## Abstract

The dynamics of gaseous detonation is revisited on the basis of analytical studies. Problems of initiation, quenching, pulsation and cellular structures are addressed. The objective is to improve our physical understanding of the development, stability and structure of gaseous detonations. New insights that have been gained from analytical investigations are emphasized. Specific problems discussed are the direct initiation of detonations in spherical geometry, the spontaneous soft initiation and quenching of detonations in a temperature gradient, the stability threshold and dynamics of galloping detonations, and the multi-dimensional instability threshold and cellular structures of both overdriven and near-Chapman–Jouguet detonations. It will be seen that, although there have been many accomplishments, some outstanding questions remain.

## 1. Introduction

According to Zeldovich, von Neumann and Döring (ZND), gaseous detonations consist of a strong inert shock followed by a reaction region. In this structure, following the shock two distinct layers are identified, an induction zone followed by an exothermic zone. The lengths of these two zones are of a similar order of magnitude and are both much larger than the shock thickness, which is therefore considered as a hydrodynamic discontinuity across which the Rankine–Hugoniot jump conditions apply. The gas flow is subsonic relative to the shock, but its velocity is sufficiently large that heat conduction and molecular diffusion can be neglected, the spatial distribution of temperature being described by a balance between advection and reaction. Conceptually challenging aspects remain, even when given these ZND simplifications.

Investigation of the dynamics of gaseous detonations then proceeds by solving the (hyperbolic) Euler equations with exothermic reactions and with two types of boundary conditions: one at the perturbed shock (a free boundary) and the other in the burned gas. Overdriven detonations are supported by a piston moving at a constant velocity, subsonic with respect to the shock. With the piston placed at infinity, the boundary condition at infinity is a boundedness condition of unstable modes. This corresponds also to a radiation condition at the exit of the reaction zone. In contrast, self-sustained detonations are characterized by the existence of a sonic surface in the burned gases, a condition called Chapman–Jouguet (CJ).

Planar detonations are usually unstable, and detonation fronts are typically cellular. The boundary condition in the burned gas of self-sustained detonations (the sonic surface) does not introduce new physical mechanisms concerning the formation of cellular structures; the results at small overdrive and at CJ conditions are similar, and the overdriven regime is more easily investigated. On the other hand, critical conditions for initiation and quenching in spherical or planar geometry are exhibited well and conveniently by using a sonic condition along with a quasi-steady-state approximation.

These hyperbolic problems may be treated numerically, either by direct numerical simulation (DNS) [1,2] or, in the linear approximation, by solving the eigenvalue problem, which can be accomplished either by performing a normal-mode analysis using shooting algorithms, as done by Short & Stewart [3], or by formulating the stability problem as an initial-value problem with the help of a Laplace transform in time, as done by Erpenbeck [4], whose pioneering work ended with Erpenbeck [5]. This latter approach was revisited recently by Tumin [6]. All of these numerical results are useful but, as is typical of DNS and numerical studies, they do not fully explain the physical mechanisms. Analytical studies promote better understanding. However, they can be completed only for idealized situations and in limiting cases. Attention is focused in this paper on some of the analytical approaches, including new results, with an emphasis on the physical aspects more than on the mathematical formalism.

## 2. Formulation

A two-dimensional, time-dependent, chemically reacting Euler flow of an ideal gas is considered, with *x* denoting the coordinate in the main flow direction, *y* the transverse coordinate and *t* time. Velocity components in the *x* and *y* directions are denoted by *u* and *w*, respectively, **u**=(*u*,*w*). The Euler equations are studied downstream from the shock, and chemistry presumed not to occur ahead of it. Expressed in terms of the density *ρ* and temperature *T*, the sound speed *a* and pressure *p* are given by
2.1where *γ* is the ratio of the specific heat at constant pressure *c*_{p} to the specific heat at constant volume *c*_{v}. The equations for conservation of mass, momentum and energy can be written, respectively, as
2.2where D/D*t*=∂/∂*t*+**u**⋅∇ is the material derivative, *Q* denotes the chemical heat release per unit mass of mixture, *t*_{r} is a representative chemical reaction time and is a non-dimensional function of state variables that describes the rate of chemical heat release. In the following, *t*_{r} will be chosen to be the induction time at the Neumann state of the unperturbed detonation, . An alternative and often more useful form of the energy conservation equation, obtainable by use of equations (2.1) and (2.2), is the entropy equation, which may be written as
2.3where the equation for mass conservation has been used to eliminate the density.

It is convenient to let *Y* denote the fraction of chemical heat that has been released. When the complex systems of equations for describing a detailed chemical kinetic scheme are discarded, one may use the short notation
2.4where is regarded as a function of *Y* and *T*, the dependence of the reaction term on the pressure being neglected for simplicity, the dependence on temperature in gaseous mixtures usually being much stronger. This hyperbolic system possesses three characteristics, two corresponding to acoustic wave propagation, and one, travelling at the velocity of a fluid element, across which entropy and (in more than one space dimension) vorticity change, the so-called entropy–vorticity wave.

### (a) Conditions at the shock

Let denote the detonation propagation velocity, an overbar identify the steady, planar solution whose stability is investigated and the subscript u represent conditions ahead of the shock. Introducing the propagation Mach number corresponding to the normal shock velocity , the pressure *p*_{N} and density *ρ*_{N} at the Neumann state (in the compressed gas just behind the inert shock) are obtained in terms of the pressure and density in the initial mixture by the Rankine–Hugoniot conditions
2.5The temperature at the front is obtained from these equations when using the perfect gas law (2.1). The additional requirements of mass conservation and transverse momentum conservation provide the flow velocity at the Neumann state, written in the linear approximation as
2.6where *u*_{N} and *w*_{N} are the longitudinal and transverse components of the flow velocity at the Neumann state, expressed in the reference frame of the unperturbed solution, and *x*=*α*(*y*,*t*) represents the perturbed shock position at transverse position *y* at time *t*.

### (b) Downstream boundary condition

For self-sustained detonations it is a difficult task in multi-dimensional problems to enforce the sonic condition [7]. Concerning stability analyses, the boundary condition in the burned gases is much simpler for overdriven detonations, even for weakly overdriven detonations very close to a sonic condition [8]. A boundedness condition at infinity for unstable acoustic modes removes one of the two acoustic eigenmodes. This leads to a compatibility condition for the perturbations of pressure and the components of the flow velocity in the burned gas. This condition corresponds to a radiation condition that prevents acoustic waves from travelling upstream from infinity to impinge the detonation. Then, as for the case of self-sustained detonations with a sonic locus, there is no disturbance coming from the burned gas that influences the lead-shock dynamics. This persuades us to think that the stability analysis of CJ detonations does not involve mechanisms other than those controlling the overdriven detonations.

## 3. Direct initiation in spherical geometry

We present in this section a transient phenomenon that is well represented by a simplified analysis using a steady-state approximation for the reaction zone of the detonation structure. Direct initiation of detonation refers to the formation of a detonation in the decay of the strong blast wave produced by a powerful concentrated ignition source. If sufficient energy is deposited, , a CJ detonation is formed at a certain distance from the source. The challenge is to predict the critical initiation energy *E*_{C} from first principles.

Assuming a spherical geometry and considering a small deposition time with a tiny size of the igniter, the self-similar solution of Sedov [9] and Taylor [10] may be considered as the initial condition, since the chemical heat release is negligible compared with *E* at early time,
3.1where *R*(*t*) and are the radius and the velocity of the shock wave at time *t*, and *ρ*_{u} is the density of the initial mixture. The first ignition criterion was provided by Zeldovich *et al.* [11] on the basis of a dimensional analysis: the time *t* taken for the blast wave to reach the CJ velocity, , must be larger than the transit time of a fluid particle across the CJ detonation, , yielding
3.2This gives for the minimal radius *R*_{C} at which the CJ detonation may be formed *R*_{C}≈*l*_{CJ}, where is the detonation thickness of the planar CJ detonation. Recalling that , where *Q* denotes the total heat released per unit mass in the detonation, the criterion in (3.2) may be written as
3.3meaning that the detonation thickness of the CJ planar wave is the length scale governing the ignition problem. This is not right; the criterion (3.2) gives values for *E*_{C} many orders of magnitude smaller than the experimental data [12].

The temperature sensitivity of the heat-release rate introduces a large non-dimensional parameter, namely the reduced activation energy *β*≈10. Therefore, unsteadiness and weak curvature may modify strongly the length and time scales. Using a length larger than *l*_{CJ} in (3.3) was considered in the past on phenomenological grounds by A. Liñán, who suggested to use *βl*_{CJ}; see Sichel [13]. Much larger values of *E*_{C} and *R*_{C} were obtained by He & Clavin [14] when the modifications introduced by the spherical geometry were taken into account. This work is worth summarizing because it illustrates how sometimes the correct underlying physics may be extracted from drastic simplifications.

The first step is to analyse the structure and propagation of weakly curved self-sustained detonations in the quasi-steady-state approximation. Owing to temperature sensitivity, nonlinear curvature effects are exhibited for weak curvature. It is thus shown that the curve of the propagation velocity versus the radius () exhibits a turning point corresponding to a curvature-induced quenching at a critical radius *R**. There is no solution for *R*<*R**. A simple expression for *R** and for the critical propagation velocity was obtained by He & Clavin [14] from an Arrhenius law in the limit (yielding the so-called square-wave model when the detonation thickness *l*_{CJ} is used as unit length),
3.4This shows how large the critical radius is when compared with *l*_{CJ} and how close to the critical velocity is when *β* is large. The result is caused by the strong variation of the induction length *l* with the Neumann temperature *T*_{N} and thus with the shock velocity ,
3.5the second approximation being valid for a strong shock. An asymptotic analysis of this problem has also been performed by Yao & Stewart [15]. The C-shape of the curve has also been obtained numerically for hydrogen–oxygen mixtures [16].

In view of these results, it becomes reasonable to assume that a detonation is initiated by a blast wave only if the radius of the shock *R* is equal to or larger than *R** when the shock velocity reaches ; so the critical energy given by (3.1) is
3.6which shows how large the ratio between the two expressions (3.6) and (3.3) for the critical energy is, namely, (*R**/*l*_{CJ})^{3}≈10^{6} for *β*=10. The same result is obtained when a reacting blast wave is considered instead of the inert blast wave (3.1).

In the limit of zero thickness of the detonation front, modifications to the inner detonation structure are neglected. In this limit, there is no critical energy; a CJ detonation is systematically reached by the reacting blast wave at a radius *R*=*R*′. This radius may be expressed in terms of the energy *E*, the heat release *Q* and the initial density *ρ*_{u} by dimensional analysis, *R*′^{3}∝*E*/*ρ*_{u}*Q*. When the internal structure of the detonation is taken into account, it is reasonable to assume that a CJ detonation is initiated when the energy is sufficiently large so that , where *R** is given by the first equation in (3.4). This leads to (3.6) again for the critical energy.

Three assumptions limit the quantitative applicability of this result, namely a blast wave, an oversimplified chemical kinetic model and a quasi-steady state of the detonation structure. The first approximation is justified if the deposition time of energy is much smaller than and if the size of the igniter is much smaller than *R**. The dependence of the critical energy *E*_{C} on the deposition time could be addressed by extending the earlier analysis of Kurdyumov *et al.* [17] to the reactive case. Concerning the quasi-steady state, DNS carried out in He & Clavin [14] shows that unsteadiness of the detonation structure does not change drastically the order of magnitude of *E*_{C}. This points out the essential role of nonlinear curvature effects that may be described in an approximation of weak curvature, thanks to the strong thermal sensitivity. A somewhat different point of view is presented by Lee & Higgins [18] and Eckett *et al.* [19], who conclude that unsteadiness is essential. The work of He & Clavin [14] was extended later by Kasimov & Stewart [20] and by Vidal [21], essentially by taking into account unsteadiness in an approximation of slow time evolution.

As explained next, addressing intrinsic unsteadiness requires considering time scales as short as the transit time at least, and it therefore cannot be correctly described by assuming a slow time evolution.

## 4. Unsteady planar detonations

In unsteady propagation, the induction length *l*(*t*) and the reaction zone both vary in time. In gaseous detonations with a large temperature sensitivity, this is mainly due to the temperature variations.

### (a) Variations of the induction length

The calculation of the variation of the induction length *l* in terms of the time-varying shock velocity is not an easy task in general. The calculation is simplified in the following limits. When the induction zone is sufficiently subsonic, the temperature fluctuations propagate across the shocked gas mainly with the entropy wave, the acoustic waves being sufficiently fast to make the pressure quasi-uniform throughout the induction zone. Then, assuming a large temperature sensitivity, *β*≫1, and considering small variations of the detonation velocity, , one finds that the induction length *l*(*t*) is given in terms of *T*_{N}(*t*) by a nonlinear equation delayed by the entropy wave,
4.1where *v*_{N} is the flow velocity relative to the shock at the Neumann state,
4.2Equation (4.1) is valid at leading order in the limit (see eqns (A 9)–(A 10) in [22]). Its relative simplicity comes from the fact that the variations of pressure and flow velocity are negligible at leading order. It shows that equation (3.5) is valid only when two conditions are satisfied—that is, when the time scale of is much longer than the transit time and when the detonation is stable. Unfortunately, ordinary detonations are unstable, with a linear growth rate and a period of oscillation of the same order as .

### (b) Galloping detonations

One-dimensional oscillations, called galloping detonations, were first observed in the 1960s in the DNS of Fickett & Wood [23]. The first numerical stability analysis goes back to [24], and detailed numerical results were presented more recently by Lee & Stewart [25] and Tumin [6] for a reaction rate governed by an Arrhenius law. A consistent asymptotic analysis of this phenomenon was carried out by Clavin & He [22] for strongly overdriven detonations and general reaction rates. The analysis was carried out for a distribution of heat release sensitive to the temperature, *β*≫1, in the Newtonian approximation (*γ*−1)≪1 and in the distinguished limit
4.3Within the approximation (2.4), *β* is simply the temperature sensitivity of the reaction rate at the Neumann state,
4.4The analysis may be extended to more complex kinetics; see [26] for one example and also (4.10). The first two relations in (4.3) along with the identity
imply that the induction zone is very subsonic, . The additional third relation in (4.3) then implies that the quasi-isobaric approximation is valid throughout the detonation thickness, including the reaction zone. As is usual in one-dimensional, unsteady problems, it is convenient to introduce a mass-weighted coordinate,
4.5and
4.6where is the density at the Neumann state of the unperturbed detonation. Considering small variations of the detonation velocity,
4.7we see that the equations for conservation of energy (2.2) and species (2.4) take, at leading order, the simple forms
4.8and
4.9with the boundary conditions *T*=*T*_{N}(*t*) and *Y* =*Y* _{u} at the Neumann state, *ξ* = 0. The solution is given by the steady-state distributions , , , in which the Neumann temperature is delayed by the time lag of the entropy wave , yielding . This relation is valid for any complex chemical scheme under the assumptions (4.3) and (4.7). In order to exhibit the thermal sensitivity more explicitly, it is convenient to rewrite the distribution of heat-release rate in the form
4.10where is the distribution of the steady, planar detonation associated with a non-dimensional Neumann temperature *Θ*_{N}, with
4.11This last quantity expresses the deformation of the distribution of heat release in steady-state, overdriven detonations when the propagation velocity is modified, .

On the other hand, continuity, the first equation in (2.2), yields at leading order
4.12with boundary conditions at infinity and at the Neumann state,
4.13where, at leading order, , since , with, according to (2.5) and (4.3), *ρ*_{u}/*ρ*_{N}=*O*(1/*β*). A nonlinear integral equation for *Θ*_{N}(*t*) is then obtained by integrating (4.12) from *ξ*′=0 to when the quasi-isobaric condition and equation (4.8) are used together with the Hugoniot condition linking and *T*_{N}(*t*). Introducing the non-dimensional quantities
4.14we find that
4.15yielding in the linear approximation
4.16where the second equation is a transcendental equation for the complex linear growth rate *σ*=*s*+i*ω* of perturbations proportional to e^{στ}, exhibiting a discrete set of solutions.

Two quantities govern the nonlinear lead-shock dynamics, namely the scalar *b* and the distribution or in the linear approximation. For a given distribution , the system is stable for sufficiently large values of *b*, but a Hopf bifurcation is encountered upon decreasing *b* in (4.16). The present result describes the oscillatory behaviour of galloping detonations with a time scale of the same order as the transit time of a fluid particle. Upon decreasing *b* further, it is observed that the nonlinear equation (4.15) describes a spontaneous quenching of the detonation, and planar detonations cannot exist for a reaction rate that is too sensitive to the temperature. Equation (4.15) qualitatively represents the DNS results well [22]. The linear growth rate and the oscillatory frequency of the unstable modes increase also when the distribution becomes stiffer. For example, the limit of a one-step Arrhenius model (a square-wave model) leads to singular dynamics, and it cannot be used to study the intrinsic dynamics of detonations, even though it may lead to interesting results for stable cases if a quasi-steady-state approximation is valid (for example, see the end of §4*c*).

In real detonations, the linear dynamics is controlled by a loop between the lead shock and the reaction zone, involving the entropy wave and two acoustic waves, one running downstream, the other forward, so that two additional time delays are involved, besides the transit time. They both are usually shorter than the transit time, since the induction zone is subsonic. They shrink to zero in the limit considered earlier. The effects of the acoustic waves have been described in perturbation by Clavin & He [27], who proved that acoustics is stabilizing. This means that the oscillatory instability of detonations does not arise from the mechanisms of thermo-acoustic instability, quite contrarily to what was often thought before.

On the basis of an analysis by Clavin & Williams [28] of the limit considered in §5*d* for a detonation close to a CJ wave with small heat release, it was conjectured that, in proceeding from strongly overdriven detonations to CJ waves, the bifurcation parameter *b* is changed to
where the scaled overdrive factor *f* is defined in that later section; see equation (5.49). Furthermore, in that limit, an additional term coming from compressible effects appears in (4.16). This term, which does not depend on *β*, is a stabilizing influence, showing that the only source of pulsating instability is the strong dependence of the heat-release rate on the Neumann temperature, described in (4.16). Interpolation of the two analytical results obtained in the two opposite limits (strong overdrive and small overdrive with small heat release) fits the numerical results for the stability threshold in a wide range of overdrive factor.

### (c) Spontaneous initiation and quenching

Spontaneous initiation of a detonation in a preconditioned medium with a temperature gradient was predicted by Zeldovich *et al.* [29], Lee *et al.* [30] and Zeldovich [31]. The key point is that a gradient of induction time is associated with the temperature gradient. For a sufficiently high initial temperature *T*_{u}, the induction time *τ*_{I}(*T*_{u}) varies strongly with the initial temperature *T*_{u}, in a way similar to the variation of the induction length with the Neumann temperature *T*_{N} in (3.5),
4.17Roughly speaking, a detonation is initiated when the velocity of the induction front, namely the inverse of the gradient of the induction time, is decreased to the velocity of pressure disturbances (sound speed *a*_{u}) so that a runaway of the pressure occurs, leading quasi-spontaneously to a CJ wave, a phenomenon reminiscent of Rayleigh's criterion for the occurrence of thermo-acoustic instability. More precisely, spontaneous ignition may occur at a temperature *T**_{u} where the temperature gradient satisfies the condition
4.18Here and are the induction time and the characteristic length of variation of the temperature at *T**_{u}, respectively. Equation (4.17) has been used in the second relation (4.18). In spherical geometry, a second condition should be satisfied for spontaneous initiation, namely that the radius at which (4.18) holds should be larger than the critical radius given by (3.4). In contrast to direct initiation, this spontaneous initiation does not result from the blast wave generated by the energy deposition. It is a soft process since it does not require a high energy deposition in a tiny pocket, but instead it just needs a particular temperature gradient at high temperature. The problem has received considerable attention in the literature; see [32] for an extensive review and a detailed numerical study.

Another phenomenon was described by He & Clavin [33]. They showed that a spontaneous quenching may occur shortly after ignition. The same temperature gradient that is responsible for spontaneous initiation at high temperatures may well also be responsible for spontaneous quenching of the detonation at lower temperature. This led to definitions of critical shapes and critical sizes of initial hot pockets of fresh mixture for igniting a detonation, as observed experimentally in photochemical initiation by Lee *et al.* [30] (see [34]).

This spontaneous quenching may be explained in simple terms using a particular form of the quasi-steady-state approximation. The numerical results of He & Clavin [33] showed that the structure of the detonation that is formed is close to that of a CJ wave, but there are small differences. Plotting the pressure *p* as a function of the specific volume 1/*ρ* throughout the reaction zone, a straight line NB is obtained, linking the Neumann state (labelled N) to the burned gas (labelled B), which is, as in CJ waves, tangent to the equilibrium curve of burned gases corresponding to the initial state at temperature *T*_{u}. This shows that the reaction zone where the chemical energy is released is in quasi-steady-state equilibrium. However, because of the changing conditions, the magnitude of the slope of this NB line is slightly less than the slope of the line NI in the *p*−1/*ρ* plane that links the Neumann state to initial state (labelled I) (the Michelson–Rayleigh line of the steady-state detonation). Therefore, the mass flux across the lead shock is slightly larger than the mass flux across the reaction zone, so that the reaction zone separates slowly from the shock. In this approximation, unsteadiness of the detonation structure concerns only processes occurring in the induction zone. This is correct if the transit time across the reaction layer is shorter than that across the induction zone, which generally is true because the flow velocity in the (higher-temperature, lower-pressure and thus lower-density) reaction layer is larger than in the induction zone, while the two thicknesses are comparable.

The difference of mass flux is proportional to the time derivative of the induction length times the density at the Neumann state, *ρ*_{N} d*l*/d*t*. The time derivative d*l*/d*t* may then be computed in two different ways, namely by a geometrical construction around the slope of the CJ Michelson–Rayleigh line, leading to expressing d*l*/d*t* as a linear function of ,
4.19or by using (3.5), leading to expressing d*l*/d*t* as a linear function of ,
4.20where , and are the detonation velocity, the flow velocity at the Neumann state and the induction length of the local CJ wave. The non-dimensional constant *c* is obtained from the relative difference of slopes when the lead-shock velocity is slightly modified from its CJ value (for which the difference of slope is zero),
where the approximation *c*≈1 holds for , . Putting together the two equations (4.19) and (4.20) leads to a nonlinear equation for the local velocity of the shock, or, more precisely, for ,
4.21where is the transit time of a fluid particle across the induction zone (the induction time at the Neumann state), and where the relation has been used, since is negligible, of order 1/*β*. The first expression in (4.21) defines a critical value for the onset of spontaneous quenching, since there is no solution for .

Since the variation of the CJ detonation velocity with the initial temperature *T*_{u} is , from the second expression in (4.21),
4.22where the characteristic length of variation of the temperature *l*_{u}≡*T*_{u}/|*dT*_{u}/*dx*| has been introduced. According to (4.18), *K* may also be written as
4.23where is the propagation Mach number of the CJ detonation at the temperature *T**_{u} at which spontaneous initiation occurs, and where the approximation holds for a constant temperature gradient, since and the propagation Mach number of a CJ detonation may be approximated as . This result relates spontaneous ignition to spontaneous quenching.

For a detonation propagating into a region of decreasing temperature, at a temperature slightly smaller than *T**_{u}, , the transit time across the induction zone of the CJ detonation is much shorter than the induction time *τ**_{I} of the reactive mixture at *T**_{u}, . Therefore *K*≪1, and the detonation propagates across the temperature gradient. However, increases quickly when the temperature of the fresh mixture decreases, and spontaneous quenching appears if *K* reaches before a uniform temperature of fresh mixture is reached. A comparison with DNS calculations shows good agreement with spontaneous quenching predicted by (4.21) [33,34].

For a constant temperature gradient, a rough estimate of the critical condition for quenching is obtained as follows: The value of the coefficient is of order unity, typically between 1.5 and 2. Therefore, spontaneous quenching occurs when the transit time increases to reach a value comparable with the induction time *τ**_{I} at spontaneous ignition. This corresponds roughly to *T*_{N} close to *T**_{u}.

The particular quasi-steady-state approximation that is used in this analysis is justified when two conditions are satisfied. The temperature of the initial mixture must be sufficiently high that the sensitivity to temperature remains well below its critical value for the instability of galloping detonations. In addition, the characteristic evolution time must be larger than the induction-zone transit time , implying, according to the relation , that the temperature gradient cannot be too large, , a condition that is typically satisfied in experiments.

## 5. Cellular structure

Gaseous detonations are known to exhibit transverse structure with cell sizes that are much larger than the detonation thickness. Cellular detonations occur because the planar waves are strongly unstable to disturbances with transverse components. This happens even when the detonations are stable to purely longitudinal disturbances. In that sense, the cellular instability is stronger than the galloping instability described earlier. However, the underlying oscillatory behaviour of the planar detonation, even when it is damped in planar geometry, is essential to cell formation.

### (a) Method

Analytical studies of cellular detonations cannot be performed in a systematic way in the general case. This observation applies, in fact, for most nonlinear patterns encountered in physics. However, analytical descriptions may be obtained for weakly unstable regimes. If there are no secondary bifurcations, these bifurcation analyses provide good physical insight into the real patterns, even if they result from a strong instability. The first step is to study the stability limits, and the second step is to proceed to a weakly nonlinear analysis of weakly unstable situations, namely those having a small linear growth rate.

This is the strategy that we have adopted to decipher the physical mechanisms of cellular detonations, taking advantage of the Newtonian approximation, (*γ*−1)≪1 for clarification purposes. We shall thus consider the cases of small heat release that characterize the transition regimes, namely those where the threshold of multi-dimensional instability of gaseous detonations appear [5]. Two opposite limits are of particular interest in this regard, namely strongly overdriven regimes and near-CJ regimes. For strongly overdriven regimes, the instability threshold appears for small heat release compared with the thermal enthalpy of the gas at the Neumann state (post-shock gas) [35,36]. In near-CJ regimes, the bifurcation appears for small heat release compared with the thermal enthalpy of the initial state [8]. The critical heat release is much smaller in the latter case than in the former, for which the density jump across the lead shock may be as large as in ordinary detonations. The perturbation analyses in both cases are quite different from that of Short & Stewart [37], who do not include the Newtonian approximation, this approximation being essential to reveal in a clear manner the main physical mechanisms of real detonations within the context of small heat release; see [8,36] for more detailed discussions.

Cellular instability results from coupling two phenomena, namely the lead-shock disturbances propagating in the transverse direction and the oscillatory behaviour of longitudinal disturbances of the reacting gases flowing out from the shock wave. It is thus worth considering first the linear dynamics of wrinkled shocks, there being intricacies encountered there that affect the detonation stability analysis.

### (b) Wrinkled shocks

The linear dynamics of wrinkled shock waves was investigated long ago by D'Yakov [38] and Kontorovich [39] for general materials (see also the textbook of Landau & Lifchitz [40]). The analysis is simplified for strong shocks in polytropic gases. In the linear approximation, the disturbances of the shocked gas flow may be decomposed into acoustic (superscript *a*) waves and an entropy–vorticity (superscript *i*) wave, which is an isobaric and incompressible flow simply convected by the unperturbed flow velocity,
5.1where, according to incompressibility,
5.2The boundary conditions are
where, to leading order in the limit , the Rankine–Hugoniot equations (2.5) and (2.6) yield
5.3In Fourier transform,
5.4acoustic waves read *σ*=i*ω*, with *ω*>0 and
5.5where . In the Newtonian limit
5.6the Rankine–Hugoniot equations (2.5) and (5.3), written in the form , imply that
hence equation (5.5) shows that the components of the acoustic velocity near the Neumann state are smaller than *δu*_{N} and *δw*_{N} at least by a factor . Therefore, to leading order, the velocity components of the entropy–vorticity wave are *δu*_{N} and *δw*_{N}, , . From the leading-order approximation , the continuity equations (5.2) and (5.3) then show that transverse disturbances of the front displacement travel without damping, approximately at the sound speed of the shocked gases,
5.7describing neutral oscillatory modes of the shock wave. Equation (5.7) is valid to leading order in the limit (5.6). The phase velocity of the neutral oscillatory modes is in fact slightly smaller than , as found at higher order.

The linear analysis in the limit (5.6) shows that the weak acoustic modes propagating in the shocked gas and generated by the oscillatory modes of the shock are quasi-parallel to the unperturbed shock and slightly non-radiating,
In other words, there is no spontaneous sound emission at leading order in the limit (5.6). In the absence of acoustic modes satisfying a radiation condition, the reflection coefficient associated with external sound waves impinging on the front in the shocked gas from infinity (incoming sound waves) never diverges. In that sense, then, the shock is stable, in accord with experimental observations. This stability criterion is specific to fronts having only neutral oscillatory modes, Re(*σ*)=0. It is quite possible that, owing to the presence of a square root in the dispersion relation, initial disturbances decrease as power laws in time, as could be checked by using a Laplace transform in time instead of a normal mode analysis.

The property of shock waves shown here in the limit (5.6) is more general. Any shock wave in polytropic gases has neutral non-radiating eigenmodes, contrarily to what is usually thought; see [3,41], where it is reported that disturbances are damped exponentially, Re(*σ*)<0. The stability limits of shock waves for general materials are plotted in figure 1, where the critical values and (*m*−1)/(*m*+1) for the parameter *r* have been obtained by D'Yakov [38] and Kontorovich [39], see also the textbook of Landau & Lifchitz [40]. These critical values for *r* correspond to the critical values 1/(1+*Γ*) and for the parameter , *Γ* being the Gruneisen coefficient. For a polytropic gas, it is readily shown that (*m*−1)/(*m*+1)<*r*<*r**; see the legend of figure 1 for the definition of *r**.

### (c) Overdriven detonations

In the linear analysis, we use the Fourier transformation (5.4) with a prescribed real wavenumber *k*_{y}, *σ* and *k*_{x} being now unknown complex numbers. As a consequence of unsteadiness of its inner structure, the detonation front may become unstable to transverse disturbances, in the sense that the linear growth rate is positive, Re(*σ*)>0. For unstable overdriven detonations, boundedness of acoustic modes at infinity in the inert burned gas, which is labelled throughout by the subscript b, removes one of the two acoustic eigenmodes, leaving one mode having Re(i*k*_{x})<0, so that, according to the d'Alembert equation satisfied by the pressure disturbances,
5.8where the real part of the square root is taken to be positive, a choice that will be made consistently. The incompressibility condition of the entropy wave associated with this selected mode then induces a compatibility condition,
5.9first written explicitly by Buckmaster & Ludford [42]. If the detonation thickness is much smaller than the wavelength, , the quantities , and may be computed at the exit of the reaction zone, from study of the inner structure of the wrinkled detonation wave, by a perturbation method. The compatibility condition (5.9) then leads to a dispersion relation *σ*(*k*_{y}). This method was used by Clavin *et al.* [43] for studying strongly unstable overdriven detonations. It has been adapted by Clavin [44] for performing a weakly nonlinear analysis of detonations close to the instability threshold; see also [45].

#### (i) The perturbation method for the threshold

Scale separation appears if one anticipates that the period of oscillation of the cellular structures is close to that of the longitudinal oscillations observed in the galloping detonations described earlier, , where is the transit time of a fluid particle across the detonation thickness. Since, according to (5.7), , the wavelength *λ*=2*π*/|*k*_{y}| is larger than the detonation thickness by a factor of the order of , . Moreover, according to (5.3), the transverse component *δw* is larger than the longitudinal component, since in the Newtonian approximation (5.6). In this limit, the multi-dimensional instability threshold of strongly overdriven detonations occurs in the regime that is defined by the small parameters
5.10According to the first relation, both (*γ*−1)=*O*(*ϵ*^{2}) and . According to (4.3), the regimes (5.10) correspond to detonations that are stable to longitudinal disturbances. The earlier work of Clavin *et al.* [43] for *q*=*O*(1) concerned detonations that are too strongly unstable to multi-dimensional disturbances for the instability threshold to be exhibited. Following the scalings discussed earlier, it is appropriate to introduce the non-dimensional variables
and the coordinates (4.14), but with the coordinate of the flame position in (4.5) depending now on the transverse coordinate in the variables
5.11

In the weakly nonlinear analysis, the non-dimensional amplitude *A* will be small, *A*=*O*(*ϵ*^{2}), and quadratic terms *A*^{2} must be taken into account, meaning that the perturbation analysis must be pursued to order *ϵ*^{4}, but higher-order terms will be neglected. Up to this order, the shock conditions (2.5) and (2.6) take the form
5.12
5.13
where , , and where we have introduced the notation *A*′_{η}=∂*A*/∂*η* and . Notice that the dominant nonlinear term in the pressure is of order *ϵ*^{6}. Up to this order, the Euler equations (2.2) and the entropy equation (2.3) reduce to
5.14and
5.15with where .

In the limit (5.10), the flow differs weakly from that for the entropy–vorticity wave of a strong shock, presented earlier. The flow Mach number in the reference frame of the lead shock is small everywhere across the detonation thickness. To leading order, the pressure is uniform across the detonation thickness, since, according to (5.14) and (5.15), *P*=*O*(*ϵ*^{4}) for both the steady state and the fluctuation if *A*=*O*(*ϵ*^{2}). Therefore, an extension of the quasi-isobaric approximation (described above for the galloping detonation) may be used. The thermal expansion resulting from heat release may then be clearly separated from the compressible effects of sound waves. This is an enlightening simplification, useful for understanding the inner structure of wrinkled detonations near the stability threshold.

#### (ii) The acoustic waves and the entropy–vorticity wave in the burned gas

The linear analysis in the burned gas follows that for wrinkled shocks. The same decomposition as (5.1) is introduced,
5.16In the limit (5.10), the modifications induced by the heat release are of order *ϵ*^{2}. According to (5.12), the leading order of the entropy–vorticity wave is thus the same as that for the pure shock,
5.17where the subscript 0 denotes the zeroth order in the expansion in powers of *ϵ*. Continuity then leads to (5.7), ∂^{2}*A*/∂*τ*^{2}−∂^{2}*A*/∂*η*^{2}=0, valid to leading order.

The pressure fluctuations at the lead shock are small, of order *ϵ*^{2}*A*, and they are fully absorbed by the sound wave in the burned gases, up to order *ϵ*^{4}. In notation similar to that of (5.4),
5.18where *κ* is a real non-dimensional transverse wavenumber, looking for the solution in the form of a power-series expansion in *ϵ*,
5.19we find from (5.8) that *μ* is of order *ϵ*^{2}, *μ*_{0}=0, meaning that the sound waves propagate in the burned gas in a direction quasi-parallel to the unperturbed front. By selecting the solution that has no divergence in the limit for unstable conditions, Re(*s*_{2})>0, the dominant order of the sound wave is fully determined in terms of *s*_{2}, *κ* and ,
5.20
5.21
where *j* is a real number of order unity, *j*≡−1+(*γ*−1+*q*)/*ϵ*^{2}, and the short notation has been used. The components *δU*^{(i)}_{b} and of the entropy–vorticity wave involve a constant of integration of second order, *ϵ*^{2}*K*_{2}, which is unknown at this stage,
5.22the leading-order term being given by (5.17). The constant *K*_{2} and the unknown function *s*_{2}(*κ*) are determined by studying the inner structure of the wrinkled detonation.

#### (iii) Flow splitting and disturbances of the distribution of heat release

In order to analyse the inner structure, it is convenient to introduce the quantities *U*^{(i)}, *W*^{(i)} and *P*^{(i)}, defined by subtracting the acoustic flow in the burned gas from the total flow field,
5.23Focusing attention on terms of order *ϵ*^{4} with *A*=*O*(*ϵ*^{2}), and anticipating that the pressure terms and *P*^{(i)} contribute only at higher order, we find that equation (5.15) reduces to
5.24In view of (5.12), this nonlinear equation shows that at leading order *U*^{(i)} and *W*^{(i)} are still given by the entropy–vorticity wave in the burned gas (5.17), . Introducing the notation *A*′′_{ηη}=∂^{2}*A*/∂*η*^{2} and , we may write the entropy equation (5.24) in a more convenient form,
5.25where the steady-state relation has been used. The nonlinear equation (5.25) is valid up to order *ϵ*^{4}. Only the linear approximation of the perturbation of the distribution of heat release appears in it. To leading order, this distribution may be affected only by the fluctuations of temperature *T*_{N} and the shear flow (5.17). The rest of the flow introduces higher-order terms, and may then be computed in terms of *A*(*η*,*τ*) without restriction to a specific description of the chemical kinetics by solving a system of equations similar to (4.8) and (4.9),
5.26and
5.27with the boundary conditions (5.12), at *ξ*=0 being and *δY* =0. The solution is the same as in Clavin *et al*. [43] expressing as a functional of *A*(*η*,*τ*) through two steady-state distributions and , with , , which characterize the unperturbed detonation and its thermal sensitivity; see (4.11). From the solution there follows a relation useful when integrating (5.25), namely
5.28with
5.29The first term on the right-hand side of (5.29) is responsible for the galloping detonation, which occurs when the parameter *β*(*γ*−1)*q* is sufficiently large; see (4.16). It involves the thermal sensitivity represented by the parameter *β*. The second term has a similar form, except that it involves the wrinkling of the front and does not depend on the sensitivity of the heat-release rate to temperature. It is responsible for the cellular instability, which here is seen to arise even in the absence of thermal sensitivity and is associated instead with the interaction of the shear flow and the density (temperature) profile.

To the same degree of approximation as in (5.24), the substantial derivative in the Euler equations (5.14) reduces to 5.30These Euler equations may then be written as 5.31and 5.32where we have used the notation . The nonlinear terms come from the Reynolds tensor associated with the shear flow of the entropy–vorticity wave. They appear as source terms in (5.31) and (5.32).

#### (iv) Linear instability threshold

In the linear analysis, since the pressure term ∂^{2}*P*^{(i)}/∂*η*^{2} in (5.32) is found from (5.31) to introduce terms of order *ϵ*^{4}*A*, higher than *ϵ*^{4} for *A*=*O*(*ϵ*^{2}), the quantity *W*^{(i)} is simply convected from the lead shock by the unperturbed flow. Using the boundary condition (5.12) and the one coming from the acoustic solution (5.21), , we obtain
5.33An integration of (5.25) from *ξ*=0 with the same boundary conditions (5.12) and (5.21), , yields an expression for . This expression contains two types of terms, namely those varying with *ξ* as and those not depending on *ξ*. Matching the terms proportional to with (5.22) determines the constant *K*_{2} in (5.22). Moreover, matching implies that the sum of the constant terms should be zero. This leads to an expression for i*μ*_{2} in terms of *s* and *κ*,
5.34and then to an equation for *s*_{2}, after using (5.20),
5.35where *Z* is related to the Fourier transform of (5.29),
5.36Notice that *μ*_{2} in (5.34) and the square root in (5.35) both come from the acoustic wave. The solution to (5.35) is
5.37where the sign ± has to be chosen so that , since, according to the definition of the square root in (5.35), ,

From now on, using the notation *κ*≡|*κ*| to simplify the presentation, we observe that the linear growth rate may then be written as
5.38where and *B* are two parameters of order unity,
see (5.10). The first term on the right-hand side of (5.38) describes a multi-dimensional instability that is due to the coupling of the entropy–vorticity wave and the heat release, while the second arises from the damping mechanism produced by the acoustic waves. Contrary to the common belief, therefore, the initial instability that results in cellular detonations for strongly overdriven detonations does not result from a thermo-acoustic instability, since the acoustic effect is damping. This, in fact, is a general conclusion, independent of the Newtonian limit, but its obscuration by algebraic complexities that arise in the absence of this limit can lead to false hypotheses based on general recollections of Rayleigh's criterion. The Newtonian limit serves to remove effects of compressibility from the entropy–vorticity wave and thereby provides a clearer physical insight into the mechanism by showing that the acoustic waves do not produce the instability.

For a sufficiently smooth spatial distribution , both and [36]. Then, the existence of the instability threshold, separating planar detonations that are linearly stable at all wavelengths from detonations that are unstable to transverse disturbances in a small domain of wavelengths around a finite critical wavelength, is readily demonstrated: for small , the damping term dominates irrespective of the wavelength,
5.39In the opposite limit, the damping term is negligible, except at small wavelengths, and the planar detonation is unstable to transverse disturbances with *κ* of the order of unity. Therefore, there must exist a threshold value of separating stable and unstable conditions. Thanks to the sound wave, the detonation is stable to disturbances with a small wavelength, *κ*≫1, since the second term on the right-hand side of (5.38) is proportional to −*κ* in the limit .

The quantity *μ*_{2} appearing in the acoustic field is obtained from (5.20), (5.34) and (5.37),
5.40where the sign ± is chosen as in (5.37) so that Re(i*μ*_{2})<0, as required by the boundedness condition downstream. For Re(*s*)>0, this corresponds to a radiation condition; see [36].

The results of the linear analysis may be written in the form of a linear integro-differential equation for the perturbed front *A*(*η*,*τ*). The resulting operator is obtained from equation (5.34) multiplied by *s*,
5.41where *C*^{2}≡1+3(*γ*−1)/2≈1. The operator is defined in (5.29); it describes the destabilizing effect of the entropy–vorticity wave. The operator is defined in Fourier space as with
5.42Since , a simplified expression arises in the unstable domain when the bifurcation parameter is sufficiently large, , . This operator represents the stabilizing effect of the radiating sound wave in the burned gas. The first two terms on the right-hand side of (5.41) constitute the wave equation for the lead shock, corresponding to *s*_{0}=i*κ* to leading order, while the following order leads to 2*ϵ*^{2}*s*_{0}*s*_{2}, as can be demonstrated by introducing two time scales *A*(*τ*,*ϵ*^{2}*τ*,*η*) into the wave equation.

#### (v) Weakly nonlinear analysis, Mach stem formation and diamond patterns

The objective of a weakly nonlinear analysis is to determine the nonlinear operator that is added to in order to describe the nonlinear patterns,
5.43The methodology follows that of the linear analysis but requires computation of the pressure *P*. An expression for *U*^{(i)} and then for (∂/∂*τ*+∂/∂*η*)*U*^{(i)} is obtained by integrating equation (5.25) from *ξ*=0 to *ξ* with the nonlinear boundary condition (5.12) and the value of the acoustic solution (5.21) at *ξ*=0 being used. The integral that appears in this expression may be eliminated by using the Euler equation (5.32). A Poisson equation is then obtained for by using (5.31),
5.44where is a nonlinear functional of *A*(*τ*,*η*) and *A*(*τ*−*ξ*,*η*), which need not be written explicitly here. The solution to (5.44) for is obtained by using the boundary condition at *ξ*=0, namely *I*=0, and a boundedness condition for .

The analysis is simplified when attention is focused on a simple wave travelling on the shock front, ∂*A*/∂*τ*±∂*A*/∂*η*≈0 to leading order. In a Fourier representation, the dominant order of the pressure term, *P*^{(i)}(*ξ*)=∂*I*/∂*ξ*, then takes the form
5.45leading to a nonlinear correction to of order *ϵ*^{5}. By noticing that, according to (5.13), the boundary condition for *P*^{(i)} is of a higher order, *P*^{(i)}(*ξ*=0)=−*ϵ*^{2}(∂*A*/∂*η*)^{2}=*O*(*ϵ*^{6}), it may be reasoned that *E*=0, meaning, according to (5.43), that
5.46

Formation of cusps representative of Mach stems is easily obtained from (5.41), (5.43) and (5.46). When the integral term (which drives the cellular instability) is disregarded, the slow time evolution yields
5.47This equation shows the same nonlinear term (wave breaking) as does the Burgers equation, but with a different damping term at small wavelength, which behaves like for smoothing out the singularity, instead of of the Burgers equation. The damping is due to the radiating sound wave in the burned gas. On the short time scale, the cusps, according to (5.41), travel in the transverse direction at nearly the sound speed in the burned gas. The pattern is fed by the quasi-isobaric instability resulting from the coupling of the heat release and the entropy–vorticity wave generated at the wrinkled shock. Numerical solutions to (5.43) with periodic conditions in *η*, such as those presented by Clavin & Denet [45], show oscillatory diamond patterns that look quite similar to those observed in experiments on cellular detonations. A nonlinear selection of the cell size in favour of large cells also is observed after a relaxation time about 100 times longer than the period of oscillation, as also observed in DNS of cellular detonations [2].

The solutions to (5.43) with pulsating cells and crossover of cusps are not simple waves. A more general treatment shows the existence of a mean streaming flow with a mean shape of the cellular front, resulting from a non-zero average of the quadratic terms [44]. Mean streaming flows have been known for a long time in fluid mechanics in the presence of forced oscillations, such as those enforced by sound waves impinging an obstacle. A main difference for detonations is that the oscillation here results from an oscillatory instability of the front.

### (d) Near-Chapman–Jouguet regimes

According to (5.10), real CJ detonations are necessarily strongly unstable. The reason is that, for ordinary conditions, is a small number, so that the value of needed to reach a sonic condition at the end of the exothermic reaction cannot be small. The multi-dimensional threshold in fact appears already for the extremely unrealistic situations in which the heat release is smaller than the thermal enthalpy of the fresh mixture 5.48According to the Rankine–Hugoniot relations, such CJ detonations near the instability threshold would then propagate at a velocity slightly larger than the sound speed, , . The lead shock would then be weak, and the flow relative to the lead shock would be transonic everywhere. Moreover, in order to preserve a temperature of the burned gas larger than the Neumann temperature, , one must have , so that the increase in temperature would be very small, . Such regimes cannot be observed in experiments since the crossover temperature for the exothermic reaction is about 1000 K for usual combustible mixtures. These regimes are, however, worth investigating with an artificial temperature cutoff, for improving understanding, since the bifurcation occurs there. They correspond to a situation that is just the opposite of that considered earlier in that the vorticity wave plays a negligible role, and the dynamics is controlled by the interaction between acoustic waves and the heat-release rate, the longest delay time being the propagation time of the upstream-running acoustic wave, rather than the transit time of the entropy wave.

The three-dimensional instability threshold of detonations near CJ regimes was investigated analytically by Clavin & Williams [8] in the limit , and *f*=*O*(1), where
5.49is a scaled overdrive factor that decreases to unity at CJ conditions. The value of *f* changes by an amount of order unity over a range of conditions where the more conventional overdrive factor, the square of the ratio of the detonation propagation velocity to its CJ value, remains near unity. As for the planar detonations studied under the same conditions by Clavin & Williams [28], the transonic condition implies that the evolution is controlled by a slow time scale associated with the upstream-running acoustic wave, which propagates in a direction quasi-perpendicular to the detonation front, at a velocity slightly higher than the velocity of the unperturbed gas flowing downstream. An analytical solution for the stability analysis is obtained when the pressure-induced modification to the heat-release rate is neglected, keeping only the sensitivity to the Neumann temperature, *β*. The result shows the existence of a Hopf bifurcation point at a finite wavelength of order . The relevant parameter of bifurcation of order unity was found to be the ratio of the thermal sensitivity *β* to . The instability is promoted by increasing the thermal sensitivity or by approaching the CJ condition (*f*=1), consistent with strong detonations being less unstable than CJ waves.

## 6. Concluding remarks and perspectives

It is noteworthy that, by considering limiting cases, analytical studies have uncovered the mechanisms of many of the phenomena observed in gaseous detonations. A notable exception is the process of transition from deflagration to detonation, which is still not fully understood. Aspects of analytical investigations of this phenomenon are addressed in the present issue by Brailovsky *et al.* [46]. What has been understood in simple physical terms by these methods, however, are both direct and spontaneous initiation or quenching in one-dimensional geometries. Our comprehension of these time-evolving phenomena thus is well developed, as is knowledge of the important role of the sensitivity of heat-release rates to temperature in galloping detonations.

It is also true that understanding has advanced for the cellular structures of overdriven detonations, thanks to weakly nonlinear analyses performed in a systematic way in the neighbourhood of the instability threshold in the Newtonian approximation. The main destabilizing mechanism results from the coupling of the finite-rate heat release with the oscillatory shear flow of the entropy–vorticity wave that is produced at the lead shock by the front wrinkling. Through this mechanism, detonations that are stable against longitudinal disturbances become unstable to transverse disturbances having a wavelength sufficiently small but still larger than the detonation thickness. The order of magnitude of the unstable wavelengths is controlled by the oscillatory frequency of the inner structure of the detonation (the inverse of the transit time) and by the phase velocity (the sound speed of the compressed gas) of the travelling waves that begin propagating on the lead shock as a result of the shear flow of the vorticity–entropy waves. The dominant compressible effects arise in acoustic waves, which are stabilizing at short wavelengths, so that the multi-dimensional oscillatory instability of gaseous detonations appears in a bounded range of wavelengths larger than the detonation thickness. Contrary to the pulsations of galloping detonations, this multi-dimensional instability occurs even when the heat-release rate is not sensitive to temperature, the bifurcation parameter being simply , defined below (5.38), for a given reduced reaction-rate distribution . This is in agreement with the results of the original numerical analyses of Erpenbeck [5,47] who first realized that detonations become unstable for heat release larger than a critical value, which is small, even if the rate of the chemical reaction is not sensitive to temperature.

According to the weakly nonlinear analysis, the dominant nonlinear term arises from the Reynolds tensor of the shocked gas flow. The resulting nonlinear term (5.46) in the front evolution equation (5.43) exhibits the same form that is found in flames for small heat release, studied by Sivashinsky [48], see (5.47), even though it is of a quite different nature. It leads to the formation of cusps representing Mach stems travelling at the sound velocity of the burned gases and producing the characteristic ‘diamond’ or ‘fish-scale’ patterns first observed in experiments more than 50 years ago. A nonlinear selection mechanism leads to cell sizes that are close to the wavelength of the largest linearly unstable mode (the marginal mode on the small-wavenumber side). This contributes to reaching a final cell size that is much larger than the detonation thickness.

In the Newtonian approximation in (5.10), the parameter is small at the instability threshold, that is, the enthalpy change associated with the heat release is small compared with the thermal enthalpy at the Neumann state. Nevertheless, the quantities (where is the gas constant) and , the ratio of the temperature rise across the lead shock to the initial temperature, are both of order unity in the analysis. For this reason, the main results of the weakly nonlinear asymptotic analysis in the limit (5.10) may be expected to be at least qualitatively relevant to ordinary detonations for which the Neumann-state Mach number is sufficiently small, provided there is no secondary bifurcation. The accuracy with which the present analysis may produce quantitative agreement with experimental data for the cell sizes of overdriven detonations, however, remains to be explored.

An outstanding problem is to determine analytically the cell size of ordinary CJ detonations propagating in a wide tube or expanding freely in three dimensions. Not only is the set of linearly unstable modes usually wide, but also it is not bounded on the large-wavelength side if the detonation is unstable against planar disturbances, as it may be under actual conditions. In this latter case, the (positive) linear growth rate of planar disturbances (*k*=0), however, is smaller than that of the most unstable disturbance with non-zero wavelength. In that case, curvature effects of the detonation front could play an important role in the cell size selection. In the former case, on the other hand, according to the nonlinear selection mentioned earlier, the upper bound of the unstable wavelengths would be a good candidate for the cell size. In either case, a promising analytical approach to such investigations of CJ detonations would be to consider using a method similar to that which proved useful in §4*c* for describing spontaneous quenching. The basic approximation would be to treat the exothermic zone of the wrinkled detonation in a quasi-steady approximation, retaining the unsteady effects only in the quasi-isobaric induction zone. This rough approximation is justified by the fact that, except in a narrow transition region, the flow velocity relative to the lead shock is higher and the transit time smaller in the high-temperature exothermic region than in the cooler induction zone. The consequent enhanced analytical tractability may lead to useful simplified cell dimension estimates having reasonable accuracy for CJ detonations.

## Acknowledgements

We thank Prof. Amable Liñán and Prof. Antonio Sánchez for stimulating discussions.

## Footnotes

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

- This journal is © 2012 The Royal Society