NASA Contractor Report 3367 

Measurements of Laminar and 

NASA 

CR 

3367 

c. 1 

Turbulent Flow in a Curved Duct 

With Thin Inlet Boundary Layers 

A. M. K. P. Taylor, J. H. Whitelaw, 

and M. Yianneskis 

CONTRACT NASW-3258 

JANUARY 198 1 

NASA 
- - 

TECH LIBRARY KAFB. NM 

OOb20L3 ~ 

NASA Contractor Report 3 367 

Measurements of Laminar and 

Turbulent Flow in a Curved Duct 

With Thin Inlet Boundary Layers 

A. M. K. P. Taylor, J. H. Whitelaw, 

and M. Yianneskis 

Imperial College of Scieiice aird Techrology 

Loizdoti, Etzgluird 

Prepared for 

Lewis Research Center 

under Contract NASW-3258 

NASA 

National Aeronautics 

and Space Administration 

Scientific and Technical 

Information Branch 

1981 
TABLE OF CONTENTS 

SUMclARY 

1. INTRODUCTION 

2. EXPERIMENTAL PROCEDURE 

2.1 FLOW CONFIGURATION 

2.2 VELOCIMETER CONFIGURATION, DOPPLER SIGNAL PROCESS1 

AND MEASUREMENT TECHNIQUE 

2.3 ACCURACY AND PRECISION 

3. RESULTS 

3.1 LAMINAR FLOW RESULTS 

3.2 TURBULENT FLOW RESULTS 

4. DISCUSSION 

5. CONCLUDING REMARKS 

APPENDIX 1 - TABULATED DATA 

TABLE I LAMINAR FLOW: REYNOLDS NUMBER 790. VELOCITY MEAS 

TABLE 11 TURBULENT FLOW: REYNOLDS NUMBER 40 000. MEAN VEL0 

TURBULENCE LEVEL AND CROSS-CORRELATION MEASUREMENT 

TABLE IIITURBULENT FLOW: REYNOLDS NUMBER 40 000. WALL PRE 

MEASUREMENTS 

APPENDIX 2 - DEFINITION OF SYMBOLS 

REFERENCES 

FIGURES 

iii 
SUMMARY 

The secondary flows in a 90° square bend of 40 mn hydraulic diameter 

and mean radius of 92 mm have been measured at Reynolds numbers of 790 and 

40 000 in a water tunnel. The boundary layers at the inlet to the bend were 

approximately 25% and 15% of the hydraulic diameter respectively. provide a better understanding of the two flows, are compared with published 

data of the flow with fully developed profiles at the bend inlet and will 

assist the evaluation of numerical prediction techniques. Laser-Doppler 

velocimetry was used to measure velocities in the plane of curvature and, in 

the turbulent flow, to measure the associated fluctuations and crosscorrelations. Data is also given for stations upstream and downstream of 

the bend. For turbulent flow, wall pressure distributions are reported. 

The results 

In the laminar flow strong streamwise vorticity is generated on 

entry to the bend, with the result that the locus of maximum streamwise 

velocity on the symnetry plane moves towards the outer radius and regions of 

high streamwise velocity are convected along the sidewalls. 

flow, smaller secondary motions at the inlet initially allow this locus to 

move towards the inner radius under the action of pressure gradients before 

the influence of the strengthening secondary motion becomes evident about 

60° around the bend. The difference in the behaviour of the secondary motion 

is related to the difference in thickness of the inlet boundary layers. 

Comparison with data for fully-developed profiles at inlet shows that 

the differences from the present laminar flow are of detail. In the turbulent 

flow, the influence of the secondary motion is apparent at earlier streamwise 

positions for the fully-developed profile. 

qualitative similarities, such as higher shear stress towards the outer radius 

and large anisotropy at the exit from the bend. Quantitatively, the turbulent 

levels in the present flow are much smaller, as expected. 

In the turbulent 

The turbulent fields show 

The measurements are presented in tabular form and are also available 

on magnetic tape for ease of comparison with predictions. 

1 
1. INTRODUCTION 

Flows in curved ducts are found in a wide range of practical configurations and, in this report, they are discussed in the context of a 90' bend 

of square cross-section and comparatively strong curvature. A distinguishing 

characteristic of such flows is the generation of streamwise vorticity, or 

"secondary motion", within the duct which can result in large redistributions 

of, for example, the streamwise velocities. The change in the flow on passing 

through the duct can be sufficient to impair the performance of plant downstream 

of the duct, for example the performance of the compressor in a curved aircraft 

intake duct. 

Information on the flow in bends has been considered in reference 1 

and, with the exception of a few measurements of local velocity in bends of 

square cross-section, relates to pressure characteristics. 

have been obtained for laminar flows in the round ducts of references 2 and 3 

and in the square duct of reference 4 which also yielded heat-transfer 

information. Turbulent flow measurements have been reported in reference 5 

for a near two-dimensional channel and in reference 6 for a channel with an 

aspectratio of 2. The maximum ratio of cross-stream to bulk velocity was 

found, in reference 3, to be of the order of 0.5 and for the turbulent flow 

of reference 6 to be of the order of 0.3. The measurements of references 7 

and 8 provide the most comprehensive velocity information for laminar and 

turbulent flow in a square duct with strong curvature. 

the bend was fully developed for both the laminar and turbulent flows and the 

development of the velocity through the ducts was markedly different in the 

two cases. 

Local measurements 

The flow approaching 

The duct geometry investigated in this report is identical to that 

of references 7 and 8,with thin inlet- boundary layers at the entrance to the 

bend. This condition is relevant to the case of aircraft intake ducts. 

Measurements of the velocity characteristics within the duct are reported for both laminar and turbulent flow and provide a basis for the understanding 

of the flow and its development in the two flow regimes. The thickness 

of the boundary layers at the inlet is expected to be important because of 

the related changein the cross-stream vorticity and this can be quantified 

by comparison with the results of references 7 and 8. A further purpose is 

to provide detailed data in a form suitable for the evaluation of numerical 

prediction techniques and this requires careful measurement of the conditions 

at the inlet of the duct for use as boundary conditions in the prediction algorithm. 

the numerical accuracy of the procedure and to provide an assessment of the 

turbulence model. 

The laminar and turbulent flow can be used to study, respectively, 

The aim of the report is to make available accurate measurements of the 

streamwise and cross-stream velocity and wall static pressure in sufficient 

detail to fulfil the purposes of the previous paragraph. 

previous investigations of refererces 7 to 9, laser-Doppler velocimetry was 

used to measure the velocity components. 

2 

In cmn with the 
The following section describes the experimental procedure and 

The results are discussed in section 4 and a sumnary of 

I includes estimates of the errors associated with the measurements presented 

I in section 3. ' conclusions is provided in section 5. 

2. EXPERIMENTAL PROCEDURE 

i 2.1 Flow Configuration 

The duct was a 90' bend of mean radius 92 mm and of radius ratio 2.3, 

identical to that of references 7 and 8. The dimensions of the cross-section 

(40 f 0.1 x 40 kO.1 mm) conform closely to those used for the straight squaresectioned duct investigated in reference 9 and used in references 7 and 8 to 

provide fully-developed flow upstream of the bend. 

in figure 1 , together with the co-ordinate system adopted in the report. The 

bend was installed in a water tunnel with the plane of synmetry located 

horizontally. Figure 2 shows a plan view of the water tunnel and the lengths l of the straight ducts upstream and downstream of the bend. 

through the bend was controlled by precision-bore flowmeters. The bulk velocities, V , were 1.98 un/s and 1.00 m/s for the laminar and turbulent flows 

respectivhy, corresponding to Reynolds numbers, Re, of 790 and 40 000 and 

Dean numbers, De, of 368 and 18 700. The temperature of the water was maintained at 20°+-20C for all the experiments. 

The geometry is illustrated 

The flow rate 

2.2 Velocimeter configuration, Doppler signal processing and measurement 

technique 

Figure 3(i) shows the optical arrangement of the laser-Doppler velocimeter with the sensitivity vector of the velocimeter in the gapwise direction. 

It is similar to an arrangement described in reference 10 except that 

frequency shifting of the laser light was not used. Figures 3(ii) and (iii) 

show photographs of the bend and optical system respectively. which was used to turn the optical axis through 9O0,was flat to one-tenth of 

the wavelength of light. 

are summarised in the following table: 

The mirror, 

The principal characteristics of the optical system 

3 
Characteristics of the optical 

arrangement 

Lami na r Turbulent 

flow flow 

Focal length of imaging lens (mn) 

h'aif-angle of intersection 

Fringe separa ti on ( 1 i ne-pai r spacing ) 

Number of fringes in measuring volume 

Inter sec ti on vol ume di ame ter ca 1 cul a ted 

Intersection volume length calculated 

Photomultiplier pinhole diameter (mm) 

Transform constant (MHz/ms-l) 

2 at l/e intensity (mn) 

at l/e2 intensity (mm) 

300 

5 .go 

3 s.lm 

52 

0.158 

1.500 

0.33 

0.324 

200 

9.3O 

2 !Jm 

86 

0.167 

1.357 

0.50 

0.510 

The Dopplersignals were detected by a photomul tiplier tube (E.M.I. 

9658B) and demodulated by a frequency tracker (Cambridge Consultants CCOl). 

The voltage analdgue of the Doppler frequency was averaged to give a mean and 

root-mean-square voltage, as indicated in figure 4. For the turbulent flow 

measurements, the flow was seeded (see reference 11) with minute quantities 

of milk to increase the scattering particle concentration and thereby increase 

the particle arrival rate. 

The local streanwise (U) and gapwise (V) components of velocity, and 

for turbulent flow the fluctuating components 6, v' and cross-correlation z, 

were obtained by making measurements of the Doppler frequency in three 

directions (Oo,+ 45O) relative to the local streamwise direction at each 

measurement point. The method of data reduction involves resolving these 

measured frequencies into streamwise and gapwise components as described in 

reference 9. The third component of velocity, W, and the associated correlations 

Q and UW were measured in a similar manner, with the three measurements being made in the X- e plane. 

Wall pressure measurements were obtained by means of pressure tappings 

let into the walls of the duct. Pressure measurements were made with a watermicromanometer. 

of water and were too small for reliable measurement. 

In laminar flow the pressure differences amounted to 20 pm 

2.3 Accuracy and precision 

error) and precision (random error) associated with each measurement presented 

in section 3. 

The following table sumarises the estimates of accuracy (systematic 

No corrections have been applied to take account of finite 

4 
transit time, mean velocity gradient and instrument noise broadening; reference 11. 

been included in the systematic errors of the table. 

23 and 3% errors associated with non-dimensional mean velocities; 

systematic errors away from the region of high velocity gradients are less 

than 1%. 

see 

Only velocity gradients have a significant effect and this has 

This accounts for the 

the 

Measurement errors 

Quantity Systematic error Random error 

xH 

e 

9 

Vc (laminar) 

V, (turbulent) 

r* , z* 

u/vc 

V/Vc 

U/Vc 

v/v, 

Wc 

w/v 

1. 

- 

k/Vc 2 

~~ 

k0.5 mn 

50.2 mm 

2 0.3O 

5 0.05' 

2 0.8% 

5 1/2% 

up to 23% 

up to 3% 

up to 3% 

up to 3% 

up to 3% 

up to 3% 

20.02 mn 

20.02 mn 

t 0.17O 

ni 1 

f 0.8% 

5 4% 

5 13% 

2 14% 

5 14% 

21 to? 3% 

213% t025X 

511% to +5% 

224% to iaaz 

524% to 284% 

5 8% 

5 
3. RESULTS 

3.1 Laminar F1 ow’ Results 

The laminar nature of the flaw was confinned by extensive visualisation 

Still and cine photography recordings were made. The 

using hydrogen bubbles preduced nn 0.375 mn Nschrome wires inserted through 

the pressure tappings. photograph presented as figure 5( i) shows the secondary velocity convecting 

fluid from the pressure (outer) wall along the sidewall to the suction (inner) 

wall, for Re = 1275. Flow visualisation did not reveal a region of longitudinal 

recirculation on the pressure wall as was observed in reference 7. Fig. 5(ii) 

shows the flow at the exit of the bend. The streak1 ines indicate large secondary motion near the suction surface and its absence along the mean radius of 

the bend. Near the pressure surface the bubbles are convected by the secondary 

motion from the symnetry plane to the sidewall. The velocity profiles presented 

in the following paragraphs quantify the magnitude and extent of this pattern. 

Preliminary measurements were obtained to evaluate the symnetry of the 

flow and to determine the extent to which the bend influenced the flow upstream. 

It was found that measurements at equivalent locations on either side of the 

symnetry plane of the duct agreed within the precision of the measurement. 

There were no discernable trends and the flow may be regarded as symmetric. 

Profiles of streamwise velocity at XH = -0.25 and -0.50 showed that the 

influence of the bend was detectable at the two stations closest to the plane of the bend entrance but was very small and is unlikely to be present at 

XH = -0.75. 

The development of the streamwise velocity is depicted in figure 6 in 

a form comparable to that of reference 7. 

defined at 95% of the maximum velocity, are a quarter of the hydrau Y ic 

diameter (40 mm). The same results are presented as isotachs of streamwise 

veloci ty in figure 7 for ease of presentation and subsequent discussion. The 

contours have been drawn using 1 inear interpolation between the measurement 

points; no extrapolation outside the measured grid has been carried out. 

isotachs in figure 7 (i) suggest that the bend has little effect on the flow 

upstream of the inlet and that as the fluid moves through the bend, 

figures 7(ii)-(v) , high velocities are found near the pressure surfaces and the 

sideyalls with a corresponding low velocity region adjacent to the suction 

surface. The development of the streamwise velocity is influenced by the 

streamwise pressure gradients. form can be expected to be similar to that calculated in reference 7, which 

showed that the gradient is strongly adverse over the suction surface for about 

450 while it is strongly favourable over the pressure surface for the same 

region. 

of high streamwise velocity fluid from the pressure surface along the sidewalls. 

Figure 8 shows profiles of the gapwise component of secondary velocity, with 

figure 9 displaying the same information as isotachs. 

The boundary layers at X = -0.25, 

The 

Although wall pressures were not measured, the 

The generation of streamwise vorticity is responsible for the convection 

At XH = -0.25 

6 
(figure 8( i)) , the velocity is small (0.05Vc) but directed towards the 

suction surface of the straight duct. This flow is evidence of a gapwise 

pressure gradient upstream of the bend. The profiles in the bend confirm 

that the fluid near the symnetry plane moves way from the siietion siirface and 

that fluid near the sidewalls moves towards the suction surface. This results 

in a secondary flow of the "first kind" (reference 12), which is a strong , 

helical motion near the suction surface. The largest velocities measured 

occur at e = 600 with values of the order of 0.6Vc, although the velocity 

near the symnetry plane and suction surface has decayed and is close to zero. 

This feature is seen to continue at e = 77.50 and XH = 0.25 and is 

associated with the accumulation of low streamvise velocity f l u i d a t the 

suction surface. 

Downstream of the bend, figure 7 (vi) shows that the streamwise velocity 

has decelerated on the pressure side of the duct. Figure 8 (vi) presents. 

the corresponding profile of gapwise velocity which has decreased greatly 

from the exit plane of the bend and has also reversed direction close to 

the pressure surface. 

was measured and is presented in figure 10. 

of two counter-rotating vortices which is in agreement with the predictions 

of reference 7,al though the predicted values are generally larger over the 

cross-section. It is associated with the rapid deceleration of strearwise 

velocity near the pressure surface causing a positive gapwise velocity near the 

syme try pl ane . 

in Appendix 1 as Table I. 

For this station, the spanwise (W) component of velocity 

The secondary flow thus consists 

The results from which the graphs have been plotted are tabulated 

3.2 Turbulent flow results 

As for the laminar flow, symetry tests were carried out with results 

similar to those for the laminar flow. The influence of the bend on the flow 

upstream was again examined and found to be more extensive than for the laminar 

flow, being present at XH = -0.50 and -0.75. 

The development of the streamwise component of velocity is shown in 

figure 11. 

the hydraulic diameter) than those of reference 8 at the inlet to the bend. 

Figure 12 shows the corresponding isotachs which, in contrast to laminar flow, 

are displaced to the suction surface of the duct, upstream of the bend. 

Measurements showed that this displacement was negligible at XH = -0.75. The 

fluid near the suction surface continues to accelerate at 0 = 30° and by 

8 = 600 the influence of the secondary motion on the contour shape is evident 

and continues to the exit of the bend, with the position of maximum velocity 

migrating from r* -0.7 to r* = 0.4. 

As for the laminar flow, the boundary layers are thinner (1 5% of 

7 
The profiles of gapwise velocity are presented in figure 13, with the 

corresponding contours in figure 14. As expected at XH = -0.25, the velocity 

is directed towards the suction surface, for the same reasons as in the laminar 

flow. The streamwise vorticity that is generated within the bend develops 

more slowly than for the laminar flow, while the positive velocities occur in 

the region very close to the sidewalls, corresponding to the thin boundary 

layers. The maximum velocities are again found at 8 = 600, and are about 

0.4Vc. In common with the laminar flow, the velocities on the symnetry plane 

near the suction surface are small, because of the small streamwise velocity 

in this region. 

' 

The wall pressures are presented in figure 15 (i) asp(8) with r* as 

Initially an adverse pressure gradient develops 

Fig. 15 (ii) 

parameter and in figure 15 (ii) as p(r*) with 0 as parameter. 

radial pressure gradient at the inlet plane confirms the influence of the 

bend on the upstream duct. 

on the pressure surface,and a favourable gradient on the suction surface, 

causing the observed deceleration and acceleration of the fluid near the 

respective surfaces. 

shows that as, expected, the largest values of gapwise pressure gradient are 

associated with the large gapwise velocities at 0 = 600 an$ 0 = 77.50. 

The pressure loss due to the bend is of the order of 0.1 pVc, which is 

comparable to the value reported in reference 1 for a similar configuration. 

The large 

These trends are reversed after 0=500. 

The development downstream of the bend is shown in figures 13 (vi) and 

14 (vi). 

pressure wall while fluid near the suction wall has accelerated. 

velocity has decayed less than was the case for the laminar flow, and only 

one sense of circulation is evident at XH = 2.50. 

velocity, W,is presented in figure 16 with the positive velocity direction 

The position of maximum streamwise velocity lies closer to the 

The gapwise 

The third component of 

, defined as for laminar flow. 

Figure 17 gives contours of the Reynolds stress E. Upstream of the 

bend 

At successive streamwise planes, large values of 

surface and near the sidewalls indicating that momentum transport is large 

in this region coinciding with large streamwise-velocity gradients. The 

change in the sign of UV is closely associated with changes in the sign of 

aU/ar*, at least for the region z* L. 0.8. 

the shear str.ess is generally small except for 8 = 600. 

Initially, high values are to be found only within the boundary layers. The 

influence of the secondary motion can be detected at 8 = 60' and, at the exit, 

-. the turbulence is anisotropic with high J near the pressure surface and high 

u near the suction surface. At XH = 2.50 the behaviour of UV is complex with 

a large area on the suction surface being occupied by positive values of - uv 

and near zero values on the pressure side. Momentum transfer in the span 

(z*) direction is accomplished through G, which is plotted in figure 20. Values 

are large near z* = 0.5, as could be expected, because of the region of large 

the values are low, corresponding to the small boundary layer thicknesses. 

are found near the pressure 

On the suction side of the bend, 

Figures 18 and 19 show the corresponding development of 3 and ?. 

- 

8 
streamwise-velo ity gradients present. The maximum values of and are 

and V also show large changes, with maximum values of the order of 0.11 Vc. 

Figure 21 shows contours of the w" component of fluctuating velocity. The 

contours of turbulent kinetic energy were constructed from G, T and i? 

measurements and are presented in figure 22. 

The results from which the figures have been plotted are tabulated in 

Appendix 1 as Table I1 for the velocity measurements and Table I11 for the 

wall pressure measurements. 

similar (0.003Vc) 5 although present in different regions. The contours of il 

4. DISCUSSION 

Comparison of the laminar and turbulent results shows that the influence 

of the bend results in a greater asymnetry about the mean radius in the 

upstream supply duct for turbulent flow and is associated with the favourable 

pressure gradient on the suction side. In contrast, the laminar flow with 

thicker boundary layers is found to develop larger gapwise velocities more 

rapidly, and over a larger region, which counteract the effect of the pressure 

gradient and result in smaller asymmetry at inlet. This mechanism is also 

evident in the fully-developed turbulent flow of reference 8 where the inlet 

asymmetry is smaller than in the present turbulent flow. 

Within the bend the locii of the maximum streamwise velocities in 

figures 7 and 12 are quite different, as a consequence of the secondary flow 

development. 

a greater pressure loss in the bend, e.g. 0.4~ Vh calculated in reference 7, 

as compared with O.lpVc for the present turbulent case. 

assessed by reference to references 7 and 8. In the case of laminar flow, 

the maximum streamwise velocity at e= 600 is larger and the deceleration of 

fluid near the suction surface at the exit is more extensive for fullydeveloped boundary layers at the inlet to the bend. 

probably reflect a more extensive streamwise vorticity wtth the thicker 

inlet boundary layer, although the locii of maximum streamwise velocity, 

for example, are similar. 

boundary layers, the profiles at the inlet are affected by secondary flows 

of the "second kind" driven by noma1 stresses which, in turn, 

influence the development of flow in the bend. 

The larger secondary flow in the 1 minar case is associated with 

The influence of the boundary layer thickness at the inlet can be 

These differences 

In the case of turbulent flow with fully-developed 

There is no correspond9 
ing effect in the present flow. The presence of an extensive irrotational 

flow at the inlet in the present data resuqts in the first 30° behaving in a 

manner consistent with a potential irrotational flow with the streamwise velocity 

maximum occurring close to the suction surface and there is little evidence 

of the effect of secondary flow until 0 = 600. This is in contrast to reference 

8, where the effects of secondary flow are evident upon entry to the bend. 

Beyond this station, both reference 8 and the present results show qualitatively similar behaviour with the locii of maximum streamwise velocity occurring near 

the mean radius of the bend. 

of the streamwise isotachs suggest differences in the streamwise vorticity. Thus, in the exit plane of the present bend, there are higher negative gapwise velocities (-0.3 V as opposed to -0.15 Vc) and smaller velocities on the 

symetry plane (0.2 Vc as opposed to 0.28 Vc). For the fluctuating velocities - 

there is qualitative agreement between the two flows; for example, - uv is 

large near the suction surface and,at the exit of the bend, 3 is large near 

the suction surface while ij is large near the pressure surface. Quantitive 

agreement is not gound and is not expected. is up to 0.0013 Vc in the present flow as compared to 0.004 V$ in reference 8. 

At exit, the respective values are 0.003 V$ and 0.007 V:. The influence of 

streamline curvature is expected to increase turbulence levels near the suction 

- surface and to decrease them neak the pressure surface and the behaviour of 

uv and ? support this expectation although the behaviour of 6 is anomalous. No 

doubt convection of fluid away from the suction surface is also influential 

in determining the fluctuation levels in this region. 

The quantitative differences in the behaviour 

For example, at the inlet UV 

5. CONCLUDING REMARKS 

1. This experimental programme bar reported in detail on the velocity and 

pressure distribution in a square duct,of strong curvature with thin inlet 

boundary layers for laminar and turbulent flows. 

2. Secondary flows of the first kind are present in both flows and are more 

pronounced in the laminar flow partly because of the thicker boundary 

layers. 

turbulent case results in greater asymmetries about the mean radius. 

Secondary velocities of up to 0.6 and 0.4 of the bulk velocity are found 

in the laminar and turbulent regimes respectively. 

The lower magnitude of secondary flows at inlet to the bend in the 

3. The position of maximum streamwise velocity migrates,in laminar flow,from 

the mean radius of the bend towards the pressure surface. 

flow, the migration is initially towards the suction surface and lies at the 

mean radius at the exit. 

potential irrotational flow in the entry region. 

In turbulent 

The turbulent flow behaviour is more akin to a 

10 
4. The pressure loss across the bend is about one fifth of the velocity head 

in the turbulent case. 

5. Comparisons have been made with laminar and turbulent flows with fullydeveloped boundary layers at the inlet to the duct. 

the differences were small and in the turbulent flow the influence of 

streamwise vorticity is apparent much later with the thin inlet boundary layers. 

similar although, as expected, the levels are not comparable. 

This report presents benchmark measurements against which the results 

of numerical calculation methods can be compared. 

For the laminar flow, 

The trends in the fluctuating velocities and shear stress are 

6. 

11 
APPENDIX 1 

TABULATED DATA 

TABLE I. 90 DEGREE BEND OF 2.3 RADIUS RATIO AND SQUARE CROSS SECTION 

LAtVINAR FLOW: REYNOLDS NUMBER 790 VELOCITY MEASUREMENTS 

(All velocities normalised 

S treamwise 

station 

xH* 

-0.50 

Spanwis e 

lo cat ion ** 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

0.9 

0.6 

Gapwi se 

locatiap 

r-1: 

r -r io 

0.1 

** * 

0 

0.5 

0.9 

q the bulk velocity V_= 0.0198 m/s) 

S tr e ami se 

ve lo c i ty 

U/Vc 

0.835 

0.823 

0.826 

0.814 

0.782 

0.736 

0.673 

0.543 

0.378 

0.208 

1.641 

1.641 

1.638 

1.635 

1.607 

1.575 

1.472 

1.280 

0.969 

0.512 

0.891 

0.866 

0.869 

0.826 

0.802 

0.761 

0.698 

0.667 

0.512 

0.279 

Gapwise 

lo cation 

r-r- ** 

ri-ro 

0.3 

0.7 

S t re amw is e 

velocity 

u/vc 

1.595 

1.594 

1.584 

1.547 

1.513 

1.464 

1.289 

1.087 

0.782 

0.363 

1.632 

1.610 

1.601 

1.590 

1.550 

1.491 

1.392 

1.174 

0.888 

0.356 

* 

** 

*** Distance from outer wall normalised by the duct gap 

Number of hydraulic diameters upstream of bend entry plane 

Distance from plane of symmetry normal’ised by the duct half-span 

12 
TABLE I. (Cont.. .) 

0.9 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

-0.25 

0.5 

0.9 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.1 

i treamwi se 

velocity 

u/vc 

0.876 

0.880 

0.863 

0.869 

0.841 

0.795 

0.729 

0.615 

0.372 

0.197 

1.693 

1.691 

1.685 

1.675 

1.645 

1.565 

1.403 

1.143 

0.815 

0.651 

0.953 

0.929 

0.945 

0.911 

0 - 887 

0.833 

0.792 

0.693 

0.542 

0.233 

sapwise 

re loci t y 

V/Vc 

0.032 

0.032 

0.019 

0.017 

0.023 

0.021 

0.017 

0.021 

- 

- 

0.053 

0.052 

0 .os9 

0.064 

0.072 

0.088 

0.080 

0.060 

0.076 

0.062 

-0.004 

0.002 

0.014 

0.006 

0.004 

0.016 

0.002 

0.008 

-0.015 

-0.007 

apwise 

1 oca t ior 

r-r - 

r -r io 

0.3 

0 

0.7 

I treamwi se 

velocity 

u/vc 

1,606 

1.616 

1.604 

1.579 

1.537 

1.446 

1.315 

1.115 

0.813 

0.633 

1.631 

1.640 

1.632 

1.618 

1.586 

1.522 

1.413 

1.221 

0.922 

0 363 

;apwise 

re loci t y 

v/vc 

0.045 

0.046 

0 -052 

0.052 

0 -049 

0.051 

0.056 

0.082 

0.066 

0 -080 

0.058 

0.045 

0 -037 

0.040 

0.065 

0.072 

0.059 

0 -065 

0 -050 

0.060 

13 
TABLE I 

treamwis 

station 

e* 

(deg.) 

30 

:Cant. . .I 

Spanwise 

location 

z/z+ 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

3.7 

D. 8 

3.9 

3.0 

3.1 

3.2 

3.3 

3.4 

1.5 

1.6 

1. 7 

1.8 

1.9 

Gapwise 

locatio1 

r-r 

r -r io 

0.1 

0 

0.5 

0.9 

Streamwise I Gapwise I Gapwise1 StreamwiJe Gapwist 

vel0 ci t 

V/Vc 

1.04& 

1.067 

1.085 

1.078 

1.052 

1.021 

0.916 

0.522 

0.216 

1.708 

1.718 

1.696 

1.676 

1.632 

1.562 

1.450 

1.232 

0.851 

0.469 

0.722 

0.707 

0 -698 

0.664 

0.667 

0.742 

0.790 

0.729 

0.562 

0.344 

0.786 

ve loc i t 

v/vc 

-0.116 

-0.114 

-0.122 

-0.107 

-0.09 3 

-0.072 

-0.028 

0 .OS2 

0.161 

0.028 

-0.410 

-0.408 

-0.390 

-0.371 

-0.331 

-0.261 

-0.153 

0.059 

0.357 

0.276 

-0.098 

-0.096 

-0.105 

-0.127 

-0.122 

-0.114 

-0.039 

0.092 

0.180 

0.160 

locatic 

r-r 

0 

r -r io 

0.3 

0.7 

velocit velocitj 

u/vc 1 v/vc 

1.624 

1'.577 

1.564 

1.558 

1.558 

1.529 

1.437 

1.248 

0.863 

0.452 

1.558 

1.552 

1.540 

1.518 

1.481 

1.419 

1.300 

1.109 

0.766 

-0.292 

-0.294 

-0.303 

-0.309 

-0.301 

-0.276 

-0.197 

-0.034 

0.261 

0.219 

-0.367 

-0.369 

-0.358 

-0.349 

-0.325 

-0.268 

-0.127 

0.107 

0.397 

Angular position from bend entry plane 

14 
TABLE I 

treamwisc 

station 

(deg. 1 

e 

60 

Cont.. .I 

Spanwi se 

location 

Z/Z$ 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

0 -9 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.6 

;apwise 

.ocation 

T-r 

0 

ri-r 

0 

0.1 

0.5 

0.9 

ltreamwi se 

velocity 

U/Vc 

1.439 

1.434 

1.439 

1.441 

1.419 

1.399 

1.351 

1.087 

0 -678 

1.173 

1.177 

1.155 

1.123 

1.183 

1.240 

1.344 

1 394 

1.338 

0.926 

0.332 

0.362 

0 -469 

0.618 

0.712 

0.723 

0.708 

0.530 

0.301 

1.287 

0.748 

Gapwise 

velocity 

v/vc 

-0.054 

-0.068 

-0.068 

-0.070 

-0.065 

-0.063 

-0.032 

-0.004 

0 -072 

0.129 

-0.259 

-0 .,25 4 

-0.236 

-0.218 

-0.186 

-0.194 

-0.123 

0.046 

0.153 

0.505 

-0.025 

-0.054 

-0.128 

-0.176 

-0.148 

-0 -082 

0.074 

0.203 

0.199 

0.084 

apwise 

Locatio1 

r-r 

r -r io 

0.3 

0 

0.7 

~ 

Streanwis 

velocity 

v/vc 

1.588 

1.579 

1.588 

1.569 

1.536 

1.551 

1.542 

1.491 

1.399 

1.021 

0.580 

0.571 

0.648 

0.773 

0.926 

1.038 

1.129 

1.181 

1.236 

0.929 

~ ~~ - 

Gapwi se 

relocity 

-0.287 

-0.265 

-0.248 

-0.232 

-0.182 

-0.15 3 

-0,101 

0 -020 

0.195 

0.296 

-0.211 

-0.231 

4.308 

-0.293 

-0.224 

-0.117 

0.021 

0.217 

0.541 

0.617 

15 
TABLE I 

Streamwis 

station 

(deg .I 

77.5 

8 

L 

:Cant.. .I 

Spanwisc 

locat io1 

z/z* 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

Gapwise 

locatio 

i-r 0 

r -r 

0.1 

io 

0.5 

0.9 

Streamwis 

velocity 

u/vc 

1.553 

1.555 

1.542 

1.549 

1.533 

1.512 

1.468 

1.390 

1.210 

0.802 

0.735 

0.745 

0.782 

0.823 

0.955 

1.087 

1.281 

1.507 

1.463 

0.986 

0 I 369 

0.326 

0.294 

0.505 

0.920 

1.169 

1.245 

1.232 

1.028 

0.503 

Gapw i se 

velociti 

V/VC 

-0.041 

-0.043 

-0.057 

-0 .OS4 

-0 -048 

-0.039 

-0.026 

-0.015 

0.046 

0.133 

-0.072 

-0.063 

-0.096 

-0.098 

-0.142 

-0.164 

-0.191 

-0.162 

0.153 

0.322 - 

-0.009 

-0.024 

-0.092 

-0.215 

-0.248 

-0.105 

0.068 

0.204 

0.289 

0.160 

;apwise 

to ca tior 

r -r 

r -r io 

0.3 

0 

0.7 

Streamwis 

velocity Q/TJ 

C‘ 

- 

1.533 

1.542 

1.545 

1.549 

1.571 

1.595 

1.632 

1.652 

1.565 

1.095 

0.562 

0 -614 

0.685 

0.760 

0.921 

0.951 

0.944 

1.044 

1.300 

0.845 

Gapwise 

velocit] 

“/VC 

-0.136 

-0.136 

-0.133 

-0.129 

-0.133 

-0.127 

-0.107 

-0.030 

0.131 

0.354 

-0.052 

-0.114 

-0.224 

-0.255 

-0.209 

-0.024 

0.127 

0.176 

0.237 

0.358 

16 
I 

5 treamwise 

station 

* 

xH 

tO.25 

Spanwise 

location 

z/z+ 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0 -0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

zapwise 

Lpcation 

r-r 

r.-r 

0.1 

0 

10 

0.5 

0.9 

; t r eamw i se 

velocity 

u/vc 

1.711 

1.712 

1.711 

1.716 

1.718 

1.716 

1.694 

1.661 

1.523 

1.093 

0.772 

0.773 

0.762 

0.761 

0.790 

0.852 

1.037 

1.300 

1.278 

0.831 

0.311 

0.331 

0.484 

0.782 

1.034 

1.126 

1.148 

1.078 

0.780 

0.382 

Gapwise 

velocity 

V/Vc 

-0.028 

-0.029 

-0 -028 

-0 -028 

-0.039 

-0.041 

-0.032 

-0.008 

0.028 

0.130 

0.014 

0 -000 

-0.024 

-0 -058 

-0.061 

-0.052 

-0.074 

-0.092 

0.087 

0.185 

-0.017 

-0.065 

-0.113 

-0.151 

-0.107 

-0.031 

0.028 

0 -085 

0.094 

0.070 

;apw i se 

.ocation 

r-r 

r.-r 

0.3 

0 

10 

0.7 

treamwise 

velocity 

u/vc 

1.471 

1.476 

1.487 

1.501 

1.511 

1.544 

1.621 

1.693 

1.650 

1.156 

0.550 

0.539 

0.681 

0.943 

1.061 

1.032 

0.969 

1.083 

1.131 

0.687 

*Number of hydraulic diameters downstream of bend exit plane. 

Gapwi se 

velocity 

V/Vc 

-0.053 

-0.082 

-0.076 

-0.089 

-0.114 

-0.138 

-0.154 

-0.082 

0.081 

0.224 

-0.016 

-0 .OS5 

-0.171 

-0.251 

-0.191 

-0.013 

0.112 

0.122 

0.142 

0.141 

' 17 
station 

xH 

a.40 

Spanwisc 

locatior 

z/z+ 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

' 0.6 

0.7 

0.8 

0.9 

Gapwise 

locatior 

r-r 

r -r 

0.1 

0 

io 

0.5 

0.9 

Streamwisc 

velocity 

u/vc 

1.704 

1.706 

1.706 

1.713 

1.707 

1.709 

1.698 

1.698 

1.580 

1 .lo9 

0.794 

0.754 

0.736 

0.733 

0.767 

0.811 

0.990 

I. 308 

1.345 

0.674 

0.272 

0.279 

0.379 

0.624 

1.049 

1.096 

1.096 

0.770 

0.365 

0.882 

lapwi se 

.ocatior 

r-r 

r -r io 

0.3 

0 

0.7 

~- 

t-seamwise 

velocity 

u/vc 

1.497 

1.523 

1.520 

1.513 

1.523 

1.547 

1.587 

1.669 

1.656 

1.131 

0.519 

0.577 

0.782 

0.984 

1.062 

1.025 

0.984 

1.124 

1.119 

0 - 588 
; treamwise 

station 

5-l 

--~ -__- ~ 

+2.50 

~~ 

Spanwise 

locat ion 

Z/Z$* 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

~~ 

Gapwise 

location 

r-r . 

r -r io 

0.1 

0 

0.3 

~~ 

Streamwise 

velocity 

u/vc 

1 - 393 

1.393 

1.403 

1.403 

1.403 

1.403 

1.383 

1.334 

1.166 

0.790 

1.512 

1.512 

1.512 

1.502 

1.482 

1.423 

1.482 

1.581 

1.581 

1.186 

~ 

Gapwise 

velocity 

2 

v/vc 

x10 

0 .Ooo 

0 .Ooo 

O.OO0 

0 -000 

1.397 

1.397 

1.397 

1.397 

1.397 

1.397 

4.193 

4.193 

2.795 

0 .om 

-4 -193 

-4.892 

-4.193 

1.397 

4.193 

0 .ooo 

Spanwise 

veiocity 

w/vc 

2 x10 

0 .ooo 

0 .ooo 

'O.Oo0. 

1.397 

1.397 

2.795 

2 -069 

1.397 

0.000 

0.000 

- 

0 -000 

0 .om 

0.000 

0 .ooo 

0 .Ooo 

0.000 

0.000 

1.398 

1.398 

2.736 

19 
> 

~ ~~~ 

Spanwise 

location 

Z/Z$ 

Streamwist 

station 

33 

+2.50 

20 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.85 

0.9 

0 .o 

0.1 

0.3 

0.4 

0.5 

0.6 

0.7 

0.80 

0.85 

0.90 

0.3 

Gapwise 

location 

r-r 

r -r 

0 

io 

0.5 

0.7 

0.9 

Streamwise 

ve loci ty 

u/vc 

1.091 

1.043 

0.984 

0.931 

0.919 

0 -946 

1.045 

1.170 

1.027 

0.575 

0.812 

0.871 

0.992 

1.075 

1.055 

3.034 

1.029 

1.023 

0.819 

0.512 

- 

0.482 

0.543 

0.662 

0.763 

0.816 

0.820 

0.765 

0.652 

0.470 

0.251 

- 

Gapwise 

velocity 

v/vc 

x102 

0.139 

-0.139 

-0.978 

-3 -075 

-2.796 

-3 -215 

-1 -677 

1.677 

1.957 

0.838 

-11.743 

-12 -023 

-10.765 

-8.668 

-4.194 

1.258 

5.732 

8.108 

6.291 

2.376 

- 

-2.516 

-2.655 

-3.075 

-2.655 

-1.258 

0.559 

1.817 

1.957 

0.838 

- 

0.276 

Spanwise 

veloc it! 

WbVc 

x102 

0.699 

3.709 

5 -730 

6.710 

6.570 

5.172 

3.355 

1.118 

0.139 

0.559 

-0.414 

-2.935 

-2.376 

-2,239 

-0.978 

-1.398 

-1.817 

-2.234 

-1.537 

-0.8 30 

- , 

0.265 

-0.419 

-2.656 

-5.732 

-7.269 

-6.990 

-6.291 

-3.215 

-1.398 

-0.559 

- 

__ - _. .- 
Table 11. 90 DEGREE BEND OF 2.3 RADIUS RATIO AND SQUARE CROSS-SECTION 

TURBULGm FLOW: MEAN VELOCITY,TURBULENCE LEVEL AND CROSSCD-Ei?E’iATiON MZASlXEmS, my~orns NIJ’MB~ 40 - 

(All quantities normalised by bulk velocity V-= 1.002 4s) 

2.762 

2.762 

treamwisc 

station 

xH* 

-0.25 4.086 

4.200 

Spanwise 

location 

z/z3** 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0 -6 

0.7 

0.8 

0.9 

0.915 

0.930 

0.950 

0 -0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0,915 

0.930 

0.950 

Gapwise 

Location 

r-r *** 

0 

r -r io 

0.1 

0.3 

I x102 

0.931 

0.929 

0.933 

0.929 

0 -914 

0.914 

0.910 

0.906 

0.847 

0.691 

0.632 

0 -535 

0.496 

1.085 

1,082 

1.082 

1.08 2 

1.082 

1.082 

1.070 

1.007 

0.902 

0.738 

0.695’ 

0.625 

0.515 

6.546 

6.485 

6.423 

6 -546 

6 -794 

6.794 

6.794 

6.794 

6.917 

9.511 

9.765 

11.328 

12.109 

0.781 

0.781 

0.781 

0.781 

0.898 

1.132 

2.773 

5.558 

7.535 

3.8~2 

10.499 

11.328 

12.500 

Gapwise ~ Gapwise 

21 2 x10 I x10 

4,143 

4.143 

4 -143 

4.419 

4.143 

4.419 

3.314 

2.009 

2.009 

3 . 599 

4.916 

5.975 

6.047 

4.538 

5 .6o8 

- 

kass cor 

:elatiop 

- uv/vz 

x1~3 

1.181 

1.181 

1 I102 

1.128 

1.128 

1.208 

1.290 

-0.292 

-0.280 

-0.297 

-0 -640 

o -918 

- 

0.014 

0 . 008 

0.207 

0.003 

0.000 

0.009 

0 .om 

0.153 

0.099 

-0 ;14 2 

-1.716 

0.088 

- 

* Number of hydraulic diameters upstream of bend entry plane 

** Distance from plane of symmetry narmalised by the duct half-span 

*** Distance from outer wall normalised by the duct gap. 

21 
TABLE 11. (Cont ... . 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.915 

0.930 

0.950 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.915 

0.930 

0.950 ' 

0-0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.915 

0.930 

0.950 

Streamwist 

- cta -- t -- i on 

xH 

-0.25 0.5 

0.7 

0.9 

~ 

Spnwise Gapwise 

location location 

r -I: 

r.-r 

0 

I 

z/z+ 

10 

1 

Streamwis 

man velo 

u 

C ,zI 

1.109 

1.105 

1 .lo5 

1.105 

1.105 

1.101 

1.074 

1.019 

0.914 

0.753 

0.718 

0.640 

0.542 

1.134 

1.128 

1.128 

1.128 

1.128 

1.128 

1.117 

1.070 

0.968 

0.812 

0.750 

0.667 

0.578 

1.035 

1.035 

1.040 

1.035 

1.031 

1.027 

1.027 

1.015 

0.933 

0.789 

0.734 

0.628 

0.558 

Streamwi s 

turb. levc 

G/Vc 

2 

x 10 

0.742 

0.742 

0.742 

0.742 

0.742 

1.230 

3.125 

5.126 

7 -164 

9.635 

10.438 

12.109 

13.281 

0.781 

0.781 

0.781 

0.820 

0.859 

1.250 

2.343 

4.690 

6.794 

8.955 

10.499 

12.402 

13.281 

5.682 

5.929 

5.929 

5.929 

6.546 

6.299 

5 -805 

5.805 

6.917 

9.264 

10.561 

12.109 

13.281 

Gapwise 

wan vel0 

v/vc 

2 

s 10 

4.695 

4.695 

4.695 

4.695 

4.975 

4.975 

4.975 

4.975 

4.975 

5.524 

4.419 

3.869 

3.033 

4.419 

4.419 

4.419 

4.143 

4.419 

4.419 

4.419 

4.419 

4.419 

4.419 

4.419 

3.314 

2.485 

3 -038 

3.038 

3.038 

3.038 

3.038 

3.038 

3.314 

3.590 

3.038 

0.552 

0.552 

0.276 

0.276 

Gapwise 

hwb. lev6 

G/Vc 

2 

x10 

0.385 

0.536 

0.591 

0.810 

0.893 

1.312 

2.854 

4.456 

5 "018 

6.214 

6.413 

6.861 

0.896 

0.930 

0.930 

1.009 

1.015 

1.005 

2.068 

3.211 

5.079 

6.044 

6.302 

3.894 

3.900 

3 .e37 

3.713 

3.837 

3.960 

4.232 

4.262 

4.332 

5.229 

5.963 

6.148 

4.690 

- 

- e--. __ 

I 

cross cor? 

relation 

- 

- uvh2 

C 

3 XlO 

-0.002 

-0.002 

-0.007 

-0.002 

-0.040 

-0.015 

0.046 

0.293 

0.305 

0,098 

0 .lo2 

-1.305 

-0.006 

-0.003 

-0.003 

-0.036 

-0.014 

-0.008 

-0 -008 

-0.148 

-0.222 

-0.037 

-0.903 

-0.701 

-0.995 

-1 -024 

-1.071 

-1.281 

-1.281 

-1.409 

-1.024 

-0.803 

-0.528 

-0.476 

-1.055 

- 

- 

- 

- 

22 
it re mist 

station 

6 

(deg .I 

* 

Gapuise 

30 

Spanwi se Cross cor 

location 

Z/Z$ 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.915 

0.930 

0 -950 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.0 

0.1 

0 -2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

;apwise 

Loca tior 

r-r 

r -r 

0 

io 

0.1 

0.3 

0.5 

Streamwise 

velocity - 

u/vc 

0.789 

0.789 

0.789 

0.785 

0.777 

0.757 

0.750 

0.746 

0.710 

0.660 

0.609 

0.546 

0.480 

1.062 

1.058 

1.058 

1.058 

1.058 

1.062 

1.055 

1.030 

0.963 

0.856 

1.142 

1.148 

1.148 

1.148 

1.148 

1.152 

1.142 

1.109 

1.022 

0.871 

Streamiss 

curbuence 

Ir level 

u/vc 

X102. 

8.984 

8.882 

8.984 

9.375 

9.635 

10.499 

10.546 

10.623 

9.765 

9.388 

8.984 

9.765 

11 -328 

1.210 

1.457 

1.484 

1.523 

1.562 

1.601 

1.992 

3.495 

5.682 

8.245 

0.703 

0.781 

0.820 

0.859 

0.976 

1.09 3 

2.265 

4.076 

6.917 

10.005 

*Angular position from bend entry plane 

Gapwis e 

ve lo c i ti 

o/v- b 

2 

XI0 

-3.314 

-4.419 

-4 -971 

-4.143 

-2.485 

-1 -657 

-1.381 

-1.381 

-0.276 

1.657 

4.971 

6.352 

7.181 

-5 - 800 

-5 -690 

-5.800 

-6 -076 

-6.284 

-6 -905 

-7.098 

-5.521 

0.550 

12.291 

-8.015 

-8.015 

-8.563 

-8.832 

-9.391 

-9.667 

-9.501 

-6.465 

3.972 

14.250 

turbulen 

level 7r 

5 v/v 

x10 

7.572 

7.264 

6.665 

7.222 

7.585 

7.559 

7.480 

7.125 

6.622 

5 -931 

5 -449 

5.290 

1.560 

1.605 

1.610 

1.579 

1.450 

1.826 

2 -657 

3.656 

5.958 

7.449 

- 

0.605 

0.615 

0.610 

0.610 

0.629 

1.222 

2.591 

4.115 

5.956 

6.285 

B relatioi 

-W/Vc2 

3 x10 

4.696 

4.568 

4.421 

4.635 

4.6 35 

4.730 

5.371 

4.223 

3.543 

2.667 

2.307 

2.260 

0 -984 

0.048 

0.048 

0.050 

0.060 

0 -060 

0 .OS1 

0.085 

0.205 

0.496 

0.295 

-0.028 

-0.028 

-0.028 

-0.028 

-0.064 

-0.027 

0.061 

0.460 

2.024 

1.900 

23 
TABLE I1 (Cont.. .) 

0.8 

0.9 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

S t r e ami sl 

station 

9 

(deg) 

30 

0.9 

1 

Spanwise 

lo ca ti on 

z/zt 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

Gapwi se 

locatior 

r-r 

r.-r 

0 

10 

0.7 

5 t r e amw i s t 

ve loc it y 

u/vc 

1.257 

1.257 

1.257 

1.255 

1.253 

1.253 

1.231 

1.191 

1.121 

0.981 

1.190 

1.191 

1.200 

1 -200 

1.209 

1.209 

1.204 

1.178 

1.117 

1.018 

Streamwi s~ 

turbul en c~ 

level 

WVc 

2 x10 

1.093 

0.937 

0.976 

0.976 

1.054 

1.328 

2.594 

4.199 

5.929 

8.893 

4.199 

4.199 

3.952 

4 -076 

4 -484 

4.570 

4.694 

4.817 

6.423 

8.152 

Gapwi s( 

veloci 

V/Vc 

2 xl 0 

-8.838 

-8.6 75 

-8.563 

-8 -838 

-9.556 

-9.865 

-9.391 

-5.415 

2.563 

15.575 

-4.529 

-4 -529 

-4.772 

-5.800 

-6.623 

-7.345 

-7.927 

-5.800 

-1.653 

8.479 

level 

S/Vc 

x102 

0.495 

0.587 

0.579 

0.629 

0.881 

1.965 

2 -947 

4.515 

6.049 

7.315 

2.215 

2.419 

2.845 

2.419 

1.895 

2.353 

2 -622 

3.386 

3.584 

4.710 

- Tiv/Vf 

3 XlO 

0 -009 

0 -009 

0.009 

0.006 

0.014 

0 -016 

0.032 

0.320 

0.732 

1.389 

0.283 

0.329 

0.256 

0.329 

0.428 

0.354 

0.225 

0.403 

0.732 

1.132 

24 
rABLE 11. (Cont.. .I 

itreamwisc 

station 

0 

(deg. I 

60 

Spnw ise 

location 

Z/Zt 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.915 

0.930 

0.950 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.915 

0.930 

0.950 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0 -6 

0.7 

0.8 

0.9 

0.915 

0.930 

0.950 

Ga pw i s e 

Location 

r -r 

r -r 

0 

io 

0.1 

0.3 

0.5 

Xreanwis 

=an veloc 

3°C 

0.839 

0.839 

0.835 

0.832 

0.812 

0.800 

0.796 

0.792 

0.781 

0.742 

0.730 

0.671 

0.625 

1.068 

1.064 

1.064 

1.064 

1.071 

1.066 

1.063 

1.062 

1.036 

0.927 

0.832 

0.816 

0.785 

1.167 

1.167 

1.167 

1.167 

1.167 

1.167 

1.164 

1.154 

1.105 

1.007 

0.941 

0.914 

0.871 

treamwi sc 

urb. level 

G/Vc 

2 

x 10 

8.770 

8.646 

8.523 

8.646 

8.770 

8 -893 

9.264 

9.388 

9.264 

9.017 

9.017 

9.246 

9.511 

2.539 

2.737 

2.773 

2.773 

2.656 

2.964 

3.458 

3.582 

4.570 

8.593 

9.758 

10.252 

10.746 

0.859 

0.871 

0.898 

0.898 

1.093 

1.390 

2.420 

3.211 

5.435 

8.276 

8.276 

8.893 

LO. 252 

Gapwis e 

Ean veloc 

“Vc 

2 

x 10 

-3.590 

-4.695 

-5.248 

-5.248 

-5.248 

-3.314 

-1.795 

-1.519 

-0.828 

I.. 657 

2.209 

3.. 209 

5.248 

-5.109 

-5.109 

-4.695 

-4.907 

-4.613 

-4.613 

-4.613 

-4.221 

-0.825 

8.700 

13.810 

14.363 

18.230 

-11.062 

-10.705 

-10.705 

-10.106 

-9.805 

-9.942 

-8.427 

-4.883 

4.615 

19.770 

28,173 

32.040 

32.593 

Gapwise 

urb. level 

G/vc 

2 x10 

8.011 

8.268 

7.782 

7.675 

8.046 

9.147 

9.179 

9.073 

8.997 

8.566 

8.566 

8.275 

5.396 

5.347 

5.320 

5.339 

5.196 

5.233 

4.932 

4 .a53 

5.365 

6.571 

8.147 

7.860 

0.945 

0.945 

0.956 

1.125 

1.270 

1,400 

1.526 

3.786 

6.947 

7.912 

8.315 

8.542 

6.916 

- 

- 

ross correlation 2 - uv/v 

x10 

C 

3 

3.967 

3.967 

3.972 

3.972 

3.710 

3.972 

4.272 

4.272 

4.272 

3 -158 

2.777 

1.922 

1.538 

1.081 

1.07 3 

1.300 

1.175 

0.851 

0.306 

1.007 

1.020 

1.089 

2 -047 

3.946 

3.735 

3.618 

0.044 

0.044 

0.103 

0.000 

-0.023 

-0.023 

-0.011 

-0.034 

-0.076 

-0.396 

1.927 

2.651 

1.398 

25 
TABLE 11. (Cont.. . ) 

Gapwise 

location 

Strearnwis 

station 

(deg. 1 

e 

60 

Streamwis 

mean velo 

Spanwi s< 

1 oca t i o i 

Z/Z+ 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.915 

0.930 

0.950 

Cross0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.915 

0.930 

0.950 

- 

r-r 0 I u/vc 

r -r io 1 

0.7 1.222 

1.222 

1.226 

1.226 

1.230 

1.226 

1.230 

1.200 

1.180 

1.085 

0.968 

0.949 I 0.917 

0.9 I 1.053 

1.035 

0.962 

0.968 

1.040 

1.097 

1.13.8 

1.164 

1.164 

1.097 

0.949 

0.906 

0.847 

i/vc 

2 

x10 

2.121 

2.148 

2.226 

2.210 

2.460 

2.773 

3.891 

4.694 

6.546 

8.646 

9.017 

9.635 

3.231 

4.205 

7.089 

6.085 

5.313 

3.660 

4.362 

4.761 

8.357 

9.570 

10.376 

10.546 

12.109 

8.770 

- 

s lo 

-14.473 

-13.251 

-13.589 

' -14.599 

-15.907 

-15.213 

-10.604 

-2.095 

12.213 

32.915 

37.845 

41.432 

43.641 

-13 -905 

-14.512 

-21.075 

-15.625 

-9,667 

-9.110 

-4.782 

1.182 

12.805 

29.102 

36.183 

38,669 

38; 669 

Gapwise 

tub. leve 

G/Vc 

2 x10 

3.110 

3 .COO 

3.056 

2.948 

2.977 

2.795 

3.806 

6.205 

7.739 

8.698 

9.256 

8.766 

7.872 

3.579 

3.731 

4.052 

3.976 

3,886 

5.150 

6.203 

7.158 

8.509 

9.718 

8.493 

'7.631 

- 

relation 

- iiir/V2 

C 

3 xi0 

0.269 

.0.219 

0.219 

0 - 205 

0.134 

0 * 102 

0.300 

0 -635 

1 .G25 

2.081 

1,880 

1.300 

- 0.058 

-0.324 

-2.195 

-1.175 

-0.402 

-0.140 

0 -000 

0.230 

0.720 

0.750 

1.977 

0.373 

-3.417 

1.855 

26 
I .- 

;apwise 

ocation 

r-r 0 

r -r io 

0.1 

~~ 

0.3 

TABLE I1 (Cont. . . 

Streamwise 

velocity 

E/Vc 

0.871 

0.867 

0.867 

0.867 

0.867 

0.859 

0.859 

0.859 

0.835 

0.789 

0.777 

0.753 

0.730 

1.066 

1.066 

1.064 

1.054 

1.046 

1.030 

1.027 

1.023 

1.027 

0.972 

0.871 

0.835 

0.816 

I 

;treamwise 

station 

8 

(deg . ) 

Gapwise 

velocity 

V/Vc 

2 x10 

-2.485 

-2.209 

-2.485 

-2.485 

-2.762 

-3.038 

-3.038 

-0.828 

-0.276 

3.314 

3.590 

3.638 

77.5 

. 

Gapwise 

,level 

turbulen 

VIV= 

x102 

8.378 

8.683 

8.683 

9.091 

9.091 

8.445 

8.306 

7.963 

8.050 

8.090 

8.740 

8.590 

jpanwise 

Lo cation 

Z/Zq 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.915 

0.930 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.915 

0.930 

0.950 

0.35q - 8.540 

4.261 

4.336 

4.502 

4.762 

5.438 

6.716 

6 -903 

8.583 

8.936 

9.598 

9.798 

9.275 

- h 

S treamwise 

turbulence 

L level 

2 

U/Vc 

x10 

7 782 

7 -782 

7 -905 

7.782 

7.782 

8.029 

8 -029 

8.029 

8.029 

8.152 

7.905 

8.152 

8.770 

2 -506 

3.166 

3.875 

4.282 

5 -473 

5.747 

5.921 

6.172 

6.429 

8.125 

9.388 

lo .005 

10.252 

5.524 

-9.521 

-8.502 

-7.599 

-6.7 39 

-5.6 35 

-4 .804 

-4.972 

-3.310 

1.933 

9 -115 

12.153 

11.600 

11.877 

koss cor- 

' relatior 

- iiii/v 

3 

2 

C 

x10 

4 -035 

3.710 

3.710 

3.348 

3.348 

2.854 

2.854 

2.883 

2.114 

1.983 

1.635 

1.338 

0.854 

0.135 

0.336 

0.508 

1.165 

1.798 

2 -636 

3.187 

3 -605 

3.830 

3 -938 

3.710 

3 .OS5 

2.114 

27 
II 11ll 1111111 11111 I I I 111 Ill II 

1 evel 

TABLE 11 (Cont. . . 

C 

' v/v 

:de 4) 

77.5 

Gapwise 

locatio. 

r- r 

r: -r 

0 

LO 

0 .o 0.5 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.7 

0.9 

Xreamwir 

ve loc i ti 

WVc 

1.179 

1.180 

1.183 

1.183 

1.183 

1.185 

1.179 

1.167 

1.144 

1.078 

1.093 

1 .lo2 

1.121 

1.135 

1.147 

1.165 

1.187 

1.213 

1.204 

1.121 

0.661 

0.669 

0.792 

0.936 

1.007 

1.079 

1.139 

1.170 

1.164 

1.053 

Streanwise Gapwist 

turbulence ve locit 

1.042 

1 .OS4 

1.132 

1.171 

1.250 

1.367 

1.835 

2.968 

4.941 

7.411 

3.333 

3.210 

3.397 

3.520 

3.582 

3.952 

4 -696 

5 -929 

7.411 

8.399 

8.314 

L0.427 

L0.615 

7.605 

7.163 

6.956 

7.231 

7.853 

8.525 

.O .405 

2 

XlO 

-15.70E 

-14.99: 

-14.20C 

-1 3.86: 

-13.321 

-12.91: 

-11.015 

-8.284 

1.67; 

14-31: 

~ 

-23.101 

-22.31E 

-20.92; 

-19.162 

-17.825 

-16.721 

-12.501 

-4.884 

9.11c 

25.334 

-5.802 

-8.596 

-16.486 

-17.602 

-12.711 

-7.015 

-2.671 

4.149 

13.119 

20.407 

I HI I IIIII 111 

Gapwise 

turbule 

I evel 

;iv c 

XlOZ 

1.935 

1.935 

2.057 

2 -060 

2.117 

2 -495 

2.668 

3.410 

6 -170 

7 -015 

3.094 

3 -094 

2 -915 

2.697 

2 -976 

3.073 

3.593 

4.645 

7.949 

11 -423 

4.466 

4.845 

6.471 

6 -710 

6.259 

5.710 

5.752 

6.827 

8.339 

9.933 

Cross cox 

:e relatic - - uv/vg 

3 x10 

-0.066 

-0.066 

-0 -029 

-0.048 

-0.06 3 

-0 -09 3 

-0.053 

0.467 

0 -686 

1.248 

-0.038 

-0.034 

-0.038 

-0.019 

-0.065 

0.190 

0.195 

0.272 

0.945 

0.987 

0.165 

-0.195 

-0.804 

-0.530 

-0.263 

0.235 

0.611 

1.296 

1.725 

0 -000 

---. ___ 

28 
Spanwise 

location 

z/z; 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.915 

0.930 

0 I950 

0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.915 

0.930 

0.950 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.915 

0.930 

0.950 

-- 

.- . 

Sapwise 

lo cat ion 

r-r 0 

r -r io 

0.1 

0.3 

0.5 

; treamwisi 

ve loc ity - 

u/vc 

1.035 

1.035 

1.035 

1.031 

1.031 

1.031 

1.031 

1.019 

1.007 

0.984 

0.976 

0.960 

0 -949 

1.144 

1.142 

1.136 

1.125 

1.111 

1.109 

1.094 

1.084 

1.083 

1.054 

0.980 

0.964 

0.953 

1.140 

1.148 

1.165 

1.165 

1.173 

1.175 

1.175 

1.179 

1.178 

1.113 

1.062 

1.023 

0.980 

S t reamwis 

turbulenc 

1 eve 1 

2 

b/VC 

x10 

6 -176 

6 -176 

6.299 

6 -917 

6.917 

6 - 546 

6 -546 

6 -670 

6 -670 

6 -670 

6 -794 

6 -917 

7 -288 

4.362 

4 -974 

5 -560 

6 -055 

6.912 

7 -113 

7.205 

7.165 

7 -288 

8 -072 

8.770 

- 

- 

3.333 

3.165 

3.030 

2.751 

2 -660 

2 -473 

2 -595 

2.914 

4 -321 

7 -421 

7.575 

8.593 

9.765 

Gapwise 

velocit 

S/Ve 

2 

XlO 

-2.761 

- 3.590 

-3.866 

-2 -485 

-1 -657 

-1.104 

-1.381 

-0.828 

1.104 

4.281 

6.067 

6.352 

6.629 

-12.306 

-11.510 

-9.805 

-6.629 

-3.507 

-2.872 

-2.486 

- 1.104 

1.858 

5.882 

11.324 

11.048 

10.772 

-17.133 

-16.814 

-16.297 

-15.178 

-1 3.816 

-10.510 

-8.75 2 

-6 -903 

2 -845 

14.106 

16.572 

17 -401 

22.649 

I 

;apwise [cross cor 

turbulence re la t io 

1 eve 1 

5 /vc 

x102 

7.398 

7.398 

7.398 

7.931 

8.325 

8.256 

7.661 

7 -687 

7.550 

8.379 

8.293 

7.798 

6.345 

6.655 

7.316 

9.562 

9.173 

8.459 

8.015 

7 -561 

7.298 

6.770 

9.356 

lo. 507 

3.157 

3.093 

3.217 

3.562 

3.603 

3 -679 

3.945 . 

4.421 

5.893 

8.667 

7.923 

7.629 

- 

- 

- 

-is/v2, 

3 XlO 

2.204 

2.204 

2.204 

2.588 

3 -422 

2.465 

2.392 

2 -087 

1 565 

1.116 

1.303 

0.817 

0.567 

1.182 

1.338 

1.160 

1.285 

2.728 

3.187 

3 -000 

2.472 

1.943 

1.742 

2.541 

2 -453 

1.460 

-0.226 

-0.305 

-0 - 305 

-0.322 

-0.395 

-0 -077 

0 .Ooo 

0.165 

1.142 

2 -656 

2.902 

2.195 

- 

* Number of hydraulic diameters downstream of bend exit plane 

29 
I 

Spanwise 

locatior 

z/zq 

'I 

Streamwis 

station 

xH 

m.25 0 .o 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

c.7 

0.8 

0.9 

0.915 

0.930 

0.950 

0.0 

0.1 

0.2 

0.3 

Oe4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.915 

0.930 

0.950 

Gapwise 

location 

r-r 

r -r 

0 

io 

0.7 

0.9 

; t reamw is i 

velocity - u/vc 

0.601 

0.636 

0.695 

0.753 

0.882 

0.378 

1.040 

i naC; 

1.152 

1.131 

1.082 

1.039 

1.007 

-0.506 

0..500 

0.617 

0.830 

0.971 

1.035 

1.070 

1.093 

1.089 

1.019 

0.996 

0.941 

0.906 

... ."a - 

Streamwis 

turbulenc 

?, level 

2 

u/vc 

XlO 

7.416 

7.537 

7.575 

7.906 

7.943 

5.103 

4.321 

r; in? 

6.210 

7.031 

7 -657 

8.593 

9.375 

8.962 

10.107 

13.730 

12.226 

8.159 

7.166 

6.862 

7.043 

7.537 

8.593 

9.015 

10.546 

12.109 

4. ...-- 

Gapwi se 

ve loc i t - v/vc 

2 x10 

-18.915 

-18.236 

-15.318 

-8.126 

-2 -485 

-6.542 

-5 -083 -1 C7n 

6.435 

18.782 

26.240 

28.450 

29.002 

-4.615 

-5.973 

-12.198 

-10.219 

-10.493 

-6.241 

1.108 

8.283 

12.291 

16.744 

28.173 

30.935 

27.068 

....-.- 

;apwise Cross cor 

turbulence relatio 

,level 

' /vc 

do2 

4 -943 

5.728 

6 -587 

5.523 

5.256 

4 -845 

4 .852 

K ln3 

6 -067 

7.509 

8.596 

8.245 

3.388 

3 -825 

4.156 

4.492 

4.763 

5'.928 

6.785 

7.477 

5.455 

4.353 

9.061 

8.939 

- .--- 

- 

- 

...~. - 

- uv/v 

C 

3 x10 

1.523 

1.396 

1.226 

0.415 

0.249 

-0.349 

-0.490 

-c!.21? 

-0.265 

1.739 

1.222 

0.984 

0 .Ooo 

-0.020 

0.521 

3.158 

2.157 

0.329 

-0.808 

-1.205 

0.403 

0.985 

-0.142 

-0.276 

-0.763 

- _. - 
-4 

d 

0 

m 

r4 

+ 

_. ............. 

............. 

mmw~~mmmm~cnrn-~ rl 

m 

0 

moo dmm 

0003000000000 

???lSY??????? 

31 
I. 

.rl r( 

wc 

In0 

........ +doo~mmcuoo I 

.- ............ .I.. ~. .... ..I ... 

.......... 

P~~J~~Q,~oJPu?-? 

m 

0 /F 

0 

Ln 

N + 
c u 

m 

N 

W 

Q1 

0 

moo m 4mm 

?"1?4Y2?????Q1? 

OdoOOOOOOOOOOO 

0 

m 

N + 33 

L 
Table IIf. 90 DEGREE BEND OF 2.3 RADIUS RATIO AND SQUARE CROSS-SECTION 

TURBULENT FLOW: REYNOLDS NUMBER 40 000 

WALL PRESSURE MElSuREMENTS 

1' I 

station location 

' 0.0 0.000 (Pref) 

1 .o -0.019 

1.0 -0 - 098 

1 .o -0.133 

1 .o -0.278 

1.0 -0.379 

0 .o 0.174 

1 .o 0.114 

1 .o -0.012 

1 .o -0.241 

1.0 -0.516 

1.0 -0 I660 

Spa nwi s e 

location 

Z/Z% 

0 

Pressure coefficient 

'-'ref 

+ vc 

cp= 2 e 

(deg. 1 

45 

r -ro 

- 

r -r io 

0.0 

0.1 

0.3 

0.5 

0.7 

0.9 

I 0.0 

~ 0.3 

~ 0.1 

0.5 

0.7 

0.9 

71 

90 

0.0 

0.1 

0.3 

0.5 

0.7 

0.9 

0.0 

0.1 

0.3 

0.5 

0.7 

0.9 

0.0 

1.0 

1 .o 

1.0 

1.0 

1.0 

0.0 

1.0 

1 .o 

1.0 

1.0 

1.0 

0.160 

0.092 

-u.w55 

-0.251 

-0.516 

-0.637 

-0.039 

-0. loo 

-0.207 

-0.327 

-0.496 

-0.548 

34 

I 11.1 I 
APPENDIX 2 

DEFINITION OF SYMBOLS 

Roman Characters 

Pressure coefficient (figure 15) 

'd 

cP 

De 

d Hydraulic diameter (40 mm) 

Dean number: De -(r) '2 1/2Re 

C 

k Turbulent kinetic energy: k 3 p(u 1 -2 + $ + W2) 

P Pressure at wall 

Pref 

r Radial co-ordinate direction (fig. 1) 

ri 

rO 

Reference value of p (0 = 00, r* = 0, z* = 0) 

Radius of curvature of suction surface 

I1 'I pressure I' II II 

V 

vc Re Reynolds number: Re 5 a 

Mean radius of curvature: Rc 0.5(ri + r ) Rc 0 

r* '0 

r - ro Normal ised radial co-ordinate: r* Z 

i 

U Mean velocity in 8 (streamwise) direction 

- 

U R.m.s. fluctuating velocity in 0 direction 

uv Cross-correlation between ij and ? 

uw Cross-correlation between 6 and 

- 

- 

V Mean velocity in r (radial) direction 

Bulk mean velocity 

R.m.s. fluctuating velocity in r direction 

VC 

- 

V 

W Mean velocity in z (spanwise) direction 

35 
ti R.m.s. fluctuating velocity in z direction 

Axial distance along straight duct, expressed in hydraulic diameters xH 

z Spanwise co-ordinate directiot! (fig. I) 

Duct half-width (20 mm) 

L I Z* Normal ised spanwise co-ordinate: Z* E - 

z1/2 

Greek Characters 

e Axial (streamwise) co-ordinate direction (fig. 1) 

V Kinematic viscosity of water 

P Density of water 

l 

I 

I @ Half-angle of beam intersection 

36 
- I 

REFERENCES 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

Ward Smith, A.J. (1971). I Pressure losses in ducted flows. 

London. 

Rowe, M. (1970). Measurements and computations of flow in pipe bends. 

J. Fluid Mech., 43, 771. 

Agrawal, Y., Talxt, L. and Gong, K. (1978). Laser anemometer study of 

flow development in curved circular pipes. Mori, Y., Uchida, Y. and Ukon, T. (1971). Forced canvective-fieat transfer 

in a curved channel with a square cross-section. Int. J. Heat Mass 

Transfer, 14, 1787. 

Eskinati, and Yeh, H. An investigation of fully-developed turbulent 

flows in a curved channel. J. Aero. Sci., 23, 23. 

Bruun, H.H. (1979). An experimental investzation of secondary flow 

losses in bends with rectangular cross-sections. 

Dept. of Eng. Report CUED/A-Turbo/TR 95. 

Humphrey, J.A.C., Taylor, A.M.K. and Whitelaw, J.H. (1977). Laminar flow 

in a square duct of strong curvature. 

Humphrey, J.A.C., Whitelaw, J.H. and Yee, G. (1980). 

square duct with strong curvature. 

Melling, A. and Whitelaw, J.H. (1976). Turbulent flow in a rectangular 

duct. J. Fluid Mech., 78, 289. 

Taylor, A.M.K.P. (1980)TConfined isothermal and combusting flows behind 

axi symne tri c baf f 1 es . Durst, F., Melling, A. and Whitelaw, J.H. (1976). Principles and practice of laser Doppler anemometry. Johnston, J.P. 19/6) I nternal flows. Bradshaw, P. (ed). furbulence. 

Springer-Yerlag! Berlin. 

Butterworths, 

J. Fluid Mech., 85, 497. 

Univ. of Cambridge, 

J. Fluid Mech., 83, 509. 

mulent flow in a 

To be published in J. Fluid Mech. 

Ph . Thesi s , University of London. 

Academic Press , London-. 
-- 

PLOW - 

Fig. 1 Dimensions of bend and co-ordinate 

definition sketch. 

38 
n 

z 

3 

Q 
40 
Fig. 3(ii) View of bend showing intersection volume. 

Fig. 3(iii) View of optical system. 

41 
P H OT 0 M U L7 I P L \ E 

H\GU PASS 

FILTER 

FPEQUENCY 

AN &LOG u E 

0uTPu-r 

Fig. 4 Block diagram of Doppler signal processing 

arrangement. 

42 
Fig. 5(i) Visualization of secondary flow at entrance to the bend. 

Re = 1275. 

Wire inserted at 0= Oo, r* = 0.3. 

Flow direction from left to top right; 

Fig. 5(ii) Visualization of flow at exit of the bend. Re = 790. 

Flow direction from left to top right; wires inserted 

at 0 = 71°, r* = 0.1, 0.5 and 0.9. 

43 
r* 

0.9 

0.7 

0.5 

0.3 

0.1 

(i) XH = -0.50 (ii) XH = -0.25 (iii) 8 = 30° (iv) 8 = 60° 

2":O. 

0. 

0. 

0. 

u. 

0. 

0.0 1.0 u/v, 

(v) e = 77.5' (vi) XH = 0.25 (vii) XH = 0.40 (viii) XH = 2.5( 

Fig. 6 Laminar flow: profiles of U/V, at successive streamwise 

stations. 

44 
I-T ! 

I. 1, I. I. I .]?El. I, ,,I. I *I, 300 

3 h) .o z z go 3 0 o*o 0 - 

- N45 
Jul-26-2005 10:53 From-CRC LIBRARY 21 643381 39 T-754 P. 003/004 F-412 

I I T v, 

46 . .. - , 
Jul-26-2005 10:53 F r om-CRC L I BRARY 21 643381 39 T-754 P. 004/004 F-412 

I 

0 

I 

r" 

! 

I 

I 

I 

I 

I 

I 

0 

b 

r-l--r- 
I0 

0 

N 

0 

0 

3000000 

3NVLDODO 

3 0 o*o 0 - -NN 

II 

L 

x 

48 
r* 

0.9 

0.7 

0.5 

0.3 

0.1 

(i) XH = -0.25 (ii) 0 = 30' (iii) 0 = 60' 

(iv) 8 = 77.5' (v) XH = 0.25 (vii) XH = 2.50 

Fig.11 Turbulent flow: profiles of U/V, at successive 

streamwise stations. ~- - - ~ ~~ 

49 
0 

-3 

c 

L 

?I 

0 

u3 

0 

-s 

0 

ui 

ru 

II 

I 

x 

In 

N 

0 

II 

x 

8 

h 

w 

u 

50 
000 

0 CJ W - o+o 0 0 - -NI 

.. 

aJ 

0 

V 

v) 

F 

0 

m 

N 

II 

1 

'h 

.r 

> 

Y 

eb 

w-- &'. .I. 

I 

0 

0 

u3 

II 

m 

N 

0 

I1 

x 

I 

h 

> 

Y 

L 

51 
I 

0 

53 

II 

Q, 

000 

G 0 O$,O 0 - 

-N0 rJ V go 

u. 

52 
I 

- 9 

r-a 

I 

F 

In 

0 J 0 

00 .. 

CP 

re 

00 

I I I 

53 
vi 

0 

p. 

.. N 

II 

I 

x 

c, 

m 

7 

\ 

3 

ct 

0 

m 

e, 

ct 

0 

L 

a 

V 

c 

.ke. 

z 

F 

Yc, 

c 

0) 

1 

L 

3 

ic 

n 

2 

m 

LL 

.I54 
/E e 

b-4 I 

I P 

0 

In 

(v 

II 

I 

x 

55 
I "1 

.- 

I I 

n / 

> 

.r I 

-I.I.I.I.I., 

56 
I 0 ‘I 

r-l 

57 
I"""'" 00 

0 

'1 c 

L 

(D 

0 

-J 

0 

ru 

0 

0 

000 GCU~ZZP 

0 0 o*o 0 - -Nu0 

L. 

30 

A 

c, 0- 

CO 

act 

0 

tn 

I 

** x 

3 

oc, 

N 

N 

r"' 

58 E-595