One-dimensional numerical simulations based on the hybrid Eulerian–Lagrangian approach are performed to investigate detonation dynamics in two-phase gas-droplet n-heptane/air mixtures with and without liquid fuel pre-vaporization. The reactive Navier–Stokes equations considering the two-way coupling for interphase exchanges of mass, momentum, energy, and species are solved with a skeletal mechanism consisting of 44 species and 112 reactions. The effects of n-heptane droplet diameter and equivalence ratio (ER) on average detonation speed and mode are studied. For pre-vaporization cases, the average detonation speed first decreases and then increases with droplet diameter ranging from 2.5 to 40 μm, which is minimum at 7.5–10 μm due to the competition between fuel vapor addition and droplet evaporative heat absorption. However, the average speed increases monotonically as the droplet ER increases from 0.2 to 1.2. A further increase in the droplet ER (e.g., 2.4) would lead to detonation suppression in the presence of large droplets (e.g., above 30 μm). The detonation is fully quenched when the droplet ER is 3.2. Similar observations are also made for the pure sprayed cases without n-heptane pre-vaporization, where the average speed increases rapidly for droplet ER of 0.2–0.8 and slowly for ER of 0.8–1.6. Various detonation modes are observed with respect to droplet diameter and equivalence ratio, either with or without fuel pre-vaporization. Generally, the pure sprayed cases show more irregular behaviors in detonation propagation. The laden droplets provide a new approach to control the intrinsically unstable or highly irregular behaviors of pure gas or pure sprayed detonations. The finite, small disturbances from the spatially non-uniform droplets, and enrichment from the droplet evaporative mass addition, are two essential mechanisms for the mitigation of the pulsating detonation.

1.
T. H.
Pierce
and
J. A.
Nicholls
, “
Time variation in the reaction-zone structure of two-phase spray detonations
,”
Symp. (Int.) Combust.
14
(
1
),
1277
1284
(
1973
).
2.
S.
Eidelman
and
A.
Burcat
, “
The mechanism of a detonation wave enhancement in a two-phase combustible medium
,”
Symp. (Int.) Combust.
18
(
1
),
1661
1670
(
1981
).
3.
G.
Yao
,
B.
Zhang
,
G.
Xiu
,
C.
Bai
, and
P.
Liu
, “
The critical energy of direct initiation and detonation cell size in liquid hydrocarbon fuel/air mixtures
,”
Fuel
113
,
331
339
(
2013
).
4.
P.
Wolański
,
W.
Balicki
,
W.
Perkowski
, and
A.
Bilar
, “
Experimental research of liquid-fueled continuously rotating detonation chamber
,”
Shock Waves
31
(
7
),
807
812
(
2021
).
5.
J. L.
Li
,
W.
Fan
,
W.
Chen
,
K.
Wang
, and
C. J.
Yan
, “
Propulsive performance of a liquid kerosene/oxygen pulse detonation rocket engine
,”
Exp. Therm. Fluid Sci.
35
(
1
),
265
271
(
2011
).
6.
M. A.
Benmahammed
,
B.
Veyssiere
,
B. A.
Khasainov
, and
M.
Mar
, “
Effect of gaseous oxidizer composition on the detonability of isooctane-air sprays
,”
Combust. Flame
165
,
198
207
(
2016
).
7.
S.
Yungster
and
K.
Radhakrishnan
, “
Pulsating one-dimensional detonations in hydrogen-air mixtures
,”
Combust. Theory Modell.
8
(
4
),
745
770
(
2004
).
8.
S.
Yungster
and
K.
Radhakrishnan
, “
Structure and stability of one-dimensional detonations in ethylene-air mixtures
,”
Shock Waves
14
(
1–2
),
61
72
(
2005
).
9.
M.
Zhao
,
Z.
Ren
, and
H.
Zhang
, “
Pulsating detonative combustion in n-heptane/air mixtures under off-stoichiometric conditions
,”
Combust. Flame
226
,
285
301
(
2021
).
10.
H. D.
Ng
,
A. J.
Higgins
,
C. B.
Kiyanda
,
M. I.
Radulescu
,
J. H. S.
Lee
,
K. R.
Bates
, and
N.
Nikiforakis
, “
Nonlinear dynamics and chaos analysis of one-dimensional pulsating detonations
,”
Combust. Theory Modell.
9
(
1
),
159
170
(
2005
).
11.
M.
Kim
,
X.
Mi
,
C. B.
Kiyanda
, and
H. D.
Ng
, “
Nonlinear dynamics and chaos regularization of one-dimensional pulsating detonations with small sinusoidal density perturbations
,”
Proc. Combust. Inst.
38
(
3
),
3701
3708
(
2021
).
12.
H. D.
Ng
,
M. I.
Radulescu
,
A. J.
Higgins
,
N.
Nikiforakis
, and
J. H. S.
Lee
, “
Numerical investigation of the instability for one-dimensional Chapman-Jouguet detonations with chain-branching kinetics
,”
Combust. Theory Modell.
9
(
3
),
385
401
(
2005
).
13.
M.
Zhao
and
H.
Zhang
, “
Rotating detonative combustion in partially pre-vaporized dilute n-heptane sprays: Droplet size and equivalence ratio effects
,”
Fuel
304
,
121481
(
2021
).
14.
Q.
Meng
,
M.
Zhao
,
Y.
Xu
,
L.
Zhang
, and
H.
Zhang
, “
Structure and dynamics of spray detonation in n-heptane droplet/vapor/air mixtures
,”
Combust. Flame
249
,
112603
(
2023
).
15.
H.
Watanabe
,
A.
Matsuo
,
K.
Matsuoka
,
A.
Kawasaki
, and
J.
Kasahara
, “
Numerical investigation on propagation behavior of gaseous detonation in water spray
,”
Proc. Combust. Inst.
37
(
3
),
3617
3626
(
2019
).
16.
H.
Watanabe
,
A.
Matsuo
,
A.
Chinnayya
,
K.
Matsuoka
,
A.
Kawasaki
, and
J.
Kasahara
, “
Numerical analysis on behavior of dilute water droplets in detonation
,”
Proc. Combust. Inst.
38
(
3
),
3709
3716
(
2021
).
17.
N. M.
Rubtsov
,
The Modes of Gaseous Combustion
, 1st ed. (
Springer
,
2016
).
18.
W. A.
Sirignano
,
Fluid Dynamics and Transport of Droplets and Sprays
, 2nd ed. (
Cambridge University Press
,
Cambridge
,
UK
,
2010
).
19.
G. B.
Macpherson
,
N.
Nordin
, and
H. G.
Weller
, “
Particle tracking in unstructured, arbitrary polyhedral meshes for use in CFD and molecular dynamics
,”
Commun. Numer. Methods Eng.
25
(
3
),
263
273
(
2009
).
20.
Z.
Huang
and
H.
Zhang
, “
On the interactions between a propagating shock wave and evaporating water droplets
,”
Phys. Fluids
32
(
12
),
123315
(
2020
).
21.
C. T.
Crowe
,
J. D.
Schwarzkopf
,
M.
Sommerfeld
, and
Y.
Tsuji
,
Multiphase Flows With Droplets and Particles
(
CRC Press
,
New York
,
1998
).
22.
M.
Pilch
and
C. A.
Erdman
, “
Use of breakup time data and velocity history data to predict the maximum size of stable fragments for acceleration-induced breakup of a liquid drop
,”
Int. J. Multiphase Flow
13
(
6
),
741
757
(
1987
).
23.
A.
Chauvin
,
E.
Daniel
,
A.
Chinnayya
,
J.
Massoni
, and
G.
Jourdan
, “
Shock waves in sprays: Numerical study of secondary atomization and experimental comparison
,”
Shock Waves
26
(
4
),
403
415
(
2016
).
24.
S.
Liu
,
J. C.
Hewson
,
J. H.
Chen
, and
H.
Pitsch
, “
Effects of strain rate on high-pressure nonpremixed n-heptane autoignition in counterflow
,”
Combust. Flame
137
(
3
),
320
339
(
2004
).
25.
E.
Hairer
and
G.
Wanner
,
Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems
, 2nd ed. (
Springer
,
Berlin
,
Heidelberg
,
1991
).
26.
Z.
Huang
,
M.
Zhao
,
Y.
Xu
,
G.
Li
, and
H.
Zhang
, “
Eulerian-Lagrangian modelling of detonative combustion in two-phase gas-droplet mixtures with OpenFOAM: Validations and verifications
,”
Fuel
286
(
2
),
119402
(
2021
).
27.
P. E.
Kloeden
and
E.
Platen
,
Numerical Solution of Stochastic Differential Equations
(
Springer
,
New York
,
1992
).
28.
M.
Sontheimer
,
A.
Kronenburg
, and
O. T.
Stein
, “
Grid dependence of evaporation rates in Euler–Lagrange simulations of dilute sprays
,”
Combust. Flame
232
,
111515
(
2021
).
29.
R.
Zhou
and
S.
Hochgreb
, “
The behaviour of laminar stratified methane/air flames in counterflow
,”
Combust. Flame
160
(
6
),
1070
1082
(
2013
).
30.
X. C.
Mi
,
A. J.
Higgins
,
S.
Goroshin
, and
J. M.
Bergthorson
, “
The influence of spatial discreteness on the thermo-diffusive instability of flame propagation with infinite Lewis number
,”
Proc. Combust. Inst.
36
(
2
),
2359
2366
(
2017
).
31.
E.
Ott
,
C.
Grebogi
, and
J. A.
Yorke
, “
Controlling chaos
,”
Phys. Rev. Lett.
64
(
11
),
1196
1199
(
1990
).
32.
Y.
Ju
,
A.
Shimano
, and
O.
Inoue
, “
Vorticity generation and flame distortion induced by shock flame interaction
,”
Symp. (Int.) Combust.
27
,
735
741
(
1998
).
33.
Z.
Huang
,
M. J.
Cleary
,
Z.
Ren
, and
H.
Zhang
, “
Large eddy simulation of a supersonic lifted hydrogen flame with sparse-Lagrangian multiple mapping conditioning approach
,”
Combust. Flame
238
,
111756
(
2022
).
34.
T. F.
Lu
,
C. S.
Yoo
,
J. H.
Chen
, and
C. K.
Law
, “
Three-dimensional direct numerical simulation of a turbulent lifted hydrogen jet flame in heated coflow: A chemical explosive mode analysis
,”
J. Fluid Mech.
652
,
45
64
(
2010
).
35.
Z.
Luo
,
C. S.
Yoo
,
E. S.
Richardson
,
J. H.
Chen
,
C. K.
Law
, and
T.
Lu
, “
Chemical explosive mode analysis for a turbulent lifted ethylene jet flame in highly-heated coflow
,”
Combust. Flame
159
(
1
),
265
274
(
2012
).
36.
W.
Wu
,
Y.
Piao
,
Q.
Xie
, and
Z.
Ren
, “
Flame diagnostics with a conservative representation of chemical explosive mode analysis
,”
AIAA J.
57
(
4
),
1355
1363
(
2019
).
37.
D. A.
Goussis
,
H. G.
Im
,
H. N.
Najm
,
S.
Paolucci
, and
M.
Valorani
, “
The origin of CEMA and its relation to CSP
,”
Combust. Flame
227
,
396
401
(
2021
).
38.
J. H.
Lee
,
R.
Knystautas
, and
N.
Yoshikawa
, “
Photochemical initiation of gaseous detonations
,”
Acta Astronaut.
5
(
11–12
),
971
982
(
1978
).
39.
K. P.
Grogan
and
M.
Ihme
, “
Weak and strong ignition of hydrogen/oxygen mixtures in shock-tube systems
,”
Proc. Combust. Inst.
35
(
2
),
2181
2189
(
2015
).
40.
J. H. S.
Lee
,
The Detonation Phenomenon
(
Cambridge University Press
,
2008
).

Supplementary Material

You do not currently have access to this content.