A study of the flow in an s shaped duct 3

29
Chapter 3

RADIAL PRESSURE GRADIENT AND
CENTRIFUGAL FORCE IN CURVED AND
S-SHAPED DUCT
♣



3.1 Introduction

Literature review in Chapter 1 shows that swirl flow in a curved duct and S-duct is
governed by a centrifugal force and radial pressure gradient force between the side walls. The
magnitude of these two forces within the duct is dependent on the curvature of the duct. For
example, in a curved duct of a sharper bend (or smaller curvature ratio), the pressure difference
between the side walls is known to be larger in magnitude than one with a gentler bend.
Therefore, the side wall pressure variations, illustrated either as plots of C
P
versus the duct’s
turning angle () or the duct’s stream-wise distance (s), generally show significant variations for
curved ducts of different curvature ratio. Since the dominant forces in a curved duct are due to
radial pressure gradient and centrifugal effects, it can be argued that plotting the experimental
data as a non-dimensional parameter consisting of a ratio of these two forces should give a better
collapse of the data, regardless of the curvature ratio. The main objective of this chapter is to
investigate this proposition by introducing a dimensionless parameter, , which is related to the
ratio of the two stated forces, and apply it to the flow in circular and square cross sectioned 90
O

curved ducts and S-shaped ducts.

♣
A major part of this chapter has been published in Physics of Fluids (2008, Vol.20, Issue 5 Art no. 055109) under
the title of “On the relation between centrifugal force and radial pressure gradient in flow inside curved and S-
shaped ducts” by Ng YT, Luo SC, Lim TT and Ho QW.

30
There are a few well known non-dimensional numbers for fluid flow involving curvature
effects, e.g. Goertler number for curved boundary layer flows and Dean number for curved pipe
flow. To the best of the author knowledge, it is relatively new to define explicitly the parameter
 and use it to study curved duct flows of different curvature ratios. A better understanding of 
and its variation in curve ducts may lead to improved design in, say, elbow flow meters (see
Hanssen (1980), Borresen (1980) and Sanchez Silva et al. (1997)) which uses the magnitude of
the radial pressure difference between the side-wall for calibration against the flow speed.
Later in this chapter, a method to compute  is shown. It is based on the known side wall
C
P
data from circular and square cross-sectioned 90
O
curved ducts and S-shaped ducts that are
available both in the literature and from the present experimental work.

3.2 Experimental Set-up and Methodology
The experimental set-up and technique used had been described in Chapter 2 and only a
very brief outline will be given here. Essentially, the S-duct wind tunnel, and Test Section 1 to 3
were used. The flow velocities were U
m
= 5 m/s and 15 m/s thus giving Reynolds number (Re),
based on hydraulic diameter (D = 0.15 m), of 4.74x10
4

and 1.47x10
5
, respectively. The inlet
boundary layer thicknesses were 0.05D at the two above mentioned Reynolds numbers. The
Scanivalve system measured the side wall surface pressure (hence C
p
) distribution. Pitot-static
tube measured the total pressure coefficient (C
PT
) distribution and crossed hot-wires measured
the normalized cross flow velocities (v/U
m
and w/U
m
) at the S-duct exit plane. Flow visualisation
using smoke wires were used to visualize the flow separation phenomenon on the near side wall
and vortex generators were used to suppress these flow separation.

31
Besides the present experimental data, C
p
data on 90
O
curved ducts and S-ducts of square
and circular cross sectioned were extracted from literature. Fig. 3.1 shows the nomenclature used
in this part of the work. The far and near side walls are labeled with the origin of the s-axis
located on the inlet of the 90
O
curved duct. From the literature, the side wall C
P

data plots were
scanned digitally into Bitmap format and the data points were acquired using a plot digitizer.
Polynomial curve fits were subsequently applied to these acquired C
P
data to obtain the
intermediate data points.
Table 1 provides a summary of the test conditions in these referenced cases, detailing
their duct geometry, curvature ratio, flow conditions and inlet boundary layer thicknesses (where
available). The 90
O
curved ducts and S-shaped ducts have square and circular constant cross-
sections, but of different curvature ratios. The selected cases are limited to moderately high
Reynolds number (Re) (i.e. Re ~ O(10
4
) to O(10
5
)) and where the side wall pressures were
measured in the plane of the bend. In these references, the side wall pressure coefficient data are
typically presented as graphs of C
P
versus  (i.e. the duct’s turning angle) or C
P
versus the duct’s
non-dimensional centre-line distance. Particular attention was paid to the different coordinate
system and nomenclature used by different authors in their work. For example, all authors define
C
P
in the usual manner as stated above except Ito (1960) who defines a loss coefficient as
g
U

H
m
2
2
,
where H is the static pressure head loss measured with respect to a reference value. Clearly, Ito’s
(1960) definition is consistent with those of other authors’ if both the denominator and numerator
are multiplied by g.

32
In the next section, the C
p
data from the present experiment and from the literature will
be presented. A method to compute the ratio of radial pressure gradient to centrifugal force will
be discussed.

3.3 Results and Discussion
3.3.1 Downstream Variation of Side Wall Cp
The C
P
data gathered from the literature are presented in Fig. 3.2 and Fig. 3.3 for 90
O

curved duct and S-duct respectively. For clarity, the present experimental C
P
data on square
cross-sectioned S-ducts are plotted separately in Fig. 3.4(a) and (b) for Re = 4.73x10
4
and Re =
1.47x10

5
respectively. These six figures show the variation of the side wall C
P
with the
normalized axial (or centre line) distance (s/S
O
) for circular and square cross sectioned 90
O

curved ducts and S-ducts of different curvature ratio. S
O
is the total centre-line coordinate
distance along the s-axis of each curved 90
O
duct or S-duct. For the ease of identification, the C
P

curves corresponding to near and far side data in Fig. 3.2 and 3.3 are circled and labeled.
From these figures, a few common trends are noted. Firstly, for the flow in those 90
O

curved ducts (in Fig. 3.2(a) and 3.2(b)), the far side wall has relatively higher C
P
values than the
near side wall, and this is also true for the C
P
distribution in the first bend of those S-shaped
ducts (Fig. 3.3(a) and 3.3(b), 3.4(a) and 3.4(b)). Secondly, for the flow in S-shaped ducts, the C
P
distribution exhibits a sinusoidal-like variation along the ducts’ centre-line distance. This implies

that the pressure difference between the side walls changes sign as the flow negotiates the bends
of opposite sense. This well known flow behavior results in a swirl development in the first bend,
which is subsequently attenuated and reversed (in direction) in the second bend. This observation
is generally true for flows in circular and square cross-sectioned S-duct and is reflected in the C
P


33
distribution displayed in Fig. 3.3(a) (for circular cross-sectioned S-duct) and in Fig. 3.3(b), 3.4(a)
and 3.4(b) (for square cross-sectioned S-duct). Thirdly, these figures also show that with
increasing curvature ratio (R
C
/D), the ducts’ geometry becomes less curved and the radial
pressure difference, p (defined as the pressure difference between the far side and near side
wall at the same s-coordinate) decreases. This is clearly observed in Fig. 3.2(a) and (b). For
example, in Fig. 3.2(a), the C
P
difference between the side walls in Briley et al.’s

(1982) data at
R
C
/D = 1.0 is larger than those at R
C
/D = 2.8. The same can be said of Ward-Smith’s (1971) data
in Fig. 3.2(b), when one compares the R
C
/D = 1.15 data with those at R
C
/D = 3.45. This is due to

centrifugal force of different magnitudes exerting on the fluid as it flows around a bend. A
sharper bend (and hence lower value of R
c
/D) gives rise to higher centrifugal force and hence, as
a reaction force, a larger pressure difference develops between the side walls. Lastly, in Fig.
3.4(a) and (b), the experimental C
P
distribution for the flow in square cross sectioned S-duct
shows a point of inflection which indicates that flow separation is present. Evidence of flow
separation will be given in a later section.
From the above discussion and the figures presented, it is clear that the variation of side
wall C
P
with s/S
O
is dependent on duct curvature ratio and cross sectional shape, and is likely to
result in the large scatter among the published C
P
data. Since the forces due to radial pressure
gradients and centrifugal effects are the dominant forces governing the physics of the flow in a
bend, it appears likely that a parameter involving the two above-mentioned forces may be able to
put ducts of different curvature on a “common ground”. If one goes back to the basic physics of
the fluid flow, for the present problem, it is reasonable to say that a force, F, experienced by the
fluid is dependent on the duct dimension D, inlet mean velocity U
m
,fluid density , dynamic
viscosity , duct radius of curvature R
c
and pressure difference between inner and outer wall p.

34
That is, F = f(D, U
m
, , , R
c
, p). Using Buckingham  Theorem, it can be shown that the
above parameters can be reduced to four dimensionless groups, namely
.
and ,
,
1
,
2
4
3
2
22
1
m
c
m
m
U
p
D
R
ReDU
DU
F
ρ

ρ
µ
ρ
∆
=Π
=Π
==Π
=Π

Here, a term that involves both the radius of curvature of the bend R
c
and the pressure
difference between the inner and outer walls would be a combination of 
3
and 
4
. In this thesis,
this new dimensionless parameter is termed  and defined as
c
m
R
U
D
p
2
43
ρ
∆
=Π×Π=Ω
. (3.1)

Note that the numerator is related to pressure gradient force between the side walls of the curved
duct while the denominator is related to centrifugal force. In terms of a better known
dimensionless group called the Dean’ s number, which is defined as De =
Re.
R
D
c
2
, it can be
shown that,
2
2
2
1






=
De
Re
U
p
m
ρ
∆
Ω
. (3.2)

A simple method of calculating
Ω
from the curved ducts’ known side wall C
P
values and its
curvature ratio is illustrated in the next section. This derived relation was subsequently applied to
all data in the present work.


35
3.3.2 Determination of
c
2
m
R
U
D
p
 =

To compute
Ω
from the available C
P
data, it is first assumed that the reference static
pressure (p

) is constant within each of the cases obtained from literature. This is a reasonable
assumption because a constant reference pressure (usually the wall static pressure) is used in
experiments. Next, the difference in C

P
between the far side wall and near side wall can be
written as,
,
2
1
2
1
22












−
−













−
=−
∞
∞
m
near
m
far
near
p
far
p
U
pp
U
pp
CC
ρρ

.
2
1
2
m
U

p
ρ
∆
= (3.3)
By multiplying a factor of
D
R
c
2
to both sides of the equation (3.3), the relation becomes,
(
)
D
R
U
p
D
R
CC
c
m
c
near
p
far
p
2
.
2
1

2
2
ρ
∆
=− .


=
∆
=
c
m
R
U
D
p
2
ρ

. (3.4)
Hence, the value of
Ω
at a particular (s/S
O
), is equal to the difference in C
P
between the
far side and near side wall at the same (s/S
O
), multiplied by half of the curvature ratio. This

mathematical operation is applied to all the C
P
data and presented as plots of (s/S
O
) versus
(s/S
O
). The clear advantage of illustrating the plots in such a manner is that it allows one to study
the variation of these two forces along the normalized curved ducts’ centerline coordinate (or

36
distance) for ducts of different curvature ratios. In addition, the scattered data of C
P
versus (s/S
O
)
for curved ducts of different curvature ratio are reduced to collapsed curves when plotted in the
proposed manner. In the following sections, salient points in the collapsed curves of (s/S
O
)
versus (s/S
O
) for the flow in square and circular cross-sectioned 90
O
curved ducts and S-ducts are
discussed.

3.3.3 Collapsed Curves of
Ω
(s/S

O
) Versus (s/S
O
)
3.3.3.1 Circular and Square Cross-sectioned 90
O
Curved Ducts
Fig. 3.5(a) and (b) show the variation of (s/S
O
) with (s/S
O
) for flows in circular and
square cross-sectioned 90
O
curved ducts. Included in each of these figures is a straight line of
unit gradient. Data points which lie close to this line imply that the force due to radial pressure
gradient is balanced by that due to centrifugal effects. In Fig. 3.5(a) and (b), the various literature
data for circular and square cross-sectioned 90
O
curved duct show that respective collapsed
curves are obtained for ducts of different curvature ratios. The solid line in each figure depicts
the averaged variation of (s/S
O
) with (s/S
O
) for the two flow cases considered. The figures also
show that the variation of averaged (s/S
O
) with (s/S
O

) remains fairly linear up to (s/S
O
)  0.6,
suggesting that the radial pressure gradient force is balanced by the centrifugal force as the flow
negotiates the bend. Beyond that, the curve attains a maximum point at (s/S
O
)  0.7-0.8 and
decreases thereafter. Increased data scatter are noted in the two figures when (s/S
O
) > 0.6. This
could be due to increased flow unsteadiness and the presence of flow separation as the flow
approaches the exit of the bend. From the points noted above, the similarity in the variation of
the averaged (s/S
O
) with (s/S
O
) for flows in square and circular cross-sectioned 90
O
curved
ducts at different curvature ratios signals the importance of the radial pressure gradient and

37
centrifugal force in influencing the flow characteristics. Comparing Fig. 3.5(a) and (b) with Fig.
3.2(a) and (b), it is clearly advantageous to plot (s/S
O
) with (s/S
O
) instead of side wall C
P
with

(s/S
O
) since the scatter in the C
P
data as depicted in the latter figures is significantly reduced (and
in the earlier part of the flow (s/S
O
<0.6), the data even collapse onto one curve). In addition, the
former figures also illustrate the relative influence of centrifugal force and radial pressure
gradient on the flow and this important piece of information is missing when side wall C
P
is
plotted against (s/S
O
).

3.3.3.2 Circular and Square Cross-sectioned S-shaped Duct
Applying the same idea to the side wall C
P
data for flows in circular and square cross -
sectioned S-shaped ducts, the plots of (s/S
O
) with (s/S
O
) are shown in Fig. 3.6(a) (for circular
cross-sectioned) and Fig. 3.6(b)-(c) (for square cross-sectioned). The data from Fig. 3.6(a) were
calculated using data from the literature, while those in Fig. 3.6(b) and (c) represent both
literature data (measured at Re = 4.0x10
4
) and the present experimental data (measured at Re =

4.73x10
4
and 1.47x10
5
). As the experiments were performed on three test sections, the data in
the figures are labeled TS 1 to TS 3. The results from Fig. 3.6(a) to (c) generally show a
collapsed curve with the expected sinusoidal-like variation of (s/S
O
) with (s/S
O
), which is due
to a change in the sign of the radial pressure gradient. Similar to the cases for the flow in a single
90
O
bend, the advantage of presenting data in such a manner is that the scatter in C
P
variation for
S-ducts of different curvature ratio (in Fig. 3.3(a), 3.3(b), 3.4(a) and 3.4(b)) becomes smaller and
in some parts non-existent. The important features of the collapsed curve and the scatter in the
data will be explained next.

38
In Fig. 3.6(a) to (c), the data points for (s/S
O
) with (s/S
O
) tend to collapse remarkably
well in the first bend of the S-duct, suggesting that the forces due to radial pressure gradient is
balanced by that due to centrifugal effects. This observation is noted in three instances, namely,
in the literature data for circular cross-sectioned S-duct (Fig. 3.6(a)) and square cross-sectioned

S-duct (Fig. 3.6(c)), and the present experimental data at the higher Re = 1.47x10
5
(Fig. 3.6(c)),
in which the data points lie along the straight line of unit gradient up to (s/S
O
)  0.30-0.35. The
exception to this is the experimental data measured at the lower Re = 4.73x10
4
where the data
points starts to deviate from the line of unit gradient at a lower value of (s/S
O
)  0.20 as shown in
Fig. 3.6(b). One possible explanation for this difference is the presence of flow separation in the
experimental work in which the S-duct test section (TS 1 to TS 3) have a lower curvature ratio (a
sharper bend) than those in the literature like Anderson et al.

(1982)

and that of Ng et al. (2006).
Since a sharper bend results in a stronger adverse pressure gradient, the existence of flow
separation in TS 1 to TS 3 is not surprising. This can be observed in Fig. 3.4(a) and (b) where a
point of inflection is present in the side wall Cp distribution for Re = 4.73x10
4
and 1.47x10
5
.
Also, by comparing the two figures, the point of inflection in the Cp distribution at the lower Re
covers a larger range of (s/So) than that at a higher Re. Hence, the local distortion of the C
P
variation at the inflection point on one side of the wall results in a decrease in radial pressure

gradient via Equation (3.4) for the lower Re = 4.73x10
4
.
In the second bend of the S-duct, some data scatter is noted in Fig. 3.6(a) to (c) and this is
most probably attributed to two sources, namely, the rapid thickening of the boundary layer
along the outer wall of the second bend in the S-duct and the existence of stream-wise vortices
along the said wall. From the earlier works involving S-shaped ducts, it is known that stream-
wise vortices are formed near the outer wall of the second bend. In the experimental work of

39
Bansod and Bradshaw (1972) for circular cross sectioned S-duct and the numerical investigation
of Sugiyama et al. (1997)

on square cross-sectioned S-duct, these near wall vortices were shown
to exist. The present measurements of the normalized cross flow velocity on the exit plane of TS
1 to TS 3 for the two Re = 4.73x10
4
and 1.47x10
5
reveal that these stream-wise vortices are also
present and formed along the outer wall of the second bend. This is shown, respectively, in Fig.
3.7(a)-(c) and Fig. 3.8(a)-(c) for the two Re investigated. The pictures are drawn with the free-
stream flow direction pointing out of page. Due to flow symmetry, only the normalized cross
flow velocities for the lower half of the duct exit are shown.
By looking at all these flow cases, the exit total pressure coefficient distributions are
fairly symmetrical about the z/D = 0.0 plane for all the test sections and Reynolds numbers
tested. Regions close to the four surrounding walls have relatively lower total pressure (and exit
velocity) than those in the core. In addition, the corresponding normalized cross flow velocity
plots shows generally that in the central region of the duct, the main bulk of the cross flow is
from the far-side wall to the near-side wall while the cross flow near the bottom floor of the test

section is in the opposite direction. The cross flow thus sets up a swirl in the lower half of the S-
duct exit. Close to the near side wall, different vortex configurations can be seen for both Re
cases. For better clarity, close up view of these near wall vortices at the two Re and three test
sections (TS 1 to TS 3) are shown in Fig 3.9(a)-(c) and Fig 3.10 (a)-(c) respectively. These
figures include normalized cross flow velocity vector plots and stream line plots from the
respective cases. The streamline plots show the presence of vortices and their configurations
more clearly. For example, for TS 1, a vortex pair is near the symmetry plane for the lower Re
case (Fig. 3.9(a)), while at a higher Re (Fig. 3.10(a)), a vortex pair is located closer to the bottom

40
corner of the duct. For TS 3, a single vortex is located near the symmetry plane at low Re (Fig.
3.9(c)) while a single vortex is also observed at higher Re case (Fig. 3.10(c)).
From these figures, it is clear that the location and configuration of these stream-wise
vortices are dependent on the duct curvature and Reynolds number. The existence of these
stream-wise vortices close to the wall alters the pressure distribution on the side walls in the
second bend of the S-duct, and one may argue that this results in flow differences in the cases
studied and hence the scattered data of (s/S
O
) with (s/S
O
) in Fig. 3.6(a) to 3.6(c). In addition,
other factors which include different inflow conditions and experimental set-up among the
different sources cited here may also contribute to the observed scatter in the data.
The explanation given above to explain the observed deviation and scatter in the data of
(s/S
O
) versus (s/S
O
) in S-duct flows is still a hypothesis. To shed more light and provide
additional concrete evidence, vortex generators (VG) are installed upstream of the flow

separation location on the near side wall of the square cross-sectioned S-ducts. Vortex generators
“ locally” control the effects of separation by mixing the high-momentum fluid in the free stream
with low-momentum fluid near the wall and thus suppressing flow separation. The installation of
vortex generators in the present investigation serves dual purposes, firstly to suppress flow
separation, and secondly to alter the flow condition and hence boundary layer thickness and the
vortex configuration of the stream-wise vortices along the outer wall of the second bend in the S-
duct. In the next section, these effects on the variation of C
P
and (s/S
O
) along the duct’ s
downstream distance will be investigated for TS 1 to TS 3 at both Re = 4.73x10
4
and 1.47x10
5
.




41
3.3.4 Effects of Vortex Generators on C
p
and
o
S
s
Ω

A row of 10 triangular-shaped VGs, arranged in “ counter-rotating” fashion as explained

in Chapter 2 was placed upstream of the separation point where the near-side surface pressure is
a minimum. This location was chosen because downstream of it, the existence of an adverse
pressure gradient is expected to cause the boundary layer to thicken rapidly. Fig. 3.11(a) and (b)
shows the side wall C
P
distribution for the three test sections, measured at both Re = 4.73x10
4

and 1.47x10
5
, with VGs installed at 
vg
= 10
O
. As indicated in the two figures, the point of
inflection is absent when the VGs are installed in the three test sections for the two Re cases. The
presence of vortex generator promotes mixing near the side wall and flow separation is locally
suppressed. To show the effects of VGs, smoke wire flow visualizations conducted close to the
near side wall for the TS 2 showed the existence of flow separation without VGs (see Fig.
3.12(a)). When VGs are installed, the flow separation is successfully suppressed (see Fig.
3.12(b)). However, it is believed that the flow separation is only locally suppressed due to the
highly three-dimensional flow in the S-shaped duct.
Using the modified C
P
distribution for all the flow cases,  was computed and as before
these results are shown as plots of (s/S
O
) versus (s/S
O
) and compared to those when no VGs are

installed. The results from TS 1, TS 2 and TS 3 at the two Re are selected to highlight the salient
points.
At the two different Re = 4.73x10
4
and 1.47x10
5
, Fig. 3.13(a) and (b) show respectively a
comparison of the variation of (s/S
O
) versus (s/S
O
) for TS 1 with and without VG installed. Fig.
3.14 (a), (b) and Fig. 3.15(a), (b) show the same but for TS 2 and TS 3, respectively. When VGs
are not installed, Fig. 3.13(a), 3.14(a) and 3.15(a) clearly illustrate that the data points of (s/S
O
)
showed different trends for the two Reynolds numbers investigated, in both the first and second

42
bend of the S-duct. In the first bend, the variation of (s/S
O
) versus (s/S
O
) for the higher Re
follows the line of unit gradient relatively better than that of the lower Re. As noted in the
previous discussion involving Fig. 3.4(a) and 3.4(b), the distortion in the side wall C
P

distribution due to a point of inflection (or presence of flow separation) for the lower Re =
4.73x10

4
is more extensive than that at the higher Re = 1.47x10
5
. Hence, this led to a smaller
magnitude of (s/S
O
) for the three test sections and the curves deviate from the line of unit
gradient at about (s/S
O
)  0.2, which is earlier than the higher Re case. When VGs are installed
upstream of the separation point, an improvement in the variation of (s/S
O
) versus (s/S
O
) in the
first bend of the S-duct is noted in Fig. 3.13(b), 3.14(b) and 3.15(b). For these cases, the data
points for TS 1, 2 and 3 closely follow the line of unit gradient at the two Reynolds numbers for
up to larger values of (s/S
O
)  0.30-0.35. Also the difference in (s/S
O
) between the two
Reynolds numbers becomes negligible up to (s/S
O
)  0.5. The improvement in the trend is
particularly obvious when one compares Fig. 3.15(a) and (b) for TS 3 at the lower Re =
4.73x10
4
. The results therefore show that, when the flow separation is artificially suppressed
using VGs, the balance between radial pressure gradient force and centrifugal force takes place

over a longer distance downstream for all the three test sections, and the influence of Reynolds
numbers is reduced. It also clearly shows the effects of flow separation on .
For the second bend in the S-duct, Fig. 3.13, 3.14 and 3.15 show slight deviations in the
curve of (s/S
O
) versus (s/S
O
) at the two Reynolds numbers, irrespective of whether VGs are
present, although generally the difference is slightly smaller when the VGs are present. This
implies that the suppression of flow separation alone does not entirely account for these observed
differences and they could also be due to fundamental differences in the flow structures. Data

43
from total pressure coefficient distribution and normalized cross flow velocity measurements at
the TS 1 to TS 3 duct exit plane can demonstrate this.
At Re = 4.73x10
4
and with VGs installed, Fig. 3.16(a), 3.16(b) and 3.16(c) show the total
pressure coefficient distribution and the normalized cross flow velocity at the S-duct’ s exit of TS
1, TS 2 and TS 3 respectively, while Fig. 3.17(a), 3.17(b) and 3.17(c) show the same
measurements for the higher Re = 1.47x10
5
.
With VGs at 
vg
= 10
O
installed in the S-duct and comparison made with the absence of
VGs, changes are noted in the exit total pressure coefficient distribution at the near-side wall of
the S-duct for the two Re and the three test sections. This can be seen by comparing Fig. 3.16

with Fig. 3.7 for the lower Re cases and Fig. 3.17 with Fig. 3.8 for the higher Re cases for TS 1
to TS 3. Without the presence of VGs, it is known that flow separation exists and leads to the
rapid thickening of the downstream boundary layer as the fluid flows through the second bend of
the S-duct. The presence of the VGs enhances the mixing process and the flow separation is
suppressed locally. This leads to a relatively thinner boundary layer at the exit and a slight
increase in magnitude of the total pressure distribution especially those near the mid height (z/D
= 0.0 horizontal plane) of the near-side wall. In addition, the normalized cross flow vector plots
indicate that the presence of VGs alters the vortex configuration and position of the near wall
longitudinal vortices. For example, in TS 1 at the low Re, a vortex pair near the symmetry plane
before the installation of VGs (Fig. 3.7(a)) is altered to a vortex pair near the bottom corner when
VGs are installed (Fig. 3.16(a)). For TS 3 at low Re, a single vortex is located near the symmetry
plane with no VGs (Fig. 3.7(c)) while a counter-rotating vortex pair is observed with the
presence of VGs (Fig. 3.16(c)). If one makes the same comparison between Fig. 3.8 (no VGs
installed) and Fig. 3.17 (with VGs installed) for the higher Re case, the vortex configuration or

44
position is different when VGs are installed in TS 1 to TS 3. Hence the same observation is
noted.
From the evidence presented in this section, it can be seen that the vortex configuration is
dependent on parameters like duct curvature, Re and the duct’ s inlet flow condition. Since
changes in vortex configuration and its position alters the velocity field near the side wall, this
leads to changes in the Cp distribution along the side wall of the second bend of the S-duct. This
can be shown by making more detailed cross flow velocity measurements using crossed hot-
wires in the vicinity of the stream-wise vortices near the wall. This is illustrated in Fig. 3.18(a)
and (b) for TS 3 at the Re = 4.73x10
4
and 1.47x10
5
, without the presence of VGs. Both figures
show a more detailed view of the near wall, stream-wise vortices and it is clear that the velocity

field close to the near-side wall is distinctively different from one another. In particular, for Re =
4.73x10
4
, Fig. 3.18(a) shows that at z/D = 0.0, the flow velocity vectors are directed away from
the side wall (as labeled in the figure) while in Fig. 3.18(b), for Re=1.47x10
5
, the flow velocity
vectors are directed towards the side wall. These differences in flow direction would almost
certainly have effects on the side wall pressure distribution.
In this work, it is plain to see that crossed hot-wire measurements are invasive and the
presence of a probe may somewhat alter the position of the vortices. In Chapter 2 and in the
published work of Ng et al. (2006), the present data are benchmarked against those of Anderson
et al (1982) (who employed LDA) and those of Sugiyama et al. (1997) (a numerical study) using
Test Section 4. It was shown that their results are reproduced fairly well. From this comparative
study, it seems to indicate that the use of probes does not have a significant effect on the
dominant vortical flow structures in the S-duct.

45
In this Chapter, the data and evidences show that large-scale, stream-wise vortices exist
along the near side wall in the S-duct. These stream-wise vortices, by virtue of their different
positions and configuration, altered the side wall pressure distribution especially in the second
bend of the S-duct. This influences the radial pressure gradient, and hence (s/S
O
) in the second
bend. The presence of these vortical structures here explains the noted data scatter in the curves
of (s/S
O
) versus (s/S
O
) for the flow in a circular and square cross sectioned S-shaped duct.

In the next Chapter, a more detailed study of the swirl development and structure of these
stream-wise vortices is performed.

3.4 Chapter Conclusion
A new non-dimensional parameter  is introduced to characterise the flows in 90
O
curved
ducts and S-shaped ducts. This parameter is based on the fact that the forces due to radial
pressure gradient and centrifugal effects play dominant roles in curved duct and S-duct flows.
Analogous to a solid particle moving in circular orbit held by an inelastic string, the solid particle
moves in circular motion because the centrifugal force and the tension (a reaction force) in the
string balance each other. For fluid particle moving in a curved path, centrifugal force and radial
pressure gradient develops to balance each other. From dimensional analysis, it was shown that
 is related to the ratio of the forces due to radial pressure gradient and centrifugal effects. A
method of computing  from the measured Cp between the side walls and the curvature ratio of
the curved duct is derived. Using data from the literature and the present experimental results,
collapsed curves are obtained when (s/S
O
) is plotted against the normalized duct centerline
coordinate, (s/S
O
), for circular and square cross sectioned 90
O
curved ducts and S-shaped ducts of
different curvature ratios. The existence of a collapsed curve for curved duct flows thus shows

46
that  is an important non-dimensional parameter governing the flow in curved and S-shaped
ducts. Some data scatter in the plots are noted and explained. These observations are presumably
traced to factors like existence of flow separation, presence of stream-wise vortices near the

curve walls and the different experimental conditions among the literature sources that were
cited here. To verify this presumption, and to shed more light on the effects flow separation and
stream-wise vortices have on (s/S
O
), vortex generators were installed to artificially change the
flow conditions. The experimental results show that the variation of (s/S
O
) with (s/S
O
) is indeed
dependent on the existence of flow separation and the vortex configuration of the near wall
stream-wise vortices and hence provides proof to the earlier presumption.
The work presented in this chapter has shown that before flow separation occurs in the
curve duct, the fluid behaves similarly to a solid particle moving in circular orbit, with radial
pressure gradient balancing centrifugal force (like tension balancing centrifugal force). When
flow separation occurs, the radial pressure gradient on the side walls and the large transverse
movement of the fluid results in fluid particles not travelling in a circular path. Therefore, the
radial pressure gradient and centrifugal force does not balance and thus resulting in the
experimental data deviating from the unit gradient line.
Finally, the work presented in this chapter could be useful for engineers who are
designing or calibrating elbow flow meters and scientist who wish to benchmark their numerical
investigations. By plotting (s/S
o
) versus (s/S
o
), engineers who wish to optimize the design of
elbow flow meters can do so such that their data follow the “ line of unit gradient” to the largest
possible (s/S
o
).