kazimardanov2018

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
AO ODK-Aviadvigatel, Perm, Russia
c
Institute of Continuous Media Mechanics, Ural Branch, Russian Academy of Sciences, Perm, Russia
*e-mail: [email protected]
**e-mail: [email protected]
***e-mail: [email protected]
****e-mail: [email protected]
b
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 characteristics of the ensemble of the droplet. The value of the most useful parameter for engineering applications, 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 decreasing 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, Sauter 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, articles on studying the breakup of jets and film via numerical simulation started to appear. Although qualitative 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.
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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 problem 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 considered. 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 considered.
The following sections describe the VOF method, the approximations and assumptions used in modeling, 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 α , which in a grid cell takes on values from 0 to 1.
If the volume fraction α = 1, the cell is filled with liquid and if α = 0 , the cell is empty. In the present work,
the volume fraction of α = 0.5 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
2
∂ αρ +
∇ j αρwv j = 0,
w
∂t
j =1
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.
∑
∂ ρv +
i
∂t
and the energy of the mixture,
2
∑
j =1
2
∇ j (ρv j vi ) = ∇i p +
∂ ρE +
∂t
2
∑∇ v
j =1
j j
∑ ∇ η(∇ v
j =1
j
i j
+ ∇ j vi ) + Fi ,
(ρE + p ) = ∇κ∇T ,
where ρ is the mixture’s density, vi are the components of the velocity vector, p is the pressure, η is the
dynamic viscosity, Fi is the surface tension force, E is the density of the internal energy, κ is the coefficient
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of thermal conductivity, T is the temperature, α is the volume fraction of the liquid, ∇ and ∇ j are respectively the gradient and its component with respect to the coordinate, i is the free index (i = 1,2 ), and j is
the coordinate index ( j = 1,2 ).
The equations linking the physical parameters of the mixture and its components take the form
ρ = αρw + (1 − α)ρa,
κ = ακw + (1 − α)κa,
η = αηw + (1 − α)ηa,
cv = α (ρw ρ) cv(w) + (1 − α) (ρa ρ) cv(a).
Here cv is the specific heat at constant volume. Air is considered an ideal gas for which the Mendeleev–
Clapeyron equation holds:
ρa = pM ( RT ) ,
where R is the universal gas constant and M is the molar mass of the air. The liquid is considered incompressible. The internal energy density is defined by the expression
E = cvT .
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 α is chosen as the marker function [21, 22].
Therefore, the expression for the surface tension force takes the following form:
Fi = σ
2
2ρk ∇i α
,
ρa + ρw
k=
∂n j
∑ ∂x ,
j =1
ni =
j
∇i α
,
∇α
where σ is the surface tension coefficient.
2.2. Boundary Conditions
The no-slip conditions
vi = 0
and the absence of heat flux conditions
2
∑n
j =1
(b )
j
∂T = 0,
∂x j
where n(jb) 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
2
∑v n
j =1
(b )
j j
= Vw ,
T = Tw ,
α = 1,
where Vw and Tw are the velocity and temperature of the liquid at the input;
—for the gas
2
∑v n
j =1
(b )
j j
= Va,
T = Ta,
α = 0,
where Va and Ta are the velocity and temperature of the gas at the input.
At the output the pressure is set:
p = pout.
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y
D
B
Air
Water
A
O
x C
Fig. 1. Geometry of problem.
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 Va = 60 m/s and a temperature of Ta = 300 K, and through hole OA the liquid is supplied at a speed of
Vw = 1 m/s and a temperature of Tw = 300 K. Borders BD and ОC are solid walls. The pressure at the
boundary CD is pout = 101325 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 (ligaments) 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): shortwave or long-wave [23]. The crossing line of the transition from one mode to another is determined by the
Weber number We = ρhU 2 (2σ), where ρ is the media density, h is the film thickness, σ is the surface
tension coefficient, and U is the speed of air. At We < 1.6875 the instability is considered long-wave; and
at We > 1.6875 , short-wave [23].
The ligament’s diameter for the short-wave mode is defined by the formula [23–25]:
(1)
dl = 2πCL K s ,
where K s = ρU 2 ( 2σ) 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 proportionality coefficient CL , 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, U ≈ 60 m/s, and σ = 71.65 × 10−3 N/m, we obtain the
Weber number We ≈ 32 (therefore the short-wave instability type occurs in this problem); the number of
waves per film length unit K s ≈ 3 × 104 ; and the expression for the ligament constant CL ≈ 5 × 103 dl from
the transformed formula (1).
Let us use the results from [26], where the dependence of the Sauter mean diameter ( D32 ) on the ligament 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, according to the formula
Vd = πD3 6,
the droplet volume Vd is calculated where D is the droplet diameter.
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D32, m
100
90
80
70
60
50
40
30
20
10
0
0.1
1255
4
2
1
3
0.2
0.3
0.4 0.5
CL
0.6
0.7
0.8
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).
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 Vw = 1 m/s and the gas was supplied at a speed of Va = 60 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
Table 1. Number of droplets in each diameter range for various grids
Diameter range D, μm
Grid cell
size Δ, μm
0–10
11–20
21–30
31–40
41–50
51–60
61–70
71–80
81–90
91–100
2
2
2
3
1
1
2
2
2
2
1
2
Number of droplets, pcs
15
7.5
3.75
1.875
0
0
61
397
0
31
103
144
2
35
38
39
7
15
14
16
5
5
7
7
3
5
4
5
2
2
2
3
Table 2. Volume of droplets in each droplet diameter range related to total number of droplets for various grids
Diameter range D, μm
Grid cell
size Δ, μm
0–10
11–20
21–30
31–40
41–50
51–60
61–70
71–80
81–90
91–100
15.2
9.5
8.3
9.9
9.7
6.1
10.6
8.5
24.2
15.2
6.6
10.6
Relative volume of droplets, %
15
7.5
3.75
1.875
0
0
1.3
6.9
0
6.2
18
20
1.7
18.9
18.0
14.7
11.7
15.7
12.8
11.7
13.8
8.6
10.5
8.4
12.3
12.8
9.0
8.9
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7.1
6.2
7.4
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Table 3. Sauter mean diameter D32 with and without consideration of droplets with size less than 10 μm for various
grids
Grid cell size Δ, μm
D32 > 0 , μm
D32 > 10 , μm
15
7.5
3.75
1.875
68.0
53.8
46.9
46.6
68.9
53.8
47.4
49.4
Table 4. Number of droplets in each diameter range related to area S (mm2) of corresponding zone, and their Sauter
mean diameter D32
Δ, μm
Droplet diameter range D, μm
Zone
11–20
21–30
31–40
41–50
D32 , μm
Relative droplet number
5D
5D
5C
4B
3B
2B
5.5
3.7
2.8
2.9
2.7
2.3
2.4
1.6
1.2
1.2
1.1
1
2.0
1.3
1
0.8
0.8
0.7
7.9
0.5
0.2
0.2
0.1
0.1
32
32
30
29
29
29
introduced by discarding the droplets with a diameter smaller than 10 μm does not exceed 7% (see Table 3). It
can be seen that even at a cell size of 3.75 μm, the Sauter mean diameter changes by less than 5% when the
grid cell size is halved.
The main interest for engineering calculations is focused on the value of the Sauter mean diameter.
Accounting for small droplets changes it insignificantly. This allows excluding such droplets from considerations, despite the fact that there are significantly more of them than large drops. Excluding the small
drops, in turn, opens the possibility of using a grid with a cell size of 3.75 μm.
6. THE EFFECT OF THE SIZE OF THE AREA ON THE MEAN CHARACTERISTICS
OF THE DROPLET’S ENSEMBLE
The mean characteristics of a droplet’s ensemble are of interest; however, to obtain them, it is necessary to carry out averaging over a certain area of space. To study the effect the averaging of the area size
has on the characteristics calculated, the area above the liquid film is split into zones (see Fig. 3). Zone
5D includes zone 5E, 5C includes zone 5D, 4B includes zone 5C, 3B includes zone 4B, and 2B includes
zone 3B.
Table 4 presents the results obtained in the calculations: the droplet concentration for each zone. Zone
5E has the smallest area. As can be seen from Table 4, the largest mean droplet concentration is observed
there for every droplet diameter range. If the size of this zone is increased (moving consistently from zone
5C
5D
5E
4B 3B 2B
2B—9.89 mm
3B—8.28 mm
4B—6.67 mm
5C—5.06 mm
5D—3.795 mm
5E—2.53 mm
Fig. 3. Schematic representation of investigated regions.
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Table 5. Droplet distribution by size (D, μm) for various zones
Droplet diameter range D, μm
Zone
11–20
21–30
31–40
41–50
Fraction of each droplet size range in zones
5C
4B
3B
2B
0.56
0.54
0.58
0.56
0.24
0.23
0.24
0.23
0.17
0.19
0.15
0.18
0.02
0.04
0.03
0.03
5E to zones 5D, 5C, 4B, 3B, and 2B), the mean droplet concentrations in each zone will decrease and the
Sauter mean diameter will change by not more than 7%. If instead of the concentration we consider the
droplet distribution by size in each zone, the dependence of the zone area disappears, as follows from
Table 5. Therefore, the area of the zone in which the droplets are examined affects the mean concentration
of the droplets in this zone but not their distribution by size or the Sauter mean diameter.
7. THE DEPENDENCE OF DROPLET ENSEMBLE CHARACTERISTICS
AVERAGED OVER SPACE ON TIME
Can the characteristics averaged over space depend not only on the averaging area size but also on
time? To answer this question, we studied the dependence of the Sauter mean diameter on time shown in
Fig. 4a, obtained by averaging over region 5E (Fig. 3). As follows from Fig. 4, the instantaneous values of
the Sauter mean diameter change randomly with time in the range from 10 to 100 μm. The changes occur
because of droplets of different diameters entering the studied region at the same time. On approaching
zone 5E the droplets can combine into a larger droplet.
The observed significant fluctuations of the Sauter mean diameter imply that the averaging over time
is required in addition to averaging over space (see Fig. 4b). From the figure we can conclude that it takes
about 6 ms to reach a stationary value.
8. THE DEPENDENCE OF THE MEAN DROPLET ENSEMBLE
CHARACTERISTICS ON THE COORDINATE
In the previous section, the Sauter mean diameter was determined from the droplets in region 5E
(Fig. 3) and averaged over the time period of about 6 ms. To determine the dependence of the characteristics of the mean droplet ensemble on the coordinate, the space above the liquid film was split into overlapping zones as shown in Fig. 5, in which zone 1 corresponded to zone 5E in Fig. 3. For each of these
zones, the characteristics presented in Fig. 6 were calculated, first averaging over the droplet ensemble in
D32, m
100
D32, m
(a)
(b)
32
80
27
60
22
40
17
20
12
0
3
4
5
6
t, ms
7
8
9
0
3
4
5
6
t, ms
7
8
9
Fig. 4. Dependence of Sauter mean diameter on time: instantaneous values (а); time-averaged values for which averaging
is carried out for different thicknesses of liquid layer ОА in Fig. 1, mm: 1 (dashed line); 1.5 (continuous line); 2 (dotteddashed line) (b).
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KAZIMARDANOV et al.
y
1 2
3
4 5 ...
60 m/s
y1
y0
1 m/s
x
O
x0
Fig. 5. Zones in which mean (averaged over coordinate, time, and ensemble) droplet characteristics were calculated.
γ, deg
(a)
3
2
1
0
4
9
v, m/s
50
14
x, mm
19
24
(b)
45
40
the zone and then over time. These dependences were calculated for three hole sizes through which the water was
supplied: 1, 1.5, and 2 mm.
From the shape of the curves in Fig. 6a, it can be concluded that on average droplets break off from the liquid
surface at a small angle and when approaching the exit
from the channel they tend to move parallel to the velocity
vector of the incoming air flow. After breaking off, the droplets accelerate and reach a speed close to the speed of air at
the exit of the channel (Fig. 6b). Note that the droplet speed
depends on the thickness of the liquid layer: the thicker the
layer the greater the mean droplet speed. The mean droplet
size increases when approaching the exit of the channel
(Fig. 6c) and takes on values of 20 to 30 μm at the exit.
This result can be compared to the value obtained by
the formula suggested by Mayer [27]:
23
30
25
20
4
9
D32, μm
27
14
x, mm
19
24
(c)
23
19
15
11
⎛ μ σ ρw ⎞
(2)
,
λmin = 2π 16 ⎜ w
2 ⎟
⎝ βρaVa ⎠
where the adjustable parameters F = 0.14 and β = 0.3
were used. Substituting the parameter values for water and
air in (2) (with the air density of ρa = 1.29 kg/m3), we
obtain a Sauter mean diameter of D32 ~ 33 μm, which is
sufficiently close to the calculated value.
Figure 7 presents a graph from [28], namely, the dependence of the mean droplet’s diameter on the diameter of
the injector’s nozzle. Consider the part of the graphs
framed by the rectangle. In this part the air speed of
106 m/s and the speed of the liquid (melted wax) of 24 m/s
are constant. The nozzle’s dimensions 0.04, 0.06, and 0.08
in inches coincide with the dimensions 1, 1.5, and 2 in millimeters of the hole ОА in Fig. 1. It can be seen that when
the diameter of the nozzle is increased the mean diameter
of the wax droplets also increases from 70 to 80 μm.
Figure 8 shows the dependence of the Sauter mean
diameter on the liquid’s layer thickness. The thicker the
layer the larger the Sauter mean diameter. From these
graphs, it follows that the modeling and the experiment are
qualitatively consistent.
D32 = 9 F λmin ,
2
35
4
9
14
x, mm
19
Fig. 6. Dependence of mean droplet characteristics: droplet break-off angle (а), velocity
modulus (b), and Sauter mean diameter (c),
on coordinate.
24
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D32, μm
100
90
80
70
60
50
40
30
D32, m
28
26
Relative
air speed = 106 m/s
24
Speed of the supplied liquid, m/s
3
6
12
24
22
20
0.5
228 m/s
20
0.04
1259
1.0
1.5
2.0
0.06 0.08 0.10
0.15 0.20
Inner tube diameter, inches
Fig. 7. Dependence of droplet diameter on
injector nozzle diameter.
2.5
h, mm
Fig. 8. Dependence of Sauter mean diameter
on liquid layer thickness.
From the points made above, we conclude that averaging over time makes it possible to obtain smooth
dependences of the characteristics of the droplet’s ensemble on the coordinate. The Sauter mean diameter
values are close to those calculated by the semiempirical formula (2).
9. CONCLUSIONS
This work studies a two-dimensional droplet breaking off from the surface of a liquid by the incoming
air flow using the VOF method. The dependences of the characteristics of the mean droplet on the coordinate are determined for various sizes of the hole supplying the liquid. The calculated value of the Sauter
mean diameter is close to that calculated by the semiempirical formula. The dependence of the Sauter
mean diameter on the thickness of the liquid layer constructed from the data calculated by the authors is
qualitatively consistent with the experiment.
The computational experiments showed that the number of the smallest droplets increases rapidly with
decreasing cell size; however, their contribution to the mean characteristics remains insignificant; thus,
there is no reason to reduce the grid cell size to account for these droplets.
ACKNOWLEDGMENTS
The authors thank the reviewers for their valuable comments and suggestions, which helped improve
the article.
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Translated by L. Trubitsyna
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