Modes of Detonation Wave Propagation in Water Vapor Concentration Gradients

ABSTRACT This numerical study investigates different combustion modes when a Chapman-Jouguet detonation propagates into H O-diluted unburnt mixture through a composition gradient layer. A time-accurate and space-adaptive compressible reacting flow solver was used to perform transient detonation simulations of stoichiometric 50%H -50%CO/air mixtures with and without H O. Concentrations of the water vapor and thicknesses of the gradient layer were varied. From the simulations, three combustion modes were observed: (1) normal detonation propagation, (2) detonation mitigation and re-initiation, (3) detonation suppression. These three modes can be well explained by the theory of shock transmission and reflection in a density-varying medium and the reduction in chemical reactivity due to the weakening of the leading shock. A regime map for limits of each mode was established showing that the mode depends on and , denoting the normalized ignition delay time including shock reflection effect, and the ratio of the gradient layer thickness to the detonation induction length, respectively. A high value of with a low indicates the separation of the leading shock and the reaction front; thus, detonation suppression is more probable. These non-dimensional parameters can be further extended to the cases with gradient layers from other gases. In addition, the normalized reactivity gradient, , was used to understand the detonation re-initiation process after the mitigation of the initial detonation.


Introduction
Prevention and suppression of detonations are extremely important, as detonations with violent pressure waves frequently cause catastrophic human casualties and property damages.The fundamental process of detonation suppression has been studied actively (Dorofeev 2011).Several methods have been proposed to suppress detonations, and one of the effective suppression methods is using water sprays (Parra et al. 2004;Thomas et al. 1990;Ye et al. 2005).Thomas et al. (1990) experimentally investigated detonation quenching by measuring pressure in a vertical tube fitted with various spray generators which inject water at the top of the tube during detonation propagation.With a proper water spray, detonation quenching was observed.The spray droplet size and loading densities were suggested as the key parameters governing the extinction of detonation wave.A fieldscale pipe experiment was done by Ye et al. (2005).Water mist was used to suppress detonation, and the suppression effect of water mist was successfully proved in a methane/ air mixture.They found that the detonation suppression depended on the density and the size of water mist suspended inside the pipe.Parra et al. (2004) performed numerical simulations on the extinction of detonation wave with a water mist.They focused on droplet break-up effect of the introduced water content and confirmed detonation quenching.
As the addition of water introduces inhomogeneity to the fuel/air mixtures, many researchers have investigated the detonation mitigation process through a compositionvarying layer of unburnt mixtures, both experimentally and numerically.An experimental research with C 2 H 2 /O 2 mixture was performed by Thomas et al. (1991) to study detonation propagation along concentration gradients of several dilution gases, and they found that the properties of transmitted detonation were rapidly adjusted to values appropriate to the local gas mixture.They also observed secondary pressure peaks due to an explosion of the shocked but unreacted gas after the decoupling of the incident shock and reaction front.The experiments by Lieberman andShepherd (2007a, 2007b) investigated ethyleneoxygen detonation interactions with interfaces separating the two different gas mixtures.The effects of detonation wave refraction, Mach stem, and turbulent mixing zone were observed and analyzed.Later, the influences of H 2 concentration gradients on detonation propagation were experimentally investigated.Boeck et al. (2016) observed velocity deficits of detonation propagation in H 2 /air mixture by H 2 concentration gradients, and Grune et al. (2017) investigated the critical conditions for detonation propagations in both H 2 /air and H 2 /O 2 mixtures with H 2 concentration gradients.Kessler et al. (2012) and Ettner et al. (2013) numerically simulated detonation propagations along equivalence ratio gradients, and explained the effects of the gradient on the cell size of CH 4 /air detonation and the Mach reflection of H 2 /O 2 detonation, respectively.The numerical study by Reynaud, Virot, and Chinnayya (2017) examined the detonation cellular structure and velocity deficit in a composition-varying layer depending on various activation energies of fictive mixtures.Similar studies have been performed for the propagation of detonations through inert gases (Gavrilenko et al. 1982;Teodorczyk and Benoan 1996).Experimental studies of the inert gas effect on detonation propagations showed similar behaviors with concentration gradient studies.The influence of acoustic impedance of an inert gas was also simulated recently (Houim and Fievisohn 2017).
Many previous studies (Boeck et al. 2016;Ettner et al. 2013;Grune et al. 2017;Houim and Fievisohn 2017;Kessler et al. 2012;Lieberman and Shepherd 2007a;Reynaud et al. 2017) focused on detonations propagating perpendicularly to the direction of concentration gradient, while relatively little attention was given to the detonations propagating parallel to the gradient.Moreover, the above-mentioned studies on detonation mitigation mainly correlated the final states of the detonation propagation with the initial mixture conditions and did not investigate the detailed transient states during the mitigation process.The effect of composition-varying layer on chemical kinetics was not considered.The behaviors of detonation waves propagating into a H 2 O-diluted mixture can be better understood by examining the transient process of the interaction between gas dynamics and chemical kinetics.Therefore, in this paper, the numerical investigation focuses on the transient process of established detonations propagating parallel through vapor concentration gradients.Specifically, the following questions are to be answered: • How does an established Chapman-Jouguet (CJ) detonation respond when passing through a H 2 O gradient layer?• What are the underlying physics of various combustion modes resulting from the gradient layers?• Can these combustion modes be classified by some parameters?If so, what kind of information is needed for such a classification?• Can these parameters be further utilized generally in the gradient layers formed by other species?
To answer these questions, detonation simulations of stoichiometric 50%H 2 -50%CO/air mixtures with H 2 O gradient layers were performed.The theory of shock transmission and reflection in a density-varying medium was introduced to elaborate the decoupling process of leading shock and reaction front, i.e., detonation suppression.Two dimensionless parameters based on the initial conditions were proposed to determine possible combustion modes, while a transient parameter was proposed to predict the detonation re-initiation after the mitigation of the initial detonation.

Numerical simulation setup
Recent two-dimensional numerical analyses (Ettner et al. 2013;Houim and Fievisohn 2017;Kessler et al. 2012;Reynaud et al. 2017) focused on the multi-dimensional detonation structure by solving Euler equations, but these studies did not investigate the effects of gradient layers on the mixture reactivity.The main goal of this work is to understand the interactions between the leading shock and chemical kinetics at the local detonation fronts.Therefore, this study is conducted in a 1-D planar domain, as it is the simplest model allowing a thorough understanding of these local interactions in real detonations.Effects of heat and friction losses through the walls, and multi-dimensional effects such as cellular structure, turbulence, and stretching could be important; however, they are outside the scope of this paper.
Figure 1 shows the schematic setup of this numerical study.A fully developed CJ detonation was initialized near the left end of the domain and propagating to the right.The left end has a wall (reflective) boundary condition to sustain the detonation wave, and the right end has a transmissive boundary condition to attenuate reflected pressure waves.H 2 O was added in the unburnt mixture upstream of the detonation wave.The location at x = 0 is marked as the beginning of H 2 O-containing region of unburnt mixture.From x = 0 to d, the concentration of H 2 O linearly increases to a certain mole fraction X H 2 O .This region is referred to as the gradient layer.On the right side of the gradient layer (x !d), the mole fraction of H 2 O is uniform throughout the domain.The CJ detonation front was initially placed at x = −0.02cm.As the mass diffusion timescale under the simulated conditions is on the order of 10 ms, there is negligible H 2 O mass diffusion before the arrival of the incident CJ detonation.
Transient detonation simulations were carried out using a parallel version of the Adaptive Simulation of Unsteady Reacting Flow, ASURF-Parallel (Shi et al. 2016(Shi et al. , 2017a)).Message Passing Interface (MPI) was implemented to parallelize the code, and the code was successfully applied to a previous study of detonation initiation and propagation (Shi et al. 2017b).The original serial version of A-SURF (Chen 2009;Chen et al. 2009) is a time-accurate and space-adaptive solver for one-dimensional unsteady compressible multi-component reactive flows using finite volume method.Species, momentum, and energy conservation equations are solved.Specifically, the stiff chemistry term and nonreactive flow terms are integrated separately by a fractional-step procedure.An implicit VODE solver is used for the chemistry term, while a first-order Euler explicit method is used for non-reactive terms with the HLLC and a second-order central difference schemes for convection and diffusion terms, respectively.Chemical kinetics and species transport coefficient calculations are handled by CHEMKIN packages (Kee et al. 1996).In addition, a multi-level dynamically adaptive mesh refinement (AMR) method based on temperature gradient is used to resolve the detonation frontal structure.For steady-state Zel'dovich-Neumann-Döring (ZND) detonation (Döring 1943;von Neumann 1942;Zel'dovich 1940) calculations, the Caltech Shock & Detonation (SD) Toolbox (Browne et al. 2004;Kao and Shepherd 2004) was used.For zero-dimensional ignition delay calculations, SENKIN (Lutz et al. 1988) was used.
A stoichiometric 50%H 2 -50%CO/air mixture at initial temperature, T, of 1200 K and pressure, P, of 30 atm was simulated.The initial temperature and pressure conditions were chosen so that the simulations can be compared to previous studies (Dai et al., 2015;Gu et al. 2003;Shi et al. 2017b).When H 2 O was added, the mixture temperature and pressure remained the same.H 2 -CO syngas was chosen as the fuel due to its relatively wellunderstood chemical kinetics.A skeletal chemical kinetic model with 11 species and 21 reactions (Hawkes et al. 2007) was used in the current study.The whole domain length of the simulation was 10 cm.The smallest induction length of the ZND structure among conditions investigated in this study was on the order of 10 À5 m, and the minimum grid size in the area of detonation front was 6.25 Â 10 À7 m to resolve the detonation structure using a 5-level AMR method.The time step was set to 6.25 Â 10 À11 s accordingly to satisfy the acoustic Courant-Friedrichs-Lewy condition at all times.A grid convergence test with different levels of AMR is provided in the Supplemental Material showing negligible changes when the grid is further refined.An example of the simulation initial state is shown in Figure 2. Note that H 2 O is one of the product species of the CJ detonation, so there should have values of X H 2 O in the region x < 0, corresponding to H 2 O in the burnt gas.However, they are omitted for clarity in Figure 2 as the plot only intends to show the H 2 O dilution on the unburnt mixture side.
Multiple numerical simulations were performed, in which the amount of H 2 O on the unburnt side and the thickness of the gradient layer were varied.Five different mole fractions of H 2 O, X H 2 O = 0.1, 0.2, 0.3, 0.4, and 0.5 were used.The thicknesses of the gradient layer, d, were chosen from 10 À5 m to 10 À2 m, while the induction length of the ZND detonation structure is on the order of 10 À5 m for this particular syngas/air mixture and the imposed thermodynamic conditions.

ZND detonation calculations
CJ and von Neumann (VN) states were first obtained from calculated ZND detonations of stoichiometric 50%H 2 -50%CO/air mixtures with and without H 2 O using the SD Toolbox.The results are presented with the ZND induction lengths, l ind , in Table 1 as a function of X H 2 O , which increases from 0 to 0.5 with an increment of 0.1.The corresponding ignition delay times of mixtures at the VN states, τ VN , were also calculated using SENKIN.These results served as references to be compared with the transient A-SURF simulation results.For the incident detonation without H 2 O dilution, the initial propagation speed of the detonation front was , 1800 m/s, and the peak pressure was about 202 atm in the A-SURF simulation.In comparison, for the corresponding ZND detonation, the CJ speed, S CJ , was 1799 m/s and the VN pressure, P VN , was 204.9 atm, which were very close to the values from the A-SURF simulation.

Modes of detonation propagation through the gradient layer
When the incident detonation propagates through the gradient layer, three combustion modes are observed depending on X H 2 O and thickness of gradient layer: (1) normal detonation propagation, (2) detonation mitigation and re-initiation, and (3) detonation suppression.In this study, detonation mitigation refers to the intermediate state where a detonation is about to transition into a deflagration, while detonation suppression refers to the process where a detonation fully transitions into a deflagration.More quantitative comparisons are provided in the following sections.Temporal snapshots of pressure, temperature, and heat release rate, _ Q, profiles are shown to explain the physical processes leading to different combustion modes.Figure 3 presents the first mode: normal detonation propagation.The imposed mole fraction of H 2 O on the unburnt mixture side is 0.1, and the thickness of the gradient layer is 0.1 cm.The location of pressure peak is defined as the leading shock (LS), while the location of the maximum heat release rate is defined as the reaction front (RF).In this simulation case, the RF followed closely with LS during the whole time when the detonation passed through the layer, although LS peak pressure was slightly weakened.The peak pressure of the detonation wave after passing through the gradient layer was around 193 atm, matching well with the calculated VN values.For this case, the transient detonation can be treated "quasi-steadily", i.e., described by a series of ZND detonations based on local mixture compositions throughout the gradient layer.A similar phenomenon was observed experimentally by Thomas et al. (1991).
When the amount of H 2 O on the unburnt mixture side is increased, it is anticipated that the detonation wave would be suppressed.However, an intermediate combustion mode was observed.Figure 4 presents the simulation results of a case with X H 2 O = 0.3 and d = 0.1 cm.When the detonation front entered the gradient layer as denoted by the time sequence 2, the RF followed closely with LS. Between time sequences 2 and 4, the RF trailed behind the LS, and LS peak pressure was reduced due to the decoupling of the two fronts.The temperature profiles reveal two regions of sharp temperature changes: the right region containing a step increase is caused by the compression from the LS, while the left region of a relatively slow increase represents the heat release from the RF.This indicates that the detonation is mitigated and about to transition into a deflagration wave.However, starting from time sequence 5, another pressure peak was seen behind the LS.The pressure peak kept growing and eventually caught up with the LS at time sequence 7.The resulting peak pressure was even higher than the VN pressure of the local mixture.In the meantime, the reaction front followed closely with the LS again.This secondary pressure peak after the separation of LS and RF was also observed in the experiments by Thomas et al. (1991).After time sequence 8, the newly generated detonation approached its   corresponding CJ and VN states of X H 2 O = 0.3 mixture.The shock pressure was about 174 atm, which was close to its VN pressure as shown in Table 1.A movie file of this case is included in the Supplemental Material.
Figure 5 presents the results of a simulation case with X H 2 O = 0.5, and d = 0.1 cm.The phenomenon of detonation suppression was observed.The LS started to separate from the RF at , 0.48 µs (time sequence 2) and the peak pressure became weaker.After a while, these two wavefronts completely separated, and the speed of reaction front propagation approached the laminar flame speed.
Propagation of LS and RF from all three above cases are compared in Figure 6.The horizontal dotted lines represent the initial position of the gradient layer.In the case of X H 2 O = 0.1, the detonation front propagated at , 1760 m/s after the gradient layer, close to the corresponding CJ speed.In the case of X H 2 O = 0.3, the temporary separation of LS and RF can be clearly seen: the LS and RF propagated at , 1540 m/s and , 1170 m/s, respectively.Eventually, the re-initiated detonation propagated at , 1700 m/s, close to the CJ speed of X H 2 O = 0.3 mixture.In the third case, X H 2 O = 0.5, the detonation was suppressed, as the LS and RF fully separated.The reaction front propagated at , 970 m/s, and the leading shock speed was , 1480 m/s after the gradient layer.Note that the reaction front speed continued to decrease until it reached the laminar flame speed (after 20 µs), but it is not shown in Figure 6 as the timescale of this case is much larger than those of other cases.

Effect of water vapor dilution
As shown in the experimental study by Thomas et al. (1991), when a detonation entered a concentration gradient layer, the LS peak pressure and the detonation propagation speed tended to adjust to the corresponding VN pressure and CJ detonation speed, respectively, based on local mixture composition and conditions, since overdriven detonation state is unstable.In this numerical study, a similar trend was observed in the first combustion mode (X H 2 O = 0.1 and d = 0.1 cm), where the detonation can be "quasi-steadily" approximated by local CJ and VN states of the gradient layer.However, large departures from such a "quasi-steady" approximation were found in the other two combustion modes when the amount of H 2 O dilution was increased.In these two cases with X H 2 O = 0.3 and 0.5, the LS peak pressures became lower than the corresponding VN pressures when the detonation passed the gradient layer.The weakened LS led to lower post-shock temperature, thus longer ignition delay time of the unreacted mixture.Such a process promoted the separation of LS and RF as the delayed heat release at the RF can no longer support and couple with the LS.Eventually, the LS fully decoupled from the RF, i.e., the detonation was mitigated and suppressed.Therefore, the key to understand detonation mitigation and suppression is to identify the conditions when the leading shock is sufficiently weakened by the gradient layer.
Note that the dilution of H 2 O not only reduces the mixture reactivity but also lowers the total mixture density.According to acoustic theory, when an ideal shock propagates from one medium to another, part of the incident shock is reflected and the rest is transmitted.The properties of transmitted and reflected shocks are determined by the densities and sound speeds of the two media.Compared to mixtures without water vapor, the unburnt mixture with H 2 O has a lower density and a higher acoustic speed.When the detonation enters the H 2 O-containing region, some portion of the leading shock is reflected, and the resulting transmitted shock is thus weakened.Assuming a chemically frozen and linear shock response through an infinitely thin gradient layer, the percentage of the leading shock transmitting into the H 2 O-containing region can be estimated using the ratio of specific acoustic impedances across the interface (Kinsler et al. 2000).The specific acoustic impedance of medium j, I j , is defined as (1) where ρ j and a j are the density and the sound speed of the medium j, respectively.When the wave propagates from medium 1 to medium 2, the pressure transmission coefficient, c t , can then be calculated by where P 0 t and P 0 i are acoustic pressures of transmitted and incident waves, respectively, and z ¼ I 2 =I 1 is the ratio of the specific acoustic impedances of the two media.In the detonation simulations, the mixtures on both sides of the gradient layer are known thus I 1 and I 2 are readily available; z and c t can then be calculated.Taking the incident shock pressure, P 0 i , as the VN pressure of the incident detonation, P 0 t can be obtained by multiplying P 0 i and c t .Values of z, c t , and P 0 t are presented for several cases with different levels of X H 2 O in Table 2.Note that when P 0 t in Table 2 are lower than P VN in Table 1, the transmitted detonation is underdriven compared to the corresponding CJ detonation.Based on P 0 t and the corresponding mixture composition, the speed of transmitted shock, S t , can be calculated using the SD Toolbox.The ratio between S t and the corresponding CJ detonation speed, S CJ , was calculated and shown in Table 2.Moreover, the ignition delay time of the mixture at P 0 t state, τ t , was calculated and compared to the corresponding ignition delay at VN states, τ VN .The ratios between the two ignition delay values are also given in Table 2.
Based on the results presented in Table 2, the speed of the transmitted shock is very close to the CJ detonation speed, as S t =S CJ is approximately 1 even at X H 2 O = 0.5.In comparison, the ignition delay can be greatly increased, e.g., when X H 2 O = 0.5, the ignition delay under transmitted shock is , 56% longer than the corresponding VN ignition delay.If the shock speed remains the same but the ignition delay time is longer, the reaction front will lag behind the shock, possibly leading to the decoupling of the leading shock and reaction front.Therefore, the chemical effect caused by the transmitted shock is believed to be dominant in determining the consequent combustion mode after the interface.Hence, the ratio of τ t and τ VN may be considered as one parameter to quantify different combustion modes: (3) Note that the above analysis is solely based on the thermodynamic proprieties of mixtures without considering any transient simulation result.In order to further explore the effects of ζ, simulations with a thin gradient layer were performed and analyzed.The thickness of the gradient layer was set to 0.001 cm, which was the same order of magnitude as the detonation induction length of the undiluted mixture.Figure 7 shows these simulation results with different levels of X H 2 O .The amplitudes of the leading shock during the course of simulations are plotted versus distance.After the detonation passed through the thin layer, the pressure of the transmitted shock became lower than the VN pressures (represented by the dashed lines), similar to the estimation in Table 2.For X H 2 O = 0.1, the pressure of transmitted shock was slightly weaker than VN pressure after the thin layer.However, the LS pressure reduction did not cause the decoupling of shock and reaction front.The LS pressure rose and eventually approached the VN pressure of X H 2 O = 0.1 mixture.For X H 2 O = 0.4 and 0.5, the reductions in transmitted LS pressure were sufficient to cause the separation of the LS and RF. of initial mitigation followed by re-initiation.With more H 2 O dilution, the transmitted shock is weakened further, causing longer ignition delay and eventually detonation suppression.

Effect of gradient layer thickness
With the same amount of H 2 O concentration, the level of concentration gradient can be altered by changing the size of the gradient layer, which also affects the resulting combustion mode.The results shown in Figure 8 were obtained from the simulations with X H 2 O = 0.2 and different sizes of the gradient layer, d = 0.001 cm, 0.01 cm, 0.1 cm, and 1 cm.In the cases of d = 0.001 cm and 0.01 cm, the LS pressure rapidly dropped due to shock reflection, and the magnitude was below the VN pressure of the X H 2 O = 0.2 mixture.The reduced LS pressure resulted in an initial detonation mitigation followed by a re-initiation process.The mechanism of re-initiation will be discussed in the later section.If the size of the layer was increased to 1 cm, the LS pressure dropped gradually and adjusted to the corresponding VN pressure.
Since the LS transmission and reflection are seen to be affected by the size of the gradient layer, comparing the gradient layer thickness to the induction length of the CJ detonation can be relevant to quantify the LS behaviors.When the RF arrives at the gradient layer, the LS is already inside the layer, ahead of the RF by a distance of the induction length.Figure 9 shows the schematics of detonation propagation through the gradient layer with different levels of the gradient layer thickness: d , l ind , and d ) l ind .If the thickness of the gradient layer is the same order of magnitude as the induction length as shown in Figure 9 (a), LS will pass through the gradient layer as if it is infinitely thin.The LS strength can be calculated on the basis of acoustic impedance change across the thin gradient layer.The ignition delay time increases substantially due to the strong LS reflection, and RF tends to separate from LS in this case.
Once LS and RF decouple, the strength of LS is weakened and the speed of RF is decelerated repeatedly.In the other scenario, as shown in Figure 9 (b), the thickness of the layer is much larger than the induction length.As the vapor level at the LS is not the maximum X H 2 O , the weakening of the transmitted LS by reflection is rather gradual.The ignition delay time increase behind the LS is not enough to decelerate the speed of RF, so the LS strength can be still supported by the following RF.Therefore, the RF is more likely to keep up with the LS.Based on the above argument, the ratio between the gradient layer thickness and the detonation induction length, is proposed to describe the gradient of acoustic impedance thus identify different combustion modes.Note that l ind in Equation ( 4) is the ZND detonation induction length of the H 2 O-containing mixture.When η is large, the effect of LS reflection is negligible, and the detonation propagation approximately follows the local CJ detonation properties.Therefore, the combustion mode can be classified as a normal detonation propagation rather than a mitigation or a suppression.

Regime map for detonation propagation modes in concentration gradient
From the results presented above, the specific mode is seen to be influenced by both the normalized ignition delay time considering shock reflection effect, ζ, and the ratio of gradient layer thickness to the induction length, η.A regime map was developed by conducting 24 cases with various X H 2 O and d values.detonation suppression mode, in which LS reflection is able to decouple RF and LS.The middle region is the re-initiation mode, in which the detonation wave is temporarily mitigated, but another detonation is initiated behind the leading shock.The bottom-right region corresponds to the normal detonation propagation mode, in which the detonation can be "quasi-steadily" approximated by the corresponding local CJ states.The result regime map reveals that both the H 2 O concentration and the size of the gradient layer play an important role in detonation suppression.
As the above analysis is based on the relative parameter values to those without shock reflection, the two non-dimensional factors, ζ and η, are not limited to H 2 O concentration gradients.Therefore, gradients of other dilution gases such as He or N 2 also can be presented in the same regime map.Several cases with gradients of He and N 2 were simulated and also presented in Figure 10.The plus sign markers represent the cases containing 0.1 cm helium gradient layers with X He = 0.1, 0.3, and 0.5.When X He = 0.1 and 0.3, the mode of detonation propagation followed the normal detonation propagation (red plus sign markers).If X He increases to 0.5, the mode becomes re-initiation (pink plus sign marker).Although the ratio of shock reflection is larger than that in the case with H 2 O gradient due to the small molecular mass of helium, the effect of shock reflection on the ignition delay time is weaker, and the value of ζ is smaller than those in the H 2 O case.From the calculation using the same analysis, the value of ζ is 1.45 for the case with X He = 0.5, while ζ = 1.56 for the X H 2 O = 0.5 case as shown in Table 2. Nitrogen dilution cases are also presented in the regime map using cross markers.The mole fractions of N 2 were 0.1, 0.3, and 0.5, and the gradient layer thickness was 0.1 cm.Note that X N 2 here does not include the amount of N 2 in the air.As the molecular mass of nitrogen is relatively larger than the average value of other gases in the unburnt mixture, the acoustic impedance increases when the mixture contains more nitrogen.In these cases, the shock reflection decreases ignition delay time due to the stronger transmitted leading shock.Therefore, the values of ζ are smaller than 1 as shown in Figure 10, and the modes of detonation propagation were the normal propagation for all N 2 concentration gradient layers (red cross markers).The above results demonstrate that both He and N 2 cases can be understood by using the same parameters as the non-dimensional analysis and regime map can be utilized for general cases.

Detonation re-initiation after mitigation
The combustion mode of mitigation and re-initiation was observed between the detonation suppression and the normal detonation propagation modes as shown in the regime map (Figure 10).After the initial mitigation, the re-initiation process is similar to the shock initiated detonations, which were actively studied by several researchers (Melguizo-Gavilanes and Bauwens 2013;Melguizo-Gavilanes et al. 2010;Sharpe and Short 2007).The detailed re-initiation process can be further understood by considering the normalized reactivity gradient (Gu, Emerson, Bradley 2003).Zel'dovich and his colleagues proposed a theory that relating reaction front propagation modes to spatial auto-ignition sequence, i.e. the reactivity gradient (Zel'dovich 1980;Zel'dovich et al. 1970).Gu et al. (2003) applied this theory to the hot spot induced deflagration to detonation transition (DDT).The occurrence of different detonation initiation modes was strongly correlated to the value of normalized reactivity gradient of the hot spot, , which compared acoustic wave velocity with ignition front propagation velocity using a spatial distribution of autoignition delay times.Similarly, this theory can be used to analyze the current re-initiation process.The region between the LS and the RF may be considered as a hot spot.The ignition delay times and sound speeds based on the properties between LS and RF can be correlated with the occurrence of the detonation re-initiation.First, the ignition front propagation velocity relative to the unburnt gas, u, is defined as: where τ is the ignition delay time, and x is the spatial location.Then, one can define the ratio of acoustic velocity to the ignition front propagation velocity as the normalized reactivity gradient: where a is the sound speed of the mixture.Both a and τ in the induction zone depend on temperature, pressure, and mixture composition.The temporal and spatial profiles of τ, u, and a for the case of X H 2 O = 0.3 with d = 0.1 cm (Figure 4) are shown in Figure 11.These variables were calculated using the corresponding local thermodynamic states from the simulation.For each time sequence, only the profiles between LS and RF are shown.As the temperature is high near the RF region, τ is much smaller near the left end of the region presented.Since the relative difference of maximum and minimum acoustic wave velocities is much smaller than that of ignition front propagation velocities, mainly depends on the spatial gradient of autoignition delay times.
If the value of is much larger or smaller than unity, the acoustic wave and the reaction front wave cannot couple; thus, the detonation initiation does not happen.If approaches unity, a detonation wave is likely to be initiated.For the same case shown in Figure 11, profiles of pressure, temperature, and over the region between LS and RF are shown in Figure 12.Note that the dashed lines in the figure provide only rough upper and lower bounds for the unity order of magnitude.At time sequence 3 (1.23 µs), the initial detonation was mitigated, and was on the order of unity so a detonation was expected to develop.The reaction and acoustic waves resonated and formed a new detonation at time sequence 6 (3.73 µs).The average value of between the reaction front and the leading shock front can be used as an indicator of the re-initiation process, and the values were 2.5, 1.5, 1.1, and 1.1 at time sequences 3-6, respectively.As is close to one, there exists a possibility of a new detonation initiation after the mitigation of initial detonation wave.
In the case of X H 2 O = 0.5 with d = 0.1 cm (Figure 5), profiles of pressure, temperature, and are presented in Figure 13 to illustrate the suppression process without re-initiation.The average values of between RF and LS were 6.8 and 7.0 at time sequences 3 and 4, respectively.As such, the reactivity gradient failed to initiate a new detonation.Again, the minimum amount of water vapor to have a detonation suppression depends on the gradient layer thickness as shown in the regime map (Figure 10).
The cases with ( 1 do not exist during the detonation mitigation process, as these are in the region of normal detonation mode.If ( 1 in the area between LS and RF, the reaction front propagation speed is fast enough to catch up with the leading shock.Hence, the reaction front attaches to the leading shock, which is basically a normal detonation propagation.

Conclusions
Numerical simulations of stoichiometric 50%H 2 -50%CO/air mixtures were performed to investigate the combustion modes when a CJ detonation propagates into H 2 O-diluted mixture through a composition gradient layer.The results were analyzed by considering the effect of shock reflection on chemical reactivity and the shock reflection effectiveness using non-dimensional variables.The following conclusions are drawn: • Three combustion modes were observed in the numerical simulations: normal detonation propagation, detonation mitigation and re-initiation, and detonation suppression.The H 2 O concentration and gradient layer thickness are believed to be the two main factors influencing the resulting modes, and can be expressed in terms of two non-dimensional variables, ζ and η. ζ is the normalized ignition delay time including shock reflection effect, and η is the ratio of the H 2 O gradient layer thickness to the induction length of the ZND detonation.• The shock transmission and reflection across a thin gradient layer were analyzed theoretically using the acoustic impedance.The weakening of the transmitted shock due to reflection was observed in transient simulation results.When the thickness of the gradient layer is relatively large compared to the detonation induction length, the impact of LS reflection is reduced thus the RF is able to readjust itself and keep up with the LS, resulting in the normal detonation propagation.• A regime map for detonation propagation modes through a H 2 O concentration gradient was established in terms of ζ and η.The incident detonation is more likely to be suppressed with larger ζ and smaller η, representing more reduction in chemical reactivity by the weakening of the LS.As the non-dimensional analysis is not dependent on the choice of diluent, this regime map can be further utilized for gradient layers from other gases such as He or N 2 .• The ratio of the acoustic wave velocity to the ignition front propagation velocity, , was used as a normalized reactivity gradient for a better understanding of detonation reinitiation behind the mitigated leading shock.Temporal and spatial reactivity gradients were estimated from autoignition delay times using the Zel'dovich theory.If between RF and LS is on the order of unity, a new detonation is expected to develop.
This study provides a detailed description of the physical mechanism and analysis tools for detonation propagation in water vapor concentration gradients.For practical applications, liquid water droplets would be used, so the interaction between the detonation wave and droplets should be further investigated.As temperature gradients in the unburnt mixture due to the vaporization are expected, the combined effect of temperature and vapor concentration gradients should be considered.

Figure 1 .
Figure 1.Schematic setup of detonation propagation through a H 2 O gradient layer in 1-D planar domain.

Figure 2 .
Figure 2.An example of H 2 O mole fraction in unburnt side, pressure, and temperature profiles at the simulation initial state.

Figure 6 .
Figure 6.Computed temporal locations of leading shock front and reaction front for detonation propagations through 0.1 cm gradient layer with X H 2 O = 0.1 (top), X H 2 O = 0.3 (middle), and X H 2 O = 0.5 (bottom).

Figure 8 .
Figure8.Amplitudes and positions of leading shock fronts through gradient layers with X H2O = 0.2 and different layer thicknesses, d = 0.001 cm, 0.01 cm, 0.1 cm, and 1 cm.Dashed lines are VN spike pressures of uniform X H2O = 0 and 0.2 mixtures.
Figure 10 presents the results of these simulations in terms of the two non-dimensional factors, ζ and η.In the figure, circular markers present detonation suppression, triangle markers indicate detonation mitigation and re-initiation, and square markers represent normal detonation propagation.The top-left region corresponds to the

Figure 9 .
Figure 9. Schematics of leading shock and reaction front passing vapor gradient layers with same amount of water concentration.

Figure 10 .
Figure10.Limits of detonation propagation modes through water vapor concentration gradient, in terms of normalized ignition delay time including shock reflection effect, ζ, and ratio of gradient layer thickness to the induction length, η.

Table 1 .
Calculated CJ speeds, VN states, ignition delay times at VN states, and ZND induction lengths of homogeneous H 2 -CO/air/H 2 O mixtures using SD toolbox and SENKIN.

Table 2 .
Estimated properties of transmitted shock to H 2 -CO/air/H 2 O mixtures from H 2 -CO/air mixture.
The cases of X H 2 O = 0.2 and 0.3 show the mode Figure 7. Amplitudes and positions of leading shock fronts through thin gradient layers with X H2O = 0.1, 0.2, 0.3, 0.4, and 0.5.Dashed lines are corresponding VN pressures (Black-dashed line is P VN without H 2 O).
Ignition delay time of mixture under transmitted shock τ VN Ignition delay time of mixture at VN state ξ Reactivity gradient dimensionless number ζ Normalized ignition delay time including shock reflection effect ORCIDJe Ir Ryu http://orcid.org/0000-0002-1098-9313