case where they coincide under constant transport properties. Explicit formulas for the Markstein numbers are provided, and the resulting new dispersion relation, linking the perturbation wave number $k$ to the growth rate $s$, takes the form $s = (a|k| - bk^2 - d|k|^3) / (1 + c|k|)$. This relation, applicable under Darcy's law, is to be compared to the classical Clavin--Garcia dispersion relation derived from the Navier--Stokes equations. The theory provides a rigorous foundation for flame dynamics in strongly confined environments, with direct applications to porous media combustion and Hele-Shaw cell experiments.

physics.flu-dyn↗

arXiv2026

Flame dynamics and Markstein numbers in Hele-Shaw cells and porous media under Darcy's law

The propagation of premixed flames in narrow Hele-Shaw cells and permeable porous media is governed by Darcy's law, leading to hydrodynamic behaviour distinct from conventional flames. This study investigates the role of confinement on flame dynamics, focusing on the associated Markstein numbers. A hydrodynamic model treating the flame as a discontinuity surface is presented, in which the burning rate depends on curvature and tangential flow strain, characterised by two Markstein numbers $\mathcal{M}_c$ and $\mathcal{M}_t$. A major finding is that $\mathcal{M}_c \neq \mathcal{M}_t$ under Darcy's law, as the law permits tangential velocity discontinuities at the flame front due to viscosity variations. Additionally, a third Markstein number $\mathcal{M}_g$ associated with gravity also emerges uniquely under Darcy's law. The Darcy-specific effects vanish in purely radial flows but are important for strained flames. In planar counterflows, for instance, the strain rate jump across the flame is dictated by the unburnt-to-burnt viscosity ratio $\mathfrak{m}$ rather than the density ratio $\mathfrak{r}$, a dramatic departure from conventional behaviour. The influence of confinement on the combined hydrodynamic instabilities of planar flames, namely Darrieus--Landau, Saffman--Taylor, and Rayleigh--Taylor instabilities, is discussed. Weakly nonlinear dynamics under strong confinement is found to follow a Michelson--Sivashinsky equation with modified coefficients (long-wave instability), while under moderate confinement, Ginzburg--Landau dynamics (finite-wavenumber instability) is found to apply. Strong confinement amplifies the Darrieus--Landau instability, enhancing hydrodynamic coupling in conjunction with augmented streamline refraction caused by tangential velocity discontinuities.

physics.flu-dyn↗

arXiv2025

Premixed flame quenching distance between cold walls: effects of flow and Lewis number

This study investigates the critical conditions for flame propagation in channels with cold walls. We analyze the impact of the Lewis number and flow amplitude ($A$) on the minimum channel width required to sustain a premixed flame. Our results span a wide range of Lewis numbers, encompassing both aiding and opposing flow conditions. Results are presented for both variable and constant density models. A combined numerical approach, involving stationary and time-dependent simulations, is employed to determine quenching distances and solution stability. We find that smaller Lewis numbers and aiding flows ($A < 0$) facilitate flame propagation in narrower channels, while opposing flows ($A > 0$) tend to destabilize the flame, promoting asymmetric solutions. For sufficiently large positive values of $A$, the quenching distance is determined by asymmetric solutions, rather than the typical symmetric ones.

physics.flu-dyn↗

arXiv2024

Hydrodynamic theory of premixed flames under Darcy's law

This paper investigates the theoretical implications of applying Darcy's law to premixed flames, a topic of growing interest in research on flame propagation in porous media and confined geometries. A multiple-scale analysis is carried out treating the flame as a hydrodynamic discontinuity in density, viscosity and permeability. The analysis accounts in particular for the inner structure of the flame. A simple model is derived allowing the original conservation equations to be replaced by Laplace's equation for pressure, applicable on both sides of the flame front, subject to specific conditions across the front. Such model is useful for investigating general problems under confinement including flame instabilities in porous media or Hele-Shaw channels. In this context, two Markstein numbers are identified, for which explicit expressions are provided. In particular, our analysis reveals novel contributions to the local propagation speed arising from discontinuities in the tangential components of velocity and gravitational force, which are permissible in Darcy's flows to leading order, but not in flows obeying Euler or Navier-Stokes equations.

physics.flu-dyn↗

arXiv2024

Hydrodynamic instabilities of propagating interfaces under Darcy's law

The hydrodynamic instabilities of propagating interfaces in Hele-Shaw channels or porous media under the influence of an imposed flow and gravitational acceleration are investigated within the framework of Darcy's law. The stability analysis pertains to an interface between two fluids with different densities, viscosities, and permeabilities, which can be susceptible to Darrieus-Landau, Saffman-Taylor, and Rayleigh-Taylor instabilities. A theoretical analysis, treating the interface as a hydrodynamic discontinuity, yields a simple dispersion relation between the perturbation growth rate $s$ and its wavenumber $k$ in the form $s=(ak - bk^2)/(1+ck)$, where $a$, $b$ and $c$ are constants determined by problem parameters. The constant $a$ characterises all three hydrodynamic instabilities, which are long-wave in nature. In contrast, $b$ and $c$, which characterize the influences of local curvature and flow strain on interface propagation speed, typically provide stabilisation at short wavelengths comparable to interface's diffusive thickness. The theoretical findings for Darcy's law are compared with a generalisation of the classical work by Joulin & Sivashinsky, which is based on an Euler-Darcy model. The comparison provides a conceptual bridge between predictions based on Darcy's law and those on Euler's equation and offers valuable insights into the role of confinement on interface instabilities in Hele-Shaw channels. Numerical analyses of the instabilities are carried out for premixed flames using a simplified chemistry model and Darcy's law. The numerical results corroborate with the explicit formula with a reasonable accuracy. Time-dependent numerical simulations of unstable premixed flames are carried out to gain insights into the nonlinear development of these instabilities.

physics.flu-dyn↗

arXiv2024

Three-dimensional diffusive-thermal instability of flames propagating in a plane Poiseuille flow

The three-dimensional diffusive-thermal stability of a two-dimensional flame propagating in a Poiseuille flow is examined. The study explores the effect of three non-dimensional parameters, namely the Lewis number $Le$, the Damk\"ohler number $Da$, and the flow Peclet number $Pe$. Wide ranges of the Lewis number and the flow amplitude are covered, as well as conditions corresponding to small-scale narrow ($Da \ll 1$) to large-scale wide ($Da \gg 1$) channels. The instability experienced by the flame appears as a combination of the traditional diffusive-thermal instability of planar flames and the recently identified instability corresponding to a transition from symmetric to asymmetric flame. The instability regions are identified in the $Le$-$Pe$ plane for selected values of $Da$ by computing the eigenvalues of a linear stability problem. These are complemented by two- and three-dimensional time-dependent simulations describing the full evolution of unstable flames into the non-linear regime. In narrow channels, flames are found to be always symmetric about the mid-plane of the channel. Additionally, in these situations, shear flow-induced Taylor dispersion enhances the cellular instability in $Le<1$ mixtures and suppresses the oscillatory instability in $Le>1$ mixtures. In large-scale channels, however, both the cellular and the oscillatory instabilities are expected to persist. Here, the flame has a stronger propensity to become asymmetric when the mean flow opposes its propagation and when $Le<1$; if the mean flow facilitates the flame propagation, then the flame is likely to remain symmetric about the channel mid-plane. For $Le>1$, both symmetric and asymmetric flames are encountered and are accompanied by temporal oscillations.

physics.flu-dyn↗

arXiv2024

Effect of a shear flow on the Darrieus-Landau instability in a Hele-Shaw channel

The Darrieus--Landau instability of premixed flames propagating in a narrow Hele-Shaw channel in the presence of a strong shear flow is investigated, incorporating also the Rayleigh--Taylor and diffusive-thermal instabilities. The flow induces shear-enhanced diffusion (Taylor dispersion) in the streamwise direction, but not in the spanwise direction and this leads to anisotropic diffusion and flame propagation. To understand how such anisotropies affect flame stability, two important cases are considered. These correspond to initial unperturbed conditions pertaining to a planar flame propagating in the streamwise or spanwise directions. The analysis is based on a two-dimensional model derived by asymptotic methods and solved numerically. These address the influence of the shear-flow strength (or Peclet number $Pe$), preferential diffusion (or Lewis number $Le$) and gravity (or Rayleigh number $Ra$). Dispersion curves characterizing the perturbation growth rate are computed for selected values of $Pe$, $Le$ and $Ra$. Taylor dispersion induced by strong shear flows is found to suppress the Darrieus--Landau instability and to weaken the flame wrinkling when the flame propagates in the streamwise direction. In contrast, when the flame propagates in the spanwise direction, the flame is stabilized in $Le<1$ mixtures, but destabilized in $Le>1$ mixtures. In the latter case, Taylor dispersion coupled with gas expansion facilitates flame wrinkling in an unusual manner. Specifically, stagnation points and counter-rotating vortices are encountered in the flame close to the unburnt gas side. More generally, an original finding is the demonstration that vorticity can be produced by a curved flame in a Hele-Shaw channel even in the absence of gravity, whenever $Pe \neq 0$, and that the vorticity remains confined to the flame preheat and reaction zones.

physics.flu-dyn↗

arXiv2024

Stability of diffusion flames under shear flow: Taylor dispersion and the formation of flame streets

Diffusion flame streets, observed in non-premixed micro-combustion devices, align parallel to a shear flow. They are observed to occur in mixtures with high Lewis number ($Le$) fuels, provided that the flow Reynolds number, or the Peclet number $Pe$, exceeds a critical value. The underlying mechanisms behind these observations have not yet been fully understood. In the present paper, we identify the coupling between diffusive-thermal instabilities and Taylor dispersion as a mechanism which is able to explain the experimental observations above. The explanation is largely based on the fact that Taylor dispersion enhances all diffusion processes in the flow direction, leading effectively to anisotropic diffusion with an effective (flow-dependent) Lewis number in the flow direction which is proportional to $1/Le$ for $Pe\gg 1$. Validation of the identified mechanism is demonstrated within a simple model by investigating the stability of a planar diffusion flame established parallel to a plane Poiseuille flow in a narrow channel. A linear stability analysis, leading to an eigenvalue problem solved numerically, shows that cellular (or finite wavelength) instabilities emerge for high Lewis number fuels when the Peclet number exceeds a critical value. Furthermore, for Peclet numbers below this critical value, longwave instabilities with or without time oscillations are obtained. Stability regime diagrams are presented for illustrative cases in a $Le$-$Pe$ plane where various instability domains are identified. Finally, the linear analysis is supported and complemented by time dependent numerical simulations, describing the evolution of unstable diffusion flames. The simulations demonstrate the existence of stable cellular structures and show that the longwave instabilities are conducive to flame extinction.

physics.flu-dyn↗

arXiv2024

Effective Lewis number and burning speed for flames propagating in small-scale spatio-temporal periodic flows

Propagation of premixed flames having thick reaction zones in rapidly-varying, small-scale, zero-mean, spatio-temporal periodic flows is considered. Techniques of large activation energy asymptotics and homogenization theory are used to determine the effective Lewis number $Le_{\mathrm{eff}}$ and the effective burning speed ratio $S_T/S_L$, which are influenced by the flow through flow-enhanced diffusion. As the flow Peclet number $Pe$ becomes large, the effective fuel and thermal diffusivities behave respectively like $(PeLe)^\sigma$ and $Pe^\sigma$, where $Le$ is the Lewis number and $\sigma\leq 2$ is a constant that depends on the flow and the flame-propagation direction. The maximal value $\sigma=2$ is achieved for steady, unidirectional, spatially periodic shear flows, while for steady 2D square vortices, we have $\sigma=1/2$. In general, the constant $\sigma$ is determined by solving a linear partial differential equation. The scaling laws for the diffusion coefficients lead to corresponding scaling laws for the effective Lewis number and the effective burning speed ratio of the form $Le_{\mathrm{eff}}\simeq Le^{1-\sigma}$ and $S_T/S_L\sim (Pe/Le)^{\sigma/2}$. Effects of thermal expansion and volumetric heat loss on the flame are also briefly discussed. In particular, it is shown that the quenching limit is enlarged by a factor $1/Le^\sigma$ for $Le<1$ and diminished by the same factor for $Le>1$, due to the flow-enhanced diffusion. Potential implications of the results for turbulent combustion are discussed. A special emphasis is placed on the dependence of the flame on $Le$ in high-intensity, small-scale flows. This dependence is intimately linked to flow-induced diffusion, rather than to the traditional molecular diffusion coupled with curvature effects. The effective Lewis number may also explain why turbulence aids ignition in $Le>1$ mixtures but hinders it in $Le<1$ mixtures.

physics.flu-dyn↗

arXiv2024

Diffusive-thermal instabilities of a planar premixed flame aligned with a shear flow

The stability of a thick planar premixed flame, propagating steadily in a direction transverse to that of unidirectional shear flow, is studied. A linear stability analysis is carried out in the asymptotic limit of infinitely large activation energy, yielding a dispersion relation. The relation characterises the coupling between Taylor dispersion (or shear-enhanced diffusion) and the flame thermo-diffusive instabilities, in terms of two main parameters, namely, the reactant Lewis number $Le$ and the flow Peclet number $Pe$. The implications of the dispersion relation are discussed and various flame instabilities are identified and classified in the $Le$-$Pe$ plane. An important original finding is the demonstration that for values of the Peclet number exceeding a critical value, the classical cellular instability, commonly found for $Le<1$, exists now for $Le>1$ but is absent when $Le<1$. In fact, the cellular instability identified for $Le>1$ is shown to occur either through a finite-wavelength stationary bifurcation (also known as type-I$_s$) or through a longwave stationary bifurcation (also known as type-II$_s$). The latter type-II$_s$ bifurcation leads in the weakly nonlinear regime to a Kuramoto--Sivashinsky equation, which is determined. As for the oscillatory instability, usually encountered in the absence of Taylor dispersion in $Le>1$ mixtures, it is found to be absent if the Peclet number is large enough. The stability findings, which follow from the dispersion relation derived analytically, are complemented and examined numerically for a finite value of the Zeldovich number. The numerical study involves both computations of the eigenvalues of a linear stability boundary-value problem and numerical simulations of the time-dependent governing partial differential equations. The computations are found to be in good qualitative agreement with the analytical predictions.

physics.flu-dyn↗

arXiv2024

Tricritical point as a crossover between type-I$_s$ and type-II$_s$ bifurcations

A tricritical point as a crossover between (stationary finite-wavelength) type-I$_s$ and (stationary longwave) type-II$_s$ bifurcations is identified in the study of diffusive-thermal (Turing) instability of flames propagating in a Hele-Shaw channel in a direction transverse to a shear flow. Three regimes exhibiting different scaling laws are identified in the neighbourhood of the tricritical point. For these three regimes, sixth-order partial differential equations are obtained governing the weakly nonlinear evolution of unstable solutions near the onset of instability. These sixth-order PDES may be regarded as the substitute for the classical fourth-order Kuramoto--Sivashinsky equation which is not applicable near the tricritical point.

nlin.PS↗

arXiv2024

A thick reaction zone model for premixed flames in two-dimensional channels

Direct interactions between the flow field and the chemical reaction in premixed flames occur when the reaction zone thickness is comparable to, or greater than flow length scales. To study such interactions, a laminar model is considered that has direct bearings to steadily propagating deflagrations in a Hele-Shaw channel with a background plane Poiseuille flow. The study employs asymptotic analyses, pertaining to large activation energy and lubrication theories and considers a distinguished limit where the channel width is comparable to the reaction zone thickness, with account being taken of thermal-expansion and heat-loss effects. The reaction zone structure and burning rates depend on three parameters, namely, the Peclet number, $\mathcal{P}$, the Lewis number, $Le$ and the ratio of channel half-width to reaction zone thickness, $\lambda_*$. When the parameter $\lambda_*$ is small, transport processes are controlled by Taylor's dispersion mechanism and an explicit formula for the effective burning speed $S_T$ is obtained. The formula indicates that $S_T/S_L \propto 1/Le$ for $\mathcal{P}\gg 1$, which interestingly coincides with a recent experimental prediction of the flame speed in a highly turbulent jet flame. The results suggest that the role played by differential diffusion effects is significant both in laminar and turbulent cases. The reason for the peculiar $1/Le$ dependence can be attributed, in our laminar model, to Taylor dispersion. Presumably, this dependence may be attributed to a similar but more general mechanism in the turbulent case, rather than to diffusive-thermal curvature effects. The latter effects play however an important role in determining the flame speed when $\lambda_*$ is large. The magnitude of heat losses at extinction, is multiplied by a factor $1/Le^2$ in comparison with those corresponding to the no-flow case in narrow channels.

physics.flu-dyn↗