Wei-jie FAN ,Wei-dong LIU? ,Hao-yang PENG? ,Shi-jie LIU ,Jian SUN

1Science and Technology on Scramjet Laboratory,College of Aerospace Science and Technology,National University of Defense Technology,Changsha 410073,China

2State Key Laboratory of Aerospace Dynamic,Satellite Control Center,Xi’an 710043,China

Abstract:To investigate the impact of combustor width on continuous rotating detonation (CRD) fueled by ethylene and air,a series of 3D simulations are conducted by changing the inner cylinder radius of an annular combustor while retaining the same outer cylinder radius.The results show that the CRD wave propagates more steadily and faster as the combustor width increases.The high-temperature zone at the backward-facing step preheats the propellants and contributes to the steady propagation of the CRD wave in 25-and 30-mm wide combustors.The highest and the lowest velocities are obtained in the 30-and 15-mm wide combustors at,respectively,1880.27 and 1681.01 m/s.On the other hand,the average thrust decreases as the combustor width increases.The highest thrust is obtained in the 15-mm wide combustor while the lowest is in the 30-mm wide combustor,at 758.06 and 525.93 N,respectively.Nevertheless,the thrust is much more stable in the 25-and 30-mm wide combustors than in the 15-and 20-mm wide combustors.

Key words:Continuous rotating detonation(CRD);Ethylene-air;Combustor width;Propagation mode;Propulsive performance

1 Introduction

At present,the engines of aerospace propulsion systems usually work in the deflagration combustion mode.After much research-led development,the pro‐pulsive performance of these systems has almost reached its maximum theoretical value.Therefore,a new propulsion system using a different combustion mode has become an important issue.In the past decade,much research has centered on another com‐bustion mode,detonation,because of its high heat release rate and thermodynamic efficiency(Bykovskii and Vedernikov,1996;Stewart and Kasimov,2006).In general,detonation-based engines in aerospace engi‐neering applications can be divided into various cate‐gories including the pulse detonation engine (PDE)(Nikitin et al.,2009),the oblique detonation engine(ODE) (Fang et al.,2017),and the continuous rotat‐ing detonation engine (CRDE) (Lin et al.,2015;Sun et al.,2018a).Because CRDE can work in a broad range of flight conditions and produce a reasonably stable thrust with a compact engine structure,it has received a lot of attention in the field of propulsion research.

In the past two decades,extensive investigations on the working processes of CRDE have been con‐ducted in annular combustors,such as the initiation(Bykovskii et al.,2007;Liu et al.,2013;Yang et al.,2016),the propagation modes (Lin et al.,2015;Liu et al.,2015;Sun et al.,2019),and the propulsive per‐formance(Yi et al.,2011;Tang et al.,2015;Yao et al.,2017;Sun et al.,2018b).However,most of these investigations focused on a hydrogen CRDE due to the high chemical activity and detonation ability of hydrogen.However,considering both practicability and safety in engineering applications,hydrocarbon fuels,especially kerosene,may be superior to hydro‐gen for CRDE.At present,it is still hard to obtain stable liquid kerosene-air continuous rotating detona‐tion(CRD)in the annular combustor owing to its low chemical activity and detonation ability (Kindracki,2015).Even with the addition of hydrogen and oxy‐gen,the performance of liquid kerosene CRD is still quite poor.Bykovskii et al (2006a) experimentally in‐vestigated liquid kerosene-air CRD in an annular com‐bustor with a 16.5-mm wide channel.The results showed that the CRD wave was hardly detonated un‐less the mass fraction of oxygen in the oxidant was higher than 50%.Le Naour et al.(2017)tried to estab‐lish a liquid kerosene detonation regime in an annular combustor with a 25-mm wide channel.Liquid kero‐sene CRD was successfully established due to the addition of hydrogen.However,the CRD wave propa‐gated at a low velocity of about 1000 m/s and the ve‐locity deficit was quite large.As the fuel composition changes from hydrogen to liquid kerosene,the chemi‐cal activity and detonation ability decrease significant‐ly.The transition investigations of CRD fueled by hy‐drocarbons with moderate chemical activity help to a better understanding of liquid kerosene CRD and ac‐celerate the application of CRDE in engineering.

Ethylene is one of the main components of ker‐osene pyrolysis products,with a higher chemical activ‐ity than kerosene but lower than hydrogen.In recent years,studies on ethylene CRD have been carried out in annular combustors (George et al.,2015;Cho et al.,2016;Wilhite et al.,2016;Andrus et al.,2017;Peng et al.,2019b).George et al.(2015)experimen‐tally investigated ethylene-air CRD in an annular com‐bustor and observed the single-wave propagation mode of the CRD wave.However,the highest propa‐gation velocity of the CRD wave was only 850 m/s with severe propagation instability.Wilhite et al.(2016) carried out a set of ethylene-air CRD experi‐ments in a 13.1-mm wide annular combustor and found the CRD wave could only be achieved with a limited range of equivalence ratios and mass flow rates.Cho et al.(2016) conducted ethylene-air CRD experiments in an annular combustor with a 7.6-mm wide channel and observed detonation waves propa‐gated in a weak counter-rotating mode with an aver‐age velocity of (994±43) m/s.Andrus et al.(2017)experimentally studied the mixing effects on ethyleneair CRD in an annular combustor with a 23-mm wide gap.The CRD wave propagated at a velocity near the speed of sound and suffered a great velocity deficit even under the premixed supply condition.Peng et al.(2019b) performed a set of ethylene-air CRD experi‐ments with the extra addition of hydrogen and oxygen in the racetrack-like combustor.The realized CRD wave propagated in a counter-rotating mode with a great velocity deficit.All in all,obtaining stable ethylene-air CRD in the annular combustor with a low velocity deficit is still challenging and needs fur‐ther study.

To enhance the propagation performance of the ethylene-air CRD,new combustor configurations of CRDE are proposed,including the hollow combustor and the cavity-based annular combustor.Anand et al.(2018) investigated the ethylene-air CRD mechanics in a hollow combustor by using two different injection schemes.During stable propagation,the velocity of the CRD wave could reach about 95% of the Chapman-Jouguet (C-J) velocity.Peng et al.(2018) obtained a stable single-wave mode of ethylene-air detonation wave in the hollow combustor and the detonation wave could rotate with a velocity as high as 1915.40 m/s.Wang et al.(2018) also experimentally proved that the ethylene-air CRD was feasible in such a combus‐tor.There was only one CRD wave rotating in the combustor under the operating conditions and the wave generally propagated at above 80% of the C-J velocity.Compared with the annular combustor,a more stable ethylene-air CRD was readily achieved in the hollow combustor.However,some deficiencies in the propulsive performance of the hollow combustor were found,which mainly resulted from the divergent flow (Tang et al.,2015) and from imperfect detona‐tion combustion (Kawasaki et al.,2019).The cavitybased annular combustor is a novel kind of CRDE scheme put forward by Peng’s team (Peng et al.,2019a,2021;Liu et al.,2020).By installing a cavity at the inner cylinder of the annular combustor,the width of the combustor increased and the combustion organization was improved.As a result,the operating range of the ethylene-air CRD was increased and the propagation velocity of the detonation wave increased to some extent.However,the velocity deficit was still in the range of 30%to 40%of the C-J velocity,which was high.The combustor configuration has great ef‐fects on the operation of the CRD and the require‐ments for stable propagation and the optimization of efficient propulsive performance deserve further study.

Only a few simulations on ethylene detonation were carried out because of its low chemical activity and the great difficulty in initiation (Khokhlov et al.,2004;Yungster and Radhakrishnan,2005;Gottiparthi et al.,2009;Schwer and Kailasanath,2013;Fujii et al.,2017).The cellular detonation structures were ana‐lyzed using 1D (Yungster and Radhakrishnan,2005)and 2D (Khokhlov et al.,2004;Gottiparthi et al.,2009)simulations.Ignoring the impacts of the com‐bustor width and using 2D simulations,Schwer and Kailasanath (2013) analyzed the flow field and the specific impulse of a CRDE fueled by the ethylene-air mixture.Fujii et al.(2017)investigated the velocity of the ethylene-oxygen CRD wave by 2D simulations.Some information on the flow field and propagation characteristics of ethylene-air CRD was obtained.Fan et al.(2021)studied the propagation characteristics of ethylene-air CRD under different injection conditions by 2D simulations.The transition process of the deto‐nation wave propagation mode was analyzed in detail.However,the combustor width was found to be key for realization of hydrocarbon CRD (Bykovskii et al.,2006b;Kawasaki et al.,2019).The propagation char‐acteristics of the ethylene-air CRD wave and the pro‐pulsive performance of the CRDE with an enlarged annular combustor are still unclear.Thus,numerical investigation on the ethylene-air CRD with variation of the combustor width can contribute to pinpointing a self-sustaining mechanism.

In this study,the impact of the combustor width on the ethylene-air CRD is studied numerically by using 3D simulations.The propagation characteristics of the CRD wave and the propulsive performance of the CRDE are discussed.In addition,the selfsustaining mechanism of the CRD wave is analyzed.This study will deepen understanding of the ethyleneair CRD and give some guidance for the design of the combustor in a CRDE fed by hydrocarbon fuels.

2 Physical model and numerical method

2.1 Physical model

The cross-sectional schematic of the CRDE com‐bustor under the premixed injection condition is shown in Fig.1.A stoichiometric ethylene-air mixture is injected axially into the combustor through an annu‐lar slot.The throat width of the slot and the length of the combustor are 1.2 and 100 mm,respectively.The outer radius of the combustor is kept constant as 65 mm,while the inner radius of the cylinder is low‐ered from 50 to 35 mm at intervals of 5 mm.Thus,the combustor width (

W

) varies from 15 to 30 mm at intervals of 5 mm.In addition,a pressure monitor point named

C

is placed in the combustor with coor‐dinates(5 mm,57.5 mm,0 mm)to record the pressure profile in the combustor caused by the CRD wave.

Fig.1 Cross-sectional schematic of the CRDE combustor.Rin is the inner radius of the cylindrical combustor

2.2 Numerical method

All cases in this investigation are solved by the commercial software ANSYS FLUENT,which is widely applied in the CRD research field (Sun et al.,2018a,2018b,2019).A stoichiometric ethylene-air mixture is injected at the inlet and the ideal gas law is used to calculate the density.Three-dimensional Reynolds-averaged Navier-Stokes (RANS) control‐ling equations are solved by the transient implicit density-based solver.The inviscid flux vector is evalu‐ated by Roe’s flux-difference splitting (Roe-FDS)method.For spatial discretization,the cell-based least squares method is used to compute the gradient,and the second-order upwind scheme is applied for flow,turbulent kinetic energy,and specific dissipation rate.The second-order implicit scheme is used for time stepping.The time step is fixed as 0.1 μs and the instantaneous pressure at the monitor point is sam‐pled at each time step to reconstruct the pressure record curves.The shear-stress transport (SST)

k-ω

turbulence model is employed to simulate the turbu‐lent flow.The laminar finite rate reaction model and the reaction rate constants are determined by the Arrhenius formulation,i.e.,

where

k

is the reaction rate constant,

A

is the preexponential factor,

T

is the temperature,

R

is the gas constant,

E

is the activation energy,and

b

is the temperature exponent.A reduced chemical mecha‐nism(Baurle et al.,1998)treating three reactions with seven reacting species is applied,including nitrogen(N) as the seventh species.The leading edge of the backward-facing step is located at

x

=0 mm.To avoid flashback,the chemical reaction is artificially shut down in the upstream (zone of

x

<0 mm) and only occurs in the combustor(zone of

x

≥0 mm).Such reac‐tion restriction has been generally adopted in simula‐tions under premixed conditions (Schwer and Kailas‐anath,2012;Sun et al.,2017).To further verify the chemical mechanism,a 1D detonation study using a grid size of 0.5 mm is conducted here.The domain is 2000-mm long and is filled with a stoichiometric ethylene-air mixture.The static pressure and static temperature of the combustible mixture are respec‐tively 100 kPa and 300 K.Twenty monitor points at intervals of 100 mm are set along the line (

x

axis) to record the static pressure during the propagation process of the detonation wave.The pressure record curves and propagation velocities of the detonation wave are shown in Fig.2.Based on the stable propa‐gation process between 950 and 1950 mm,the aver‐age detonation propagation velocity is 1863 m/s with a 2.17% relative difference from the theoretical C-J velocity.This velocity difference is acceptable,and thus the reduced chemical mechanism is deemed reliable.

Fig.2 Pressure record curves and propagation velocities of the 1D detonation

The mass flow inlet boundary condition is adopted at the mixture inlet.The total mass flow rate and the total temperature of the stoichiometric ethylene-air mixture are 800.31 g/s and 300 K,respectively.The pressure outlet condition boundary is used at the exit of the combustor and the backpressure is set to 100 kPa.All the walls are non-slip and adiabatic.

The computational domain is meshed with hexa‐hedron cells using the commercial software ICEM CFD.When the research focuses on the main propa‐gation characteristics and propulsive performance rather than the detailed structure of the CRD wave,grids with a size of 0.5 mm were widely used and reli‐able results could be obtained (Tang et al.,2015;Sun et al.,2018b).In this study,grids with an average axial size of 0.25 mm are used in the domain of 0≤

x

≤30 mm while grids with an average axial size of 0.50 mm are adopted in the remaining computational domain.The average size of grids is 0.50 mm in the radial and cir‐cumferential directions.To further confirm grid inde‐pendence,some 2D simulations are carried out.The rectangular domain is 300-mm long and 100-mm wide and it is discretized by grids with the accuracy of 0.5,0.25,and 0.125 mm correspondingly.A stoi‐chiometric ethylene-air mixture is injected from the bottom boundary with a total pressure of 500 kPa and a total temperature of 300 K.The detonation products are discharged from the top boundary with a backpres‐sure of 100 kPa.The two side borders are set as peri‐odic boundaries which ensure that the detonation wave propagates continuously.As Fig.3 displays,similar flow field structures are well captured by all three grids including the CRD wave,oblique shock wave,slip line,triple point,and fresh mixture layer.The results show that grids of the size of 0.5 mm are fine enough to obtain the basic flow field characteris‐tics of a CRD wave.The grids used in this study are thought to obtain reliable results for ethylene-air CRD.

Fig.3 Temperature contours of three grid sizes:(a)0.50 mm;(b)0.25 mm;(c)0.125 mm

The initiation setup is demonstrated in Fig.4,in which the black solid line and green solid line respec‐tively indicate the locations of

x

=0 mm and

x

=10 mm.The preset flow field can be divided into three zones including combustible mixture zone (1) (marked in red),ignition zone (2) (marked in green),and down‐stream zone (3) (marked in blue).Zone (1) and zone(3) are both cylindrical in area and their axial coordi‐nate ranges are respectively ?18 mm≤

x

≤10 mm and 10 mm<

x

≤100 mm.Zone(2)is a fan-shaped area and its coordinates are 0≤

x

≤10 mm,?5 mm≤

y

≤5 mm,and

z

ranging from the corresponding inner wall to the outer wall.The detonation is expected to be ignited in zone (2) with high pressure (2000 kPa) and high tan‐gential speed (2000 m/s).More detailed information about the preset flow field is listed in Table 1.The ignition zone with high pressure and tangential veloc‐ity is expected to ignite the combustible mixture and form the CRD wave.

Fig.4 Preset flow field in the initiation setup

3 Results and discussion

The stoichiometric ethylene-air CRD in combus‐tors of different widths is simulated at a constant total mass flow rate of 800.31 g/s.The CRD wave propaga‐tion process shows great differences as the combustor width varies.In addition,the characteristics of the flow field are revealed by analyzing contours on axial slices and radial-azimuthal slices.Finally,the propul‐sive performances of different combustors are also quantitatively studied.

3.1 Effects of combustor width on the CRD wave propagation process

The propagation processes of the CRD wave in different combustors are analyzed in this section.When the CRD wave propagates circumferentially passing by the monitor point,the high pressure result‐ing from the CRD wave will be captured.The pres‐sure curves obtained by the monitor point

C

are shown in Fig.5.The propagation processes of the CRD wave clearly vary with the different combustor widths.

Fig.5 Pressure record curves of monitor point C1 in different combustors:(a)W=15 mm;(b)W=20 mm;(c)W=25 mm;(d)W=30 mm

When the combustor width is 15 mm,the propa‐gation process of the CRD wave is quite unstable and the propagation mode changes many times over time,as shown in Fig.5a.The propagation mode can be classified as either a single-wave mode,an unsteady multiple-wave mode,a co-rotating two-wave mode,or a co-rotating three-wave mode.In addition,the amplitude of the pressure peak varies greatly,which means the intensity of the CRD waves changes vio‐lently.To show the evolution of the CRD wave in the annular combustor more intuitively,cylindrical surfac‐es with a radius of 57.5 mm are unrolled into 2D sur‐faces colored by temperature contours as shown in Fig.6.It can be seen that the preset ignition zone spreads unidirectionally along the tangential velocity direction,and then the combustible mixture is strongly compressed and ignited.At the time of 0.50 ms,the CRD wave is first established and it propagates in the single-wave mode,i.e.,there is only one CRD wave propagating circumferentially in the combustor.How‐ever,this CRD wave cannot maintain a single-wave mode for long as the hot combustion product heats the fresh combustible mixture and induces a new detonation wave.As illustrated in Fig.7,the hotspot behind the CRD wave induces two new CRD waves that propagate in opposite directions at the time of 1.00 ms.At the time of 1.04 ms,the induced CRD wave collides with the previous one resulting in a high-temperature and high-pressure zone.Then,the CRD waves decay into transmitted shock waves at the time of 1.06 ms.When there is enough fresh com‐bustible mixture in front of the transmitted shock wave and the intensity of the shock wave is high enough to ignite the combustible mixture,the shock wave can re-evolve into a CRD wave,otherwise it will get extinguished as shown at the time of 1.10 ms.Due to the generation of a new CRD wave by the hot‐spot and the extinction of the CRD wave by the colli‐sion,there are multiple CRD waves propagating in an unsteady multiple-wave mode.This mode can be re‐garded as an intermediate mode of transition.After the adjustment in the unsteady multiple-wave mode,a co-rotating two-wave mode is observed in Fig.8,i.e.,two CRD waves propagate in the same direction in the combustor.Before long,the co-rotating two-wave mode may transform into a co-rotating three-wave mode as the hot combustion products induce another new CRD wave.As Fig.9 shows,the circumferential distance between the CRD waves on the left side is not long enough to establish a stable combustible mix‐ture layer,which is mainly attributed to the high pres‐sure behind the CRD waves blocking the injection of the fresh mixture.Soon afterwards,the induced CRD wave gets extinguished.For the same reason that the continuous propagation of the detonation wave is greatly affected by the refilled fresh mixture,the deto‐nation wave propagates again in single-wave mode at the time of about 7.00 ms.However,the propagation mode of the CRD wave may change again in the sub‐sequent propagation process.Above all,the propaga‐tion process of the CRD wave in the 15-mm wide combustor is unsteady.

Fig.6 Single-wave propagation mode after ignition(W=15 mm)

Fig.7 Unsteady multiple-wave propagation mode(W=15 mm)

Fig.8 Co-rotating two-wave mode(W=15 mm)

Fig.9 Co-rotating three-wave mode(W=15 mm)

In the 20-mm wide combustor,the CRD wave still cannot sustainably maintain a stable propagation mode.Five kinds of propagation modes are observed including a counter-rotating two-wave mode,an un‐steady multiple-wave mode,a single-wave mode,a co-rotating two-wave mode,and a co-rotating threewave mode.Firstly,the ignition zone spreads to both sides and two counter-rotating CRD waves are formed as shown in Fig.10.After the collision of the CRD waves,the propagation mode transforms into an unsteady multiple-wave mode.Affected by these unstable factors,including the unstable injection state of the premixed mixture and the induced combustion of the detonation product,the extinction and genera‐tion of the CRD wave occur from time to time.This causes frequent changes in the CRD wave propaga‐tion mode in the 20-mm wide combustor.

Fig.10 Counter-rotating two-wave mode(W=20 mm)

The instantaneous propagation modes of the CRD waves in the 25-and 30-mm wide combustors are carefully identified by an analysis both of the pressure record curves in Figs.5c and 5d and of the unrolled 2D temperature contours as in Fig.7.In the 25-mm wide combustor,the CRD waves propagate in counterrotating two-wave mode after ignition.However,the mode changes into a co-rotating two-wave mode after the collision of the CRD waves.The extinction and reignition of the CRD waves are shown in Fig.11.Due to the interaction between the CRD wave propa‐gation and the injection of fresh mixture,the deto‐nation waves get extinguished for lack of fresh mix‐ture in front of the waves at the time of about 1.64 ms.After the combustible mixture layer is re-established and balanced,the CRD waves are reignited and finally propagate steadily in a co-rotating two-wave mode.For

W

=30 mm,the CRD waves propagate in co-rotating two-wave mode steadily and sustainably after a short adjustment in counter-rotating two-wave mode.As Fig.12 displays,the propagation direction of the det‐onation waves in the steady propagation process is opposite to the initiation direction in the 25-and 30-mm wide combustors.After reignition or colli‐sion,newly generated detonation waves usually prop‐agate in a direction where the combustible mixture is sufficient.Thus,the propagation direction of the CRD wave may get changed in the propagation process.A similar phenomenon was also found and analyzed in(Sun et al.,2017).

Fig.11 Extinction(a)and reignition(b)of the CRD waves(W=25 mm)

Fig.12 Steady propagation of the CRD waves in the combustors with W=25 mm(a)and W=30 mm(b)

In addition,the time interval required for the det‐onation wave to propagate in a cycle can be obtained by the pressure peaks of the pressure record curves.Combining those with the coordinates of the monitor point

C

,the instantaneous velocity of the detonation wave in a cycle can be calculated.Since there is reig‐nition or extinction of the CRD wave in the counterrotating two-wave mode and unsteady multiple-wave mode,the propagation velocity is hard to calculate in such modes.Therefore,only the velocities in steady propagation modes are taken into consideration,in‐cluding the single-wave mode,the co-rotating twowave mode,and the co-rotating three-wave mode.The average propagation velocity of the CRD wave,

V

,can be obtained (Sun et al.,2018a,2018b,2019).As Table 2 lists,the average propagation veloc‐ity increases with the increase in the combustor width.The highest velocity is obtained in the 30-mm wide combustor and reaches 1880.27 m/s,while the lowestvelocity is acquired in the 15-mm wide combustor at 1681.01 m/s.

From the above-mentioned discussions,the criti‐cal combustor width for the steady propagation of the stoichiometric ethylene-air CRD wave is found to be 25 mm in the tests under a constant mass flow rate of 800.31 g/s.When the combustor width is smaller than 25 mm,the CRD wave cannot propagate steadily in a single mode but it can self-sustain steadily in corotating two-wave mode when the combustor width is 25 or 30 mm.Moreover,the average propagation velocity of the CRD wave increases as the combustor width increases.

3.2 Effects of the combustor width on the flow field characteristics

To investigate the characteristics of the flow field in the combustors,the temperature contours on the

x

=10 mm surfaces and the radial-azimuthal surfaces in front of the CRD wave are chosen for analysis.In addition,ethylene (CH) reacts with oxygen (O) to produce hydrogen (H) and carbon monoxide (CO) in the primary reaction (Baurle et al.,1998).The inter‐mediate products are more chemically active than eth‐ylene and may contribute to the steady propagation of the CRD wave and,therefore,the contours of the mass fraction of hydrogen(H)are also analyzed.Fig.13 shows the temperature contours and Hmass fraction contours on the

x

=10 mm surfaces of different combustors.As the propagation mode in the 15-and 20-mm wide combustors changes frequently,only the flow field in the co-rotating three-wave mode is chosen for analysis.As Fig.13a shows,the fresh mixture layer forms a low-temperature zone in front of the CRD waves.Obviously,the detonation waves are non-uniformly distributed along the circumferen‐tial direction when the width is 15 or 20 mm.There is a greater interaction on the injection of fresh combus‐tible mixture between two closer waves.The injection of the fresh mixture is more likely to be blocked by the high-pressure products as shown in Fig.13b,and thus the mixture layer in front of the waves is quite short along the circumferential direction.For the lack of fresh mixture,the following CRD wave may get extinguished and the propagation mode of the CRD wave may be transformed.As the combustor width increases to 25 and 30 mm,the CRD waves in the combustors are evenly distributed along the circum‐ferential direction with central angles of 179.0° and 178.3°,respectively.The long circumferential distance between the CRD waves allows the fresh mixture to be fully injected so that a long fresh mixture layer can be established.This contributes to the long-term sta‐ble propagation of the CRD waves.Fig.13c illus‐trates the Hmass fraction contours in different com‐bustors.Obviously,in the 15-and 20-mm wide com‐bustors,hydrogen is mainly distributed in front of the CRD waves due to the chemical reactions caused by detonation.However,when the width is 25 or 30 mm,there is a large amount of hydrogen on the interface between the fresh mixture layer and the detonation product.The high-temperature detonation product induces the preliminary chemical reactions of the pro‐pellants,which produce high-activity intermediate products,such as hydrogen.The induced time and length of hydrogen are both shorter than those of eth‐ylene,indicating that hydrogen has higher detonation ability than ethylene.

Fig.13 Contours on the x=10 mm surfaces:(a) temperature contours;(b) pressure contours;(c) H2 mass fraction contours

For a better understanding of the flow field along the axial direction,temperature contours and Hmass fraction contours on the radial-azimuthal surfaces in front of the CRD waves are also selected for analysis.When the combustor width is larger than 15 mm,a backward-facing step is formed at the forepart of the combustor due to the sudden expansion of the crosssectional area of the channel there,as shown in Fig.1.As Fig.14a displays,a high-temperature zone filled with detonation product is formed at the backwardfacing step especially in the 25-and 30-mm wide combustors.The residence phenomenon of the hot detonation products at the backward-facing step was also experimentally observed by Hsu et al.(2020)using megahertz-rate OH planar laser-induced fluores‐cence imaging.The high-temperature zone works as a pilot flame and preheats the freshly injected mix‐ture.As a result,the induced length and time of the propellant are greatly reduced with higher initial tem‐perature (Peng et al.,2019a).Moreover,the primary combustion products with high chemical activity,such as hydrogen,are produced on the interface between the accumulation layer of combustible mixture and the high-temperature zone,as Fig.14b illustrates.The mass flow rates of ethylene and intermediate product hydrogen along the axial direction are calculated by

Fig.14 Contours on the radial-azimuthal surfaces in front of the CRD waves:(a) temperature contours;(b) H2 mass fraction contours

where

ρ

is the density,

u

is the velocity along the

x

direction (axial direction),and

φ

represents the mass fraction of species

i

referring to ethylene or hydrogen.

As Fig.15 shows,the mass flow rate of ethylene decreases as the axial distance increases.However,the distances that ethylene is advected downstream in different combustors are different.The ethylene is advected to the axial location of 20 mm in the 15-mm wide combustor but is transported further downstream to near the axial location of 30 mm in the other three combustors.The mass flow rate of hydrogen along the axial direction is plotted in Fig.16.The mass flow rate of hydrogen firstly increases to a maximum value and then drops with some fluctuations as the axial distance increases.The increase results from the pre‐heating effects of the hot detonation products on the fresh mixture,as mentioned above,while the drop is due to the consumption of the detonation wave.The incomplete detonation leads to deflagration down‐stream and thus results in the fluctuation of the mass flow rate of hydrogen downstream.The maximum mass flow rate of hydrogen in the 15-mm wide com‐bustor is significantly lower than that in the other three combustors.That is due to the more severe pre‐heating effects of the hot detonation products at the backward-facing step in the other three combus‐tors.Moreover,the maximum value of the mass flow rate of hydrogen in the 15-mm wide combustor is obtained at the axial location of 14 mm,while those in the other three combustors are obtained at the axial location of 32 mm.Above all,the reactants and inter‐mediate products can be transported further along the axial direction in the wider combustors,which estab‐lishes a suitable fresh mixture layer in front of the detonation wave.All these favorable factors contrib‐ute to the steady propagation of detonation waves in the wider combustors.

Fig.15 Mass flow rates of ethylene along the axial distance

Fig.16 Mass flow rates of hydrogen along the axial distance

3.3 Effects of combustor width on the propulsive performance

Propulsive performance is of great importance in the CRDE design process.In this section,the effects of combustor width on the propulsive performance are analyzed.The thrust

F

can be calculated by Eq.(3),where

A

is the exit area of the combustor,

p

is the static pressure,and

p

is the ambient pressure which is set at 0.1 MPa.Moreover,the mass-average pressure and the mass-average axial velocity along the axial dis‐tance are calculated by Eqs.(4)and(5),respectively.

The thrust record curves of different combustors are illustrated in Fig.17.In the 15-and 20-mm wide combustors,the thrust fluctuates widely,especially in the unsteady multiple-wave mode.In the co-rotating propagation mode,the number of detonation waves has little impact on the magnitude of the thrust.How‐ever,as Figs.17a and 17b show,the extinction of the detonation wave may cause fluctuation of the thrust while an increase in the number of the detonation waves will not.As the combustor width increases to 25 and 30 mm,the thrust keeps quite stable with small fluctuations due to the steady propagation of the CRD waves in co-rotating two-wave mode.

Fig.17 Thrust record curves of different combustors:(a)W=15 mm;(b)W=20 mm;(c)W=25 mm;(d)W=30 mm

Based on the thrust record curves,the average thrust and relative standard deviation (RSD) of the thrust of the different combustors are calculated and listed in Table 3.The average thrust decreases consid‐erably from 758.06 to 525.93 N as the combustor width increases from 15 to 30 mm.Nevertheless,the RSD of the thrust is as high as 16.92%and 19.69%in the 15-and 20-mm wide combustors,respectively,while it is just 5.49%and 5.87%in the 25-and 30-mm wide combustors,respectively.This is beneficial by giving a steadier propagation process of the CRD wave in the 25-and 30-mm wide combustors.It can be inferred from Eq.(3)that the pressure at the exit of the combustor and the axial velocity both affect the thrust of the CRDE.The mass-average pressure and the mass-average axial velocity are plotted in Figs.18 and 19,respectively.The pressure first rises and then falls sharply due to the quick heat release of the deto‐nation and the rapid expansion behind the CRD wave,which occurs within 20-mm long axial distance at the forepart of the combustor.When the axial distance is larger than 20 mm,the pressure decreases slowly as the axial distance increases.The highest pressure is obtained in the 15-mm wide combustor due to the smallest cross-sectional area of the channel.It is worth noting that the pressure peak in the 25-mm wide combustor is much higher than that in the 20-mm wide combustor.This results from the more efficient detonation combustion in the 25-mm wide combustor as discussed in Section 3.2.Due to the larger volume of the combustor,the pressure behind the CRD wave in the 25-mm wide combustor drops more quickly than that in the 20-mm wide combustor.Unsurpris‐ingly,the lowest pressure is obtained in the 30-mm wide combustor owing to its largest combustor vol‐ume.As the axial distance increases,the velocity first rises sharply and then increases slowly with fluctuations as shown in Fig.19.As the width of the combustor increases,the restriction on the flow direction of the combustion product by the inner cyl‐inder decreases.Therefore,the axial velocity decreases as the combustor width increases and results in a great kinetic energy loss in the circumferential and radial directions.This is not conducive to increasing the thrust of the CRDE.

Fig.18 Mass-average pressures along the axial distance

Fig.19 Mass-average axial velocities along the axial distance

All in all,increasing the width of the combustor is not conducive to increasing the magnitude of the thrust although it is beneficial to improving its stability.Furthermore,enlarging the channel at the forepart of the combustor where the detonation occurs while nar‐rowing the downstream channel where the combustion products accelerate may lead to a relatively high and stable thrust of the CRDE.This proposed combustor configuration exactly matches the cavity-based annu‐lar combustor.

5 Conclusions

To reveal the impacts of combustor width on the ethylene-air CRD,four cases with different combus‐tor widths are carried out using 3D simulations.Based on the pressure record curves,contours of the flow field,and thrust record curves,simulation results have been analyzed in detail.The main conclusions can be drawn as follows:

1.The critical combustor width to obtain steady propagation of the ethylene-air CRD wave is 25 mm.When the combustor width is smaller than 25 mm,an unsteady multiple-wave mode is observed and the CRD waves do not reach a long-term steady mode.As the combustor width is increased to 25 and 30 mm,the CRD waves can remain steadily and sustainably in a co-rotating two-wave mode.

2.The average propagation velocity of the CRD wave increases as the combustor width is increased.The highest propagation velocity is obtained in the 30-mm wide combustor,reaching 1880.27 m/s,while the lowest propagation velocity is acquired in the 15-mm wide combustor at 1681.01 m/s.

3.A high-temperature zone filled with primary detonation products is observed at the forepart of the wide combustors and its effects are revealed.The high-temperature zone works as a pilot flame and pre‐heats the propellants.This can greatly improve the detonation ability of the propellants and produce chemically active preliminary products.

4.As the combustor width increases,the average thrust of the CRDE decreases while the stability of the thrust increases.With the increase in combustor width,the volume of the combustor increases and the combustion products flow more divergently.This re‐sults in a considerable pressure drop in the combustor and great kinetic energy loss in the circumferential and radial directions.

5.The cavity-based annular combustor is sup‐posed to obtain a good propulsive performance of the CRDE.By applying the cavity to the annular combus‐tor,the channel is enlarged at the forepart of the com‐bustor where the detonation occurs,while the down‐stream channel is narrowed where the combustion products accelerate.This may lead to a relatively high and stable thrust of a CRDE fed by hydrocarbon fuels.

This paper may contribute to understanding the realization mechanism of ethylene-air CRD and enrich‐ing the combustor design theory for CRDE.More‐over,how to obtain both steady propagation of the CRD wave and high thrust of the CRDE fed by hydro‐carbon fuels is worth further study.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No.51776220) and the Postgraduate Scientific Research Innovation Project of Hunan Province,China.

Author contributions

Wei-dong LIU and Hao-yang PENG designed this numer‐ical study.Wei-jie FAN carried out the simulations and ana‐lyzed the results under their guidance.Shi-jie LIU and Jian SUN provided important suggestions on the improvement of the simulations.All authors reviewed and revised the manu‐script carefully and approved the content of the manuscript.

Conflict of interest

Wei-jie FAN,Wei-dong LIU,Hao-yang PENG,Shi-jie LIU,and Jian SUN declare that they have no conflict of interest.