A study of the flow in an s shaped duct 4

Vertical wire smoke visualization was used close to the near-side wall in the first bend of the S-duct to visualize the flow near that wall due to swirl development in the first bend.. I

Chapter 4 ♠

SWIRL DEVELOPMENT IN SQUARE

CROSS SECTIONED S-SHAPED DUCT

4.1 Introduction

As discussed in the Chapter 3, curved duct flows are influenced predominately by two related forces, namely, the centrifugal force and the radial pressure gradient force that exist between the outside and inside walls of the curved duct, resulting in the formation of secondary flow In that chapter, a non-dimensional parameter , defined as the ratio of radial

pressure gradient to centrifugal force, was established and its relation with C p and curvature

ratio was shown explicitly The variation of (s/S O ) with normalised downstream distance, (s/S O ) results in collapsed curves independent of curvature ratio Using the data from square

cross-sectioned S-shaped duct, it was also shown that data scatter in the collapsed curved was attributed to differences in inlet condition, the presence of flow separation and stream wise vortices In this Chapter, the swirl development of square cross sectioned S-shaped duct is investigated in more detail, with attention focused on the formation mechanism of these stream-wise vortices in the second bend of the S-duct and to propose a simple flow model to clearly show the vortical flow topology

♠ A major part of this chapter has been published in Experiments in Fluids (2006, Vol 41 Issue: 6 pp 975-989) under the title “On swirl development in a square cross-sectioned, S-shaped duct” by Ng YT, Luo SC, Lim

TT, and Ho QW

4.2 Experimental Set-up and Methodology

As described in detail in Chapter 2, the investigations here were conducted at U m = 5

m/s and 15 m/s with corresponding Re = 4.73x104 and 1.47x105, respectively Using three

square cross-sectioned, S-shaped ducts (termed TS 1 to TS 3) with curvature ratio R C /D =

2.422, 1.933 and 1.667, and a duct turning angle =33.4O, 43.6O and 53.1O, respectively The inlet boundary layer thickness, measured using a single hot-wire at a flow speed of 15 m/s,

was 7.5 mm (or 0.05D)

The side wall C p was measured using the Scanivalve system and the total pressure and cross flow velocity at the S-duct’s exit were measured using Pitot-static tube, and cross-hot wire respectively A single hot wire was also used to measure the turbulence intensity at the S-duct’s exit The probes were mounted on the linear traversing device, and the spatial resolution for all probe measurements is 5 mm In all cases, a 5 sec waiting time was allowed for the flow to stabilise before acquiring data at each new probe position For total pressure and single wire measurement, the entire exit plane was measured while for cross wire measurements, only the lower half plane was surveyed

Smoke wire visualization was used to visualize the flow separation phenomenon Vertical wire smoke visualization was used close to the near-side wall in the first bend of the S-duct to visualize the flow near that wall due to swirl development in the first bend Finally, surface flow visualization on the entire bottom wall of the S-duct was also conducted to give

an overall picture of swirl development in the duct

4.3 Results and Discussion

4.3.1 Side wall C p distribution

The data for side wall C p distribution and flow visualization were presented in

the side wall C P distribution versus s/D (where s is measured from the center of the duct inlet plane along the curved duct centerline), for the three S-shaped test sections investigated In all cases, the C P distribution on the near and far-side wall displayed a general sinusoidal-like variation, and the pressure difference between the two side walls increases with the duct curvature In addition, all the three pressure distributions of the near-side wall display an inflection point, which indicates the presence of flow separation This was confirmed by smoke wire flow visualization near the separation point as shown in Fig 4.2(a) to (c) for Test Section 1 to 3, respectively This clearly shows that the presence of flow separation led to a

local distortion of the C P distribution on the near-side wall resulting in the inflection point The strong adverse pressure gradient due to the high curvature side wall in the duct led to flow separation along the near-side wall The adverse pressure gradient is defined in the present investigation as the difference between the minimum near-side wall pressure and the pressure at the inflection point for each test section Denoting the adverse pressure gradient

along the near side wall for the three test sections as 1 s'

C p

∆

∆

, 2 s'

C p

∆

∆

and 3 s'

C p

∆

∆

as

shown in Fig 4.1(a) to (c) respectively, and together with s ’ = s/D, which is the

curvilinear distance taken over the two end points in the adverse pressure gradient region, the

estimated magnitudes of 1 s'

C p

∆

∆

= 0.2053, 2 s'

C p

∆

∆

= 0.2878 and 3 s'

C p

∆

∆

= 0.3748 Expectedly, this shows that the increase in curvature leads to a corresponding increase in the magnitude of the adverse pressure gradient

In addition to the presence of flow separation, the side wall pressure distribution clearly shows that the pressure difference between the near and far side walls changes sign

along the axial flow direction of the S-duct Compared to the works of Anderson et al (1982) and Taylor et al (1982a) on the gentle curvature S-shaped duct of R C /D = 7.0, it is quite well

known that swirling flow develops in the first bend and the direction of swirl is largely

reversed after the second bend, due to both the change in curvature and the change in sign of the pressure difference between the two side walls Therefore, the present surface pressure measurements are consistent with those reported in the literature In the present study, where the ducts have sharper curvature than those reported in literature, surface oil flow visualization on the bottom floor of the duct and smoke wire flow visualization on the near-side wall of all the three S-duct test sections were conducted to better understand the flow and the swirl development This is discussed in the next section

4.3.2 Surface and Smoke Flow Visualization

Fig 4.3(a) to (c) show the surface oil flow patterns at the bottom floor of Test Section

1 to 3 respectively All three test sections show the distinctive feature of a dividing line (where the white powder is accumulated) that emanates from the end of the first bend at the near-side wall, and leaves the duct exit close to the far-side wall of the second bend The presence of this line marks two flow regions denoted as region A and B in the figures On closer inspection of the streaks in region A in Fig 4.3(a) to (c), it is noted that the flow, upon entering the first bend, develops a swirl such that the flow component, next to the bottom floor, moves from the far-side wall to the near-side wall However, on entering the second bend, a reversal of swirl direction begins, which can be seen from the streaks in region B In addition, the streaks in region A in the second bend, moves from the near-side wall to the far-side wall The existence of this dividing line, which arises due to the swirling flows of opposite direction, was not previously reported in the literature

To complement the surface flow visualization study and provide more insight to swirl development in S-duct flows, smoke wire flow visualization on the near-side wall was also conducted As pointed out in Chapter 2.7, a vertical smoke wire was placed upstream of the inflection plane and close to the near-side wall of the S-duct Due to flow symmetry about the

mid-plane of the S-duct (the z/D = 0.0 plane), the flow visualization was conducted for the

upper half of the near-side wall only, and the results are shown in Fig 4.4(a) to (c) for Test Section 1 to 3, respectively Of interest is the change in flow direction of the smoke trails especially those near the top wall of each test section In all cases, it can be seen that the smoke trails near the top wall exhibit an initially downward trajectory before turning upward Although not visualized, it can be conjectured that the smoke trail at the bottom half of the test section, will follow an initially upward trajectory before turning downward, essentially a mirror image trend The change in direction of the smoke trails is due to the change in swirl directions, close to the near-side wall, as the smoke is convected from region A into B as labeled in Fig 4.3(a) to (c)

In addition to the flow visualization study, which provides a qualitative understanding

of the swirl development in a square cross-sectioned S-duct, total pressure distribution and cross flow velocity measurements at the S-duct exit were also conducted to obtain a quantitative appreciation of the flow features The results are discussed in the next section

4.3.3 Total Pressure Distribution and Cross Flow Velocity

The data for total pressure distribution and cross flow velocity had been presented in Chapter 3.3.3 and are repeated here for convenience of further discussion and presentation Fig 4.5(a) to (c) show the total pressure distribution (using a Pitot-static tube) and the normalized cross flow velocity (using a cross hot-wire) at the S-duct exit for Test Section 1 to

3, respectively The Reynolds number was Re = 1.47x105 In these Figures, the s-y-z coordinate axes are adopted, and Fig 4.5 is plotted with the positive s-axis (the exit flow direction) pointed out of page, positive y-axis pointed to the right and positive z-axis pointed

upwards On a more general note, the figures show that in all three test sections, the exit total

pressure distributions are fairly symmetrical about the z/D = 0.0 plane and the regions close

to the four surrounding walls have relatively lower total pressure (and exit velocity) than those in the core Furthermore, the normalized cross flow vector plots showed 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 at the duct exit, in the lower half of the duct exit plane

With the combined normalized cross flow vector plots in Fig 4.5(a) to (c) and the surface oil flow visualization in Fig 4.3(a) to (c), the swirl developments in Test Section 1 to

3 can be studied in more detail By comparing the respective results from Figs 4.3 and 4.5, one can deduce that the location of flow reversal on the bottom floor of the S-duct (in Fig 4.5) corresponds approximately to the location of the dividing line near the S-duct exit (in Fig 4.3) The oil streaks in region B of Fig 4.3(a) to (c) indicate the existence of a cross flow velocity component near the bottom wall of the S-duct, which is directed from the near-side wall to the far-side wall This observation is supported by the cross-wires measurements shown in Fig 4.5(a) to (c) Since the oil streaks in region A are directed from the far-side wall to the near-side wall, it can thus be said that close to the bottom floor of the test section, the flow in region A and B meet and subsequently separate along the dividing line Hence, this dividing line is actually a flow separation line, resulted from the “meeting” of swirling flows of opposite direction

The overall flow behavior can be seen more clearly in a 3-D flow model presented in Fig 4.6 for the lower half of a typical square cross-sectioned, S-shaped duct It is expected that the upper half would behave in a similar but mirror image way It can be seen in Fig 4.6

in the flow model that the flow develops a swirl (in Region A) after passing through the first bend of the S-duct, while in the second bend of the duct, the swirl of the opposite direction (in Region B) develops near the inflection plane of the duct, close to the near-side wall

While the swirl in Region B grows in size and strength due to the S-duct’s curvature in the second bend and the increasing transverse pressure gradient, the swirl in region A undergoes

a corresponding decrease in size as it approaches the duct exit The two swirling flows of opposite direction meet along a dividing or separation line as shown in Fig 4.6, which is consistent with the surface oil flow visualization presented in Fig 4.3(a) to (c) In addition, the 3-D flow model is in agreement with the vertical smoke wire flow visualization presented

in Fig 4.4 As noted and discussed previously, the smoke trails close to the near-side wall on the lower half of the S-duct followed an initially upward trajectory before turning downward

as the smoke crosses the dividing line This observation is also reflected in the flow model in Fig 4.6(b), and is labeled as C, near the entrance to the inflection plane of the duct and close

to the near side wall Just upstream of C (in the first bend of the S-duct), the flow carries the smoke in an upward trajectory away from the bottom wall However, upon entering the second bend and crossing the dividing line close to the near-side wall, the smoke trails follow

a downward path towards the bottom wall Hence, the change in the direction of the smoke trails is a manifestation of the change in swirl direction From Fig 4.5(a) to (c), the normalized cross flow vector plots show that stream-wise vortices are present close to the near-side wall of the S-duct, and they exist either as a counter rotating vortex pair or as just a single vortex For illustration purposes, the presence of stream-wise vortices along the near-side wall of the second bend is also illustrated in Fig 4.6 as a counter-rotating vortex pair The formation mechanism for these stream-wise vortices is quite different from that of the

swirl development discussed earlier

It should be noted that these stream-wise vortical structures were also observed in the

experimental work of Anderson et al.(1982), Taylor et al (1982a) and the numerical work of Sugiyama et al.(1997) (in Japanese) Using an S-duct with curvature ratio R C /D = 7.0, duct

turning angle = 22.5O and Re = 4.0x104, they found only a single vortex at the near-side

wall of the second bend and showed that the axial vorticity developed in the first bend is

locally reinforced at the near-side wall of the second bend Anderson et al (1982) and Taylor

et al (1982a) explained the formation mechanism of such vortices by considering the Squire

and Winter formula (Squire and Winter (1951) and Scorer (1997)) for the production of axial vorticity from radial vorticity in turning flows It was shown by Squire and Winter (1951) that, for flows in a bend, a redistribution of vorticity takes place, such that the axial vorticity

in the direction of flow is equal to the velocity gradient (or∂ u ∂ z) in the approaching stream multiplied by twice the angle of deflection That is,

z

u z

v r

w

∂

∂

−

=

∂

∂

−

∂

where u ,v and w = velocity components,

r = the radial co-ordinate directed towards the centre of curvature (same as y

axis at the exit plane of the duct),

z = the vertical co-ordinate and = the turning angle of a bend measured in radians

The derivation of the Squire and Winter formula is shown in Appendix A and B of this thesis To test whether the Squire and Winter formula is applicable in the present work,

which was conducted at a Re = 1.47x105 and with S-shaped ducts of sharper curvature,

z

u ∂

∂ was deduced from the velocity measurements from a single hot wire and this is shown

in Fig 4.7(a) and (b) In Fig 4.5(a) and (b), it can be seen that the center of the

counter-rotating vortex pair occurs approximately along the line y/D = -0.4 and from Fig 4.7(a) and

(b), the rate of change of normalized ∂ u ∂ z shows two sign changes as z/D increases for Test Section 1 and 2 In other words, with increasing z/D along the line y/D = -0.4, normalized

z

u ∂

∂ is initially positive, and followed by a negative value before turning positive again as

the opposite sign which is consistent with the counter-rotating vortices seen in Fig 4.5(a) and (b) The same argument can be applied to Test section 3 in Fig 4.7(a) and (b) where normalized ∂ u ∂ z follows a positive and then a negative trend Here, with only one sign change, only single vortex is seen in Fig 4.5(c) The explanation based on Squire and Winter formula suggests that the initial swirl developed in the first bend of the duct leads to a redistribution of the stream-wise velocity (or isotachs) close to the near-side wall of the second bend This in turn leads to velocity gradients, which results in the formation of stream-wise vortices

To further study the influence of swirl on the stream-wise vortices near the wall, measurements of exit-plane total pressure distribution, normalized stream-wise and cross

flow velocities were repeated for Test section 1, 2 and 3 at a lower Re = 4.73x104 (or U m = 5 m/s) These results are shown in Fig 4.8(a) to (c) respectively for the three above mentioned test sections Fig 4.9(a) and (b) shows the changes in normalized ∂ u ∂ z for the three test

sections Fig 4.8(a) to (c) illustrate that the stream-wise vortices are also present for the low

Re case, in the form of a pair of counter-rotating vortices for Test section 1 and 2 and a single

vortex for Test Section 3 Therefore, for these test sections, the vortex configuration remains

the same as that in the higher Re cases However, the positions of the vortex cores in these low Re cases are located closer to the duct centre plane than in the higher Re cases If the Squire and Winter formula is applied one at a time to Fig 4.8(a) along y/D = - 0.35 for Test Section 1, to Fig 4 8(b) along y/D = 0.35 for Test Section 2, and to Fig 4.8(c) along y/D =

-0.3 for Test Section 3, one notes that the variation of normalized∂ u ∂ z for Test Section 1 and

2 undergoes two sign changes as z/D increases, while for Test Section 3 only a single sign

change is present, as shown in Fig 4.9(a) to (b) The above illustration thus shows that the Squire and Winter formula is applicable at both the Reynolds numbers and provides a qualitative explanation of the formation mechanism of the stream-wise vortices near the wall

In addition, the use of the formula allows one to predict the vortex configuration of these stream-wise vortices based only on the sign changes in velocity gradient

4.3.4 Exit Turbulence Intensity

The turbulence intensity at the S-duct exit was measured for Test section 1, 2 and 3

for Re = 1.47x105 and Re = 4.73 x 104 The normalized turbulence intensity (u ’ /U m)

distribution plots are shown in Fig 4.10 for Test Section 1 to 3 at Re = 1.47x105 and in Fig

4.11 for Re = 4.73x104 The turbulence intensity in each figure is normalized by their respective flow velocities The two figures showed that the core flow has relatively lower turbulence intensity (about 3%) than the near wall, low momentum flow region (typical values of 7% to 12% and above) Comparing Fig 4.10 with Fig 4.5 and Fig 4.11 with Fig 4.8, it is clear that the near wall vortical regions correspond to regions of relatively high turbulence intensity Given the relatively high turbulence intensity, one may question the applicability of the Squire and Winter formula on explaining the formation mechanism of the

near wall longitudinal vortices This can be addressed by looking at the data from Taylor et

al (1982a) In their work, Taylor et al (1982a) performed an investigation at a low Re = 790

(laminar case) and at comparatively higher Re = 4.0x104 (or turbulent case) and reported that

the turbulence intensity for Re = 4.0x104 near the wall is 13% This is in good agreement

with the present data for Re = 4.73x104 (typically about 12% and above) The Squire and Winter formula was applied to the two stated cases and was shown to be applicable to both

laminar and turbulent flow in Taylor et al.’s work Even though the turbulence intensity in their work is relatively high near the wall (about 13%) for Re = 4.0x104, the Squire and Winter formula can still explain the formation of the stream-wise vortices In the present investigation, the Squire and Winter formula also correctly predicted the number of stream-wise vortices Hence it can be argued that the Squire and Winter formula can be extended to