High-fidelity simulations for polydispersed sprays in the Eulerian–Lagrangian framework need to incorporate subgrid-scale effects in the particle evolution equations. Although the quasi-linear evaporation rate formulation captures evaporating droplet statistics, further improvement is required when subgrid-scale velocity effects become essential. The subgrid dispersion model strongly affects droplets spatial distribution, and subsequently net evaporation rate, depending on how rapidly they are dispersed into the dry air region. The main original contribution of this study is to assess the performances of a number of commonly used dispersion models in a consistent manner, against a reference direct simulation results. The models considered are (i) discrete random walk, (ii) approximate deconvolution method, (iii) stochastic model based on the Langevin equation, and (iv) combined approximate deconvolution method with the Langevin equation. Mass and enthalpy transfer source terms together with droplet diameters and particle distributions were compared against corresponding direct numerical and large-eddy simulations without a model as reference cases. Numerical results at low Stokes and moderate Reynolds numbers indicate that the dispersion model choice does not affect Eulerian field averages or fluctuations. However, proper dispersion models are essential to capture droplet distributions in the far-field region after jet breakup for Stokes number smaller than unity. The unclosed Lagrangian momentum equation without any dispersion model most accurately reproduces direct numerical simulation in the near field.

1.
D. P.
Schmidt
and
M.
Corradini
, “
The internal flow of diesel fuel injector nozzles: A review
,”
Int. J. Engine Res.
2
,
1
22
(
2001
).
2.
L.
Bayvel
and
Z.
Orzechowski
,
Liquid Atomization
(
Routledge
,
2019
).
3.
P.
Jenny
,
D.
Roekaerts
, and
N.
Beishuizen
, “
Modeling of turbulent dilute spray combustion
,”
Prog. Energy Combust. Sci.
38
,
846
887
(
2012
).
4.
F.
Dalla Barba
and
F.
Picano
, “
Clustering and entrainment effects on the evaporation of dilute droplets in a turbulent jet
,”
Phys. Rev. Fluids
3
,
034304
(
2018
).
5.
P.
Ciottoli
,
F.
Battista
,
R. M.
Galassi
,
F.
Dalla Barba
, and
F.
Picano
, “
Direct numerical simulations of the evaporation of dilute sprays in turbulent swirling jets
,”
Flow, Turbul. Combust.
106
,
993
1015
(
2021
).
6.
J.
Wang
,
F.
Dalla Barba
, and
F.
Picano
, “
Direct numerical simulation of an evaporating turbulent diluted jet-spray at moderate Reynolds number
,”
Int. J. Multiphase Flow
137
,
103567
(
2021
).
7.
S.
Tenneti
,
R.
Garg
,
C.
Hrenya
,
R.
Fox
, and
S.
Subramaniam
, “
Direct numerical simulation of gas–solid suspensions at moderate Reynolds number: Quantifying the coupling between hydrodynamic forces and particle velocity fluctuations
,”
Powder Technol.
203
,
57
69
(
2010
).
8.
R. N.
Dahms
,
G. A.
Paczko
,
S. A.
Skeen
, and
L. M.
Pickett
, “
Understanding the ignition mechanism of high-pressure spray flames
,”
Proc. Combust. Inst.
36
,
2615
2623
(
2017
).
9.
D.
Noh
,
S.
Gallot-Lavallée
,
W. P.
Jones
, and
S.
Navarro-Martinez
, “
Comparison of droplet evaporation models for a turbulent, non-swirling jet flame with a polydisperse droplet distribution
,”
Combust. Flame
194
,
135
151
(
2018
).
10.
S.
Aggarwal
,
Y.
Xiao
, and
J.
Uthuppan
, “
Effect of stokes number on particle dispersion
,”
Atomization Sprays
4
,
223
236
(
1994
).
11.
Y.
Pei
,
M. J.
Davis
,
L. M.
Pickett
, and
S.
Som
, “
Engine combustion network (ECN): Global sensitivity analysis of spray a for different combustion vessels
,”
Combust. Flame
162
,
2337
2347
(
2015
).
12.
N.
Maes
,
S. A.
Skeen
,
M.
Bardi
,
R. P.
Fitzgerald
,
L.-M.
Malbec
,
G.
Bruneaux
,
L. M.
Pickett
,
K.
Yasutomi
, and
G.
Martin
, “
Spray penetration, combustion, and soot formation characteristics of the ECN spray C and spray D injectors in multiple combustion facilities
,”
Appl. Therm. Eng.
172
,
115136
(
2020
).
13.
M.
Manish
and
S.
Sahu
, “
Analysis of droplet clustering in air-assist sprays using Voronoi tessellations
,”
Phys. Fluids
30
,
123305
(
2018
).
14.
C.
Law
, “
Multicomponent droplet combustion
,”
Combustion Physics
(
Cambridge University Press
,
New York
,
2006
), pp.
585
602
.
15.
T.
Poinsot
and
D.
Veynante
,
Theoretical and Numerical Combustion
(
RT Edwards, Inc.
,
2005
).
16.
F. A.
Williams
,
Combustion Theory Benjamin
(
Cummings
,
Menlo Park
,
1985
).
17.
K. K.
Kuo
,
Principles of Combustion
(
U.S. Department of Energy
,
2005
).
18.
J.
Warnatz
,
U.
Maas
, and
R.
Dibble
,
Combustion: Physical and Chemical Fundamentals, Modeling and Simulation, Experiments, Pollutant Formation
(
Springer
,
Berlin
,
2001
), p.
298
.
19.
S. B.
Pope
,
Turbulent Flows
(
Cambridge University Press
,
2001
).
20.
G.-N.
Dixon-Lewis
, “
Flame structure and flame reaction kinetics ii. transport phenomena in multicomponent systems
,”
Proc. R. Soc. London, Ser. A
307
,
111
135
(
1968
).
21.
W. A.
Sirignano
, “
Fuel droplet vaporization and spray combustion theory
,”
Prog. Energy Combust. Sci.
9
,
291
322
(
1983
).
22.
W. A.
Sirignano
,
Fluid Dynamics and Transport of Droplets and Sprays
(
Cambridge University Press
,
2010
).
23.
B.
Abramzon
and
W.
Sirignano
, “
Droplet vaporization model for spray combustion calculations
,”
Int. Journal Heat Mass Transfer
32
,
1605
1618
(
1989
).
24.
R.
Miller
,
K.
Harstad
, and
J.
Bellan
, “
Evaluation of equilibrium and non-equilibrium evaporation models for many-droplet gas-liquid flow simulations
,”
Int. J. Multiphase Flow
24
,
1025
1055
(
1998
).
25.
M.
Yuen
and
L.
Chen
,
On Drag of Evaporating Liquid Droplets
(
Taylor & Francis
,
1976
).
26.
M.
Bini
and
W.
Jones
, “
Large-eddy simulation of particle-laden turbulent flows
,”
J. Fluid Mech.
614
,
207
252
(
2008
).
27.
W. P.
Jones
,
A. J.
Marquis
, and
D.
Noh
, “
An investigation of a turbulent spray flame using large eddy simulation with a stochastic breakup model
,”
Combust. Flame
186
,
277
298
(
2017
).
28.
D. B.
Spalding
, “
Combustion of liquid fuels
,”
Nature
165
,
160
160
(
1950
).
29.
G.
Godsave
, “
Studies of the combustion of drops in a fuel spray-the burning of single drops of fuel
,” in
Symposium (International) on Combustion
(
Elsevier
,
1953
), Vol.
4
, pp.
818
830
.
30.
N.
Frossling
, “
Uber die verdunstung fallender tropfen
,”
Gerlands Beitr. Geophys.
52
,
170
(
1938
).
31.
W.
Ranz
and
W. R.
Marshall
, “
Evaporation from drops. Part I
,”
Chem. Eng. Prog.
48
,
141
146
(
1952
).
32.
W.
Ranz
and
W. R.
Marshall
 et al., “
Evaporation from drops. Part II
,”
Chem. Eng. Prog.
48
,
173
180
(
1952b
).
33.
G.
Faeth
,
Current Status of Droplet and Liquid Combustion: Progress in Energy and Combustion Science
(
Pergamon Press
,
1977
), Vol.
3
, p.
4
.
34.
I.
Langmuir
, “
The evaporation of small spheres
,”
Phys. Rev.
12
,
368
(
1918
).
35.
F.
Dalla Barba
,
J.
Wang
, and
F.
Picano
, “
Revisiting D 2-law for the evaporation of dilute droplets
,”
Phys. Fluids
33
,
051701
(
2021
).
36.
Z.-G.
Feng
and
E. E.
Michaelides
, “
Heat and mass transfer coefficients of viscous spheres
,”
Int. J. Heat Mass Transfer
44
,
4445
4454
(
2001
).
37.
C.
Chiang
,
M.
Raju
, and
W.
Sirignano
, “
Numerical analysis of convecting, vaporizing fuel droplet with variable properties
,”
Int. J. Heat Mass Transfer
35
,
1307
1324
(
1992
).
38.
J.
Bellan
and
K.
Harstad
, “
Analysis of the convective evaporation of nondilute clusters of drops
,”
Int. J. Heat Mass Transfer
30
,
125
136
(
1987
).
39.
R.
Clift
,
J. R.
Grace
, and
M. E.
Weber
,
Bubbles, Drops, and Particles
(
Courier Corporation
,
2005
).
40.
J.
Smagorinsky
, “
General circulation experiments with the primitive equations
,”
Mon. Weather Rev.
91
,
99
164
(
1963
).
41.
S.
Ghosal
,
T. S.
Lund
,
P.
Moin
, and
K.
Akselvoll
, “
A dynamic localization model for large-eddy simulation of turbulent flows
,”
J. Fluid Mech.
286
,
229
255
(
1995
).
42.
D.
You
and
P.
Moin
, “
A dynamic global-coefficient subgrid-scale eddy-viscosity model for large-eddy simulation in complex geometries
,”
Phys. Fluids
19
,
065110
(
2007
).
43.
F.
Menter
and
Y.
Egorov
, “
The scale-adaptive simulation method for unsteady turbulent flow predictions. Part 1: Theory and model description
,”
Flow, Turbul. Combust.
85
,
113
138
(
2010
).
44.
J.-P.
Minier
,
S.
Chibbaro
, and
S. B.
Pope
, “
Guidelines for the formulation of Lagrangian stochastic models for particle simulations of single-phase and dispersed two-phase turbulent flows
,”
Phys. Fluids
26
,
113303
(
2014
).
45.
J.
MacInnes
and
F.
Bracco
, “
Stochastic particle dispersion modeling and the tracer-particle limit
,”
Phys. Fluids A
4
,
2809
2824
(
1992
).
46.
C. W.
Gardiner
 et al.,
Handbook of Stochastic Methods
(
Springer
,
Berlin
,
1985
), Vol.
3
.
47.
G. A.
Voth
,
A.
La porta
,
A. M.
Crawford
,
J.
Alexander
, and
E.
Bodenschatz
, “
Measurement of particle accelerations in fully developed turbulence
,”
J. Fluid Mech.
469
,
121
160
(
2002
).
48.
I.
Pesmazoglou
,
A.
Kempf
, and
S.
Navarro-Martinez
, “
A dynamic model for the Lagrangian stochastic dispersion coefficient
,”
Phys. Fluids
25
,
125108
(
2013
).
49.
F.
Doisneau
,
M.
Arienti
, and
J. C.
Oefelein
, “
A semi-Lagrangian transport method for kinetic problems with application to dense-to-dilute polydisperse reacting spray flows
,”
J. Comput. Phys.
329
,
48
72
(
2017
).
50.
S.
de Chaisemartin
,
F.
Laurent
,
M.
Massot
, and
J.
Reveillon
, “
Evaluation of Eulerian multi-fluid versus Lagrangian methods for ejection of polydisperse evaporating sprays by vortices
,” https://hal.archives-ouvertes.fr/hal-00169721/document (
2007
).
51.
S.
Stolz
,
N. A.
Adams
, and
L.
Kleiser
, “
An approximate deconvolution model for large-eddy simulation with application to incompressible wall-bounded flows
,”
Phys. Fluids
13
,
997
1015
(
2001
).
52.
C.
Van Cittered
, “
Zumeinflub der spaltbreite auf die intensitatsverteilung in spektrallinien
,”
Z. Phys.
65
,
547
563
(
1930
).
53.
J.
Kuerten
, “
Subgrid modeling in particle-laden channel flow
,”
Phys. Fluids
18
,
025108
(
2006
).
54.
M. J.
Cernick
,
S.
Tullis
, and
M.
Lightstone
, “
Particle subgrid scale modelling in large-eddy simulations of particle-laden turbulence
,”
J. Turbul.
16
,
101
135
(
2015
).
55.
Q.
Wang
,
X.
Zhao
, and
M.
Ihme
, “
A regularized deconvolution model for sub-grid dispersion in large eddy simulation of turbulent spray flames
,”
Combust. Flame
207
,
89
100
(
2019
).
56.
H.
Li
,
C.
Rutland
,
H.
Im
, and
F. H.
Perez
, “
Large-eddy simulation of turbulent dispersion effects in direct injection diesel and gasoline sprays
,”
SAE Int. J. Adv. Curr. Pract. Mobility
1
,
675
690
(
2019
).
57.
T.
Holzmann
,
Mathematics, Numerics, Derivations and OpenFOAM®
(
Holzmann CFD
,
Loeben
,
2016
).
58.
P. P.
Ciottoli
,
A.
Petrocchi
,
L.
Angelilli
,
F. E.
Hernandez Perez
,
R.
Malpica Galassi
,
F.
Picano
,
M.
Valorani
, and
H. G.
Im
, “
Uncertainty quantification analysis of RANS of spray jets
,” AIAA Paper No. 2020-3882,
2020
.
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