The paper presents new experiments performed in the developing flow region of two-dimensional water jets discharging into air. The results indicate that the distributions of void fraction follow closely an analytical solution of the diffusion equation, and that the transfer of momentum between the water jet and the surrounding air is negligible for x/do < 20. An analogy with self-aerated open channel flows is further developed. The distributions of air bubble frequency have the same shape for both flow situations suggesting some similarity in the air-water flow structure

Brattberg, T.

Chanson, Hubert

Toombes, L.

University of Queensland eSpace

Proceedings of FEDSM'98:
1998 ASME Fluids Engineering Division Summer Meeting
June 21-25, 1998, Washington, DC
FEDSM98-4805
DEVELOPING AIR-WATER SHEAR LAYERS OF TWO-DIMENSIONAL WATER
JETS DISCHARGING INTO AIR
T. BRATTBERG
Department of Civil Engineering
The University of Queensland
Brisbane QLD 4072
Australia
L. TOOMBES
Department of Civil Engineering
The University of Queensland
Brisbane QLD 4072
Australia
H. CHANSON
Department of Civil Engineering
The University of Queensland
Brisbane QLD 4072
AustraliaABSTRACT
The paper presents new experiments performed in the
developing flow region of two-dimensional water jets
discharging into air. The results indicate that the distributions
of void fraction follow closely an analytical solution of the
diffusion equation, and that the transfer of momentum
between the water jet and the surrounding air is negligible for
x/do < 20. An analogy with self-aerated open channel flows is
further developed. The distributions of air bubble frequency
have the same shape for both flow situations suggesting some
similarity in the air-water flow structure.
Keywords : water jets, air entrainment, developing flow,
experimental measurements, analogy with self-aerated flows.
INTRODUCTION
Turbulent water jets discharging into the atmosphere are
often characterised by a substantial amount of air
entrainment. Applications include water jets at bottom outlets
to dissipate energy, jet flows downstream of a spillway ski
jump, mixing devices in chemical plants, spray devices, fire-
fighting equipment and jet cutting equipment (e.g. coal
mining). A related case is the ventilated cavity flow, observed
downstream of blunt bodies, on the extrados of foils and
turbine blades, at spillway aeration devices, behind a body
impacting into a pool.
With a turbulent water jet, aeration occurs along the air-
water jet interfaces. The effects can be detrimental (e.g. loss
of jet momentum) or beneficial (e.g. mixing enhancement). In
any case, knowledge of the air entrainment mechanisms is
essential for an optimum design.
Some experimental results are available on the free-
surface aeration of circular water jets (e.g. HERAUD 1966,ERVINE and FALVEY 1987, RUFF et al. 1989, TSENG et
al. 1992) but there is little information on the free-surface
aeration of two-dimensional water jets (see review in
CHANSON 1997a). This paper describes new experiments
performed with two-dimensional water jets discharging into
air. The study is focused on the air-water flow properties of
the developing flow.
EXPERIMENTAL APPARATUS
Experimental channels
Three experimental configurations were used (Fig. 1,
Table 1). Experiment No. 1 is basically a water wall jet.
Experiment No. 2 is an air-water free-shear layer at an abrupt
drop. The third experiment is a vertical free-falling jet issuing
from a slot nozzle and discharging downwards.
Discharges were measured with a Venturi-type device
(i.e. Dall™ tube) in experiments No. 1 and No. 2, and with
orifice meters in experiment No. 3. The error on the discharge
measurement was less than 2%.
For experiments No. 1 and 2, the vertical probe
translation was controlled by a fine adjustment travelling
mechanism connected to a MitutoyoTM digimatic scale unit.
The error on the vertical position of the probe was less than
0.025 mm and the accuracy on the longitudinal position of the
probe was less than 1 cm. In experiment No. 3, the
displacements of the probe in the direction normal to and
along the jet support were controlled by two fine adjustment
travelling mechanisms (made in-house) and the positions
were measured with two Lucas Schaevitz Magnarules Plus.1 Copyright © 1998 by ASME
The error in the longitudinal and normal positions of the
probes was less than 0.1 mm in each direction.
Instrumentation
The air-water flow properties were recorded using
conductivity probes made at the University of Queensland
(CHANSON 1995a, CUMMINGS and CHANSON 1997).
A single-tip conductivity probe (inner electrode  = 0.35
mm, outer electrode  = 1.42 mm) was used to perform air
concentration measurements in experiments No. 1 and 2. A
two-tip conductivity probe was used to record simultaneously
air concentration, air-water velocity and bubble frequency in
experiment No. 3. Each tip was identical with an internal
concentric electrode ( = 25 m) and a 8-mm tip spacing.
Both conductivity probes were excited by an electronic
system designed with a response time less than 10 s. The
measurements were recorded with a scan rate ranging from 10
to 20 kHz per channel.
In addition clear water jet velocities were measured with
a Pitot tube (Ø = 3.3 mm).
AIR-WATER FLOW CHARACTERISTICS
In the water jets, the air concentration distributions
follow closely a solution of the diffusion equation :
C  =  12 * 







1  -  erf







y
2 * 
Dt
Vo
 * x
(1)
where x is the distance in the flow direction, y is the distance
normal to the flow, Vo is the nozzle velocity, Dt is the
turbulent diffusivity, assumed independent of the transverse
direction y (CHANSON 1996) and the function erf is defined
as :
erf(u)  =  2

  *  
0 
 u
 exp(- t2) * dt (2)
Equation (1) was developed and validated for a two-
dimensional free-shear layer. Note that it may be applied to
all three investigated flow configurations (fig. 1), suggesting
that the gradual diffusion/advection of air bubbles is nearly
independent of the flow situations. The result (eq. (1)) may be
extended to the developing flow region of a two-dimensional
water jet discharging into the atmosphere but it is not valid
when the jet core become aerated.
Air-water velocity and bubble frequency distributions
were recorded in experiment No. 3 (fig. 2). In the near-flow
field (i.e. x/do < 20), the transfer of momentum from the
water jet to the air is small and the velocity distributions are
basically not affected by the advective diffusion of air
bubbles. The velocity distribution is nearly constant across
the developing air-water layer. For a free-falling jet
(experiment No. 3), it yields :
V  =  Vo * 1  +  
2*g*x
Vo
2 for C < 0.90 and x/do < 17 (3)where do is the jet nozzle thickness and g is the gravity
acceleration.
The variation of air bubble frequency gives some
information on the air-water flow structure. Experimental data
show a maximum bubble frequency for 50% air content. At a
given cross-section (i.e. x constant), the relationship bubble
frequency-void fraction follows closely a parabolic shape
(fig. 3) :
f  =  FFM
  =  1  -  4 * (C - 0.5)2 (4)
where F is the air bubble frequency and FM is the maximum
frequency which is best correlated by :
FM  =  1.242 * 
x*do
Vo
for w*
Vo*x
w
 > 1.5E+5  (5)
where w and w are the water density and dynamic viscosity
respectively.
Bubble chord length distributions are presented in figure
4. The data (fig. 4) are cumulative bubble chord length
distributions for 0 < C < 0.90 at various distance x from the
nozzle. The histogram columns represent each the probability
of a bubble chord length in 0.5-mm intervals: e.g., the
probability of a chord length from 2.0 to 2.5 mm is
represented by the column labelled 2.5. The last column (i.e.
100-mm) indicates the probability of bubble chord lengths
larger than 100-mm.
The results (fig. 4) show the broad spectrum of bubble
chord lengths : i.e., from less than 0.5-mm to larger than 100-
mm. The distributions are skewed with a preponderance of
small bubble sizes relative to the mean. The probability of
bubble chord lengths is the largest for bubble sizes between 0
and 2.0 mm. Close to the jet nozzle (fig. 4A), most entrained
air bubbles have small sizes. Further downstream the
distribution of chord length is redistributed toward larger
sizes (fig. 4B).
INTERFACIAL AERATION : SIMILARITY WITH SELF-
AERATED FLOWS
Presentation
There is some similarity between air entrainment in high-
velocity water jets and in supercritical open channel flows
(fig. 5). Both flow configurations are high-speed turbulent
flows with interfacial aeration rather than local aeration as in
hydraulic jumps and plunging jet flows. In water jets and
open channel flows, the air bubbles are gradually
diffused/dispersed within the mean flow and the distributions
of air concentrations have the same shape (fig. 2 and 6). The
channel invert is analog to the jet centreline and the solution
of the diffusion equation yields at equilibrium :
C  =  1  -  tanh2


K'  -  
y'
2 * D' Open channel flow (6)
where K' = tanh-1( 0.1) + 0.5/D',
D' = Dt/((ur)Hyd*cos*Y90), (ur)Hyd is the rise velocity in
hydrostatic pressure gradient,  is the invert slope, y' =
y/Y90, y is the distance normal to the invert and y = Y902 Copyright © 1998 by ASME
when C = 0.9 (CHANSON 1997a). The dimensionless
turbulent diffusivity D' is a function of the mean air content
only.
Further identical bubble frequency distributions (i.e. eq.
(4)) are observed in both free-jets and fully-developed
supercritical flows, suggesting that the air-water flow
structure might be similar (CHANSON 1997b).
Discussion
A comparison of the turbulent diffusivity Dt, used in
equations (1) and (6), shows that Dt/(Vo*do) ranges from
1.5E-3 to 1.7E-2 for the present study (water jets) while
Dt/(V90*Y90) is between 2.7E-3 and 1.6E-2 for the data of
STRAUB and ANDERSON (1958) (open channel flow, fig.
6). The results are of the same order of magnitude, suggesting
again some similarity in the advection/diffusion process.
The major differences between the two flow
configurations are the shear layer characteristics. Self-aerated
open channel flows are fully-developed shear flows while the
developing air-water region of water jets is a free shear layer.
In addition gravity effects (i.e. buoyancy) are significant in
open channel flows.
CONCLUSION
The paper presents new experiments performed in the
developing flow region of two-dimensional water jets
discharging into air. With three different configurations (fig.
1), the results indicate that the distributions of void fraction
follow closely an analytical solution of the diffusion equation.
Further the transfer of momentum between the water jet and
the surrounding air is negligible for x/do < 20 and it is not
affected by the air bubble diffusion process.
An analogy with self-aerated open channel flows is
further developed. The distributions of air bubble frequency
have the same shape for both flow situations suggesting some
similarity in the air-water flow structure. Both flow situations
are characterised by a gradual diffusion of air bubbles (in the
longitudinal direction) although the characteristics of the
shear flow are significantly different.
ACKNOWLEDGMENTS
The authors acknowledge the helpful comments of
Professor Colin J. APELT (The University of Queensland)
and Dr J. MACINTOSH (Water Solutions).
REFERENCES
CHANSON, H. (1995). Report CH46/95, Dept. of Civil
Engineering, University of Queensland, Australia, June, 368
pages (ISBN 0 86776 611 5).
CHANSON, H. (1996). "Air Bubble Entrainment in Turbulent
Water Jets Discharging into the Atmosphere." Aust. Civil Engrg
Trans., I.E.Aust., Vol. CE39, No. 1, Sept., pp. 39-48 (ISSN
0819-0259).
CHANSON, H. (1997a). "Air Bubble Entrainment in Free-surface
Turbulent Shear Flows." Academic Press, London, UK, 401
pages.CHANSON, H. (1997b). "Air Bubble Entrainment in Open
Channels. Flow Structure and Bubble Size Distributions." Intl Jl
of Multiphase Flow, Vol. 23, No. 1, pp. 193-203 (ISSN 0301-
9322).
ERVINE, D.A., and FALVEY, H.T. (1987). "Behaviour of
Turbulent Water Jets in the Atmosphere and in Plunge Pools."
Proc. Instn Civ. Engrs., London, Part 2, Mar. 1987, 83, pp. 295-
314. Discussion : Part 2, Mar.-June 1988, 85, pp. 359-363.
HERAUD, D. (1966). "Dispersion des Jets Liquides; Influence des
Rugosités de Paroi." ('Dispersion of Liquid Jets; Influence of the
Wall Roughness.') Ph.D. thesis, University Grenoble 1, France
(in French).
RUFF, G.A., SAGAR, A.D., and FAETH, G.M. (1989). "Structure
and Mixing Properties of Pressure-Atomized Sprays." AIAA Jl,
Vol. 27, No. 7, pp. 901-908.
STRAUB, L.G., and ANDERSON, A.G. (1958). "Experiments on
Self-Aerated Flow in Open Channels." Jl of Hyd. Div., Proc.
ASCE, Vol. 84, No. HY7, paper 1890, pp. 1890-1 to 1890-35.
TSENG, L.K., RUFF, G.A., and FAETH, G.M. (1992). "Effects of
Gas Density on the Structure of Liquid Jets in Still Gases." AIAA
Jl, Vol. 30, No. 6, pp. 1537-1544.3 Copyright © 1998 by ASME
Table 1 - Experimental flow conditions (University of Queensland)
Ref. Slope

Deflector and nozzle Vo do Comments
(deg.) geometry (m/s) (m)
(1) (2) (3) (4) (5) (7)
Experiment No. 1 4.0 Elliptical convergent (10:1 contraction in
flow thickness and 2.2:1 in jet width).
5.0 0.03 Water wall-jet discharging into
air. W = 0.5 m.
Experiment No. 2 0 Elliptical convergent (10:1 contraction in
flow thickness and 2.2:1 in jet width).
5.0 0.03 Nappe flow at an abrupt drop
(z = 0.131 m). W = 0.5 m.
Experiment No. 3 89 S-shaped convergent (12.5:1 jet contraction
followed by a 50-mm straight section).
1.43 to
7.88
0.012 Vertical free-falling supported
jet. W = 0.269 m.
Fig. 1 - Air entrainment in a two-dimensional jet discharging into atmosphere4 Copyright © 1998 by ASME
Fig. 2 - Air-water flow characteristics in free-falling jets: distributions of air concentration C, dimensionless
mean air-water velocity V/Vo and dimensionless bubble frequency fab = F* x*do/Vo
(A) Vo = 2.65 m/s, do = 0.012 m, x/do = 8.3 (B) Vo = 6.86 m/s, do = 0.012 m, x/do = 8.3
Fig. 3 - Dimensionless bubble frequency  = F* x*do/Vo as a function of the air content C
0
20
40
60
80
100
0.4 0.6 0.8 1 1.2 1.4
0
0.5
1
1.5
C Data
C Theory
V/Vo Data
V/Vo Theory
fab Data
fab EQ.
y/do
C (%) V/Vo
fab
0
20
40
60
80
100
0.6 0.8 1 1.2 1.4 1.6
0
0.5
1
1.5
y/do
C (%) V/Vo
fab
0
0.5
1
1.5
0 20 40 60 80 100
Vo = 4.8 m/s
Vo = 5.8 m/s
Vo = 6.86 m/s
Vo = 7.9 m/s
EQ.
do = 0.012 m
C
x = 0.1 m
5 Copyright © 1998 by ASME
(A) x = 0.05 m
(B) x = 0.15 m
Fig. 4 - Bubble chord length distributions for Vo = 3.75 m/s and do = 0.012 m (Chord length interval: 0.5 mm)
Fig. 5 - Sketch of self-aerated flows
0
0.05
0.1
0.15
0.5 5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 50.5 55.5 60.5 65.5 70.5 75.5 80.5 85.5 90.5 95.5
Pr(Chab)
Chab (mm)
Chord lengths > 100 mm
0
0.05
0.1
0.15
0.5 5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 45.5 50.5 55.5 60.5 65.5 70.5 75.5 80.5 85.5 90.5 95.5
Pr(Chab)
Chab (mm)
Chord lengths > 100 mm6 Copyright © 1998 by ASME
Fig. 6 - Air concentrations in self-aerated flows (Data : STRAUB and ANDERSON 1958)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Theory
Cmean=0.16
Cmean=0.24
Cmean=0.31
Cmean=0.41
Cmean=0.62
C
y/Y90
Data : Straub and Anderson 19587 Copyright © 1998 by ASME


Developing Air-Water Shear Layers Of Two-Dimensional Water Jets Discharging Into Air

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Developing Air-Water Shear Layers Of Two-Dimensional Water Jets Discharging Into Air

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