 Open Access
 Total Downloads : 231
 Authors : Sumit Kumar, Somnath Chakrabarti
 Paper ID : IJERTV3IS060105
 Volume & Issue : Volume 03, Issue 06 (June 2014)
 Published (First Online): 09062014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
A Numerical Study on Variation of Streamline Contours, Reattachment Length, Wall Pressure, Static Pressure and Stagnation Pressure with the Configuration of a Sudden Expansion: Viewed from BioMedical Application
S. Kumar *
*Department of Mechanical Engineering, Heritage Institute of Technology, Kolkata, Pin700107, W.B, India
S. Chakrabarti1
1Department of Mechanical Engineering,
Indian Institute of Engineering Science and Technology, Shibpur, Pin 711103, W.B, India
P
P
s
Abstract : In this work, an attempt has been taken to carry out a numerical study on fluid flow through a sudden expansion. Twodimensional steady differential equations for conservation of mass and momentum have been solved for the Reynolds number ranging from 50 to 250 and aspect ratio ranging from 1.5 to 6.0 when fully developed velocity profile has been considered at the inlet. Stream functions in the flow domain are computed and several streamline contours are generated by MATLAB software. The effect of aspect ratio and Reynolds number on the streamline contours, reattachment length, wall pressure, average static pressure, distance of maximum average static pressure rise from throat, L *, variation of maximum average static pressure rise (p*) at location L *, effectiveness, average stagnation pressure and maximum average stagnation pressure drop (p *) at Lp* of the sudden expansion configuration have been extensively studied.
Keywords: Sudden expansion, Streamline contour, Atherosclerosis, Backwardfacing step

INTRODUCTION
Suddenly expanded geometry has immense importance to researchers involved in bioengineering and biomedical area due to its rich features such as recirculation, separation and reattachment etc. These types of phenomena may occur in vein and arterial system due to some disease. One of the most important diseases of veins and arterial system is atherosclerosis. Early development of atherosclerosis often occurs in regions of arterial branching and sharp curvature.
Many early events in
the pathogenesis of atherosclerosis are linked, at least initially, to complex homodynamic forces unique to atherosclerosis prone regions of the vasculature [1]. Flow
characteristics such as flow separation, recirculation and reattachment, as it occurs close to arterial bifurcations, may directly contribute to the initiation of focal atherogenesis [2,3]. An established in vitro model used to simulate flow bifurcating arterial regions in the backwardfacing step flow chamber [4]. In this model, fluid flows from a narrow channel over a step expansion into a wider channel. The asymmetric expansion of the flow path leads to a separation of the flow. Close to the expansion step, there is a recirculating eddy with a flow direction against the main flow. Farther downstream, the flow reattaches and eventually reestablishes a unidirectional parabolic flow profile. At physiological Reynolds numbers, the flow is laminar in the entire chamber. In the backwardfacing step flow chamber, endothelial cells exposed to the separated flow streams experience large spatial shear stress gradients, especially close to the second stagnation point. However, if the onset of flow in sudden expansion in the backward facing step flow chamber, the steady state of the flow requires several milliseconds to develop fully [5], subjecting the cells to a large temporal gradient of shear stress.
Extensive research work on sudden expansion configurations viewed as a diffuser as well as industrial applications has been carried out by several researchers [6 18].
As per brief literature review, it has been observed that a number of researchers have mainly done numerical and experimental works on the sudden expansion flow chamber from industrial viewpoint. Very few researchers have studied the configuration with respect to medical applications. Therefore, in this work, flow characteristics of a sudden expansion geometry have been studied in connection with different biomedical applications. In this
paper, an attempt has been made to investigate the effect of important parameters such as aspect ratio and Reynolds number on the streamline contours, reattachment length, average wall pressure, average static pressure, and average stagnation pressure. Our own FORTRAN code is developed for this study.

MATHEMATICAL FORMULATION
A schematic diagram of the computational domain is illustrated in Figure 1. The flow under consideration is
Boundary Conditions
In the present work four different types of boundary conditions are applied. They are as follows,

At the walls: No slip condition is used, i.e., u* = 0, v* = 0.

At the inlet: Fully developed velocity profile is prescribed and the transverse velocity is set to be zero, i.e., u* = 1.5 1 2 2 , v* = 0.

At the exit: Fully developed condition is assumed and
hence gradients are set to zero,
assumed to be steady, twodimensional and laminar. The
fluid is considered to be Newtonian and incompressible.
i.e.
= 0,
= 0.
The following dimensionless variables are defined to obtain the governing conservation equations in the non dimensional form;
Lengths:

At the line of symmetry: The normal gradient of the axial velocity and the transverse velocity are set to zero, i.e., = 0, v* = 0.
=
, = , = =


NUMERICAL PROCEDURE
1
1
1 ,
= =
The partial differential equations (1), (2) and (3) have been
1 ,
1 ,
1 ,
discretised by a control volume based finite difference
1
1
Velocities; =
, =
method. Power law scheme has been used to discretise the
Pressure,
1
= 2
convective terms (Patankar [19]). The discretised equations
have been solved iteratively by SIMPLE algorithm, using linebyline ADI method. The convergence of the iterative
With the help of these variables, the nondimensional mass
and momentum conservation equations are written as follows,
+ = 0 (1)
scheme has been achieved when the normalized residuals for mass and momentum equations summed over the entire calculation domain will fall below 107. The distribution of grid nodes has been considered nonuniform and staggered in both coordinate directions allowing higher grid node
1
concentrations in the region close to the step and the walls
+
=
+
+
(2)
of the duct.
In the present computation work, the flow is assumed to be
+
=
1
fully developed at the exit and hence the exit is chosen far
away from the throat. For aspect ratio of 1.5 to 3.0, the exit length, L * is considered to be 50 and the inlet length, L *
+ +
(3) ex i
has been considered to be 1 due to assumption of fully developed velocity profile at the inlet of the computational domain. During computations, the numerical mesh
Where, the flow Reynolds number, Re = 11 .
L ex
LR 2
comrised of 17 Ã— 13 grid nodes in the inlet section and 85
Ã— 31 in the exit section in x and y directions respectively have been considered (Chakrabarti et al. [8]).
The size of numerical mesh in x and y directions for aspect ratio ranging from 4.0 to 6.0 has been fixed after grid independence test. In the present computation, the exit
length, L * is considered to be 120 and the inlet length, L *
e Secondary duct ex i
B
i,1 Li
Primary duct
A
C D
W2 /2
has been considered to be 1 due to assumption of fully developed velocity profile at the inlet of the computational domain. The numerical mesh comprised of 17 Ã— 13 grid nodes in the inlet section and 205 61 in the exit section in x and y directions respectively provides grid
W1 /2
y, y*
Inlet
i,1
x, x*
Line of Symmetry
e
Exit
2
independent results.

VALIDATION OF COMPUTATIONAL
RESULTS
In order to validate the accuracy of the numerical model,
Figure 1. Schematic Diagram of the computational domain
comparisons are made between the numerical results and the experimental data reported by Durst et al [20]. The experimental geometry used in Durst et al. [20] is
considered in the simulation, which is shown in Fig. 1. The computations have been carried out for Reynolds number of 56 for aspect ratio of 3. Axial locations of presented velocity profile in the paper (Durst et al. [20]) have been converted to the locations in accordance with our consideration, and accordingly the results have been placed in Fig. 2. It shows the variations of axial velocity profiles at different locations of downstream of the sudden expansion, along with the experimental measurements of Durst et al. [20]. As observed, the numerical results are in good agreement with the experimental data.

RESULTS AND DISCUSSIONS
The important results of the present study are reported in this section. The parameters those affect the flow characteristics are indentified as:

Reynolds number, 50 Re 250.

Aspect ratio, 1.5 A* 6.0.

Inlet velocity distribution Fully developed.
1.0
diseases, atherosclerosis is one of the most common manifestations of arterial disease and is characterized by deposits of yellowish plaques containing cholesterol, lipid material, and lipophages formed within the intima and inner media of arteries. The recirculation zone in arteries is considered to be an important phenomenon for the formation and propagation of atherosclerosis. The physiological significance of the recirculation zone is that blood stream stagnates locally in this region and allows platelets and fibrin to form a mesh at the inner wall in which lipid particles become trapped and eventually coalesce to form atheromatous plaque, this may tend to cause a more severe stenosis and hardening of the arteries. Apart from that, the reattachment point is having also significance on the formation and propagation of atherosclerosis. The high cell turnover rate takes place near the reattachment point due to high cell division and low cell density near that region. For this, a leaky junction may develop which is considered to be the possible pathway for transport of lowdensity lipoprotein through the arterial wall. Flow characteristics such as flow separation, recirculation and reattachment, as it occurs close to arterial bifurcations, may directly contribute to the initiation of
0.9
0.8
x*=7 (Our)
x*=7 (Durst, 1974) x*=11 (Our)
focal atherogenesis.

Variation of wall pressure
0.7 x*=11(Durst, 1974)
0.6
The wall pressure plays an important role in the arterial systems of the human organs. Since low pressure at the
0.5
0.4
0.3
0.2
0.1
0.0
Re=56, AR=3.0
stenosis zone correlates the tearing action of endothelium layer with subsequent thickening of plaque. Therefore, the chances of tearing action and plaque deposition are very high in the downstream away from the throat and very near to the throat and it increases either with increase of Reynolds number or aspect ratio. The effect of flow Reynolds number on the variation of nondimensional wall pressure at the upstream and downstream of the sudden
0.20 0.00 0.20 0.40 0.60 0.80 1.00
u*
Fig. 2 Axial velocity profiles at different axial locations
5.1. Variation of streamline contours and reattachment length
The effect of Reynolds number and aspect ratio on the variation of streamline contours has been investigated. From the presented streamline contours (Fig. 3 and Fig. 4), it is observed that with increase in Reynolds number, reattachment point moves downstream of the throat. The size of recirculating bubble increases with increase in Reynolds number for a fixed aspect ratio. For a fixed Reynolds number, the recirculating bubble size also increases with increase in aspect ratio. This is also supported by the Fig. 5 (a) and Fig. 5 (b) respectively. The size and strength of recirculating bubble formed due to negative velocity zone in this configuration plays an important role for different biomedical applications.
In the artery of human body, the configuration of sudden expansion encounters frequently. In coronary artery
expansion configuration for five Reynolds number, namely 80, 100, 150, 200 and 250 for a typical aspect ratio of 5 (Figure 6 (a)), has been presented. Figure6 (b) shows the effect of aspect ratio on the variation of nondimensional wall pressure at the upstream and downstream of the configuration for 4.0, 5.0 and 6.0 for a typical Reynolds number of 200. From both the graph it has been seen that nature are very similar. From the overall study of the effect of aspect ratio and Reynolds number, it is noted that the axial location of the maximum pressure recovery points moves towards the downstream with Re and A*. An interesting feature is that the maximum pressure rise occurs earlier for lower values of Re. This can be explained readily by noting that for very low low Reynolds number flow, the fluid will almost adhere to the wall without showing any tendency to separation and the point of maximum pressure rise will be sufficiently close to the throat. As the flow Reynolds number is increased, it is natural to expect a larger reattachment length; consequently pressure rise will be continued over a large length, before a maxima is reached. From the graph, it is also observed that the maximum pressure rise occurs earlier for lower values of aspect ratio. This can be explained by the fact that for very low aspect ratio, the size of recirculating bubble is
very small resulting in lower diffusion due to which the fluid will almost adhere to the wall without showing any tendency to separation and the point of maximum pressure rise will be sufficiently close to the throat.

Variation of average static pressure along the axial length
The initiation and progression of atherosclerosis is dependent on the accumulation of low density protein in the artery wall. One of the biomechanical forces of the chances of the deposition is depending on transmural fluid flux. Since the fluid flux depends on average pressure of the blood, therefore, the average static of blood at any section of the coronary artery may be considered to be an important parameter in assessing the extent of growth of stenosis. The rise in static pressure is one of the important parameters in assessing the performance of sudden expansion used as a diffuser. A properly designed sudden expansion used as a diffuser should ensure a high static pressure rise while having minimum stagnation pressure loss. Therefore, in this section, an attempt has been made to study the effect of Reynolds number and aspect ratio separately on the average static pressure in the post throat zone. In the present work, the average static pressure at any crosssection is determined by the folloing expression:
pdA
more because of higher diffusion. Figure 7 (b) shows the variation of average static pressure along the length of sudden expansion configuration for three 4.0, 5.0 and 6.0 for a typical Reynolds number of 200. The general trend of the curve of Fig. 7 (b) is more or less same as that of the previous cases. Initially in the postthroat region, the overall crosssectional area of the recirculating bubble is high and so the static pressure rise is small. Further in downstream there is more positive pressure and also increased kinetic energy diffusion and so the average static pressure rises resulting in significant pressure recovery. After a certain maximum pressure, there is a drop in the average static pressure. The decrease in average static pressure in sudden expansion configuration may be explained by the fact that size of recirculating bubble increases with increase in aspect ratio, since this large eddy is associated with larger wall length, a substantial amount of pressure head, recovered due to diffusion, is used up for providing the frictional drop along the sudden expansion configuration wall.
Also, it is seen that for lower aspect ratio, there is a sharp increase of average static pressure, in terms of distance along the axis in the postthroat region i.e., the peak average static pressure point is progressively shifted to the right as the aspect ratio increases; however the pressure
pavg
dA
(4)
recovery is more for higher aspect ratio.

Variation of average stagnation pressure along the
The effect of different Reynolds number and different aspect ratio on the average static pressure at downstream of the configuration has been studied Figure 7 (a) shows the variation in average static pressure for five Reynolds number ranging from 80 to 250 for a typical aspect ratio of
5. The general trend of these curves of Fig. 7 (a) is noted to be more or less same as that of the cases of wall pressure. From the graph, it is observed that the average static
axial length
In this section, an attempt has been made to compute the average stagnation pressure at any section by the following expression:
e 2
e e e
p 1 V 2 u dA
A
pressure steeply drops at the throat region, thereafter the
p *s,avg
e
u dA
(5)
pressure again rises. The drop is due to fact that across the throat region, there is an abrupt change in crosssectional area of the sudden expansion configuration. This increases the denominator of the equation (4) to an extent that even a static pressure recovery in this zone cannot adequately compensate and hence the steep pressure drops. At the postthroat region, the numerator of equation (4) is influenced by both positive pressure zone (created by the fluid in the main stream that does not undergo recirculation) and negative pressure zone (created by the recirculation fluid). Initially in the postthroat region, the overall crosssectional area of the recirculating bubble is high and so the static pressure rise is small. Further in downstream there is more positive pressure and also increased kinetic energy diffusion and so the average static pressure rises resulting in significant pressure recovery. After a certain maximum pressure, there is a drop in the average static pressure as the viscous dissipative effects supersede at this stage. Also, it is seen that for lower Reynolds number flow, there is a sharp increase of average static pressure, in terms of distance along the axis in the postthroat region i.e., the peak average static pressure point is progressively shifted to the right number flow is
e e
Ae
Where the subscripts erefers to the plane of measurements. The direction of the velocity vector, particularly for a recirculating type flow situation, has been taken into account during the calculation of average stagnation pressure at any given section. The details have been described in Chakrabarti et al. [8]. Figure 8 (a) shows the variation in average stagnation pressure for typically five Reynolds number, 80, 100, 150, 200 and 250 for the aspect ratio of 5.0. It is obvious that along a streamline only, the stagnation pressure should remain constant or there may be some decrease (in the absence of energy transfer). By using equation (5) for average stagnation pressure along the length, the expected behaviour of variation of average stagnation pressure has been obtained. It has been observed that the average stagnation pressure increases with increase in Reynolds number. This can be explained as, for higher Reynolds number flow, the kinetic energy contribution towards the working fluid at a section will be higher at that section. Figure 8 (b) shows the variation in average stagnation pressure for three aspect ratio of 4.0, 5.0 and 6.0 for a typical Reynolds number of
200. From the figure, it is seen that the average stagnation pressure increases with increase in aspect ratio. This can be explained by using the equation (5). The increase in average stagnation pressure may be attributed to the increasing nature of recirculating bubble into the development of static pressure (pe) which increases the numerator of equation (5), and as a result Ps avg increases. The stagnation pressure along the axial distance is gradually decreasing in nearthroat zone mainly and then it shows the asymptotic behavior throughout the length of the configuration. The generation of high pressure near the postthroat zone may compress the stenosis, formed inside the artery which will lead to opening of the blockage in the artery.

Variation of maximum average stagnation pressure drop (P *) at location L *
6. CONCLUSION
The present work, the flow characteristics of plain sudden expansion in low Reynolds number regime with fully developed velocity profile at the inlet has been studied. The effect of Reynolds number and aspect ratio on streamline contours, reattachment length, wall pressure, average static pressure, average stagnation pressure and stagnation pressure drop have been investigated in detail and this leads to the following important conclusions:

The size of recirculating bubble increases with increase in Reynolds number for a fixed aspect ratio and it also increases with increase in aspect ratio for a fixed Reynolds number. It is expected that the rate of heat transfer as well as temperature of the flowing
fluid will increase as the recirculation zone increases.
S P
Figure 9 (a) shows the variation of stagnation pressure drop with Reynolds number of 80, 100, 150, 200 and 250 for
typical aspect ratio of 1.5, 2.0, 3.0, 4.0, 5.0 and 6.0 respectively. From the graph, it is observed that stagnation pressure drop remains nearly asymptotic with the increase of Reynolds number for a fixed aspect ratio. Initially, static pressure rises due to diffusion of kinetic energy. This gain is considerably greater than eddy losses, therefore, in very low Reynolds number regime the stagnation pressure drop decreases. Even at higher Reynolds number the diffusion of kinetic energy occurs, but the gains are offset by the eddy losses. Thus, in Reynolds number regime the stagnation pressure drop curve shows asymptotic behaviour. Figure 9

shows the variation of stagnation pressure drop with aspect ratio of 1.5, 2.0, 3.0, 4.0, 5.0 and 6.0 for typical Reynolds number of 80, 100, 150, 200 and 250 respectively. From the presented graph, it is revealed that there is a sharp rise in the stagnation drop with increase in aspect ratio for a fixed value of Reynolds number. This can be explained by equation (5). With the increase in aspect ratio, the shape and size of the recirculating bubble increases and since this large eddy is associated with larger wall length, a substantial amount of static pressure, recovered due to diffusion is used up for providing the frictional drop along the sudden expansion configuration and hence nuerator of equation (5) decreases which increases the average stagnation pressure drop.
It occurs mainly due to conversion of kinetic energy to the flowing fluid into heat energy. Due to the increasing recirculation zone, retention of blood at a particular area will increase as a result of which, toxide medicine, if injected at that location, will get time to be sedimented on the required area, without affecting the important part of the body like heart, kidney, liver, lung etc.

The pressure at the wall of the configuration is found to change quite sharply in the region of throat.

The average static pressure steeply drops at throat region, thereafter pressure again rises. Initially in the postthroat region, the static pressure rise is small but further in downstream the average static pressure rises resulting in significant pressure recovery.

For all the cases, it is observed that average stagnation pressure gradually decreases.

Reynolds number has no strong effect on the variation of maximum average stagnation pressure drop, but aspect ratio has significant effect on it. The maximum average stagnation pressure drop increases with increase in aspect ratio. The generation of high pressure at the expanded zone may compress the stenosis, formed inside the artery which will lead to opening of the blockage in the artery.
Re = 50
Re = 100
Re = 200
Figure 3. Streamline Contour plotting at A* = 5.0 for different Reynolds number
A* = 2.0
A* = 4.0
A* = 6.0
Figure 4. Streamline Contour plotting at Re = 200 for different aspect ratio
100
95
90
85
80
75
70
65
60
55
L*
R
50
45
40
35
30
25
20
15
10
A*_1.5 A*_2.0 A*_3.0 A*_4.0 A*_5.0 A*_6.0
Fully Developed
(a)
0.30
0.25
0.20
Pw*
0.15
0.10
0.05
0.00
0.05
(b)
Fully Developed
Re_200
A*_4.0 A*_5.0 A*_6.0
5
0
60 80 100 120 140 160 180 200 220 240 260
Re
10 0 10 20 30 40 50 60 70 80 90 100 110 120
S*
Fig. 6 Variation of average static wall pressure
100
Fully Developed
80
Re_80
Re_100 Re_150
60 Re_200
Re_250
L*
R
40
20
(b)
0.25
0.20
0.15
P*
avg
0.10
0.05
(a)
A*_5.0
Re_80 Re_100 Re_150 Re_200 Re_250
Fully Developed
0
1 2 3 4 5 6
A*
0.00
10 0 10 20 30 40 50 60 70 80 90 100 110 12
X*
Fig. 5 Variation of reattachment length
0.30
0.25
Fully Developed
(b)
0.20
0.30
0.25
0.20
Pw*
0.15
0.10
0.05
0.00
(a)
Fully Developed
A* = 5.0
Re_80 Re_100 Re_150 Re_200 Re_250
0.15
P*
avg
0.10
0.05
0.00
Re_200
A*_4.0 A*_5.0 A*_6.0
10 0 10 20 30 40 50 60 70 80 90 100 110 120
X*
0.05
10 0 10 20 30 40 50 60 70 80 90 100 110 120
S*
Fig. 7 Variation of average static pressure
0.2
0.1
0.0
0.1
P*
s avg
0.2
0.3
0.4
0.5
0.6
(a)
Fully Developed A*_5.0
Re_80
Re_100 Re_150 Re_200 Re_250
1.1
Maximum average stagnation pressure drop
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
(b)
FULLY DEVELOPED
Re_80 Re_100 Re_150 Re_200 Re_250
10 0 10 20 30 40 50 60 70 80 90 100 110 120
X*
1 2 3 4 5 6
A*
Fig. 9 Variation of maximum average stagnation pressure drop
0.2
0.1
0.0
0.1
P*
s avg
0.2
0.3
0.4
0.5
0.6
Fully Developed Re_200
(b)
A*_4.0 A*_5.0 A*_6.0
10 0 10 20 30 40 50 60 70 80 90 100 110 120
X*
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Fig. 8.Variation of average stagnation pressure along the axial length
1.10
maximum average stagnation pressure drop
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p * = Nondimensional average stagnation pressure drop Re = Reynolds number
s
S* = Nondimensional distance along the wall
u, v = Velocity components in x and y directions of a Cartesian coordinate
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V1 = Average velocity

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W1, W2
= Width of inlet and exit section

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Nomenclature
A* = Aspect ratio or area ratio, given by A2/A1 dA= Elemental area
A1 = Crosssectional area at inlet section A2 = Crosssectional area at exit section
Li = Inlet length (i.e., length between inlet section and throat)
Li* = Nondimensional inlet length
Lex = Exit length (i.e., length between throat and exit section)
Lex* = Nondimensional exit length
LP = Distance of maximum average static pressure rise
LP* = Nondimensional distance of maximum average static pressure rise
LR = Reattachment length
LR* = Nondimensional reattachment length
Âµ = Dynamic viscocity = Density
p = static pressure
p* = Nondimensional static pressure
Pavg* = Nondimensional average static pressure Pw* = Nondimensional wall pressure
Ps* = Nondimensional stagnation pressure
x, y = Cartesian coordinates
1, i = Inlet