A study of the flow in an s shaped duct 3

Chapter 3 RADIAL PRESSURE GRADIENT AND CENTRIFUGAL FORCE IN CURVED AND 3.1 Introduction Literature review in Chapter 1 shows that swirl flow in a curved duct and S-duct is governed by

Chapter 3

RADIAL PRESSURE GRADIENT AND

CENTRIFUGAL FORCE IN CURVED AND

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 90O 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-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 90O 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 4.74x104 and 1.47x105, 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

Besides the present experimental data, C p data on 90O 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 90O 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 90O 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(104) to O(105)) 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

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 90O

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 = 4.73x104 and Re =

1.47x105 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 90O

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 90O 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 90O

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

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

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

2 2 1

m

c m m

U p D R

Re D U

D U 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 4

∆

= Π

× Π

=

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

∆

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

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

−

−

−

=

m

near m

far near

p far p

U

p p U

p p C

C

ρ ρ

2

m

U

p

ρ

∆

By multiplying a factor of

D

R c

2 to both sides of the equation (3.3), the relation becomes,

U

p D

R C

m

c near p far p

2 2

1

∆

=

∆

=

c

m

R

U D

p

2

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

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 90O curved ducts and S-ducts are discussed

3.3.3 Collapsed Curves of Ω (s/SO ) Versus (s/S O)

3.3.3.1 Circular and Square Cross-sectioned 90OCurved 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 90O 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 90O 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 90O curved ducts at different curvature ratios signals the importance of the radial pressure gradient and

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 = 4.0x104) and the present experimental data (measured at Re =

4.73x104 and 1.47x105) 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

90O 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

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 = 1.47x105 (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 = 4.73x104 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 = 4.73x104 and 1.47x105

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 = 4.73x104

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