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Numerical Simulations and Measurements of a1
Droplet Size Distribution in a Turbulent Vortex2
Street3
Ellen Schmeyer∗, Robert Bordas†, Dominique Thevenin‡, Volker John§¶4
Abstract5
A turbulent vortex street in an air flow interacting with a disperse6
droplet population is investigated in a wind tunnel. Non-intrusive mea-7
surement techniques are used to obtain data for the air velocity and the8
droplet velocity. The process is modeled with a population balance system9
consisting of the incompressible Navier–Stokes equations and a population10
balance equation for the droplet size distribution. Numerical simulations11
∗Weierstrass Institute for Applied Analysis and Stochastics Leibniz Institute in
Forschungsverbund Berlin e. V. (WIAS), Mohrenstr. 39, 10117 Berlin, Germany,
[email protected]†University of Magdeburg ”Otto von Guericke”, LSS/ISUT, Universitatsplatz 2, 39106
Magdeburg, Germany, [email protected]‡University of Magdeburg ”Otto von Guericke”, LSS/ISUT, Universitatsplatz 2, 39106
Magdeburg, [email protected]§Weierstrass Institute for Applied Analysis and Stochastics Leibniz Institute in
Forschungsverbund Berlin e. V. (WIAS), Mohrenstr. 39, 10117 Berlin, Germany, and Free
University of Berlin, Department of Mathematics and Computer Science, Arnimallee 6, 14195
Berlin, Germany, [email protected]¶corresponding author
1
are performed that rely on a variational multiscale method for turbulent12
flows, a direct discretization of the differential operator of the population13
balance equation, and a modern technique for the evaluation of the coales-14
cence integrals. After having calibrated two unknown model parameters,15
a very good agreement of the experimental and numerical results can be16
observed.17
In einem Windkanal wird eine turbulente Wirbelstraße in einer Luft-18
stromung mit einer dispergierten Tropfchenpopulation untersucht. Um19
Daten bezuglich der Luft– und Tropfchengeschwindigkeiten zu gewinnen,20
werden nichtintrusive Messtechniken verwendet. Der zu Grunde liegende21
Prozess wird mit einem Populationsbilanzsystem modelliert, welches aus22
den inkompressiblen Navier–Stokes–Gleichungen und einer Populations-23
bilanzgleichung fur die Tropfchenverteilungsdichte besteht. Numerische24
Simulationen werden durchgefuhrt, welche eine variationelle Mehrskalen-25
methode fur turbulente Stromungen, eine direkte Diskretisierung des Dif-26
ferentialoperators der Populationsbilanzgleichung und ein modernes Ver-27
fahren zur Berechnung der Koaleszensintegrale verwenden. Nachdem zwei28
unbekannte Modellparameter kalibriert worden sind, kann eine sehr gute29
Ubereinstimmung der experimentellen und numerischen Ergebnisse beob-30
achtet werden.31
1 Introduction32
An active field of research in meteorology is the evolution of clouds. Since an33
enormous difficulty consists in obtaining reliable data from nature, a straight-34
forward approach is to perform wind tunnel experiments that represent some35
important properties of clouds. Based on available experimental data, models36
2
and numerical methods can be studied for their ability to reproduce these data37
and therefore to be potential candidates for the modeling and simulation of38
clouds. This paper presents a study that includes experiments, modeling, and39
numerical simulations.40
Clouds can be thought of as consisting of small water droplets moving in41
air. The motion of the air is often turbulent and it causes frequent droplet-42
droplet interactions. The performed wind tunnel experiments focused exactly43
on this feature. In first experiments, described in detail in Bordas et al. (2011,44
2012, 2013b), a turbulent channel flow was considered. This type of flow is one45
of the simplest turbulent flows and it is characterized by the absence of large46
coherent eddies. In Bordas et al. (2012, 2013a), a research program containing47
experiments, modeling, and numerical simulations was performed for this type48
of flow. In particular, a number of numerical methods were identified such that49
a good agreement of experimental and numerical results could be obtained.50
However, large coherent eddies might be present in turbulent flows in clouds.51
For this reason, a second experiment was designed, which led to the formation52
of large coherent eddies (large with respect to the diameter of the droplets). To53
this end, a cylinder was inserted into the wind tunnel such that the formation54
of a turbulent Karman vortex street was caused. The current paper considers55
this configuration.56
Water droplets were injected into a Gottingen-type wind tunnel with a closed57
test section. In between the injection point and the test section, the cylin-58
der was inserted. Experimental data of both phases, i.e., velocities of air and59
droplets, were determined with non-intrusive measurement techniques. These60
time-averaged data were used as boundary conditions for the model and for com-61
3
parison with the results from the numerical simulations. All measurement data62
are collected in an online database accessible at http://www.ovgu.de/isut/lss/metstroem.63
The considered process can be found as M3 in this database. The raw measure-64
ment results were further post-processed, so that all required data are in a65
suitable form for comparisons and validation.66
The behavior of the air-droplet flow was modeled by means of a population67
balance system. The incompressible Navier–Stokes equations describe the mo-68
tion of the turbulent flow field. They are coupled with a population balance69
equation for the droplet size distribution (DSD). Whereas the velocity and the70
pressure of the flow field depend on time and space, the DSD depends in addi-71
tion on an internal coordinate, which is here the diameter of the droplets. In the72
used model, the transport of the droplets with the flow field, the growth of the73
droplets in supersaturated air, and the coalescence of droplets were considered.74
The selected collision kernel has two unknown parameters.75
There are several, principally different, approaches for the numerical simula-76
tion of population balance systems. Applying some time-stepping scheme, one77
has to solve in each discrete time a population balance equation that is defined78
in a 4D domain, spanned by the spatial coordinates and the internal coordinate.79
Popular approaches for dealing with this difficulty are momentum-based meth-80
ods, in particular the direct quadrature method of moments (DQMOM) from81
Marchisio and Fox (2005). Also operator-splitting schemes have been proposed82
in Ganesan and Tobiska (2012). These two approaches reduce the complexity83
of the numerical solution of the 4D equation by solving problems in lower-84
dimensional domains instead. However, this reduction of complexity introduces85
some (unknown) error. With the rise of computer power in the last decade, also86
4
direct discretizations of the 4D equation became possible. Errors resulting from87
complexity reduction are avoided in this approach. Direct discretizations of the88
equation for the DSD will be studied in the numerical simulations presented in89
this paper.90
In recent years, we gained some experience with the approach of using direct91
discretizations, based on finite difference and finite element methods, see Bordas92
et al. (2012, 2013a); Hackbusch et al. (2012). In the numerical simulations pre-93
sented in this paper, the best methods from these studies were applied. A main94
goal of the numerical simulations was the identification of the unknown model95
parameters from the collision kernel. This goal was achieved by calibrating96
experimental and numerical data. The basis of this calibration was a space-97
time averaged DSD at the outlet of the test section. In addition, the impact of98
different kinds of flows on the same DSD will be investigated numerically.99
The paper is organized as follows. Section 2 describes the setup and the100
procedure for the measurements. The model of the process is introduced and101
discussed in Section 3. A short description of the numerical methods is provided102
in Section 4. The experimental and numerical results are presented in Section 5.103
The paper concludes with a summary and an outlook in Section 6.104
2 Measurement setup105
A wind tunnel capable of generating a disperse two-phase flow, available at the106
University of Magdeburg “Otto von Guericke”, has been used for the present107
measurements, roughly corresponding to conditions found in cumulus clouds as108
discussed in Bordas et al. (2011, 2013b), see also Table 1. This wind tunnel is109
5
a fully computer-controlled, so-called Prandtl or Gottingen-type wind tunnel,110
shown in Fig. 1. The test section itself is of size height × width × length =111
0.5× 0.6× 1.5 m. It includes transparent windows of size 0.45× 0.5 m. In this112
manner, non-intrusive optical measurements are possible, which is essential for113
high-quality experimental investigations of such flows.114
Table 1: Comparison of physical properties in cumulus clouds and in the wind
tunnel flow, from Bordas et al. (2011).
cumulus clouds wind tunnel
relative humidity saturated saturated
(vertical) velocities a few m/s ∼ 2.5 m/s
LWC 0.1 . . . 7 g/m3 ∼ 2 g/m3
mean droplet diameter 10 . . . 20 µm ∼ 12.5 µm
number density 1 . . . 7000 no./cm3 ∼ 2000 no./cm3
The present paper considers a configuration with a bluff body. It is a cylinder115
with a diameter of D = 0.02 m, fixed at the height of z = +0.09 m perpendicular116
to the main flow, right before the measurement section as shown in Fig. 1,117
right. For the reasons described in Bordas et al. (2012), the measurement area118
is finally restricted to the lower half of the cross-section in the wind tunnel. The119
longitudinal coordinate of the beginning of the measurement section was set120
to be x = 0 m. Different measurement planes perpendicular to the main flow121
direction were investigated, at x = 0 m, x = 0.2 m, and x = 0.4 m. The first122
plane at x = 0 m was measured particularly thoroughly, since it provided the123
information needed for the boundary condition of the numerical simulations.124
The disperse phase was generated with the help of a spray system, actuated125
6
by means of eccentric screw pumps. The number of revolutions per minute of126
the pumps (rpm), needed to inject the required water volume flow rate, was127
adjusted by a frequency regulator including a Proportional Integral Derivative128
(PID) regulation coded in LabView. In order to generate a typical cumulus129
cloud droplet spectrum, a full cone air-assisted atomizing nozzle was used (type130
166.208.16.12, co. Lechler). The applied air gauge pressure was 1.2 bar.131
Experimental measurements were systematically carried out by means of132
non-intrusive measurement techniques. Therefore, a small quantity of suitable133
tracer particles had to be added to the flow. Such particles follow the structures134
of the continuous phase much better than the considered droplets, allowing an135
indirect measure of the gas flow properties. For this reason, the velocities of136
both phases were measured in two separate steps.137
3 The Model of the Process138
The process was modeled via a population balance system consisting of the in-139
compressible Navier–Stokes equations to describe the flow field and a population140
balance equation which describes the behavior of the droplet size distribution.141
Considering first the flow field, the incompressible Navier–Stokes equations142
are given by143
∂tu−∇ · 2νD(u) + (∇ · u)u +1
ρ∇p = 0 in (0, te)× ΩNSE,
∇ · u = 0 in (0, te)× ΩNSE,
(1)
where u (m/s) is the velocity field, p (Pa) the pressure, ρ = 1.2041 kg/m3 the den-144
sity of dry air at T = 293.15 K, ν = 15.031 · 10−6 m2/s the kinematic viscosity,145
ΩNSE the flow domain, and te (s) denotes the final time. The velocity deforma-146
tion tensor, which is the symmetric part of the velocity gradient, is denoted by147
7
D(u). Because of the low Mach number in the experiments, the density can be148
assumed to be constant. The term describing the gravitational acceleration was149
absorbed into the pressure. In our experience, as well in the experiments as in150
the numerical simulations, the considered measurement section is too short for151
the gravity to possess a noteworthy influence. The motion of the droplets with152
the air flow dominates their motion caused by gravity.153
The computational domain has to contain the cylinder and the measurement154
section. Measurements were performed only in the lower half of the wind tunnel155
such that the consideration of this half is sufficient. The inlet has to be chosen156
sufficiently far away from the cylinder. In addition, a boundary condition for157
the air velocity at the last measurement plane was not known. In the numerical158
simulations, a standard outflow condition was applied. In order that this con-159
dition does not distort the computational results at this plane, the outlet of the160
computational domain was chosen to be somewhat behind this plane. Taking161
all these constraints into account and setting the origin of the coordinate system162
in the center of the wind tunnel at the location of the first measurement plane,163
the domain for the flow simulation was chosen to be164
ΩNSE = (−0.3, 0.5)× (−0.225, 0.225)× (−0.18, 0) m3.
Appropriate boundary conditions and an initial condition are necessary to165
close the Navier–Stokes equations (1). As initial condition, a fully developed166
flow field, which was computed in a preprocessing step, was used. The boundary167
condition at the inflow boundary Γin = −0.3 × (−0.225, 0.225) × (−0.18, 0)168
was based on experimental data. Before reaching the cylinder, the flow is a169
channel flow as it was considered in Bordas et al. (2012, 2013a). Therefore, the170
experimental inlet data for the flow from Bordas et al. (2012, 2013a) were used171
8
also in the simulations presented here. For this purpose, a time-averaged ex-172
perimental velocity uexp(−0.3, y, z) and the corresponding time-averaged stan-173
dard deviation σexp(−0.3, y, z) were available. In the simulations, these data174
were disturbed with noise calculated with the method proposed in Klein et al.175
(2003). This method enables the calculation of a spatially and temporally cor-176
related random variable, where the fluctuations of a point correlate with those177
of the same vortex. The integral length scales serve as measures for the vortex178
lengths. For the considered example, the most important integral length scale,179
which is the length scale in flow direction, could be determined in the experi-180
ments (Lx = 0.065 m). For the two other length scales, Ly and Lz, experimental181
data were not available. In front of the cylinder, a nearly isotropic flow field is182
given where all length scales are the same. Past the cylinder, the turbulence183
has been increased mainly in flow direction. But the length scales Ly and Lz184
should be still roughly of the same size as Lx. Numerical simulations showed185
that setting them equal to Lx led to similar flow fields like obtained with white186
noise (which one wants to avoid using the method from Klein et al. (2003)).187
Based on numerous numerical studies, we found that Ly = Lz = 0.02 m were188
appropriate values. In this way, a field Rx of correlated random numbers was189
obtained and the inflow condition was set to be190
u(t,−0.3, y, z) = uexp(−0.3, y, z)+Rx(t,−0.3, y, z)σexp(−0.3, y, z) in (0, te)×Γin.
Values at the inlet away from the measurement points were computed using191
bilinear interpolation. The second and third component of the inlet velocity192
were set to be zero.193
Since measurement data for the flow field at the outflow boundary were194
not available, a standard outflow boundary condition, the do-nothing boundary195
9
condition,196
(2νD(u)− pI)n∂Ω = 0 in (0, te)× Γout
was used, where n∂Ω denotes the unit normal vector and Γout = 0.5 ×197
(−0.225, 0.225) × (−0.18, 0). On all other boundaries, a free slip with pene-198
tration condition was applied in (0, te)199
n∂Ω · (2νD(u)− pI)n∂Ω = 0, τ i,∂Ω · (2νD(u)− pI)n∂Ω = 0, i ∈ 1, 2.
Here, τ 1,∂Ω, τ 2,∂Ω is a system of orthonormal tangential vectors at the bound-200
ary. This choice of the boundary condition differs from the simulations without201
cylinder studied in Bordas et al. (2012, 2013a), where on the top boundary a202
symmetry condition was used. Clearly, for the turbulent vortex street, symme-203
try is not given.204
The second equation of the population balance system models the droplet205
behavior206
∂tf+∇x ·(fudrop)+∂d
(adf)
= C+ +C− in (0, te)×ΩDSD×(dmin, dmax), (2)
where f (no./m4) is the droplet size distribution, d ∈ (dmin, dmax) (m) is the207
internal coordinate (diameter of the droplets), udrop= udrop(t, x, y, z) (m/s) is208
the droplet velocity, a = 5.0613 ·10−12 m2/s the growth rate, C+ the coalescence209
source, C− the coalescence sink, and ΩDSD the spatial domain in which the DSD210
was simulated.211
For the well-posedness of (2), it is necessary to prescribe boundary conditions212
at the inlet. Since measurement data, which could be used for this purpose, were213
available only at the first measurement plane x = 0 m, the domain for simulating214
the DSD was chosen to be215
ΩDSD = (0, 0.5)× (−0.225, 0.225)× (−0.18, 0) m3.
10
At this plane, also data for the time-averaged air velocity uair,exp(0, y, z) and216
the time-averaged droplet velocity udrop,exp(0, y, z) were available. Usually, one217
would assume in the considered model setup that the droplets are so small that218
they move with the flow field and flow velocity and droplet velocity coincide,219
i.e., udrop(t, x, y, z) = uair,sim(t, x, y, z). However, in the experimental setup,220
the spray system is so close to the measurement section that this situation is221
not yet given and the droplets are slower, in main flow direction, than the air222
flow in this section.223
The experimental air velocity uair,exp(0, y, z) was utilized for a comparison
with the time-averaged air velocity that is obtained from the simulations, see
Section 5. In (2), the droplet velocity udrop(t, x, y, z) is needed for all times
and in the whole domain ΩDSD. For constructing an appropriate function,
both experimental data uair,exp(0, y, z) and udrop,exp(0, y, z) were extrapolated
constantly into ΩDSD, e.g., udrop,exp(x, y, z) = udrop,exp(0, y, z) for all (x, y, z) ∈
ΩDSD. Then, the first component of the droplet velocity was defined by
(udrop)1(t, x, y, z) :=(udrop,exp(x, y, z) + uair,sim(t, x, y, z)− uair,exp(x, y, z)
)1.
For the other two components of udrop, the model is used that droplet velocity224
and air velocity coincide, i.e., these components of udrop were set to be equal225
to the components coming from the numerical solution of (1). In this way, the226
simulated turbulent flow field determines the motion of the droplets.227
Also equation (2) for the DSD has to be equipped with appropriate boundary228
and initial conditions. The choice of the boundary condition for the droplets229
at the inlet of ΩDSD was based on experimental data. Measurements were230
performed at a grid of nodes at the first measurement plane x = 0 m. For a231
detailed description of the conversion of the experimental data to values for the232
11
DSD in these nodes, denoted by fin,exp(x, d), and the corresponding standard233
deviation σf (x, d), it is referred to Bordas et al. (2012). The values in the nodes234
were interpolated to the whole measurement plane and the boundary condition235
was set to be236
f(t,x, d) =
fin,exp(x, d) + randnormal(t,x)σf (x, d), for d ∈ [dmin, dmax],
0, for d ∈ [dmin,art, dmin),
(3)
with x = (0, y, z), t ∈ [0, te], and randnormal denotes a normally distributed237
random variable. In (3), dmin,art = 0 m is an artificial smallest diameter for238
the droplets that was introduced to define the necessary boundary condition239
with respect to the internal coordinate because of the positive growth rate, see240
Bordas et al. (2012) for a discussion of this topic. In the numerical simulations,241
dmin = 10−6 m was used. An initial distribution is not known from the ex-242
periments. Since the simulations were started with a fully developed flow field,243
the extrapolation of the experimental DSD from the first measurement plane244
to ΩDSD was considered to be a reasonable approach for defining an initial con-245
dition. It should be noted already here that the time averaging of the data246
from the simulations started not at t = 0 s but after having allowed the system247
to develop for a while. Hence, the influence of the actual choice of the initial248
condition of the DSD on the time-averaged numerical results is expected to be249
negligible.250
The coalescence was modeled using the volume V of the droplets. Let fV
be the DSD and C+,V , C−,V be source and sink, all with respect to the volume
of the droplets. Then, the source term has the form proposed in Hulburt and
12
Katz (1964)
C+,V =1
2
∫ V
0
κcol(V − V ′, V ′)fV (V − V ′)fV (V ′) dV ′
and the sink term251
C−,V = −∫ Vmax
0
κcol(V, V′)fV (V )fV (V ′) dV ′
= −fV (V )
∫ Vmax
0
κcol(V, V′)fV (V ′) dV ′.
The choice of an appropriate collision kernel was discussed in detail in Bordas252
et al. (2013a). In particular, it was emphasized that in the wind tunnel ex-253
periments, unlike as, e.g., in clouds, a gravitational kernel can be neglected.254
Therefore, the combination of a standard Brownian motion kernel and a shear255
kernel was used256
κcol(V, V′) = Cbrown
2kBT
3µ
(3√V +
3√V ′)( 1
3√V
+1
3√V ′
)+Cshear
√2∇udrop : ∇udrop
(3√V +
3√V ′)3
, (4)
where kB = 1.38 10−23 J/K is the Boltzmann constant and µ = 18.15 ·10−6 kg/ms257
is the dynamic viscosity. The model parameters Cbrown and Cshear were un-258
known and they will be determined by calibrating numerical results against259
experimental data.260
A main mechanism of the studied process is the turbulence of the flow.261
Because of this feature, the use of the shear kernel, which models turbulent262
shear coagulation, see Saffman and Turner (1956), is a straightforward choice.263
The scaling factor of this kernel has been studied in a number of papers, see264
de Boer et al. (1989) for a brief overview. For the case of homogeneous isotropic265
turbulence, values of order 1 were found. Since in the considered process the266
turbulence is neither homogeneous nor isotropic, the factor Cshear is a priori267
13
unknown. It is known (Pruppacher and Klett, 1997, Chap. 11.6.2.1), and we268
found it also in our numerical studies, that the shear kernel affects droplets that269
are larger than a few microns. However, the studied process involves also smaller270
droplets, such that a complementary model is needed for these droplets. We271
chose the Brownian motion kernel for this purpose. One finds in the literature272
that it affects droplets with radii . 1 µm, e.g., see (Pruppacher and Klett,273
1997, Chap. 11.5), where one finds also a detailed discussion of this kernel. We274
could not find in the literature collision (coagulation) coefficients which take275
into account a flow field. For this reason, the parameter Cbrown was introduced.276
277
4 Numerical Methods278
A number of numerical methods for solving the population balance system279
(1), (2) were explained in detail and studied comprehensively in Bordas et al.280
(2013a). Therefore, only a short presentation of the methods will be provided281
here and it is referred to Bordas et al. (2013a) for a detailed description.282
All methods, which were studied in Bordas et al. (2013a), were also studied283
for the experiment considered in this paper, see Schmeyer (2013). For the sake284
of brevity, only the results for the best methods from Bordas et al. (2013a) will285
be presented here.286
The general solution strategy for the population balance system was as fol-287
lows. In each discrete time, first the Navier–Stokes equations (1) were solved288
and the flow field was computed. Then, the equation for the DSD (2) was solved,289
where the coalescence terms were always treated explicitly with respect to the290
14
DSD. With this approach, the problem in 4D became linear.291
The turbulent flow field was simulated with a projection-based variational292
multiscale (VMS) method, which was introduced in John and Kaya (2005).293
More precisely, the version of the method which utilizes an adaptively chosen294
projection space presented in John and Kindl (2010) was applied. The Crank–295
Nicolson scheme was used as time stepping scheme and the inf-sup stable pair296
of finite element spaces Q2/Pdisc1 as spatial discretization.297
For the discretization of the differential operator on the left-hand side of (2),298
three schemes were studied in Bordas et al. (2013a): a total variation diminish-299
ing essentially non-oscillatory (TVD-ENO) finite difference method from Shu300
and Osher (1988) and two linear flux-corrected transport methods with Crank–301
Nicolson time stepping (CN-FCT, CN-GFCT) proposed by Kuzmin (2009). The302
conclusion of Bordas et al. (2013a) was to recommend the TVD-ENO scheme,303
because it was the most efficient scheme and the finite element schemes were304
shown to introduce too much numerical diffusion if the convection field and the305
grid are aligned. The strong alignment of the flow field and the grid is a main306
feature in the channel flow studied in Bordas et al. (2013a). But this situation is307
not given for the turbulent vortex street studied here. Therefore, we think that308
it is of interest to present also results for the finite element schemes. Both con-309
sidered schemes are of very similar accuracy, see John and Novo (2012). Thus,310
results will be presented only for the more efficient scheme, which is the group311
finite element version CN-GFCT.312
A number of approaches were studied in Bordas et al. (2013a) for the eval-313
uation of the integral terms on the right-hand side of the population balance314
equation (2). The recommendation of Bordas et al. (2013a) was a method pro-315
15
posed in Hackbusch (2006, 2007, 2009). This method requires the use of special316
grids with respect to the internal coordinate. They have to be piecewise equidis-317
tant with respect to the volume of the droplets. In these grids, the nodes did318
not coincide with the internal coordinates for which experimental data were319
available. Thus, an interpolation of the experimental data became necessary.320
In Bordas et al. (2013a), a linear interpolation and a log-normal interpolation321
were studied. It turned out that the experimental data could be reproduced322
better with the linear interpolation.323
5 Results324
5.1 Experiments325
The velocity distribution of the air phase at the inlet plane (x = 0 m) was326
measured by means of Laser-Doppler Velocimetry (LDV). Measurements were327
carried out with a main flow velocity of U = 2.32 m/s, corresponding to a328
cylinder-based Reynolds number of Re = UD/ν = 3 · 103. During these mea-329
surements in the continuous phase, the nozzle was operating at the same pressure330
as in normal (spray) operation, but only with air and without water. Since the331
mass flow rate of air and water entering the nozzle are similar for normal oper-332
ation conditions, only minor flow changes should be induced by this necessary333
operation.334
In order to define the locations of the measurement points for the Laser-335
Doppler Velocimetry and the Phase-Doppler Anemometry (PDA) techniques, a336
measurement grid was generated with 874 (19 in z-direction × 46 in y-direction)337
measurement points, with 0.01 m distance in both directions between them.338
16
LDV and PDA measurements led to a high temporal resolution. Thus, the ve-339
locity components measured in the mean flow direction, Fig. 2, left, included340
the temporal fluctuations as well. In this way, the determination of the turbu-341
lence intensity was also possible, Fig. 2, right. The measured mean velocity of342
the air flow was U = 2.45 m/s. Based on U and on the hydraulic diameter of343
the wind tunnel (DH = 0.5454 m), the Reynolds number of the flow is 8 · 104.344
The measured fluctuation of the air flow velocity in main flow direction was in345
the average u′ = 0.33 m/s. This leads to a mean turbulence intensity of 15%.346
The energy cascade of a turbulent flow can be estimated by post-processing the347
measurement results. The resulting Kolmogorov length scale is 2 · 10−4 m, as348
shown in Bordas et al. (2013b).349
The properties of the disperse phase (water spray) were then measured sep-350
arately in the three vertical planes at x = 0 m, x = 0.2 m, and x = 0.4 m, using351
the same measurement grid as previously. Velocities measured by PDA are based352
on the same principles as LDV. However, using PDA, the simultaneous mea-353
surement of the diameter and the velocity values was possible. Calculating the354
droplet Stokes number St from the droplet properties, the Kolmogorov length355
scale and the RMS velocity fluctuations, the most frequent value of St = 2 ·10−3356
in a range of St = 10−3 . . . 102 was found.357
At the outlet boundary (x = 0.4 m), experimental data were needed as well358
for comparison purposes. Therefore, a corresponding post-processing of the359
PDA measurements was necessary to obtain values for the number density or360
droplet concentration. These calculations were performed in the same manner361
as described, e.g., in Bordas et al. (2012), resulting in the droplet concentration362
as a function of the droplet diameter, together with the corresponding standard363
17
deviation.364
5.2 Numerical Simulations and Comparison with the Ex-365
perimental Data366
All simulations were performed in the time interval [0, 1] s. In [0, 0.5] s, the367
system was allowed to develop and time-averaging of data was performed in368
[0.5, 1] s. In particular, all droplets injected at t = 0 s had already left the369
domain before the start of the time-averaging.370
The grid for ΩNSE is presented in Fig. 3. On this grid, there were 973 728371
degrees of freedom for the velocity and 156 032 degrees of freedom for the pres-372
sure. The grid is refined in the domain around and behind the cylinder since373
most of the turbulence can be expected in these regions. The grid for ΩDSD was374
just the right part of the grid for ΩNSE, starting at x = 0 m. All measurement375
points were nodes of the grid. For the internal coordinate, an appropriate grid376
for the method for evaluating the coalescence integrals was chosen which had 94377
nodes. Altogether, there were 4 086 650 degrees of freedom for the TVD-ENO378
and CN-GFCT discretizations of the DSD equation. The length of the time379
step was always set to be ∆t = 10−3 s.380
The simulations were performed with the research code MooNMD, see John381
and Matthies (2004).382
First, the flow field will be considered. Behind the cylinder, a turbulent383
Karman vortex street is developing, see Fig. 4. As already mentioned above,384
experimental data for the time-averaged velocity component in flow direction385
were available at the plane x = 0 m, see Fig. 2. A reduction of the velocity can386
be observed in the regions behind the cylinder and behind the spray injection387
18
system (z < 0.7 m,−0.4 m < y < 0.4 m). In Fig. 5, the time-averaged velocity388
that was obtained in the simulations at x = 0 m is shown. One can observe a389
very good agreement. The values are in the same range and the slight asymmetry390
observed experimentally is well reproduced. Thus, one can conclude that the391
choice of the inlet condition as described in Section 3 was appropriate and392
also that the simulation of the turbulent flow field with the VMS method was393
accurate, at least with respect to the time-averaged velocity.394
Concerning the DSD, the first goal was the determination of suitable pa-395
rameters Cbrown and Cshear in the collision kernel (4). Because of the turbulent396
nature of the flow, the time-averaged experimental data were different in differ-397
ent measurement points. Therefore, a spatial averaging was applied. The same398
strategy was used in the numerical simulations for calibrating the parameters.399
The calibration of the model parameters was performed by trial and error400
using the method TVD-ENO. It turned out that a good fit between experimental401
and numerical results was obtained for Cbrown = 105 and Cshear = 30, see402
Fig. 6. Comparing these parameters with the parameters found for the channel403
flow in Bordas et al. (2012), one can observe some differences. For the channel404
flow, Cshear = 0.1 was found to be appropriate, but it was also noted that405
this parameter has only little influence in those setting. We checked that with406
Cshear = 30 one obtains for the channel flow almost the same results as for407
Cshear = 0.1. Thus, Cshear can be calibrated in the same way for both situations.408
For the parameter Cbrown, it should be noted that the value of the Brownian409
motion part in the collision kernel for droplets of around the same size is of order410
10−11 m3/s (with Cbrown = 105). Since we found in the numerical simulations411
that generally√
2∇udrop : ∇udrop ∈ (100, 200), one can conclude that the shear412
19
part in the collision kernel (4) starts to dominate the Brownian motion part if the413
diameter of the droplet is around 11−15 µm. For the channel flow, it was found414
that Cbrown = 1.5 ·106 is an appropriate value. Thus, there is a difference of one415
order of magnitude for Cbrown between the two settings. There is most probably416
some uncertainty in the experimental data which might contribute to having417
obtained different parameters. Furthermore, difficulties with the measurement418
of small droplets, with diameter less than 15 µm, and the development of an419
appropriate collision kernel have been reported also recently in Siewert et al.420
(2014). However, the modeling question of an appropriate collision kernel is an421
active field of research but outside the scope of the present paper.422
Results obtained with the two numerical methods for discretizing the left-423
hand side of the DSD equation (2) are presented in Fig. 7. One can observe that424
the curve obtained with CN-GFCT is somewhat below the curve computed with425
TVD-ENO. This result is similar to the observation for the turbulent channel426
flow from Bordas et al. (2013a). Thus, even if the vortex street is not that closely427
aligned to the grid as the channel flow, CN-GFCT behaves similarly for both428
situations. This method seems still to introduce too much numerical diffusion,429
a feature which was identified in Bordas et al. (2013a) for the case that grid and430
convection are aligned.431
Figure 8 presents results which are spatially resolved and only temporally432
averaged. For each node at the outlet plane x = 0.4 m, the temporally averaged433
DSD is a curve of the form presented in Figs. 6 and 7. In the top row of434
Fig. 8, the maximal values of these curves are shown and in the bottom row435
the corresponding abscissas. One can see that the numerical results (right-436
hand side) are more equilibrated (smeared) compared with the experimental437
20
data (left-hand side). The maximal values of the experimental data are taken438
in (1.5, 3.5) · 1014 no./m4 whereas for the numerical DSD we obtained values in439
(1.75, 2.75) · 1014 no./m4. The maximal experimental data are obtained mostly440
for d = 7 µm and d = 9 µm (note that experimental data in between are not441
available) and for the numerical results, the maxima are taken for d ∈ (7, 8) µm.442
One reason for the numerical results to be more equilibrated is certainly the443
numerical viscosity which is introduced by the turbulence model. Also, the444
use of different meshes for the internal coordinate in the experiment and the445
simulations seems to have some influence. But despite the differences in detail,446
there is a good overall agreement of experimental and numerical results.447
The simulations were performed on a compute server HP BL680c G7 2xXeon,448
Ten-Core 2400MHz. For one time step, solving the Navier–Stokes equations449
took around 125 seconds, computing the coalescence integrals nearly 30 seconds,450
applying TVD-ENO around 5 seconds, and CN-GFCT about 45 seconds.451
Finally, the impact of different turbulent flows on the same DSD is studied452
in a model situation. To this end, the turbulent channel flow considered in453
Bordas et al. (2012, 2013a) and the turbulent vortex street presented in this454
paper were used. Concerning the DSD, three isolated diameters were chosen,455
dsmall = 9 × 10−6 m, dmiddle = 19 × 10−6 m, and dlarge = 39 × 10−6 m. At456
the inlet, the value of the DSD was set to be 103 no./m4 for these diameters457
in all nodes. All other inlet values of f were set to be zero. The simulations458
were performed with TVD-ENO, Cbrown = 105, and Cshear = 30. The resulting459
DSD at the outlet is presented in Fig. 9. One can observe that the number of460
droplets for the considered diameters is smaller for the turbulent vortex street.461
On the other hand, one can see clearly that for dmiddle and dlarge more droplets462
21
with larger diameters are created, due to coalescence, for the turbulent vortex463
street. This observation corresponds to the expectation that in the turbulent464
vortex street there are more collisions of droplets and therefore eventually more465
successful coalescence processes occur.466
6 Summary and Outlook467
This paper studied the behavior of a disperse droplet population in a turbulent468
air flow. Inserting a cylinder into a wind tunnel, a turbulent vortex street was469
generated. Data of the velocities of air and droplets were obtained by non-470
intrusive measurement techniques. The process was modeled with a population471
balance system. Simulations of this model were performed by using modern472
numerical methods. Two unknown model parameters had to be calibrated. Very473
good agreements between experimental and numerical data could be observed474
for the mean velocity in a cut plane and for the space-time averaged droplet size475
distribution at the outlet.476
A possible improvement of the collision kernel for the coalescence integrals477
should be considered in future. The calibration of one of the parameters re-478
sulted in a different order of magnitude for the considered vortex street and479
a channel flow in previous simulations. Concerning the numerical methods, a480
better understanding and improvement of CN-GFCT would be desirable. This481
method can be applied, in contrast to TVD-ENO, not only on tensor-product482
meshes. From this point of view, CN-GFCT would be a potential method to be483
used if general spatial domains are considered.484
22
Acknowledgments485
The funding of this project in the framework of the priority program SPP 1276486
MetStrom: Multiple Scales in Fluid Mechanics and Meteorology, by the Ger-487
man Research Foundation (DFG) is gratefully acknowledged. E. Schmeyer was488
founded under grant number Jo329/8-3 and R. Bordas under grant number489
Th881/13-3.490
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Figure 1: Two-phase Prandtl wind tunnel with closed test section (left), mea-
surement section with counter-flow droplet injection and a cylindrical bluff body
right before the test section (right).
−200 −100 0 100 2000
50
100
150
1.6
2
2.4
2.8
−200 −100 0 100 2000
50
100
150
0
0.2
0.4
0.6
Figure 2: Mean longitudinal velocity distribution in the plane x = 0 m (left),
turbulence intensity distribution in the plane x = 0 m (right).
Figure 3: Grid for the simulations.
27
Figure 4: Snapshot of the simulated flow field, iso-surfaces of the pressure for
two values.
Figure 5: Time-averaged first component of the velocity from the simulations.
Figure 6: Calibration of the model parameters Cbrown (left, with Cshear = 30)
and Cshear (right, with Cbrown = 105), simulations with TVD-ENO.
28
Figure 7: Comparison of results with different discretizations of the DSD equa-
tion (2), Cbrown = 105, Cshear = 30.
Figure 8: Spatially resolved and temporally averaged DSD at the outlet plane
x = 0.4 m; maximal values of the DSD (top) and corresponding abscissas
(bottom); experiment (left) and numerical simulation with Cbrown = 105 and
Cshear = 30 (right).
29
Figure 9: Effect of different turbulent flows on certain droplets with different
diameters, TVD-ENO, Cbrown = 105, Cshear = 30.
30
