Daniel Fernández-Galisteo

Anne Dejoan

Josué Melguizo-Gavilanes

Vadim N Kurdyumov

A three-dimensional study of the influence of momentum loss on hydrodynamically unstable premixed flames

Keywords: Hele-Shaw, confined flow, hydrodynamic instability, premixed flame, single-cusp structure

The propagation of an isobaric premixed flame into a quiescent gas mixture of fuel and oxidizer contained between two parallel plates is investigated numerically. The plates are separated by a small distance h and considered as adiabatic. The mixture is assumed to be lean in fuel and the combustion model includes a single-step Arrhenius-type reaction, constant heat capacity and unity fuel Lewis number. Transport properties are considered to be temperature dependent or constant, which allows us to decouple two different instability mechanisms of hydrodynamic nature: (i) Darrieus-Landau (associated with the density change due to thermal expansion) and (ii) Saffman-Taylor (associated with the viscosity contrast). The overall propagation rate is analyzed as a function of the dimensionless parameter a = h/δT , where δT is the thermal thickness of the planar flame front. Results show that, as the distance between the plates decreases, loss of momentum enhances the hydrodynamic instability in comparison with that of a freely (unconfined) propagating flame. Likewise, the viscosity contrast across the flame brings about an additional destabilizing mechanism. It is also shown that, near the limit a → 0, the problem can be effectively described by a two-dimensional formulation, in which momentum conservation is reduced to a linear equation for the velocity similar to Darcy's law.

1 1. Introduction
The instability produced by the deformation of a planar premixed flame front was first discovered independently by [START_REF] Darrieus | Propagation d'un front de flamme, unpublished work presented at La Technique Moderne[END_REF] [START_REF] Darrieus | Propagation d'un front de flamme, unpublished work presented at La Technique Moderne[END_REF] and Landau (1944) [START_REF] Landau | On the theory of slow combustion[END_REF]. This hydrodynamic instability stems from the deflection of streamlines through the flame front produced by gas expansion, giving rise to a positive growth rate of any small flame wrinkle larger than the characteristic flame thickness in absence of other effects. Later, insights on stabilizing/destabilizing influence of diffusion processes and gravity on the linear evolution was reported in further theoretical works [START_REF] Pelce | Influence of hydrodynamics and diffusion upon the stability limits of laminar premixed flames[END_REF][START_REF] Matalon | Flames as gasdynamic discontinuities[END_REF][START_REF] Frankel | The effect of viscosity on hydrodynamic stability of a plane flame front[END_REF][START_REF] Frankel | On effects due to thermal expansion and Lewis number in spherical flame propagation[END_REF].

The long-term dynamics of hydrodynamically unstable flames makes its study more difficult due to the presence of nonlinearities. In particular, at large times after the initial stages of the linear development of the instability, the flame front reaches a large cellular shape controlled by the geometrical domain size. The nonlinear Michelson-Sivashinsky equation [START_REF] Michelson | Nonlinear analysis of hydrodynamic instability in laminar flames. II. Numerical experiments[END_REF] was shown to qualitatively reproduce relevant features of these unstable flames [START_REF] Searby | Comparison of experiments and nonlinear model equation for spatially developing flame instability[END_REF][START_REF] Radisson | Nonlinear dynamics of flame fronts with large-scale stabilizing effects[END_REF][START_REF] Creta | Turbulent propagation of premixed flames in the presence of Darrieus-Landau instability[END_REF][START_REF] Radisson | Nonlinear dynamics of premixed flames: from deterministic stages to stochastic influence[END_REF]. However, its restriction to small thermal expansion has motivated the use of more realistic approaches built on the complete Navier-Stokes equations [START_REF] Denet | Laminar premixed flame dynamics: a comparison of model and complete equations[END_REF][START_REF] Sharpe | Linear stability of planar premixed flames: reactive Navier-Stokes equations with finite activation energy and arbitrary Lewis number[END_REF][START_REF] Rastigejev | Nonlinear evolution of hydrodynamically unstable premixed flames[END_REF][START_REF] Altantzis | Hydrodynamic and thermodiffusive instability effects on the evolution of laminar planar lean premixed hydrogen flames[END_REF][START_REF] Creta | Strain rate effects nonlinear development of hydrodynamically unstable flames[END_REF]. In agreement with experiments, both theoretical and numerical works show that the nonlinear growth of the Darrieus-Landau instability results in the formation of a steady-propagating monocellular flame with a cusp-like structure (a curved flame with a sharp peak pointing toward the burned gas). In moderate-size domains (of about 40 times the flame thickness), this structure, that fills the entire domain, remains stable. However, for larger domains, secondary structures may appear on the flame. These structures propagate in form of small wrinkles along the flame surface giving the flame evolution an unsteady character. The very recent experiments [START_REF] Radisson | Nonlinear dynamics of premixed flames: from deterministic stages to stochastic influence[END_REF] carried out in a large device (150-cm long × 50-cm wide) reports a merging/splitting cell process occurring at the longtime evolution of the nonlinear regime. This unsteady process, also reported in numerical studies, has been explained as a result of stochastic influence of background noise [START_REF] Creta | Turbulent propagation of premixed flames in the presence of Darrieus-Landau instability[END_REF][START_REF] Radisson | Nonlinear dynamics of premixed flames: from deterministic stages to stochastic influence[END_REF][START_REF] Denet | Laminar premixed flame dynamics: a comparison of model and complete equations[END_REF][START_REF] Rastigejev | Nonlinear evolution of hydrodynamically unstable premixed flames[END_REF].

A relevant experimental facility for examining flame instabilities is the Hele-Shaw burner. In this device, the flame propagates between two parallel plates separated by a small distance, h. When the plates are made of transparent material, direct visualization of the flame front is possible in a quasi-two-dimensional (quasi-2D) form. To our knowledge, first experimental observations of flame instabilities in Hele-Shaw burners can be traced back to the work of Sharif et al. [START_REF] Sharif | Premixed-gas flame propagation in Hele-Shaw cells[END_REF]. Since then, good experimental progress has been made [START_REF] Radisson | Nonlinear dynamics of premixed flames: from deterministic stages to stochastic influence[END_REF][START_REF] Almarcha | Experimental two dimensional cellular flames[END_REF][START_REF] Sarraf | Darrieus-Landau instability and Markstein numbers of premixed flames in a Hele-Shaw cell[END_REF][START_REF] Veiga-López | Experimental analysis of oscillatory premixed flames in a Hele-Shaw cell propagating towars a closed end[END_REF][START_REF] Shen | Flame propagation in quasi-2D channels: stability, rates and scaling[END_REF][START_REF] Alexeev | Experimental study of cellular premixed propane flames in a narrow gap between parallel plates[END_REF][START_REF] Veiga-López | Thermoacoustic analysis of lean premixed hydrogen flames in narrow vertical channels[END_REF][START_REF] Veiga-López | Unexpected propagation of ultra-lean hydrogen flames in narrow gaps[END_REF][START_REF] Tayyab | Experimental and numerical Lattice-Boltzmann investigation of the Darrieus-Landau instability[END_REF]. Nevertheless, heat and momentum losses inherent to Hele-Shaw burners may interplay with the intrinsic flame instabilities. This influence was first reported by Joulin and Sivashinsky [START_REF] Joulin | Influence of momentum and heat losses on the large-scale stability of quasi-2D premixed flames[END_REF] who performed a linear stability analysis modeling the flame as a thin discontinuity. The authors con-cluded that friction-induced pressure gradients, coupled with viscosity contrast, enhance the hydrodynamic instability. The latter viscosity effects are referred to as the Saffman-Taylor instability [START_REF] Saffman | The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid[END_REF] and is originally associated with the formation of fingerlike structures at the interface between two fluids with different viscosity driven by an imposed pressure gradient. If the less viscous fluid displaces the more viscous one, the interface is unstable. The analogy with flames rests upon the viscosity change present between the unburnt and the burnt gas across the front that is also subject to pressure gradients.

The effect of confinement in Hele-Shaw burners has been studied in numerical works [START_REF] Kang | A computational study of Saffman-Taylor instability in premixed flames[END_REF][START_REF] Kang | Effects of heat and momentum losses on the stability of premixed flames in a narrow channel[END_REF][START_REF] Fernández-Galisteo | Analysis of premixed flame propagation between two closely-spaced parallel plates[END_REF][START_REF] Martínez-Ruiz | The role of conductive heat losses on the formation of isolated flame cells in Hele-Shaw chambers[END_REF][START_REF] Fernández-Galisteo | Impact of the gravity field on stability of premixed flames propagating between two closely spaced parallel plates[END_REF] through averaged flow properties across h, thus reducing the problem to a quasi-2D form. Note, however, that in addition to the above mentioned confinement effects the flame behavior can also be affected by curvature effects along the direction perpendicular to the plates. The latter effects can evidently not be captured with a quasi-2D model. The natural way to improved understanding of these phenomena is thus to carry out three-dimensional (3D) simulations by varying the distance between the plates, h, as in [START_REF] Melguizo-Gavilanes | Threedimensional simulations of lean H 2 -air flames propagating in a narrow gap: Validity of the quasi-twodimensional approximation[END_REF]. The dimensionless parameter a = h/δT , that can be interpreted as a Peclet number, emerges as functional parameter for the study, with δT the thermal thickness of the planar flame front. Furthermore, two additional set of simulations are performed in this work: (i) quasi-2D simulations, as in [START_REF] Fernández-Galisteo | Analysis of premixed flame propagation between two closely-spaced parallel plates[END_REF][START_REF] Martínez-Ruiz | The role of conductive heat losses on the formation of isolated flame cells in Hele-Shaw chambers[END_REF][START_REF] Fernández-Galisteo | Impact of the gravity field on stability of premixed flames propagating between two closely spaced parallel plates[END_REF] and (ii) pure 2D simulations. The quasi-2D simulations can be interpreted as the high wall-friction limit (a → 0). In the pure 2D simulations gas friction with the walls is absent.

General formulation

We consider a laminar premixed flame propagating in a Hele-Shaw burner. A sketch of this configuration is given in Fig. 1. The chemical reaction is modeled by a global irreversible step of the type Fuel + Oxidizer → Products. The corresponding reaction rate (moles of fuel per unit time per unit volume) is given by an Arrhenius law and takes the form ω = Bρ 2 YF exp(-E/RT ) for a mixture deficient in fuel, where E is the overall activation energy, T is the temperature, ρ is the density, YF is the fuel mass fraction, R is the universal gas constant, and B is an appropriately defined pre-exponential factor that includes the mass fraction of the oxidant. Primes ( ) denote dimensional quantities.

The formulation is derived using the low-Mach number approximation and constant mixture heat capacity cp, Lewis, Le = D/DT , and Prandtl, P r = µ /(λ /cp), numbers, where D is the mass diffusivity of the fuel; DT , µ and λ are the thermal diffusivity, viscosity and thermal conductivity of the gas mixture, respectively. The transport properties are considered to vary with temperature, µ /µ0 = λ /λ0 = (T /T0) σ , with σ an exponent equals to 0 or 0.7, see [START_REF] Smooke | Formulation of the premixed and non-premixed test problems[END_REF]. Subscript 0 denotes fresh unburnt mixture.

3D formulation (finite h)

The full set of governing equations are scaled using the temperature T0, mass fraction YF 0 , density ρ0 and viscosity µ0 of the unburnt gas. The thermal thickness of the planar flame δT = DT 0 /SL, with SL the planar burning velocity, is used as the reference length scale, and DT 0 /S 2 L as the reference time scale. The nondimensional variables are

x = (x /δT , y /δT , z /δT ), u = (u /SL, v /SL, w /SL), t = t S 2 L /DT 0 , ρ = ρ /ρ0, p = (p -patm)/ρ0S 2 L , θ = (T -T0)/(Ta -T0), Y = YF /YF 0 ,
where Ta = T0 + QYF 0 /cp is the adiabatic temperature, with Q the heat of combustion. The dimensionless equations, written in a reference frame moving at flame velocity u f with respect to the wall, read

∂ρ ∂t + ∇ • [ρ(u -u f )] = 0, (1)
∂(ρu) ∂t + ∇ • [ρu(u -u f )] = -∇p + P r ∇ • [µ(∇u + ∇u T ) -2/3µ(∇ • u) I], (2)
∂(ρθ) ∂t + ∇ • [ρθ(u -u f )] = ∇ • (µ∇θ) + ω, (3)
∂(ρY ) ∂t + ∇ • [ρY (u -u f )] = 1 Le ∇ • (µ∇Y ) -ω, (4)
with the ideal gas equation of state

ρ(1 + qθ) = 1, (5)
and the reaction rate given by

ω = β 2 2u 2 p Le (1+q) 2-σ ρ 2 Y exp β(θ -1) (1 + qθ)/(1 + q) . (6)
The remaining parameters are: the Zel'dovich number β = E(Ta -T0)/RT 2 a , the thermal expansion coefficient q = (Ta -T0)/T0, and the exponent σ of the viscosity law µ = (1 + qθ) σ . In this work we assume β = 10, q = 5, Le = 1, and P r = 0.7, as representative of hydrocarbon combustion; σ = 0 or σ = 0.7 for constant and temperature dependent viscosity, respectively.

The factor up = SL/UL in ( 6) is introduced to account for the difference between the asymptotic (β 1) value of the planar burning velocity, UL, and the value of SL for finite β, and ensures that the velocity of the planar flame equals one in the computations. The values of up are 1.0547 for σ = 0 and 0.9997 for σ = 0.7 (see [START_REF] Fernández-Galisteo | Analysis of premixed flame propagation between two closely-spaced parallel plates[END_REF] for further details).

The normalized flame velocity u f is determined by

u f = 1 aLy Lx 0 Ly 0 a 0 ω dx dy dz. (7)
The term u f is also known as the overall propagation rate, Sc/SL, with Sc the consumption speed.

The mathematical problem described in Eqs. ( 1)-( 6) is solved with the general boundary conditions

z = 0, a : u = ∂θ/∂z = ∂Y /∂z = 0, (8)
x → -∞ : ∂u/∂x = ∂θ/∂x = ∂Y /∂x = 0, (

x → +∞ :

u = θ = Y -1 = 0, (10)
and periodic conditions in the y-direction.

The initial condition is a planar flame to which a weak harmonic perturbation in the form cos (ky) exp (-|x -xω|) is added in the temperature field. The amplitude of the perturbation is set to 10 -2 , with xω the position of the maximum reaction rate and k the wavenumber. In all simulations presented below, the wavenumber of the perturbation to k = 2πn/Ly is fixed, with n = 2, corresponding to the maximum growth rate obtained from the linear stability analysis [START_REF] Fernández-Galisteo | Impact of the gravity field on stability of premixed flames propagating between two closely spaced parallel plates[END_REF], in the limit a = h/δT → 0.

Quasi-2D formulation (h δT )

In the limit a = h/δT → 0, the governing equations ( 1)-( 4) can be reduced to a 2D set of equations, referred to as the quasi-2D form hereafter. Indeed, averaging flow quantities across the gap distance h reduces the momentum equation ( 2) to a linear relation for the velocity similar to Darcy's law

u = - 1 12µP r ∇p, (11)
where the operator ∇ is defined in the xy-plane only.

The derivation of the quasi-2D formulation is out of the scope of this paper; the reader is referred to [START_REF] Fernández-Galisteo | Analysis of premixed flame propagation between two closely-spaced parallel plates[END_REF][START_REF] Martínez-Ruiz | The role of conductive heat losses on the formation of isolated flame cells in Hele-Shaw chambers[END_REF] for further details. This formulation results in a drastic reduction of the computational cost compared with the full 3D description. Although the limit a → 0 has shown reasonable agreement with experimental observations [START_REF] Fernández-Galisteo | Analysis of premixed flame propagation between two closely-spaced parallel plates[END_REF], its range of validity has not been tested explicitly.

Pure 2D formulation (unconfined geometry)

This case corresponds to a classical 2D formulation where the vector operators in Eqs. ( 1)-( 4) are defined in the xy-plane only. Pure 2D simulations have been used extensively in the context of hydrodynamic (and diffusive-thermal) theory [START_REF] Denet | Laminar premixed flame dynamics: a comparison of model and complete equations[END_REF][START_REF] Sharpe | Linear stability of planar premixed flames: reactive Navier-Stokes equations with finite activation energy and arbitrary Lewis number[END_REF][START_REF] Rastigejev | Nonlinear evolution of hydrodynamically unstable premixed flames[END_REF][START_REF] Altantzis | Hydrodynamic and thermodiffusive instability effects on the evolution of laminar planar lean premixed hydrogen flames[END_REF][START_REF] Creta | Strain rate effects nonlinear development of hydrodynamically unstable flames[END_REF].

Numerical treatment

Transient computations are carried out in a finite domain of dimensions Lx × Ly × a in the respective longitudinal, transversal and normal directions. The equations are integrated using the Open source Field Operation And Manipulation (OpenFOAM) toolbox [START_REF] Weller | A tensorial approach to computational continuum mechanics using object-oriented techniques[END_REF]. The code is based on the finite volume method formulated in a collocated grid arrangement. A firstorder Euler scheme is used for temporal discretization whereas for spatial discretization a second-order scheme (upwind for convective terms and central for diffusive terms) is used. A grid resolution of ∆x = ∆y = ∆z = 0.1 was shown to be sufficient to correctly capture the flame evolution. Doubling the resolution did not result in appreciable changes in the unsteady flame dynamics. For the smallest values of a, the resolution in the normal direction was reduced to ∆z = 0.006 to capture the development of the Poiseuille flow profile. Finally, Lx is taken to be twice Ly to avoid any influence of boundary conditions on the solution, and a fixed time step equal to ∆t = 5 × 10 -5 is used for numerical stability. The total computational cost of the study is approximately 1.25 × 10 6 CPU hours.

Length scale considerations

The temperature dependence of viscosity modifies the flame thickness compared to that obtained assuming constant viscosity [START_REF] Kang | A computational study of Saffman-Taylor instability in premixed flames[END_REF]. The length scale δT used to normalize Eqs. ( 1)-( 6) does not maintain identical domain size to flame thickness ratio between the variable and the constant viscosity cases. A more appropriate reference scale to analyze variable viscosity effects is the flame thickness based on the maximum temperature gradient definition δ f = (Ta -T0)/ max (dT /dx ). For a meaningful comparison ST-SL_fix_le-1_bt-10_q-5_G-0_sg-0_b-0_wn-0-314_80 between constant and variable viscosity cases, δ f is thus used to re-scale the size of the computational domain. Based on planar flame computations, the ratio δ f /δT is 1.52 and 4.34 for σ = 0 and 0.7, respectively. This implies that any domain employed for σ = 0 must be re-scaled by a factor 2.86 when computing cases using σ = 0.7. To simplify the discussion below, the hat symbol is used to refer to variables normalized with δ f , i.e. x = x/δ f , â = h/δ f , Ly = Ly/(δ f /δT ).

Results and discussion

The influence of momentum loss and viscosity contrast is first analyzed in Section 5.1 by varying â for constant and variable viscosity keeping Ly fixed. In Section 5.2, the influence of Ly is analyzed. The effect of flame curvature along the z-direction is investigated in Section 5.3. A brief summary is given in Section 5.4.

Momentum loss and viscosity contrast effects

Figure 2 compares the time evolution of the overall propagation rate, Sc/SL, for â → 0 (quasi-2D) and finite values of â ranging from 0.033 to 1.978, with results obtained from the pure 2D formulation. The size the transversal domain is fixed to Ly = 26.3. In all cases, the planar flame initially destabilizes into two cusps (due to the two-wave mode of the initial perturbation, i.e. n = 2) to finally reach, at large times, a single cusp that fills the entire domain and remains steady (see the flame structure in Fig. 3). The peaks in the Sc/SL curve are associated with merging flame_shape_fix_w-1_le-1_bt-10_q-5_G-0_sg-0_wn-0-314_801x401xx_80x40xx_oc_pbc_open-fo of the two cusps. For â ≤ 0.659 all cases converge to the same final value of Sc/SL ≈ 1.56, but for â = 1.978 the propagation rate decreases by 5%. The figure also shows that the propagation rate is larger in the presence of confinement than in the case where the geometry is unconfined (pure 2D simulation).

Figure 3 illustrates the flame front topology (represented by the reaction rate isocontour ω = 1) for different values of â and for the conditions given in Fig. 2. The flame propagates from left to right. As expected, the increase in the propagation rate is due to the increase in the flame surface area. Because of the gas friction with the walls, the motion of the hot-gas products in the left side induces a higher strain near the trough (the region that is concave toward the fresh mixture) as h decreases, significantly elongating the flame surface. By reference to the pure 2D simulation, in which flame curvature manifests only through thermal expansion, Figs. 2 and 3 also indicate that momentum loss, related to confinements effects, enhances the instability mechanism associated to thermal expansion, and that 3D simulations are in close agreement with the quasi-2D formulation for â 1. For larger â, contribution of wall friction to longitudinal pressure gradient decreases, which in turn, reduces the flame elongation and thus flame propagation speed. Note that as â increases curvature effects may also occur in the direction perpendicular to the plates (along z); see Section 5.3.

The time evolution of the propagation rate, Sc/SL, for variable viscosity (σ = 0.7) is shown in Fig. 4. Similarities with the dynamics described in Fig. 2 for â ≥ 0.659 are evident. However, for â ≤ 0.066, spo-ST-SL_fix_le-1_bt-10_q-5_G-0_sg-0.7_b-0_wn-0-109_ Fig. 4: The variation of the propagation rate with time for variable viscosity (σ = 0.7) and â = 0, 0.033, 0.066, 0.659, and 1.978. Transversal size is Ly = 26.3. Pure 2D simulation is plotted with dashed curve. radic oscillations are obtained. This unsteady behavior results from a merging/splitting process between the cusps (no shown here but similar to those observed in previous studies [START_REF] Creta | Turbulent propagation of premixed flames in the presence of Darrieus-Landau instability[END_REF][START_REF] Radisson | Nonlinear dynamics of premixed flames: from deterministic stages to stochastic influence[END_REF][START_REF] Denet | Laminar premixed flame dynamics: a comparison of model and complete equations[END_REF][START_REF] Rastigejev | Nonlinear evolution of hydrodynamically unstable premixed flames[END_REF]). The drops in the Sc/SL curve are associated with the splitting of the cusp. The oscillations appear in cases where the confinement effect is higher. For â ≥ 0.659 the 3D simulations yield steady solutions for long times. Note that these oscillations seem to be physical as they persist when the grid resolution is increased up to a factor of 10 for â → 0 cases (quasi-2D simulation). We do not exclude, however, that for larger Ly unsteady flame dynamics may appear for â ≥ 0.659, but this is beyond our current computational resources.

The main consequence of introducing variable viscosity is an increase in the overall propagation rate. This increase is significant for small â. For instance, for â = 0.659, Sc/SL increases from ≈ 1.55 to ≈ 2.13 at late times when variable viscosity is included in the model. The comparison of the corresponding steady flame topologies is illustrated in Fig. 5. The flame surface is further stretched when variable viscosity is considered, indicating that viscosity changes across the flame front induce an additional destabilizing mechanism. In absence of confinement, however, variable viscosity plays a minor role. This is clearly seen when comparing the pure 2D simulation in Figs. 2 and4. In both cases, Sc/SL ≈ 1.23 at the final evolution.

Transversal domain size variation

The oscillations observed in Fig. 4, associated with the merging/splitting process of the cusps, have been discussed extensively in the context of pure 2D simulations [START_REF] Denet | Laminar premixed flame dynamics: a comparison of model and complete equations[END_REF][START_REF] Rastigejev | Nonlinear evolution of hydrodynamically unstable premixed flames[END_REF][START_REF] Altantzis | Hydrodynamic and thermodiffusive instability effects on the evolution of laminar planar lean premixed hydrogen flames[END_REF]. Our interest here is on determining the onset of unsteadiness when momentum loss is present. Figure 6 shows the propagation rate found during the long-time evolution as a function of Ly for both constant and variable viscosity. Quasi-2D and pure 2D simulations are compared. As expected, the propagation rate corresponds with the planar velocity (Sc/SL = 1) up to a critical size, Lyc . Above this critical length, the flame destabilizes into a single cusp and the propagation rate increases to an upper limit. There is agreement between the value of Lyc found in our simulations with those reported in linear stability analyses. For instance, the value of Lyc = 11.7 for the pure 2D simulation (σ = 0) agrees well with the value reported by Sharpe et al. [START_REF] Sharpe | Linear stability of planar premixed flames: reactive Navier-Stokes equations with finite activation energy and arbitrary Lewis number[END_REF]; and the value of Lyc = 8.3 for the quasi-2D simulation (σ = 0) coincides with that reported in [START_REF] Fernández-Galisteo | Impact of the gravity field on stability of premixed flames propagating between two closely spaced parallel plates[END_REF]. The upper limit of the propagation velocity for the pure 2D problem is also in agreement with previous studies in the context of the nonlinear development of hydrodynamic flames [START_REF] Rastigejev | Nonlinear evolution of hydrodynamically unstable premixed flames[END_REF][START_REF] Altantzis | Hydrodynamic and thermodiffusive instability effects on the evolution of laminar planar lean premixed hydrogen flames[END_REF][START_REF] Creta | Strain rate effects nonlinear development of hydrodynamically unstable flames[END_REF][START_REF] Bychkov | Nonlinear equation for a curved stationary flame and the flame velocity[END_REF]. Altantzis et al. [START_REF] Altantzis | Hydrodynamic and thermodiffusive instability effects on the evolution of laminar planar lean premixed hydrogen flames[END_REF] reported an average value of Sc/SL ≈ 1.25, close to that found in the present study (Sc/SL ≈ 1.23).

In Fig. 6 unsteady solutions are represented with empty circles, for which the propagation rate is calculated as a time-averaged value. It can be seen that variable viscosity and/or confinement trigger the onset of oscillations at smaller Ly than constant viscosity and/or unconfined geometries. This figure clearly shows that the quantitative effect of wall-friction and viscosity contrast (in terms of the propagation velocity) is domain-size dependent . The hydrodynamic mechanism at the origin of the cusp-like structure in the xy-plane can also alter the flame shape in the xz-plane if the plates are sufficiently separated. Fig. 7 illustrates the flame topology that emerges in the xz-plane for σ = 0.7 and increasing values of â. Clearly, as the distance between the plates is increased, the curvature of the flame plays a role in this direction. Of interest here is that there is a critical gap distance, âc, above which the symmetric flame destabilizes to the non-symmetric shape, as shown in Fig. 7 (top). According to [START_REF] Dejoan | Critical conditions for non-symmetric flame propagation in narrow channels: Influence of the flow rate, the thermal expansion, the Lewis number and heat-losses[END_REF], where the flame propagation in a narrow planar channel was investigated, non-symmetric flame solutions were found to be stable and the corresponding symmetric solutions were unstable. A symmetric solution can be forced by halving the computational domain in the h direction (i.e. Lx × Ly × h/2) and imposing a symmetry boundary condition on the z = h/2-plane. Expectedly non-symmetric shapes result in a significant increase of the flame surface area and consequently in the propagation velocity. This is illustrated in Fig. 8, where the variation of the propagation rate with â is shown. In this figure, triangles correspond to non-symmetric (full domain) flame solutions and circles represent forced symmetric solutions. The critical gap distance of âc = 3.9 reported in [START_REF] Dejoan | Critical conditions for non-symmetric flame propagation in narrow channels: Influence of the flow rate, the thermal expansion, the Lewis number and heat-losses[END_REF] is in accordance with Fig. 8. Note that when high flame curvature takes place in the xz-plane, the elongation of the cusp in the xy-plane is significantly reduced or even suppressed (not shown here) yielding nearly planar solutions; this holds true for of Ly = 13.15 considered to construct Fig. 8. This outcome suggests that curvature effects along h compete with those present in the xy-plane and therefore larger Ly may be required to form cusp-like topologies.

Curvature along the gap distance

Summary

Figure 8 summarizes nicely our results. For small â the difference in the propagation velocity between variable and constant viscosity is appreciable. Clearly, wall-friction and variable viscosity have a strong influence on the hydrodynamic stability. The propagation velocities, and the difference between σ = 0 and σ = 0.7, diminish as the gap distance increases, approaching the pure 2D formulation propagation rate; confinement effects may thus be considered negligible. For â ≥ 4, the propagation velocities increase again (denoted with dashed-dotted lines) as a result of flame curvature effects in the xz-plane. Note that in Fig. 8 Ly = 13.15 domain size for which all flame solutions are steady during their long time evolution (see Section 5.2).

Conclusions

Confinement effects on the large-scale hydrodynamic instability of a flame propagating between two adiabatic plates was analyzed in three dimensions by varying the distance separating the plates, h. The limit a → 0 (high friction limit) and the pure 2D simulations (no confinement) were also considered and compared. The present numerical study confirmed the analytical results anticipated by Joulin and Sivashinsky [START_REF] Joulin | Influence of momentum and heat losses on the large-scale stability of quasi-2D premixed flames[END_REF]: high wall-friction and viscosity contrast reinforce the hydrodynamic instability of flames propagating in narrow gaps. However, our results show that if the distance between the plates is large enough (h/δ f ≈ 4), flame curvature in the xz-plane (nonsymmetric flames) can compete with the large-scale single-cusp structure that develops in the xy-plane, resulting in a stabilizing effect. The emergence of oscillatory behavior during their long-time evolution was also reported. The domain size ( Ly) above which this unsteady behavior is triggered decreases in presence of high confinement and/or variable viscosity. Finally, the quasi-2D formulation was found to provide very satisfactory results up to a ≈ 1, when compared to 3D simulations.

Aknowledgments

Fig. 1 :

 1 Fig. 1: Schematic view of a curved flame propagating from left to right into a quiescent gas in a Hele-Shaw burner where plates are separated by the distance h.

Fig. 2 :

 2 Fig. 2: The variation of the propagation rate with time for constant viscosity (σ = 0) and â = 0 (quasi-2D), 0.033, 0.066, 0.659, and 1.978. Transversal size is Ly = 26.3. Pure 2D simulation is plotted with dashed curve.

Fig. 3 :

 3 Fig. 3: Illustration of the single-cusp structure represented by the reaction rate isocontour (ω = 1) in the plane ẑ = 0 for different values of â and for the pure 2D simulation. Same conditions as given in Fig. 2. The flame propagates from left to right.

Fig. 5 :

 5 Fig. 5: Illustration of the single-cusp structure represented by the reaction rate isocontour (ω = 0.2) in the plane ẑ = 0 for constant viscosity (σ = 0) and variable viscosity (σ = 0.7). Calculated for Ly = 26.3 and â = 0.659.

Fig. 6 :

 6 Fig. 6: The variation of the propagation rate with transveral domain for constant (σ = 0) and variable (σ = 0.7) viscosity. Empty circles stand for simulations where unsteady long-time flame regime appears.

Fig. 7 :

 7 Fig. 7: Illustration of the flame curvature represented by the reaction rate isocontour (ω = 0.2) in the plane ŷ = 0, for σ = 0.7 and â = 0.659, 3.3, and 4.6.

Fig. 8 :

 8 Fig. 8: The variation of the propagation rate with gap distance â. Calculated for Ly = 13.15. Circles stand for symmetric flame solutions in the xz-plane and triangles with non-symmetric solutions. Dashed lines represent pure 2D formulation values.

This work was financed by project #PID2019-108592RB-C42/ AEI/ 10.13039/501100011033. The authors acknowledge the Spanish Supercomputing Network, and the technical support provided by UAM (RES-IM-2019-3-0004, RES-IM-2020-1-0003, RES-IM-2020-2-0021, RES-IM-2021-1-0011, and RES-IM-2021-2-0008).