Determination of Impurities Generation in 10–DAB by XRD, 1HNMR and 13C–NMRon Storage for 10 Years Original Research Article Journal of Chemistry and Materials Research Vol. 1 (2), 2014, 44–51 Omprakash H. Nautiyal*

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© (2014) Copyright ORIC Publications
Journal of Chemistry and Materials Research
Vol. 1 (2), 2014, 35–39
JCMR
Journal of Chemistry and
Materials Research
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Original Research
Numerical steady of laminar flow over two–dimensional obstacle
Salim Gareh*
Department of Mechanical Engineering, University of Biskra, Biskra 07000, Algeria
Received 02 July 2014; received in revised form 05 August 2014; accepted 09 August 2014
Abstract
A numerical investigation is conducted in a rectangular channel with obstacle mounted on the lower wall. Time-independent two dimensional
laminar flow with constant thermophysical properties is assumed for air at different values of the Reynolds number (100, 400, 700, 1200 and
1800). A detailed analysis is carried out to investigate flow pattern and Nusselt number. It is also found that a travelling wave generated by the
vortex shedding contributes mainly to heat transfer enhancement.
Keywords: laminar flow; obstacle; finite volume simulation; convective heat transfert.
1. Introduction
The equations for the former class of flows result
from the Navier-Stokes equations by introducing
simplifying assumptions. Such assumptions were found
necessary in the past to permit flow predictions to be
carried out with the reduced computational capabilities that
were available and/or were introduced to allow analytical
treatments of flows. In recent years, advances in
numerical techniques for solutions of partial differential
equations (PDEs), together with the development of larger
digital computers, have provided a good basis for
computing elliptic flows, i.e. to provide numerical
solutions to the full set of Navier-Stokes equations. The
numerical techniques developed and the digital computers
available, however, are not yet developed far enough to
yield accurate solutions of flow problems relevant to
engineering.
Computations of laminar flow over an obstacle were
carried out in previous studies [1–10], e.g. by D.
* Corresponding author. Tel.: +213796651689.
E-mail address: garehsalim@gmail.com (S. Gareh).
All rights reserved. No part of contents of this paper may be reproduced or
transmitted in any form or by any means without the written permission of
ORIC Publications, www.oricpub.com.
Greenspan [1] who provided flow results within his general
study of solutions of the Navier–Stokes–equations. Ghia
and Davis [2] applied conformal transformations to present
solutions to the flow past a semi-infinite obstacle. Young
and Vafai [3] investigated the forced convective heat transfer
of individual and array of multiple two-dimensional obstacles
for a Reynolds number ranging from 800 to 1300. The effect
of a change in the channel height and input heat power was
investigated and an empirical correla-tion established. In
another study, Wang and Vafai [4] studied the mixed
convection and pressure losses in a channel with discrete
flush-mounted and protruding heat sources. In the same work,
the effect of obstacle geometry and flow rate was considered.
An empirical correlation for both pressure drop and Nusselt
number was presented. The result in good heat transfer
enhancement. Bilen and Yapici [5] carried out experimental
investigations on the effect of orientation angle and
geometrical position of wall mounted rectangular blocks. Their
results indicated that the most efficient parameters were the
Reynolds number and orientation angle. The maximum heat
transfer rate was obtained at 450°C orientation angle value.
Kim and Anand [6] looked at the effect of using a slot behind
each of six heated blocks on heat transfer from these blocks.
Their results showed an enhancement factor ranging from 4.2
to 27.2 depending of slot size and Reynolds number.
The present work represents a two-dimensional numerical
investigation of forced laminar convection in a rectangular
channel containing obstacle, attached to the lower wall.

2. 36 S. Gareh / Journal of Chemistry and Materials Research 1 (2014) 35–39
2. Mathematical Formulation
Two-dimensional flow around a surface mounted obstacle
can be described by means of the classical continuity,
momentum and convection diffusion equations.
To achieve the calculations, these equations and the
corresponding boundary conditions are made dimensionless.
The characteristic scales
At the inlet section, p0 = ρ u02
The non-dimensional number.

 0u
Re 

v
Pr 
And the non–dimensional quantities
u*=u/u0, v*=v/v0, p*=p/ρ*u02
are used for that purpose. Therefore, the governing equations
in non–dimensional form can be written as:
Continuity :
0*v.* 
X-momentum:


































2
2
2
2
*
*
*
*
Re
1
*
*
*
*
*
*
*
*
Y
U
X
U
X
P
Y
U
V
X
U
U
(1)
Y–momentum:


































2
2
2
2
*
*
*
*
.Re
1
*
*
*
*
*
*
*
*
Y
V
X
V
Y
P
Y
V
V
X
V
U
(2)
Energy:































2
2
2
2
*
*
*
*
Pr
1
*
*
*
*
*
*
Y
T
X
T
Y
T
V
X
T
U
(3)
The applied boundary conditions expressed in the
dimensionless form are:
Inflow
u = 1, v = 0, T = 0
Non-slip wall
y = 0 : u = v = 0
T = 0
y=1: u = v = 0
T=1
Obstacle
u = v = 0,T = 0
Outflow
∂u/∂x = 0, ∂v/∂x =0, ∂T/∂x = 0, ∂p/∂x = 0
3. Numerical solution
In the numerical solution of the Navier–Stokes and energy
equations, Eqs.(1) – (4), obtained by integrating over an
element cell volume. The staggered type of control volume for
the x– and y–velocity components was used, while the other
variables of interest were computed at the grid nodes. The
discretized forms of the governing equations were numerically
solved by the SIMPLER algorithm of Patankar [9–11].
Numerical solutions were obtained iteratively by the Gauss
seidel method (Fig. 1). Numerical calculations were performed
by writing a computer program in MATLAB 7.1. The
convergence criterion was assumed to have been achieved
when the values of residual terms in the momentum and energy
equati-ons did not exceed 10–3
. As the result of grid tests for
obtaining the grid-independent solution, an optimum grid of 21
x 15 is determined in the x– and y–directions, respectively.
The grid is concentrated close to the obstacles, in order to
ensure the accuracy of the numerical solution. Another hand,
we are based in this paper on the Code of the Gareh [12].
Fig. 1. Geometry and boundaries conditions dimensionless.
4. Results and discussion
The dimensionless parameters to be considered and which
characterize the flow field and heat transfer are as follows: The
Reynolds number based on channel height is taken equal to
100, 400,700,1200 and 1700, the obstacle dimensions (h,w)
and the obstacle stream wise spacing (L) are taken as h =0.5, w
= h/2 and L= 1.
4.1.Dynamical study
Fig. 2. shows the results for the u–v streamlines directly
above the channel floor Reynolds numbers ranging from Re
=100, 400 to Re = 700. It can be seen that for higher Reynol-
ds numbers, both upstream and downstream vortex structures
(2)
(3)

3. S. Gareh / Journal of Chemistry and Materials Research 1 (2014) 35–39 37
0.1
0.1
0.1
0.1
0.1
0.10.1
0.1
0.2
0.2
0.2
0.2
0.2
0.20.2
0.2
0.3
0.30.3
0.3
0.3
0.3
0.3
0.4
0.40.4
0.4
0.4
0.40.4
0.4
0.5
0.50.5
0.5
0.5
0.50.5
0.5
0.6
0.60.6
0.6
0.6
0.6
0.6
0.7
0.70.7
0.7
0.7
0.8
0.80.8
0.8
0.8
0.9
0.9
0.9
0.9
0.90.9
0.9
0.9
11
1
111.11.1
y
velocity (u)
Re=100
2468101214
2
4
6
8
10
12
14
16
18
20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.4
0.40.4
0.4
0.4
0.4
0.4
0.6
0.60.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
0.8
0.8
0.81
1
1
11
11
11
1
1
1.2
1.21.2
y
Velocity (u)
Re=400
2468101214
2
4
6
8
10
12
14
16
18
20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0.2
0.20.2
0.2
0.2
0.20.2
0.4
0.40.4
0.4
0.4
0.40.4
0.6
0.60.6
0.6
0.6
0.60.6
0.8
0.80.8
0.8
0.8
0.8
0.8
1
1
1
11
1
1
1
11
11
11
11
1.21.2
1.2
1.21.41.41.4
y
Velocity (u)
Re=700
2468101214
2
4
6
8
10
12
14
16
18
20
0.2
0.4
0.6
0.8
1
1.2
Fig. 2. Streamlines for three values of Reynolds number: (a) Re =100, (b) Re = 400, (c) Re = 700.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0
20
40
60
80
100
120
LR
= 0.059 * Re
0.9457
LR
Re
0 2 4 6 8 10 12 14 16 18 20
0,75
0,80
0,85
0,90
0,95
1,00
Re = 1800
Re = 1200
Re = 400
Re = 100
Dimensionlesspressure
x
Fig.3. Logarithmic plot of the recirculation flow length Fig. 4. Dimensionless pressure distribution in the
vs the Reynolds number. channel Re = 100, 400, 1200 and 1800.
increase in size. For Re = 100, the downstream circulation em-
erges from the upstream vortex. For Re = 400, it emerges from
both the upstream vortex and the flow that is detached at the
upstream obstacle edges. For Re = 700 and higher, the downst-
ream circulation emerges from the detached flow at the obstac-
le edges. The horseshoe flow bends around these structures.
The rear side of the obstacle is characterised by small
values of the wall shear stress. Down–stream from the
obstacle, a recirculating region corresponding to a vortex is
formed. Its size can be characterised by the position of the
reattachment point (x = LR). It is obvious that the vortex size
increases with the increasing Reynolds number (Fig. 3). After

4. 38 S. Gareh / Journal of Chemistry and Materials Research 1 (2014) 35–39
0.320.32
0.320.32
0.340.34
0.340.34
0.36
0.36
0.36
0.36
0.380.380.38
0.38
0.380.38
0.380.38
0.40.40.4
0.40.4
0.4
0.40.4
0.4
0.420.42
0.42
0.420.42
0.42
0.42
0.42
0.44 0.44 0.44
0.46 0.46 0.46
0.48 0.48 0.48
0.5
y
x
Re=100
24681012
2
4
6
8
10
12
14
16
18
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
0.48
0.5
0.320.32
0.320.32
0.340.340.34
0.340.34
0.340.34
0.360.360.36
0.36
0.36
0.36
0.360.36
0.38
0.380.38
0.380.38
0.38
0.380.38
0.38
0.4
0.4
0.40.4
0.4
0.42
0.42
0.42
0.42
0.42
0.44
0.44
0.46
0.46
0.48
0.48
0.5
y
x
Re=700
24681012
2
4
6
8
10
12
14
16
18
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
0.48
0.320.320.32
0.320.32
0.320.320.34
0.340.34
0.34
0.34
0.340.34
0.36
0.36
0.36
0.36
0.36
0.38
0.38
0.380.38
0.38
0.4
0.40.40.4
0.4
0.42
0.42
0.42
0.42
0.42
0.44
0.44
0.46
0.360.36
0.36
0.48
y
x
Re=1200
24681012
2
4
6
8
10
12
14
16
18
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
0.48
0.320.320.32
0.320.32
0.320.32
0.34
0.340.34
0.34
0.34
0.34
0.340.34
0.36
0.36
0.360.36
0.38
0.38
0.380.38
0.38
0.4
0.4
0.4
0.4
0.4
0.42
0.42
0.42
0.42
0.42
0.44
0.46
0.48
y
x
Re=1800
24681012
2
4
6
8
10
12
14
16
18
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
Fig. 5. Isotherms contours at steady state for different Reynolds number: (a) Re=100, (b) Re=700, (c) Re = 1200, (d)
Re=1800.
the flow reattachment, a new boundary layer is formed and the
wall shear stress tends to its initial value. Fig. 4 shows the
distribution of the dimensionless pressure with different
Reynolds number, it is obvious that the pressure decrease with
the increasing Reynolds number.
4.2.Thermal Study
Fig. 5. shows the representation of the isotherm contours
obtained for different values of the Reynolds number. It is
observed that the isotherm lines are denser close to the
upstream and the top faces of the obstacles. Furthermore, the
isotherm contours near the downstream face of the obstacles
are denser than those obtained in channels without obstacle on
the upper wall. The latter results are not presented here for
conciseness but if needed, the reader may consult the related
references [9, 11]. It is also observed that when the Reynolds
number increases, a vortex appears inside the inter–obstacle
cavities and the isotherm contours become thoroughly denser
especially near the faces. This yields to the removal of higher
quantities of energy from both the right and the left obstacle
faces.
5. Conclusion
The laminar flow around a cubic obstacle placed on the
floor of a bottom channel has been studied numerically. As a
first step a base solution is described, it consists of a flow at
Re= solved on a grid of 21 x 15 nodes. Four main flow zones
are identified: (i) a horseshoe-vortex system, (ii) inward
bending flow at the side walls of the obstacle, (iii) a vortex
with a horizontal axis at the downstream upper half of the
obstacle and (iv) a downstream wake containing two counter-
rotating vortices with vertical axes.
References
[1] Greenspan, D. (1969). Numerical studies of steady, viscous,
incompressible flow in a channel with a step, Journal Engineering
Mathematic , pp. 21-28.
[2] Ghia, U., and Davis. R. I. (1974). Navier-Stokes solutions of How
past a class of two-dimensional semi-infinite bodies,. AIAA J 12,
pp. 1559-1665.
[3] Young T. J., Vafai K., (1999). Experimental and numerical
investigation of forced con-vective characteristics of array of channel
mounted obstacles, ASME Journal Heat Transfer 121, pp 34–42.
[4] Wang Y., Vafai K., (1999). Heat transfer and pressure loss
characterisation in a channel with discrete flush-mounted and
protruding heat sources, Experimental Heat Transfer 12, pp 1–16.
[5] NgoBouma G.B., Martemianovb S., Alemanya A., (1999).
Computational study of lami-nar flow and mass transfer around a
surface-mounted obstacle, International Journal of Heat and Mass
Transfer 42, pp 2849-2861.
[6] Tripolia G.J., Smith E. A., (2014). Introducing variable-step topography
(VST) coordinates within dynamically constrained nonhydrostatic
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[8] Kim. S.H., Anand. N. K., (2001). Use of slots to enhance forced
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[10] Corporation, Taylor & Francis Group, New York.
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