A study of the flow in an s shaped duct 1

Besides the swirling helical secondary flow mentioned above, a pair of counter-rotating Dean vortices appears along the outside wall in the bend of circular and square cross sections.. L

Chapter 1

INTRODUCTION AND LITERATURE

REVIEW

1.1 Introduction

The study of flow in curved ducts has received constant attention from researchers due

to its wide applications in the industry The layout of any practical piping system necessarily includes bends and the accurate prediction of pressure losses, flow rate and pumping requirements demands knowledge of the character of curved duct flows

Curved duct flows are also very common in aerospace applications Many military aircraft have wing root or ventral air intakes and the engine is usually located in the centre of the aircraft’s fuselage Air entering these intake ducts must be turned through two curves (of opposite sign) before reaching the compressor face Such a configuration results in an S-shaped air intake duct and therefore the engine performance becomes a strong function of the uniformity and direction of the inlet flow and these parameters are primarily determined by duct curvature

The present introductory chapter intends to provide a review of the flows in curved ducts and S-shaped ducts Discussion is focused on the mechanism of vortex formation, the vortex topology, surface pressure measurements and duct’s exit flow conditions

1.2 Flow in Curved Ducts

Due to the centre-line curvature, flows in a bend are influenced predominantly by two related forces, namely, the centrifugal force and the radial pressure gradient that exist between the outside and inside walls of the curved duct A helical secondary flow is present

in the duct bend and Miller (1990) provided a good explanation for the origin of this helical secondary flow His figures are reproduced in this thesis as Fig 1.1 (a) and (b)

As shown in the figures, the faster moving fluid near the axis of the duct travels at the highest velocity (Fig 1.1(a)) and is therefore subjected to a larger centrifugal force than the slower moving fluid in the neighbourhood of the duct walls This results in the superposition

of a transverse (or radial) motion onto the primary axial flow, in which the fluid in the central region of the duct moves away from the centre of curvature and towards the outer wall of the bend As this core fluid approaches the outside wall of the bend, it encounters an adverse pressure gradient as shown in Fig 1.1(b) and begins to slow down This energy deficient fluid approaching the outside wall is unable to overcome the adverse pressure gradient, and instead moves around the walls towards the low static pressure region on the inside of the bend The movement of low energy fluid towards the inside of the bend, combined with the deflection of the high velocity core region towards the outside of the bend, sets up two cells

of counter rotating secondary flow as shown in Fig 1.1(a) at the end of the first bend Thus, for ducts of symmetrical cross section with respect to the plane of the curvature, a secondary flow exists which consists of a pair of helical vortices

The flow structure described above is generally true for curved duct flows and is termed the “two vortex secondary flow” structure in literature However, as the flow velocity increases, additional vortical flow structures appear which show a strong dependence on

Reynolds number (Re), curvature radius ratio (D/2R c where D and R c are the hydraulic diameter and radius of curvature respectively), the cross sectional shape and the angle of turn ( ) To account for the effects of the first two parameters, a non-dimensional number, the

Dean number (De), is usually used It is defined as,

R

D c

Flows at different De range result in the formation of additional vortical structures

Besides the swirling helical secondary flow mentioned above, a pair of counter-rotating Dean vortices appears along the outside wall in the bend of circular and square cross sections An interesting feature about these Dean vortices is that they are present for a certain

(intermediate magnitude) range of Dean number That is, they are absent in flows of low De and again disappear for higher values of De In a square cross-sectioned, = 180O curved

duct of curvature ratio 6.45, Hille et al (1985) found that the Dean vortex pair began to develop along the outside wall only within a De range of approximately 150 < De < 300.The

vortex pair was found in the region between = 108O and 171O The structure of these Dean vortices was captured clearly in the experimental flow visualization and LDV measurements

of Bara et al (1992) who conducted their experiments in a = 270O square cross-sectioned

curved duct of curvature ratio 15.1 and at De = 50 to 150 Fig 1.2 shows the results at De =

150 and the Dean vortex is noted to form at = 100O and continues to develop until =180O

(as labelled) Beyond = 180O, the Dean vortex seems to attain a fully developed state At

the lower De = 137, Bara et al (1992) noted that the Dean vortex, though present, does not

attain fully developed state for the entire axial length of the curve duct They stated that for

De in the range of 50 to 175, the development axial length decreases with increasing De The

presence of the Dean vortex is commonly termed the “four vortex or four cell flow” configuration in the literature

The critical Dean number for the onset of Dean vortices seems to vary widely among

researchers Bara et al (1992), Ghia and Sokhey (1977) and Hille et al (1985) have tried to

detect the critical Dean number for square cross section ducts, finding it to be respectively

137, 143 and 150 These discrepancies seem to arise from the different curvature ratio of the curved duct Thangam and Hur (1990) showed that this hypothesis is true and demonstrated that the critical Dean number increases with an increase in curvature ratio (that is, a sharper

bend) The result is reproduced in Fig 1.3 which shows the variation of critical Dean number with curvature ratio for square curved duct flows

The discussion thus far has highlighted that Dean vortices are present in curved duct flow To explain its physical mechanism, Winters (1987) showed that the transition from a two-vortex flow to a four-vortex flow within a critical Dean number range is a symmetry breaking bifurcation process Both the two-vortex flow and four-vortex flow are stable to symmetric disturbances while the four-vortex flow is unstable to asymmetric perturbations

Experiments by Bara et al (1992) and stability analysis by Daskopoulos and Lenhoff (1989)

suggested that when perturbed asymmetrically, the four-vortex flow might evolve to flows with sustained spatial oscillations further downstream These oscillations in the Dean vortices

are shown to exist in the flow visualisation of Mees et al (1996a) and Wang and Yang (2005)

and these are reproduced in Fig 1.4 and 1.5 respectively In Fig 1.4, the time series flow

visualisation shows the unsteady oscillation of Dean vortex at De = 220 and = 200O in the

curved bend Mees et al (1996a) commented that a characteristic feature of the wavy flow is

the oscillating in-flow region between two Dean vortices The upper or lower Dean vortex alternate in size and during a cycle, they can be mirror image of each other (e.g compare image 2 and 5 in Fig 1.4) Also the stagnation point near the center of the outer wall does not move In Fig 1.5, Wang and Yang (2005) noted that between Dean numbers of 192 and

375, the flow in a square cross-sectioned 270O curved duct develops a steady spatial and temporal oscillation that switches between symmetric/asymmetric 2-vortex flows and

symmetric/asymmetric 4-vortex flows as shown They also found that for De < 192 or De >

375, the flow pattern attains a symmetric 2-vortex solution only, without the presence of Dean vortex

The discussion on Dean vortex development shows that it occurs at a relatively low

Re (or De) and at slightly higher De, the flow returns to a two-cell flow Humphrey et al

(1977) and Taylor et al (1982b) investigated flow in a 90O square cross-sectioned curved duct in this regime and did not detect the presence of Dean vortices Using LDA and flow

visualisation, they noted that at De = 368, the location of maximum axial velocity (or

commonly called the “fluid core”) moves from the centre of the duct towards the outer wall

as the flow negotiates around the 90O bend At its exit plane, the fluid core is located around 85% of the duct width from the inner wall and the flow attains its distinctive two-cell secondary flow with velocities at 65% of the bulk axial velocity The generation of stream-wise vorticity (or secondary flow) along the curve duct is responsible for the convection of the fluid from the outer wall along the sidewall towards the inner wall of the duct Low momentum boundary layer fluid accumulates along the inner wall and hence near that wall, a much thicker boundary layer exists as compared to the outer wall As an example, Fig 1.6 shows the flow structure at the duct exit plane by Miller (1990) for 90O duct of circular and rectangular cross sectional shape The flow near the inner wall of the bend has a thicker boundary layer while the opposite occurs near the outer wall The “core fluid” is displaced towards the outer wall of the bend

If one is to further increase the Dean number (or Reynolds number) and to compare the flow structure in the laminar flow case with that in the turbulent flow case in 90O bend, one finds that the flow structure at the duct exit stays relatively invariant as that described above That is a thicker boundary layer exists along the inside wall and a thinner boundary layer at the outside wall of the bend, coupled with secondary flow for both flow regimes But

minor differences exist upon comparison and Taylor et al (1982b) and Enayet et al (1982b)

performed such studies on a square and circular cross sectioned 90O bend respectively The

former made comparisons at De = 368 and 18700 while the latter at about 211, 462 and

18170 Similar comparative work was also done by Humphrey et al (1977 and 1981) Their

results show the development of secondary flow in the form of a pair of counter rotating

vortices in the stream wise direction in both the laminar and turbulent regime But the differences lie in the magnitude of the swirling flow velocities which were found to depend strongly on the inlet boundary layer thickness, since the laminar flow case has a thicker inlet

boundary layer than the turbulent one Taylor et al (1982b) reported the secondary velocities attained in these two flow regimes as 0.6U m and 0.4Um for the thick boundary layer (laminar) and thin boundary layer (turbulent) case respectively

In addition to the stated differences, the presence of a bend affects the upstream inlet flow differently for these two flow regimes, and this determines the subsequent secondary flow development in the bend downstream A side-by-side comparison taken from the work

of Taylor et al (1982b) helps to clarify this In Fig 1.7(a) in the laminar flow case, the flow

at the inlet of the bend shows little variation with distance and a fully developed velocity profile is maintained at the inlet At the bend inlet, the velocity profile shows a velocity maximum that is closer to the outer wall of the bend The opposite occurs for the turbulent flow case where upstream effects due to the presence of the bend can be felt As the flow approaches the bend, the “core flow” migrates towards the inner wall of the bend At the bend inlet, the velocity profile thus shows a velocity maximum that is closer to the inner wall (see Fig 1.7(a)) In Fig 1.7(b) to 1.7(e) which shows the velocity contours within the bend, the

“core fluid” for the laminar case is found to migrate progressively to the outer wall, and with

a corresponding low velocity region adjacent to the inner wall Contrast that for the turbulent case, and within the first 60O of the bend, the velocity maximum stays relatively close to the inner wall of the bend with little evidence of the accumulation of low velocity fluid along the inner wall Thereafter, there is a rapid creation of a region of low momentum fluid along the inner wall and the flow continues to exit the bend with a corresponding migration of “core fluid” towards the outer wall The boundary layer thickness at the inner wall is thicker for the laminar flow case than the turbulent flow case as noted in Fig 1.7(e)

As stated previously, the accumulation of low momentum fluid along the inner wall was due to secondary flow which transports fluid from the outer wall of the bend towards the

inner wall Sudo et al (1998 and 2001) shows this clearly in their work in 90O circular and

square cross-sectioned duct at high Re = 6x104 and 4x104 respectively The three component velocity and Reynolds stresses were measured at several stations along the pipe Fig 1.8 shows the measured cross flow velocities and axial mean velocity contours at different stations for a square cross sectioned 90O duct At the inlet of the bend, the fluid is slightly accelerated towards the inner wall and decelerated near the outer wall which induces a cross

stream towards the inner wall in the central part of the cross section At the = 30O station, secondary swirling flow appears in the cross section and it forms two counter-rotating vortices which circulate outwards (away from the center of curvature) in the central part of the duct and inwards (towards the center of curvature) near the upper and lower walls At

stations = 30O to 60O, the secondary flow grows and the fast fluid near the inner wall is convected by the secondary flow to the outer wall through the central portion of the duct At the same time, the slow moving fluid near the upper and lower walls is transported towards the inner wall by the secondary flow Due to fluid transportation to the inner wall, low momentum fluid begins to accumulate and decelerate along the inner wall The boundary

layer along the inner wall thus thickens considerably when the fluid exits at = 90O and at the same time, the core fluid moves further towards the outer wall

The above discussion on curved duct flow attempted to provide a broad overview of the various vortical structures and their formation mechanism that are present in different Dean number regimes In the next section, a discussion of the flow development in S-shaped ducts will be presented

1.3 Flow in S-shaped Ducts

As stated in the previous section, the internal flows in curved S-shaped ducts are often found in various aerodynamics and fluid mechanics applications, where a combination of bends is employed to re-direct the flow A good example is an aircraft jet engine intake (Guo and Seddon (1983)) Similar to the flow in a simple curved duct, flows in S-ducts are influenced predominately by two related forces, namely, the centrifugal forces and the radial pressure gradients that exist between the outside and inside walls of the curved duct resulting

in the formation of secondary flow The description in the previous sections for swirling flow

in a single bend also occurs in the first bend of an S-duct Hence, a pair of helical vortical flow exists in the first bend of the S-duct Additional complex flow structures form when the flow enters the second bend of opposite curvature In the second bend, the swirl that developed initially in the first bend of an S-duct is attenuated and reversed in the second bend due to the opposite curvature and the reversed radial pressure gradient Since an S-duct consists of bends of opposite curvature and the radial pressure gradient changes sign along the S-duct, the side wall pressure distribution is sinusoidal-like in shape An example of such

a distribution is shown in Fig 1.9 from Kitchen and Bowyer (1989) where at the inlet of the duct, the high and low pressure side walls are respectively at the outer wall and inner wall of the first bend The high and low pressure side changes in the second bend to reflect the change in duct curvature

The observation on swirl development in S-duct is generally true for both circular and

non-circular geometry Previous work by Bansod and Bradshaw (1972) and Taylor et al (1984) (both for circular cross-sectioned S-duct) and Sugiyama et al (1997) and Taylor et al

(1982a) (both for square cross sectioned S-duct) drew similarities between them Firstly, it was noted that the swirl flow in the second bend still retains a distinctive symmetrical,

two-velocity) in the S-duct migrates towards the outside wall of the first bend and exits near the inside wall of the second bend Lastly, large scale, vortical structures exist along the outside wall of the second bend in both the circular and square cross-sectioned S-ducts This finding

is consistent with many other works, for example Anderson et al (1982) and Cheng and Shi

(1991) (for square cross-sectioned S-ducts) In particular, the development of these longitudinal vortices near the wall is accompanied by the rapid thickening of the boundary layer at the outside wall of the second bend Fig 1.10, taken from Bansod and Bradshaw (1972) on circular S-shaped duct, shows the total pressure distribution on the S-duct exit and the thickened boundary layer along the outside wall of the second bend The core flow exists close to the inside wall of the second bend Bansod and Bradshaw (1972) pointed out that since a favourable longitudinal pressure gradient exists on the outside wall of the second bend, the flow near this wall is accelerating and hence these longitudinal vortices could undergo “ vortex stretching” , thus intensifying vorticity This causes the boundary layer near that wall to thicken rapidly due to enhanced entrainment and accumulate into a region of low-momentum fluid

Another variant in S-duct flows is the S-shaped diffuser which can be found in many applications In addition to the combined effects of centrifugal forces and radial pressure gradient in S-duct flows, flow separation is another important factor that influences the flow structure in an S-duct diffuser Due to the increasing cross sectional area, a stream-wise adverse pressure gradient is also present The combined effects may result in increased flow non-uniformity and total pressure loss at the duct exit as compared to a uniformed cross-sectioned S-duct In a circular cross cross-sectioned S-duct diffuser, flow separation results in a comparatively large pair of contra-rotating stream-wise vortices, which occupy about a third

to a half of the S-duct exit area Such problems were investigated by Whitelaw and Yu

(1993a, b) and Yu and Goldsmith (1994), Anderson et al (1994) for circular cross sectioned

diffusers The flow in S-duct diffusers of rectangular or square cross section was studied by

Rojas et al (1983), Sullerey and Pradeep (2003), and Pradeep and Sullerey(2004)

Among these works on constant cross sectioned S-duct or diffusers, the effects of inlet

boundary layer play an important role in the swirl development Anderson et al (1982) (for constant square S-duct), Rojas et al (1983) (for square S-duct diffuser) and Whitelaw and Yu

(1993a) (for circular S-duct diffuser), investigated the effects of boundary layer thicknesses

in their respective studies Similar to the case of a single 90O bend, flows with a thicker inlet boundary layer result in a larger magnitude swirl generated in the first bend The difference in

swirl magnitude can be 0.22U m and 0.15U m for the thick and thin boundary cases noted by

Anderson et al (1982) Rojas et al (1983) and Whitelaw and Yu (1993a) quoted respectively 0.4U m versus 0.15U m and 0.16U m versus 0.12U m for the corresponding thick and thin boundary layer cases However, the details in flow topology within the second bend are

dependent on whether flow separation is present For the work of Anderson et al (1982) and Rojas et al (1983) where flow separation is not present, increasing the inlet boundary layer

thickness led to a corresponding increase in boundary layer development along the outer wall

of the second bend However, for Whitelaw and Yu (1993a) where flow separation is present,

they found that an increase in inlet boundary layer thickness led to a corresponding decrease

in boundary layer thickness along the outer wall of the second bend Whitelaw and Yu (1993a) also found that increasing inlet boundary layer led to a reduction in separation region

in their S-duct study and argued that the earlier reattachment of separated flow for the thick inlet boundary layer case led to a corresponding decrease in outlet boundary layer along the outer wall of the second bend Their LDA measurements show this very clearly

The above review has concentrated mainly on S-duct with limited turning angle at

relatively high Re (or De) where Dean vortices are not present It is of interest to study flows

within the critical Dean number range and where the turning angle in the first bend of the