Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
306 addition to the separation at a sharp edge a pressure
driven separation region upstream of the obstacle. The boundary layer along the plate and the upstream
separation region will interact with the separation region downstream [6].
Depending on the ratio of the object height to its length in the streamwise direction, a secondary
separation region may form on top of the obstacle itself [7].
The aim of this work is the passive control of the flow around the obstacle, including point of separation, due to
the upstream edge of the obstacle and recirculation zone downstream area by changing locally the geometry
where instead of taking a upstream sharp edge we take a rounded edge having a radius of 0.2 times the obstacle
height as shown in Figs. 1.
Figs. 1. Both models of obstacles a Obstacle to sharp edge
b Obstacle to upstream edge rounding
Some car companies have the objective to develop control solutions able to reduce at least 30 of the
aerodynamic drag of the vehicles without constraints on the design [8].
Thus, it is necessary to modify locally the flow, to remove or delay the separation position or to
reduce the development of the recirculation zone at the back of the obstacle and of the separated swirling
structures [9]. This can be mainly obtained by controlling
the flow with or without additional energy using active or passive devices. In the later case we can use various
adapted devices which change locally the geometry. Several authors have added small obstacles upstream,
on or downstream of the obstacle .
Neumann and H. Wengler [10] use a second obstacle thin fence located
above the obstacle in this way, the turbulent kinetic energy generated downstream of the obstacle
recirculation zone will weakened by withdrawing some of that energy and dissipated to below the fence on top of
the obstacle. In their study on processes of increase in heat transfer generated by one or more obstacle, as is
the case of electronic components, Y.-L. Tsay et al. [11] use a fence downstream of the obstacle and on the upper
wall of the canal. It studied particularly the dimensions effect of the fence as well as the distance between the
obstacle and the fence. In their studies with an interest in reducing pollution from areas downstream of the
buildings, Cheng-Hsin Chang and Robert N. Meroney [12] use a rough surface located upstream of the
obstacle. The authors investigate the dimensions influence of
the rough surface. They come to show that the use of rough surface
significantly reduces the recirculation zone downstream of the obstacle, especially when the width of the street
the channel is smaller. For a street having a larger width, the roughness tends to have a larger frame
roughness.
Mortazavi and Bruneau [13] - [14] - [15] proposed a technique of passive control, using a porous material
located in carefully chosen places of the surface of the obstacle. Thus, they result in local control of flow and a
significant decrease in the recirculation zone.
II. Mathematical Formulation
II.1. Turbulence Modeling
Reynolds-Averaged Navier-Stokes Equations are the governing equations for the problem analyzed
momentum balance, with the continuity equation. For a two-dimensional incompressible flow of a newtonian
fluid, mass and momentum equation become, respectively:
j J
U X
∂ =
∂
1 and:
, ,
i i
j i
j j
i J
J
U U
P U
U U X
X X
X
ρ µ
ρ
⎛ ⎞
∂ ∂
∂ ∂
= − +
− ⎜
⎟ ∂
∂ ∂
∂ ⎝
⎠
2 where
ρ is the fluid density constant, p the pressure, µ dynamic viscosity, U
i
and U
j
are mean velocity components in X
i
and X
j
directions. In equation 2,
i j
U U
ρ is the additional term of
Reynolds stresses due to velocity fluctuations, which has to be modeled for the closure of equation 2.
The classical approach is the use of Boussinesq hypothesis, relating Reynolds stresses and mean flow
strain, through the eddy viscosity concept. Hinze [16] in its general formulation, as proposed by Kolmogorov,
Boussinesq hypothesis is written as:
, ,
2 3
j i
i j
t ij
j i
U U
U U k
X X
ρ µ
ρδ
⎛ ⎞
∂ ∂
− =
+ −
⎜ ⎟
⎜ ⎟
∂ ∂
⎝ ⎠
3 where µ
t
is the eddy viscosity, δ
ij
the Kroenecker Delta and k the kinetic energy of the turbulence, defined as:
1 2
i i
k U U
=
4 Successful turbulence models are those based on the
eddy viscosity concept, which solve two scalar transport
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
307 differential equations. The most well known is the k-
ε model [17].
The so called “standard” k- ε model is a semi-
empirical one, based on the conservation equation of the kinetic energy k and its dissipation rate
ε. The basis of the model is the Boussinesq’s hypothesis,
that the Reynolds stresses
i j
U U
ρ are proportional to
the strain rate of the mean flow, by means of the eddy viscosity concept, thus:
2 t
k C
µ
µ ρ ε
=
5 Balance equations for the kinetic energy k and its
dissipation rate ε for the model are, respectively:
j j
t k
j k
j
k k
U t
X k
G X
X
ρ µ
µ ρε
σ
⎛ ⎞
∂ ∂
+ =
⎜ ⎟
⎜ ⎟
∂ ∂
⎝ ⎠
⎡ ⎤
⎛ ⎞
∂ ∂
= +
+ −
⎢ ⎥
⎜ ⎟
∂ ∂
⎢ ⎥
⎝ ⎠
⎣ ⎦
6
2 1
2 t
j j
j j
k
U t
X X
X C
G C
k k
ε ε
ε
µ ε
ε ε
ρ µ
σ ε
ε ρ
⎛ ⎞
⎡ ⎤
⎛ ⎞
∂ ∂
∂ ∂
+ =
+ +
⎜ ⎟
⎢ ⎥
⎜ ⎟
⎜ ⎟
∂ ∂
∂ ∂
⎢ ⎥
⎝ ⎠
⎝ ⎠
⎣ ⎦
+ −
7
In equations 6 and 7, C
µ
, C
1 ε
and C
2 ε
are model constants 0.09, 1.44 and 1.92 respectively.
These constant values are often used for a wide range of turbulent flows [18]-[19].
G
k
represents the production rate of the kinetic energy due to the energy transfer from the mean flow to
turbulence, given by:
, ,
j k
i i
i
U G
U U X
ρ ∂
= − ∂
8 The production term G
k
is modelled according to Boussinesq’s:
2 k
t
G S
µ
=
9 where S is the modulus of the mean strain tensor, given
by: 2
ij ij
S S S
= 10
and the strain tensor is: 1
2
j i
ij i
i
U U
S X
X ⎛
⎞ ∂
∂ ⎜
⎟ =
+ ⎜
⎟ ∂
∂ ⎝
⎠ 11
which is the symmetrical part of the velocity gradient. The presence of a wall influences the velocity field in
its vicinity through the non slip boundary condition. Considering the effects of the wall for the standard k-
ε model, based on Launder and Spalding
[18], a “law of the wall” for the mean velocity distribution is given by:
1 U
ln E y k
= 12
where y is the dimensionless distance to the wall is given by:
1 1
2 4
P P
C k
y y
µ
ρ µ
= 13
and: k, von Kármán constant = 0.42
E, empirical constant = 9,81 k
P
, kinetic energy of turbulence at position P y
P
, distance from position P to the wall II.2.
Numerical Method The governing equations are solved by means of a
collocated grid system using the finite volume method, and a segregated method is used to solve the discretised
equations. Pressure-velocity coupling is solved by means of the semi-implicit method for pressure
-linked equations algorithm SIMPLE, based on an iterative process that
includes a solution to a pressure-correction equation to ensure mass conservation. This algorithm is described in
detail in [20].
The advection-differencing scheme applied is based on the quadratic upstream interpolation for
convective kinetics QUICK scheme described in [21].
The effect of the grid size on streamlines and mean axial velocity was tested for various grid systems. The
choice of the grid distribution 112; 67 was sufficient to provide a grid-independent and produces satisfactory
accuracy.
The mesh is too fine near the solid boundary. This refinement was necessary to resolve the strong velocity
and pressure gradients in that region. Typically, it took about 1500-2500 iterations to reach
convergence. In creasing the number of nodes up to 112; 117 does not affect much on the mean axial
velocity. The minimum relative error defined by Er = Ф
grid1
– Ф
grid2
Ф
grid1
between the solutions of different grid studied less than 0.1559 for the mean axial
velocity.
II.3. Problem Definition and Boundary Conditions
The problem to be analyzed is the turbulent flow inside a channel of rectangular section, provided with
obstacles. A schematic view of the physical problem is shown in Fig. 2. The obstacle has a height h=1m which
is equal to its length l. The height of the channel is worth
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
308 2h and its length is of 12h. Obstacles are placed 2h far
from channel entry. The obstacle edge rounded upstream has a curvature radius which is about a ratio of 0.2 to the
height of the obstacle. The Reynolds number of the
experiments is Re = 5×10
5
, defined as:
h
D U Re
ρ µ
=
14 where U
is the entrance reference velocity, and D
h
is the hydraulical diameter of the channel. The total length
of the channel is sufficient for the flow development. Therefore, no influence will result from the side walls, so
that the flow can be considered as being two- dimensional. Kinetic energy of turbulence and
dissipation rates are prescribed, respectively, as:
2
0 005 k
, U
=
15 and:
2
0 1 , k
ε
=
16 For the upper and lower walls it is imposed:
k n
∂ = ∂
17 for the kinetic energy, where n is the coordinate normal
to the wall, and ε is computed in the volume P adjacent
to the wall as:
3 3
2 4
P p
p
C k
k y
µ
ε
=
18 Besides, non slip and impermeability boundary
conditions are imposed at the walls. In the channel outlet it is prescribed the atmospheric pressure.
l X
R
h H
X,
u
Y,
v
Fig. 2. Sketch of the obstacle geometry in a channel flow
III. Results and Discussion
