Effect of surface friction on ultrafast flame acceleration in obstructed cylindrical pipes

The Bychkov model of ultrafast flame acceleration in obstructed tubes [Valiev et al., “Flame Acceleration in Channels with Obstacles in the Deflagration-to-Detonation Transition,” Combust. Flame 157, 1012 (2010)] employed a number of simplifying assumptions, including those of free-slip and adiabatic surfaces of the obstacles and of the tube wall. In the present work, the influence of free-slip/non-slip surface conditions on the flame dynamics in a cylindrical tube of radius R, involving an array of parallel, tightly-spaced obstacles of size αR, is scrutinized by means of the computational simulations of the axisymmetric fully-compressible gasdynamics and combustion equations with an Arrhenius chemical kinetics. Specifically, non-slip and free-slip surfaces are compared for the blockage ratio, α, and the spacing between the obstacles, ∆Z, in the ranges 1/3 ≤ α ≤ 2/3 and 0.25 ≤ ∆Z/R ≤ 2.0, respectively. For these parameters, an impact of surface friction on flame acceleration is shown to be minor, only 1∼4%, slightly facilitating acceleration in a tube with ∆Z/R = 0.5 and moderating acceleration in the case of ∆Z/R = 0.25. Given the fact that the physical boundary conditions are non-slip as far as the continuum assumption is valid, the present work thereby justifies the Bychkov model, employing the free-slip conditions, and makes its wider applicable to the practical reality. While this result can be anticipated and explained by a fact that flame propagation is mainly driven by its spreading in the unobstructed portion of an obstructed tube (i.e. far from the tube wall), the situation is, however, qualitatively different from that in the unobstructed tubes, where surface friction modifies the flame dynamics conceptually. © 2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5087139


I. INTRODUCTION
Among the geometries associated with fast flame acceleration and the deflagration-to-detonation transition (DDT) scenarios, [1][2][3][4][5][6][7][8][9] obstructed cylindrical tubes provide fastest acceleration. 10 While flame propagation through the obstacles is oftentimes associated with turbulence or shocks 11 or hydraulic resistance, 12 Bychkov et al. [13][14][15][16] identified a conceptually laminar, shockless mechanism of ultrafast acceleration in semi-open channels or cylindrical tubes equipped with a comb-shaped array of obstacles. The Bychkov mechanism is illustrated in Fig. 1, and it is devoted to a powerful jet-flow along the centerline of a channel or a tube, generated by a cumulative effect of delayed combustion in the "pockets" between the obstacles. According to the analytical formulation, 14 substantiated by the comprehensive numerical simulations, 14,17 a premixed flame front accelerates exponentially as Utip/SL ≈ Θ exp(στ), where Utip ≡ dZtip/dt is the velocity of the flame tip in the laboratory reference frame, SL is the unstretched laminar burning velocity, Θ ≡ ρ f /ρ b is the thermal expansion ratio, τ ≡ tSL/R is the scaled time, R is the radius of the tube, and the scaled exponential acceleration rate σ in the cylindrical axisymmetric geometry is given by 14

ARTICLE
scitation.org/journal/adv FIG. 1. The Bychkov mechanism of ultrafast flame acceleration in obstructed pipes (a half of the pipe is shown).
where α is the blockage ratio. This acceleration is extremely powerful indeed, exceeding that due to wall friction 18 or combustion instabilities. 6 Moreover, the quantity σ grows with α and Θ, thereby promoting flame acceleration; σ drastically depends on α, but it does not depend on R, which makes this acceleration mechanism scaleinvariant (Reynolds-independent) and, thereby, relevant to various scales: from micro-combustors to very large mining and subway tunnels. The Bychkov model 13-17 adopted a set of simplifications, such as free-slip walls. However, slip walls are relevant for a noncontinuum flow, while the physical boundary conditions are nonslip as far as the continuum assumptions are valid. Consequently, the Bychkov model needed to be validated in terms of its applicability to the practical reality. Indeed, it was shown that both surface friction 18 and thermal (cold and hot) wall conditions 19 play an enormous role in unobstructed pipes. Will it be the case in the obstructed ones? Here we are answering this question. While Ugarte et al. 16 have recently shown a minor effect of the outer isothermal walls as compared to the adiabatic ones, in obstructed pipes; in the present work we have compared slip and nonslip surfaces and came to the same conclusion.

II. NUMERICAL METHOD
We have performed the computational simulations of the hydrodynamic and combustion equations including transport processes (thermal conduction, diffusion, and viscosity), full compressibility and an Arrhenius chemical kinetics. The basic equations in the cylindrical geometry read: where Y is the mass fraction of the fuel mixture, ε = ρ(QY + CvT) + ρ u 2 z + u 2 r 2 is the total energy per unit volume, with the energy release in the reaction Q = CpT f (Θ − 1), specific heats at constant pressure and volume, Cp and Cv, pressure P, temperature T, density ρ and the radial and axial velocity components, ur and uz. Equation (6) describes a one-step irreversible Arrhenius reaction of the first order, with the activation energy Ea and the constant of time dimension τR. The stress tensor ζ αβ is given by and the energy diffusion vector qα takes the form where µ ≡ ρν is the dynamic viscosity, being µ f = 1.7 × 10 −5 kg/(m⋅s) in the fuel mixture, and Sc and Pr are the Schmidt and Prandtl numbers, respectively, with the Lewis number being their ratio, Le = Sc/Pr. To avoid diffusional-thermal instability, similar to Refs. 13-16, in the present work we took Le = Sc = Pr = 1. The fuel mixture has initial temperature, T f = 300 K, pressure, P = 1 bar, and density, ρ f = 1.16 kg/m 3 . The thermal expansion ratio is taken to be Θ = 8, with the laminar flame speed being SL = 0.347 m/s, which represents a near-stoichiometric methane-air mixture. The initial speed of sound in the fuel mixture is c 0 = 347 m/s, which exceeds SL by a factor of 10 3 , thereby making gasdynamics almost incompressible at the initial stage of burning, with the Mach number associated with flame propagation being M 0 ≡ SL/c 0 = 10 −3 . Then the thermal flame thickness can be defined, conventionally, as

III. RESULTS AND DISCUSSION
We have compared the cases of free-slip and non-slip boundary conditions for various R, ∆Z, and α. Specifically, the scaled flame tip position Z f /R versus the scaled time τ = tSL/R is shown in Figs. 2-4, with the free-slip and non-slip boundary conditions depicted by the solid and dashed lines, respectively, in all plots. It is seen that the impact of free-slip and non-slip surface boundary conditions is minor as long as the obstacles spacing is small, ∆Z ≤ R/2, and this is true for all α considered. Indeed, both the curves almost coincide in the cases studied. A reasonable way to measure a relative deviation between the plots is calculating the following quantity (in %): The result of Eq. (10)  in obstructed pipes, which employs the free-slip surfaces of obstacles and walls. The similarity of the color snapshots of Fig. 5 taken at the scaled time instant τ = 0.15 leads to the same conclusion. This result can be explained by the fact that the flow is mainly driven in the axial direction such that the small obstacles spacing mitigates a potential effect of surface fiction (if any). In fact, Ref. 16 suggested the same conclusion with the same explanation when studying the thermal boundary conditions at the walls and obstacles in obstructed channels.
We also investigated what happened when the obstacles spacing is increased. While in Figs. 6-8 we noticed that the results for both free-slip and non-slip conditions almost coincide for a small ∆Z, ∆Z ≤ R/2, as discussed previously, the difference between the results is observed when ∆Z = R. Additionally, the color snapshots have been taken at time scaled instant τ = 0.225 from the simulations for the free-slip and non-slip boundary conditions, as presented in Figs. 9-11. It is seen that there is only a minor difference between the free-slip and non-slip surfaces, with the flame tip being ∼1.5 R ahead for the non-slip case. This result can potentially be attributed to the fact that vortices in the pockets are barely in contact with the walls.
We have also considered a wider spacing between the obstacles, ∆Z = 2R, with a remarkable difference between the free-slip and non-slip conditions observed in that case, as shown in Figs Fig. 15. It is clearly seen that surface friction promotes flame acceleration, substantially, as compared to the free-slip condition: the deviation given by Eq. (10) is ∼24% in that case. Comparing Fig. 14b to 14a, one can attribute such a discrepancy to the formation of high vorticity in the pockets between the obstacles in the case if large ∆Z. Obviously, vorticity evolves differently with free-slip and non-slip surfaces, with a stronger flow distortion in the latter case, leading thereby to faster flame acceleration. It is recalled, in this respect, that the Bychkov model does not consider vorticity and, therefore, it is probably not fully applicable here. Moreover, the very approach of tightly-packed obstacles, ∆Z ≪ R, Fig. 1, considered in the Bychkov model, is definitely broken when ∆Z exceeds R. In fact, an inapplicability of the Bychkov formulation for ∆Z > R has been shown even in the pilot studies 13,14 as well as later, in the detailed analysis. 16 Qualitatively, we arrive to the same conclusions by means of the color snapshots in Fig. 15. Indeed, while a resemblance between Figs. 5a and 5b was evident, with the flame tip at Z f ∼ 27 R in both figures, a serious difference between the flame tip positions in Figs. 15a and 15b is clearly seen, with the flame tip being ∼19 R ahead for the non-slip case.

IV. CONCLUSIONS
We have investigated flame propagation in obstructed cylindrical tubes. It is shown that for small and moderate spacing between the obstacles, an impact of surface friction on flame acceleration is minor, 1∼4%, slightly facilitating acceleration in a tube with ∆Z/R = 1/2 and moderating acceleration in the case of ∆Z/R = 1/4. However, the situation is different in the case of a wide spacing. Earlier, a minor effect of the isothermal surfaces as compared to the adiabatic ones was also demonstrated. 16 With the fact that the Bychkov model employed slip walls whereas the physical boundary conditions are non-slip as far as the continuum assumption is valid, the present work thereby justifies the Bychkov approach and makes its wider applicable to the practical reality; but only within its validity limit. Indeed, at a large spacing, ∆Z ≫ R, the Bychkov model is not applicable anyway, and vorticity comes to play and show different behaviors for the free-slip and non-slip surfaces. In fact, we anticipated such an outcome that the effect of surface friction is generally minor (but only as long the spacing between the obstacles is small). It can be explained by a fact that flame propagation is mainly driven by its spreading in the unobstructed portion of an obstructed tube (i.e. far from the tube wall). However, there was a need to justify this hypothesis because surface friction may potentially play a significant role as it does in other geometries such as unobstructed tubes. 18 Here we proved this is not the case for an obstructed tube.