## Abstract

The effects of hydraulic resistance on the burning of confined/obstacle-laden gaseous and gas-permeable solid explosives are discussed on the basis of recent research. Hydraulic resistance is found to induce a new powerful mechanism for the reaction spread (diffusion of pressure) allowing for both fast subsonic as well as supersonic propagation. Hydraulic resistance appears to be of relevance also for the multiplicity of detonation regimes as well as for the transitions from slow conductive to fast convective, choked or detonative burning. A quasi-one-dimensional Fanno-type model for premixed gas combustion in an obstructed channel open at the ignition end is discussed. It is shown that, similar to the closed-end case studied earlier, the hydraulic resistance causes a gradual precompression and preheating of the unburned gas adjacent to the advancing deflagration, which leads (after an extended induction period) to a localized autoignition that triggers an abrupt transition from deflagrative to detonative combustion. In line with the experimental observations, the ignition at the open end greatly encumbers the transition (compared with the closed-end case), and the deflagration practically does not accelerate up to the very transition point. Shchelkin's effect, that ignition at a small distance from the closed end of a tube facilitates the transition, is described.

## 1. Overview

Although the importance of hydraulic resistance for confined/obstacle-laden gaseous combustion and gas-permeable solid combustion has long been recognized [1–4], a systematic exploration of its specific impact is a relatively recent activity [5,6]. The present paper is intended as a wide brush discourse on the theoretical efforts in this area that have tangibly improved our understanding of a number of first-order effects rooted in the very fundamentals of combustion fluid dynamics. Apart from covering the earlier developments, the paper contains some novel, and as yet unpublished, material.

A self-sustained wave of the exothermic chemical reaction spreading through a homogeneous explosive premixture is known to occur either as a subsonic deflagration (premixed flame) or as a supersonic detonation. In unconfined obstacle-free systems, the concrete realization of the specific propagation mode is, as a rule, controlled by the ignition conditions. Normally, deflagrations are initiated by a mild energy discharge, i.e. by an electric spark, whereas detonations are provoked by shock waves via localized explosion. A shockless (soft) initiation of detonation is also feasible. It may occur in appropriately preconditioned, for example, non-uniformly preheated, gas. Thus, both deflagration and detonation appear to be stable attractors, each being linked to their own base of initial data, a rather uncommon feature in dynamical systems. It is known, however, that, in the presence of obstacles or confinement, the initially formed deflagration undergoes slow acceleration abruptly ending up as a detonation.

Premixed gas combustion in smooth-walled tubes is the simplest system for studying the deflagration-to-detonation transition (DDT). Yet, even in this geometry, the emerging dynamical picture is extremely complex for the straightforward identification of the mechanisms involved. The ultimate goal of the theory is to represent the transition event as an interaction of elementary building blocks (well-conceived model problems), each describing a simple causal process.

In the traditional attempt to explain the transition, the role of confinement is reduced exclusively to a generation of hydrodynamic disturbances (turbulence). The latter promotes extension of the flame interface resulting in the flame acceleration and, hence, enhancement of the flame-supported compression waves, which allegedly leads to formation of hot-spots triggering localized explosions and transition to detonation [7]. Yet recently, in the course of studying obstacle-affected deflagrative combustion [8–10], it was realized that there is a complementary aspect of the flame–confinement interaction that somehow has escaped proper attention and that seems to be of crucial importance. Apart from inducing hydrodynamic disturbances, the confinement also exerts resistance to the gas flow, causing reduction of its momentum, which appears to be an agency perfectly capable of provoking the DDT event even if the predetonation acceleration due to the flame folding is deliberately suppressed and the system is regarded as effectively one-dimensional. In this approach, the confinement is accounted for phenomenologically through the velocity-dependent drag-force term added to the momentum equation, while leaving the equation for energy conservation unaltered (Fanno's hydraulic resistance model [11]). To illustrate the basic idea underlying Fanno's model, consider a decelerating flow of a non-reactive viscous gas in a thermally insulated loop-like duct. Moreover, assume that there are no external (e.g. gravity) forces involved. The evolving flow, owing to no-slip boundary conditions, obviously loses its momentum. Yet, being thermally insulated and not affected by external forces, the system preserves its total (mechanical plus thermal) energy. In a one-dimensional description, justified by the duct's extended geometry, the effect of momentum loss is simulated by an appropriate sink term (drag force). On the other hand, in order not to violate conservation of energy, characterizing the original system, the one-dimensional equation of energy should be free of any sources. In the case of reactive systems, other conditions being as above, the only acceptable source of energy is the chemical heat release.

For tubes closed at the ignition end, the hydraulic resistance causes a gradual precompression and hence preheating of the unburned gas adjacent to the advancing deflagration. Upon some extended induction period, this development may lead to a localized autoignition (thermal explosion) triggering an abrupt transition from deflagrative to detonative propagation (figures 1 and 2). The thermal explosion is caused by a mild spatial gradient of the induction time in the precompression-induced extended preheat zone formed ahead of the advancing deflagration. In turn, the extended precompression zone is a product of the resistance-induced pressure diffusivity that significantly exceeds the thermal diffusivity of the system.^{1}

The nature of the results obtained—the explosive character of the transition; velocity and pressure overshoots; predetonation expansion and shrinking of the reaction zone; increase of the predetonation distance with the tube diameter; reduction of the predetonation time by the ignition at some distance from the closed end (all these effects are well known experimentally)—gives serious grounds for the belief that a good deal of the actual DDT dynamics is captured quite adequately. The DDT problem therefore turns out to be physically related to the shockless (soft) initiation of detonation in non-uniformly preheated gas, a topic that has enjoyed much attention in the recent past (see [5,13] and references therein). Unlike the latter systems, however, in the DDT case, the non-homogeneous environment that provides the required spatial gradient of the induction time is not prescribed but arises as a product of the flame–confinement interaction.

In the resistance-based concept, extension of the flame interface is regarded as a prominent by-product of the processes occurring in the boundary layer, where the flow–wall interaction results in a local autoignition. Arguments in favour of this interpretation are the experimental evidence that the detonation is generally conceived in the boundary layer [14–18], and that the transition can be reproduced within a quasi-one-dimensional Fanno model [10] where the interface extension is not one of the model's ingredients.

As the Fanno model is not a rational approximation of the real-world multi-dimensional system, its success should be perceived only as a good argument in support of the resistance concept, rather than a definite proof. In the real-world system, the interface extension and hydraulic resistance are invariably entangled, which makes the cause–effect analysis very difficult. A rational isolation of elementary mechanisms is often attained if some of the system's parameters approach their limiting values. It transpires that some insight into resolution of the flame folding versus resistance controversy may be acquired if one considers the classical limit of small heat release (SHR), which has proved itself previously in analysing different aspects of subsonic combustion. Our recent explorations of the SHR limit produced an unexpected picture. While the developing flow field is two-dimensional (owing to no-slip boundary conditions), the associated temperature and pressure fields turn out to be nearly one-dimensional, provided the tube is thermally insulated. As a result, the well-settled flame appears to be almost planar (figure 3). The subsequent transition thus occurs without the interface extension, thereby validating the quasi-one-dimensional approach, at least for the SHR limit. Similar suppression of the flame folding and predetonation acceleration may be achieved by artificially making the system diffusively anisotropic. Specifically, one keeps the kinematic viscosity isotropic while setting transverse thermal and molecular diffusivities large compared with their longitudinal counterparts and viscosity.

Another argument in favour of the resistance concept is provided by the familiar piston–flame analogy [19]. Owing to thermal expansion of hot products, the flow field formed ahead of the flame propagating from the closed end of a tube is similar to the flow field induced by a moving piston. The flow driven by the piston is obviously a more transparent and better-controlled process than the flow induced by the evolving flame, especially when instabilities and turbulence are involved. Yet, for all its conceptual simplicity, the piston flow system appears to be dynamically rich enough, and involves interactions that are likely to be relevant to the DDT event. Consider a piston moving uniformly through a long tube filled with a non-reactive gas initially at rest. Owing to compression, the pressure and temperature at the piston disc rise and settle at some constant value controlled by the piston velocity. Yet, this picture is valid only in the interior of the disc, where the impact of the near-wall drag does not manifest itself. However, near the piston edge, the gas is found to undergo an unrestricted precompression and preheating. The net effect therefore is similar to that of a gas enclosed in a finite isentropically shrinking volume, although in the piston case the vessel is open at one end and, owing to resistance, the entropy is not conserved. In the case of an explosive premixture, this development may well lead to a localized autoignition triggering detonation. Prior to the autoignition, the bulk of the reaction occurs near the piston surface. Upon autoignition, the reaction zone breaks off from the piston, resulting in a self-sustained detonation occurring (upon overshooting and merging with the precursor shock) in a slightly overdriven Chapman–Jouguet (CJ) regime. One may therefore speak of the DDT-like transition from subsonic burning, controlled by the advancing piston, to supersonic burning, where the reaction zone is attached to the shock. We emphasize that the transition here occurs without acceleration of the reaction zone, which is often considered to play a key role in the DDT. It is interesting that, for wide tubes, the predetonation time and distance are effectively independent of the tube diameter, as is occasionally observed also in conventional DDT systems [20,21].

Further experimental/numerical evidence in support of the resistance concept comes from the transition in obstructed channels open at the ignition end [22,23]. In this geometry, the detonation emerges abruptly after an extended period of nearly constant-velocity deflagrative burning. Such a dynamical pattern and its good qualitative correlation with predictions of the quasi-one-dimensional model (see §2) seems to challenge the acceleration as a primary mechanism of the transition. Moreover, because it is highly unlikely that the transitions in open-end and closed-end systems are controlled by physically different mechanisms, the above observations suggest that the incipient flame folding and acceleration occurring in closed-end systems are likely to be merely prominent facilitating features of the transition rather than its key cause. Reduction of the acceleration occurs also in open-end smooth tubes. Yet, according to Akkerman *et al*. [24] and our own tests, the effect is not so dramatic, at least in narrow tubes.

The outlined resistance concept of the dynamic transition proved to be of relevance also for explaining the parametric (gradual or hysteretic) transition from sub-CJ supersonic detonation to the near-sonic pressure-driven convective/choked regime [12,25,26] occasionally observed in obstacle-laden/porous systems [27–29] as the initial pressure/permeability is decreased (figure 5).

The resistance concept of the parametric transition in obstructed gaseous systems proved to be helpful also for interpretation of the overpressure-induced abrupt transition from conductive to convective burning in gas-permeable solid explosives [30–33]. As soon as the incipient flame penetrates into the interior of the explosive, pressure gradients develop because of the resistance to the gas flow in the interstices between the explosive particles. One thus ends up with a situation very similar to that occurring in gaseous systems. The transition is triggered by a localized autoignition, provided the pressure difference between hot gas products and gases deep inside the pores of the unburned solid exceeds a certain critical level. In line with observations, the critical overpressure increases with diminishing permeability. The transition occurs when the temperature profile formed ahead of the slowly advanced reaction zone fails to maintain the unburned explosive in its chemically frozen state.

In both gaseous and gas-permeable solid explosives, the convective mode displays properties common to both deflagrative and detonative combustion. Similar to conventional deflagration, the reaction wave is driven by the dissipative agency, resistance-induced diffusion of pressure. The pressure diffusivity may, however, exceed the thermal diffusivity by several orders of magnitude, thus allowing propagation both at high subsonic and at supersonic velocities [6,12]. In the latter case, the shock emerges not as a driving agency but rather as a by-product of supersonic propagation. If the reaction zone is driven supersonically, then it will inevitably be accompanied by a shock.

Similar to the high-velocity sub-CJ detonation, the near-sonic choked/convective regimes are controlled by the ‘ignition’ temperature arising at the entrance to the reaction–diffusion zone rather than at its exit as occurs in conventional deflagrative combustion.

As in the CJ detonation, the near-sonic mode is prone to the oscillatory/spinning instability [34] with similar structures of instability criteria. Presumably, this is due to the fact that the main reason for the instability is the strong disparity between the rates of mass and pressure transfer, irrespective of whether the latter is diffusive or acoustic in nature.

Because the subsonic convective mode may be perceived as a continuous extension of the supersonic one (figure 5), it may well be labelled as *subsonic detonation*, to emphasize the role of adiabatic compression as a key physical ingredient unifying all pressure-driven regimes.^{2} Moreover, as in conventional detonation and unlike deflagration, in the subsonic mode the speed vector of the products is in the direction of the propagation of the reaction front.

The quasi-one-dimensional model is clearly capable of describing the transition only in some crude average sense where predetonation acceleration due to flame folding is suppressed and the details of the hot-spot nucleation and autoignition are glossed over. To obtain a more resolved picture, one needs to invoke the spatial dynamics where the hydraulic resistance is determined directly by boundary conditions rather than through the effective drag force. Yet, since the real-world DDT is hardly ever laminar, its direct numerical simulation not unexpectedly meets with formidable difficulties owing to insufficient computational power unable to provide the required spatial and temporal resolution in realistic multidimensional systems. The theoretical findings based on the quasi-one-dimensional model suggest, however, that the complexities due to turbulence are likely to be largely irrelevant to the transition that is presumably triggered by the flow deceleration in the boundary layer, irrespective of whether the bulk of the flow is turbulent or not. Hence, in order to reproduce the spatial picture of the transition, it would seem natural to begin with a two-dimensional thermally insulated channel, narrow enough to ensure the laminar character of the developing flow with all the technical advantages this entails. The pertinent numerical simulations showed that even extremely narrow (10 flame-width wide) channels are perfectly capable of capturing both the incipient flame acceleration [36] as well as the transition [37–41] (figure 4). These predictions have recently been corroborated experimentally by Wu and co-workers in their studies on the transition in capillary tubes [42,43].

Whatever its role in the transition mechanism, the acceleration is definitely one of the salient features of the DDT and naturally enjoys much attention (see Valiev *et al*. [41] and references therein). A remarkable product of the acceleration is that, even in narrow channels, it may induce supersonic propagation arising prior to the DDT event (figure 4). The question is what is the limiting mode of the flame spread when the reaction ahead of the evolving flame is deliberately suppressed, as in the familiar flame-sheet approximation? In a more comprehensive formulation, this mode is likely to become a precursor of the DDT. Figures 6 and 7 show the outcome of our recent simulations of the model based on the fictitious temperature-independent kinetics with a non-zero heat release. This model obviously rules out the unburned gas autoignition, thereby making the CJ detonation dynamically unfeasible. Yet, owing to thermal expansion and wall friction, the model allows for the flame folding, which may become strong enough to support fast, even supersonic, propagation. Figure 6*c* depicts the well-settled reaction wave composed of the planar shock and the dome-shaped reaction zone, both moving supersonically and as one flame–shock complex. Note a markedly stronger preheating of the unburned gas at the wall (figure 7*a*), which—under normal temperature-dependent kinetics—should create a favourable environment for the autoignition and tuliping (figure 4*a*). It is interesting that, unlike the temperature, the pressure does not vary over the channel's cross section (figure 7*b*). The pressure, however, does vary along the channel's axis, undergoing some elevation between the shock and the trailing edge of the flame brush.

Realistic confined systems are invariably affected by heat losses. The hydraulic resistance and heat losses exert opposite effects on the transition. The resistance raises the local temperature (through adiabatic compression) and thereby promotes autoignition. The heat loss tends to reverse this trend by reducing the temperature. In smooth-walled channels, owing to the Reynolds analogy, the two factors are of comparable influence. Therefore, one cannot be certain about the final outcome of the competition. Experimentally, however, an often successful transition even in capillaries is an undeniable fact. As reported in Kagan *et al*. [40,44], with the channel walls maintained at the ambient temperature, and the reaction kinetics assumed monomolecular, the transition does not occur, at least within the parameter range explored. However, for the bimolecular kinetics (other conditions as in the monomolecular case), the transition proved readily feasible. Higher molecularity implies a higher sensitivity of the explosive mixture to the pressure change, which in these processes is quite significant. Recently, Wang & Wu [45] successfully simulated the transition in hydrogen–oxygen mixtures in 1 mm wide capillary tubes under both thermally insulating and isothermal boundary conditions.

## 2. Transition in open-end systems

The near-absence of the incipient acceleration [22,23] renders open-end obstructed systems particularly advantageous for quasi-one-dimensional modelling. In open-end systems, accounting for multi-dimensional effects through a lump drag-force term seems to be less of a compromise than in closed-end systems, where the flame markedly accelerates virtually from the very point of its inception (figure 4). In the present study, as in Brailovsky & Sivashinsky [10], to reduce the number of parameters involved, the effects due to heat losses are discarded. The adiabatic limit may serve as quite a legitimate approximation for the description of combustion in rough-walled/obstructed tubes or inert porous beds where, away from the propagability threshold, the Reynolds analogy is violated and the momentum loss emerges as a dominating influence. The reaction rate is assumed to be one-step, irreversible, first-order and with Arrhenius temperature dependence. In suitably chosen units, the set of governing equations thus reads:

*continuity and state*,
2.1
*momentum*,
2.2
*heat*,
2.3
*concentration*,
2.4
*chemical kinetics*,
2.5
Here, is the scaled pressure in units of the initial pressure, is the scaled concentration of the deficient reactant in units of its initial value, is the scaled temperature in units of *T*_{p}=*T*_{0}+*QC*_{0}/*c*_{p}, the adiabatic temperature of burned gas under constant pressure, *P*_{0}; *T*_{0} is the initial temperature of unburned gas; *Q* is the heat release; *σ*_{p}=*T*_{0}/*T*_{p}; *γ*=*c*_{p}/*c*_{v}; *c*_{p},*c*_{v} are specific heats; , where *ρ*_{p}=*P*_{0}/(*c*_{p}−*c*_{v})*T*_{p} is the density of combustion products in free-space deflagration; is the scaled flow velocity; is the sonic velocity at *T*=*T*_{p};*N*_{p}=*E*/*RT*_{p} is the scaled activation energy; , where the reference time is defined as is the thermal diffusivity at *T*=*T*_{p};*u*_{p} is the velocity of the free-space deflagration relative to the burned gas, regarded as a prescribed parameter; is the scaled thermal diffusivity; *Pr* and *Le* are the Prandtl and Lewis numbers, respectively; and is a normalizing factor to ensure that at *N*_{p}≫1 the scaled deflagration velocity relative to the burned gas is close to .

In the adopted formulation, the molecular transport coefficients as well as specific heats are assumed to be constants specified by their values at *T*=*T*_{p}. As may be readily shown, the reference length scale , where is the flame width.

The scaled drag force is specified as
2.6
where *κ*_{p} is the resistance intensity. Outside the tube . Inside the tube , where *c*_{f} is the drag coefficient assumed to be constant, and *d* is the hydraulic diameter.

For the open-end system, equations (2.1)–(2.6) are considered over the infinite interval, , and the pertinent solution is required to meet the following initial and boundary conditions: 2.7 and 2.8 Here, is the scaled width of the initiation hot-spot.

The extension of the one-dimensional formulation over the resistance-free half-space presumes that the hydraulically resisted half-space is occupied by a tight bundle of obstructed tubes or an inert porous bed.

In numerical simulations, the original infinite interval is replaced by a finite one (−*L*_{−}<*x*<*L*_{+}), and conditions (2.8) are applied to the new boundaries. The intervals (−*L*_{−},0),(0,*L*_{+}) should be wide enough to ensure that, within the time period considered, the emerging compression waves generated by combustion waves and ejected products do not reach the boundaries.

To avoid too large a disparity between the spatio-temporal scales involved and facilitate visualization, yet hopefully without much detriment to the generic qualitative picture, the numerical simulations are conducted for somewhat reduced values of the scaled activation energy (*N*_{p}), and elevated values of the scaled diffusivity (*ϵ*) and the resistance parameter (*κ*_{p}), compared with those typical of real-world systems. Specifically, we set (as in Brailovsky & Sivashinsky [10]): *N*_{p}=5.635; *ϵ*=0.04; *σ*_{p}=0.245; *κ*_{p}=0,0.34; *Pr*=0.75; *Le*=1; *γ*=1.3; ; *L*_{±}=±750.^{3}

In view of the relatively low activation energy, in order to suppress the reaction in the fresh premixture, the Arrhenius exponent in equation (2.5) is modified by .

The results of numerical simulations may be summarized as follows. At zero friction (*κ*_{p}=0) soon after ignition one ends up with two diverging shocks (confined and unconfined) propagating at near-sonic velocities (figure 8). The confined shock is followed by a markedly slower deflagration spreading at the visible velocity . Owing to the counter-flow, the latter is smaller than , corresponding to the deflagration velocity relative to the burned gas. Note the change in the direction of the gas flow across the deflagration. At one also observes a step-like temperature wave as well as a minor pocket of the leaked reactant—both advected by the flow ejected from the confinement.

At *κ*_{p}=0.34, similar to the friction-free case, one observes a readily identified trend to form a subsonic deflagration. This time, however, owing to friction, the confined shock forms a non-uniform pressure and temperature profiles undergoing rapid expansion and the resistance-induced pressure buildup (figure 9). As a result, the emerging deflagration spreads over a non-uniformly precompressed and preheated background gradually enhancing the deflagration speed, compared with its friction-free counterpart. Upon an extended induction period, the unburned gas adjacent to the advancing reaction zone undergoes a nearly instantaneous autoignition, triggering an abrupt transition from subsonic to supersonic propagation (figure 2). In contrast to the rapid evolution of the basic hydrodynamic profiles, the visible deflagration velocity, up to the very point of DDT, remains virtually constant. This picture is quite in line with the recent experimental/numerical studies on the transition in vented obstructed channels [22,23].

Unlike the closed-end problem (figure 1), up to the transition point, the open-end pressure coincides with that of the friction-free situation (figure 8). As a result, the ignition at the open end greatly encumbers the transition, compared with the closed-end case.

Note the experimentally well-known velocity and pressure overshoots occurring immediately after the transition before the wave settles to a galloping or a constant velocity detonation. The latter takes place at somewhat lower activation energies, e.g. at *N*_{p}=4, other parameters being fixed.

## 3. Minimal model: Shchelkin's effect

The hydraulic resistance models discussed in this paper may obviously be extended to incorporate more ingredients that are likely to be of quantitative importance. Much has been performed in this direction particularly in the field of gas-permeable energetic materials. Here, however, we would like to stress the potential of the opposite strategy where one deliberately discards many aspects in order to elucidate the impact of those that are kept. The latter approach is of great value where the goal is not so much to obtain numbers for comparison with experimental data, but rather to gain physical insight by making the problem tractable. In this sense, the quasi-one-dimensional formulation of §2 is actually not the simplest framework for displaying the transition event. As mentioned in §1, at strong hydraulic resistance, the compression-supported burning may spread at a high but subsonic velocity (figure 5). In this parameter range, the dynamically simplest situation emerges for the simultaneous limit of small Mach number and small heat release, where one ends up with a compact quasi-linear model governed by a set of parabolic equations [8,9],
3.1
3.2
3.3
and
3.4
Equations (3.1) and (3.2) represent partially linearized conservation equations for energy and the deficient reactant mass fraction. Equation (3.3) is a linearized continuity equation, incorporating the equation of state and momentum (Darcy's law). A detailed derivation is presented in Brailovsky *et al*. [8]. The reference scales employed here are somewhat different from those of §2. Specifically, *Θ*=(*T*−*T*_{0})/(*T*_{v}−*T*_{0}), *Π*=(*P*−*P*_{0})/(*P*_{v}−*P*_{0}), *Φ*=*C*/*C*_{0}, *Ω*=*W*/*W*_{0}, *ξ*=*x*/*x*_{0}, *τ*=*t*/*t*_{0}, where *T*_{v}=*T*_{0}+*QC*_{0}/*c*_{v} and *P*_{v}=(*c*_{p}−*c*_{v})*ρ*_{0}*T*_{v} are the final temperature and pressure of combustion products upon adiabatic homogeneous explosion; are the reaction rate and spatio-temporal scales, respectively; *D*_{p} is the resistance-induced pressure diffusivity (see footnote 1 in §1); *σ*_{v}=*T*_{0}/*T*_{v} and *N*_{v}=*E*/*R*^{0}*T*_{v} are the temperature ratio and scaled activation energy based on *T*_{v}; *A* is a normalizing factor to ensure that at *N*_{v}≫1 the scaled velocity of the well-settled subsonic detonation is close to unity; *ε*=*D*_{th}/*D*_{p} is the thermal diffusivity/pressure diffusivity ratio. For many realistic porous systems, *ε* varies within the range *ε*∼10^{−4} to 10^{−7}, which makes it a natural small parameter of the problem.

At small *ε*, one may single out two distinct modes of combustion: (i) fast wave sustained by the diffusive transfer of pressure (subsonic detonation) and (ii) slow wave sustained by the diffusive transfer of heat (deflagration). In the subsonic detonation regime for the leading-order asymptotics, the original equations (3.1)–(3.3) simplify to
3.5
3.6
and
3.7
This shortened system admits to the travelling wave solutions spreading at velocity *λ*∼1.

In the deflagrative regime, , , and for the leading-order asymptotics, equations (3.1)–(3.3) yield 3.8 3.9 and 3.10 This system is obviously nothing but a conventional constant-density model for the free-space deflagration. The associated travelling wave solution spreads at a velocity proportional to the square root of the thermal diffusivity, .

The higher-order approximation for subsonic detonation—that is, incorporation of the thermal diffusivity—does not produce any significant change in the overall dynamic picture. There still exists a steady travelling wave solution with *λ*∼1. For the deflagration, the picture is different. Here, the higher-order approximation—that is, accounting for the hydraulic resistance—leads to the local elevation of pressure. The latter, however, does not stabilize at some low level but rather keeps growing as . This gradual buildup of pressure results in the formation of an extended preheat zone ahead of the advancing and slightly accelerating flame. The fresh mixture adjacent to the flame undergoes a low gradient precompression and preheating. This slowly proceeding development ultimately ends up as an adiabatic explosion that abruptly converts the burning from deflagrative to detonative, yet subsonic propagation ([8,9]; figures 10 and 11*a*).

Chronologically, it is precisely the outlined minimal model that happened to be a precursor of the resistance concept of the transition, and we believe that the model still has much to offer as a guide for further exploration of the transition phenomena in gaseous and gas-permeable systems.

As a new example, let us take a look at how the above model reproduces the well-known Shchelkin's observation that the ignition at a small distance from the closed end facilitates the transition [46–48]. In this case, equations (3.1)–(3.4) may be considered jointly with the following set of boundary and initial conditions:
3.11
and
3.12
Here, *Δ* is the site of the initiation hot-spot. For the conventional closed-end ignition, *Δ*=0 (figure 10).

As one would expect, at *Δ*>0 the incipient dynamics involves two deflagration waves spreading to the left and to the right (figure 12). At a certain point, the fuel adjacent to the wall undergoes compression-induced autoignition, resulting in a dramatic elevation of the pressure. The overpressure rapidly spreads over the rest of the precompressed fuel, thereby reducing the predetonation time (figure 11). Yet, for large *Δ*, one ends up with a situation effectively similar to that of the closed-end ignition with the wall located at *ξ*=*Δ*. This explains the non-monotonic character of the predetonation time dependence on *Δ* (figure 13).

Our exploratory numerical tests have shown that the described effect holds also for fully nonlinear quasi-one-dimensional (figure 14) as well as two-dimensional formulations allowing for shocks and conventional supersonic detonations. The plots of figure 14 are qualitatively in line with recent experimental studies by Blanchard *et al.* [48], thereby providing a new argument in support of the resistance concept of the DDT.

## 4. Concluding remarks

One of the difficulties for a theoretical exploration of the DDT is its essentially dynamic time-dependent structure involving several transient stages. The problem, however, does not seem to be rationally intractable, and presumably may be tackled along the basic lines of a multiple-scale analysis, i.e. interpreting DDT as a failure of the slowly evolving deflagrative propagation, where the unburned gas undergoes a quasi-steady precompression and preheating.

Another attractive possibility is to convert the DDT problem from dynamic to parametric as occurs in the transition from conductive to convective burning in gas-permeable solid explosives [32,33]. One of the crucial differences between burning of gas-permeable and confined gaseous explosives is that, in the former case, the solid phase gasifies and hence the products are not subjected to hydraulic resistance. In the light of this, it would be interesting to explore an imaginary situation where the confinement of the gaseous explosive vanishes behind the reaction front. In this formulation, the burned gas pressure becomes independent of the pressure deep inside the confinement and therefore may be used as a free parameter controlling the character of the emerging steady-state combustion waves (low-velocity deflagration, near-sonic choked/convective burning, high-velocity near-CJ detonation) and transitions between different regimes.

## Acknowledgements

The authors thank Elaine Oran and Forman Williams for inviting this paper, and Michael Kuznetsov and Ming-Hsun Wu for fruitful discussions and correspondence. These studies were supported in part by the Bauer-Neumann Chair in Applied Mathematics and Theoretical Mechanics, the US–Israel Binational Science Foundation (grant 2006-151), and the Israel Science Foundation (grant 32/09).

## Footnotes

One contribution of 12 to a Theme Issue ‘The physics, chemistry and dynamics of explosions’.

↵1 The simplest system to observe the pressure diffusivity is the isothermal near-constant-density flow in porous media with the linear Darcy's law as the momentum equation. In this case, the pressure (

*P*) is governed by the diffusion equation ∂*P*/∂*t*=*D*_{p}∇^{2}*P*, where*D*_{p}=*Ka*^{2}/*γν*is the pressure diffusivity;*K*is permeability;*ν*is kinematic viscosity;*a*is sonic velocity; and*γ*=*c*_{p}/*c*_{v}is the ratio of specific heats. For non-isothermal compressible flows with the quadratic Forchheimer's law as the momentum equation, the expression for the pressure diffusivity is somewhat more complicated [6,12].↵2 The idea of extending the concept of detonation is not new. In the 1987 survey by Mitrofanov [35], while discussing the non-classical combustion waves, the author wrote, ‘It would seem appropriate to extend the notion of detonation over a certain subsonic range of wave velocities (

*D*<*a*_{u}) where there is a continuous in*D*passage to this domain with the preservation of pressure and density peaks within the front.’ It is interesting that this issue was raised by the experimentalists 10 years ahead of the theoretical substantiation of the phenomenon.↵3 A realistic set of parameters would be

*ϵ*=2.5×10^{−3},*σ*_{p}=0.125 and*κ*_{p}=0.14, which would correspond to*c*_{f}=0.55 (porous bed),*d*=0.25 cm,*D*^{p}_{th}=8 cm^{2}s^{−1},*u*_{p}=50 m s^{−1}and*a*_{p}=10^{3}m s^{−1}.

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