## Abstract

The origins of autoignition at hot spots are analysed and the pressure pulses that arise from them are related to knock in gasoline engines and to developing detonations in ducts. In controlled autoignition engines, autoignition is benign with little knock. There are several modes of autoignition and the existence of an operational peninsula, within which detonations can develop at a hot spot, helps to explain the performance of various engines. Earlier studies by Urtiew and Oppenheim of the development of autoignitions and detonations ahead of a deflagration in ducts are interpreted further, using a simple one-dimensional theory of the generation of shock waves ahead of a turbulent flame. The theory is able to indicate entry into the domain of autoignition in an ‘explosion in the explosion’. Importantly, it shows the influence of the turbulent burning velocity, and particularly its maximum attainable value, upon autoignition. This value is governed by localized flame extinctions for both turbulent and laminar flames. The theory cannot show any details of the transition to a detonation, but regimes of eventually stable or unstable detonations can be identified on the operational peninsula. Both regimes exhibit transverse waves, triple points and a cellular structure. In the case of unstable detonations, transverse waves are essential to the continuing propagation. For hazard assessment, more needs to be known about the survival, or otherwise, of detonations that emerge from a duct into the same mixture at atmospheric pressure.

## 1. Autoignition in engines and ducts

In this paper, autoignition is considered in two contexts. In the first, the necessary high pressures and temperatures are achieved through compression in an engine; in the second, they are achieved through compression in the shock wave generated ahead of a flame with a high flame speed. ‘Knock’ in internal combustion engines has been defined as ‘objectionable noise outside the engine’ [1]. Its origin can be traced to sudden pressure rises during the combustion processes. One of the earliest investigators of knock was Bertram Hopkinson at the University of Cambridge, UK, who, between 1904 and 1907, was assisted by a young student, Harry Ricardo [2]. Sufficiently severe knock could lead to extensive engine damage and Hopkinson attributed this to a detonation. Subsequently, Ricardo tested many fuels for the Shell group and, in 1919, developed the variable compression ratio, E35 research engine. This enabled the highest useful compression ratio to be measured, indicating the threshold of knock.

In the early 1930s in the USA, with the aim of rating the fuel rather than the engine, the Cooperative Fuel Research Committee standardized knock testing by using a Waukesha variable compression ratio engine. Standardization was achieved on the basis of the octane scale for gasoline fuels that had been proposed in 1927 by Edgar [3]. Despite the inherent difficulty in obtaining an engine-independent fuel rating, this mode of knock testing has remained essentially unchanged to the present day. The octane number (ON) is the percentage by volume of *iso*-octane with *n*-heptane in a primary reference fuel (PRF) mixture of these, with the same knock characteristics as the commercial fuel. There are two standardized measurement techniques: one with an inlet mixture temperature of 325 K measures the research octane number (RON); the other with an inlet temperature of 422 K measures the motor octane number (MON). There are a number of deficiencies in this approach: real gasolines are a complex mixture of about a hundred species, of which usually less than 6 per cent are *iso*-octane and *n*-heptane; some fuels may be more knock resistant than *iso*-octane; operational temperatures and pressures are currently rather higher than those in the RON and MON tests; autoignition delay times of PRFs have a greater pressure dependency than gasolines [4]; and the compression pressure depends partly upon the burning velocity of the mixture.

The autoignition delay time, *τ*_{i}(*T*,*P*), which depends upon the temperature and pressure of the mixture, is a more basic parameter for characterizing autoignition and knock than the RON or MON. It has been found that the attainment of a value of unity by the Livengood–Wu [5] integral, *I*, given below, is a good indication of when autoignition will occur in an engine,
1.1The integral is evaluated during mixture compression and any expansion in the engine during the time *t*^{′}, allowing for the changing value of *τ*_{i}(*T*,*P*). Values of *τ*_{i}(*T*,*P*) for various mixtures have been found from measurements in rapid compression machines, shock tubes and engines and from detailed chemical kinetic models. Such values, for PRFs of 0, 60, 80, 90 and 100 ON, obtained from the shock tube measurements reported in Fieweger *et al.* [6], are plotted against reciprocal temperatures at 4 MPa in figure 1. The non-Arrhenius nature of the variations is apparent, as is the similarity of the low values of *τ*_{i} for all ONs, at high temperatures.

In a spark-ignition engine, autoignitions occur in the end gas, between the propagating turbulent flame and the cylinder liner. The unburned mixture is inhomogeneous with gradients of both temperature and composition and with the result that autoignition occurs first at isolated, comparatively small, hot spots. This mode of autoignition is not confined to engines. It occurs more generally and can be a hazardous precursor to a detonation. Observation of the autoignition and propagation of an autoignitive flamefront at hot spots requires good spatial and temporal resolution. Schlieren images of a turbulent flame in an engine and the onset of autoignition in the end gas ahead of it, at several hot spots [7], is shown in figure 2.

The present paper examines, in a generalized way, the nature of autoignition at a hot spot and how this depends upon the characteristics of the fuel and the gradient of reactivity in the mixture. Important aspects are: how the rate of autoignitive combustion influences the amplitude of the pressure wave it creates; whether this is further increased in a transition to a detonation; and whether such a detonation can propagate beyond the hot spot. With this background, autoignition and the conditions for transition to detonation will be discussed in the two contexts previously mentioned. The first covers knock in different types of engine, the second the transition from a high-speed flame to a detonation in a duct that is closed at one end.

## 2. Autoignition velocities

Flammable mixtures are seldom completely homogeneous, and Zeldovich and co-workers [8] have shown how gradients of reactivity create propagating autoignitive reaction fronts initiated at hot spots. Such gradients of reactivity on the threshold of autoignition are characterized by gradients of *τ*_{i}. At a point, the localized autoignitive propagation velocity, relative to the unburned mixture, *u*_{a}, is given by
2.1where *r* is the distance normal to the gradient. If the composition were to be homogeneous and the heterogeneity were to arise entirely from temperature gradients, then
2.2As will become apparent, it is convenient to employ a dimensionless parameter, *ξ*, embodying the acoustic speed in the unburned mixture, *a*, in
2.3The last term on the right-hand side was evaluated at 4 MPa from the profiles of *τ*_{i} in figure 1 for the different values of ON. The gradient, d*T*/d*r*, was assumed to be −1 K mm^{−1}, and the resulting values of *ξ*, with values of *a* from the gas equation code [9], are plotted against *T* in figure 3. Any change in (d*T*/d*r*) would result in a proportional change in *ξ*. With values of *a* in the region of 550 m s^{−1}, it is clear that very high autoignitive velocities are generated at the lower values of ON of 0, 60 and 80. At a hot spot, the value of (d*T*/d*r*) is negative and so usually are values of (d*τ*_{i}/d*T*). However, at the lower values of ON, there can be some ranges of *T* within which (d*τ*_{i}/d*T*) is positive. In such regions of negative temperature coefficient, autoignition would first occur at a ‘cold spot’.

Where the local global activation energy, *E*, for autoignition is small, the values of (d*τ*_{i}/d*T*), given by
2.4can approach zero, with *u*_{a} apparently tending to infinity. This implies that a thermal explosion would occur almost instantaneously throughout the mixture. In practice, if (d*τ*_{i}/d*T*) is close to zero, both (d*τ*_{i}/d*T*) and (d*T*/d*r*) at a given location will fluctuate about zero, giving a mean value of *ξ* also close to zero. At autoignition, this would generate a near-homogeneous thermal explosion. In this connection, the studies of Meyer & Oppenheim [10] of autoignition of stoichiometric H_{2}–O_{2} mixtures induced by a planar reflected shock in a rectangular shock tube are of interest. They observed near-instantaneous autoignitions across the whole cross section of the tube, at a low value of (d*τ*_{i}/d*T*), after reaction had been initiated in the gas layer adjacent to the wall. This near-planar reaction shock front then moved towards the reflected shock and coalesced with it, creating a fine detonation-like multi-wave structure. This phenomenon was termed a ‘strong ignition’. In contrast, a ‘mild ignition’ occurred at higher values of *τ*_{i} and (d*τ*_{i}/d*T*). Here, autoignition at the wall was followed, not by homogeneous initiation of reaction across the whole cross section, but by initiations at distinct centres, in what has been termed the ‘multi-spot’ mechanism. The strong ignition boundary for stoichiometric H_{2}–O_{2} mixtures was defined as [10]
2.5With this value, *a*=500 m s^{−1} and d*T*/d*r*=−1 K mm^{−1}, then, from equation (2.3), *ξ*=1. The mild regime is identified with higher numerical values of (d*τ*_{i}/d*T*)_{P} and of *ξ*. The effects of diluents on this boundary have been studied by Oran & Boris [11].

## 3. Pressure pulses, knock and developing detonation

The relationship between values of *ξ* and engine knock is now briefly reviewed. The mini-explosion at a small hot spot drives a compression wave that moves into the surrounding mixture at the local acoustic speed. This overpressure originates from the rapid rate of change of the rate of volume generation, (d*V*/d*t*), arising from hot spot reactions. At a sufficient distance, *d*, from the autoigniting hot spot, the pressure pulse approximates that from a monopole sound source. Simple acoustic theory gives the associated instantaneous sound pressure, *P*(*t*), above the ambient at time *t* as [12]
3.1where *t*_{a} is the time for the sound wave to propagate the distance, *d*, from the source to its measurement point, through non-reacting gas of density *ρ*.

If *r* is the reaction front radius and 4*πr*^{2} is the area of the reaction front propagating relative to the unburned gas at a velocity *u*_{a}, then the volumetric rate of consumption of unburned gas volume is 4*πr*^{2}*u*_{a}. The corresponding volumetric rate of production of burned gas is greater in the ratio of unburned to burned densities, *σ*, hence
3.2The overpressure above the ambient generated by the propagation of the reaction front through the hot spot is found by differentiating equation (3.2) with respect to *t* and substituting it in equation (3.1). Ultimately, this leads to a dimensionless expression in terms of *ξ* for the overpressure [13],
3.3Here, the acoustic speed , *γ* is the ratio of specific heats and , where *t*_{ro}=*r*_{o}/*a* is the time for an acoustic wave to traverse the original hot spot radius of *r*_{o}.

Equation (3.3) is valid only for the relatively mild knock that occurs in the absence of shock waves and detonations. For these conditions, the final term in can be relatively small. The maximum pressure will occur at the maximum radius of the reacting hot spot. The multiplying term for *ξ*^{−2} in the dominant first term is approximately 3.5 for an engine bore diameter of 86 mm. Consequently, in engines, is approximately of the order of *ξ*^{−2}. The smaller the value of *ξ*, the greater the strength of the pressure pulse.

Values of *ξ*^{−2} at 4 MPa with d*T*/d*r*=−1 K mm^{−1}, derived from figure 3 for the five PRFs, are shown plotted against *T* in figure 4. Bearing in mind the approximation
3.4it can be seen that, for the ONs of 0, 60 and, to some extent, 80, values of *P*(*t*)_{max}/*P* would be unrealistically high. Temperature ranges for ‘strong ignition’ defined by equation (2.5) are indicated for the different ONs in the figure caption. Values of can only be related to *ξ*^{−2} for mild knock with *ξ*≫1. Although values of *ξ*≤1 cannot be realistic, they can be indicative of high over-pressures. Not surprisingly, values of *ξ* increase as ONs increase. For all fuels, there is evidence in figure 4 of low- and high-temperature knock regimes. For ON=100, only the latter is significant. Some high values of *ξ*^{−2} might be indicative of a thermal explosion, as discussed above. Others are indicative of the transition of the acoustic wave to a developing detonation, for which equations (3.1) and (3.3) are invalid.

A detonation can begin to develop when the value of *u*_{a} approaches that of *a* (*ξ*≈1), provided sufficient energy is fed into the wave to generate the characteristic high-pressure ratio of a detonation front [8]. When sufficient chemical energy is fed into the acoustic wave, and increases its amplitude significantly, a new entity is created, in which the two are coupled into a developing detonation wave.

To trigger the detonation, the heat released by autoignition must be unloaded sufficiently rapidly into the acoustic wave as it moves through the hot spot during the time, *t*_{ro}. The duration of the heat-release rate is measured by the excitation time, *τ*_{e}, with a different global activation energy from that associated with *τ*_{i} in equation (2.4) [14]. In the present computations, this time extended from 5 per cent to the maximum of the heat-release rate. In the range of interest, *τ*_{i} is usually measured in milliseconds, and *τ*_{e} in microseconds. The rapidity of the unloading into the acoustic wave is quantified approximately by the number of excitation times that can be contained within *r*_{o}/*a*, and is given by Gu *et al.* [15] as
3.5When equations (3.1) and (3.4) are inapplicable, only direct numerical simulations can reveal the details of any transition to a developing detonation and give values of . Such simulations for equimolar mixtures of CO and H_{2} with air, at different temperatures, pressures and equivalence ratios, *ϕ*, have been reported previously [15,16]. These fuels were chosen because their detailed chemical kinetics were reasonably well established. Values of *τ*_{i} and *τ*_{e} were computed for a range of different conditions. As suggested above, development of a detonation depended not only on the value of *ξ* but also on that of *ε*. A variety of hot spot radii and temperature elevations at their centres were studied, and it was assumed that the initial temperature declined linearly with radius, up to *r*_{o}. From these simulations over a variety of initial boundary conditions, and in the absence of reflected shock waves, it was possible to plot values of *ξ* against those of *ε* at the onset of autoignition.

Figure 5 shows the upper and lower limits of a peninsula, indicated by *ξ*_{u} and *ξ*_{l}, within which detonations were developing within the hot spot. Outside the peninsula, regimes of supersonic and subsonic autoignitive wave propagations are indicated, respectively, by P and B. An instantaneous thermal explosion is represented by values of *ξ* close to zero outside the peninsula. In this regime, the reaction wave is faster than the acoustic wave, there is no coupling between them and the maximum pressure ratio is less than in a detonation. Within the peninsula, the strong ignition regime seems to extend into the toe of the peninsula from *ε*=2 to *ε*=8, with values of *ξ* lying between 2 and 7. It will be shown in §7 that, for detonations in ducts, low values of the product (*ξε*) are associated with detonation stability, while high values are associated with instability.

## 4. Engine operational regimes

On the assumption that such generalized regimes are applicable to other fuels, engine operational regimes for a variety of conditions can be plotted as shown in figure 5. The labelled bold curve shows the locus of end-gas conditions at the onset of autoignition, with increasing compression in gasoline engines after turbo-charging to an inlet pressure of 0.2 MPa, as reported by Kalghatgi & Bradley [17]. The end-gas pressures and temperature were increased as a result of unwanted, and ever earlier, pre-ignitions in the engine cycles. For the same intake conditions, this resulted in autoignitions and knock being initiated at progressively higher values of *P* and *T*, designated by *P*_{a} and *T*_{a}, with associated decreases in *τ*_{i}, *τ*_{e} and *ξ*, and increases in *ε*. All these values, along with the measured initial values of at the onset of knock and the acoustic speed, are given for five different conditions in table 1. The symbol N represents the maximum pressure conditions in a normal non-knocking cycle, and K2 represents the conditions at the onset of knock in a cycle with fairly heavy knock. The onset of ‘super-knock’ is indicated by S, the average of the two high-pressure conditions, S1 and S2, while E is an extrapolation to higher values of *P* and *T*. Values of *T* for the unburned mixture were obtained from the measured values of *P* and a polytropic compression law in which *T*^{−1}*P*^{(1.35−1)/1.35} is constant.

Generated pressure waves are reflected at the confining walls and further autoignitions are initiated. Consequently, pressure oscillations arise with as the initial amplitude. To plot these points at the initiation of knock in figure 5, in the absence of direct measurements, *r*_{o} was assumed to be 5 mm and d*T*/d*r* to be −2 K mm^{−1}. The values of *τ*_{i} and *τ*_{e} were those computed by Kalghatgi & Bradley [17] at *P*_{a} and *T*_{a} using detailed chemistry for the gasolines employed in the engine tests. The computations were for an appropriate surrogate fuel comprising 62 per cent *i*-octane, 29 per cent toluene and 9 per cent *n*-heptane [18]. Values of *τ*_{i}, *τ*_{e}, *ξ* and *ε* are given in table 1. The locus of points on figure 5 shows that any knock at N2 would be almost undetectable, with given by *ξ*^{−2}. The fairly heavy knock at K2 would probably be just within the developing detonation peninsula, while the ‘super-knock’ at S would be well within it, with lower values of *ξ* and higher values of *ε*. Clearly, knocking in present turbo-charged engines can become severe at low values of *ξ* and high values of *ε*. It is aggravated by unwanted pre-ignitions at hot surfaces and also by earlier autoignitions in the gas phase. The latter might possibly be associated with long-chain lubrication oils or additives, rather than solely the fuel [17]. Further increases in pressure, as for the conditions at E, would further increase *ε* and the intensity of the knock. Reference to table 1 also shows that the values of (d*τ*_{i}/d*T*) for the S and E conditions are close to the ‘strong ignition boundary’, given by equation (2.5).

In the numerical simulations of Bradley *et al.* [16] the pressure ratio, , at the detonation front was found to increase with *ε*, as , within the detonation peninsula. In general, the absence of a reactivity gradient and the mass divergence prevented a developing detonation from propagating outside the hot spot. Only when *ε* reached 85 at a low value of *ξ* did the detonation continue to propagate into this region of zero reactivity gradient in the study of Bradley *et al.* [19]. This only occurred at significantly higher temperatures than the present values and with larger hot spots. It would be very damaging if the propagation of the detonation from a hot spot were to continue into the main mixture. However, almost equally damaging would be the strongly developing detonations within a multitude of hot spots. The experiments by Pan & Sheppard [20] show that, as autoignition fronts from adjacent hot spots propagate towards each other, pressure fronts ahead of them increase the temperature and pressure of the reactants. This promotes further autoignitions, violent reaction and the generation of shock waves. The further coupling of reaction fronts and shock waves could create a self-sustaining detonation in the regime of low *ξ* and high *ε*.

A very different regime of benign autoignition is in evidence in controlled autoignition engines, in which combustion occurs, often ‘spottily’, in benign autoignitive reaction fronts. Engine knock is avoided by using lean mixtures, with larger values of *ξ* and, importantly, smaller values of *ε*. The ‘spotty’ nature of this type of autoignitive burning is shown by the laser diagnostic flame images in the study of Plessing *et al.* [21]. Data on this mode of autoignition, provided by a single-cylinder engine running at 1200 r.p.m. with a PRF of 84 ON at *ϕ*=0.25, are given in table 2 for three different conditions in the study of Bradley & Head [4]. The air-inlet pressure could be boosted up to 0.2 MPa and autoignition was induced for different combinations of either high *P*, low *T* or vice versa. Data cover the conditions after the original autoignition, at the instant when 10 per cent of the heat release had occurred, at pressure *P*_{10} and temperature *T*_{10}. The corresponding crank angle after top centre is CA_{10}. Values of *τ*_{i} and *τ*_{e} for this weaker mixture were not available directly and had to be estimated. Following the study of Bradley *et al.* [22], the values of *τ*_{i} at *ϕ*=0.25, which were necessary to find values of *ξ*, as in figure 3, were obtained from the stoichiometric values in Fieweger *et al.* [6] by multiplying them by 0.8/*ϕ*, with *τ*_{i} varying as *P*^{−1.7} [4]. No values of *τ*_{e} were available, so those of Kalghatgi *et al.* [18] were employed, with the same allowance being applied for different *ϕ* and *P* as for *τ*_{i}. The new values of *ξ* and *ε* for this lean mixture are given in table 2, again for *r*_{o}=5 mm and d*T*/d*r*=−2 K mm^{−1}.

These three autoignitive, yet knock-free, operational points are plotted as shown in figure 5, labelled with the pressure normalized by that of the atmosphere. All three points are outside the peninsula of developing detonation, with the intermediate pressure of 48.4 atm burning close to it. There were no indications of even moderate knock from the pressure record. This might suggest that the point is outside the toe of the peninsula. In this study, no attempt was made to optimize the engine thermal efficiency. Despite the necessity of having to assume both a value for the hot spot radius and the temperature gradient within it, figure 5 gives at least a qualitative picture of different regimes, ranging from benign autoignitive burning to severe knock.

## 5. Flame propagation along a duct and the transition to detonation

The configuration for the studies of autoignition in a duct is shown diagrammatically in figure 6. In this system, autoignition and propagation along reactivity gradients are induced, as in engines, by compression of the reactants, but this time entirely as a result of a sufficiently high burning velocity. If a flame is initiated at the closed end of the long duct, it accelerates by a turbulent feedback mechanism, creating a shock wave ahead of it. This, and the confining walls, can compress the mixture between the flame and the shock plane to the point where it autoignites. When this occurs and gives rise to a detonation, it is termed a deflagration-to-detonation transition (DDT).

Shear at the duct wall induces turbulence, which by a feedback mechanism increases the turbulent burning velocity, *u*_{t}. This increases the gas flow ahead of it, and hence further increases the turbulence. The turbulent boundary layer extends into the duct, but the turbulent kinetic energy and potential burning velocity are less towards the centre [23]. Lindstedt & Michels [24], with stoichiometric fuel–air mixtures, increased the surface roughness and wall turbulence through the insertion of Shchelkin spirals, to enhance the feedback mechanism. This enabled them to attain more rapid development of the turbulent flame and better penetration of the turbulence into the centre of the duct and to obtain steady, quasi-stable deflagrations, with different flame speeds for different fuels. It was possible to tailor high-speed deflagrations that accelerated to detonations.

The seminal studies of Urtiew & Oppenheim [25] traced the progress of the flame and planar shock fronts along a duct of rectangular cross section (25.4×38.1 mm) closed at one end. These provide further quantitative insights and we shall concentrate in some detail on their equimolar mixtures of H_{2} and O_{2}, which were less reactive than the stoichiometric mixtures of Meyer & Oppenheim [10]. Initially, the mixture was at room temperature and at 10.74 kPa. This was sufficiently reactive to generate a shock wave strong enough to cause a DDT, solely owing to boundary layer turbulence, without the need for additional turbulence generators. Figure 7 shows three very high-speed stroboscopic schlieren images, at 5 μs time intervals. High shear rates can be generated in viscous sub-layers [23], which might initiate autoignition of the reactants [26,27], in this case located between the two fronts. In their studies of the autoignitive ‘explosion in the explosion’, Urtiew & Oppenheim [25] found that it was usually initiated at the boundary layer of the duct wall and was variously aided by weak reflected transverse waves originating from initiating hot spots, the turbulent flame, the main shock front and coalescing shock waves. After such explosions, reaction fronts swept through the unburned mixture, penetrating the leading shock and generating a self-sustained detonation there, often creating triple-wave interactions.

Figure 8 shows three rather more detailed later images of the internal explosion, from the same sequence as figure 7. As the flame propagated along the duct, one of the three pressure gauges traced the development of the planar shock wave ahead of the flame. From these, the planar shock pressure just prior to autoignition was estimated by the present author to be 151.9 kPa. The images trace a series of events that culminated in a detonation. They began with the formation of a secondary transverse shock wave at the wall surface close to the turbulent flame. This shock wave is visible on the image at 50 μs in figure 8, between the turbulent flame and the principal planar shock front, which is approximately 60 mm ahead of the centre of the flame's leading edge. Another shock wave is visible, also originating at the top wall at the upper lip of the flame. On the following image at 60 μs, an autoignitive leading front has developed at the head of the original flame and just behind the secondary shock wave. The presence of this shock facilitates the further development of the autoignition wave. It also can be seen on the next image, at 70 μs, that it has overtaken the shock fronts originally ahead of it. Finally, not shown in the figure, a planar self-sustained detonation is created. The images show how the autoigniting gas acquires a cell-like structure, characteristic of a self-sustained detonation wave.

The pressure readings and schlieren images [25] are now analysed using the one-dimensional steady-state theory in Bradley *et al.* [28]. This relates the turbulent burning velocity *u*_{t} to the strength of the leading shock, prior to autoignition. A parameter, *c*, expresses the dimensionless volumetric expansion owing to combustion, at the post-shock elevated temperature and pressure, *T*_{2} and *P*_{2}, by
5.1Here *a* is the cross-sectional area of the duct and *A* is the area of the flamefront associated with *u*_{t}. The first three terms on the right-hand side constitute the gas velocity along the duct, ahead of the flame. The ratio *A*/*a* was estimated to be 1.44 from measurements of the schlieren images in Urtiew & Oppenheim [25]. Images in only one direction of view were presented and the assumption was made that the orthogonal view would be similar. The ratio of specific heats in the shock is *γ*, the acoustic speed in the cold mixture ahead of the shock wave is *a*_{1} and *σ* is the ratio of unburned to burned gas density in constant pressure combustion at *P*_{2}. The gas velocity of the reactants into, and relative to, the shock wave is *u*_{1}. There is no gas velocity ahead of the shock wave and *u*_{1} is numerically equal to the shock velocity along the duct, with a Mach number, *M*_{1}, equal to *u*_{1}/*a*_{1} and given by Bradley *et al.* [28] as
5.2The planar shock equations for *P*_{2}/*P*_{1}, *T*_{2}/*T*_{1} and *u*_{2}/*u*_{1}, in which *u*_{2} is the gas velocity relative to the shock wave, away from it and towards the flame, are expressed in terms of *γ* and *M*_{1} in the study of Bradley *et al.* [28]. For the equimolar H_{2}–O_{2} mixture, the value of *γ* was 1.4 [9] and, with this value, these three ratios and *M*_{1} are shown as a function of *c* in figure 9.

In the experiments associated with figures 7 and 8, the pressure ratio measured at the peak of the principal planar shock wave just prior to autoignition was 14.14. For this value, the theoretical expressions gave *c*=3.21, *M*_{1}=3.50 and *u*_{2}/*u*_{1}=0.23. With *a*_{1}=453 m s^{−1} [9], the velocity of the planar shock along the duct is 1585 m s^{−1}. This compares with a value measured from the shock images, just prior to autoignition, of 1404 m s^{−1}.

No data for *τ*_{i} at the shocked pressure, *P*_{2}, and temperature, *T*_{2}, of 151.9 kPa and 995 K could be found directly in the literature, but correlations and interpolations of data for the same mixture at higher temperatures and for a stoichiometric mixture at 1000 K led to an estimated value of 2 ms [29]. With *u*_{1}=1585 m s^{−1}, the theoretical value of *u*_{2}/*u*_{1} of 0.23 yields *u*_{2}=365 m s^{−1}. The distance between the planar shock from the centre of the turbulent flame, measured from the images in the early stage in Urtiew & Oppenheim [25], is 0.06 m. Consequently, the residence time of the reactants as they moved this distance towards the flame, where the first autoignition occurred at the wall, is approximately this distance divided by *u*_{2}, which is 0.2 ms. This is one-tenth of the estimated value of *τ*_{i} for the mixture. However, unlike the autoignition at a single initiating hot spot discussed in §3 and §4, in this case the precursor secondary transverse wave would raise the pressure and temperature of a proportion of the reactants above *P*_{2} and *T*_{2}. This probably explains this lower deduced value of *τ*_{i} of 0.2 ms.

The theoretical turbulent flame speed, *S*_{f}, in the duct, when normalized by *a*, is given by Bradley *et al.* [28] as
5.3This also is plotted in figure 9, for the value of *σ*=2.62 from Morley [9] for constant pressure combustion at 151.9 kPa, and the planar shocked temperature of 995 K. With *a*_{1}=453 m s^{−1}, *S*_{f}=1963 m s^{1}. This compares well with a measured value of 1930 m s^{−1} obtained from the full sequence of turbulent flame images in Urtiew & Oppenheim [25], just before autoignition.

It is clear that conditions are not uniform in the unreacted mixture, which is subjected to stochastic, non-equilibrated, physico-chemical rate processes. These often result in reaction front speeds that overshoot the Chapman–Jouguet (CJ) values. These could arise from high reaction rates prior to the attainment of the equilibrated CJ conditions [30]. In the present analyses, the time from the first autoignition (not always easy to identify) to a steady detonation front, largely owing to the influence of the secondary shock wave in inducing further autoignition, was as low as approximately 0.1 ms. This compares with a similar time observed in the study of Babkin & Kozachenko [31] and of 0.04 ms in the study of Wu & Wang [32], all of which are less than might have been anticipated for a uniform, homogeneous, autoignition.

In the study of Bradley *et al.* [28], the criteria adopted to assess whether a DDT might occur following an autoignition was that the turbulent flame speed should attain the CJ value and that the reactants between the flamefront and shock wave would autoignite in the time available. Because of the vigour of the non-uniform autoignition, it would appear that the second condition is always attained. With regard to the first condition, the turbulent flame would remain in a supportive role to maintain the shock wave, until the detonation had become established.

The leading autoignitive front shown at 60 and 70 μs in figure 8 propagated at a speed of 3040 m s^{−1}, significantly greater than both the measured turbulent flame speed of 1930 m s^{−1} prior to autoignition and the CJ speed of 2237 m s^{−1}. The flame speed, *S*_{f}, in this case an autoignitive front speed, is related to an associated autoignitive burning velocity, *u*_{t}, by Bradley *et al.* [28]
5.4This yielded a value of *u*_{t}, an autoignitive velocity relative to the unburned reactants, *u*_{a}, of 806 m s^{−1}. With an acoustic speed in the reactants, behind the planar front of 808 m s^{−1} [9], *ξ*=1. It is difficult to quantify an associated value of *ε*, as the autoignitive front is not spherical and is larger than the envisaged initiating hot spot sizes. However, with *r*=10 mm, *a*_{2}=808 m s^{−1} at the shock pressure and temperature of 151.9 kPa and 995 K, and *τ*_{e} estimated from the study of Browne *et al.* [33], as 10^{−4} *τ*_{i}, with *τ*_{i}=1 ms, then *ε*>100. These values of *ξ* and *ε* are well within the peninsula of developing detonations as shown in figure 5, and suggest that a localized detonation might be maintained.

## 6. Values of the turbulent burning velocity

Returning to the turbulent burning velocity of the initiating flame, from equation (5.1)
6.1With *A*/*a*∼1.44 and *σ*=2.62, values of *u*_{t}/*a*_{1} are expressed as a function of *c* in figure 9. For these conditions, just prior to autoignition (when *c*=3.21, *P*_{2}=151.9 kPa and *T*_{2}=995 K), *u*_{t}/*a*_{1} is equal to 1.15. With *a*_{1}=453 m s^{−1}, then *u*_{t}=521 m s^{−1}.

Values of *u*_{t} are frequently normalized by the laminar burning velocity, *u*_{l}. However, no data are available for *u*_{l} for the equimolar H_{2}–O_{2} mixture under these conditions. Gray *et al.* [34] give a value of 5.18 m s^{−1} at 9.33 kPa and 335 K. Because of the high reactivity of the H_{2}–O_{2} mixture, it has not been possible to measure *u*_{l} close to 1000 K at any pressure. High-temperature measurements have been made at atmospheric pressure with equimoles of H_{2}–CO with air. These show an approximately eightfold increase in *u*_{l} when the temperature is increased from 300 to 700 K [35]. It is probable that, for a temperature of 1000 K, the increase could even be more than 15-fold. The value of *P*_{2} of 151.9 kPa is higher than the pressure of 9.33 kPa at which *u*_{l} was measured [34]. In the study of Bradley *et al.* [36], it is suggested that, for an equimolar mixture of H_{2}–O_{2}, but within a H_{2}–air mixture, the value of *u*_{l} varies with *P*^{−0.29}. With this relationship, at *P*_{2}, *u*_{l} would be reduced to 2.31 m s^{−1}. However, a 15-fold increase owing to temperature would give a value of *u*_{l} of 34.6 m s^{−1}, and an approximate value of *u*_{t}/*u*_{l} of 15.

Many turbulent flames have significantly higher ratios of *u*_{t}/*u*_{l} than this. The reason why autoignition and detonation readily occurred in the present case [25] is that the generated turbulent kinetic energy was able to accelerate the flame sufficiently to attain the CJ speed and create a shock sufficiently strong to autoignite a mixture with a sufficiently small *τ*_{i}. More fundamentally, values of *c* given by equation (5.1) must be high enough to generate combined pressure and temperature ratios sufficient for autoignition (figure 9). For mixtures with higher values of *τ*_{i} and smaller values of *u*_{l}, for autoignition to occur, the flame would have to be accelerated to higher values of *u*_{t} and *u*_{t}/*u*_{l} to generate a strong enough shock.

Spirals, orifices and baffles across the duct have been employed as generators of strong turbulence to accelerate flames rapidly to a maximum steady-state value [24,37–39]. However, the flame cannot accelerate indefinitely, because of the eventual onset of localized flame extinctions owing to the increasing flame stretch rate. Mixtures exhibit different maxima in values of *u*_{t}/*u*_{l}, indicated by (*u*_{t}/*u*_{l})_{m}. Attempts have been made to calculate such values semi-theoretically in earlier studies [28,40]. Whether autoignition of the less reactive mixture occurs will depend upon the value of (*u*_{t}/*u*_{l})_{m}. There are many theoretical problems in attempting this calculation; here, an attempt is made through the use of experimental data.

Maximum values of *u*_{t} have been measured in experimental studies of isotropic turbulent flame propagation in fan-stirred bombs. As the root mean square (r.m.s.) turbulent velocity, *u*′, increases, so does *u*_{t} up to a maximum value. This then declines with further increases in *u*′ as a result of increasing flame extinctions [41,42]. But whereas increasing turbulence can decrease the burning velocity and flame speed to the point of extinction in a fan-stirred bomb, the feedback mechanism in the duct will hold these at values close to the maximum attainable value for the mixture.

Measured values of *u*_{t}/*u*′ and the onset of flame extinctions have both been expressed in terms of the Karlovitz stretch factor, *K*, and the strain rate Markstein number of the mixture, *Ma*_{sr} [43]. The former can be regarded as a ratio of a chemical to eddy lifetime
6.2in which the laminar flame thickness, *δ*_{l}, is given by *ν*/*u*_{l}, where *ν* is the kinematic viscosity. The reciprocal of an ‘eddy lifetime’ is *u*′/*λ*, the r.m.s. strain rate, in which *λ* is the Taylor length scale. The turbulent Reynolds number based on this scale is related to that based on the rather more practical integral length scale, *L*, by . It is readily shown that
6.3A useful re-formulation of this expression introduces *u*_{t}/*u*_{l} in
6.4Experiments in fan-stirred bombs suggest that a probability of approximately 0.8 for an initial flame kernel continuing to propagate is close to the maximum attainable value of *u*_{t}/*u*_{l} for a given mixture [44]. The value of *K* for this condition is indicated by *K*_{0.8}. This was measured by increasing the fan speed until flame extinctions occurred in a fan-stirred bomb with mixtures of CH_{4}, C_{3}H_{8} and *i*-octane with air and very lean H_{2}–air mixtures, all in the pressure range of 0.1–1.5 MPa and at room temperature [45]. Values of *K*_{0.8} could be expressed in terms of *Ma*_{sr}, by
6.5Extensive measurements of *u*_{t}/*u*′ have been correlated by an expression of the form [43]
6.6in which the values of *α* and *β* depend upon *Ma*_{sr} and its sign. With values of *K*_{0.8} from equation (6.5) substituted into equation (6.6), and values of *α* and *β* from the study of Bradley *et al.* [43], maximum values of *u*_{t}/*u*_{l}, indicated by (*u*_{t}/*u*_{l})_{m}, were obtained from equation (6.4) for three different values of *u*_{l}*l*/*ν* over a range of *Ma*_{sr}. These are shown in figure 10. High values of *u*_{l}*L*/*ν* and high negative values of *Ma*_{sr} are clearly conducive to a DDT. In the study of Bradley *et al.* [28], the one-dimensional analysis predicts how the higher value of *L*, used in a larger experimental duct [46], lowers the limiting value of *φ* which could support a DDT in H_{2}–air mixtures, as observed experimentally.

Limitations to the application of these correlations lie in the limited data available at high temperatures and pressures for laminar burning velocities, Markstein numbers and, for autoignition, autoignition delay times. These aspects determine the lower limit values of *φ* that can support DDTs. Other important limitations of the data in figure 10 are that quenching data could not be obtained for the more reactive mixtures. Also, it was not possible to obtain experimental turbulent burning velocities and flame-quenching data in the presence of strong shock waves. On the other hand, experiments suggest that, for values of *u*′/*u*_{l}>3, the influences of flamelet cellularity and Rayleigh–Taylor instabilities are much diminished [47]. Equation (6.5), and consequently the data in figure 10, is most valid for the less reactive mixtures. However, it is such mixtures upon which attention is focused in many problems of potential detonations.

Of course, autoignition of a mixture can occur at ratios less than its value of (*u*_{t}/*u*_{l})_{m} and this is the case in most DDT studies, including the one in §5. In this case, the value of *u*_{l}*L*/*ν* was estimated to be approximately 2000 and of *Ma*_{sr} as approximately −7 [36]. For these values, figure 10 suggests values of (*u*_{t}/*u*_{l})_{m} greater than 140, well in excess of the actual approximate value of *u*_{t}/*u*_{l} of 15. The usefulness of the maximum value is in assessing marginal autoignitions at the lower values of *u*_{l}. An important aspect when (*u*_{t}/*u*_{l})<(*u*_{t}/*u*_{l})_{m} is that any sideways venting in the duct would reduce the gas velocity ahead of the flame, thus reducing *u*_{t}. But because of the potential for (*u*_{t}/*u*_{l}) to increase, the flame might still accelerate sufficiently for a DDT.

For a known maximum value of *u*_{t}, *u*_{t}/*a*_{1} can be evaluated for known values of *A*/*a* and *σ*. The associated maximum pressure and temperature ratios at the planar shock can be obtained for these values, as indicated by figure 9. The important influence of the area ratio, *A*/*a*, is indicated in the plots of these ratios against *u*_{t}/*a*_{1} in figure 11, for values of *A*/*a* of 1.44 and 3. Not surprisingly, a higher value of *A*/*a* increases the shock ratios of pressure and temperature for a given *u*_{t}/*a*_{1}. Whereas, when turbulent flames are accelerated, the flame surface is wrinkled in an extending flame brush [48], with an accelerating laminar flame there is no such wrinkling but only a direct increase in the area of the laminar flamefront through its elongation.

Detonations have been created in capillary tubes, with diameters of a few millimetres, in which the original flame is laminar or near laminar. Wu & Wang [32] have reported DDTs in stoichiometric ethylene–oxygen mixtures, under initially atmospheric conditions. Contrary to what is suggested by equations (5.4) and (6.4), the CJ speed was more rapidly attained in the smaller diameter tubes. Neither pressures nor the progress of shock waves were measured, but chemiluminescence from the laminar-like flames was recorded by a high-speed camera. From these measurements of flame propagation in a 2 mm diameter tube, the present author estimated the value of *A*/*a*, for an assumed parabolic flame, as approximately 42. In the absence of significant turbulence, the pressure generated by combustion produced an elongated laminar, or near-laminar, flame. The theoretical CJ speed was 2379 m s^{−1} [9]. Just prior to autoignition, for tube diameters ranging from 0.5 to 3 mm, the measured flame speeds were close to 1600 m s^{−1}, from which they rose very rapidly to 2760 m s^{−1} and then declined rapidly to the CJ speed, all in a time of 0.4 ms.

As discussed in §5, following an autoignition, for a DDT to develop the flame speed it must attain the CJ value, and this condition was applied to this transition for a laminar flame. In equation (5.4), *S*_{f} was set equal to the CJ speed of 2379 m s^{−1}, with *A*/*a*=42. The necessary values of *c*, given by equation (5.1), with *a*_{1}=328 m s^{−1}, were essentially constrained by this value of *S*_{f} and equation (5.4). From the values of *c*, the corresponding values of *M*_{1} yielded the shocked temperatures *T*_{2}, pressures *P*_{2}, and mean values of *γ* [9]. Values of *σ* were calculated for the subsequent constant pressure combustion. The converged solutions for the CJ speed are given in table 3. Also shown are the solutions for the flame speed of 1600 m s^{−1} at the start of autoignition.

Comparisons of the computed values of *u*_{t}, in this case a laminar burning velocity, in table 3 were hindered by the lack of data on stoichiometric ethylene–oxygen laminar burning velocities, particularly at high temperatures and pressures. In the study of Wu & Wang [32], *u*_{l} was taken to be 5.5 m s^{−1} under atmospheric conditions. Some data on the variations of *u*_{l} for ethylene–air with temperature are given by Kumar *et al.* [49] and with pressure by Egolfopoulos *et al.* [50] and Jomaas *et al.* [51]. Extrapolating the expressions for these variations to much higher temperatures and pressures, and assuming that they apply to ethylene–oxygen mixtures, gave a value of *u*_{l} of 52 m s^{−1} corresponding to the CJ speed, and one of 31 m s^{−1} corresponding to the initial autoignition speed. Both values are significantly higher than the computed values in table 3. Apart from errors in the extrapolation of data on *u*_{l} and the error band on the value of *A*/*a*, another explanation for the differences is the possibility of a reduction in burning velocities in the experiments owing to high flame stretch rates. An assumed parabolic velocity ahead of the flame suggested very high radial velocity gradients at the wall in excess of 10^{5} s^{−1}.

## 7. Detonation waves

Although the combined one-dimensional turbulent flame and shock wave theory is a useful guide to the possibility of autoignition and detonation, it cannot predict either the details of the transition to detonation or the structure of the detonation. With regard to the former, in the analyses of §5, a localized autoignitive flame speed as high as 3040 m s^{−1} was attained in the ‘explosion in the explosion’. This reveals something of the nature of the ‘quasi-detonation’ regime, prior to the establishment of a coherent detonation front. In this regime, reactivity gradients lead to autoignitions at hot spots, with shock waves over- or under-running reaction waves, or combining with them in developing detonations which might subsequently extinguish. In addition, localized extinctions and re-ignitions in the turbulent flame induce pressure pulses.

This is particularly so in the disordered experimental turbulent and autoignitive flames in highly baffled ducts, designed to give high turbulence intensities and flame speeds [37–39]. The pressure pulses from these transient events are reflected at baffles and walls and can induce Rayleigh–Taylor flame instabilities. In Chao & Lee [38] and Teodorczyk & Lee [52], the effects of shock waves were reduced by lining walls and obstacles with acoustic-absorbing material. In the experiment analysed in §5, the autoignitive front coalesced first with the secondary shock wave, then with the main shock to form a detonation front. The extensive numerical simulations of Oran and co-workers have demonstrated clearly how localized detonations in ducts, which may or may not survive, can arise at randomly localized regions of appropriate reactivity gradients [53,54].

With regard to the structure of an established detonation, a one-dimensional planar detonation is fundamentally unstable to transverse waves, but can be stabilized by a complex three-dimensional structure in which transverse waves interact with the leading shock. Shock waves are reflected from the duct walls and triple points are created at the intersection of incident and reflected shock waves. A Mach stem is formed as the intersection is forced away from the wall. The paths of intense combustion close to the triple point create a cellular structure. A criterion employed for the propagation of a detonation is that the size of a cell should be less than that of the duct cross section. The shock reflections cause the leading shock front to pulsate quite strongly in the direction of propagation, alternating between strong Mach stems and weaker incident shocks. In spite of these complexities, CJ velocities and the CJ-equilibrated state are eventually attained and a stochastic steady, one-dimensional representation of detonation waves would seem to be attainable [55].

With regard to the stability of the detonation structure, at the lower values of *E*/*RT*, with *ξ* close to unity in the strong ignition regime, the detonation front appears flat, stable and one-dimensional, with relatively weak transverse waves and an associated cellular pattern [56,57]. The transverse waves have no significant role in the propagation, which depends upon the main shock front [58]. When lower values of *E*/*RT* are also combined with relatively small values of *τ*_{i}/*τ*_{e}, the reaction is spatially fairly uniform, with separate power pulses overlapping in time and contributing to the overall stability. Instabilities can develop within this structure for two reasons. First, as the value of *E*/*RT* increases, because of the sensitivity of *τ*_{i} to *T* (see equation (2.4)), the reaction zone begins to decouple from the shock front. It becomes irregular, with a large range of fluctuating length scales, and the shock velocity drops below the CJ value [57,59]. Second, at the larger values of *τ*_{i}/*τ*_{e}, the power pulses are no longer uniformly spread, the heat release is spiky and this leads to instabilities in the reaction zone structure [60]. Values of *τ*_{e}, which are less than those of *τ*_{i}, decrease less sharply with an increase in *T* than those of *τ*_{i}, and there is an associated decrease in *τ*_{i}/*τ*_{e} [14,16,18,33]. Consequently, on both counts, stability tends to be jeopardized at the higher temperatures.

It has been proposed that the product, (*E*/*R*)(*τ*_{i}/*τ*_{e}), provides a figure of merit for stability [61,62]. The smaller values of the product are associated with the more stable detonations; the larger values with unstable detonations. From equations (2.3), (2.4) and (3.5),
7.1This suggests that the strong ignition, stable, regime is probably associated with the toe of the peninsula in figure 5. The last term is the reciprocal of a reactivity gradient. A reaction zone that has become completely uncoupled from the shock front might degenerate to a turbulent flame capable of further acceleration along the duct, initially aided by decaying vortices created at the detonation front.

An important practical issue is the extent to which a detonation that has become fully developed in a duct or ‘bang box’ might continue to propagate through the same mixture, but outside the confinement. This is clearly important in environments like chemical plants, refineries and fuel-storage facilities, and is an issue of pressing concern [63]. The fundamental studies by Radulescu & Lee [58] of detonation failure mechanisms in ducts are important in this context. They arranged that detonations entered a part of the duct with porous walls, where the transverse waves were damped upon reflection. Some new triple points were generated, but eventually the detonation front failed when new triple points were insufficiently amplified.

For the more stable detonations, although there was a regular cellular structure, the transverse waves were weaker and were less significant for the propagation mechanism. When the detonation wave reached the porous walls, its propagation velocity decreased and transverse waves could not be generated to overcome the losses owing to the mass divergence into the wall leading to a curved shock front, and the detonation front failed. For the less stable detonations, the transverse waves proved to be an essential part of the ignition and detonation–propagation mechanisms. Failure of this mechanism occurred when the rate of transverse wave attenuations at the wall exceeded the generation of new transverse waves in the reaction zone, The unstable detonations propagated a shorter distance than the stable ones, before the detonation failed, and had a smaller failure diameter.

## 8. Conclusions

The pressure pulse generated at a single hot spot has been evaluated for different conditions for a range of PRFs. The pulses diminish at a decreasing rate with increases in ON, which for a number of reasons is becoming a less informative parameter for current fuel and automotive design. The results of direct numerical simulations, with reasonably complete chemical kinetics, give rise to a peninsula, within which detonations can develop but often cannot be sustained, when *ξ* is plotted against *ε*, as in figure 5. Such a diagram is useful in assessing engine knock; the regime of controlled autoignition lies outside the peninsula, while that of super-knock in highly turbocharged engines lies inside it at rather low values of *ξ*. Although a detonation does not readily propagate outside a single hot spot, the multiple effect from several hot spots can be very damaging.

In the DDT context, low values of (*ξε*), particularly those within the toe of the peninsula, are indicative of stable detonation in a duct, while high values are indicative of an unstable detonation, in which the reaction zone becomes detached from the leading shock wave and the detonation fails.

The roles of hot spots and shock waves in the complex transition from a turbulent flame to autoignition and a developed detonation have been examined in some detail. Despite this complexity, the one-dimensional theory which combines the acceleration of turbulent flames, potentially up to maximum values of burning velocity with planar shock wave theory, provides a useful guide to the probability of autoignition followed by detonation. In both turbulent and near-laminar deflagrations, flame quenching controls the maximum burning velocities, including laminar DDTs in capillary tubes. Autoignition and probable detonation occur when the turbulent flame speed is close to the CJ speed and the unburned mixture between the flame and the shock autoignites. In many cases, the flame speeds are significantly less than the maximum possible flame speeds.

The turbulent flame conditions that are conducive to high turbulent burning velocities, of relevance to marginal mixtures, are high values of *u*_{l}, large ducts and small, particularly negative, values of *Ma*_{sr}. In the presence of sideways venting, stable detonations are more likely to survive than unstable ones. The former are associated with the toe of the detonation peninsula and occur at low *E*/*R*, high *T* and low *τ*_{i}/*τ*_{e}.

One of the outstanding problems is understanding the conditions under which a detonation that has become fully developed in a duct or ‘bang box’ might continue to propagate through the mixture outside the region of confinement. Improvement of basic understanding in this area is vital for assessing the safety of fuel-storage facilities and chemical plants. The complexity of operational conditions and geometrical configurations in such large plants militates against detailed reliable mathematical modelling of the deflagrations and detonations that might ensue after leakage and ignition of flammable gas. Of greater importance is the detailed attention to all hazardous eventualities at the design stage.

## Acknowledgements

The author thanks Gary Sharpe and Slava Babkin for helpful discussions on many aspects at Leeds and Novosibirsk, respectively.

## Footnotes

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

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