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Technical note
Numerical simulation of high-speed turbulent water jets in air
ANIRBAN GUHA, Department of Mechanical, Automotive and Materials Engineering, University of Windsor,Windsor, ON, Canada N9B3P4. Present address: Department of Civil Engineering, University of British Columbia,Vancouver, BC, Canada V6T1Z4.Email: [email protected] (author for correspondence)
RONALDM. BARRON,Department of Mechanical, Automotive and Materials Engineering, University of Windsor,
Windsor, ON, Canada N9B3P4.Email: [email protected]
RAM BALACHANDAR (IAHR Member), Department of Civil and Environmental Engineering, Universityof Windsor, Windsor, ON, Canada N9B3P4.Email: [email protected]
ABSTRACTNumerical simulation of high-speed turbulent water jets in air and its validation with experimental data has not been reported in the literature. It istherefore aimed to simulate the physics of these high-speed water jets and compare the results with the existing experimental works. High-speedwater jets diffuse in the surrounding atmosphere by the processes of mass and momentum transfer. Air is entrained into the jet stream and theentire process contributes to jet spreading and subsequent pressure decay. Hence the physical problem is in the category of multiphase flows, forwhich mass and momentum transfer is to be determined to simulate the problem. Using the Eulerian multiphase and the k–e turbulence models,plus a novel numerical model for mass and momentum transfer, the simulation was achieved. The results reasonably predict the flow physics ofhigh-speed water jets in air.
Keywords: CFD, jet cleaning, multiphase flow, numerical modeling, turbulence, water jet
1 Introduction
High-speed turbulent water jets having velocity of 80–200 m/s
in air are extensively used in industrial cleaning operations. They
exhibit a high velocity coherent core surrounded by an annular
cloud of water droplets moving in an entrained air stream.
Leu et al. (1998) discussed the anatomy of these high-speed
jets (Fig. 1). Much like Rajaratnam et al. (1994, 1998), theydivided the jet into three distinct regions:
(1) Potential core regions close to the nozzle exit, instabilities
cause eddies resulting in a transfer of mass and momentum
between air and water with air entrainment breaking up
the continuous water into droplets. There remains a
wedge-shaped potential core surrounded by a mixing layer
in which the velocity is equal to the nozzle exit velocity.
(2) Main regionwhere air dynamics and continuous interaction of
water with surrounding air results in the break-up of the water
jet stream into droplets. There is a high degree of air entrain-
ment and the size of water droplets decreases with the increase
of radial distance from the axis. Due to momentum transfer
to the surrounding air, the mean velocity of the water jet
decreases and the jet expands. The jet region close to the
jet-axis is called the water droplet zone. Between the latter
and the surrounding air, there is a water mist zone in which
drops are very small and the velocity is almost negligible.
(3) Diffused droplet region, where extremely small droplets
of negligible velocity are produced by complete jet
disintegration.
Although the characteristics of submerged high-speed water
jets were thoroughly studied (Long et al. 1991, or Wu et al.1995), few experimental studies on high-speed water jets in air
have been reported in the literature. Leach et al. (1966) studiedthe pressure distribution on a target plate placed at a given
axial distance from the nozzle. They demonstrated that the
Journal of Hydraulic Research Vol. 48, No. 1 (2010), pp. 124–129
doi:10.1080/00221680903568667
# 2010 International Association for Hydro-Environment Engineering and Research
Revision received 27 August 2009/Open for discussion until 31 August 2010.
ISSN 0022-1686 print/ISSN 1814-2079 onlinehttp://www.informaworld.com
124
normalized pressure distribution along the centreline of a jet
depends on the nozzle geometry while it is independent in the
radial direction. The normalized pressure becomes equal to
the ambient pressure at a distance of around 1.3 times the
nozzle exit diameter D from the centreline. Outside this
region, the shear stress is too small to clean the target surface.
They also found that the normalized pressure distribution was
similar for both various inlet pressure conditions and nozzle
geometries.
Yanaida and Ohashi (1980) did similar work and developed
a mathematical expression for the centreline pressure.
Unfortunately, their curve did not provide satisfactory results
for the relevant axial distances in cleaning operations.
Rajaratnam et al. (1994, 1998) used a converging-straight
nozzle of D ¼ 2 mm and nozzle exit (subscript “0”) velocity
of around V0 ¼ 155 m/s. They found that the centreline jet
velocity remains constant and equal to V0 for more than 100Dand then linearly decays to 0.25V0 at about 2500D. Surprisingly,
severe air entrainment causes the water (subscript “w”) volume
fraction aw is the ratio of volume of a particular phase to
sum of all phases present in the mixture to fall drastically.
Measurements along the centreline indicate that aw at 20D is
20%, at 100D is 5% and at 200D is just 2%.
To the best of our knowledge, numerical simulation of this
problem has been reported in the literature in only one instance
yet the results were not validated against test results (Liu et al.2004). Also their results do not simulate the actual physics
as experimentally observed by Rajaratnam et al. (1994). Thus
the flow physics of high-speed turbulent water jets in air are
simulated. The next step will be to validate the results with the
available test data.
2 Novel mass-flux model
Due to Leu et al. (1998), the potential core and the water droplet
zones (Fig. 1) are of prime importance for industrial cleaning,
since these zones have a significant momentum to clean a
surface. Yanaida and Ohashi (1980) analysed the problem by
dividing the jet flow according to radial distance from the
centreline (Fig. 1). The inner region corresponds to a continuous
flow region, of which the radial width Ri varies as
Ri ¼ k1ffiffiffix
pþ k2 (1)
Outside of this region is the droplet flow region, of which the
radial width Ro varies as
Ro ¼ Cxþ k2 (2)
where k1 and C are spread coefficients related as
k1 ¼ 1:9C (3)
and k2 is the parameter depending on nozzle radius. Subscripts
“i” and “o” relate to inner and outer, respectively.
According to Erastov’s experiment (Abramovich 1963), the
mass flow rate of these water jets follow
_M ðx; rÞ_M ðx; 0Þ
¼ 1�r
R
� �1:5� �3
(4)
where _M ðx; rÞ is the mass flux in the axial direction of water
droplets given by
_M ðx; rÞ ¼ awðx; rÞ � rw � Vwðx; rÞ (5)
and x and r are the axial and radial coordinates of a point in the
jet. Further, rw is the density of water; aw(x,r), the volume
fraction; Vw(x,r), the axial velocity of water droplets,
respectively. According to the mass conservation principle, the
mass flow rate at any cross-section of the jet is equal to the
mass flow rate at the nozzle exit. If the droplet flow is assumed
to be a continuum, then this principle can be represented as
_M 0p R0ð Þ2¼ 2p
ðR0
_M ðx; rÞrd rð Þ (6)
where R0 is the nozzle radius and _M0, the mass flux of water dro-
plets at nozzle exit. Using Eqs. (4) and (6), a relation between the
centreline mass flux and the nozzle exit mass flux is obtained as
_M ðx; 0Þ ¼5:62 _M0R2
0
R2(7)
The mass flux of water droplets at any point in the jet can be
expressed in terms of the nozzle exit mass flux by substituting
Eq. (4) in Eq. (7), resulting in
_M ðx; rÞ ¼5:62 _M 0R2
N
R21�
r
R
� �1:5� �3
(8)
Let
_M 0 ¼ rw0 � aw0 � Vw0 (9)
Figure 1 Anatomy of high-speed water jets in air (Leu et al. 1998)
Journal of Hydraulic Research Vol. 48, No. 1 (2010) Numerical simulation of high-speed turbulent water jets in air 125
where aw0 is the volume fraction and Vw0 the axial velocity of
water droplets at the nozzle exit. aw0 is assumed to be 100%.
Substituting Eq. (9) into Eq. (8) gives
_M ðx; rÞ ¼5:62� rw � aw0 � Vw0 � R2
0
R21�
r
R
� �1:5� �3
(10)
Equation (10) is the polynomial function based on an empirical
mass-flux model. If the nozzle exit velocity is properly known,
this model can be used to estimate the flow characteristics of
high-speed water jets in air.
3 Numerical simulation
The objective is to perform numerical simulations of high-speed
turbulent water jets in air and to compare the results with
published test data of Rajaratnam et al. (1994, 1998) and
Leach et al. (1966). Equation (10) needs therefore to be
coupled with the continuity and momentum equations of
turbulent multiphase flows.
The computational domain (Fig. 2) and a structured grid
system were created in the commercial mesh generation
package GAMBIT. Since this problem involves circular jets,
only half of the domain was simulated in a two-dimensional
axis-symmetric space. The computational space was 1000 mm �
500 mm, and a tightly clustered grid was ensured in the regions
where larger flow gradients are expected. The radial extent of
the domain was large enough to ensure that the pressure outlet
boundary condition (set to atmospheric pressure) and the wall
boundary conditions can be accurately applied, i.e. without
adversely affecting the flow field. The radial width of the velocity
inlet boundary (set at 155 m/s) was 1 mm as per the test conditions
of Rajaratnam et al. (1994, 1998).FLUENT was applied as the flow solver. The Eulerian
multiphase model and the standard k–1 turbulence model with
standard wall functions were used to capture the flow physics.
Water was treated as the secondary phase. The drag coefficient
between the phases was determined by the Schiller–Naumann
equation (Schiller and Naumann 1935). The continuity and
momentum equations for the water phase in the Eulerian
model for multiphase flows are, respectively,
@ðawrwÞ
@tþ r � awrw~vw
� �
¼Xi¼w;a
ð _ma!w � _mw!aÞ þ Sw(11)
@ðawrw~vwÞ
@tþ r � awrw~vw~vw
� �¼ �awrpþ r � ��tw
þ awrw gQ
þ
Xi¼w;a
Kwað~vw � ~vaÞ þ _ma!w~va!w � _mw!a~vw!a
� þ ~Fw
(12)
The term _mw!a is the mass transfer from the water phase to the
air (subscript “a”) phase. In the physical problem, the surround-
ing air is entrained into the jet and the mass of air in the jet
increases. To implement this process numerically, both _ma!w
and Sw are mass source terms for the water phase, were set to
zero, leaving _mw!a as the only mass source term at the right
hand side of Eq. (11). Note that physically there is no mass
transfer between air and water; it is used because of the ease in
numerical implementation in FLUENT. Since the mass flux of
the water phase at all points in the domain is known from the
empirical mass flux model by Eq. (10), it was incorporated
into the continuity equation (11) as
_mw!a ¼ r � ð _M ; 0Þ (13)
The source term due to the momentum transfer ( _mw!a~vw!a) in
Eq. (12) is automatically handled by FLUENT once the mass
transfer is specified, namely by
~vw!a ¼ ~va if _mw!a . 0
~vw!a ¼ ~vw if _mw!a < 0 (14)
The term Kwað~vw � ~vaÞ in Eq. (12) represents the inter-phase
interaction force and Kwais the inter-phase momentum exchange
coefficient. The incorporationofEq. (13) in the continuity equation
is accomplished, using user defined functions in FLUENT.
The k–1 mixture turbulence model was used for turbulence
modelling. The transport equations are:
@ðrmkÞ
@tþ r � rmk~vm
� �¼ r �
t;m
skrk
�þ Gk;m � rm1 (15)
@ðrm1Þ
@tþ r � rm1~vm
� �
¼ r �t;m
s1
r1
�þ1
kðC11Gk;m � C21rm1Þ;
(16)
Figure 2 Computational domain, boundary conditions and meshing
126 A. Guha et al. Journal of Hydraulic Research Vol. 48, No. 1 (2010)
where rm is the mixture density and ~vm, the mixture velocity. The
turbulent viscosity mt,m and the production of turbulent kinetic
energy Gk,m are calculated as
mt;m ¼ rmCm
k2
1(17)
Gk;m ¼ mt;m r~vm þ r~vm� �T� �
(18)
The model constants are the standard values C11 ¼ 1.44, C21. ¼
1.92, Cm ¼ 0.09, sk ¼ 1.0, s1 ¼ 1.3. Standard wall functions
were used to model near wall flows. For brevity, the description
of standard wall functions is not discussed. Interested readers
refer to the FLUENT 6.3.26 user manual for details.
Pressure–velocity coupling was achieved using the phase-
coupled SIMPLE algorithm. All the residuals tolerances were
set to 1026 and the time step size was 1025 s. The program
was run for a time long enough to attain quasi-steady state.
The default under-relaxation parameters of FLUENT were
used in the computation. The discretization schemes used in
the simulation are listed in Table 1.
4 Results
Figures 3–5 compare the simulation results with that of the pub-
lished test data of Rajaratnam et al. (1994, 1998) and Leach et al.(1966). Rajaratnam et al. found that the jet centreline velocity V0
remains constant for more than 100D and then decays linearly to
about 0.25V0 at about 2500D. Severe air entrainment causes the
water volume fraction aw to fall drastically from 20% at 20Dto 5% at 100D. Figures 3 and 4 confirm that the simulation
accurately predicts the centreline characteristics.
Figure 5 shows the velocity profiles for x/D ¼ 100, 200 and
300. In comparison to Rajaratnam et al. (1994), the velocity distri-bution gives good results within a radial width of 5D. Outside
this region, the water mist zone is more prominent. Since the mist
zone is formed of sparse droplet flows, the continuum hypothesis
as a basic assumption of Eulerian model becomes invalid; hence,
the model is no longer suitable to capture the flow physics.
Note that the mist zone has little effect in cleaning appli-
cations; hence its modelling is not a major concern. Thus, we
can conclude that the simulation results match reasonably well
with the test data of Rajaratnam et al. (1994, 1998).Figures 6 and 7 show the velocity and volume fraction
contours of the water-phase up to x/D ¼ 10. These figures are
drawn to the same geometric scale, giving a quantitative
comparison between the two contours. The volume fraction
contour shows that the water-phase volume fraction decays
sharply with increased radial distance while the velocity
contour indicates that the velocity magnitude remains almost
constant for considerable radial distance. The velocity contour
is much wider than the volume fraction contour. This observation
is in agreement with Rajaratnam and Albers (1998) yet they did
not provide the results of volume fraction distribution in the
radial direction. Thus, it can be concluded that a considerable
Figure 5 Velocity distribution at x/D ¼ 100, 200, 300 and comparisonwith experimental results of Rajaratnam et al. (1994)
Table 1 Discretization schemes for jet flow
Variable Discretization scheme
Time First-order implicit
Momentum QUICK
Volume fraction QUICK
Turbulent kinetic energy Second-order upwind
Turbulent dissipation rate Second-order upwind
Figure 4 Numerical simulations of normalized centreline water-phasevelocity and comparison with experimental results of Rajaratnam et al.(1994)
Figure 3 Numerical simulation of decay of centreline water-phasevolume fraction and comparison with experimental results ofRajaratnam et al. (1998)
Journal of Hydraulic Research Vol. 48, No. 1 (2010) Numerical simulation of high-speed turbulent water jets in air 127
amount of air is entrained within the jet. Near the outer jet region,
the co-flowing air carries the water droplets (of negligible
volume fraction) and has considerably high velocity. Near the
centreline, the entrained air has a relatively high volume fraction
increasing radially, and moving with identical velocity as the
water phase.
The radial distribution of the volume fraction and the water-
phase velocity within x/D ¼ 30 is of major importance in
cleaning and cutting applications. Figures 8 and 9 respectively
show the water-phase velocity and volume fraction distributions
at various axial locations. From Fig. 8, it is obvious that the
potential core still exists at x/D ¼ 30. Figure 9 shows that the
volume fraction of water drops from 0.43 at x/D ¼ 10 to 0.21
at x/D ¼ 30, indicating the amount of air entrainment along
the centreline. The distribution of water-phase volume fraction
is expected to be Gaussian (Rajaratnam and Albers 1998), but
the simulation results show a distribution close to Gaussian
with a bulge at the jet–air interface. Since it is impossible to
predict the mist region with an Eulerian approach, the volume
fraction of water actually lost as mist numerically accumulates
near the jet–air interface and produces the erroneous bulging
effect. The bulging effect flattens out with increased axial
distance. The entrained air flows with the same velocity as the
water-phase, but owing to low air density in comparison to
water (�1:815); the momentum delivered to cutting or cleaning
surface is significantly reduced.
From an application point of view, the pressure distribution on
a target (subscript “T”) plate PT placed perpendicularly to the jet
flow field is of prime concern. Since the jet loses a sufficient
amount of centreline pressure PT(x,0) as it travels, the target
plate should be kept near the nozzle exit to ensure efficient
cutting or cleaning. It is essential for the simulation to predict
the pressure distribution at the target plate accurately, hence
the test conditions with a jet velocity of 350 m/s and nozzle
radius of 0.5 mm of Leach et al. (1966) were numerically
implemented. Figure 10 compares the simulation results with
the experiment.
The numerical simulation matches well near the centreline
but deviates slightly toward the edge. Leach et al. (1966) useda third-order polynomial curve fit for their test data to represent
Figure 7 Contour of water-phase velocity in jet (within x/D ¼ 10)
Figure 6 Contour of water-phase volume fraction in jet (within x/D ¼ 10)
Figure 9 Water-phase volume fraction at x/D ¼ 10, 20 and 30
Figure 8 Water-phase velocity at x/D ¼ 10, 20 and 30Figure 10 Normalized pressure distribution on a target plate placed at76D and comparison with Leach et al. (1966)
128 A. Guha et al. Journal of Hydraulic Research Vol. 48, No. 1 (2010)
the radial pressure distribution. According to Guha (2008),
the test results for different nozzle exit velocities indicate
that the Gaussian fit is more appropriate (Fig. 11). Since the
present simulation results resemble the Gaussian distribution,
the flow physics are more accurately predicted than by the
experiments.
5 Conclusions
Numerical simulations were performed to capture the entrain-
ment of surrounding air into high-speed water jets. The
simulation reasonably predicts velocity, pressure and volume
fraction distributions of high-speed water jets in air. The results
accurately describe the centreline characteristics, but under-
predict the velocity and over-predict the volume fraction
distribution near the jet edge. Since the near-edge region is
predominantly a sparse droplet flow region, the Eulerian models
fail to accurately capture the physics. The proposed simulation
methodology is helpful for predicting the flow behaviour of jets
used in industrial cleaning applications since these focus on the
near-field jet region.
Notation
D ¼ diameter of nozzle
F ¼ momentum source term
G ¼ production of turbulent kinetic energy
k1, C ¼ spread coefficients_M ðx; rÞ ¼ axial mass flux of water droplets
_m ¼ mass transfer
P ¼ pressure
r ¼ radial distance
R ¼ radial width of jet droplet zone
S ¼ mass source term
x ¼ axial distance
Greek symbols
1 ¼ turbulent dissipation rate
m ¼ viscosity
r ¼ density
Subscripts
a air
i inner
m mixture
o outer
t turbulent
w water
0 nozzle outlet
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Journal of Hydraulic Research Vol. 48, No. 1 (2010) Numerical simulation of high-speed turbulent water jets in air 129