kazimardanov2018

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ISSN 0021-8944, Journal of Applied Mechanics and Technical Physics, 2018, Vol. 59, No. 7, pp. 1251–1260. © Pleiades Publishing, Ltd., 2018.
Original Russian Text © M.G. Kazimardanov, S.V. Mingalev, T.P. Lubimova, L.Yu. Gomzikov, 2018, published in Vychislitel’naya Mekhanika Sploshnykh Sred,
2017, Vol. 10, No. 4, pp. 416–425.
Simulation of Primary Film Atomization
Due to Kelvin–Helmholtz Instability
M. G. Kazimardanova,b,*, S. V. Mingalevb,**, T. P. Lubimovaa,c,***, and L. Yu. Gomzikovb,****
a Perm State University, Perm, Russia
b AO ODK-Aviadvigatel, Perm, Russia
c Institute of Continuous Media Mechanics, Ural Branch, Russian Academy of Sciences, Perm, Russia
*e-mail: kazimardano[email protected]
**e-mail: smingalio[email protected]
***e-mail: lubimov[email protected]
****e-mail: [e-mail]
Received June 15, 2017; in final form, December 30, 2017
Abstract—Liquid film atomization under a high-speed air flow (water was considered as the liquid)
due to the Kelvin–Helmholtz instability is studied using the volume of fluid (VOF) method. We
develop an approach for modeling the primary breakup and use it to investigate the grid convergence,
choose the optimal size of the grid cell, and calculate the primary breakup of the film in the channel.
The dependences of the mean break-off angle, the velocity modulus, and the Sauter droplet diameter
on the longitudinal coordinate of the channel are obtained. The step-by-step averaging over the
ensemble of droplets and over time allows us to get smooth coordinate dependences of the character-
istics of the ensemble of the droplet. The value of the most useful parameter for engineering applica-
tions, the mean Sauter diameter D32 (equal to the ratio of the mean droplet volume to its mean area)
is close to that obtained using a semiempirical formula from the literature, based on the experiment
where hot wax is atomized by a high-speed airflow. The dependence of the Sauter mean diameter on
the thickness of the liquid layer agrees qualitatively with the experimental dependence. The study of
the grid’s convergence showed that the number of the smallest droplets increases rapidly with decreas-
ing cell size. Their contribution to the average characteristics of the droplet’s ensemble, however,
remains insignificant; nonetheless, their input to the mean characteristics remains insignificant; thus,
there is no reason to decrease the grid cell size to account for small droplets.
Keywords: Kelvin–Helmholtz instability, volume of fluid (VOF) method, 2D flow, atomization, Sau-
ter mean diameter.
DOI: 10.1134/S0021894418070064
1. INTRODUCTION
This paper focuses on the atomization of a liquid surface under an incoming air flow in 2D using the
volume of fluid method (VOF) [27]. Currently, there are a number of articles on modeling the secondary
breakup of droplets in a turbulent flow. The continuous media approaches to the secondary breakup of
droplets and droplet ensembles, suggested by Luo and Svendsen [1] and Lehr et al. [2], have been used in
the ANSYS CFX and ANSYS Fluent software for years. The Taylor Analogy Breakup (TAB) [3] and
Rayleigh–Taylor Hybrid Model (KHRT) [4] methods are widely used for describing the dispersed phase
within the Lagrange approach. These methods allow accounting for the secondary droplet breakdown
with sufficient accuracy for engineering calculations; however, their suitability for modeling the primary
jet break-off remains debatable.
Direct numerical simulation of a droplet’s break-off from the surface of a jet or a film using the VOF
method appears to be the simplest solution of this problem. This approach was barely used in the past
because it requires considerable processing power. However, with the advances made in technology, arti-
cles on studying the breakup of jets and film via numerical simulation started to appear. Although quali-
tative results prevailed in the papers written five years ago (see, for example, [5, 6]), the latest papers [7–
10] present a quantitative comparison of the droplet distribution by size with the experimental data.

JOURNAL OF APPLIED MECHANICS AND TECHNICAL PHYSICS Vol. 59 No. 7 2018
1252 KAZIMARDANOV et al.
As the object of studying the features of a droplet’s break-off from the liquid surface, this work chose
a two-dimensional channel. One of its walls has two adjacent holes. Through one hole air is conducted at
a speed of 50 to 70 m/s and through the other hole water is conducted at a speed of 1 to 2 m/s. This prob-
lem setup was considered the simplest example of a system with the Kelvin–Helmholtz instability over
many years [11–14]. The causes of this type of instability, leading to the film’s breaking up into droplets,
were researched in [15], where the authors were interested in the conditions of the droplet’s break-off.
They also described the mechanism of droplet formation because of the development of the Kelvin–
Helmholtz instability. Later in [16], the dependence of the maximum angle at which the droplets move
after breaking off the liquid surface on the air speed was obtained based on the experiment. Using the
direct numerical simulation, the dependence of this angle of the air and liquid density ratio was obtained
in [17]. In the same work, the influence this ratio value has on the time interval between the beginning of
the Kelvin–Helmholtz wave’s origination and its destruction, leading to the droplet formation, was con-
sidered. In [18] the influence of the channel parameters and the relative air speed on the total volume of
the droplets breaking off the surface of the liquid per unit of time was analyzed.
The distribution of droplets by size was studied in [9], where it was discovered that the distribution peak
shifts with a decreasing grid cell size. Even with the finest grid used, the authors of [9] were unable to
obtain the distribution in the small droplet size region. In this case, the mass density of the droplets inside
the channel changed slightly with transition from one grid cell size to another.
The present paper is aligned with [9] and [17]; however, here, in addition to studying the features of the
atomization modeling using the VOF method, the influence of the problem’s parameters on the Sauter
mean diameter (equal to the ratio of the average droplet volume to their average area), the mean angle of
the droplet’s break-off from the surface of the liquid, and the mean velocity magnitude are also consid-
ered.
The following sections describe the VOF method, the approximations and assumptions used in mod-
eling, and the geometry of the problem. The results of the research on the solution’s convergence when
the grid step is changed are presented. The influence of the size of the area over which the averaging is
done on the mean characteristics of the droplet’s ensemble is studied. The pulsations of the characteristics
averaged over the droplet’s ensemble and the dependence of the characteristics of the droplet’s ensemble
on the problem parameters are considered.
2. THE VOLUME OF FLUID METHOD
2.1. The System of Equations
The VOF method uses the volume fraction of a liquid , which in a grid cell takes on values from 0 to 1.
If the volume fraction , the cell is filled with liquid and if , the cell is empty. In the present work,
the volume fraction of corresponds to the interface between the media.
The VOF method requires storing only a single variable in a cell, which makes it more economical in
terms of processing power compared to the similar methods described in [19]. The VOF method is not
reduced solely to solving the equation
as it also requires accurate algorithms to account for the transfer of the volume fraction function in which
the mass conservation law holds [20].
In the VOF model in ANSYS Fluent, a system of partial differential equations is solved, consisting of
the momentum equations of the air–water mixture.
and the energy of the mixture,
where is the mixture’s density, are the components of the velocity vector, is the pressure, is the
dynamic viscosity, is the surface tension force, is the density of the internal energy, is the coefficient
α
α=
1
α=
0
α=
0.5
=
∂
αρ + ∇ αρ =
∂∑v
2
1
0,
wjwj
j
t
==
∂
ρ+ ∇ρ =∇+ ∇η∇ +∇ +
∂∑∑
vvv vv
22
11
() ( ),
ijjii jijjii
jj
pF
t
()
=
∂
ρ+ ∇ ρ+ =∇κ∇
∂∑v
2
1
,
jj
j
EEpT
t
ρ
v
i
p
η
i
F
E
κ

JOURNAL OF APPLIED MECHANICS AND TECHNICAL PHYSICS Vol. 59 No. 7 2018
SIMULATION OF PRIMARY FILM ATOMIZATION 1253
of thermal conductivity, is the temperature, is the volume fraction of the liquid, and are respec-
tively the gradient and its component with respect to the coordinate, is the free index ( ), and is
the coordinate index ( ).
The equations linking the physical parameters of the mixture and its components take the form
Here is the specific heat at constant volume. Air is considered an ideal gas for which the Mendeleev–
Clapeyron equation holds:
where is the universal gas constant and is the molar mass of the air. The liquid is considered incom-
pressible. The internal energy density is defined by the expression
The surface tension force is modeled in the framework assuming the continuity of the surface force. In
this case, a two-component mixture is presented as one medium with the properties defined by the marker
function. In this work, the volume fraction of the liquid is chosen as the marker function [21, 22].
Therefore, the expression for the surface tension force takes the following form:
where is the surface tension coefficient.
2.2. Boundary Conditions
The no-slip conditions
and the absence of heat flux conditions
where is the normal to the region boundary, are set on the solid boundaries of the studied region.
At the input of the mixture’s components to the region, the following conditions are met:
—for the liquid
where and are the velocity and temperature of the liquid at the input;
—for the gas
where and are the velocity and temperature of the gas at the input.
At the output the pressure is set:
T
α
∇
∇
j
i
=
1, 2
i
j
=
1, 2
j
ρ=αρ + −αρ
(1 ) ,
wa
κ=ακ + −ακ
(1 ) ,
wa
η=αη + −αη
(1 ) ,
wa
() ()
=α ρ ρ + −α ρ ρ
vv v
() ()
(1 ) .
wa
wa
cc c
v
c
(
)
ρ=
,
a
pM RT
R
M
=
v
.
EcT
α
=
∂
ρ∇α ∇α
=σ = =
ρ+ρ ∂ ∇α
∑
2
1
2
,,,
j
ii
ii
j
aw j
n
k
Fkn
x
σ
=
v
0
i
=
∂
=
∂
∑
2
()
1
0,
b
j
jj
T
nx
()
b
j
n
=
==α=
∑v
2
()
1
,,1,
b
jj w w
j
nVTT
w
V
w
T
=
==α=
∑v
2
()
1
,,0,
b
jj a a
j
nVTT
a
V
a
T
=
out
.
pp

JOURNAL OF APPLIED MECHANICS AND TECHNICAL PHYSICS Vol. 59 No. 7 2018
1254 KAZIMARDANOV et al.
3. THE GEOMETRY OF THE PROBLEM
The studied region is a two-dimensional channel (Fig. 1). Through hole AB air is supplied at a speed
of 60 m/s and a temperature of K, and through hole OA the liquid is supplied at a speed of
m/s and a temperature of K. Borders BD and ОC are solid walls. The pressure at the
boundary CD is Pa. The liquid in the channel moves along its lower wall slower than air.
Due to the substantial difference in speed, the Kelvin–Helmholtz instability is originated and it leads to
the liquid film breaking up into ligaments and then into droplets.
4. APPROXIMATIONS AND ASSUMPTIONS USED IN MODELING
In this article the behavior of the droplet’s ensemble is studied in 2D. Cylinders of infinite length (lig-
aments) are considered. The diameters of ligaments are directly related to the diameters of the droplets
into which each of them would have broken up with time had the problem been three-dimensional. The
diameters of the ligaments, into which the film breaks up, depend on the instability mode (its type): short-
wave or long-wave [23]. The crossing line of the transition from one mode to another is determined by the
Weber number , where ρ is the media density, h is the film thickness, σ is the surface
tension coefficient, and is the speed of air. At the instability is considered long-wave; and
at , short-wave [23].
The ligament’s diameter for the short-wave mode is defined by the formula [23–25]:
(1)
where is the number of waves per unit of film length.
For the short-wave instability, the ligament’s diameter is related to the number of waves by the propor-
tionality coefficient , called the ligament constant [23]. For this type of instability, the ligaments break
off from the crests of the waves running along the film [26].
Let us make an estimation to discern the type of instability occurring in the two-dimensional case. For
the parameters ρ = 1.29 kg/m3, h = 1 × 10–3 m, m/s, and σ N/m, we obtain the
Weber number (therefore the short-wave instability type occurs in this problem); the number of
waves per film length unit ; and the expression for the ligament constant from
the transformed formula (1).
Let us use the results from [26], where the dependence of the Sauter mean diameter ( ) on the liga-
ment constant (see Fig. 2) is obtained for two turbulence models. Now we tabulate the expression for the
ligament constant and plot the corresponding graph (see line 4) and compare both dependence curves to
it. As can be seen from the figure, droplets with a diameter of about 30 μm agree with the mean ligament
diameter obtained below. The difference between the diameters of a droplet and a ligament in the range
from 30 to 100 μm is not above 25%, which is quite acceptable for engineering calculations; therefore, let
us further assume the ligament diameters to be the droplet diameters.
In the present paper, the ligament (droplet) diameter is obtained during the calculation. Then, accord-
ing to the formula
the droplet volume is calculated where is the droplet diameter.
=
a
V
=
300
a
T
=
1
w
V
=
300
w
T
=
out
101325
p
We
=ρ σ
2
(2 )
hU
U
We < 1.6875
We > 1.6875
=π
2,
lLs
dCK
()
=ρ σ
2
2
s
KU
L
C
≈
60
U
−
= 71.65 ×10
3
We 32
≈
×≈
4
310
s
K
×≈3
510
Ll
Cd
32
D
=π 3
6,
d
VD
d
V
D
Fig. 1. Geometry of problem.
y
B
A
O
C
D
x
Air
WaterWater
Water

JOURNAL OF APPLIED MECHANICS AND TECHNICAL PHYSICS Vol. 59 No. 7 2018
SIMULATION OF PRIMARY FILM ATOMIZATION 1255
5. THE RESULTS’ CONVERGENCE WHEN THE GRID STEP IS CHANGED
At the first stage of the work, the effect of the grid step on the calculation results was checked. The size
of hole ОA was taken to be 1 mm, hole АB was taken to be 3 mm, and the channel length was 12 mm. The
liquid was supplied at a speed of m/s and the gas was supplied at a speed of m/s. To estimate
the grid’s convergence, uniform meshes with square cells were considered, with their size changing in the
following order: 15, 7.5, 3.75, 1.875 μm. Table 1 presents the calculated number of droplets in each range
of droplet diameters (D) for various grid cell sizes Δ.
Based on the data in this table, we can conclude that with mesh refinement, the number of droplets
smaller than 10 μm increases significantly. A similar result was obtained in [9]. However, this raises the
question on how the droplet breaking off from film can be studied if the decrease of the grid cell size is not
followed by an independent number of droplets.
The answer is to exclude droplets smaller than 10 μm from consideration. As follows from Table 2,
droplets of this size make up less than 10% of the total number of droplets; however, the systematic error
=
1
w
V
=
60
a
V
Fig. 2. Dependence of Sauter mean diameter on ligament constant for two turbulence models from [26]: standard (line 1),
realizable (line 2), and experiment (line 3), and dependence chosen by authors of present paper (line 4).
0.80.70.60.5
CL
0.40.30.20.10
3
1
4
2
100
D32, m
90
80
70
60
50
40
30
20
10
Table 1. Number of droplets in each diameter range for various grids
Grid cell
size Δ, μm
Diameter range D, μm
0–10 11–20 21–30 31–40 41–50 51–60 61–70 71–80 81–90 91–100
Number of droplets, pcs
15 0027532212
7.5 0 31 35 15 5 5 2 2 1 2
3.75611033814742221
1.8753971443916753322
Table 2. Volume of droplets in each droplet diameter range related to total number of droplets for various grids
Grid cell
size Δ, μm
Diameter range D, μm
0–10 11–20 21–30 31–40 41–50 51–60 61–70 71–80 81–90 91–100
Relative volume of droplets, %
15 0 0 1.7 11.7 13.8 12.3 11.4 15.2 9.7 24.2
7.5 0 6.2 18.9 15.7 8.6 12.8 7.1 9.5 6.1 15.2
3.75 1.3 18 18.0 12.8 10.5 9.0 6.2 8.3 10.6 6.6
1.875 6.9 20 14.7 11.7 8.4 8.9 7.4 9.9 8.5 10.6