Numerical Study of Spray Combustion Effects on Detonation Propagation

Abstract

Considering the recent interest in the use of liquid fuel in rotating detonation engines, there is a need to understand the interactions of a detonation wave with liquid fuel.
Detonation propagation in a three-dimensional periodic channel is simulated in this work using Eulerian-Lagrangian reactive simulations.
To model the non-homogeneity of combustion, discrete injectors for gaseous hydrogen fuel are used, and the liquid spray is injected along with the air from a continuous plenum.
The results show that when the hydrogen injection rate is reduced to a certain condition, the detonation wave is unable to sustain, but the injection of the kerosene spray helps it sustain and the system transitions from a pure gaseous detonation to a hydrogen-driven kerosene-sustained detonation.
Effect of droplet injection diameter and the fuel mass-flow rate are also studied.
Hydrogen promotes the vaporization and the burning of kerosene droplets.
Kerosene vaporization is a relatively slow process and the vapor burns as either a weak detonation or through the post-shock region, which in turn provides sufficient energy for detonation propagation.
Therefore, the contributions of both fuels are interlinked and responsible for sustaining the continuous propagation of the detonation wave.

Full text

Numerical study of spray combustion effects on detonation
propagation
Marc Salvadori ∗
HyPerComp, Inc., Westlake Village, CA, 91361, United States
Achyut Panchal †, and Suresh Menon ‡
Georgia Institute of Technology, Atlanta, GA 30332, United States
Considering the recent interest in the use of liquid fuel in rotating detonation engines, there
is a need to understand the interactions of a detonation wave with liquid fuel. Detonation
propagation in a three-dimensional periodic channel is simulated in this work using Eulerian-
Lagrangian reactive simulations. To model the non-homogeneity of combustion, discrete
injectors for gaseous hydrogen fuel are used, and the liquid spray is injected along with the
air from a continuous plenum. The results show that when the hydrogen injection rate is
reduced to a certain condition, the detonation wave is unable to sustain, but the injection of the
kerosene spray helps it sustain and the system transitions from a pure gaseous detonation to a
hydrogen-driven kerosene-sustained detonation. Effect of droplet injection diameter and the
fuel mass-flow rate are also studied. Hydrogen promotes the vaporization and the burning of
kerosene droplets. Kerosene vaporization is a relatively slow process and the vapor burns as
either a weak detonation or through the post-shock region, which in turn provides sufficient
energy for detonation propagation. Therefore, the contributions of both fuels are interlinked
and responsible for sustaining the continuous propagation of the detonation wave.
I. Introduction
A detonation wave is foreseen as an attractive possibility for aerospace vehicle propulsion or ground-based power
generation [
1
] due to high energy release. In fact, the detonation mode of combustion constitutes a potential alternative
to the subsonic and quasi-isobaric combustion regime currently employed in conventional combustion chambers because
the detonation front generates high pressures and temperatures without the need to add complex components to the
system [
1
,
2
]. Combustion devices that utilize a detonation wave have therefore multiple advantages such as mass and
Presented as Paper 2022-0394 at the AIAA SciTech Forum, San Diego, California and Virtual, 3–7 January 2022; received 8 March 2023;
revision received 6 July 2023; accepted for publication 3 September 2023; published online XX epubMonth XXXX. Copyright ©2023 by Marc
Salvadori. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. All requests for copying and permission
to reprint should be submitted to CCC at www.copyright.com; employ the eISSN 1533-385X to initiate your request. See also AIAA Rights and
Permissions www.aiaa.org/randp.
∗Member of Technical Staff; msalvadori13@hypercomp.net.(Corresponding Author)
†Research Engineer II, Daniel Guggenheim School of Aerospace Engineering; apanchal7@gatech.edu.
‡Hightower Professor, Daniel Guggenheim School of Aerospace Engineering; sm53@gatech.edu. AIAA Associate Fellow.

size reduction, greater simplicity, and the potential of increased robustness and reliability [
1
]. One of the most prominent
concept is the Continuous Detonation Wave Engine (CDWE), also known as the Rotating Detonation Engine (RDE)
[
3
,
4
]. The principles of operation for an RDE consist of the initiation and formation of one or multiple detonation
waves that continuously propagate within a cylindrical chamber [
1
,
5
,
6
] at very high frequencies. Many studies have
employed the injection of gaseous fuels such as hydrogen [
3
,
6
–
8
] and hydrocarbons [
3
,
9
]; however, the use of liquid
fuels [
10
–
13
] is of particular interest for practical applications due to its higher energy release, safety, and storage
benefits.
The use of liquid fuel introduces complex and interlinked physical phenomena such as atomization, breakup,
dispersion, evaporation, mixing, partially premixed burning, which have an important influence on the structure and
dynamics of a detonation wave [
10
,
14
]. The focus of this work is on understanding the later half of this process, where
the dilute spray vaporizes, disperses and burns to result in a sustained detonation wave. The earliest experimental studies
involving liquid detonations can be traced back from the work of Webber [
15
] and Cramer [
16
] in which it was shown
that larger droplets result in larger detonation velocity deficits up to 40
%
of the ideal Chapman-Jouguet (CJ) value.
Bar-Or et al. [
17
] found that high vapor pressure fuels have a higher tendency to sustain a detonation wave compared
to low pressure fuels. Eidelman and Burcat [
18
] numerically showed that smaller droplet size (
𝑑𝑝<100𝜇𝑚
) had
negligible effects on the detonation propagation with speeds close to ideal CJ; however significant velocity deficits were
observed for
𝑑𝑝>2000𝜇𝑚
. Smaller droplets are faster to vaporize [
19
] which can potentially provide prevaporized
fuel to react ahead or near the detonation wave, although their vaporization and acceleration can also take energy and
momentum out of the propagating wave [
20
]. Chang and Kailasanath [
20
] concluded through their numerical studies
that to sustain a detonation wave, the energy release due to combustion must overcome the attenuation due to the
liquid phase such that the pressure waves emanating from the combustion can couple with the front shock. These early
fundamental studies characterized detonation propagation in terms of droplet size, temperature, vaporization properties,
etc., but they were limited to 1D and homogeneous droplet mixtures.
Although most of the RDE research relied on the use of gaseous fuels [
4
], the experimental work by Bykovskii et al.
[
3
] was the first to achieve a stable propagating wave within a kerosene and enriched oxygen annular chamber. It was
also found that whenever air was used as the oxidant the detonation cannot self-propagate and the addition of gaseous
fuel such as hydrogen was necessary. Similarly, Kindracki et al. [
11
,
21
] and Zhou et al. [
22
] investigated the effects of
liquid kerosene injection in air with small amounts of hydrogen and were able to achieve steady operation. The inclusion
of a more reactive gas not only helped in the initiation of the detonation, but also allowed for rapid evaporation of the
droplets to facilitate the propagation of the front inside the combustor. The dynamic of initiation and stabilization of a
detonation within a liquid-fueled RDE was investigated by both Zhen et al. [
23
] and Ma et al. [
24
] suggesting that
pre-heating the oxidizer tends to result in a successfull RDE operation. Zhong et al. [
25
,
26
] attempted to study the
injection of pre-vaporized kerosene showing that the performance of the device and speed of the front were improved
2

when oxygen-enriched air was used. Based on these studies it is clear that the use of oxygen tends to increase the
detonability of using liquid fuel for RDE applications and the use of highly reactive gaseous fuel can assist in steady
operation when air is employed as the oxidizer. Therefore, both hydrogen and kerosene are used in this work as fuels
with air as an oxidizer.
As the physical insights on the coupled combustion processes occurring between the phases within the extreme
environments found in typical RDEs are difficult to characterize, numerical simulations can be used to gain an in-depth
understanding of the time-dependent and spatial processes involved. Many numerical investigations involving two-phase
RDEs [
27
–
31
] have been limited to simplified two-dimensional (2D) domains which neglect the three-dimensional
nature of the unsteady flow-field and the injection is modelled with either premixed gas or a combination of fuel vapor
and droplets. Compared to detonation propagation in an idealized homogenuous mixture, in a real system injected
droplets would have to disperse, mix, vaporize and burn in a 3D reactive flow-field for a detonation wave to sustain
[
12
,
13
]. In this study a new canonical configuration is modeled based on RDE designs with radial injection strategy
[
6
,
7
,
32
] to allow the exploration of the interaction between injected mixture, liquid droplets, and detonation in order to
gain insights on the underlying physical processes. A 3D periodic channel with a radial injection design composed of 25
discrete hydrogen injectors and a continuous air/kerosene slot is used. This reduced configuration represents a suitable
domain to study and characterize the flow physics typically found in RDEs. The setup was recently used to analyze the
effects of mixing on the propagation of a detonation wave in gaseous hydrogen-air [
33
]. Spray combustion introduces
additional modelling challenges to capture the interaction of the liquid phase with the gas phase [
34
,
35
]. Either an
Eulerian-Eulerian (EE) or an Eulerian-Lagrangian (EL) approach are possible [
34
] but EL is used here as it provides a
more accurate representation of the droplets dynamics for dilute two-phase flows involving extreme environments such
as detonation or explosions [36, 37].
The primary objective of this work is to understand whether liquid fuel can sustain continuous propagation of a
detonation wave and combustion processes involved, and if so, then how. We recently showed the possibility of this
using EL simulations for a full-scale RDE at one particular condition [
13
], however, the current 3D periodic channel
setup enables us to conduct additional simulations and analyses in a computationally feasible manner to gain a deeper
understanding of the underlying physical processes, such as, interlinked burning of hydrogen and kerosene, and effect of
droplet dispersion and vaporization on detonation survival. Since the focus here is on interactions between the droplets
and the detonation wave within the combustion chamber, the near-injector atomization of the liquid droplets is not
modeled. Starting from a hydrogen-air system, the gaseous hydrogen mass flow rate is lowered to lean limits where the
detonation would not survive on its own, and kerosene droplets are introduced from the oxidizer stream to study their
effect.
The EL model requires sub-models for handling interphase mass, momentum and energy transfer. In terms of
interaction of a single droplet with a detonation wave, only limited studies are available [
38
]. In absence of any such
3

well-established sub-models, conventional drag and vaporization laws are used in this work [
19
], but it is noted that
the validity of such sub-models may need further evaluation in the future, either through experiments or via modeling
resolved droplets.
To understand the mechanism of detonation propagation in presence of spray, two reduced hydrogen injection rates
and two injected droplet sizes are simulated. In terms of analyses, first, a one-to-one comparison is provided between a
pure gaseous configuration that fails to sustain a continuous detonation propagation at lean injection conditions and
another configuration at the same gaseous injection condition but now with additional spray injection that helps sustain
the detonation propagation. Next, the obtained two-phase quasi-steady detonation structure is analyzed to understand
the nature of hydrogen and kerosene burning, their interlinking, and their effect on the detonation propagation. The
effect of the droplet size and the gaseous injection rate are evaluated, as they assist in understanding these mechanisms.
The two-phase detonation propagation is understood as a coupled process that involves droplet vaporization, gaseous
hydrogen burning and kerosene vapor burning. The results are referenced against our previous work on pure gaseous
detonation [6] (stoichiometric injection) with stable operation.
The paper is organized as follows. The governing equations and numerical methods employed in this study are
summarized in Section II. Section III provides the details of the computational setup. Section IV provides a preliminary
zero-dimensional (0D) analysis to characterize the coupling between various physical processes and their time-scales.
In Section V, the 3D simulation results are discussed. Finally, the outcome of this study and avenues for future work are
summarized in Section VI.
II. Governing Equations and Numerical Formulation
A. Gas Phase
Compressible, unsteady, multi-species and multiphase 3D Navier-Stokes equations for reacting gas flow that are
solved in this work in the limit of negligible volume fraction (i.e, dilute gas-droplet mixture) are as follows:
𝜕𝜌
𝜕𝑡 +𝜕
𝜕𝑥 𝑗𝜌 𝑢 𝑗=¤𝜌𝐷(1)
𝜕𝜌𝑢𝑖
𝜕𝑡 +𝜕
𝜕𝑥 𝑗𝜌𝑢𝑖𝑢𝑗+𝑝𝛿𝑖 𝑗 −𝜏𝑖𝑗 =¤
𝐹𝐷,𝑖 (2)
𝜕𝜌𝐸
𝜕𝑡 +𝜕
𝜕𝑥 𝑗h(𝜌 𝐸 +𝑝)𝑢𝑗+𝑞𝑗−𝑢𝑗𝜏𝑖 𝑗 i=¤
𝑄𝐷+¤
𝑊𝐷(3)
𝜕𝜌𝑌𝑘
𝜕𝑡 +𝜕
𝜕𝑥 𝑗h𝜌𝑌𝑘𝑢𝑖+𝑉𝑖, 𝑘 i=¤𝜔𝑘+¤
𝑆𝐷, 𝑘 (4)
Here,
𝜌
correspond to the gas density,
𝑢𝑖
is the velocity vector,
𝐸
is the total specific energy, and
𝑌𝑘
is the mass fraction
4

associated to the
𝑘th
species. The thermodynamic pressure,
𝑝
, is computed through the use of the ideal gas equation of
state
𝑝=𝜌𝑅𝑇
, where
𝑇
is the gas temperature and
𝑅
is the gas constant. The terms,
𝜏𝑖𝑗
,
𝑞𝑖
, and
𝑉𝑖,𝑘
, are shear-stress
tensor, heat conduction, and diffusion flux of
𝑘th
species, respectively. The
𝜏𝑖𝑗
is computed for a Newtonian fluid with
the local viscosity computed using a Sutherland law [
39
]. The heat conduction is computed using a constant Prandtl
number assumption, and the species diffusion flux are computed using a unity Lewis number (
𝐿𝑒
) assumption. The
terms with subscript 𝐷are interphase coupling terms and the are discussed further later.
B. Liquid Phase
As the focus here is to study interaction of liquid droplets with a propagating detonation wave, and the dense
liquid injection is typically located away from the region of detonation propagation [
11
,
21
], the liquid droplets are
modeled as dilute point-particles with the effects of collisions, deformation, and breakup neglected. Primary breakup
and atomization are not modeled in this work as the liquid fuel is not directly injected into the chamber, but instead,
liquid droplets travel and enter the chamber in with the air flow in a dilute manner (volume loading
<0.02
%). The
droplets could still undergo secondary breakup depending primarily on their Weber number, however, our previous
study [
13
] using the same injector configuration showed the Weber number (We) to be
<
12 for 50
%
of the injected
droplets, for which, a secondary breakup is not expected to occur [
40
]. Consistently, the current study also does not
include secondary breakup modeling. Furthermore, development of breakup models in presence of strong shocks and
detonation waves is still an active area of research, and the validity of the conventional models in such conditions is
unclear.
To track the droplet mass
𝑚𝑝
, position
𝑥𝑝,𝑖
, velocity
𝑢𝑝,𝑖
, and temperature
𝑇𝑝
of
𝑝th
droplet, a Lagrangian approach
is used. The liquid-phase governing equations are given as
𝑑𝑚 𝑝
𝑑𝑡 =− ¤𝑚𝑝(5)
𝑑𝑥𝑝, 𝑖
𝑑𝑡 =𝑢𝑝 ,𝑖 (6)
𝑑𝑢 𝑝 ,𝑖
𝑑𝑡 =𝜋
2𝜌𝑟2
𝑝𝐶𝐷|𝑢𝑔.𝑖 −𝑢𝑝 ,𝑖 |(𝑢𝑔, 𝑖 −𝑢𝑝,𝑖 ) − 4
3𝜋𝑟3
𝑝
𝜕 𝑝𝑔
𝑑𝑥𝑖
,(7)
𝑚𝑝𝐶𝑝
𝑑𝑇𝑝
𝑑𝑡 =2𝜋𝑟 𝑝𝜅 𝑁 (𝑇𝑔−𝑇𝑝) − ¤𝑚𝑝𝐿𝑣(8)
Here,
𝑢𝑖
,
𝑇
,
𝑝
, and
𝜅
are gas-phase velocity, temperature, pressure, and conductivity, respectively, and they have to be
interpolated at the droplet location
𝑥𝑝,𝑖
from the gas-phae solution. The droplet radius is denoted as
𝑟𝑝
, and it relates to
the droplet mass as
𝑚𝑝=(4/3)𝜋𝑟3
𝑝𝜌𝑝
with
𝜌𝑝
being the liquid droplet density. Well-established empirical models [
19
]
are used for modeling the drag coefficient
𝐶𝐷
, Nusselt number
𝑁𝑢
, and the mass-transfer term
¤𝑚𝑝
. The latent heat of
5

vaporization is denoted as 𝐿𝑣. The drag coefficient is computed in this work as
𝐶𝐷=









24
𝑅𝑒 1+1
6𝑅𝑒 2
3𝑅𝑒 < 1000
0.424 otherwise
(9)
This expression was derived for a solid sphere in low and high Reynolds number flow [
41
]. In addition, the
Lagrangian evolution equation (Eq. 7) also includes a pressure drag term that may play a crucial role in presence of
shocks. More advanced forms of
𝐶𝐷
, e.g., [
42
,
43
], that include the effect of shock formation ahead of a droplet in a
supersonic or hypersonic flow are available, but are not used here as the current simulations show more than 95% of the
droplets with a slip Mach number (
𝑀𝑠𝑙𝑖 𝑝
) lower than the critical value of 0.6. The vaporization rate is computed using
a thin film model for spherical droplets [
19
]. The model includes convective heat and mass transfer effects through
empirical correlations that are mostly valid for
𝑅𝑒 < 2000
[
19
]. Stefan flow effects are not included in the current study.
We note that advanced models, e.g., [
44
], can be employed in the future to include this effect, however, evaluating their
validity in high-speed and high temperature/pressure flows caused by a detonation wave is still an open area of research
requiring further studies and that is out of the scope of this work.
Due to the presence of detonations and strong shocks, it is expected that strong pressure gradients exist within
the flow. Hence, along with the conventional viscous drag, a pressure gradient acceleration term is also considered
in this work. The importance of this term has been demonstrated in the past studies for highly compressible flows
[
37
]. The corresponding coupling source terms such as interphase mass (
¤𝜌𝐷
,
¤
𝑆𝐷, 𝑘
), momentum (
¤
𝐹𝐷
), and energy
transfer terms (
¤
𝑄𝐷
,
¤
𝑊𝐷
) are added into the gas-phase equations (Eqs.
(1)
-
(4)
). The interphase work term
¤
𝑊𝐷
includes
contributions from the pressure gradient acceleration as well. Further details about these terms can be found elsewhere
[
37
]. Finally, to make these simulations computationally affordable, the droplets are combined in groups of eight, known
as computational ‘parcels’ [19, 45].
C. Numerical method
A well-established multi-block finite-volume solver LESLIE (Large Eddy Simulation with LInear Eddy model) is
used to solve the governing equations [
6
,
13
,
35
]. The gas-phase equations are evolved in time using a second-order
predictor-corrector approach, and the fluxes are modeled using a formally third-order accurate monotonic upstream-
centered scheme for conservation law (MUSCL)[
6
]. Viscous terms are modeled using a Sutherland law. A unity Lewis
number transport is assumed for the species and our previous work has shown this to be a valid assumption for this
system [33].
A finite-rate kinetics approach is used for modeling the combustion, wherein a two-step Kerosene chemistry [
46
]
is combined with a seven-step hydrogen (
H2
) chemistry [
47
]. The two-step kerosene kinetics is validated against
6

experiments for auto-ignition delay and 1D detonation propagation with a prevaporized mixture in supplementary
material Section 1 and 2, respectively. A verification for 1D detonation propagation in a two-phase mixture is provided
in Section 3 of the supplementary material. The hydrogen kinetics is the same that we have used in our earlier works
with pure gaseous detonation [
6
]. The combined chemistry was validated against a detailed high-pressure kerosene
kinetics [
48
] for auto-ignition delay times over a range of kerosene/
H2
ratio [
13
]. The Lagrangian equations are solved
using fourth order Runge-Kutta scheme, and the inter-phase coupling terms in Eqs.
(1)
-
(4)
are volumetrically averaged
over the computational cell. More information regarding the implementation, validation, and application of the current
multi-phase solver can be found in previous work [34, 37].
III. Computational Setup
(b) Injector near-field grid and
geometrical features
Air (g) + Kerosene (l)
Air/droplets injection plenum 25 H2 injectors
(a) Overview of periodic linear geometry during steady operation
H2(g)
Detonation front
H2 gas
Liquid droplets
Fig. 1 Schematic of the non-premixed periodic channel. 3D flow-field under steady operation is shown in (a)
along with the injectors and the air plenum. Computational grid in the injector near-field is shown in (b).
A periodic channel composed of 25 discrete injectors (with a throat diameter of 0.89 mm) for non-premixed hydrogen
injection and a continuous slot (throat height of 0.55 mm) for air and kerosene droplets injection is modeled as shown
in Fig. 1). The injectors and the plenum use the same geometrical features as the RDE designed by the Air Force
Research Laboratory (AFRL) (gaseous injection [
7
], multiphase injection [
49
]), that was also studied numerically in the
past [
6
,
13
]. Given that the propagation of a detonation wave inside the RDE chamber is repetitive with approximately
constant speed and period [
3
,
50
], its annulus is unrolled into a channel with periodic boundary conditions corresponding
to a single period of propagation at gaseous
H2
operation. This reduced model does not account for the curvature of the
RDE configuration but those effects can be neglected if the diameter of the RDE is sufficiently larger than the chamber
width [
51
,
52
]. The droplets are injected at the beginning of the converging section within the air plenum (see Fig.
1b) with their velocity matching that of the air. This procedure is designed to avoid the droplets getting trapped in
7

re-circulation zones.
Unlike a majority of multiphase detonation wave propagation studies in an unwrapped reduced configuration that are
2D [
27
–
31
], 3D discrete injectors are modeled here for capturing realistic non-premixed interactions of the injected fuel
and the detonation wave. Unlike the past experiments of detonation propagation in a channel that focused on propagation
of a single detonation pass over the discrete injectors [
52
–
54
], periodic boundary conditions are used here to study
continuous propagation and also to include the effect of products from a previous cycle as it would occur in a real device.
To evaluate the validity of the reduced configuration, the profiles of pressure, temperature, heat release rate and fuel
concentration were compared across the detonation wave between the numerical simulations of the full-scale RDE and
the unwrapped reduced configuration. Both show substantial similarities in terms of the peaks and the shape (not shown
here). Furthermore, the overall observations for multiphase operation, i.e., lean hydrogen is unable to survive on its
own and kerosene injection helping it sustain, are the same between the full-scale RDE simulation [
13
] and the current
reduced configuration. The reduced configuration allows us to focus on the physics of the detonation propagation in
presence of spray combustion and related interactions in a computationally feasible manner.
The computational configuration is meshed using 27 different multi-block structured grids that are joined through a
non-conformal grid interface method which is second-order accurate [
6
]. The total mesh contains approximately 38
million grid points with the finest resolution located in the injector near-injector field (
Δ𝑥= Δ𝑦= Δ𝑧
= 100
𝜇
m) which
is exponentially coarsened toward the outflow. The same grid resolution was used in our previous work [
6
,
12
,
13
,
33
]
and found to be sufficient to capture the relevant features of a detonation wave propagating in a non-premixed mixture.
For both the injection inflows and the outflow, characteristic non-reflective boundary conditions [
55
] are used. Walls are
modeled as no-slip and adiabatic.
Since both the gas-phase hydrogen and the liquid kerosene are simultaneously injected, two equivalence ratio are
defined correspondingly.
𝜙𝑔=( ¤𝑚H2/ ¤𝑚air)𝑠𝑡
( ¤𝑚H2/ ¤𝑚air), 𝜙𝑙=( ¤𝑚KERO/ ¤𝑚air)𝑠𝑡
( ¤𝑚KERO/ ¤𝑚air),(10)
where
¤𝑚H2
,
¤𝑚KERO
and
¤𝑚air
are hydrogen, kerosene, and air injection rate, respectively. The subscript
𝑠𝑡
refers to their
stoichiometric values with respect to air. Here,
𝜙𝑔
refers to the gaseous fuel equivalence ratio and
𝜙𝑙
refers to the liquid
fuel equivalence ratio. Unless specifically mentioned,
𝜙𝑔+𝜙𝑙=1
to set the global mixture stoichiometric. The flow
conditions simulated in this study are summarized in Table 1. Here,
𝑇H2
and
𝑇air
are their gaseous injection temperatures,
𝑑𝑝,0
is the initial droplet diameter, and
𝑇𝑝,0
is droplet temperature. The value for volume-fraction
𝛼𝑝
is estimated
based on the liquid fuel injection rate and is approximately
9×10−5
. Even though realistic RDEs would employ liquid
injectors which result in a wide distribution of droplets size and velocity [
56
], to fundamentally understand the effect of
8

injected droplet size on mixing, vaporization, and detonation propagation, a monodisperse injection is considered here.
Table 1 Summary of conditions for the 3D simulations in periodic reduced configuration.
𝜙𝑔¤𝑚H2[kg/s] ¤𝑚KERO [kg/s] ¤𝑚𝑎𝑖𝑟 [kg/s] 𝑇H2[K] 𝑇air [K] 𝑇p[K] 𝑑𝑝,0[𝜇m]
0.5 0.0073 0.0170 0.5 300 300 300 20, 5
0.7 0.0102 0.0102 0.5 300 300 300 20, 5
The initialization procedure of the detonation wave in the periodic channel is carefully designed to minimize any
numerical artifacts. Furthermore, the same exact procedure is used for all simulations shown in this work and this
allows for a one-to-one comparison to understand propensity of detonation propagation in different mixtures. The 3D
detonation in the two-phase periodic channel is initiated using a gas-phase pre-detonator as follows. This process is
schematized in Fig. 2.
1)
A non-reacting flow with target gaseous and liquid injection rates corresponding to
𝜙𝑔
and
𝜙𝑙
is established in the
periodic channel with a pre-detonator attached to it. This initial step is performed in order to allow the individual
injector plenums to converge to their corresponding injection mass flow rates.
2)
A charge with high-pressure (50 atm) and high temperature (3000 K) is placed within the premixed
H2
-Air
pre-detonator with
𝜙𝑔=1
to develop a detonation front. Once a fully three-dimensional detonation front has
propagated within the first few injectors of the chamber, the simulation is interrupted. The initial charge is not
expected to affect the wave propagation in the periodic channel as the 3D detonation is allowed to be established
first in the pre-detonator.
3)
The reactive solution from the previous step is then projected onto the periodic geometry as initial condition.
This technique allows for a fast transition to an established detonation. The simulation is resumed and the
droplets injection applied.
Apart from the transient analysis in Section V.B that is used to understand the propensity of detonation propagation
in different mixtures, multiple cycles of stable detonation propagation are established before analyzing and comparing
any simulation results to reduce any effects of the initial transients. The previously studied full-scale RDE [
13
]
employed the same injection method as the current study. However, it only considered
𝜙𝑔=0.5
,
𝜙𝑙=0.5
, and
𝑑𝑝
=
20
𝜇
m. Simulation of a single detonation cycle within the current linear channel configuration requires around 7,000
single-processor hours compared to 76,000 on the full-scale domain. The current simulations were carried out on 1440
processors, as opposed to 3200 for our previous work.
9

Projected 3D solution
Initial 3D solution
Fig. 2 The initialization process to obtain a detonation in a periodic channel. The detonation is shown using the
mid-slice of temperature (300-3000 K).
IV. Combustion and vaporization characteristics
Several physical processes are interlinked during the detonation propagation in a two-phase mixture. These are
analyzed in this section using 0D models to understand their connection on each other. A schematic is provided in
Fig. 3 that shows the dependencies between them. During a sustained detonation propagation, both the Hydrogen and
the Kerosene burning generate the thermochemical energy needed to sustain a reactive shock propagation (process A).
The shock in turn elevates the pressure and the temperature for both the fuels to keep reacting (process B) and this
increased temperature also promotes the droplet vaporization (process C). Fuel vaporization is a necessary prerequisite
for kerosene burning (process D). In addition to the shock, the high temperature products also helps the liquid fuel to
vaporize (process E). Finally, both the Hydrogen and Kerosene contribute to continuously ignite and burn directly via
creation of high temperature products (process F).
The detonation sustainability of a
H2
-Kerosene vapor mixture is characterised via constant-volume auto-ignition
delay times (
𝜏𝑖𝑔𝑛
), that are computed using the same reduced reaction kinetics employed in this study. These are
computed first at unburnt conditions of 300 K and 1.0 MPa (representative of high chamber pressure), and they are
reported in Fig. 4(a) as lines. For same unburnt conditions, kerosene vapor at
𝜙𝑙=1
is significantly difficult to ignite
compared to
H2
at
𝜙𝑔=1
and this explains the difficulty of a kerosene-air detonation to self-sustain. On the other hand,
the addition of
H2
, i.e.,
𝜙𝑔=0.5, 𝜙𝑙=0.5
, provides almost the same auto-ignition characteristics as
H2
,
𝜙𝑔=1.0
due
to hydrogen’s higher reactivity and this explains how
H2
can help kerosene sustain a detonation wave. This can be
understood as process F in Fig. 3.
10

Hydrogen Burning
()
Kerosene vapor
burning ()
Kerosene
vaporization
()
Continued shock-
reaction coupling
(   
 
)
A
A
B
B
C
DE
F
E
 󰇛󰇜 burning sustain shock/flame coupling
High T/P promote vaporization
Kerosene products help vaporization
Coupled 󰇛󰇜 burning Hydrogen products help vaporization
Fig. 3 Coupling mechanism of sustained spray detonation in
H2
-Kerosene-Air mixture. Further details about
processes A-F can be found in the text.
On the other hand, for a fuel-mixture of
H2
(without kerosene) at
𝜙𝑔=0.5
and
𝜙𝑔=1.0
similar auto-ignition
characteristics are obtained, and therefore, this does not fully explain the low propensity of self propagation for a lean
H2
detonation, i.e.,
𝜙𝑔=0.5
(see next section for 3D results). To understand this, a chemical equilibrium analysis
(CEA) [
57
] is conducted for all the representative fuel mixtures and the Chapman-Jouguet (CJ) detonation speed (
𝐷𝐶 𝐽
),
pressure (
𝑝𝐶 𝐽
), temperature (
𝐷𝐶 𝐽
), along with post-shock pressure (
𝑝𝑝𝑠
), and temperature (
𝑇𝑝𝑠
) are shown in Table
2. These are a result of process A in Fig. 3. Here, lean
H2
, i.e.,
𝜙𝑔=0.5, 𝜙𝑙=0.0
, shows a lower
𝐷𝐶 𝐽
and also
lower
𝑝𝐶 𝐽
,
𝑇𝐶𝐽
,
𝑝𝑝𝑠
, and
𝑇𝑝𝑠
compared to the other fuel mixtures with
𝜙𝑔+𝜙𝑙=1
. For detonation propagation, the
auto-ignition at post-shock conditions is of relevance (process B in Fig. 3), and the corresponding auto-ignition delay
times, i.e., 𝜏𝑖𝑔𝑛, 𝑝 𝑠, are reported in Table 2 and in Fig. 4 as symbols. These show that because of the reduced 𝑝𝑝 𝑠 , and
𝑇𝑝𝑠
for
𝜙𝑔=0.5, 𝜙𝑙=0.0
, the auto-ignition delay is larger, leading to a lower propensity to sustain a detonation wave
in this mixture. However, addition of kerosene as a fuel to this lean
H2
mixture, i.e.,
𝜙𝑔=0.5, 𝜙𝑙=0.5
, supports the
detonation propagation by strengthening the post-shock conditions. In summary,
H2
helps Kerosene to detonate via
reducing the auto-ignition delay time, i.e., via process F, whereas, the Kerosene helps lean
H2
to detonate via increasing
the post-shock pressure and temperature, i.e., via process A and B.
The liquid fuel vaporization process is characterized using a single spherical droplet placed in quiescent environment
at the post-shock conditions (
𝑝𝑝𝑠
,
𝑇𝑝𝑠
). This is represented by process C in this Fig. 3. After an initial period during
which the temperature of the droplet rises, the normalized squared droplet diameter (
(𝑑𝑝/𝑑𝑝,0)2
) reduces linearly as
expected [
19
]. The time that it takes for 90% of the droplet mass to evaporate is reported in Table 2 as
𝜏90
𝑒𝑣 𝑎 𝑝, 𝑝 𝑠
. Even
though
𝜏90
𝑒𝑣 𝑎 𝑝, 𝑝 𝑠
is affected by
𝑝𝑝𝑠
,
𝑇𝑝𝑠
, and
𝑑0, 𝑝
, it is more than two orders of magnitude larger than
𝜏𝑖𝑔𝑛, 𝑝𝑠
, suggesting
that the two processes will be decoupled. The vaporization is expected to occur at a much slower rate, over multiple
11

cycles, however, the available vaporized fuel is expected to burn during the detonation propagation. Compared to the
vaporization analysis discussed here which is only considered at representative post-shock conditions, the combustion
chamber is expected to be filled with hot products moving at high-speed (represented by process E in Fig. 3), but even at
those conditions, 𝜏90
𝑒𝑣 𝑎 𝑝, 𝑝 𝑠 >> 𝜏𝑖𝑔𝑛 , 𝑝𝑠 and such numbers are not reported here since the conclusions remain the same.
(a) Ignition (b) Evaporation
𝜙𝑔= 1.0, 𝜙𝑙= 0.0
𝜙𝑔= 0.5, 𝜙𝑙= 0.0
𝜙𝑔= 0.0, 𝜙𝑙= 1.0
𝜙𝑔= 0.5, 𝜙𝑙= 0.5 x
Fig. 4 Representations of a) autoignition and b) vaporization characteristics of mixture. Lines in Fig. 4a
correspond to constant-volume autoignition (
𝜏𝑖𝑔𝑛
) computed for the corresponding
H2
/kerosene vapor mixtures
at 1 MPa and 300 K. Symbols denote autoignition delay times computed at postshock corresponding conditions
(
𝑝𝑝𝑠
,
𝑇𝑝𝑠
). Figure 4b characterizes the normalized droplet diameter (
𝑑𝑝/𝑑𝑝,0
) reduction for the kerosene
vaporization process in a dilute mixture at corresponding postshock gas-phase conditions (𝑝𝑝𝑠 ,𝑇𝑝𝑠 ).
The three insights that are obtained from these 0D analyses are: 1) The contribution of hydrogen is to lower the
ignition time delay of the mixture, leading to an increased reactivity (process F). 2) The burning of hydrogen and
kerosene maintains a continuous shock–reaction coupling, allowing the evaporation of the liquid fuel (processes A and
B). 3) The longer evaporation timescale as compared to ignition delay 11 results in a decoupled process (processes C, D,
and E), are further corroborated by the 3-D results in the next section.
Table 2 For various representative
H2
-Kerosene mixtures considered in this study, Chapman-Jouguet (CJ)
detonation speed (
𝐷𝐶 𝐽
), CJ pressure (
𝑝𝐶 𝐽
), CJ temperature (
𝐷𝐶 𝐽
), post-shock pressure (
𝑝𝑝𝑠
), and post-shock
temperature (
𝑇𝑝𝑠
) computed using a chemical equilibrium analysis (CEA) tool are provided.
𝜏𝑖𝑔𝑛, 𝑝𝑠
and
𝜏90
𝑒𝑣 𝑎 𝑝, 𝑝 𝑠
are constant-volume auto-ignition delay time and the time it takes for 90% mass of a spherical droplet to vaporize
(for 𝑑𝑝,0=5𝜇𝑚 ), respectively, at the corresponding post-shock conditions (𝑝𝑝 𝑠,𝑇𝑝𝑠).
Case 𝐷𝐶 𝐽 𝑝𝐶 𝐽 𝑇𝐶 𝐽 𝑝𝑝𝑠 𝑇𝑝 𝑠 𝜏𝑖𝑔𝑛, 𝑝 𝑠 𝜏90
𝑒𝑣 𝑎 𝑝, 𝑝 𝑠
[m/s] [MPa] [K] [MPa] [K] [𝜇𝑠] [𝜇𝑠]
𝜙𝑔=1.0,𝜙𝑙=0.01976.6 1.571 2964.0 2.805 1540.5 0.348 129.7
𝜙𝑔=0.5,𝜙𝑙=0.01618.5 1.186 2222.4 2.166 1269.0 1.167 167.2
𝜙𝑔=0.0,𝜙𝑙=1.01800.3 1.881 2858.8 3.502 1555.2 2.428 128.2
𝜙𝑔=0.5,𝜙𝑙=0.51887.5 1.698 2910.2 3.091 1546.1 0.345 129.1
12

V. Simulation results and discussion
A. Global behavior
In order to understand the global burning behavior during propagation, a volume-integrated heat release rate (
𝑞𝐻 𝑅𝑅
)
is recorded during the simulations and is shown for all cases in Fig. 5. For reference, the same conditions without the
liquid fuel injection are also shown. The time is normalized by
𝜏𝑐𝑦𝑐 𝑙𝑒
which corresponds to the detonation propagation
time over the entire periodic channel. For the condition of
𝜙𝑔
= 0.7, the detonation wave is able to propagate with
(
𝜙𝑙=0.3
) and without (
𝜙𝑙=0.0
) the droplet injection showing minor differences. On the contrary, as the hydrogen mass
flow rate is lowered to achieve
𝜙𝑔=0.5
(with
𝜙𝑙=0.0
), an oscillatory behavior is obtained with
0.5≤𝑞𝐻 𝑅𝑅 ≤1.5
MW where local extinction and ignition occur similar to the instabilities found in full-scale RDE studies [
13
]. As
shown later, this case does not show a sustained detonation propagation. With the liquid fuel injected into the channel
(
𝜙𝑙=0.5
), an increase in
𝑞𝐻 𝑅𝑅
is observed with a reduction of oscillations and a stable detonation propagation is
observed after the initial transients (𝑡/𝜏𝑐𝑦𝑐𝑙 𝑒 >2).
(a) (b)
Extinction/re-ignition
only
   
  
Fig. 5 Time-varying volume-integrated heat release rate (𝑞𝐻𝑅𝑅 ) for (a) 𝜙𝑔= 0.7, and (b) 𝜙𝑔= 0.5.
The front speed (
𝑈𝐷
) is a well known parameter used to characterize the strength of the wave during self-sustained
propagation [
58
]. In order to track the position of the shock front, a threshold of both pressure and heat release rate
based on past studies [
6
] is used. The time-averaged values over
2< 𝑡/𝜏𝑐𝑦𝑐𝑙 𝑒 <6
are computed in order to neglect the
initial transients. For the pure gaseous case of
𝜙𝑔=0.7
,
𝜙𝑙=0.0
, we obtained a
𝑈𝐷
= 1562.04 m/s, whereas due to the
instability observed prior to quenching it was not possible to track the front for
𝜙𝑔=0.5
,
𝜙𝑙=0.0
. On the other hand,
when droplets are injected (to maintain
𝜙𝑔+𝜙𝑙=1
), the detonation wave is able to self-propagate even for
𝜙𝑔=0.5
.
Even though kerosene vapor is reactive, to evaluate its effect as a purge gas, an additional simulation was conducted at
𝜙𝑔=0.5
,
𝜙𝑙=0.5
,
𝑑𝑝=5𝜇𝑚
, where the liquid kerosene vaporized but does not burn (not shown here). This case
showed a similar unstable behavior as that of
𝜙𝑔=0.5
,
𝜙𝑙=0.0
, suggesting that the kerosene burning helps stabilize
13

the detonation propagation. However, further evaluation of how a purge gas can affect the detonation propagation in a
RDE is out of the scope of the current work.
Overall, the measured front velocities for all cases that inject the spray are
1450 < 𝑈𝐷<1590
m/s. These are
between 60
%
and 70
%
of the theoretical CJ values at corresponding equivalence ratios. Past experimental [
3
,
21
–
24
]
and numerical [
13
,
27
–
31
] studies of liquid fueled RDEs have shown the detonation speed to vary between 70-90
%
of
the theoretical CJ values, and they attributed their findings to the incomplete fuel-oxidizer mixing and liquid evaporation
phenomena occurring during steady operation of the device. The simplified domain considered here is capable of
capturing these effects as observed in typical annular combustors.
B. Detonation extinction and mechanisms for stable propagation
Stage-1 (  )Stage-2 (    )Stage-3 (  )
       
   
Temperature [K]
HRR [W/m3]
Temperature [K]
HRR [W/m3]
HRR HRR
HRRKERO HRR HRR 
HRR  HRR HRR HRR
HRR HRR HRR
Fig. 6 Evolution of the detonation wave starting from the initial transient to a stabilized burning (only achieved
for the cases with liquid injection). The contours of
∇𝑃
are superimposed on the temperature fields to identify
the structure and location of the shock front.
To understand how the liquid droplets help sustain the propagation of the detonation wave when a pure gaseous
mixture is not sufficient, a one-to-one comparison of the transient solutions is provided in this section between the two
cases: pure gaseous injection at
𝜙𝑔=0.5
,
𝜙𝑙=0.0
, where the detonation is unable to sustain, and another case with the
same gaseous injection, i.e.,
𝜙𝑔=0.5
, but with kerosene spray at
𝜙𝑙=0.5
with initial
𝑑𝑝=5𝜇
m injected which helps
the detonation sustain. Both simulations were started from the exact same initial conditions with the only difference in
the spray injection. Figure 6 shows the evolution of temperature and heat release rate (HRR) fields taken at mid-chamber
14

at three representative instances.
To further understand the contributions between the two fuels, the heat release rate
HRR
(total) is decomposed into
HRRH2(hydrogen), and HRRKERO (kerosene vapor). The total heat release rate is computed as
𝐻𝑅𝑅 =−
𝑁

𝑘=1
Δℎ0
𝑓 ,𝑘 ¤𝜔𝑘(11)
where
ℎ0
𝑓 ,𝑘
and
¤𝜔𝑘
correspond to the standard heat of formation and reaction rate of species
𝑘
. The heat release rate due
to the combustion of the kerosene vapor (having chemical formula
C10H20
) only is obtained from the reaction rates
of the species pertaining to the two chemical reactions: (R1)
C10H20
+
10 O2−−−→ 10 CO
+
10 H2O
, and (R2)
CO
+
0.5 O2←−→ CO2as follows:
HRRKERO =−Δℎ0
𝑓 , 𝐾 𝐸 𝑅𝑂 ¤𝜔𝐾 𝐸 𝑅𝑂 −Δℎ0
𝑓 ,𝑂2¤𝜔𝑂2, 𝑅1−Δℎ0
𝑓 ,𝑂2¤𝜔𝑂2, 𝑅2
−Δℎ0
𝑓 ,𝐶𝑂 ¤𝜔𝐶𝑂 −Δℎ0
𝑓 ,𝐻2𝑂¤𝜔𝐻2𝑂, 𝑅1−Δℎ0
𝑓 ,𝐶𝑂2¤𝜔𝐶𝑂2,
(12)
where,
¤𝜔𝐻2𝑂, 𝑅1
, and
¤𝜔𝑂2, 𝑅1
correspond to the reaction rates of
H2O
and
O2
for the first reaction of kerosene combustion
(R1), and
¤𝜔𝑂2, 𝑅2
represents the reaction rates of
𝑂2
for the second reaction (R2). The
HRR
due to hydrogen burning is
then simply HRRH2=HRR −HRRKERO.
To identify the effects of spray injection on detonation propagation, the detonation establishment process is
understood here in three stages. During the first stage (
𝑡 < 𝜏𝑐𝑦 𝑐𝑙𝑒
) of propagation, both cases show the same initial
transients leading to a detonation mainly dominated by the hydrogen/air injection. Major changes begin to occur during
the second stage (
𝜏𝑐𝑦𝑐 𝑙𝑒 < 𝑡 < 2𝜏𝑐𝑦 𝑐𝑙𝑒
). A consistently higher post-detonation temperature is observed in presence of
kerosene, possibly due to the burning that occurs over a wide region behind the wave front at both lower and higher
heights. This is also consistent with the 0D analyses in the previous section that showed a higher
𝑇𝐶𝐽
and
𝑇𝑝𝑠
in presence
of kerosene. At this stage, as the heat release rate contours show, both hydrogen and gaseous kerosene are burning to
enable the detonation to self-propagate. Even though HRRKERO is not stronger than HRRH2, the overall HRR is much
weaker in absence of the kerosene injection for
𝜙𝑔=0.5
. This suggests that the contribution of kerosene burning is to
maintain the coupling of the shock with the flame by providing additional energy, and in turn it helps the hydrogen to
burn stronger too. Without kerosene injection, on the other hand, hydrogen combustion occurs at the near-injection field
where most of the fuel is consumed upon the passage of the front. In this scenario the mixture is highly lean leading to a
more distributed regions of high and low temperature spots resulting in a weak shock front. During stage-3 (
𝑡 > 5𝜏𝑐𝑦𝑐𝑙 𝑒
),
the detonation for the pure gas case (
𝜙𝑔=0.5
,
𝜙𝑙=0.0
) quenches; however the injection of liquid droplets (
𝜙𝑔=0.5
,
𝜙𝑙=0.5
) allows for a continuous detonation propagation, driven by a mixture of kerosene-
H2
and air. The kerosene
combustion occurs at the shock front, with a similar magnitude of the heat release rate as that due to the hydrogen,
15

keeping the propagation stable. For the pure gas condition, the leading shock and flame are completely decoupled and
the detonation is extinguished leaving the newly injected mixture to burn partly through deflagration.
C. Detonation structure during self-sustained propagation
The three-dimensional (3D) structure of the detonation front as it propagates through the injected droplets is shown
in Fig. 7 for all cases. The iso-surfaces of pressure (1-5 MPa) colored by temperature are used to identify the location
and behavior of the detonation wave. The results shown here correspond to representative instantaneous snapshots
during self-sustained propagation; the solutions at other instances in time have also been analyzed and they also lead to
the same conclusions discussed next.
(a) (b)
(c) (d)
Oblique shock
Detonation front
Droplets refill
Droplets
evaporated
Propagation
direction
Fig. 7 Instantaneous 3D snap shots of pressure iso-surfaces (1-5 MPa) colored by temperature and liquid
droplets colored by diameter for (a)
𝜙𝑔
= 0.7,
𝜙𝑙
= 0.3 -
𝑑𝑝,0
= 5
𝜇
m (b)
𝜙𝑔
= 0.7,
𝜙𝑙
= 0.3 -
𝑑𝑝,0
= 20
𝜇
m, (c)
𝜙𝑔
=
0.5, 𝜙𝑙= 0.5 - 𝑑𝑝, 0= 5 𝜇m, and (d) 𝜙𝑔= 0.5, 𝜙𝑙= 0.5 - 𝑑𝑝, 0= 20 𝜇m.
The 3D front is highly corrugated with the bow-shock extending from the refill region to the outflow. The structure is
similar to waves found in typical full-scale RDEs [
6
,
8
,
9
,
13
]. This is further visualized by extracting a two-dimensional
(2D) slice taken at mid-chamber. These results in Fig. 8 clearly show a strong coupling between high pressure of the
shock front with high temperature of the combustion processes. The detonation wave extends up to approximately y =
10 mm for all cases until a bow-shock emanates extending to the chamber outflow. Overall, the 3D wave structure is
similar across all cases, but the droplet fields do show noticeable differences depending on the injected droplet size. For
instance, for the the cases with
𝑑𝑝=5𝜇
m (see Fig. 7a and Fig. 7c), many liquid droplets are completely evaporated
16

in the post-detonation region behind the shock front. Furthermore, a large spatial variation in the droplet diameter is
observed ahead and behind of the detonation wave indicative of evaporation due to the high temperature products from
previous cycles. On the other hand, for
𝑑𝑝=20 𝜇
m, the variation in droplet diameter is observed primarily along
the chamber height and certain small-sized droplets are still seen leaving the domain. All simulated cases with spray
injection show similar overall features; however for a more quantitative comparison, a spatial averaging of the flow
properties is conducted to understand the effects of
𝜙𝑔
and
𝑑𝑝,0
. This process is further quantified and understood in the
next sections.
T [K] 𝑌
𝐾𝑒𝑟𝑜
(a)
(b)
(c)
(d)
𝑌
𝐻2
𝝓𝒈= 𝟎. 𝟕
,
𝝓𝒍= 𝟎. 𝟑
𝒅𝒑,𝟎 = 𝟓 𝝁𝒎
𝝓𝒈= 𝟎. 𝟓
,
𝝓𝒍= 𝟎. 𝟓
𝒅𝒑,𝟎 = 𝟓 𝝁𝒎
𝝓𝒈= 𝟎. 𝟓
,
𝝓𝒍= 𝟎. 𝟓
𝒅𝒑,𝟎 =𝟐𝟎 𝝁𝒎
𝝓𝒈= 𝟎. 𝟕
,
𝝓𝒍= 𝟎. 𝟑
𝒅𝒑.𝟎 =𝟐𝟎 𝝁𝒎
Fig. 8 Instantaneous 2D snap shots of temperature (left), hydrogen mass fraction (middle), and kerosene vapor
(right) for (a)
𝜙𝑔
= 0.7,
𝜙𝑙
= 0.3 -
𝑑𝑝,0
= 5
𝜇
m (b)
𝜙𝑔
= 0.7,
𝜙𝑙
= 0.3 -
𝑑𝑝,0
= 20
𝜇
m, (c)
𝜙𝑔
= 0.5,
𝜙𝑙
= 0.5 -
𝑑𝑝,0
= 5
𝜇
m, and (d)
𝜙𝑔
= 0.5,
𝜙𝑙
= 0.5 -
𝑑𝑝,0
= 20
𝜇
m taken at mid-channel (x-y plane). The contours of pressure (1-5
MPa) are shown in black to identify location of the propagating detonation.
To gain a quantitative understanding, the data at a representative instant is averaged over the channel width and
profiles are extracted at various heights (
0< 𝑦 < 20
mm). First, the pressure and the temperature profiles are presented
in Fig. 9. It is observed that these are similar for all the cases and only the results for a representative case (
𝜙𝑔=0.5
,
𝜙𝑙=0.5
with
𝑑𝑝=5𝜇
m) are described here. They are also compared to a pure gas case with
𝜙𝑔=1.0
to identify any
multiphase effects. Both show strong coupling between peak pressures and high temperatures of the reaction zone
indicative of a detonation mode of combustion which clearly extends up to a height of 10 mm. This region extends
over an appreciable length and a decay is found to occur after approximately 1-20 mm behind the shock front (
𝑥𝑑𝑒𝑡
),
indicating a distributed induction zone of the detonation in an averaged sense. Within the region of high kerosene vapor
which exists 10 mm behind the detonation wave (see Fig. 8), the temperature is around 1800 K (see Fig. 9). In this
17

region a complex coupling exists where the rapid evaporation of droplets in presence of hot product gas causes the gas
temperature to drop as energy is taken to support evaporation. The kerosene vapor at heights above the detonation
height continue to burn through deflagration, but it is unable to support the propagation of the wave directly. Past
simulations of full-scale non-premixed RDE combustors [
6
,
8
,
9
,
13
] have shown that the detonation thickness extends
to a wider length compared to a front propagating in a premix mixture. For
𝑦 > 10
mm, the jumps in the pressure are
smaller indicating the existence of a weaker shock, e.g., an oblique shock.
(b)        
(a)    
Fig. 9 Time-averaged profiles of pressure and temperature along the propagation direction at various chamber
heights for two representative cases: (a) pure gas-phase at
𝜙𝑔
= 1, and (b) two-phase at
𝜙𝑔
= 0.5,
𝜙𝑙
= 0.5 and
𝑑𝑝,0
= 5 𝜇m.
In order to clearly identify the relative effects of kerosene and hydrogen burning, phase-averaged heat release rates
due to the hydrogen and the kerosene burning are plotted in the shock normal direction in Fig. 10 and 11, respectively.
The heat release rate due to both fuels peak right behind the detonation front, and even though the heat release rate
due to kerosene vapor is lower than that of the hydrogen, both are of the same order of magnitude suggesting their
contribution toward sustaining the detonation wave. Furthermore, unlike the kerosene, the hydrogen heat release rate
drops off quickly at higher heights as also identified before.
18

(a)        (b)      
(c)        (d)      
Fig. 10 Time-averaged profiles of the heat release rate contribution due to hydrogen combustion (
𝐻𝑅𝑅𝐻2
) along
the propagation direction and at various chamber heights for all cases.
19

(a)        (b)      
(c)        (d)      
Fig. 11 Time-averaged profiles of the heat release rate contribution due to vapor kerosene combustion
(𝐻𝑅𝑅𝐾 𝐸 𝑅𝑂 ) along the propagation direction and at various chamber heights for all cases.
D. Fuel distribution in two-phase mixture
The gaseous and the vapor fuel distributions at the same representative instant are shown also in Fig. 8. Here,high
localized concentrations of gaseous
H2
are observed near the injector face, due to not only the discrete injectors, but
also suggesting poor mixing which can result in a locally weakened detonation wave. This poorly mixed state is further
corroborated by the existence of pockets of hydrogen fuel that still remain behind the detonation wave and are not
fully consumed. This remaining hydrogen fuel mixes with the high-temperature detonation products over a spatially
broad distance of about 20 mm behind the front where it ultimately burns in a deflagrative mode (as confirmed from
reaction rate contours later). Similar fuel injection characteristics have also been recently observed in detonation channel
experiments [59, 60].
Upon reducing
𝜙𝑔
from 0.7 to 0.5 (and increasing the kerosene injection), the penetration of
H2
drastically reduces
from
𝑦≈15
mm to
𝑦≈7
mm. If this was a pure
H2
detonation with
𝜙𝑔=1
, then the
H2
penetration would have been
higher, i.e.,
∼
20 mm [
33
], however, for the lean extreme of
𝜙𝑔=0.5
, a continuous propagating detonation could not
survive without the addition of liquid fuel. Regarding this, Bykovskii et al. [
3
] proposed a relation between mixture
20

ayer height (
ℎ
) and detonation cell size (
𝜆
) to estimate the critical value needed for self-sustained detonation. Assuming
purely premixed gaseous mixture, it was shown that
ℎ≈ (12 ±5)𝜆
. However, we note that this does not correlate well
with our findings here due to both the non-premixed nature of the injection and the combination of two fuels from
which the experimental data for
𝜆
is not available. For instance, experimental data [
61
] has shown that a detonation
propagating in premixed mixtures of
H2
-air at ambient conditions (
𝑃0
,
𝑇0
= 1 atm, 300 K),
𝜙
= 1.0, 0.7, 0.5 has
𝜆≈
8, 20, 95 mm, respectively, resulting in
ℎ≈
96, 240, and 1140 mm. It is clear from Fig. 8 that the detonation wave
propagates in a much smaller
ℎ
through a highly stratified mixture of both reactants and products, resulting in strong
cell size variations [
33
,
60
,
62
]. Hence, the correlation proposed by Bykovskii et al. may need to be revisited for
non-premixed injection with and without droplets including the effects of mixing variations.
A large concentration of kerosene vapor is found ahead of the detonation wave at higher heights, and behind the
detonation wave primarily at lower heights. Ahead of the detonation wave, the temperature is low at lower heights
due to the freshly injected mixture, but at higher heights the detonation products from the previous cycles have mixed
and this promotes vaporization. Behind the detonation wave, the hot temperature products cause a faster vaporization
of the spray, resulting in significant concentrations of kerosene vapor, while the
H2
is either consumed here due to
the detonation wave or due to the post-detonation deflagrative burning. As many previous works [
6
,
8
,
9
,
13
] have
shown, deflagration burning due to the high-temperature post-detonation products from previous cycles can result in the
consumption of freshly injected mixture. Although it can represent a source of inefficiency for the overall performance
of an RDE [
6
,
8
], pre-burning for two-phase mixtures could be beneficial as it allows for the droplets to evaporate prior
to the detonation passage. This phenomena is equivalent to the process E and C mentioned earlier. It is observed that
some amount of it is present even in the near-injection region due to pre-vaporization occurring within the plenum prior
to the droplets reaching the interior of the channel.
The axial profiles of both hydrogen and gaseous kerosene averaged along the depth of the channel at a representative
time are depicted in Fig. 12 and Fig. 13. As noted earlier, with the reduction of
𝜙𝑔
from 0.7 to 0.5, the hydrogen
fuel penetration is drastically reduced. As Fig. 12(a,b) show, for
𝜙𝑔
= 0.7,almost up to a height of
𝑦=10
mm the
averaged
H2
concentration stays above stoichiometry (
𝑌𝐻2,𝑠𝑡 𝑜𝑖 𝑐
= 0.0283), whereas for
𝜙𝑔
= 0.5, this is for
𝑦 < 5
mm.
Furthermore, across the detonation wave, the
H2
concentration drastically reduces, and this suggests that the detonation
wave is primarily sustained by the the
H2
fuel near the bottom of the chamber. This is the case for
𝜙𝑔=0.7
up to almost
𝑦=10 mm, whereas, for 𝜙𝑔=0.5it is for 𝑦 < 5mm.
As shown in Fig. 13, the kerosene vapor mass-fraction increases soon after the detonation wave in a gradual manner,
and this suggests fuel vaporization due to the high-temperature post-detonation products (process C,E). This is most
pronounced at
𝑦=0
mm, but it is also observed at higher heights. Similar to
H2
, the kerosene vapor also reduces at
the detonation front, suggesting a simultaneous burning. The overall kerosene vapor concentration reduces and the
variation along the detonation wave is also more gradual at higher chamber heights. This gradual variation across the
21

detonation wave (unlike
H2
) could be due to two possible reasons. It may suggest a decoupled and or a weaker burning
of kerosene vapor behind the shock as compared to a detonation wave, or it could be due to the vaporization of liquid
kerosene behind the detonation wave, that increases the kerosene vapor concentration in turn. These effects are further
analyzed in the next section.
(a)         (b)       
(c)         (d)       
Fig. 12 Time-averaged profiles along the propagation direction of hydrogen mass fraction (
𝑌𝐻2
) at various
chamber heights for all cases. The pressure profile (symbol line) taken at y = 0.0 mm is used to identify the
detonation location.
22

(a)         (b)      
(c)         (d)       
Fig. 13 Time-averaged profiles along the propagation direction of vapor kerosene mass fraction (
𝑌𝐾 𝐸𝑅 𝑂
) at
various chamber heights for all cases. The pressure profile (symbol line) taken at y = 0.0 mm is used to identify
the detonation location.
E. Droplets vaporization and fuel burning
To understand the fuel vaporization process and its effects on the burning, the evaporation rates of kerosene are
spanwise averaged and they are shown in Fig. 14. A significant variation is obtained along the chamber length for the
smaller droplets compared to the larger ones where the profiles are relatively uniform. This observation can be understood
through a time-scale analysis of the liquid vaporization, that is estimated as
𝜏𝑒𝑣𝑎 𝑝 =𝜌𝑝𝑑𝑝 ,0/(8𝜌 𝐷 ln(1+𝑋𝑟) )
[
63
],
where
𝜌𝑝
,
𝑑𝑝,0
,
𝐷
, and
𝑋𝑟
are the droplet density, initial diameter, vapor diffusivity, and molar ratio, respectively.
The time-scale of liquid vaporization is estimated to be
𝜏𝑒𝑣𝑎 𝑝 =0.8
ms (for
𝑑𝑝,0=20 𝜇
m) and
𝜏𝑒𝑣𝑎 𝑝 =0.05
ms (for
𝑑𝑝,0=5𝜇
m) based on a single droplet evaporation in a representative quiescent environment at 1273.5 K. Additionally,
the detonation cycle time is used to normalize the vaporization time-scale resulting in a
𝜏𝑒𝑣𝑎 𝑝 /𝜏𝑐𝑦 𝑐𝑙𝑒
of
∼0.625
for
𝑑𝑝,0=5𝜇
m, and
𝜏𝑒𝑣𝑎 𝑝 /𝜏𝑐𝑦 𝑐𝑙𝑒 =10
for
𝑑𝑝,0=20
micron. For the smaller diameters, this suggests that the droplets
would vaporize within a single detonation wave cycle, but for larger sizes, the droplets would only vaporize after multiple
cycles of the detonation wave, possibly leading to an incomplete vaporization of the injected liquid fuel. This is critical
in realistic liquid fueled RDEs as it will be of utmost importance to maximize the amount of vaporized liquid prior to
the passage of a detonation wave. The phase-averaged droplet Sauter Mean Diameter (SMD) is computed and shown
along the shock normal direction in Fig. 15. Consistent with the evaporation profiles,
𝑑𝑝=5𝜇𝑚
shows a significant
23

and gradual reduction in the droplet size behind the detonation wave due to the evaporation process, whereas, the
𝑑𝑝=20 𝜇𝑚
droplets mostly show a reduction in the SMD along the chamber height and not in the direction of the
detonation propagation.
(a)         (b)       
(c)         (d)       
Fig. 14 Time-averaged profiles along the propagation direction of droplets evaporation rate (
¤𝑚𝑑
) at various
chamber heights for all cases. The pressure profile (symbol line) taken at y = 0.0 mm is used to identify the
detonation location.
24

(a) 𝜙𝑔= 0.7, 𝑑𝑝= 5 𝜇𝑚 (b) 𝜙𝑔= 0.7, 𝑑𝑝=20 𝜇𝑚
(c) 𝜙𝑔= 0.5, 𝑑𝑝= 5 𝜇𝑚 (d) 𝜙𝑔= 0.5, 𝑑𝑝=20 𝜇𝑚
Fig. 15 Time-averaged profiles along the propagation direction of droplet Sauter Mean Diameter (SMD) (
𝑑𝑝
) at
various chamber heights for all cases. The pressure profile (symbol line) taken at y = 0.0 mm is used to identify
the detonation location.
To further understand the vaporization process and determine the amount of available fuel prior to the detonation
arrival, the concentration of hydrogen (
𝑌𝐻2
), kerosene vapor (
𝑌𝐾 𝐸𝑅 𝑂
), and liquid volume fraction (
𝛼𝑑
) are averaged
along the span of the channel and plotted along the height. Each profile, shown in Fig. 16, is extracted ahead of the
front based on its location (
𝑥𝑑𝑒𝑡
) and averaged over a single propagation cycle. Because of the fast vaporization, the
liquid volume-fraction profile for the case with
𝑑𝑝,0=5𝜇
m reduces from an initial value of
10−2
to approximately zero
at a channel height of about 12 mm (
𝜙𝑔=0.7
) and 15 mm (
𝜙𝑔=0.5
). This quantity can be defined as the droplet
penetration or dispersion height [
64
] and it corresponds to a refill distance, which, for this case, is similar to the height
of the detonation front. On the contrary, the large sized droplets show a decrease in
𝛼𝑑
from
10−1
to
10−3
by
𝑦=15
mm
and a further reduction to
10−4
by
𝑦=40
mm (see Fig. 16). Again, this shows the negative impact that large droplets
can have on the overall performance of an RDE as large quantity of fuel cannot be mixed and burned in a detonation
mode of combustion. As opposed to the hydrogen, increased values of kerosene vapor concentration are obtained at
larger channel heights
𝑦 > 15
mm as also presented previously in Fig. 8. Upon entering the channel, droplets interact
with the post-detonation high-temperature gases from the previous cycle and begin to vaporize along the chamber height.
25

(a)    
 
   
KERO

(b)    
Fig. 16 Time-averaged axial profiles ahead of the detonation wave of mass fraction of gaseous hydrogen (
𝑌𝐻2
),
kerosene (𝑌𝐾 𝐸𝑅 𝑂), and liquid volume fraction (𝛼𝑑).
Corresponding to this vaporization process, now to identify the location of burning during the self-sustained
detonation propagation, the 2D contours of reaction rates for kerosene and
H2
are shown at the same instant as Fig. 8 in
Fig. 17. Unlike the
H2
, which burns in a concentrated manner near the bottom at the detonation front, the kerosene vapor
shows a weak and uneven burning that is predominately present along the oblique shock. Another notable characteristic
in Fig. 17 is that a noticeable amount of heat release is found ahead of the detonation front. In the region ahead of the
wave, the fuel that is refilled mixes with high-temperature products from previous cycles and any remaining oxidizer
in order to react and burn via deflagration. Because of this, the burning of kerosene vapor predominately occurs in
the post-detonation region for
𝑦 > 10
mm. Profiles across the pressure front (see Fig. 12 and Fig. 13) suggest that
the burning of kerosene might not be occurring at the shock front, but its burning still increases the post-detonation
conditions to strengthen the existing detonation wave propagation (process A, B). This results in a coupled feedback
loop between the heat release and the shock compressed ignition.
26

𝝓𝒈= 𝟎. 𝟕
,
𝝓𝒍= 𝟎. 𝟑
𝒅𝒑,𝟎 = 𝟓 𝝁𝒎
𝝓𝒈= 𝟎. 𝟕
,
𝝓𝒍= 𝟎. 𝟑
𝒅𝒑,𝟎 =𝟐𝟎 𝝁𝒎
𝝓𝒈= 𝟎. 𝟓
,
𝝓𝒍= 𝟎. 𝟓
𝒅𝒑,𝟎 = 𝟓 𝝁𝒎
𝝓𝒈= 𝟎. 𝟓
,
𝝓𝒍= 𝟎. 𝟓
𝒅𝒑,𝟎 =𝟐𝟎 𝝁𝒎
ሶ𝜔𝑘[𝑘𝑔
𝑚3𝑠]
Fig. 17 Instantaneous 2D snap shots of hydrogen
¤𝜔𝐻2
(left) and vapor kerosene
¤𝜔𝐾 𝐸 𝑅𝑂
(right) reaction rates
for all cases.
In order to quantify the location and amount of vaporization and combustion occurring during self-sustain
propagation, volume-integrated evaporation rate (
¤𝑚𝑑
), hydrogen reaction rate (
¤𝜔𝐻2
), and kerosene vapor reaction rate
(
¤𝜔𝐾 𝐸 𝑅𝑂
) are extracted for different zones based on the location of the detonation front (
𝑥𝑑𝑒𝑡
). The zones are: upstream
(
𝑥𝑑𝑒𝑡 < 𝑥 < 𝑥𝑑 𝑒𝑡 +20
mm), downstream (
𝑥𝑑𝑒𝑡 −40 < 𝑥 < 𝑥𝑑 𝑒𝑡 −20
mm), and across (
𝑥𝑑𝑒𝑡 −20 < 𝑥 < 𝑥𝑑 𝑒𝑡
mm)
the detonation wave. The values are normalized by their respective volume-integrated quantities across the entire
computational domain.
27

(a)
Upstream
Downstream
Across
(b) (c)
Fig. 18 Normalized volume-integrated (a) vaporization rate (
¤𝑚𝑑
), (b) hydrogen reaction rate (
¤𝜔𝐻2
), and
(c) kerosene vapor reaction rate (
¤𝜔𝐾 𝐸 𝑅𝑂
) distributions in different zones. The three zones are upstream
(
𝑥𝑑𝑒𝑡 < 𝑥 < 𝑥𝑑 𝑒𝑡 +20
mm), downstream (
𝑥𝑑𝑒𝑡 −40 < 𝑥 < 𝑥𝑑 𝑒𝑡 −20
mm) and across the detonation wave
(𝑥𝑑𝑒𝑡 −20 < 𝑥 < 𝑥𝑑𝑒𝑡 mm).
The results are presented in Fig. 18 as bar charts. As indicated by the fractional amounts, most vaporization (see
Fig. 14(a,c)) and fuel consumption (see Fig. 8) occurs over the detonation location region. The distribution of kerosene
burning across the various zones is approximately similar between all the cases. The relative amount of hydrogen that
burns across the detonation wave is higher for
𝜙𝑔=0.5
case as compared to
𝜙𝑔=0.7
, since there is a lesser amount of
hydrogen available overall, and its burning away from the detonation wave reduces as a result. The relative vaporization
rates also shows differences, and
𝑑𝑝=5𝜇𝑚
vaporizes more strongly across the detonation wave as droplets have shorter
evaporation time-scales. This is consistent with our observations in both Fig. 7 and Fig. 8 where a small number of
droplets are present and the amount of both fuels is already consumed either over the detonation or upstream of the
wave. Due to the longer time-scale of the larger droplets, a significant portion of the overall vaporization still occurs
downstream of the front. It is interesting to notice the effect of hydrogen and kerosene reaction rates in the three zones.
Most of the hydrogen is consumed over the detonation front and ahead of it (i.e deflagration). The kerosene vapor reacts
largely over the detonation front but also partly upstream and downstream of it. Upon stable propagation, the flow
features and the observed interlinking of kerosene and H2burning are summarized using a schematic in Fig. 19.
28

Oblique
Shock
Detonation
Wave
Evaporated
Kerosene
Refill H2 fuel
Unburned
kerosene fuel
zone
Y
Z
X
Unburned H2
fuel
Fig. 19 Illustration of spray/detonation features and evaporation/burning location as studied in this article.
In order to gain an understanding into the droplet sizes across the detonation wave, droplet size probability density
functions (PDFs) were computed ahead of and behind the detonation wave, and these are plotted in Fig. 20. For all
the cases, the droplet PDF behind the detonation wave approximately maintains the same shape but shifts towards the
smaller droplet sizes suggesting a gradual vaporization. The droplet PDFs for
𝑑𝑝=5𝜇𝑚
show a monotonic behavior,
however,
𝑑𝑝=20 𝜇𝑚
shows an oscillatory behavior and this is hypothesized due to the remaining droplets from the
previous wave cycle as the vaporization process for 𝑑𝑝=20 𝜇𝑚 has a larger time-scale.
29

(a)        (b)      
(c)        (d)      
Upstream
Downstream
Fig. 20 Probability density function (PDF) of droplets diameter distribution upstream (
𝑥𝑑𝑒𝑡 < 𝑥 < 𝑥 𝑑𝑒 𝑡 +20
mm) and downstream (𝑥𝑑𝑒𝑡 −20 < 𝑥 < 𝑥𝑑𝑒𝑡 mm) of the detonation wave for each case.
VI. Conclusions
Detonation propagation in a 3D channel with discrete gaseous
H2
injection and a continuous slot injection of liquid
kerosene-air is simulated to understand interactions of a detonation wave with spray dispersion, vaporization, and
burning. The simulations are conducted using a reactive compressible Eulerian-Lagrangian (EL) approach.
The simulations show that for lean 𝜙𝑔=0.5, without any liquid fuel injection, the detonation wave cannot survive
and it results in extinctions and re-ignitions until quenching at
𝑡 > 5𝜏𝑐𝑦𝑐𝑙 𝑒
. The shock front decouples from the reactions
as a result. On the other hand, when liquid kerosene droplets are injected, the additional heat release from the kerosene
vapor combustion provides sufficient energy to the shock, allowing the detonation wave to continuously propagate
through the domain. The detonation velocities are predicted to be between
60%
and
70%
of the CJ value which is
consistent with undergoing experimental research in liquid fueled RDEs [3, 21–24].
Hydrogen combustion occurs predominately near the injectors where a strong shock-reaction coupling is maintained.
The injected liquid droplets vaporize behind and ahead the front in presence of hot products from the current and the
previous cycles, respectively. In this manner, the propagating
H2
detonation helps the droplets to continuously vaporize
and burn. The kerosene vapor then burns primarily at higher chamber heights (
𝑦=10 −15
mm) and provides enough
30

energy for sustained propagation of the detonation wave. Conducted 0D/1D analyses for vaporization and combustion
result in a similar conclusion; that
H2
helps kerosene to vaporize and burn due to its faster reactivity, whereas, kerosene
helps lean
H2
to detonate by providing enough energy for a sustained front propagation. However, the 3D simulation
results provide further insights into this process and show that
H2
detonates primarily near the bottom of the chamber,
whereas, the kerosene burns at higher heights, behind the bow shock, in an uneven and a relatively weak manner. The
contribution of kerosene combustion is to raise the post-detonation temperature behind both the detonation wave and
the bow-shock such that a continuous coupling between the shock front and the reactions is maintained. Through the
extraction of the heat release rate contribution of both hydrogen and vapor kerosene combustion, it was shown that the
kerosene and the H2burning are closely interlinked.
Through quantification of various properties across of the detonation wave, it is found that the use of smaller droplets
(
𝑑𝑝,0=5𝜇
m) results in a much larger variation of the vaporization rates across the detonation wave as compared to
larger droplets (
𝑑𝑝,0=20 𝜇
m). A vaporization timescale analysis shows that for smaller diameters, the droplets will
completely evaporate within a single cycle of the detonation passage; but larger droplets, on the other hand, may require
almost 10 cycles. This could result in a negative impact on the performance of a realistic RDE because some liquid fuel
may not even burn [
13
]. Volume-integrated quantities in different zones show that a large portion of the vaporization
and combustion occurs across the detonation front as expected. However, larger droplets show about
10%
of the total
vaporization downstream of the detonation front and about
10%
upstream of it as compared to a nearly negligible
amount of those for
𝑑𝑝,0=5𝜇
m. The probability density functions of the droplet size distribution across the detonation
wave provide insights on the process of slow vaporization for both smaller and larger droplets.
In summary, the addition of liquid kerosene to a lean hydrogen/air system has substantial effects on the propensity of
a detonation wave. The hydrogen combustion improves the evaporation rate and the reactivity of the liquid fuel; and
these in turn release energy to sustain the detonation propagation. Although a lean gaseous hydrogen/air detonation
propagation was not achievable, the addition of kerosene as a liquid fuel helped. Similarly, only liquid kerosene as a fuel
in air will also not be able to result in a sustained detonation propagation, but the addition of hydrogen helps. This
overall phenomenon could be described as a H2-driven, kerosene-sustained two-phase detonation.
Supplementary Material
See the supplementary material for animations of the flow fields associated with the transient propagation of the
detonation wave when liquid droplets are injected.
Acknowledgments
This work is supported in part by the Georgia Tech Foundation funds for the Hightower Professorship. The second
author is supported in part by NASA Glenn Research Center grant. The computational resource provided by the Georgia
31

Tech Partnership for an Advanced Computing Environment (PACE) is greatly appreciated.
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