Vol.2, No.6, 635-640 (2010) Natural Science
http://dx.doi.org/10.4236/ns.2010.26079
The dynamic field in turbulent round jet discharging into
a co-flowing stream
Mohamed Hichem Gazzah1*, Nejmiddin Boughattas2, Hafedh Belmabrouk1, Rachid Said2
1Département de Physique, Faculté des Sciences de Monastir, Monastir, Tunisie; *Corresponding Author:
2UREMIR, Institut Préparatoire aux Etudes d’Ingénieurs de Monastir, Monastir, Tunisie
Received 22 February 2010; revised 16 March 2010; accepted 7 April 2010.
ABSTRACT
The effects of a co-flow on a spreading and en-
trainment rate of turbulent round jets have been
studied numerically. The first and second order
closure models are used and have been comp-
ared with existing experimental data. The influ-
ence of theses models on the dynamic fields is
examined. The results of the models in general
agree well with the trends observed experiment-
tally. The co-flowing imposed noticeable restri-
ctions on the spreading and the turbulent mix-
ing. Finally, an entrainment hypothesis has been
introduced to describe the development of tur-
bulent jets issuing into a stagnant or co-flowing
air. It relates the mass flow rate of the surround-
ing fluid entrained into the jet to the character-
istic velocity difference between the jet and the
co-flow. It is obvious that the co-flow decreases
considerably the entrainment of air.
Keywords: Co-Flow; Turbulence; Jets; Models;
Entrainment
1. INTRODUCTION
The experimental and numerical studies concerning the
aerodynamics of co-flowing turbulent round jets have
lately observed an increasing interest as they are encou-
ntered in several industrial applications, such as turbul-
ent diffusion flames of combustion chambers where a
fuel jet flow is commonly injected into a co-flowing str-
eam jet. The important parameters that influence the
mixing characteristics of a jet are the presence of density
difference and a co-flowing between the jet and its sur-
roundings.
The effect of the density variation in the turbulent ro-
und free jet has been investigated experimentally and
numerically by several authors like [1-6]. The results of
these works show that the mean and turbulent quantities
are strong functions of density ratio. This confirms a
higher mixing efficiency when the density ratio between
the jet and the quiescent air decreases. It is shown that
the density effects are affected by the buoyancy terms in
the similarity region of the jet. The influence of the
emission initial conditions on the evolution of the dy-
namic and scalar fields is also studied.
The present study is a continuation of the work started
by the authors [7,8] to investigate numerically the influ-
ence of the co-flow surrounding the turbulent jets. In this
numerical study, several traditional scalar dissipation
rate models are examined for scalar transport modeling
in mixing turbulent round jets with co-flowing air.
Therefore, it is not intended here to review previous
work detail [9-11] on a co-flowing jet, since a review on
the subject matter is given by the references [7,8]. The
experimental and numerical investigations of this type of
flow are also relatively scarce.
In his recent work, Hakem et al. [12], have studied
experimentally the mixing characteristics of an elliptical
jet with large varying aspect ratio in a co-flow current, to
verify that the Elliptic jet with varying large aspect ratio
has also much higher dilution in a co-flow than an
equivalent round jet under the same flow conditions.
Wang et al. [13] have also studied the variable-density
turbulent round jets discharging into a weakly confined
low-speed co-flowing air stream with the aid of large-
eddy simulation. Thus, the majority of this work shows
that the effects of the co-flow variation on the structure
of the reactive and non-reactive turbulent jets are com-
plex problems and remain very interesting.
The study of the jets with co-flow shows the presence
of three essential areas: an initial area, a principal area
and an area of transition. When the flow is confined, the
process of the co-flow driven by the jet is modified and
the mixing process depends strongly not only on the
velocity ratio, but also on the interaction between the
boundary layer, the mixing layer and the main flow.

M. H. Gazzah et al. / Natural Science 2 (2010) 635-640
636
A considerable pressure gradient can appear and gen-
erates phenomena of recirculation. Curtet [14] is inter-
ested in a parameter of similarity, called parameter of
Craya-Curtet which is formulated in the literature in
variable density by Steward and Gurus [15]. He showed
that, for a value of this number higher than 0.8, the phe-
nomenon of recirculation is avoided, irrespective of the
fluid considered.
The major objective of this paper is to determine the
effect of a co-flowing on the dynamic fields of a turbu-
lent round jet. Therefore, we investigate a turbulent
round jet into a co-flowing air with various co-flow to
fuel velocity ratios. We have thus used first and second
order closure methods to investigate and compare their
performances. The influence of the co-flow on various
physical parameters of the jet is analyzed in comparison
with the experimental data of Djeridane [16].
2. TURBULENCE MODELS
The equations which govern the turbulent flow are de-
rived from the conservation laws of mass and momen-
tum. All variables are conventionally averaged. These
conventional averaged variables are denoted by an
overbar Φ. Conventional fluctuations are indicated by
Φ.
ΦΦΦ   (1)
2.1. The Mean Equations
We suppose that, the mean motion is steady, the turbu-
lent Reynolds numbers are high enough and the molecu-
lar diffusion effects are neglected.
The continuity equation is given by
0
j
j
x
U


 (2)
The momentum equation is:
 
iji
ji
ji
j
guu
xx
p
UU
x








 

 (3)
As a consequence of the nonlinearity (2), the averag-
ing process used introduces unknown correlations which
are modelled through turbulence models. In order to
solve transport equation for mean velocity in the turbu-
lent jets, the turbulent Reynolds stress shown in these
equations is computed using two turbulence closure
models, called the k-ε model and the second order model.
Details of theses models can be found in Schiestel [17].
2.2. The First-Order k-ε Model
The turbulent fluxes are approximated with k-  model.
The Reynolds stresses tensors are related to the strain
rate by the following equation:
ij
i
j
j
i
tji k
x
U
x
U
uu 





3
2









 (4)
where t

is the turbulent viscosity, which is obtained
from the turbulent kinetic energy k and its dissipation
rate
 using the relation:

 
2
k
C
t (5)
The turbulence model consists of equations for the
turbulent kinetic energy and their dissipation. These are
The kinetic energy equation
 








 













j
i
ji
jk
t
j
j
jx
U
uu
x
k
x
k U
x (6)
The energy dissipation rate

k
Ce
k
PCe
x x
U
xk
j
t
j
j
j
2
21























 (7)
The model constants used in the present study are
given in Table 1.
The k-
 model has been used with success in the cal-
culation of various turbulent jets. However, in flows
with significant streamline curvature, the isotropic eddy
viscosity assumption may not be able to describe the
turbulent diffusion effects adequately.
2.3. The Second-Order Model
The second turbulence model considered in this study is
a Reynolds Stress Model (RSM). The Reynolds stress
equation is:

ij
i
j
j
i
ijijjik
kx
u
x
u
pDPuuU
x




3
2


















 (8)
The first term on the right hand side is the production
term due to the mean strain:
k
i
k
j
k
j
kiij x
U
uu
x
U
uuP









 (9)
The diffusion term is modelled as:









 
l
jj
lk
k
sij x
uu
uu
k
x
CD






 (10)
Table 1. Turbulence constants for the first order k-  model,
where the value of C

is adapted for the axisymmetric jet case.

C 1,e
C 2,e C k



0.06 1.44 1.92 1.00 1.30

M. H. Gazzah et al. / Natural Science 2 (2010) 635-640
637
637
The pressure-strain correlation is:
II
ij
I
ij
i
j
j
i
x
u
x
u
p
 















 (11)









 ij
jj
I
ij k
uu
C

3
2
1: The return to isotropy
term.






 ijkkij
II
ij PPC

3
2
2: The rapid term.
The model constants used in the present study are
given in Table 2.
3. NUMERICAL APPROACH
The computations for the governing equations can be
made using a parabolic marching procedure if the radial
pressure gradients are small and the axial diffusion is
neglected. Such a situation occurs if velocities in the two
streams are comparable. Assuming parabolic conditions,
a numerical solver has been developed using finite vol-
ume of Patankar [18].
The computations are performed up to an axial dis-
tance of approximately 100D with an axial forward step
size of 0.01 times the local jet half width u Lx 01.0


and 80 grid points in the radial direction are used. The
radial expansion dx
dL
xru
 is so small that it does
not affect the assumption of an orthogonal grid. This
means that the grid expands in the radial direction fol-
lowing the jet expansion and this is sufficient to obtain a
grid independent numerical solution.
No boundary conditions are prescribed due to the
parabolic nature of the flow. The computation progresses
from section to section, and its implementation requires
only the profiles at the jet nozzle. The boundary condi-
tions at the nozzle exit are those of a fully developed
pipe flow [19]. The radial velocity is zero at the nozzle
and in the ambient. All variables at the radial jet bound-
ary are equal to those in the ambient. At the axis of
symmetry, the radial velocity and the radial gradients of
other variables are set to zero. For all calculations, a
small co-flow velocity value is used. For the turbulence
quantities this implies a value of zero or a negligible
small value. The kinetic energy and the Reynolds stress
profile are used to derive the energy dissipation through
the following relationship:
Table 2. Turbulence constants in the second order model,
where the value of C1 is adapted for the axisymmetric jet case.
C1 C2 Cs
2.3 0.6 0.22












r
U
uv
kC 2

 (12)
4. RESULTS AND DISCUSSION
The influence of co-flow is investigated using the ex-
periments data of Djeridane [16]. The jet is ejected from
a round nozzle of an internal diameter D of 26 mm with
various co-flow to jet velocity ratios of Uco /Uj = 0.0 and
0.1. The experimental details are given in Table 3. The
far field behavior of the computed quantities such as the
velocity decay constant, the turbulence intensity, the
spreading rate and the turbulent flux are presented and
discussed.
Figure 1 shows, the jet centreline axial mean velocity




coccojUUUU  / with and without co-flow as a
function of the normalized distance x/D. It is seen that
the velocity is somewhat overpredicted with experiment
values. The predicted results obtained, using the two
turbulence models, agree quite well with the experimen-
tal data of Djeridane [16]. The major visible effect of the
co-flow is the jet decay rate reduction, in comparison
with the free jet case.
Therefore, the jet without co-flow, tends to mix more
rapidly with the ambient air than the co-flowing jets. In
the far region, a hyperbolic decrease of the mean veloc-
ity is observed. However, for 20 < x/D < 50, the velocity
decay constant Ku, is defined by the following,



jco
u
cco
UU
x
K
D
UU

 (13)
Table 3. Properties of the investigated turbulent co-flowing
jets.
Jet/co-flow Ujet U
co S  j
Re
Air/air 12 0.00 1 21 000
Air/air 12 1.20 1 21 000
01020 30 40 50
x
/D
0
2
4
6
8
10
RSM
k-

Djeridane [16] no co-flow
Djeridane [16] with co-flow
with co-flow
no co-flow



jco
cco
UU
UU


Figure 1. Centreline values of the axial mean velocity.

M. H. Gazzah et al. / Natural Science 2 (2010) 635-640
638
This constant is closely related to the spreading rate,
and the asymptotic value of the turbulence intensity.
The comparison of this decay constant with those
found in the literature is a good validation tool for the
developed computational code. The results of the present
investigation give for the case with a co-flow, a value of
Ku = 0.188 with the second order model, and a value of
Ku = 0.174 with the first order model, while the experi-
mental value is 0.176 Djeridane [16]. However, in the
case of the jet without a co-flow, the value for Ku is
found to be 0.199 with the second model and 0.201 with
the first order model. The experimental value of the
slope Ku is 0.218 Djeridane [16]. Table 4 recapitulates
the asymptotic values of the mean velocity decay for
various co-flow strengths of the different model predic-
tions and compared to the experimental values of Djeri-
dane [16].
Using the similarity law proposed by Chen and Rodi
[20] ()/(/ equcj DxKUU), the influence of the veloc-
ity ratio on the decay Ku is also described quite well by
the most recent experimental study of Wang et al. [13]
(shows Ku = 0.158, with velocity ratios of Uco/Uj = 0.075)
and Antoine et al. [11] (gives Ku = 0.146, with velocity
ratios of Uco/Uj = 0.05). The major visible effect of the
co-flow, when the velocity ratio increases, is the jet de-
cay rate reduction.
Figure 2 features a comparison of the computational
and experimental jet spreading rates of the velocity field
based on the mean velocity half radius Lu. It is noticed
that the half-width Lu in the jets without a co-flow are
much larger than those of the corresponding jets with a
co-flow. Furthermore, with the first model, the predicted
half-width of the jet obtained by the two jets cases
agrees much better with the experimental data of Djeri-
dane [16]. Here, the mean velocity half-width is defined
by
Table 4. Comparison between the model predictions and meas-
urements of the velocity flow field for asymptotic values of
mean velocity decay, the spreading rate and the velocity fluc-
tuation intensities at various co-flow strengths.
Authors Uco /Ujet S
 Ku 2/1
u
S

2
c
cco
u
UU

max
2
cco
u
UU
RSM 0.10 1.0 0.1880.100 0.286 0.252
RSM 0.0 1.0 0.1990.192 0.251 0.244
k-  0.10 1.0 0.1740.096 0.305 0.269
k-  0.0 1.0 0.2010.195 0.283 0.267
Djeridane [16] 0.10 1.0 0.1760.132 0.261 0.219
Djeridane [16] 0.00 1.0 0.2180.184 0.253 -
Antoine et al. [11] 0.075 1. 0.1460.128 - -
Wang et al. [13] 0.05 1.0 0.1580.12 0.25-0.30 -
60
j
u
D
L2
0
2
4
6
8
10
RSM
k-

Djeridane [16] no co-flow
Djeridane [16] with co-flow
with co-flow
no co-flow
01020 30 40 50
x
/
D
Figure 2. Centreline values of the mean velocity halfwidth.
D
x
S
D
L
u
u2/1
2 (14)
where 2/1
u
S is the mean velocity spreading rate. In the
absence of a co-flow, both models predict a velocity
spreading rate Su192.0
2/1  with the second model and
195.0
2/1 
u
S with the first model. These rates are close
to the experimental value 184.0
2/1 
u
S of Djeridane
[16]. In presence of a co-flow, the half-width is no
longer a linear function of x, so 2/1
u
S can not be easily
determined. Therefore 2/1
u
S depends on the axial dista-
nce and is not a useful concept in a co-flowing jet. How-
ever, the spreading rates 10.0
2/1 
u
S and 096.0
2/1 
u
S,
obtained by the two models with a co-flow are about
22 smaller than that the average experimental values
of Djeridane [16] 1320
21 .S /
u, the air-air jet of Wang
et al. [13] 12.0
2/1 
u
S, and the water-air jet of Antoine
et al. [11] 128.0
2/1 
u
S. This low value of the velocity
spreading rate can be attributed to the presence of the
higher co-flowing stream and to the side walls position
of the enclosure, compared to those of the latter authors,
tending to reduce the jet expansion.
Figure 3 shows the axial profiles of the velocity fluc-
tuation intensities


cocc UUu /
2 on the jet centerline
with and without a co-flow. The predicted results ob-
tained by the two turbulence models agree very well
with the experimental data of Djeridane [16], and espe-
cially with the second order model. The asymptotic
value of


coccUUu/
2 is apparently strongly influ-
enced by the used turbulence model. It is seen that the
velocity fluctuation intensity with and without a co-flow
is slightly overpredicted by the second order model,
while for the first order model, the velocity fluctuation
intensity is highly overpredicted. This is obvious since
the k-  model is an isotropic model which then overes-
timates the velocity fluctuation intensity. Gharbi et al.
[21] and Sanders et al. [22] have observed this same
behaviour and concluded that this deviation is not due to
the fact that the k-ε model gives unsatisfactory results,
rather it is the anisotropy that is badly predicted.

M. H. Gazzah et al. / Natural Science 2 (2010) 635-640
639
639
Figure 3 also features a tendency toward a constant
velocity fluctuation intensity value of 0.286 at x/D > 20
with the second order model, and a value of 0.305 with
the first order model, which are both close to the value
0.261 obtained by Djeridane [16]. Additionally, it is no-
ticed at x/D  20 that, without a co-flow, the velocity
turbulence intensity increases faster with x/D than for
the co-flow case. It is interesting to note that the predi-
cted velocity fluctuation intensity shows an approximate
asymptotic behaviour which increases in value with the
increasing co-flow velocity.
Figure 4 shows the radial profiles of the mean veloc-
ity for the downstream section x/D = 20, situated in the
affinity region of the jet. It is noticed that both models
agree reasonably well with the experimental data of
Djeridane [16] for the co-flow case. Figure 5 presents
the radial velocity fluctuation intensities profiles at x/D
= 20. Qualitative agreement is obtained in the sense that
both models predict a local maximum which is also ob-
served experimentally. It should be mentioned again that
the observed difference, between the experimental and
the numerical values on the jet axis, is due to the chosen
initial conditions. These values are of the order of 15%.
The axial mean velocity should decrease faster, and thus
0.00
0.10
0.20
0.30
0.40

2
c
cco
u
UU
600 10 20 30 40 50
x
/
D
with co-flow
no co-flow
Djeridane [16] with co-flow
Djeridane [16] no co-flow
k-

RSM
Figure 3. Centreline values of the velocity fluctuation intensi-
ties.
0 2 46810
2r/D
-0.10
0.00
0.10
0.20
0.30
0.40


co
jco
UU
UU


no co-flow
with co-flow
RSM
k- 
Djeridane [16] with co-flow
Figure 4. Radial profile of the normalized velocity fluctuation
intensities at x/D = 20.
0.0 0.51.0 1.5 2.0 2.5 3.03.5
r/L
u
0.00
0.10
0.20
0.30

2
cco
u
UU
RSM
k- 
Djeridane [16] with co-flow
no co-flow
with co-flow
Figure 5. Radial profile of the mean velocity at x/D=20.
0
5
10
15
20
()
j
j
QQ
Q

60010 2030 40 50
x
/D
RSM
k-

Djeridane [16] with co-flow
no co-flow
with co-flow
Figure 6. Evolution of the axial entrainment of air.
more efficient turbulent mixing is required. Furthermore,
it should be noticed that the jet radial expansion is re-
duced when the co-flow is present.
The amount of air entrainment by the jet is determined
by the time-average radial profiles of velocity. It relates
the mass flow rate of the surrounding fluid entrained into
the jet to the characteristic velocity difference between
the jet and the co-flow.



 UcoUr
co drrUUQ 0
2  (15)
Based on this last definition, Figure 6 shows the axial
evolution of the air entrainment. It is obvious that the co-
flow decreases considerably the air entrainment. A qua-
litative analysis would suggest that a co-flowing stream
would restrict the radial in flow of air into the jet. More-
over, the free jets entrain from 30 to 75% more air than
the co-flowing jets at any given axial location.
5. CONCLUSIONS
A turbulent jet with and without a co-flowing air has be-
en theoretically and numerically investigated, using the
first and the second order turbulence closure models.
The calculation results show that both models qualita-
tively predict the behavior of jets with or without co-flo-
wing air. An investigation of the asymptotic values for

M. H. Gazzah et al. / Natural Science 2 (2010) 635-640
640
the mean velocity decay constant Ku, the spreading rate
2/1
u
S and the centerline value of the velocity fluctuation
intensities has been presented. The predictions agree
reasonably well with the very recent experimental study
in the literature for axisymmetric jets. The major visible
effect of the co-flow is the jet decay rate reduction, in
comparison with the free jet case. However, based on
entrainment definition, it is mainly shown that the co-
flow reduce the air entrainment.
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