V Rodriguez

C Jourdain

P Vidal

R Zitoun

An experimental evidence of steadily-rotating overdriven detonation

This experimental work reports on detonation behaviors in a curved chamber without inner wall after diffraction of a Chapman-Jouguet (CJ) detonation from a straight channel tangent to the chamber outer wall. The upper and lower faces of the chamber receive either soot foils for recording the history of the transmission dynamics, or optical windows for schlieren high-speed visualizations. Tests were carried out with the stoichiometric propane-oxygen mixture at initial temperature

T 0 = 288 K and initial pressures ranging between 8 kPa and 15 kPa. The primary observation is the existence of an initial pressure range (8 kPa -12 kPa) for which, after diffraction transients, a Mach detonation can rotate normal to the outer wall with a constant angular velocity such that the normal front velocity at the wall is larger than the CJ value. The height of the Mach front decreases and its tangential velocity increases with increasing initial pressure. This over-driven detonation results from the irregular reflection of the initially-oblique front as the outer wall tilts with respect to the front, similarly to shock propagation in a continuouslyconverging channel. The Mach triple-point moves away from the wall and stabilizes its trajectory parallel to the wall. This Mach front shows detonation cells parallel to the wall with a constant mean width very small compared to that on the initial CJ front. A detonation can thus smoothly propagate in a curved chamber without center body, though the Mach front does not sweep the entire equivalent cross-section of the chamber. These observations are consistent with experimental results for RDE chambers without center body, or with linearly-increasing cross sections, that show improved properties of rotating detonation, compared to chambers with constant cross sections. This supports the interest of further studies on the hollow-chamber configuration of RDEs.

Introduction

Understanding detonation wave behaviors in curved confinements interests several applications such as propulsion, defence, and safety of industrial facilities. In particular, rotating detonation is currently considered as an alternate to isobaric combustion commonly implemented in turboprops. However, the uncertainties on the actual wave dynamics and the interplay of several fundamental phenomena still make the optimum choice of the inner geometry of the curved chamber a subject of investigations. Chambers of rotating detonation engines (RDE) can be annular, i.e., with a cylindrical centerbody [START_REF] Canteins | Etude de la Detonation Continue Rotative : Application à la Propulsion[END_REF][START_REF] Frolov | Large-scale hydrogenair continuous detonation combustor[END_REF][START_REF] Nicholls | Feasibility studies of a rotating detonation wave rocket motor[END_REF], hollow, i.e., without centerbody [START_REF] Tang | Three-dimensional numerical investigations of the rotating detonation engine with a hollow combustor[END_REF][START_REF] Anand | Hollow Rotating Detonation Combustor[END_REF], or semi-hollow, i.e., with a conical centerbody [START_REF] Bykovskii | Continuous detonation combustion of an annular gas-mixture layer[END_REF][START_REF] Lin | An experimental study on CH 4 /O 2 continuously rotating detonation wave in a hollow combustion chamber[END_REF][START_REF] Zhang | Experimental investigations on H 2 -Air rotating detonation wave in the hollow chamber with Laval nozzle[END_REF][START_REF] Hansmetzger | A study of continuous rotation modes of detonation in an annular chamber with constant or increasing section[END_REF]. Although only a limited number of experiments is available, the hollow and semi-hollow configurations appear to produce larger pressures and velocities of the rotating detonation [START_REF] Hansmetzger | A study of continuous rotation modes of detonation in an annular chamber with constant or increasing section[END_REF]. Getting some fundamental physical insight into this feature is the main motivation of the present study.

Propulsion by detonation has been studied for the last seventy years because its theoretical thermal efficiency is higher than that of the isobaric combustion mode [START_REF] Roy | Propulsion par Statoréacteur à Détonation[END_REF]. The main two implementations are pulsed detonations in tubes and rotating detonations in cylindrical chambers. Since the pioneering works by Voitsekhovskii [START_REF] Voitsekhovskii | Spinning maintained detonation[END_REF] and Nicholls [START_REF] Nicholls | Feasibility studies of a rotating detonation wave rocket motor[END_REF], numerous experimental works have demonstrated the feasibility of rotating-detonation chambers. Nicholls et al. [START_REF] Nicholls | Feasibility studies of a rotating detonation wave rocket motor[END_REF] studied the rocket configuration of an annular RDE, with either hydrogen or methane as the fuel, and oxygen as the oxidizer. They reported a Chapman-Jouguet (CJ) detonation regime, at least for the first cycle. Edwards et al. [START_REF] Edwards | Shocks and detonation in channels with 90 • bends[END_REF] experimentally studied the detonation propagation in curved channels with rectangularcross-sections. By varying the initial Mach number M 0 and the channel curvature, they obtained several Mach-stem growth rates and several Mach-number ratios M/M 0 at the wall, with M the wave-front Mach number at the wall. They observed that the ratio M/M 0 is maximal along the outer (concave) wall and minimal along the inner (convex) wall. They successfully compared their experimental results against numerical modelling by means of Whitham's Geometrical Shock Dynamics [START_REF] Whitham | Linear and nonlinear waves[END_REF] and the Three-Shock Theory [START_REF] Ben-Dor | Shock Wave Reflection Phenomena[END_REF], which suggests that global geometric effects predominates over the local properties of the detonation reaction zone. Thomas et al. [START_REF] Thomas | Detonation interaction with wedges and bends[END_REF] investigated detonation behaviors along wedges in curved channels and in pipes bends. They used schlieren and soot recordings to characterize the Mach stem initiation and propagation. For the wedge case, they identified a minimum angle for observing the Mach stem. In the case of the two-dimensional curved channel, at given initial pressure, they observed more stable detonation transmissions for larger bend curvatures, and, for the same geometry, that the larger the initial pressure is, the more stable the detonation transmission is. They also observed a Mach stem along the outer wall of their curved channel. Frolov et al. [START_REF] Frolov | Shock wave and detonation propagation through U-bend tubes[END_REF] performed experimental and numerical studies on detonation and deflagration in tubes connected by one or two U bends. They investigated how the U curvature and the tube diameter may promote detonation transmission and re-inititiation. Nakayama et al. and Kudo et al. [17,[START_REF] Nakayama | Stable detonation wave propagation in rectangular-cross-section curved channel[END_REF][START_REF] Nakayama | Front shock behavior of stable curved detonation waves in rectangular-cross-section curved channel[END_REF] carried out experimental studies on detonation propagation in curved channels with con-stant rectangular cross sections. They classified their observations into three modes of detonation transmission, i.e., (1) a continuous multicellular front structure (stable mode), or with (2) successive but limited local extinctions and re-ignitions close to the inner wall (critical mode), or with (3) successive full quenching and re-ignitions (unstable mode).

They collected enough data to propose the criterion that the ratio R i /λ, where R i is the channel inner radius and λ is the mean width of the detonation cells, should be larger than about 23 so that the stable mode can be achieved. Additional works by Vidal et al. [START_REF] Vidal | Recent experimental efforts at Pprime on detonation dynamics in gases for application to propulsion[END_REF] showed that this classification also extends to the transmission of an initially marginal detonation. Gwak et al. [START_REF] Gwak | Effect of multi-bend geometry on deflagration to detonation transition of a hydrocarbon-air mixture in tubes[END_REF] performed numerical calculations on detonation initiation and propagation in multi-bends channels. They investigated how the combination of bends and obstacles may enhance transition from deflagration to detonation. Kuznetsov et al. [START_REF] Kuznetsov | Formation and structure of quasi-detonation oblique reaction front in annular channel[END_REF] carried out experiments on combustion and detonation in an annular chamber with optical access. They observed a complex structure composed of an oblique shock and a deflagration and propagating with the CJ velocity. Short et al. [START_REF] Short | Steady detonation propagation in a circular arc: a Detonation Shock Dynamics model[END_REF][START_REF] Short | Detonation propagation in a circular arc: reactive burn modeling[END_REF][START_REF] Short | Propagation of a stable gaseous detonation in a circular arc configuration[END_REF] studied experimentally and numerically the dynamics of steady detonation propagation in a constant-cross-section, two-dimensional circular arc of high explosive. They observed that the angular speed of the detonation increases monotonically with increasing arc thickness before limiting to a constant. They also observed, as the arc thickness increases, that a Mach stem forms on the detonation front but disappears for sufficiently wide arcs. They implemented numerical modelling by means of the Detonation Shock Dynamics technique [START_REF]Shock Wave Science and Technology Reference Library[END_REF], and numerical simulations to interpret their experiments. They provide an analysis that accounts for different wave front configurations depending on the boundary conditions (the degree of confinement at the inner and outer arc surfaces).

In contrast, the present work addresses the specific configuration of a curved chamber that has no inner wall (no kernel), a different configuration from that where the chamber has the same section as the entry channel. A CJ detonation is first diffracted from a straight channel into the curved chamber, so the detonation that may propagate along the curved outer wall of the chamber after diffraction is semi-confined inward, i.e., confined by the rigid curved outer wall, and, inward by the fresh gases pre-compressed by the shock resulting from detonation diffraction. This geometry can be regarded as a model of the hollow configuration of a RDE chamber. Attention focuses on the case where a CJ multi-cellular detonation is transmitted from the entry-channel to the curved chamber.

Diffraction then shows successive quenchings and ignitions near the outer wall that may result in a detonation that steadily rotates along the outer wall, with a velocity larger than the CJ value, depending on initial pressure. The next section details the experimental setup and the observation techniques. The results are presented in Section 3 and discussed in Section 4, together with a conclusion and some perspectives.

2 Experimental set-up, observation and measurement techniques Figure 1 shows the schematic of the experimental set-up. The main three elements are the curved chamber (I), a detonation tube (II) and expansion vessel (III). The detonation was generated at one end of the tube, entered the chamber from the other, ran into it, and finally expanded into the vessel.

The detonation tube was 2-m long, and 51-mm i.d. An exploding wire or a spark plug located at the tube end opposite to the chamber entry was used for ignition, after filling the tube and the chamber with the chosen mixture. A 500-mm long Shchelkin spiral was positioned immediately ahead of the ignition source to promote transition of the initial deflagration to detonation. The detonation regime in the ignition tube was characterized before entering the chamber by means of four Kistler 603B pressure transducers (1µs response time, 300 kHz natural frequency, each coupled with a Kistler 5018A electrostatic charge amplifier with 200 kHz band width) positioned along the tube upper generatrix (P 1 to P 4 ) in order to obtain the wave average velocities. The first transducer was located 140 mm before the chamber entry, and the other three were spaced 250-mm from each other. Curved soot foils were inserted in the lower part of the tube, opposite to the pressure transducers, in order to record the cellular structure of the detonation before the chamber and compare it with that in the chamber.

The chamber was positioned horizontally. The CJ detonation in the ignition tube entered the curved part of the chamber after propagation in a straight channel in the continuation of the ignition tube. The upper and lower faces of the chamber received either steel plates lined on their inner sides with mirror-quality steel foils with soot deposit, for recording the history of the transmission dynamics, or borosilicate-crown-glass optical windows (BK7) for schlieren vizualisations. The chamber was machined into a 20-mm thick aluminum plate, so the straight channel and the chamber had the same height. The chamber outer-wall radius was 80 mm, the entry channel was 30-mm wide, 20-mm height and 200-mm long, and the radius of its rounded inner end was either 30 mm or 50 mm.

The outer walls of the channel and the chamber were tangent to each other. Transmission resulted from the balance between the detonation transient dynamics, along the chamber outer wall, and the diffraction effects at the inner-wall rounded-end.

The combustion products expanded into a 0.08 m 3 expansion vessel at the chamber exit to protect the chamber from shock reflection. A 18-µm thick Mylar foil separated the chamber from the vessel so as to minimize the gas volume to vacuum out before injecting the reactive mixture in the ignition tube and the chamber. Two observation techniques were implemented for characterizing the transmission dynamics. The first one was recordings on soot foils lining the upper and lower inner faces of the chamber. The soot foils covered the entire surface of the chamber faces, so each recording captured the whole dynamics in a single shot. The second one was schlieren vizualisations coupled with a Shimadzu HPV-2 high-speed camera. The acquisition frequency was either 500 kfps or 1 Mfps, and the exposure time was 250 ns and kept constant, independently of the frame rate.

Results

The reactive gas was the stoichiometric mixture of propane and oxygen C 3 H 8 + 5 O 2 initially at room temperature (T 0 = 288 K ± 2 K). The parametric study was carried out by varying initial pressure p 0 from 8 kPa to 15 kPa. Table 1 gives the CJ properties of the mixture as a function of p 0 . The CJ velocities D CJ were calculated by means of the NASA computer program of chemical equilibrium CEA [START_REF] Gordon | computer program for calculation of complex chemical equilibrium compositions and application[END_REF].

For convenience, the observation field was divided into 4 zones chosen for their physical relevance. Figure 2 is a typical soot foil recording with the 4 zones indicated by the yellow dashed lines. Zone 1 is the straight channel at the chamber entry, and in which a CJ or a quasi-CJ detonation propagates. The detonation was considered CJ when the mean cell width was the same as in the detonation tube just before the channel entry, and quasi-CJ when this mean width was only slightly larger than the CJ one. Both types are indicated by a continuous cellular pattern with constant cell mean width. A continuous cell pattern characterizes the existence of the detonation regime in gases [START_REF] Desbordes | Shock Wave Science and Technology Reference Library[END_REF]. In this work, the cell width given in Table 1 were obtained from measurements in the initiation tube and in the channel entry of the chamber, and cross-checked against those in the Caltech Database [START_REF] Kaneshige | Detonation Database Technical Report[END_REF]. The pattern of the C 3 H 8 + 5 O 2 mixture in this work can be described as irregular according to the classification by Libouton et al. [START_REF] Libouton | Cinétique, structure et entretien des ondes de détonation[END_REF]. Zone 2 covers the diffraction domain where the initial detonation is subject to changes in the lateral boundaries. An expansion wave originates from the point on the inner wall of the entry channel where rounding off begins, and propagates on the detonation front towards the chamber outer wall, thereby quenching detonation. This is indicated on the soot recording by the disappearance of the cellular structure (from the right in zone 2, in Figure 2). Close to the outer wall, depending on p 0 , the cellular detonation keeps propagating until either full quenching by the expansion wave or beginning of transmission transients. Zone 3 is where most of transients take place, until either transmission or final quenching occurs. Zone 4 is where final detonation may re-install. In the latter case, two transmission modes were observed depending on p 0 , and a limiting initial pressure p lim separates the observations of these two modes. Below p lim , an irregular 3-waves transmission of the Mach type is obtained, characterized by the steady rotation of a detonation front along the chamber outer wall.

The detonation front is normal to the wall, with a constant front height smaller than the chamber section. The triple point is the intersect of the detonation, the diffracted shock and the reflected shock. The diffracted shock does not intersect the outer wall, and the triple point is maintained at a constant distance to the wall as the stem rotates. Above p lim , a regular 2-waves transmission is obtained, characterized by detonation re-initiation in the whole chamber. The detonation front is oblique acute with respect to the curved wall, and the reflected shock is positioned at the wall, at its intersect with the detonation front reflection of the diffracted shock at the outer wall.

Typical examples are shown in Figure 3 with the irregular and regular modes on the left and right soot recordings, respectively. They were obtained with initial pressures p 0 values 12 kPa and 15 kPa, respectively, so the separating pressure p lim is comprised between these two values for the C 3 H 8 + 5 O 2 mixture in our chamber. These soot recordings show how the two modes qualitatively differ from each other though they share the same transients in Zone 3. These transients are successive initiations and quenching of diverging detonation waves. They are identified on the recordings by the limiting lines (shown as dashed yellow lines on the Figure 3) that originate from the vicinity of the chamber curved wall, and separate domains with and without cells. Close to the re-ignition points, very small detonation cells can be distinguished, which grow as the detonation front expands with a decreasing velocity, and disappears when velocity becomes too low or the detonation quenches. Below p lim , the transients result in a final, successful reinitiation of detonation which then propagates along the curved wall with a constant velocity. The yellow diamonds represent a typical detonation cell for each mode, and the yellow arrows indicate the direction of the re-initiated detonation which is parallel to the cell length. The behavior of detonation after initial diffraction at the rounded end of the entry channel is analog to that observed at the supercritical transmission of detonation from a tube to an unconfined space, i.e., when the tube diameter is larger than the critical value ≃ 13λ [START_REF] Desbordes | Aspects stationnaires et transitoires de la détonation dans les gaz : relation avec la structure cellulaire du front[END_REF]. Figure 4 compares the supercritical transmission (left) and the diffraction observed in this work (right), which duplicates Figure 3-left for comparison easiness. The blue and yellow dashed lines indicate the quenching and re-initiation limit paths, respectively. The red dot-dashed line on the left recording indicates the symmetry axis, which is the chamber wall before it starts bending on the right recording. The reinitiation line (yellow) is independent of the chamber wall, regardless of the geometric configuration. The recording in Figure 3-left indicate unambiguously that the limiting line in Zone 4 for the irregular transmission is curved and continuously parallel to the chamber wall. The band between the chamber wall and this limiting line has a constant width with detonation cells also continuously parallel to the curved wall, as illustrated by the yellow diamond. These cells have a constant mean width that is very small compared to that on a CJ front (Table 1). Indeed, given the mixture and the initial conditions, the cell mean width λ is approximately proportional to the characteristic length ℓ c of the average reaction zone of detonation, and this characteristic length is smaller for the larger wave front velocities D, i.e., for the larger temperatures T shock at the detonation leading shock. This can be easily inferred from the classical result of reactive hydrodynamics applied to detonation physics that ℓ c varies essentially like T 2 shock exp(E a /RT shock ), with E a the activation energy of the effective process that drives the beginning of the chemical evolution, at the shock. Also considering that T shock varies essentially like D 2 according to the Rankine-Hugoniot relations, the mean width λ is thus smaller if D is larger. The The three pictures (g), (h) and (i) in Figure 5, and the four pictures (f), (g), (h) and (i) in Figure 6 essentially belong to Zone 4, and they correspond to the irregular and regular transmission modes, respectively. The images for the irregular transmission show that the front close to the chamber wall is very thin and normal to the chamber wall. The soot recordings in Figure 3 at these positions show a very small cellular structure, and therefore confirms that this front is a detonation. The analysis indicates that the distance between the wall and the limiting line between domains with and without detonation cells is practically constant, so the front height remains the same as the detonation rotates along to the wall that are much smaller than those on a front that propagates at D CJ , and as explained above, given the mixture and the initial conditions, the larger D n is, the smaller the cell mean width is. However, in these experiments, the cells on the Mach front are too small and too irregular, so it is difficult to observe a variation of their mean width with the position on the front. Nevertheless, the analysis indicates that the cells on the overdriven Mach front are 2 to 10 times smaller than those on the CJ front in the entry straight channel.

With regards to the regular transmission mode, the four pictures (f), (g), (h) and (i)

in Figure 6 confirm the information from the soot recordings that the final re-initiation in Zone 4 is simply the last, but successful, of the transient waves in Zone 3. The front pattern is that of a regular oblique-shock configuration, i.e., without triple point, and with an incident wave and a reflected shock that intersect at the wall to ensure flow continuity at this wall. The incident wave is again that resulting from the diffraction of the initial CJ detonation. The flow behind it appears to be more spread out than behind the initial CJ detonation, but the soot recordings nevertheless shows a cellular structure with a cell mean width larger than that of the cells on a CJ-detonation front. The trend is that the cell width decreases as the front expands. Therefore, the schlieren images and the soot recordings indicate unambiguously that the wave front for the regular transmission is a detonation that expands towards the wall with velocity smaller than, but increasing to, the CJ velocity. In contrast, the irregular transmission mode shows a wave front propagates parallel to the wall with constant velocity larger than D CJ and a constant cell width smaller than on a CJ-detonation front.

Discussion

Physical interpretation

A physical interpretation of the Mach detonation described in the previous section can be given based on an analogy with that of an incident shock that propagates normal to a flat surface and comes to interact with a wedge [START_REF] Hornung | Regular and Mach reflection of shock waves[END_REF]. Depending on the wedge angle, a Mach stem may form at the wedge tip, with a triple point moving away from the tip, along the planar incident shock. In this work, the schlieren pictures in Figures 5-g the interaction point of the incident and reflected waves positioned at the wall. In the relative frame moving with the rotating front, this phenomenon can also be regarded as similar to that of a shock propagation in a continuously-converging channel, so the front is continuously overdriven.

Comparisons of the transmission modes in curved chamber and constant rectangular cross-section channel

According to the experimental criterion by Kudo et al. and Nakayama et al. [17,[START_REF] Nakayama | Stable detonation wave propagation in rectangular-cross-section curved channel[END_REF][START_REF] Nakayama | Front shock behavior of stable curved detonation waves in rectangular-cross-section curved channel[END_REF],

the stable, multicellular propagation of detonation in a curved channel with constant rectangular cross-section can be achieved if the channel inner radius R i is large enough compared to the cell mean width λ,

R i λ > 23. ( 1
)
When this criterion is met, no detonation quenching is observed at diffraction at the entry channel exit, the initial detonation is smoothly transmitted in the constant-section curved channel, without instabilities, transients or quenching at the inner wall. The wave front rotates at about the CJ velocity and occupies the whole channel cross section. The experiments in the present work were carried out in a chamber with a very small radius for the inner wall, to the extent that the detonation at the outer wall was considered as semi-confined inward (cf. Introduction). In addition, the small-radius center body and the outer wall have different curvature centers, so the distance e from the inner wall to the outer one decreases with increasing angular position. Therefore, this configuration is different from Kudo's and Nakayama's, so the results cannot be directly interpreted by means of criterion 1. It should nevertheless be noted that this criterion can only apply for large enough values e of the channel width (and height) so as to eliminate marginal propagation effects. In straight tubes, the minimum value of e is approximately 3 times the cell mean width λ [START_REF] Desbordes | Aspects stationnaires et transitoires de la détonation dans les gaz : relation avec la structure cellulaire du front[END_REF], and it can be assumed that this first estimate also applies to curved geometries provided that the tangential velocity of the wave front at the inner wall is close the CJ value, e.g., [START_REF] Kudo | Oblique detonation waves stabilized in rectangular-cross-section bent tubes[END_REF][START_REF] Nakayama | Stable detonation wave propagation in rectangular-cross-section curved channel[END_REF][START_REF] Nakayama | Front shock behavior of stable curved detonation waves in rectangular-cross-section curved channel[END_REF]. In the present work, the distance e from the inner wall to the outer one varies from the maximum value e max = 70 mm at angular position π rad. (approximately where most Mach detonations install) to the minimum value e min = 57 mm at angular position 3π/2 rad. (the curved chamber exit). Using the values 6.5 mm and 13 mm for λ at p 0 = 12 kPa and 8 kPa, respectively, the ratio e/λ has minimum value 4.5 at 8 kPa and angular position 3π/2 rad., and maximum value 11 at 12 kPa and angular position π rad. Therefore, the equivalent channel width in our work is larger than the minimum value 3 for multicellular detonation propagation. An equivalent inner radius can then be defined by R i = R e -e, with e varying from e max to e min with increasing angular position from π rad. to 3π/2 rad. This gives ratios R i /λ with minimum value 0.8 at 8 kPa and position 3π/2 rad. and maximum value 3.5 at 12 kPa and position π rad. Also, at the exit of the channel entry, the value of R i is that of the center body and is equal to 30 mm, so the ratios R i /λ at angular position 0 rad. are about 2 and 5 at 8 kPa and 12 kPa, respectively. Thus the largest value of R i /λ in this work is always smaller than the limiting value 23 of criterion 1, and indeed, quenching is always observed at diffraction and around the center body. Nevertheless, the experiments indicate that a detonation can smoothly propagate along the outer wall at a larger velocity than D CJ but only in a limited part of the equivalent cross section of the chamber. The condition for smooth detonation propagation in a constant-cross section channel is thus different from that in a curved chamber without a small centerbody.

A series of experiments was carried out in a constant rectangular cross-section channel

with width e = 30 mm and the same outer radius R e = 80 mm as the curved chamber without inner wall. Figures 12 and13 show soot recordings that compare the detonation transmission with (left) and without (right) center body at p 0 values 12 kPa and 15 kPa, respectively. For comparison easiness, Figure 12 and 13 duplicate Figure 3, left and right, respectively. The left recording in Figure 12 shows an unstable transmission at 12 kPa with a succession of domains without detonation cells along the inner wall, and a thin but continuous cellular-detonation band along the outer wall. In contrast, the right recording does not show any transverse instability or local quenching after the Mach detonation is installed. Importantly, the width of the Mach detonation band is much larger than the thin continuous cellular-detonation band on the left recording. Similar observations can be made from the soot recordings in Figure 13. The left one shows again an unstable transmission at 15 kPa though the quenching domains have a more limited extension than at 12 kPa, an identical observation to those in [START_REF] Kudo | Oblique detonation waves stabilized in rectangular-cross-section bent tubes[END_REF][START_REF] Nakayama | Stable detonation wave propagation in rectangular-cross-section curved channel[END_REF][START_REF] Nakayama | Front shock behavior of stable curved detonation waves in rectangular-cross-section curved channel[END_REF], i.e., the smaller the detonation cell is (the larger the initial pressure), the easier the propagation of the detonation is.

The right recording shows the regular transmission that installs around angular position π rad. and does not exhibit any instability, in contrast to the left recording at the same position.

These comparisons support the idea that a curved chamber without center body, or one with a very small radius, is also suited to multicellular detonation propagation, since, after being installed, the Mach detonation propagates without transverse instabilities, and with a larger velocity than the CJ one. However, this Mach detonation only sweeps a limited part of the chamber cross section, while magnitude reactivity behind the diffracted shock that sweeps the other part appears to be low, so the actual combustion efficiency of this phenomenon requires further investigations.

Effect of an obstacle on the transmission modes

A series of experiments was carried out to investigate the effect of an obstacle positioned at the diffraction point of the inner wall on the two transmission modes described in the previous section. The obstacle was chosen with the same height as that of the chamber (20 mm), and its length from the channel wall was selected so as to partially obstruct the entry channel. Blockage was defined by the ratio BR of the obstructed section to the full section of the channel, and the two values BR = 1/4 and BR = 1/2 were considered. Therefore, in our conditions, the limiting initial pressure p lim depends on the diffraction conditions at the curved chamber entry.

Conclusions

This work reported on experiments on detonation re-initiation and behaviors in a curved chamber after transmission and diffraction from a straight channel. The primary observation is the existence of an initial-pressure range for which a detonation rotates with a constant angular velocity, a tangential velocity at the wall larger than the CJ value, and a constant front height. This detonation front is normal to the chamber wall, the detonation cells on this front are parallel to the wall and their mean width is constant and very small compared to those of the cells on a CJ-detonation front. The comparative analysis of soot and schlieren recordings show that, after some re-initiation transients resulting of diffraction, a triple-shock configuration forms at the outer wall, breaks away from it, and, depending on initial pressure, stabilizes its triple-point trajectory parallel to the outer wall. This phenomenon is proposed to be analog to the irregular, i.e., Mach, reflection of a shock on a wedge, with, here, the curved wall moving towards the triple point with the same relative speed as the triple point moves away from this wall.

The analysis indicates that stable rotation of detonation can be achieved in a hollow curved chamber with an outer diameter smaller than the minimum value necessary for stable rotation in a constant cross-section curved channel, according to the experimental criterion by Nakayama et al. and Kudo et al. [START_REF] Kudo | Oblique detonation waves stabilized in rectangular-cross-section bent tubes[END_REF][START_REF] Nakayama | Stable detonation wave propagation in rectangular-cross-section curved channel[END_REF][START_REF] Nakayama | Front shock behavior of stable curved detonation waves in rectangular-cross-section curved channel[END_REF]. These observations are also consistent with experimental results that RDE chambers without centerbody, or with linearly-increasing sections, e.g. [START_REF] Hansmetzger | A study of continuous rotation modes of detonation in an annular chamber with constant or increasing section[END_REF] and references therein, show improved properties of rotating detonation, compared to chambers with constant sections. This supports the interest of further studies on the hollow-chamber configuration of RDEs.

The upper value p lim of the initial-pressures range for observing this Mach detonation was found to be dependent on the diffraction condition. In this work, the smaller values of The quantity Φ is the tube diameter value which should be larger than about 10-13 λ so that re-ignition occurs without full quenching. Right: soot-foil recordings at p 0 = 15 kPa.

Figure 3

 3 Figures 5 and 6 show sequences of schlieren images of detonation transmission at p 0

Figure 7 .

 7 Figure 7. Importantly, they do not show any perturbation ahead of the Mach detonation,

Figure 9

 9 Figure 9 presents measurements of the detonation velocity D n normal to the front, at

  -h-i, 7 and 8-a-b thus show a typical 3-waves Mach configuration, with the diffracted curved shock, above the flat detonation, that represents the analog to the incident shock in the generic wedge configuration. The planar detonation then represents the analog to the Mach shock. The wedge is the curved wall, and its slope constantly increases relative to the wave-front normal, from the diffraction domain in Zone 2. After the transients in Zone 3, the speed at which the triple point moves away from the outer wall can then be equal to the relative speed at which the wall would move away from the reference line tangent to the wall at the current front position so that the Mach band has a constant width. Increasing initial pressure at fixed outer-wall radius is equivalent to decreasing the outer-wall radius at fixed initial pressure, and results in the transition from the irregular Mach transmission with the triple point away from the wall. Larger diameters or larger initial pressures are thus equivalent to smaller wedge angles, and favour the regular reflection mode, with

Figures 14 and 15

 15 Figures 14 and 15 compare soot recordings without and with obstacle which is indicated

Figure 14

 14 Figure14shows the transmission features without (left) and with (right) obstacle,

Figure 15 shows

 15 Figure15shows the effect of the blockage ratio BR on the transmission mode at

p

  lim were obtained with an obstacle positioned at the end of the channel, which generated a stronger expansion process at diffraction. No specific investigation was carried out to determine precisely the lower bound of the initial-pressures range for observing the Mach detonation. It is actually a difficult issue to assess precisely what effects the diffraction configuration have on the conditions and the properties of the Mach re-initiation, and this is the object of the current efforts. However, though this Mach detonation propagates at velocity larger than the CJ value, and the obtained pressures are significantly larger than the CJ value, the detonation front does not sweep the whole cross section of the chamber. More experimental work is thus necessary to obtain quantitative information on reactivity behind the curved diffracted shock and thus on the combustion efficiency of this specific regime. This cannot be obtained from the schlieren technique in this work. Soot recordings are capable to indicate the detonation domains from the recovery of the cellular patterns, but they are inefficient at giving information on slower combustion processes, such as deflagrations that may take place above the limiting line between the flows induced by the Mach detonation band and the diffracted shock. Chemiluminescence and PLIF should be considered for this investigation, which should also be supplemented by high-resolution numerical simulation such as[START_REF] Short | Propagation of a stable gaseous detonation in a circular arc configuration[END_REF]. Similarly, specific experiment should be carried out to obtain the maximum radius of the center body below which the Mach detonation regime can be obtained at the outer wall, in order to maximize the combustion efficiency. The four zones selected for describing the detonation transmission dynamics from the straight channel (Zone 1) to the curved chamber (Zones 2, 3 and 4). Initial pressure for this recording is p 0 = 12 kPa. . . . . . . . . . . 36 3 Soot recordings showing the dependence of the detonation transmission mode on initial pressure p 0 . Left: critical transmission (p 0 = 12 kPa). Right: Supercritical transmission (p 0 = 15 kPa). . . . . . . . . . . . . . . . 37 4 Analogy of the semi-confined curved-channel configuration with the supercritical transmission of a detonation from a straight channel to a large volume [32] (left). The quantity Φ is the tube diameter value which should be larger than about 10-13 λ so that re-ignition occurs without full quenching. Right: soot-foil recordings at p 0 = 15 kPa. . . . . . . . . . . . . . . . . . . 38 5 Sequence of schlieren images showing details of the irregular (Mach) detonation transmission mode (p 0 = 12 kPa). . . . . . . . . . . . . . . . . . . . 39 6 Sequence of schlieren images showing details of the regular detonation transmission mode (p 0 = 15 kPa). . . . . . . . . . . . . . . . . . . . . . . . 40 7 Triple-point configuration from schlieren visualization. (F): fresh gas, (B): burnt gas, (T): triple point, (MD): Mach detonation, (DS): diffracted shock, (DF): deflagration (p 0 = 12 kPa). . . . . . . . . . . . . . . . . . . . . . . . 41 8 Sequence of schlieren pictures comparing details of the Mach detonation (a, b) and the regular re-initiation (c, d). p 0 = 8, 12, 13, 15 kPa for (a), (b), (c), and (d), respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 42 Ratio D n /D CJ of the normal velocity of the Mach detonation at the wall to the CJ velocity versus the angular position θ (Zone 4) from the entry channel exit at p 0 = 8 kPa and 12 kPa. Symbol diameters represent error bars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 10 Normal velocity D n of the Mach detonation at three distances d from the outer wall versus the angular position θ (Zone 4) from the entry channel exit at p 0 = 12 kPa. Symbol diameters represent error bars. . . . . . . . . 44 11 Rotation frequency f of the Mach detonation and band thickness δ M ach versus the angular position θ from the entry channel exit (Zone 4) at p 0 = 8 kPa and 12 kPa. Symbol heights represent error bars. . . . . . . . . . . . 45 12 Soot recordings comparing the influence of the inner geometry on detonation transmission. Left: center body radius R i =50 mm, right: center body radius R i =30 mm (p 0 = 12 kPa). . . . . . . . . . . . . . . . . . . . . . . . 46 13 Soot recordings comparing the influence of the inner geometry on detonation transmission. Left: center body radius R i =50 mm, right: center body radius R i =30 mm (p 0 = 15 kPa). . . . . . . . . . . . . . . . . . . . . . . . 47 14 Soot recordings comparing the influence of the blockage ratio BR on detonation transmission. Left: BR = 0.5 (no obstacle), right: BR = 0.5 (p 0 = 8 kPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 15 Soot recordings comparing the influence of the blockage ratio BR on detonation transmission. Left: BR = 0, middle: BR = 0.25, right: BR = 0.5 (p 0 = 12 kPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 p 0 (kPa) D CJ (m/s) D exp (m/s) λ (mm)

p 0 ,

 0 initial pressure, D CJ and D exp , Chapman-Jouguet and measured velocities of the detonation, respectively, and λ measured cell size.

Figure 1 :

 1 Figure 1: Schematics of the experimental setup and chamber configuration.

Figure 2 :

 2 Figure 2: The four zones selected for describing the detonation transmission dynamics

Figure 3 :

 3 Figure 3: Soot recordings showing the dependence of the detonation transmission mode

Figure 4 :

 4 Figure 4: Analogy of the semi-confined curved-channel configuration with the supercritical

Figure 5 :

 5 Figure 5: Sequence of schlieren images showing details of the irregular (Mach) detonation

Figure 6 :

 6 Figure 6: Sequence of schlieren images showing details of the regular detonation trans-

Figure 7 :

 7 Figure 7: Triple-point configuration from schlieren visualization. (F): fresh gas, (B): burnt

Figure 8 :

 8 Figure 8: Sequence of schlieren pictures comparing details of the Mach detonation (a,

Figure 9 :

 9 Figure 9: Ratio D n /D CJ of the normal velocity of the Mach detonation at the wall to the

Figure 11 :

 11 Figure 11: Rotation frequency f of the Mach detonation and band thickness δ M ach versus

Figure 12 :

 12 Figure 12: Soot recordings comparing the influence of the inner geometry on detonation

Table 1 :

 1 Detonation properties of the stoichiometric C 3 H 8 + 5O 2 mixture at T 0 = 290K.

	8	2250	2232 ± 2	13 ± 3
	10	2259	2236 ± 2	9 ± 2
	12	2267	2240 ± 2 6.5 ± 2
	13	2271	2262 ± 2	6 ± 2
	15	2277	2262 ± 2 5.5 ± 1

Acknowledgement

This work is part of the CAPA research program on Alternative Combustion Modes for Air-breathing Propulsion supported by SAFRAN Tech, MBDA France and ANR (French National Research Agency).

List of Tables