Journal of Modern Physics, 2013, 4, 1000-1012
http://dx.doi.org/10.4236/jmp.2013.47135 Published Online July 2013 (http://www.scirp.org/journal/jmp)
Influence of Rotating Speed Ratio on the Annular
Turbulent Flow between Two Rotating Cylinders
M. Raddaoui1,2
1Faculty of Science of Gafsa, University of Gafsa, Zarroug City, Tunisia
2Unit of Materials Energy and Renewalble Energy, Zarroug City, Tunisia
Email: maherzohra@yahoo.fr
Received March 7, 2013; revised April 11, 2013; accepted May 9, 2013
Copyright © 2013 M. Raddaoui. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Rotating flows represent a very interesting area for researchers and industry for their extensive use in industrial and
domestic machinery and especially for their great energy potential, annular flows are an example that draws the atten-
tion of researchers in recent years. The best design and optimization of these devices require knowledge of thermal,
mechanical and hydrodynamic characteristics of flows circulating in these devices. An example of hydrodynamic pa-
rameters is the speed of rotation of the moving walls. This work is to study numerically the influence of the rotating
speed ratio of the two moving cylinders on the mean and especially on the turbulent quantities of the turbulent flow in
the annular space. The numerical simulation is based on one-point statistical modeling using a low Reynolds number
second-order full stress transport closure (RSM model), simulation code is not a black box but a completely transparent
code where we can intervene at any step of the calculation. We have varied from 1.0 to 1.0 while maintaining al-
ways the external cylinder with same speed . The results show that the turbulence structure, profiles of mean veloci-
ties and the nature of the boundary layers of the mobile walls depend enormously on the ratio of speeds. The level of
turbulence measured by the kinetic energy of turbulence and the Reynolds stresses shows well that the ratio is an in-
teresting parameter to exploit turbulence in this kind of annular flows.
Keywords: Rotating Flows; Annular Flows; Speed Ratio; Numerical Simulation; RSM Model; Boundary Layers;
Turbulence
1. Introduction
Former Work
Rotating flows are met in several industrial applications
like turbo-machines, thermal motors and especially in
turbines [1]. The annular rotating flows are met in re-
volving jets [2] and devices of combustion in order to
increase the mixture between the reagents and to stabilize
the flame or to obtain advantages of better mixing [3]. To
begin this study, it would be interesting to recall previous
numerical and experimental work treating rotating flows
specifically, annular flow object of this study. On the
numerical level, it is about the years 1970 that the first
models for turbulence in the rotating flows were born,
Morse [4,5] employed a model of the k-ε type in small
Reynolds number, but the author noted the existence of
an abnormally important laminar zone. He then proposes
a modified version which takes account of the anisotropy
of turbulence close to the walls. Choukairy et al. [6] have
studied numerically and analytically the transient laminar
free convection in a vertical cylindrical annulus filled
with air Pr = 0.71. For modelling of this kind of flows,
we can mention the work of Gharbi et al. [7] who deter-
mined the average heat transfer coefficients for forced
convection air flow over a rectangular flat plate and the
work of Dhakal and Walters [8] who proposed a
three-equation variant of the SST k-ω model and also the
work of Hayat et al. [9] who propose an analytic solu-
tion for the magnetohydrodynamic rotating flow. For
annular flow, the combined forced and free convection
flows in a horizontal annulus are studied numerically by
Kotake and Hattori [10]. For smooth fixed walls, we can
cite the work of Neto et al. [11] who presented a nu-
merical modelling of hydrodynamics and mass transfer in
developing laminar axial flow and the work of Cadiou et
al. [12] who studied the stability of natural convective
flows in narrow horizontal annuli. Farinas et al. [13]
have also worked on the same flows but the walls had
wings. Some other authors have worked with moving
walls like Lin [14] and Poncet et al. [15] who consid-
C
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M. RADDAOUI 1001
ered turbulent flows in a differentially heated Taylor-
Couette system with an axial Poiseuille flow, numerical
approaches were based on the Reynolds Stress Modeling
(RSM). This RSM model applied to rotating flows has
also been used by Raddaoui [16] for the same geometry
as the study here but the author has dealt with the mod-
eling and simulation of the influence of the height wings
bonded to the cylindrical walls on the characteristics of
the annular flow. However, the work proposed here has
the importance of treating a study that was not done be-
fore; it is the influence of the rotating speed ratio of the
rotating cylinders on the distribution of the turbulence in
the annular space and especially the relation between this
parameter and the level of turbulence in this kind of
flows. The same study has done by Poncet et al. [17] but
this study has done for rotating discs and not for ro-
tating cylinders as we propose in our present study. The
study of Iacovides and Toumpanakis [18], in which four
models of turbulence are examined, proves that the
model of transport of the tensions of Reynolds is the
suitable level of closing to study such complex flows. All
these works have shown that this level of closure is the
most appropriate to describe rotating flows with or with-
out through flow (centripetal or centrifugal), while the
classical k-ε model, which is blind to any rotation effect
presents serious differences. On the experimental level,
we can quote the work of Ivanic et al. [19] as well as
work of Loiseleux et al. [20] which studied the configu-
ration of the central jet turning. The dynamics of the
flows made up of an annular jet turning was studied by
Adjovi and Foucault [21], they carried out an inventory
of the swirling structures by laser tomography. Then,
they concerned the profiles speeds by LDV and PIV
more close possible of the tail pipe of the jets. Rotating
flows are exploited experimentally also in pump-turbine
by Hasmatuchi et al. [22]. For annular flow, we cite the
work of Seban and Hunsbedt [23] who used fixed walls
and Ball et al. [24] and Pfitzer et al. [25] who have ex-
perimented with moving walls. The annular flow has
been treated also by researchers who compared the cal-
culations with experimental results like the work of Hei-
kal et al. [26]. Ould-Rouis et al. [27] have determined the
hydrodynamic characteristics and the mass transfer in the
entrance region of an annulus with simultaneous de-
velopment of velocity and concentration fields when the
walls were fixed. Other authors were interested at similar
flow but the walls are moving like Bouafia et al. [28]. All
these previous works, numerical or experimental, dealing
with rotating flows have occupied several important as-
pects, but we have not seen studies on the influence of the
moving walls speed ratio on the nature of the annular flow.
For this reason, this work can be considered to be an
innovation and also a confirmation of the RSM model
to study this kind of complex flows. In particular, we
highlighted the study of the relation between turbulent
quantities and the rotating speed ratio of the rotating cyl-
inders.
This paper is divided as follows: Section 2 is the de-
scription of the geometry of our out-flow, whereas Sec-
tion 3 is devoted to the differential Reynolds Stress Mo-
del RSM. In Section 4, Numerical method is presented,
in Section 5, we have presented and interpreted the nu-
merical simulation of the effect of rotating speed ratio on
the flow structure, the mean and turbulent quantities be-
fore concluding in Section 6.
2. Geometry of the Flow
The system of the Figure 1 is the same as that used by
Raddaoui [16], it represents a device with an annular
space between two cylinders in uniform rotation move-
ment; the fluid considered in annular space is income-
pressible. The value of interior ray is R1 = 100 mm and
the external ray is R2 = 200 mm, the two cylinders are of
the same length h = 200 mm and can turn in the same
direction or in the opposite direction. The axis of z cor-
responds to the axis of rotation. We note the rotating
speed ratio of the interior cylinder by the external one
and we vary from 1.0 to 1.0 by taking the following
values: 1.0, 0.8, 0.6, 0.4, 0.2, 0.0, 0.2, 0.4, 0.6, 0.8
and 1.0, we maintained always the cylinder external with
same speed corresponding to a Reynolds number of
the rotation 2
2
R
based on the external ray of the
cavity constant equal to , ν is the kinematic vis-
cosity of water. The level of Reynolds number corre-
sponds to a steady flow, the work of Poncet et al. (2008)
[17] has showed that the phenomena of unsteady prob-
lems and instability start have to appear only for one very
high rotation corresponding to a Reynolds number of
rotation higher than 106.
5
210
3. Statistical Modelling
To simulate numerically the rotating turbulent flow in the
annular space, we have used the RSM model (Reynolds
Stress Model), this model is an improved version of sev-
eral other former versions like that of Elena and Schiestel
[29] who used a model of transport of the tensions of
Reynolds derived from the model from Launder and
Tselepidakis [30]. This model is more satisfactory than
that of Hanjalic and Launder [31] and the ASM model
Fixed walls
R
2
Fluid
Mobile walls
R
1
h
Figure 1. Schematic representation of the geometry out-flow
and notations.
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developed by Schiestel et al. [32]. But the authors note a
too large laminarisation of the flow compared with the
experimental results of Itoh et al. [33]. Many improve-
ments appeared like those of Elena and Schiestel [34]
who proposed an alternative to the model of transport
equations of the tensions of Reynolds (RSM), which
takes into account the implicit effects of rotation on the
turbulent field and which they compare with more tradi-
tional models. Iacovides and Theofanopoulos [35] used
an approach based on an algebraic model of the tensions
of Reynolds in the zone of fully developed turbulence
and an assumption length of mixture close to walls. A
complete review of the various studies concerning the
models of turbulence for the flows in rotation was made
by Schiestel [36] and Elena [37]. Taking into account the
complexity of the rotating flows, Elena [37] successively
used three models of the second order of the type RSM
(Reynolds Stress Model). The last version is a model of
the second order to small Reynolds number, which ac-
counts better for the effects of the rotation and the phys-
ics of these flows. This model will thus be selected for all
our numerical study. It is based on the resolution of the
following transport equation of Reynolds stresses:
,1,2 ,,
,
3,,4,
,
j jijijij ji
i
jkijli kli
i
k
VcRVcfcR
kk
cRVV ck
kk







 






,kij kijij
VR Pijij ij
D T

,PD
(1)
where ijij ijij
,,
T
,P
and ij denote respectively, the
production, pressure-strain correlation, dissipation, diffu-
sion and extra terms. For the terms ij
,
ij ij
and ij ,
the reader can refer to previous papers by Elena and
Schiestel [34] or Poncet et al. [38] or by Elena [37] in his
PhD thesis. For the term Tij, many researchers like Cam-
bon and Jacquin [39] and Cambon et al. [40-42] and
Bertoglio et al. [43] and Reynolds [44] contributed to
the modeling of the implicit effects of rotation on turbu-
lence.
D

RR
ijij ij
BJ

In the present approach, the extra term ij
T, which
takes account of the implicit effects of the rotation on the
turbulence field, contains four parts:

ij ij
TD
 (2)
R
ij
is a term resulting from the correlation pressure-
deformation, its modelling is deduced from a model of
spectral tensor of Schiestel and Elena [45].
The second term

R
ij is an inhomogeneous term of
diffusion, which, in the presence of walls, slows down
the tendency to bidimensionalisation.
D
The Bij term acts only in the event of strong rotation
and its role is to produce spectral jamming (angular dis-
persion). It was taken into account in particular by Cam-
bon et al. [40] for the modeling of the homogeneous
turbulence subjected to a fast rotation.
Jij is a corrective term added to εij [35].
The equation of dissipation rate is proposed by Laun-
der and Tselepidakis [30]:
(3)
is the isotropic part of the dissipation rate
12 12
2kk
 

3
1,1.92, 0.15,2,CCC C
,,
ii
. 12
 
 
0.92C
4
are four empirical constants.
fε is given by Elena and Schiestel [34] as:
f
2
1 10.63AA
 (4)
where A is anisotropy parameter defined by:

23
9
18
AA 
0.5
A2 and A3 are the second and the third stress-anisot-
ropy invariants.
The equation of the kinetic energy of turbulence can-
not be solved in the RSM model but it can control the
convergence while comparing it to
j
j
R:
,, ,,
,
0.22
2
jj
j jijijij ji
i
Tk
VkRVRk k


 


(5)
To numerically treat the equations of transportation of
the RSM model, we must work with dimensionless va-
riables. So we used the height of the cavity (h) and the
angular speed () as parameters for dimensionless vari-
ables.
4. Numerical Method
4.1. Stabilizing Techniques
In order to overcome stability problems, several stabiliz-
ing techniques, such as those proposed by Huang and
Leschziner [46], were introduced in the numerical pro-
cedure. For all flows studied here, the configuration is
always axi-symmetrical, the problem is then bi-dimen-
sional and as we work in cylindrical coordinates, the
transport equations of a variable are then made to the
form:
rz
rV rrV rS
rrzz





 



(6)
is the total diffusion coefficient. It regroups for
example the molecular viscosity ν and the turbulent vis-
cosities resulting from the Reynolds stresses written in
the following type:

;,, ,
i
ij ijij
V
Rijrz
j

(7)
where ij is the anisotropic pseudo-viscosity.
To still stabilize calculation, in addition to the tech-
nique of the anisotropic pseudo-viscosity ij which gives
a diffusive formulate to the RSM model, we have solved
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M. RADDAOUI 1003
the equations of the Reynolds stresses per block. This
technique is based on writing the six transport equations
of Reynolds stresses in the form of a linear system, thus
allowing a resolution per block of a tri-diagonal matrix.
The discretization is done on a non-shifted grid and all
the components of the tensor of Reynolds are evaluated
at the points of pressure. This resolution is carried out by
a direct method of the Gauss type (per block tri-diagonal
algorithm). It stabilizes the algorithm in the presence of
rotation because it maintains the couplings between the
equations.
The last technique of calculation’s stabilization is to
introduce coefficients of under-relieving. The sharp vari-
ations of the variables at the beginning of calculation are
very destabilizing and they thus should be attenuated. We
then introduce regular under-relaxation for any variable
. By noting k the value of to the iteration k and *
the computed value during the iteration k + 1, to attenu-
ate the variations of , we introduce a factor of relieving
, such as 0 < α < 1 and:
k

,,: 0V
1kk
 (8)
The value of α depends on the variable considered and
the degree of convergence reached. A low value of has
a stabilizing effect but slows down convergence. Here
the values of , which were used in the majority of our
calculations:
For .3
rz
VV
For P: 0.6
For .2
ij
,, :0kR
4.2. Boundary Conditions and Grid
Arrangement
All the variables are set to zero at walls except for the
tangential velocity Vθ, which is set to –r on cylinder 1,
+r on cylinder 2 and zero on the stationary discs. The
usual value

,, 2kjkj k
1.0

is imposed at the wall
for the dissipation rate of the turbulence kinetic energy.
As we use confined geometry, we do not have to make
inlet or outlet conditions. As the flow is axisymmetry, the
field of study is 2D (r, z) corresponding to the higher half
of the cavity of Figure 1. The computational procedure
is based on a finite volume method using staggered grids
for mean velocity components with axisymmetry hy-
pothesis in the mean. The computer code is steady ellip-
tic and the numerical solution proceeds iteratively. We
have verified that a 120 × 120 mesh in the (r, z) frame is
sufficient for all cases to get grid-independent solutions,
this identical number of elements for the two directions,
was checked in first for a ratio , then the con-
verged file of this calculation was used as a beginning of
calculation for the other ratios, but we have always
checked that the grid is sufficiently fine close to the walls
to describe correctly the viscous sublayers. Some tests
were made to confirm that for all ratios, we show on
Table 1 some results.
The abrupt variations of the variables close to the
walls require tightening the grid close to them. The field
is cut out into 3 areas according to each direction r and z,
with a coarser grid in the centre and an increasingly
dense area when approaching the walls.
In the zones close to the walls, the grid is generated
using geometrical series allowing a very weak step (the
first mesh is: 4
1.61410rR
 and
4
1.532 10zh
 
and
z
vz
).
To check that the grid is sufficiently fine close to the
walls so that the viscous sub layers are described cor-
rectly, we define, on the level of the two rotating cylin-
ders,
given by:
2
22
1
,
rzz
z
V
VVzv
vz
zzz




 


 


 



(9)
z1 is the size of the first mesh according to the axial
direction
4
1.532 10zh

and
r
vr
1
On the level of fixed walls, we define
given by:
2
22
1
,
rz
r
r
V
VV
vrrr
rv
r


 

 


 

(10)
r1 is the size of the first mesh according to the radial
direction
4
1.614 10rR

z
r
1
Figure 2 presents radial profile of (Figure 2(a))
and axial profile of
.
(Figure 2(b)) for = 1.0, 0.4,
0.4, 1.0 and and for a grid 120 × 120. For
any wall, mobile or fixed, we notice that z+ and r+ does
not exceed 0.7 and is thus well below the value limits y+
= 1. A grid 120 × 120 is thus sufficient to obtain a solu-
tion independent of the grid but it allows, moreover, a
good description of the viscous under-layers.
5
210R
To control the convergence of all calculations, we
Table 1. % of variation compared to solution 120 × 120, R
= 2 × 105.
100 × 80 100 × 140140 × 140
Mean tangential velocity ( = 1.0) 2.6 1.8 1.7
Turbulence kinetic energy ( = 1.0) 1.9 0.8 0.7
Mean tangential velocity ( = 0.4) 2.4 1.7 1.2
Turbulence kinetic energy ( = 0.4) 1.6 0.9 0.4
Mean tangential velocity ( = 0.4) 2.2 1.4 1.1
Turbulence kinetic energy ( = 0.4) 1.3 0.6 0.4
Mean tangential velocity ( = 1.0) 2.5 1.3 1.3
Turbulence kinetic energy ( = 1.0) 1.4 0.7 0.3
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Figure 2. Radial profiles of z+ (a) and axial profiles of r+ (b)
for Γ = 1.0, 0.4, 0.4, 1.0, R = 2.0 × 105.
have studied the evolution of the residues according to
the number of the iterations for and =
1.0. We have noted that the convergence of various cal-
culations required an iteration count from approximately
40,000, all the calculations were made with a personal
machine; the time necessary for convergence for this
kind of calculation is of approximately five days.
5
210R
5. Influence of Rotating Speed Ratio on the
Characteristics of the Flow
5.1. Model Performances
In order to show the performance of the RSM model, it is
interesting to compare the simulation obtained by the
RSM model to experimental results. As we do not have
measurements for annular flows treated in this work, we
compare the RSM model simulation to experimental
LDA measurements done by Raddaoui [16] for rotating
flows between two contra-rotating discs (Figure 3) very
similar to ours shown in Figure 1. This comparative
study between the numerical simulation and the experi-
ment is to compare the numeric results from the RSM
model to the k-ε model and the experimental measures
for turbulent Von Karman flow in a device exploited by
Raddaoui [16] at IRPHE (Institute of Research on the
Phenomena out of Equilibrium) in Marseille in France.
Z
Z
+
-
h
Mobile wall Fixed wall Mobile wall
R
Figure 3. Schematic representation of raddaoui experimen-
tal set-up.
Figure 4 presents axial profiles of the dimensionless
radial velocity r
Vr , dimensionless tangential velocity
Vr
dimensionless turbulence kinetic energy
22
kr for RSM model, k-ε model and experimental
results. Figure 5 presents axial profiles of the dimen-
sionless Reynolds stresses. We have chosen six sections:
0.1, 0.2, 0.4, 0.6, 0.8rRrR rR rR rR
 and
0.9, for the given aspect ratio 2.0GhR
5
3.7 10R
and the
Reynolds number . For mean velocities Vr
and Vθ, we note that the RSM model results are in a no
far from the experimental measures and this is valid for
all the flow sections, unlike the k-ε model which falls
short of describing correctly this kind of flow especially
on the level of the median plane where turbulence is
maximum because of strata torsion. However, we notice
that this difference between the RSM model and the k-ε
model is reduced near to the walls where the effect of
torsion is inexistent. For the turbulence kinetic energy,
Figure 4 shows that the k-ε model over-estimates turbu-
lence compared to the RSM model, especially, on the
level of the torsion zone where for example, the RSM
model notes a value of the dimensionless turbulence ki-
netic energy equal to approximately 0.024 for 0.1rR
and 0.5zh
(Figure 4(c1)) whereas the k-ε model
multiplies this value by ten. This value given by RSM
model is more correct because the Reynolds stresses of
Figure 5 show a very good agreement between the RSM
model and the experimental data for all the sections of
the flow, this proves the capacity of the RSM model to
describe very well this kind of flow. Figure 4 also shows
that for all sections, the value of the dimensionless radial
velocity r
Vr
is very weak

0.02 0.005Vr
r,
confirming the nature of the flow that is essentially gov-
erned by the phenomenon of rotation. On the level of the
layer of BÄodewadt, the kinetic energy of the turbulence
evaluated by the model k-ε is near to that calculated by
model RSM. For the various tensions of Reynolds, the
numerical simulation made by model RSM shows a very
good agreement between calculation and measurements
by LDA and those for all the cross-sections even on the
level of the median plane and the layers of Ekman and
BÄodewadt (Figure 5). If we evaluate the value of the
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Figure 4. Comparison of dimensionless mean velocity and turbulence kinetic energy profiles between Raddaoui experimental
results, RSM model and k-ε model for R = 3.7 × 105, G = 2.0 at six radial locations: r/R = 0.10 (a1, b1, c1), r/R = 0.20 (a2, b2, c2),
r/R = 0.40 (a3, b3, c3), r/R = 0.60 (a4, b4, c4), r/R = 0.80 (a5, b5, c5) and r/R = 0.90 (a6, b6, c6).
M. RADDAOUI
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Figure 5. Comparison of dimensionless Reynolds stress profiles
ur
222
(a),
vr
222
(b), 
uv r
22
(c) between Rad-
daoui experimental results, RSM model and k-ε model for R = 3.7 × 105, G = 2.0 at six radial locations: r/R = 0.10 (a1, b1, c1),
r/R = 0.20 (a2, b2, c2), r/R = 0.40 (a3, b3, c3), r/R = 0.60 (a4, b4, c4), r/R = 0.80 (a5, b5, c5) and r/R = 0.90 (a6, b6, c6).
Copyright © 2013 SciRes. JMP
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normal tensions of Reynolds 222
ur
and 222
vr
given by the model
k-
, we notice that
2 222 2222
23ur vrkr

 

0
1
0
5
210
5.3. Influence of the Rotating Speed Ratio on the
Tangential Velocity
For the mean quantities we chose to represent the radial
variation tangential velocity on the median plane (Figure
7(a)) and the axial variation of this velocity in the centre
of annular space (Figure 7(b)) for all values of the
speed ratio specified previously because these regions are
most significant for better observing the various phe-
nomena especially those related to turbulence. The Rey-
nolds number is always fixed equal to and the
aspect ratio 1
the model k- over-estimates these tensions compared to
model RSM. RSM model provides good results even in
the boundary layers and models in a precise way the ten-
sions of Reynolds. The behavior of the cross tensor is
less well predicted in the layer of BÄodewadt.
The second order model can now be used confidently
to carry parametric studies even for different values of
rotating speed ratio, as is the case in this study. We can
then use this model to study the influence of rotating
speed ratio on the structure, the level and the distribution
of the turbulence of the flow in the annular space.
GhR
fixed at 2.0. For radial profile of
tangential velocity (Figure 7(a)), we notice that there is
not much effect if > 0, on the other hand for < 0 we
note a little difference close to the interior cylinder. We
can nevertheless note the following remarks: the boundary
layer on the interior cylinder is crushed than that on the
external cylinder and this is for all the ratios , we also
notice that the profiles tangential velocity for
5.2. Influence of the Rotating Speed Ratio on the
Structure of the Flow 1.0

and 1.0 don’t present a symmetry plane at the level of
2
All the cases studied correspond to a Reynolds number
equal to 2.105 and the aspect ratio G = h/R1 fixed at 2.0.
Figure 6 shows that the total structure of the flow de-
pends enormously on the ratio number of revolutions ,
we notice that the swirls are more present if the two cyl-
inders turn in the opposite direction with a parameter
close to 1.0 (Figures 6(a) ( = 1.0), 6(b) ( = 0.8),
6(c) ( = 0.6), 6(d) ( = 0.4)), the swirls are definitely
less present in the contrary case (Figures 6(a’) ( = 1.0),
6(b’) ( = 0.8), 6(c) ( = 0.6), 6(d’) ( = 0.4)) and to be
completely absent for the very weak ratios (Figures
6(e) ( = 0.2), 6(f) ( = 0.0), 6(e’) ( = 0.2), 6(f’) ( =
0.0)). For the contra rotating cases , we notice
the concentration of the maximum of swirls close to the
interior cylinder, which is explained by the presence of a
turbulence related to the phenomena of torsion but this
turbulence is not maximum in the middle of annular
space (for ), as we can think it, but rather near to
the interior cylinder. More the number of revolutions of
this cylinder decreases more these swirls disperse until
disappearing completely for close to zero. We also
notice that acts on the distribution of turbulence
throughout the axis of the cylinders. Indeed, the swirls
are more numerous on the level of the median plane for
close to 1 and concentrates on the level of the fixed
disks for a ratio close to 0.4. For the co rotating cases
, turbulence is obviously much weaker, we notice
that the swirls are distributed in a more uniform way on
all the flow contrary to the contra rotating case. We also
note that turbulence in this case is generated by gradient
speed between the two cylinders than by the level of this
speed. Finally we can also say that for the same ratio in
absolute value, turbulence is more present at the level of
the median plane in the co rotating case than in the contra
rotating one.

0.75
rR
as we can think it, because for these two
cases, two cylinders turn with the same number of revo-
lutions but the product r is not the same because the
distance separating the cylinders from the axis of rotation
is not the same one. For the axial evolution of tangential
velocity (Figure 7(b)), we notice that the total form of
these profiles depends enormously on the ratio . All
these profiles are symmetrical compared to the median
plane but the velocity is more uniform on all the space
separating the two fixed disks for the ratio
close to
0.4. For the values of
close to 1.0, the profile of tan-
gential velocity presents a maximum quit marked at the
level of median plane. We notice also that the level of
tangential velocity depends on the ratio, this level in-
creases with this parameter: more this parameter increases
more tangential velocity profiles is flattened, we can note
also a presence of a central core for which is
completely absent for other values of .
0.8
Concerning the tangential mean velocity we can say
then that the ratio number of revolutions of the two cyl-
inders is an important parameter for the study of the an-
nular flows at the same time for the total structure of
these flows and especially the nature of the boundary
layers and the position of the maximum of this velocity
in annular space. This kind of parameters appears very
important then in the dimensioning and the optimization
of certain machines utilizing the revolving flows in an
annular space.
5.4. Influence of the Rotating Speed Ratio on the
Turbulence Kinetic Energy
The importance of this work appears especially by
studying the influence of the speed ratio of rotation on the
turbulent quantities, one as of these quantities is the tur-
M. RADDAOUI
1008
Figure 6. Streamlines ψ* = ψ/(h2) patterns for R = 2.0 × 105, G = 2.0: Γ = 1.0 (a); Γ = 0.8 (b); Γ = 0.6 (c); Γ = 0.4 (d); Γ =
0.2 (e); Γ = 0.0 (f); Γ = 1.0 (a’); Γ = 0.8 (b’); Γ = 0.6 (c’); Γ = 0.4 (d’); Γ = 0.2 (e’); Γ = 0.0 (f’).
Copyright © 2013 SciRes. JMP
M. RADDAOUI 1009
of the two cylinders. Figure 8(b) shows that the turbu-
lence kinetic energy is distributed in the entire cavity for a
ratio close to ±0.6. We note, like the profile of tan-
gential velocity, the presence of the central core for =
0.8.
The study of the influence of the ratio on the turbu-
lence kinetic energy assures us the importance of this
parameter in the energy optimization of the revolving
machines, this study provides us with a true data base
allowing the researchers and the industrialists to make use
of it to choose the parameters which are necessary ac-
cording to the level and of the structure of turbulence
requested.
Vθ/r
(a) 5.5. Influence of the Rotating Speed Ratio on the
Reynolds Stresses
For normal stresses, we note that on the median plane
0.5zh, the radial evolution of the normal stresses
shows that in the center annular space

0.75rR

1.0

1.0
2
these tensions decrease with (Figure 9), they are most
important for the contra-rotating case and
weakest for the co-rotating case , however near
to the walls in rotation this phenomenon is not very ob-
vious especially on the level of the interior cylinder. We
notice then that the gradient speed between the two walls
can feel on the level of the fluctuation speeds especially
that the tangential velocity which is most important
compared to the others (Figure 9(a3)). In the center of
annular space
Vθ/r
(b)
0.75rR
2, the axial evolution of the
normal tensions shows that all along the axis these ten-
sions always decrease with (
Figures 9(b1)-(b3)), the
level of these tensions are very different if the cylinders
turn in contrary direction
Figure 7. Dimensionless tangential velocity profiles for R =
2.0 × 105, G = 2.0 at axial location: z/h = 0.50 (a) and at radial
location: r/R2 = 0.75 (b).
0
whereas they are very
close if the cylinders turn in the same direction, it can be
explained by the fact that there is more velocity fluctua-
tions when the cylinders turn in the contrary direction.
For the crossed stresses, the radial and axial evolutions
show too low levels except for the tension
bulence kinetic energy which presents a fundamental role
in the comprehension and the perfection of certain energy
installations. For that, the axial and radial evolution of the
turbulence kinetic energy is discussed; Figure 8 shows
that the level of this energy depends enormously on the
ratio number of revolutions. The radial evolution of the
turbulence kinetic energy for position
2
uw
which
increase with , we also note that, as waited, 2
uw
5
210R
is of
the same sign than (Figures 9(a4) and (b4)).
0.5zh

1
(Figure
8(a)) shows that this energy is obviously maximum for the
case where there is the maximum of torsion , in
more we notice that for this case, turbulence extends on all
annular space separating the two revolving cylinders. This
property can be exploited by the industrialists who often
seek a strong turbulence but also occupying all space
offered. For the co rotating case , turbulence is,
of course, weaker, it seems concentrated near to the walls
in rotation. For the other intermediate cases, the turbu-
lence kinetic energy is the weakest; it remains about of the
same order of level while going from the cylinder external
to that interior. The axial evolution of the turbulence ki-
netic energy informs us more about the structure of tur-
bulence in annular space and more precisely in the middle
1.0
6. Conclusions and Prospects
In this work we numerically simulated the influence of
the ratio of rotating speed on the mean and turbulent
quantities of an annular steady flow. The study was made
by maintaining the cylinder external in rotation uniform
and varying the number of revolutions of the interior
cylinder so that varies from 1.0 to 1.0 for a Reynolds
number given and an aspect ratio fixed to
2.0. The numerical model used is a statistical model in a
point using the closing of the second order of the trans-
port equations of the tensions of Reynolds (Reynolds
Stress Model: RSM).
Copyright © 2013 SciRes. JMP
M. RADDAOUI
1010
Figure 8. Dimensionless turbulence kinetic energy profiles for R = 2.0 × 105, G = 2.0 at axial location: z/h = 0.50 (a) and at
radial location: r/R2 = 0.75 (b).
Figure 9. Dimensionless Reynolds stress profiles for R = 2.0 × 105 and G = 2.0 at axial location: z/h = 0.50 (a) and at radial
location: r/R2 = 0.75 (b).
Copyright © 2013 SciRes. JMP
M. RADDAOUI 1011
We have found that the streamlines show clearly the
influence of the parameter on the structure of the flow.
For the contra rotating cases
0 , the concentration
of the maximum of swirls is close to the interior cylinder,
more the number of revolutions of this cylinder decreases
more these swirls disperse until disappearing completely
for close to zero. For the co rotating cases
0
,
turbulence is obviously much weaker, we note that the
swirls are distributed in a more uniform way on all the
flow contrary to the contra rotating case. We have found
that the boundary layer on the interior cylinder is crushed
than that on the external cylinder and this is for all the
ratios .
For mean quantities, we have noted that the level of
tangential velocity increases with the parameter , we
have noted also the presence of a core for and
the boundary layer on the interior cylinder is crushed
than that on the external cylinder and this is for all the
ratios . For the turbulence kinetic energy, we have
noted that the level of this energy increases with the ratio
and reached its maximum for the contra-rotating case,
on the other hand this level is lowest for the case of the
fixed interior cylinder and the co-rotating case.
1.0

1.0

1.0
For turbulent quantities, contrary to the crossed Rey-
nolds stresses, we have found that the level of the normal
Reynolds stresses decreases with in the center annular
space, these stresses are most important for the contra-
rotating case and weakest for the co-rotat-
ing case .
The exploitation of the capacity of the RSM model can
be very beneficial in other complex situations where the
experimental conditions are very difficult to achieve and
where the experimental mechanism is very expensive to
put in place such as the rotating flows of the compressi-
ble fluids or the rotating flows with transfers of heat or
non stationary flows.
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Nomenclatures
u
R: Radius of disk (m)
h: Height of cylinder (m)
: Rotating speed ratio
G: Aspect ratio
R: Rotational Reynolds number
Vz: Mean axial velocity
Vr: Mean radial velocity
Vθ: Mean tangential velocity
: Axial velocity fluctuation
v: Tangential velocity fluctuation
w
: Radial velocity fluctuation
P: Pressure
p: Pressure fluctuation
: Density of fluid (kg·m3)
: Kinematic viscosity of fluid (m2·s1)
: Rotating velocity of disks (rad·s1)
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