Journal of Modern Physics, 2011, 2, 392-397
doi:10.4236/jmp.2011.25048 Published Online May 2011 (
Copyright © 2011 SciRes. JMP
Modeling and Numerical Simulation of Wings Effect on
Turbulent Flow between Two Contra-Rotating Cylinders
Maher Raddaoui
Unit Materials, Energy and Renewable Energies: MÉÉR”, Faculty of Science, University of Gafsa,
Zarroug, Tunisia
Received September 20, 2010; revised March 6, 2011; accepted March 28, 2011
Many industries in the world take part in the pollution of the environment. This pollution often comes from
the reactions of combustion. To optimize these reactions and to minimize pollution, turbulence is a funda-
mental tool. Several factors are at the origin of turbulence in the complex flows, among these factors, we can
quote the effect of wings in the rotating flows. The interest of this work is to model and to simulate numeri-
cally the effect of wings on the level of turbulence in the flow between two contra-rotating cylinders. We
have fixed on these two cylinders eight wings uniformly distributed and we have varied the height of the
wings to have six values from 2 mm to 20 mm by maintaining the same Reynolds number of rotation. The
numerical tool is based on a statistical model in a point using the closing of the second order of the transport
equations of the Reynolds stresses (Reynolds Stress Model: RSM). We have modelled wings effect on the
flow by a source term added to the equation tangential speed. The results of the numerical simulation showed
that all the average and fluctuating variables are affected the value of the kinetic energy of turbulence as
those of Reynolds stresses increase with the height of the wings.
Keywords: Pollution, Turbulence, Combustion, Wing, Modeling, Numerical Simulation, Contra-Rotating
Cylinders, Reynolds Stress Model, Source Term
1. Introduction
1.1. Former work
It is just in the years 1970’s that the first models for tur-
bulence in rotation saw the day, Morse [1,2] used a
model of the k- type to small Reynolds number, but the
author noticed the existence of an abnormally important
laminar zone. He proposes then a modified version
which takes into account the anisotropy of turbulence
close to the walls. The study of Iacovides and Toum-
panakis [3], in which four models of turbulence are
tested, shows that the model of transport of the tensions
of Reynolds is the suitable level of closing to envisage
such complex flows. Schiestel, et al. [4] used an alge-
braic model (ASM) in the central heart of the flow. Later,
Elena and Schiestel [5] used a model of transport of the
tensions of Reynolds that was derived from the model
from Launder and Tselepidakis [6]. This model is more
satisfactory than that of Hanjalic and Launder [7] and
than model ASM. But there too, the authors note a too
large laminarisation of the flow compared with the ex-
perimental results of Itoh, et al. [8]. Hence, many im-
provements are appeared like those of Elena and Schi-
estel [9] which proposed an alternative of the model of
transport equations of the tensions of Reynolds (RSM),
who takes into account the implicit effects of rotation on
the turbulent field and whom they compare with more
traditional models. Iacovides and Theofanopoulos [10]
used an approach based on an algebraic model of the
tensions of Reynolds in the zone of fully developed tur-
bulence and an assumption length of mixture close to
walls. A complete review of the various studies con-
cerning the models of turbulence for the flows in rotation
was made by Schiestel [11] and Elena [12]. Taking into
account the complexity of the flows which we study (ro-
tation, containment, flow, laminarized zone, anisotropy…),
it is about the adequate level of closing. Elena [12] suc-
cessively 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,
who accounts better for the effects of the rotatio n and the
physics of these flows. This model will be thus selected
for all our numerical study.
The experimental study of the flows confined between
revolving discs interested these last years several re-
searchers, in Marseilles (group of P. Le Gal in the IR-
PHE), in Lille (group of D. Buisine to the LML), in Paris
area (group of P. Queré to the LIMSI, and of O. Stew at
the University of Evry) and laboratory FAST in Orsay
which studied in experiments the instability of this kind
of flow, it proved the existence of two typ es of instab ility
which exist only for ratios of very weak aspect and
which start to appear if the number of rotation increases:
an instability in time making the flow non stationary and
an instability in space destroying the axisymètrie. In
2008, Poncet, Schiestel and Monchaux [13] studied the
flow of Von Karman corresponding to the two contra-
rotating discs for smooth discs and discs with wings,
these authors primarily were interested in the average
sizes without taking several values height of the wings.
Our present numerical study supplements this old work
by taking many heights of wings and especially it puts
the point on the turbu lent sizes lik e th e turbulence k inetic
energy and the Reynolds stresses.
2. Experimental Installation
The geometrical characteristics of rotating cylinders are
displayed in Figure 1, it consists in two fixed discs en-
closed by two moving cylinders.
These cylinders move in opposite side. 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, which corresponds to an aspect ratio G = 2.0. The
axis of z corresponds to the axis of rotation. We define
the Reynolds number of the rotation R based on the
external ray of the cavity by 2
 . All the simu-
lations considered here correspond to a Reynolds number
of rotation R = 2 105. On each cylinder, we placed 8
wings uniformly distributed with a thickness equal to 2
mm and a height a whi ch c o rresp onds to a parameter h* =
a/h, the length of each wing is equal to the height of the
walls Wings
Figure 1. Experimental installation.
cylinder. Our study is made with 6 different heights of
wing correspondent to h* = 0.01, 0.02, 0.04, 0.06, 0.08
and 0.1. The flu id cons idered in annular space is income-
pressible and the flow is steady and isotherm.
3. Numerical Scheme
3.1. Statistical Modeling
The instantaneous velocity (and respectively the pres-
sure) is decomposed into a mean part Vi (P) and a fluctu-
ating part vi (p), for turbulent mean flow prediction, the
governing equations for the conservation of mass and
momentum are deduced by statistical averaging from
Navier-Stokes equations using the Reynolds decomposi-
ijijij ij
VVV vv
The equations fluctuating speeds are then written with
difference of this last equation and Navier-Stokes equa-
,,, ,
ijijjijjijiji j
VvvV vvvvv
 (2)
with iji j
Rvv denotes the Reynolds stress t ensor.
Our approach is based on one-point statistical model-
ling using a low Reynolds number second-order full
stress transport closure derived from the Launder and
Tselepidakis model [6] and sensitized to rotation effects
[5,14]. Previous works [15] have shown that this level of
closure was adequate in such flow configurations, while
the usual k- model, which is blind to any rotation effect,
presents serious deficiencies. This approach allows for a
detailed description of near-wall turbulence and is free
from any eddy viscosity hypothesis. An example of these
models is the RSM model based on the resolution of the
following transport eq uation of Reynolds stress:
ij ijijij ijij ij
  (3)
where Pij, ij, ij, ij, Dij and Tij respectively denote the
production, the source term of Coriolis, pressure-strain
correlation, dissipation, diffusion and extra terms of rota-
The equation of dissipation rate is proposed by Laun-
der and Tselepidakis [6]:
1,2 ,,
diji jijjii
jkijli klii
cRVcf cR
tk k
 
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is the isotropic part of the dissipation rate
. C1 = 1, C2 = 1.92, C = 0.15, C3 = 2, C4
= 0.92 are four empirical constants and f i s defi ned by:
1 10.63.
A1 and A2 are respectively the first and the second in-
variant of the anisotrop y tensor.
The equation of kinetic turbu lent Energy is
iji jijjii
 
3.2. Numerical Method
The computational procedure is based on a finite volume
method using staggered grids for mean velocity compo-
nents (Vr, Vz) with axisymmetry hypothesis and a grid not
shifted for mean velocity co mponent (V), for pressure P,
for the kinetic energy of turbulence k, for the dissipation
rate and for components of the tensor of Reynolds. The
transport equations of the Reynolds stresses are solved in
block to preserve the forts couplings. The coupling
speed-pressure is solved using algorithm SIMPLER. The
computer code is steady elliptic and the numerical solu-
tion proceeds iteratively. A 80 120 mesh in the (r, z)
frame proved to be sufficient in most of the cases consid-
ered in the present work to get gr id-independent solution s
3.3. Modeling of the Wing Effect
We can not implement real straight wings in our two-
dimensional code. So we limit to model their most im-
portant effect, which is to increase the efficiency of the
disks in forcing the flow. Thus, we add a volume wing
force f in the equation of the tangential velocity compo-
nent V. If we consider n straight wing s, the volume drag
force f can be written as:
rel rel
where is the fluid density, CD the dimensionless wing
coefficient and Vrel = irV the relative tangential ve-
locity on cylinder i = 1, 2. The force is designed to make
the fluid velocity closer to the local cylinder velocity
nearcylinders. This form is close to the one proposed by
Boronski [1 6] for spectral co de.
4. Results and Discussions
4.1. Effect of Wings on the Mean Velocity
For a radial evolution, Figure 2 shows that the wings do
not have a great effect on the tangential mean velocity in
the center of annular space where the effect of torsion is
most significant. Indeed, far from this zone, torsion is
weak what highlights in a clearer way the influence of the
wings. This kind of profile is almost absent in the literature
because the totality of the research works always suppose
the cylinder wall fixed and the discs m oving or even of t he
annular flows with sm ooth wal ls .
Contrary to the radial tangential mean velocity evolu-
tion where the effect of the wings is dominated by the
effect of torsion, Figure 3 shows well, through the axial
tangential mean velocity evolu tion, a remarkable effect of
wings especially far from the zone of torsion where the
value of this velocity increases when the height of the
wings decreases, this is explained by the source term
added to the tangential mean velocity transport equation
which is proportional to the height of the wings. For ex-
ample, the tangential velocity decreases from 0.08 for a
height of wing equal to 4 mm (a/h = 0.02) to 0.01 for a
height of wing s equal to 2 0 mm.
4.2. Effect of Wings at Turbulent Kinetic Energy
Figure 4 shows for example, that the turbulence increases
very quickly in the zone of action of the wings especially
in the level of the interior cylinder because the number of
wings is distributed on a perimeter weaker than that of the
external cylinder. To find a similar effect it would be
Figure 2. Tangential velocity profiles at z/h = 0.5.
Figure 3. Tangential velocity profiles at r/R2 = 0.75.
Copyright © 2011 SciRes. JMP
interesting to do uble the number of wings on the external
cylinder. We also notice that, c ontrary to the mean velocity
which decreased with the height of the wings, the kinetic
energy of turbulence, coming from the agitation cause d by
the wings, increases with their height, for example the
kinetic energy of turbulence is thirty times larger for a
height of wings passing from 0.01 to 0.1 at a dist ance of 15
mm of the interior wall. In the center of annular space
(Figure 5), the axial evolution of the turbulence kinetic
energy shows that, in additio n to the turbulence g enerated
by torsion, another turbulence, caused by the wings, is
added which increases with their height. This new turbu-
lence is very clear near the walls where the flow is laminar.
We can then conclude that the wings, by their height,
increases the level of turbulence, thus this parameter can
then be a very interesting factor for the industrialists in
order to exploit t he level of turbulence for at the sam e time
optimizing the energy level in certain installations and also
seeking to lower pollution in others.
4.3. Effect of Wings at Reynolds Stresses
(Figure 6 - Figure 13) show that al l the tens ions of Rey-
nolds increase with the height of the wings, particu- larly
the normal stresses (Figure 6 - Figure 11) which are more
sensitive to the effect of the wings than the cross ones
(Figure 12, Figure 13) because the velocity fluctuation
caused by the wings is in the same direction. The most
affected Reynolds stresses is corresponding to the tangent-
ial velocity fluctuation; this is well coherent with the
modeling of th e wings effect used in this work. I ndeed, the
Figure 4. Turbulent kin etic en ergy profiles at z/h = 0.5.
Figure 5. Turbulent kin etic en ergy profiles at r/R2 = 0.5.
effect of the wings is model ed by a source term only added
to the tangential mean velocity transport equation. This
same phenomenon is also noted for the cross Reynolds
stresses uw
and vw
which contain the tangential
velocity fluctuation. We note also that the total pace of the
Figure 6. Reynolds stress profiles 22
ur at z/h = 0.50.
Figure 7. Reynolds stress profiles 22
ur at r/R2 = 0.75.
Figure 8. Reynolds stress profiles 22
at z/h = 0.5.
Figure 9. Reynolds stress profiles 22
at r/R2 = 0.75.
Copyright © 2011 SciRes. JMP
Figure 10. Reynolds stress profiles 222
wr at z/h = 0.5.
Figure 11. Reynolds stress profiles 222
wrat r/R2 = 0.75.
uw r
Figure 12. Reynolds stress profiles 222
uwrat r/R2 = 0. 75.
vw r
Figure 13. Reynolds stress profiles 222
vwr at r/R2 = 0.75.
normal stresses is of the same type as that o f the turb ulent
kinetic energy; this confirms well the contribution of the
normal Reynolds st resses in the expression of the turbulent
kinetic energy. For a radial evolution, Figures 6, 8 and 10
show that the contribution of the ings effect is especially
on the level of the tangential fluctuation 222
. On
the other hand, for the axial evol ution, Figures 7, 9 and 11
show that the cont ri but i on of the wings effect is especially
on the level of the axial fluctuation 222
5. Conclusion
In this work we have studied the influence of wings on
the structure of the flow, the employed method is to
model and to simulate the effect of the wings on the mean
variables and especially on the turbulent variables, the
flow considered is generated by the rotation of two op-
posite cylinders, the cavity considered is closed and the
fluid considered is incompressible taken at the ambient
temperature. For a given Reynolds number R = 2 × 105,
and a given aspect ratio G = 2.0, we have considered six
heights of wing a/h = 0.01, 0.02, 0.04, 0.06, 0.08 and 0.1.
The model of Reynolds stresses RSM with his level of
closing showed its capacity to predict this kind of com-
plex flows. The idea to m odel the effect of the wings by a
force of trail placed as an additional source term in the
transport equa tion tangential speed is very satisfactory.
The numerical simulation allowed us to quantify the
effect of the w ings at the sam e tim e on the mean variables
and especially on the turbulent variables. All these vari-
ables increased with the height of the wings, but the ki-
netic energy of turbulence and the tensions of Reynolds
are most affected. This present study provided us a true
database for the rotating confined flows between two
rotating cylinders with wings. Indeed, the dimensioning
of the machines in the industrial installations using the
rotating flows could thus be treated efficiently in the ob-
jective to control pollution and to make these installa-
tions most powerful possible.
This study can be extend ed to the case of the co-rotatif
cylinders and to apply it to other flows as in the non sta-
tionary case. It would be also important to seek other
geometrical parameters which can have a direct affect on
the level of turbulence as for example the grids.
Finally, this study can also be to apply to flows with
transfers of heats and the compressible fluids.
6. References
[1] A. P. Morse, “Numerical Prediction of Turbulent Flow in
Rotating Cavities,” ASME Journal of Turbomachinery,
Vol. 110, No. 2, 1988, pp. 202-212. doi:10.1115/1.3262181
[2] A. P. Morse, “Application of a Low Reynolds Number
k- Model to High Speed Rotating Cavity Flows,” ASME
Journal of Turbomachinery, Vol. 113, No. 1, 1992, pp.
98-105. doi:10.1115/1.2927743
[3] H. Iacovides and P. Toumpanakis, “Turbulence Model-
ling of Flow in Axisymmetric Rotor-Stator Systems,” 5th
Copyright © 2011 SciRes. JMP
Copyright © 2011 SciRes. JMP
International Symposium on Refined Flow Modelling and
Turbulence Measurements, Presses de l’Ecole Nationale
des Ponts et Chaussées, Paris, September 1993, pp. 7-10.
[4] R. Schiestel, L. Elena and T. Rezoug, “Numerical Mod-
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1, 1995, pp. 45-65. doi:10.1080/10407789308902602
[5] L. Elena and R. Schiestel, “Turbulence Modeling of Con-
fined Flow in Rotating Disk Systems,” AIAA Journal,
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[6] B. E. Launder and D. P. Tselepidakis, “Application of a
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[11] R. Schiestel, “Les Ecoulements Turbulents,” 2nd Edition,
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R1 : Radius of interior cylinder (m)
R2 : Radius of external cylinder (m)
h : Height of cylinder (m)
a : Height of wings (m)
h’ : Length of wings (m)
b : Thickness of wings (m)
G : Aspect ratio of the cavity
R : Rotational Reynolds number
U : Mean axial velocity
V : Mean radial velocity
W : Mean tangential velocity
u’ : Axial velocity fluctuation
v’ : Radial velocity fluctuation
w’ : Tangential velocity fluctuation
P : Pressure
p’ : Pressure fluctuation
h* : Height of wings
Greek letters
: Density of fluid (kg·m-3)
: Kinematics viscosity of fluid (m2·s–1)
: Rotating velocity of disks (rad·s–1)