Entrainment in Jets in Crossflow

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22 Φεβ 2014 (πριν από 3 χρόνια και 5 μήνες)

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2th

International Engineering Conference, Cairo Univ
ersity, Cairo, Egypt, Dec. 2005.

www.energyandeconomy.com

Entrainment in Jets in Crossflow


M. M. El
-
Khayat*,

S.
H. El
-
Emam*
*,
H. Mansour*
*,

and
A. R.
Abdel
-
Rahim
**

*
Mohamed
.
elkhayat
@
yahoo
.
com

*
*F
aculty of Engineering, Mansoura University, Mansoura, Egypt.



Abstract

The entrainment process occurred due to

the in
teraction of

two flow regimes

has been studied
theoretically and experimentally.
Flow regimes obtained from three
-
dimensional

jet
s

in
crossflow have been
recorded using a high speed digital camera.
The obtained images have
been analyzed using image digitiz
ing technique. The following results have been obtained, a

new correlation to estimate the entrainment fraction either in the near
-

or far
-
field
of the jet
.

The considered modification for Coelho
-
Hunt model shows
fair

results. It

has been concluded
that en
trainment fraction in the near
-
field has to be taken locally, not as a constant average
value.


Nomenclature

A

empirical dimensionless constant, Eq.(
6
)


U

x
-
direction velocity component,
m/s

B

empirical dimensionless constant, Eq.(
6
)


U
c

V
elocity

at je
t centerline
, m/s

C

empirical dimensionless constant, Eq.(
6
)


R
i

Richardson number

E
f

Entrainment fraction,


V
R

velocity ratio, jet velocity to crossflow velocity

i

Represents the cross section number


X
~

dimensionless horizontal Cart
esian coordinates

K
u

an experimental constant



J


jet flow density, kg/m3

m
J

mass flux at the jet exit


U
J


jet flow velocity, m/s

m
t

Accumulated mass flow rate along the jet





crossflow density, kg/m3

P

pressure,
N/m
2





1) Introduction

The e
ntrainment hypothesis in its original form can be stated very simply. In fluid
mechanics many situations occur where a turbulent region of flow is bounded by a flow,
which is non
-
turbulent and substantially irrotational. In the interaction between the two
flow
regimes the turbulence spreads with time into the neighboring fluid. Also, due to turbulent
mixing the region into which it spreads partakes of the general motion of the turbulent flow.
This process is generally referred to as entrainment (of the non
-
turbulent by the turbulent
flow) and is perhaps, most obvious in the case of the turbulent jet, where quite substantial
flows may be generated towards the jet due to its entrainment proprieties. Entrainment

is
a
very
important

process

in many practical sit
uations; for example it controls the flow patterns
in combustion chambers and furnaces; and many mixing devices of the chemical industry
rely on entrainment for their effectiveness.

Ricou and Spalding (1961) presented an experimental technique for measurin
g the axial
mass flow rate in the turbulent jets formed when a gas is injected into a reservoir of stagnant
air at uniform pressure. They found their technique applicable to jets of uniform and non
-
uniform density and to larger distances far from the jet.
In addition, they reported that the
new experimental technique for measuring the rate of entrainment by a turbulent jet was
found to be easy to use. Also, it is applicable to jets of non
-
uniform density and to larger
horizontal distances. Ricou and Spladin
g (1961) defined the entrainment fraction for a
certain location as, the ratio between the entrained mass to the accumulated mass along the
jet at the end of the near
-
field.

Benatt and Paul (1969) studied ambient gas entrainment into hollow cone sprays
pro
duced from spray nozzles. Turner (1986) studied the entrainment into free turbulent flows
such as jets. Turner mentioned that if there is a density step, with and without a mean shear


2

flow, the most important additional parameter on which the entrainment d
epends is a
Richardson number, R
i
. It is based on the density difference and the velocity and length
scales of the largest turbulent motions near the interface. It is the response of the fluid
outside the turbulent region that determines the rate of entrai
nment in this case.

El
-
Emam (1981) and Cantwell and Coles (1983) mentioned that, the ability of a turbulent
flow to entrain more viscous fluid depends on the response of the external fluid. Inertial
forces in the turbulent flow must be able to generate suf
ficient large normal stresses in the
surrounding fluid to overcome the viscous stresses and allow the interface to deform in the
manner necessary for mixing.

Dahm and Dimotakis (1987) mentioned that experiments over the past decade have
demonstrated that t
here are different formulae have been proposed to calculate entrainment
fraction. Also, entrainment and mixing in fully turbulent shear layers are characterized by an
organization that results from the dynamics of nearly periodic large
-
scale vortical motio
ns.
This large
-
scale organization has been shown to transport unmixed fluid from free streams
across the entire extent of the layer. As a consequence of this organization, the mixed fluid
probability density function is essentially uniform across the entir
e transverse extent of the
layer. On the basis of these results and many others, it is now generally recognized that
models of entrainment and mixing in turbulent shear layers probably need to incorporate
features of these large
-
scale motions.

Also, it can

be speculated that a similar large
-
scale organization of entrainment and
mixing may also be present in other free turbulent shear flows, including the axi
-
symmetric
free jet. Entrainment and, mixing in the jet far
-
field are classically viewed as stochasti
c
processes, involving transport by eddies whose scales are presumed to be small relative to
the overall lateral extent of the flow and to characteristically lack any persistent large
-
scale
organization.

Coelho and Hunt (1989), supported Dahm and Dimotaki
s (1987) point of view. They
showed that turbulent entrainment and the transport of the transverse component of vorticity
largely control the dynamics of the jet and its
bounding shear layer in the near
-
field of a JICF. In
particular, entrainment is the ma
in mechanism that
leads to the deflection of strong turbulent jets in the
direction of the stream. This analysis and the
experiments performed show that the external flow
around a strong jet is, to a first approximation,
potential flow around a cylinder wi
th suction, caused
by the entrainment into the jet. The diffusion of
vorticity into the wake is weak and therefore the jet
does not act on the external flow like a solid bluff
body. So, the ‘pressure
-
drag’ mechanism is negligible
compared with the effects
of entrainment on the
deflection of the jet. But the entrainment fraction used
with this model has been taken as a constant average
value for the near
-
field.

Meanwhile the role of the roll
-
up shear layer
vortices will be explained later, as a part of this
study,
the role of wake structure in entrainment process is
explained by Fric and Roshoko (1994) as follows.
Consider the sketches in Fig. (1), which show
outlines of jet trajectories at three values of velocity ratio as determined from the smoke
visualiza
tions. These three velocity ratios were used to represent three different regimes of
the wake. The arrows in the sketches, located in the down stream region, indicate an
idealized entrainment flow pattern at/or near the Y=0 plane. At velocity ratios near 2
, the jet
Fig.(1): Trajectories of jets and their entrain
-
ment patterns at different velocity ratios
(After Fric and Roshoko (1994))







(A): V
R

= 2



(B): V
R

= 4



(C): V
R

= 8



Jet Source



Jet Source



Jet Source





3

remains very close to the crossflow wall, Fig. (1.A). Even though its entrainment at the wall
is felt very strongly. The close proximity of the jet to the wall makes it difficult to
distinguish between jets, boundary layer, and wake fluid. In a s
ense, the jet is too close to
produce well defined wake structures, and it is not clearly separated from the wall.

Some stretching of the wake structures is required to define them well. Near V
R
=4, Fig.
(1.B) the jet is now far enough from the wall to indu
ce significant turning of the separated
boundary layer vorticity. But it is close enough to still strongly and efficiently pull the
separated fluid away from the crossflow wall. This moderate distance between the deflected
jet and wall allows some stretchi
ng of the wake vortices as they form, thus defining them
even better.

At higher velocity ratios, close to 8, Fig.(1.C), the crossflow boundary layer, although
separated, is not easily nor efficiently entrained by the jet, for the distance of the jet from t
he
near
-
wake wall region quickly becomes large.

Also, the near
-

and far
-
field break
-
up and atomization of a water jet by a high
-
speed
annular air jet are the interest of some studies. Head (1958), Benatt and Paul (1969), and
Lasheras et al (1998) examine
d the flow field by means of high
-
speed flow visualizations
and phase Doppler particle sizing techniques.

Visualization of the near
-
field and measurements of the frequencies associated with the
gas
-
liquid interfacial instabilities are used to study the und
erlying physical mechanisms
involved in the primary break
-
up of the water jet. This process is shown to consist of the
stripping of water sheets, or ligaments, which subsequently break into smaller lumps or
drops. An entrainment model of the near
-
field str
ipping of the liquid is proposed, and
showed to describe the measured liquid shedding frequencies. This simplified model
explains qualitatively the dependence of the shedding frequency on the air/water momentum
ratio in both initially laminar and turbulent

water jets. The role of the secondary liquid
break
-
up in the far
-
field atomization of the water jet is also investigated, and an attempt is
made to explain qualitatively the measurement of the far
-
field droplet size distribution and
its dependence on the
water to air mass and momentum ratios. Models accounting for the
effect of the local turbulent dissipation rate in the gas in break
-
up of the droplets are
developed and compared with the measurements of the variation of the droplet size along the
jet’s cen
terline.

From the above mentioned studies, it can be concluded that Coelho
-
Hunt model used a
constant value of the entrainment fraction. So, one of the objectives of the present study is to
modify this model by using a variable local value of the entrainme
nt fraction, applying the
definition of the entrainment fraction presented by Ricou and Spalding (1961).



2) Experimental Rig Arrangement

An arrangement of the experimental test rig is shown in Fig. (2). A Plexiglas 1
′ × 1′
square cross section of 2′ length mounted in a 12
-
inch sub
-
sonic wind tunnel is used as a test
section to study the concentration variation at different cross sections in both near
-

and far
-
field. The test section is painted with a black color to av
oid external light reflection. Jet
source is mounted at the center of the test section, and is fitted with a heat exchanger to
reduce temperature of the paraffin smoke produced from a smoke generator to the ambient
temperature. In addition, a sheet of ligh
t with 0.25 mm in width is produced from a lighting
system aligned to the center of the jet, and images are captured using a high
-
speed camera.
Pressure and temperature of the smoke have been measured by digital micro
-
manometer, and
digital non
-
contact the
rmometer, respectively. Meanwhile a digital illumenance meter has
been used to check test section darkness. Jet diameter used in this study is 4 mm, jet
velocities are 2, and 3 m/s , and crossflow velocity is fixed at 1m/s. In addition the side
-
view
images

taken in this study have been discussed and analyzed using image digitizing
technique introduced by El
-
Khayat (2002).



4

3) Results

The following sections present an explanation for the entrainment mechanism, in addition
jet deformation has been discussed
w.r.t. calculating jet mass flow rate. Also, a modification
for Coelho and Hunt model has been applied and results for this model before and after
modification has been put into comparison with other experimental and theoretical results.


3.1 Entrainment M
echanism

An explanation for the entrainment hypothesis is introduced by Coelho and

Hunt (1989).
As a result of a JICF, the surrounding fluid is sucked in, which means that the jet centerline
acts as a line sink for the surrounding fluid. Since crossflow
do not affect the entrainment
fraction into jets. On contrary with increasing crossflow velocity, it has been found that the
external flow is too strong to be completely entrained downwards in the jet.

Turner (1986) introduced another entrainment mechanism

based on jet roll
-
up shear layer
vortices. These vortices may occur as a result of jet and crossflow interaction. This creates a
strong non
-
equilibrium turbulent field in the forward region of the jet. Also, the momentum of
the crossflow affects their def
ormation.

Fig.(3.A) presents the interaction between the jet and crossflow using recent technique
used. Instabilities due to this interface are clearly shown on the jet boundaries. Figs. (3.B) to
Fig. (2): Schematic of the experimental rig arrangement


Pitot
tube


Air

Vessel


Bottle
cover


Smoke
generator

CCD Camera



5

X


Y

Jet

C
0

Fig.(4): Schematic diagram for the
concentration distribution

Fig. (4):

Schemati
c diagram for the


conc
entration distribution
.

Jet Source

(3.D) introduce the mechanism presented by Turner (1986), w
here vortices have been
affected, slightly,

by the crossflow as shown in Fig.(3.B), then reach the pocket
-
like shape in Fig.(3.C), and the
engulfing process has been reached in Fig.(3.D).


3.2 Jet deformation

As mentioned before, the effect of the
cro
ssflow on the jet produced from a jet source
of a circular cross section is sensitive to the
velocity ratio. So it is expected that jets keep
their circular cross section with a little deviation
near the jet source exit as the velocity ratio
increases.

Con
centration distribution at any location
along the jet can be obtained from the
experimental work. Density distribution will be
determined according to the obtained
concentration distribution. Mass flux, m, can be
estimated by integrating the velocity dist
ribution
at various radial distances as shown in Fig.(4).

All distributions are assumed to be function
of the radial position at any given cross section
(i.e. the jet is assumed to be circular). The mass
flux is estimated by the following equation;

rdr
r
U
m


2
.
0
.






(1)

This was made numerically at different
sections along the jet centerline and presented in
Fig.(5). This indicates that mass flow rate
reaches its asymptotic state in the far
-
field at
(X/d
J

>= 8)
. This conclusion confirms that jet

trajectory has to reach its saturation limit in the
far
-
field as shown by
Abdel
-
Rahim et al (2004)

(2004).

From these data, the entrainment

fraction can be calculated according to the definition
introduced by Ricou and Spalding (1961). Also, jet centerlin
e velocity at different distances,
0
5
10
15
20
25
30
D
i
s
t
a
n
c
e
,

X
/
d
j
0
0.2
0.4
0.6
0.8
1
R
e
l
a
t
i
v
e

a
c
c
u
m
e
l
a
t
e
d

m
a
s
s

f
l
o
w

r
a
t
e
VR = 1
VR = 2
VR = 3
VR = 4
F
i
g
.
(
6
.
1
6
)
:
R
e
l
a
t
i
v
e

a
c
c
u
m
l
a
t
e
d

m
a
s
s

f
l
o
w

r
a
t
e
.





d
j
=
4
m
m
,

f
o
r

d
i
f
f
e
r
e
n
t

v
e
l
o
c
i
t
y

r
a
t
i
o
s
Fig.(3): The interface between crossflow and roll
-
up sh
ear layer vortices, (A) obtained by the
recent technique used, and (B), (C) and (D) suggested by Turner (1986). d
J
=4mm, V
R
=2


(A)

(B)

(C)

Y/d
J

X/d
J

(D)

Fig (5): Relative accumulated mass flow rate d
J

= 4mm,



or different velocity ratios



6

can be estimated according to the following formula that recommended by Hasselbrink
(1999).

3
2
)
*
(
*
*
*
1
.
1



J
R
R
J
J
C
d
V
X
V
U
U










(2)


Because of the symmetry of concentration profil
e with respect to the transverse direction.
Beer and Chigier (1974) mentioned that, this is also valid for the velocity distribution, this
concept has been confirmed in this study, through the two
-
dimensional work. The
corresponding equation assuming Gauss
ian distribution is;

)
)
(
*
exp(
2
X
r
K
U
U
u
c












(3)


As a result of the above mentioned discussion, density distribution and velocity
distribution can be obtained. Therefore, entrainment fraction, E
f
, can be calculated as follows;

t
m
i
m
i
m
E
f



1










(4)






S
i
i
J
t
m
m
m
0










(5)

Entrainment fraction distribution will be estimated along the jet centerline in the near
-
field. An average value of the entrainment distribution is compared with Ricou and Spalding
value
. The obtained entrainment distribution along the jet centerline is implemented in
Coelho
-
Hunt model, rather than using a constant value as their assumption.

The definition presented by Ricou and Spalding (1961) has been chosen because it is
widely appreci
ated. The average entrainment fraction presented by them is 0.32 in the case
when jet and crossflow are gases of the same density. This value nearly matched the obtained
results in this study as will be described later.

Figure (6) shows the entrainment fr
action for different velocity ratios (d
J
=4mm). The
following equation has been used to fit the obtained results;

C
X
V
E
B
A
R
f


~












(6)

The empirical constants “B”, and “C” have been found to take the values 4.15, and
2.5*10
3

respectively
. Meanwhile, “A” has been found varies with velocity ratio, as shown in
table (1).


Table (1): Relation between coefficient A,


and V
R


Also, it has been found that at V
R

= 1, the
term
A
R
V
= 1.91*1
0
3
. Therefore, the final form
for the Eq.(6) will be as follows;

3
15
.
4
10
*
5
.
2
~


X
V
E
A
R
f


(7)


Equation(7) indicates that entrainment
fraction depends on two main parameters,
velocity ratio, and distance from the jet exit.
This correlation has been int
roduced to fit the
experimental results of the entrainment
fraction either in near
-

or far
-
field regions.

According to Fig.(5) it can be concluded that entrainment has its maximum values at the
near
-
field, meanwhile in the far
-
field it may be neglected, w
here its value is approximately
V
R

2

3

4

A

10.7

6.6

5.1

0
10
20
30
D
i
s
t
a
n
c
e
,

X
/
d
j
0
20
40
60
80
E
n
t
r
a
i
n
m
e
n
t

f
r
a
c
t
i
o
n
,

%
VR = 1
VR = 2
VR = 3
VR = 4
F
i
g
.
(
6
.
1
7
)
:
E
n
t
r
a
i
n
m
e
n
t

f
r
a
c
t
i
o
n
.
d
j

=
4
m
m
,




f
o
r

d
i
f
f
e
r
e
n
t

v
e
l
o
c
i
t
y

r
a
t
i
o
s
Fig. (6): Entrainment fraction. d
J

= 4mm, for different


velocity ratios
.



7

zero. With decreasing velocity ratio, the maximum entrainment limit increases. This can be
explained as follows, with increasing velocity ratio, the jet becomes stronger than the
crossflow. Therefore, crossflow could penetra
te weak jets (V
R
=1, and 2) easier than strong
jets (V
R
=3, and 4). Accordingly, entrainment fraction decreases with increasing velocity ratio.

Also, the obtained entrainment fraction
has been averaged in the near
-
field and
found to be ranged from 0.305 to
0.431 for
the investigated velocity ratios.


3.3 Modifications of Coelho
-
Hunt model

The obtained entrainment fraction
correlation has been introduced in Coelho
-
Hunt model to calculate the entrainment
fraction at each section, instead of
considering it as
a constant value.

Figure (7) shows the horizontal cross
sections for a JICF calculated at different
heights with respect to jet exit. These cross
sections were obtained using Coelho
-
Hunt
model when using entrainment fraction as a
constant value; Fig.(7.A)
. While Fig.(7.B)
for the case when using the entrainment
fraction correlation (Eq.7). As shown in
Fig.(7.B) deformation exists gradually till
reaches the end of the near
-
field and CVPs
occur. This is clearly due to the effect of
entrainment, which has bee
n taken at each
section with its corresponding value, not as
a constant value for the near
-
field.
Moreover, the obtained horizontal cross
sections after modifications, have an
agreement with those proposed by Kelso et
al (1996); Fig.(8).

Since these result
s seem valid and more
realistic, where deformation done
gradually, it can be assumed for layers of
small thickness results obtained from
Coelho
-
Hunt mode can be put into
comparison with these results obtained
from the same model after modification.
Doing s
o finds, marked differences among
results.

The combined effects of the influx of
fluid to the mixing layer due to turbulent
entrainment, and its convection from the
upwind side to the downwind side of the jet, lead to a higher rate of increase of


on the
downwind face.

Turbulent entrainment also provides an influx of momentum from the external stream to
the layer.

The entraining fluid further increases the overall momentum of mixing layer in the X
-
direction, increasing u
s

as z increases. This leads to a m
ore intensive concentration of

s

at the
downwind side of the jet.

Fig.(8): Isometric view of the jet shear
-
layer vortex

rings.
(After Kelso et al (1996))

Y
=
0
Y

=

1
V
R
d
J
Y

=

3
V
R
d
J
Y

=

4
V
R
d
J
Y

=

2
V
R
d
J
F
i
g
.
(
6
.
1
8
)
:
E
x
t
e
r
n
a
l

c
r
o
s
s

s
e
c
t
i
o
n
s

f
o
r

a

J
I
C
F
,

(
A
)

r
e
s
u
l
t
s

o
f





C
o
e
l
h
o
-
H
u
n
t

m
o
d
e
l
,

a
n
d

(
B
)

r
e
s
u
l
t
s

a
f
t
e
r

r
e
c
e
n
t

m
o
d
i
f
i
c
a
t
i
o
n
s
,





d
j

=

4
m
m
.


V
R
=
3
(
A
)
(
B
)
C
r
o
s
s
f
l
o
w
C
r
o
s
s
f
l
o
w
C
r
o
s
s
f
l
o
w
C
r
o
s
s
f
l
o
w
C
r
o
s
s
f
l
o
w
Fig. (7): External cross sections for

a JICF, (A) results of


Coelho
-
Hunt model, and (B) results after recent


modifications, d
J
=4mm.

Crossflow

Crossflow

Crossflow

Crossflow

Crossflow



8



4) Conclusions


-

A new correlation to estimate the entrainment fraction either in the near
-

or far
-
field has
been introduced

-

Image digitizing technique is able to be used to study the ent
rainment phenomena.

-

A

fair agreement
between the current results and

Ricou and Spalding (1961)

results

has
been reached.

-

The considered modification for Coelho
-
Hunt model shows fair results.

-

It has been

concluded that entrainment fraction in the near
-
field

has to be taken locally, not
as a constant average value.


5) References

Abdel
-
Rahim
, A. A., Mansour, H., El
-
Emam, S. H., and El
-
Khayat, M. M. (2004),

“Three
-
dimensional jets in crossflow”

Ben慴aⰠI⹓⸠慮d⁐慵氬⁅⸠⠱㤶㤩V

“Gaseous entrainment into axi
-
symmetric liquid sprays,”
J. of the Institute of Fuel, 309.

Cantwell, B. and Coles, D. (1983),

“An experimental study of entrainment and transport in
the turbulent near wake of a circular cylinder,” J. of Fluid Mech., 136, 321


374.

Coelho, S.L. and Hu
nt, J.C.R. (1989),

“The dynamics of the near field of strong jets in
crossflow,” J. of Fluid Mech., 200, 95


120.

Dahm, W.J.A., and Dimotakis, P.E. (1987),

“Measurements of entrainment and mixing in
turbulent jets,” AIAA J., 25 (9), 1216


1223.

El
-
Em
am, S.H. (1981),

“A study of the flow field and combustion of sprays,” Ph.D. thesis,
Osaka University.

El
-
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