AUSTRALIAN ATOMIC ENERGY COMMISSION · australian atomic energy commission research establishment lucas heights research laboratories developed single-phase turbulent flow through

AUSTRALIAN ATOMIC ENERGY COMMISSIONRESEARCH ESTABLISHMENT

LUCAS HEIGHTS RESEARCH LABORATORIES

DEVELOPED SINGLE-PHASE TURBULENT FLOW THROUGH

A SQUARE-PITCH ROD CLUSTER FOR AN EXTENDED

RANGE OF REYNOLDS NUMBERS

by

*J.D. HOOPER

**D.H. WOOD

W.J. CRAWFORD

* CSIRO Division of Mineral Physics, Lucas Heights Research Laboratories

** Department of Mechanical Engineering, Newcastle University, NSW

June 1983

ISBN 0 642 59773 1

AAEC/E558

AUSTRALIAN ATOMIC ENERGY COMMISSIONRESEARCH ESTABLISHMENT

LUCAS HEIGHTS RESEARCH LABORATORIES

DEVELOPED SINGLE-PHASE TURBULENT FLOW THROUGH

A SQUARE-PITCH ROD CLUSTER FOR AN EXTENDED

RANGE OF REYNOLDS NUMBERS

by

*J.D. HOOPER**D.H. Wood

W.J. CRAWFORD

ABSTRACT

The mean velocity profiles, wall shear stress distribution and allcomponents of the Reynolds stress tensor have been determined frommeasurements for developed single-phase flow through a square-pitch rodcluster. For a rod pitch-to-diameter ratio of 1.107, four Reynolds numbers,

0 Oin the range 22.6 x 10 to 207.6 x 10 , were investigated. The experimentaltechnique, which involved a rotatable inclined hot-wire anemometer probe,allowed the measurement of secondary flow components of the order of 1 percent of the local velocity. No evidence was found for secondary flows in the

* CSIRO Division of Mineral Physics, Lucas Heights Research Laboratories.

** Department of Mechanical Engineering, Newcastle University, NSW.

open rod gap area. The highly anisotropic nature of the turbulence,particularly for the interconnecting rod gap region, was shown by the level ofthe azimuthal turbulent shear stress. The mean velocity profiles weregenerally consistent with the logarithmic region of the universal velocityprofile, using the Patel values for the profile constants. The wall shearstress distribution, measured by Preston tubes, was shown to be symmetricalaround the central rods of the array.

National Library of Australia card number and ISBN 0 642 59773 1

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EXPERIMENTAL DATA; FLOW RATE; FUEL ELEMENT CLUSTERS; FUEL RODS; REACTORLATTICE PARAMETERS; REYNOLDS NUMBER; SHEAR; SQUARE CONFIGURATION; STRESSES;

TURBULENT FLOW; VELOCITY

CONTENTS

1. INTRODUCTION 1

2. EQUATIONS OF MOTION 2

3. SQUARE-PITCH ROD SUBCHANNEL TEST SECTION 3

4. MEAN FLOW RESULTS 54.1 Wall Shear Stress Distribution 5

4.2 Mean Velocity Profiles 6

5. TURBULENCE MEASUREMENTS 75.1 Reynolds Stresses 7

5.1.1 Axial turbulence intensity u' 7

5.1.2 Pxadial turbulence intensity v 1 85.1.3 Azimuthal turbulence intensity w1 8

5.1.4 Radial turbulent shear stress -p uV 95.1.5 Azimuthal turbulent shear stress -p Uw 10

5.1.6 Transverse Reynolds shear stress -p "vw 10

5.2 Axial Momentum Balance 115.3 Turbulent Kinetic Energy and the Ratio A2 11

6. CONCLUSIONS 12

7. REFERENCES 13

8. NOTATION 15

Table 1 Distributed parameter measurements of developed single- 17

phase turbulent flow in a bare rod bundle

Table 2 Hydraulic diameters and flow development lengths 18

Figure la Cross-section of six-rod cluster 19

Figure Ib General arrangement of test rig 19

Figure 2a Wall shear stress distributions, normalised by average 20

stress, for top centre-line rod of the arrayFigure 2b Wall shear stress distribution. Re = 133 x 103, 21

p/d = 1.107(Continued)

Figures 3a-4b Mean velocity profilesFigures 5a-5b Axial turbulence intensity u ' / v * (? )Figures 7a-8b Radial turbulence intensity v ' /v* (0)Figures 9a-10b Azimuthal turbulence intensity w ' /v* (e)

oFigures lla-12b Radial turbulent shear stress Uv/v* (e)

2Figures 13a-14b Azimuthal turbulent shear stress, uw/v* (e)

oFigures 15a-16b Transverse turbulent shear stress, W/v* (f t)

Figures 17a-18d Axial momentum balance

Figures 19a-20b Turbulent kinetic energyFigures 21a-22b Reynolds shear stress ratio A2

22-2526-2930-3334-3738-4142-4546-4950-57

58-6162-65

1. INTRODUCTION

The problem of calculating the coolant velocity distribution, wall shearstress distribution and forced convective neat transfer rates at tne surfaceof heat generating rods is of major interest to the nuclear power industry.Long cylindrical fuel rods are employed in the majority of power reactorcores, and developed flow for single-phase coolants may be expected to applyover most of the rod length. Because of the intractable nature of turbulentflow, prediction of the fluid velocity distribution is a difficult part of thecoupled fluid mechanics and heat transfer problem.

Limitation of the upper fuel rod cladding temperature is an importantdesign constraint for all power reactors. Moreover, the advent of the liquidmetal-cooled fast breeder reactor, with rod spacing reduced because of neutronmoderation and capture, has potentially increased the azimuthal variation ofthe rod wall shear stress and the associated heat transfer coefficient. Theanisotropy of the heat removal process in the rod gap region is of particularimportance in closely spaced rod arrays.

Axially developed turbulent single-phase flow is the simplest example ofthe coolant conditions in a bare rod array. An understanding of this isrequired before the effect of entry conditions, fuel element grids andspacers, or partial channel blockages can be considered. Relatively fewworkers have published detailed information on mean velocity distributions andcomponents of the Reynolds stresses for even basic rod bundle flow.

A summary of the published work for bare rod arrays based on a modifiedsurvey of Bartzis and Todreas [1977], is given in Table 1. The majority ofthe experimental studies are for triangular pitch rod arrays. Detection ofsecondary flow velocity components has been shown to be experimentallydifficult for rod bundle geometries, with only one study by Kjellstrom [1974]indicating their presence. Kjellstrom showed'a net circulation around one rodof the array, but the secondary flow components were most likely associatedwith the lack of flow symmetry in the test section. The extensive studies ofRehme [1977a,b; 1978a,b; 1980a,b] for closely spaced rod arrays reported nomeasurable mean secondary flow velocities. !.

The study of Seale [1979] is the only distributed parameter experiment tomodel the heat transfer process in the rod gap area. Here, a longhorizontally mounted rectangular duct containing a single row of rods

regularly spaced at three p/d ratios was used. Heat transfer was not from the

rod surface to the air coolant, but from the heated top wall of the duct to

the water-cooled lower wall. Insulated rod walls reduced the rod conduction

to about 2 per cent of the total heat flew. Seale [1979] found the effective

eddy heat diffusivities in the rod gap to be strongly anisotropic, with no

evidence of mean secondary flow.

The objective of the present work was to investigate experimentally the

flow structure of axially developed single phase turbulent flow through a

square-pitch rod array having a pitch-to-diameter (p/d) ratio of 1.107. The

dependence of the mean velocity distribution, wall shear stress distribution

and Reynolds stresses was investigated for the Reynolds number range 22.6 x

103 to 207.6 x 10. Typical results are discussed in this report, and the

complete data bank is given in Hooper et al. [1983].

2. EQUATIONS OF MOTION

The continuity equation and the fluid momentum or Navier-Stokes equationsfor a constant viscosity and density fluid in polar coordinates are [Hinze1975:28] :

1U + 1 3 (rV) + 1 3W _ n m9z r 3r + r 96 u U)

z axis momentum:

DU .. 3P 2,, 3u p 3 (ruy) p 5uw' " + u V U - ~ ~ + F

r axis momentum:

DV w 3P , 2., V 2 3Wp TJf - r - ~W - - a?

__ _} p__8vw 8 uv , w , r- ~ - — + + F

0 axis momentum:

1 3P , . I r2u W ^ 2

2p 3 w 9 vw 9 uw o vw , c / . x

" r W~ - p W~ - p JT~ - 2P — + F9 (4)

The ? operate^ represents

3z2 3r2 r 9r

and the f l u i d total der ivat ive

_ _TPE ~ 9t 9z 9r r S6

For the hor izonta l ly mounted six-rod cluster air r ig used in t h i s study,the body forces ( F Z > Fr and FQ) may be taken as zero.

3. SQUARE-PITCH ROD SUBCHANNEL TEST SECTION

The square-pitch red subchannel test section was designed to model thefluid mechanics of axially developed turbulent single phase flow through aclosely spaced rod cluster. A square-pitch geometry was selected for thestudy; most of the reported work (Table 1) has been for triangular arrays. Anumerical study of rod bundle flow by the ROFLO code [Hooper 1975] chowed thatfor the same (p/d) ratio and Reynolds number, the wall shear stressdistribution in an axial plane is substantially higher for the square-pitchthan for the triangular-pitch array.

A large variation in the wall shear stress distribution was considered tobe associated with a significant departure of the turbulent flow structurefrom axisymmetric pipe flow conditions.

The cross-section of the test section is shown in Figure la. The twointerconnected subchannels were intended to represent, for the -45 to 45°segment about the top and lower centre-line rods, the repeated symmetricalzone of an interior subchannel in a large square-pitch array. The presence ofthe walls in the rod gap is shown for laminar flow to introduce an error ofless than 1 per cent in the mean velocity at the subchannel centre, calculatedby the ROFLO [Hooper 1975] code for p/d ratios less than 1.20. For turbulentflow, neglecting the effect of secondary flows, the effect of the rod gapwalls would be further reduced by the higher mean velocity gradients near wallboundaries.

The scale of the rig was made as large as possible, consistent with theneed to have at least 80 hydraulic diameters for flow development. Subbotinet al. [1971a] have suggested that between 13 and 17 hydraulic diameters arerequired to stabilise the axial pressure gradient, and up to 50 hydraulicdiameters to establish a developed mean velocity profile. The test sectionlength was 9.14 m, and the outside diameter of the rods forming the testsection 140 mm. Measurements were normally made 50 or 100 mm upstream of therig exit. A general arrangement of the rig is shown in Figure lb, with theopen loop system being powered by a 45 kW centrifugal blower. The test-section entry was attached directly to the 1.20 m diameter by 1.50 m long flowsettling drum; this drum had a fine mesh internal screen, and its entry pipeelbow had turning vanes to ensure more uniform flow entry conditions to thetest-section. An orifice plate was located in the 0.10 m i.d. pipe connectingthe blower to the settling drum.

The axial static pressure gradient was determined by 19 static tapstations, spaced equally at 457 mm intervals along the test-section. Eachstatic tap location consisted of three 1.20 mm diameter holes locatedcentrally in the strips forming the rod gap walls, and interconnected by apiezometric ring (Figure la). The replacement of the aluminium strips inslots machined along the test-section rods allowed the rig to operate atdifferent p/d ratios, although only one value (1.107) was examined in thiswork. The equivalent infinite square-pitch rod array hydraulic diameter isgiven by Equation 5, and the subchannel hydraulic diameter by Equation 6:

dh = 4 d ((p/d)2 - J)/ir (5)

CD

dh = 4 d ((p/d)2 - J)/(ir + 3((p/d) - 1)) (6)

The hydraulic diameters and flow development lengths are given in Table2. The infinite array hydraulic diameter and the average velocity for the -45to 45° segment about the top and lower central rods of the array were used tocalculate experimental Reynolds numbers.

The surfaces of the aluminium rods were highly polished, and the threesegments of each rod joined axially were aligned to ensure that no steps orwall roughness elements higher than 0.02 mm were present. A polar scanningsystem, capable of motorised traverse in both the radial and azimuthaldirections, was designed to be located in any of the six rods at the test-section exit. Geared Slo-Syn pulse-operated motors were used for both

traverse directions. When under computer control, the resolution for theradial traverse was ± 0.02 mm and the azimuthal traverse 0.3°. Both traversedirections could be operated manually with the same accuracy.

The computerised measurement of the six terms of the Reynolds stresstensor, even when limited to the symmetry zone about the top and lowercentre-rods of the array, is a task requiring approximately 150 hours for eachrod and Reynolds number. Automatic measurement, using a two element hot-wireanemometer probe as sensor, and a PDP11/10 for rig control and dataprocessing, are described by Hooper [1980] and Hooper and Harris [1981].

4. MEAN FLOW RESULTS

Complete experimental data of the mean velocity distribution, wall shearstress distribution and Reynolds stresses for the four Reynolds numbers (22.6x 103, 46.3 x 103, 133.0 x 103 and 207.6 x 103) are presented in tabular formby Hooper et al. [1983], and typical results are given in the present report.The Re = 46.3 x 103 study was conducted with the air flow through the testsection reversed, and the measurement plane located 25 mm upstream of thetest-section exit into the flow settling drum. Additional studies atapproximately the same Reynolds number [Hooper 1980] for the normal flowdirection showed the same experimental results for the mean flow and Reynoldsstress measurements. The data for the three lower Reynolds numbers weremeasured from the top centre-line rod. Antisymmetric components of theReynolds stresses, with respect to the symmetry line at 0° from the top andlower central rods of the array, change sign when the cylindrical coordinatesystem is relocated from the top to lower rod.

4.1 Wall Shear Stress Distribution

The wall shear stress distribution for the centre-line rods of the array,normalised by the average rod shear stress, is shown in Figure 2(a) for thefour Reynolds numbers. The shear stress was measured by Preston tubes, usingthe correlations of Patel [1965]. There is some evidence that the azimuthalshear stress variation depends on Reynolds number, since the normalised shearstresses are slightly lower in the rod gap at 0° for the lower meanvelocities. In common with the experimental results of Fakory and Todreas[1979], using a triangular pitch rod array spaced at a p/d ratio of 1.10, andRehme [1978a; 1979a; 1980a,b], using a square-pitch rod array spaced at a p/d

ratio of 1.07, the wall shear stress increases monotonically from the centralrod gap at 0° to the subchannel diagonal at ± 45°.

The symmetry of the shear stress distribution about the top and lowercentral rods of the array for Re = 133.0 x 10° is shown in Figure 2b, whichalso demonstrates the consistency of the Pate! [1965] correlations for a rangeof Preston tube diameters. The normalised shear stress distribution for thelower left-hand rod on the outside of the test-section is also shown in Figure2b. The variation for this outer rod shows a local minimum in the wall shearstress at -45°. The lack of symmetry of the outer rod shear stressdistribution, in contrast to the central rods of the array, is an effectassociated with the rod gap walls. Radial traverses at ±45° from the centralrods of the array, which form lines of symmetry in large rod arrays, thereforeonly approximate lines of symmetry in the test-section.

4.2 Mean Velocity Profiles

The dimensionless mean velocity profiles for the turbulent core at Re =o

22.6 x 10 are shown in Figures 3a and b. Similar profiles for Re = 207.6 x10 are shown in Figures 4a and b. For comparison, the logarithmic velocity

profiles given by

U+(r,9) = 1 £n y+ + C (7)K^

are also shown in these figures.

The value of the Von Karman constant K has been taken as 0.4187 (assuggested by Patel [1965]), and the corresponding value of the constant C is5.45. It is apparent that for Re = 22.6 x 103 the mean velocity points arebelow the logarithmic distribution for all radial traverses. However, for Re= 207.6 x 103, the data are well described by the Patel [1965] version of thelogarithmic profile. This agreement of the mean velocity data with thelogarithmic profile is common to all studies except that performed at thelowest Reynolds number. Rehme [1978a; 1980a,b] used the Nikuradse values of0.40 and 5.5 for K and C respectively, and his data points fall slightly belowthe logarithmic distribution. The Nikuradse values were also used byKjellstrom [1974] and Subbotin et al. [1971b].

There are no universally agreed numerical values for K and thecorresponding constant C [Hooper 1980]. The use of the Patel [1965] values

is, however, consistent with the use of the Patel [1965] interpretation ofPreston tube data. Agreement between Equation 7 and the experimentalvelocities supports the use of these correlations in a geometry very differentto the flat plate boundary layer and axisymmetric pipe flow used to establishthem.

The rotatable, inclined hot-wire anemometer probe used to measure allterms of the Reynolds stress tensor, was also used to establish the directionof the mean velocity vector. Assuming a cosine response for the probe [Hooper1980; Hooper and Harris 1981], it can be shown that the average anemometerbridge voltage must be measured to an accuracy of ± 0.15 per cent to resolve asecondary flow component of velocity V or W which is 1 per cent of the localaxial velocity U. Using ensembles of the anemometer bridge voltage filteredby an 8 second passive iow pass filter and averaged over approximately twominutes, a computer-calculated mean enabled this level of accuracy to beachieved. There was no evidence of non-zero values for V and W in the openrod gap for any of the radial traverses or Reynolds numbers investigated.This is consistent with the result of Rehme [1977a,b; 1978a,b; 1980a,b], whoreported no significant secondary flow components for any of his studies in asquare-pitch array.

5. TURBULENCE MEASUREMENTS

5.1 Reynolds Stresses

The Reynolds shear stress data at each Reynolds number and radialtraverse were normalised by dividing by the local wall shear stress;similarly the turbulent intensities u ' , v 1 and w1 were normalised to the localfriction velocity v* (0 ) . The turbulence-intensity data of Laufer [1954] andLawn [1971] for axisymmetric developed pipe flow are also shown for comparisonwith the present rod cluster results. It' should be noted however thatalthough Lawn's data indica'ed no dependence of turbulence intensity onReynolds number, this was not so with Laufer1 s data. Only the Laufer studyfor Re = 500 x 10 is used for comparison.

5.1.1 Axial turbulence intensity u'

The axial turbulence intensity u1 for Re = 22.6 x 103 is shown in Figureso

5a and b, and for Re = 207.6 x 10 in Figures 6a and b. The elevation of u1

above levels typical of pipe flow for the rod gap region is a feature commonto both Reynolds numbers. The location of the maximum level of u 1 for thetraverses at ± 15° is also evident in both studies. The data of Hooper et al.[1983] for the intermediate Reynolds numbers of 46.3 x 103 and 133.0 x 103

also confirm this observation. However, there is evidence of a progressiveincrease in the general level of the axial turbulence intensity as theReynolds number is increased. This may be partly due to the scaling effectsshown by the mean velocity results, and the over-prediction of the wall shearstress by the Patel [1965] correlations for the lowest Reynolds number of 22.6x 103.

The traverse at 0° is geometrically a symmetry line of the test-section.Within reasonable limits, the reflection of the experimental results aboutthis radial traverse shows the same symmetry. Analytically, the axialturbulence intensity is a symmetric function in rod subchannels. Thedistribution at 45°, particularly for the high Reynolds number study, issimilar to the results obtained by Laufer [1954] and Lawn [1971] foraxisymmetric developed pipe flow.

5.1.2 Radial turbulence intensity v'

Figures 7a and 8a show that the normalised radial turbulence intensitiesv 1 in the rod gap are of the same order of magnitude as the pipe flow levelsestablished for Re = 22.6 x 103 and 207.6 x 103. The limiting values of bhev'/v*(0) component in the rod gap may be a result of the small distancebetween the opposing rod walls acting as a constraint on the turbulence scalein the r-z plane.

The distributions at 45° for both Reynolds numbers approximate pipe flowresults (Figures 7b and 8b), although the data for the lower Reynolds numberare again reduced in magnitude. The symmetry of the v'/v*(6) component aboutthe 45° radial traverse is reasonable at both Reynolds numbers, the resultsfor the 40° traverse being approximately equal to those at the 50"' traverse.

5.1.3 Azimuthal turbulence intensity w'

The azimuthal turbulence intensity w 1, when scaled by the local wallfriction velocity, reaches a local maximum for the traverse at 0° for bothReynolds numbers investigated (Figures 9a and lOa). The distribution iseffectively independent of the wall distance for the traverses in the rod gap

area, and values of w1 are considerably greater than those for thecorresponding pipe flow. This result is an indication of a significantmomentum interchange between the two subchannels. In common with the axialand radial turbulence intensity, the azimuthal component w1 is bothanalytically, and approximately experimentally, symmetric about the riggeometrical symmetry lines. There is, however-, some lack of symmetry in theexperimental results of w1 'bout the 0° traverse. The distributions at theother line of symmetry, i.e. 45°, (Figures 9b and lOb) again approximate thedistributions typical of developed pipe flow, particularly for the higherReynolds number study.

5.1.4 Radial turbulent shear stress -p uV

The radial turbulent shear stress -p W has a linear distribution in therod gap at both Reynolds numbers (Figures lla and 12a), and approaches thelocal wall shear stress (or unity when normalised by Tw(

9)) as the wall isneared. From the axial momentum equation (Equation 2) for this line ofsymmetry, it can be seen that there are two extra terms to the axisymmetricdeveloped pipe flow form of the equation; the advection term pV(3U/3r); andthe azimuthal gradient of the Reynolds shear stress -p/r(37jw/90). Directmeasurement of the secondary flow component V showed its magnitude to be lessthan 1 per cent of the local axial velocity U, but the advection term maystill contribute to the axial momentum balance. The azimuthal gradient of-p W is also not zero for the 0° radial traverse. However, the radialturbulent shear stress for the 0° line of symmetry is similar to thedistribution of that component of the Reynolds shear stresses for developedpipe flow.

Away from this line of symmetry, the normalised -p uv~ distributiondeparts markedly from the linear distribution, which passes through unity aty/ymav

= 0 (Figures lib and 12b). The other possible line of symmetry (45°)IllClA

does not have a linear distribution at either of the Reynolds numbers shown inFigures lib and 12b, or in the complete data bank [Hooper et al. 1983]. Thenormalised radial turbulent shear stress is analytically a symmetric functionabout the subchannel diagonal at 45° for a large array, but the experimentalresults for traverses at 40° and 50° are not identical. The result isassociated with the rod gap walls, and shows that the presence of these wallssignificantly changes the flow structure in the rod gap.

10

5.1.5 Azimuthal turbulent shear stress -p uw

Analytically, the normalised azimuthal turbulent shear stress -f uw isantisymmetric for lines of symmetry. The experimental results for Re 22.6 x10 (Figure 13a) and Re = 207.6 x 10 (Figure 14a) show this feature about the0° measurement plane. The magnitude of -p TJW is very close to the expectedvalue of zero for the 0° traverse angle. The -p liw component of the Reynoldsshear stresses increases in a remarkable manner away from the rod gap; at the

ocentre of the 15° traverse, it becomes 1.2 T (e) for Re = 22.6 x 10 , and 1.6

o ."T (0) for Re = 207.6 x 10. As with other components of the Reynoldsstresses, part of the Reynolds number dependence of the magnitude of -p iw maybe related to an over-estimate of the wall shear stress by the Preston tubemeasurements for the lowest Reynolds number. A similar increase in the levelof -p uw" near the centre of the rod gaps was noted by Rehme [1977a; 1978a;1980a,b] in his studies of a rod array with a p/d ratio of 1.07. This effectis linked to the energetic momentum interchange process between thesubchannels of closely spaced rod arrays. The normalised values of -p uw" arealmost zero for the subchannel diagonal at 45° (Figures 13b and 14b).Additional ly, the behaviour of -p uw" is approximately antisymmetric withrespect to this traverse angle.

5.1.6 Transverse Reynolds shear stress -p vw'

The transverse Reynolds shear -p vw is difficult to measure accuratelywith the rotatable inclined hot-wire probe, since essentially it is a measureof the difference of two large quantities [Hooper 1980]. However, the non-zero level of -p vw for the rod gap area, and its antisymmetric behaviouracross the 0° traverse, were shown by the data for all four studies. Theresults at Re = 2.2.6 x 10 (Figures 15a and b) show that -p "uw reaches amaximum level for the 15° traverse angle. The data are not, however,symmetrical with respect to the traverse angle at 0°. For traverse anglesgreater than approximately 35°, -p vw~is effectively zero.

The same features are present in the study at Re = 207.6 x 103 (Figures16a and b) and in the complete data bank for the four Reynolds numbers [Hooperet al. 1983].

11

5.2 Axial Momentum Balance

The axial momentum equation (Equation 2) may be integrated radially fromthe rod wall at R to a wall distance y. If the viscous term is ignored,except for its magnitude at the rod wall, where t (8) is equal to y(3U/3r),the integral becomes:

n / r(\l 3U + W 3U\ , _ 3£ y + 2Ry Rp / 0 r ( V 87 + r W d r ~ " - 5 z V — 2 - /' R

' K

R+y •4fdr

R

The advection terms of the momentum integral equation for the axial directionthus become the unknown or balance terms of Equation 8, in which all otherquantities are known or may be calculated. The axial momentum equation maytherefore be used to assess the importance of the secondary "elocitycomponents V and W, and for the symmetry line at 0°, H may be assumed to bezero. A further analysis of the possible distribution of V and W is discussedby Wood [1981].

The calculated azimuthal gradient of the azimuthal shear stress containsthe most uncertainties. However, a central finite difference scheme for thisterm was used [Hooper 1980] and, in the radial integral, the term was assumedto vary linearly from zero at the wall to the first measured value. The axial

omomentum balance for Re = 22.6 x 10 and the azimuthal angles 0, 15, 25 and45° is shown in Figures 17a to d and for Re = 207.6 x 10 in Figures 18a to d.

It is apparent that the advection term contribution, D, to the axialmomentum balance is small for all angles. Also, the radial integral of theazimuthal gradient of the Reynolds shear stress -p uw~, C, is most important at0°. For the latter traverse angle, the term becomes larger at the ductcentre-line than either the pressure term, B, or the radial Reynolds shearstress -p U7, A. The terms of Equation 8 have been normalised by T (0) .

5.3 Turbulent Kinetic Energy and the Ratio A2

The turbulent kinetic energy, defined as 1/2 (u1 + v'2 + w ) andnormalised by v* (8), is shown for Re = 22.6 x 10 in Figures 19a and b and

ofor Re = 207.6 x 10 in Figures 20a and b. The lower magnitude of the

12

turbulent kinetic energy for the lower Reynolds number at all radial traverseangles is apparent. The elevation of the turbulent kinetic energy above thelevel determined by Lawn [1971] for axisymmetric developed pipe flow in therod gap is also shown at both Reynolds numbers; this effect is consistentwith the convection of higher energy fluid from the subchannel centre to therod gap. The distribution for the 45° traverse is similar to the Lawn [1971]results for the highest Reynolds number investigated.

The ratio of the algebraic sum of the radial and azimuthal Reynolds shearstresses to the normal Reynolds stress is shown in Figures 21a and b at Re =22.6 x 103, and in Figures 22a and b at Re = 207.5 x 103. The ratio A2 isdefined as

') 9uv -f uw

"2 - 0 „ „ , gX

w

and is used as a constant in some numerical models of single-phase turbulentflows. As can be seen, however, the ratio is far from constant for the rodbundle geometry.

6. CONCLUSIONS

The experimental results for the wall shear stress variation, andcomparisons of the axial velocity profiles with a standard form of logarithmicdistribution showed tha* the normalised mean flow structure is substantiallyindependent of Reynolds number for Reynolds numbers of 46.3 x 10°, 133.0 x 10

oand 207.6 x 10 . There was, however, some discrepancy between the logarithmicdistribution and the mean axial velocity data for the lowest Reynolds numberof 22.6 x 103.

The Reynolds stresses were approximately independent of Reynolds numberfor the three highest Reynolds number studies, but the magnitudes of all sixcomponents of the Reynolds stresses were lower for Re = 22.6 x 10 .

There was no direct experimental evidence of secondary flow components Vand W within the level of accuracy of measurement, approximately ± 1 per centof the local axial velocity U. The relative unimportance of the secondaryflow components to the axial momentum balance was shown by numericalintegration of the axial momentum equation. The contribution of the advectionterm, containing the secondary flow velocities V and W, to this balance was

13

shown to be insignificant.

7. REFERENCES

Bartzis, J.G. and Todreas, M.E. [1977] - Hydrodynamic behaviour of a bare rodbundle. ERDA COO-22445-48TP..

Carajilescov, P. and Todreas, N.E. [1975] - Experimental and analytical studyof axial turbulent flows in an interior subchannel of a bare rodbundle. Paper HT/51, ASME Winter Annual Meeting.

Chieng, C.C. and Lin C. [1979] - Velocity distribution in the peripheralsubchannels of the CANDU-type 19-rod bundle. Nucl. Eng. Des. ,55:389.

Eifler, W. and Nijsing, R. [1967] - Experimental investigation of velocitydistribution and flow resistance in a triangular array of parallelrods. Nucl. Eng. Des., 5:22.

Fakory, M. and Todreas, N.E. [1979] - Experimental investigation of flowresistance and wall shear stress in the interior subchannel of atriangular array of parallel rods. J. Fluids Eng., 101:429.

Hinze. J.O. [1975] - Turbulence (2nd ed.). McGraw H i l l , New York.

Hooper, J.D. [1975] - The calculation of fully developed turbulent and laminarsingle-phase flow in four rod subchannels. AAEC/E351.

Hooper, J.D. [1980] - Fully developed turbulent flow through a rod cluster.Ph.D Thesis, School of Nuclear Engineering, University of NSW.

Hooper, J.D. and Harris, R.W. [1981] - Hot wire anemometry techniques for anautomated rig. AAEC/E516.

Hooper, J.D., Wood, D.H. and Crawford, W.J. [1983] - Data bank of developedsingle-phase flow through a square-pitched rod cluster for fourReynolds numbers. AAEC/E559.

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Kjellstrom, B.[1974] - Studies of turbulent flow parallel to a rod bundle oftriangular array. AE-487.

Laufer, J. [1954] - The structure of turbulence in fully developed pipe flow.NACA 1174.

Lawn, C.J. [1971] - The determination of the rate of dissipation in turbulentpipe flow. J. Fluid Mech., 48(3)477.

Patel, V.C. [1965] - Calibration of Preston tube and limitations on its use inpressure gradients. J. Fluid Mech., 23(1)185.

Rehme, K. [1977a] - Measurements of the velocity, turbulence and wall shearstress distributions in a corner channel of a rod bundle. KfK-2512.

Rehme, K. [1977b] - Turbulent flow through a wall subchannel of a rod bundle.KfK-2617.

Rehme, K. [1978a] - The structure of turbulent flow through a wall subchannelof a rod bundle. Nucl. Eng. Des., 45:311.

Rehme, K. [1978b] - The structure of turbulent flow through a wall subchannelof a rod bundle with roughened ribs. KfK-2716.

Rehme, K. [1980a] - Experimental investigations of turbulent flow through anasymmetric rod bundle. KfK-3047.

Rehme, K. [1980b] - Experimental investigation on the fluid flow through anasymmetric rod bundle. KfK-3069.

Rowe, D.S. [1973] - Measurement of the turbulence intensity scale and velocityin rod bundle flow channels. BNWL-1736.

Seale, W.J. [1979] - Turbulent diffusion of heat between connected flowpassages. Nucl. Eng. Des., 54:183.

Subbotin, V.I., Ushakov, P.A., Leuchenko, Yu. D and Bibkov, L.N. [1971a] -Study of the velocity profiles in the inlet section of denselypacked bundles of rods. Heat Transfer - Soviet Res., 3(5)1.

15

Subbotin, V.I., Ushakov, P.A., Leuchenko, Yu. D. and Aleksandrov, 'A.M. [1971b]- Velocity field in turbulent flow past rod bundles. Heat Transfer- Soviet Res., 3(2)9.

Trupp, A.C. and Azad, R.S. [1975] - The structure of turbulent flow in atriangular array rod bundle. Nucl. Eng. Des., 32:47.

Vouka, V. and Hoornstra, J. [1979] - A hydraulic experiment to supportcalculations of heat mixing between reactor subchannels. 2ndSymposium on Turbulent Shear Flows, London, July.

Wood, D.H. [1981] - The equations describing secondary flow in cylindricalpolar coordinates. University of Newcastle, NSW, TN-FM61.

8. NOTATION

A2 ratio of algebraic sum of radial and azimuthal Reynolds shearstresses to algebraic sum of normal Reynolds stresses

C constant in logarithmic law of wall

d, hydraulic diameter

d rod diameter

F fluid body force

I length of rig

p rod pitch

P pressure

p/d rod pitch/diameter ratio

q turbulent kinetic energy

R pipe radius

16

Re Reynolds number

U,V,W mean velocity components in z,r,6 direction

u,v,w fluctuating velocity components in z,r,u direction

u ' , v ' , w ' normalised axial, radial and azimuthal turbulence intensity

v* friction velocity

y wall distance

y+ dimensionless wall distance

y distance from rod wall to subchannel centre-line of symmetrymax

Greek Symbols

TW wall shear stress

v kinematic viscosity

p dynamic viscosity

p air density

K Von Karman constant

Subscripts

z,r,e component resolved along designated polar coordinate axis

w wall value

s test section value

value for large symmetrical array

Superscripts

' r.m.s. quantity

time-averaged quantity

17

TABLE 1

DISTRIBUTED PARAMETER MEASUREMENTS OF DEVELOPED SINGLE-PHASETURBULENT FLOW IN A BARE ROD BUNDLE

Experimenter

Eiflor &Nijsing(1967)

Subbotinet a.1.(1971b)

Rowo (1973)

Kjellstrom(1974)

Trupp &Azad (1975)

Cara jilescovS Todreas(1975)

Bartzie STodreas(1977)

Rehme (1977a)(1977b)(1978a)(1978b)(1980a)(1980b)

Fakory &Todreas(1979)

Vouka sHoornstra(1979)

Chieng sLin (1979)

4.Seale(1979)

Hooper(1980)

p/d

1.051.101.15

1.051.101.20

1.251.125

1.217

1 . 501.351.20

1.123

1 . 124

1 . 071.151 . 071.451 . 071.07

1.10

1.30

1.149

1.8331.3751.10

1.1071.194

i/dn

1399065

15410072

85-

81

203051

77

77

17714317766177177

182

117

-

216469996

Re/103

15.0,30.0,50.0" " "

18.8-31.0""

50-200

149-373

12-84

27

9,26. 5 ,65

59. 71238718210775.7

9.1136.2

140

_

34.4-29945.8-18946.2-91.1

4848-156

Mean Velocity

AxialU

*

*

A

,*A

*

*

*

*

*

*

*

*

A

*

A

A

*

*

*

*

Secondaryv,w

Intensities

u ' , v ' , w *

* ** fi-

ii

A

* * ** * A

Reynoldsstresses

-pUV , -pUW, -f';VW

*

A

* * *

* * *

Technique

Pitot tube

Pitot, Prestontubes

Laser-D-i; pier

Trf(0)

A

a:io.iiui:i*jLei

Preston tubeH.W. anemometer

H.W. anemometerPitot, Prestontubes

Laser-Doppleranemometer

Laser-Doppler

Pitot , PrestontubesH.W. anemometer

Pitot , Prestontubes

Laser-Doppleranemometer

Laser-Doppleranemometer

Pitot probeand thermo-couple survey

Pitot , PrestontubesH .W . anemometer

A

A

*

*

*

*

*

Array

Triangular-

Triangular

Square/ ;mixeageometry

Triangular

Triangular

Triangular

Triangular

Square-edge

Triangular

Triangular/mixedgeometry

Mixed"*"

Square-edgechannel

Square

t Data of Seale includes temperature fields

+ Mixed array denotes a combination of square, triangular and edge subchannels

18

TABLE 2HYDRAULIC DIAMETERS AND FLOW DEVELOPMENT LENGTHS

(P/d) dh dh £/dh £ /dhCO S CO S

mm mm

1.107 78.44 71.17 117 128

19

STATIC TAPPING

LSPACER STRIPS -6LPIEZOMETRIC RING

ALUMINIUM TUBES-6

- S T A T I C TAP POINTS

FIGURE la. CROSS-SECTION OF SIX-ROD CLUSTER

JEST SECTION

45kW BLOWER iRIFICE PLATE

FIGURE Ib GENERAL ARRANGEMENT OF TEST RIG

20

COCOLLJCC.J

CO

DCCTLJJ1CCO

crus

I rD

1,5

- - - ^

1,3

1,2

LI

1,0

0,9

0,8

0,7

0,6

0,5

n i.

1 i ' i ' i ' i ' i ' i ' i ' i ' i '

REYNOLDS NO, DEPENDENCE OE TU.- -

SYMBOLSR a 22.6 x 10

3

B o 46-3 x 1C|3

L 133 x io3

_ D ^ 207 x 103

_ ° o o"°

§ O^nA Qh-4 rK Q r-i

A °^ VO^ @ ^ a

§ O •S n~ fj ^ —

O ra W/\ rY @" ~

a $ ft &

$ Q ^t? v/— —

o

- D° 6D -o

8 8* "

-

A 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1

-80,0 -0,0

RNGLE THETR

80,0

FIGURE 2a WALL SHEAR STRESS DISTRIBUTIONS, NORMALISED BYAVERAGE STRESS, FOR TOP CENTRE-LINE

ROD OF THE ARRAY

21

i - D

i. - J

1A

-i '••[ . j

1 . r-J

CO " "~CO' , 1 '.. I—Icr i ii — I -cocra: 1 - 0•!_u

^ 0,9icr

0,8

0,7

0,6

0,5

OA

— i — | — i — | — i — | — i — | — i — | — i — | i — | i — | — i — | — i —. . PrestonLocation ^ , ,

c v M D n i ^ tube (mm)• J 1 1 1 LJ 5̂

K w g. ^ti

!! W a ^a ^- ^ w w w -

w w a- ^ z:!'H ^ ^

_, ^~*

ent—IDn:i—DCI ;

-f

>~II—i

O_JLU>

00COUJ

oI—1CO2LU

i—iO

40.0

35.0

30.0

25.0

20.0

15.0

10.0

5.0

0.010

LRU OF URLL

I

D

THETR

- 10

- 5

0

5

10

15

20R CON3TRNT OF 2.0 IN U+ DISPLRCESSUCCESSIVE VELOCITY PLOTS.

10 10DIHENSIONLESS URLL DISTRNCE Y +

10

FIGURE 3a MEAN RADIAL VELOCITY PROFILES. RADIAL TRAVERSE AT -10°NOT DISPLACED BY CONSTANT. Re = 22.6 x io 3

or

oc

_O_JLU

COCOLU

Ot—H

CO2LU1^H—(

Q

35.0

30.0

25.0

20.0

15.0

10.0

5.0

0.0

O

oQ

X

+

LflU OF URLL

O

o

a

R CONSTRNT OF 2.0 IN U+ DISPLACESSUCCESSIVE VELOCITY PLOTS.

RNGLE THETR

X

a

o

25

30

35

50

INDGO

10 10 10DInENSIONLESS UflLL DISTRNCE Y +

FIGURE 3b MEAN RADIAL VELOCITY PROFILES. R A D I A L TRAVERSE AT 25C

NOT DISPLACED BY CONSTANT. Re = 22.6 x J Q 3

10

ori—LUn:i—oc

COCOUJ

ot—ICO•z.LJrr>—!

a

35.0

30.0

25.0

20.0

15.0

10.0

5.0 -

0.010

LflU OF URLL

fl CONSTRNT OF 2.0 IN U+ DTSPLflCESSUCCESSIVE VELOCITY PLOTS.

10 10DIHENSIONLESS URLL DISTRNCE Y +

FIGURE 4a MEAN RADIAL VELOCITY PROFILES. R A D I A L TRAVERSE AT -20°

NOT DISPLACED BY CONSTANT. Re = 207.6 x l o 3

10

CE\—LUn:i—

B.

cc

oCD_1UJ

COCOLlJ

Oi—iCO

40.0

35.0

30.0

25.0

20.0

15.0

10.0

5.0 -

0.010

LflU OF URLL

f l CONSTflNT OF 2 . 0 IN U+ DISPLRCESSUCCESSIVE VELOCITY PLOTS.

flNGLE THETR

X

a

o

O

20253035

10 10DIHENSIONLESS URLL DISTRNCE Y +

FIGURE 4b MEAN RADIAL VELOCITY PROFILES. RADIAL TRAVERSE AT 20°NOT DISPLACED BY CONSTANT. Re = 207.6 x 10 3

IV)en

10

26

en•^LLU

3,0

2=5

1 1

c,J.

1 1 , 1 1 1 • 1

J.I RUN

LflUFFR

I ' i

'-

RNGLE: J

i n1 !J *L; —

— : .~r~

2,0

1,5

1 ,0

0/5

0,0

\

0,0

o

N \

1 j J l__ J ,

0,8

Y / Y NRX

0r •~[J

10

x i

° 1o 1

I

FIGURE 5a AXIAL TURBULENCE INTENSITY u'/v*(6).Re = 22.6 * 103, p/d = 1.107

27

CO

3 = 0

2 ,0

1.5

1 ,0

0,5

0,0

RNGLEC.J.LRUN

J.LRUFER

o x\x x x x x x

•„ nD

D_0

30 +35 x

50

0.0 0.8

Y / Y MRX

1,2

FIGURE 5b AXIAL TURBULENCE INTENSITY uYv*(6).Re = 22.6 x 10 3; p/d = 1.107

1.6

I / I I I • I / \

FIGURE 6a AXIAL TURBULENCE INTENSITY u' /v*(0).Re = 207.6 x 103; p/d = 1.107

29

CO

ÛJ

3 = 0

2,5

2,0

L5

1,0

0,5

0,0

C,J,LRUN

J.LRUFER

* * * * * * * * * * *+ + + + + + + +

X

D

RNGLE

20 *25 +30 x35 o

o

O50 *

0,0 0,8

Y / Y HRX

FIGURE 6b AXIAL TURBULENCE INTENSITY u' /v*(9) .Re = 207.6 x 10 3; p/d = 1.107

30

1-8

1 61 r \— '

1,1,

£ 1|2CO

| l .O1 — 1

> 0,8

0,6

0,4

0,2

n nu, u0

i | i | i | . | . | i i i | i

1flNGLE -|

r. JJ RUN 10

J.LRUFER b + -0 x

^ a10 0 ~

1 b A -j20 ;i 1 — '-* /̂N^

-

~^~"^^\

^ r̂-%^^^ a -*x&^ x /x"^x xxx

sfc i M/ ' lî*— -J- *y %^ ^> ."A" ^^ !

-f ^

1 , 1 , 1 , 1 , 1 ,

0 0,4 0,3 1 2 1.

Y / Y P1RX

FIGURE 7a RADIAL TURBULENCE INTENSITY v/v*(9).Re = 22.6 x 10 3; p/d = 1.107

31

CO

2 = 0

1.8

1.- 6

1.2

1,0

0,8

0.6

0,2

0,00,0

C.J.LflUN

J.LRUFER

1

0,8Y / Y

flNGLE

3ll

35 xn

^5 o50 o

1.2 1.6

FIGURE Ib. RADIAL TURBULENCE INTENSITY v/v'(9).Re = 22.6 * 103 , p/d = 1.101

32

en:z:LU

2,0

1,8

1.6

1.2

1,0

0,8

0.6

0.2

0,0

C,J.LRUN

J.LRUFER

0,0

RNGLE20151050

10

0,8

Y / Y ilRX1.2

x

xD

O

1,6

FIGURE 8a. RADIAL TURBULENCE INTENSITY v/v*(9).Re = 201.6 x 103; p/d = 1.107

33

en

2,0

1,8

1.6

\,2

L O

0,8

0,6

0,4

0,2

0,0

C,J.LRUN

J,LRUFER

flNGLE

303540

50

xQ

O

O

0,0 0,4 0,8

Y / Y HflX

1.2 1.6

FIGURE 8b RADIAL TURBULENCE INTENSITY \/v*(Q).Re = 207.6 x \Q3. p/d = 1.107

34

CO

3,3

2,5

1-

_

c.J,

, 1 1 1 1 1 . 1

JJ RUN

LflUFER

i - i •• i i -,i

RNGLE ]- m -1* ]

- 5 + 1

2,0

1 ,0

0,5

0.00,0

^ o o

OA

c^ U

0,8

Y / Y HRX

x -

_O

~

H

i - b

FIGURE 9a. AZIMUTHAL TURBULENCE INTENSITY w/v*(9).Re = 22.6 * 1()3; p/d = 1.107

35

2.5

2.0

1,5

1.0 -

0,5

0,00,0

C.J,LRUN

J.LflUFER

o

flNGLE

253035

50

o o o

0,8

Y / Y nnx1.2 1.6

FIGURE 9b AZIMUTHAL TURBULENCE INTENSITY w/v*(6).Re = 22.6 x 103; p/d = 1.107

36

CO

3,0

2,5

2 = 0

1,5

1,0

0.5

0.00.0

C.J,LflUN

J.LRUFER

flNGLE

201510505

1015

0.8

Y / Y nRX

1,2

X

D

O

O

w

1.6

FIGURE lOa AZIMUTHAL TURBULENCE INTENSITY w/v*(9).Re = 207.6 x 103; p/d = 1.107

37

CO~ZLLU

3,0

2,5 -

2 = 0

1,5

1,0

0,5

0,00,0

C.J.LflUN,.

J,LflUFER_

flNGLE

20

30 x35 DH J O

U 5 050 ^

0,8Y / Y nnx

1.2 1.6

FIGURE lOb. AZIMUTHAL TURBULENCE INTENSITY w/v*(9).Re = 207.6 x 103; p/d = 1.107

38

! - U

0.8

0,6

OA

0.2(VI

*0 ,-\I |J *-

£D

-0.2

-(U

-0,6

-0/8

_ 1 n

i | i i • | - | , • | , , , ( ,•

RNGLt -

- 10 *

o o r " "f ~

/!\ ^- ,•"• y— \/ '/VV*'vl.'̂ ^ ' i . *

*O ^^« y ^ y A

° ,- ^ ̂ ' A S ^

'̂ X ? ̂ ° 0 O ̂ it; -j

+ XrxJ ^ » ^ J

T X X HS3

_

-

_ _

-

-

l , l , | . l , l , l , i ,

0,0 0,8

Y / Y flRX

1.2 i , l 5

FIGURE l l a RADIAL TURBULENT SHEAR STRESS uv7v*2(6).Re = 22.6 x 103. p/d = 1.107

39

.: -J

0,8

0,6

0,4

0,2OJ-#

0,0^ID

-0,2

-0,6

-0,8

-1,0

1 i ' i ' i ' i ' i ' i ' i '

£ RNGLE -o^ o 9C-

X f -J M/7f\O

%y- c* * ^

*V^>7f+ , V/ , M/ -10 X" + * + ^ ' V -o n x v ^ ̂ •" * f n

A v X +X * ao - x X x + * +

n x 5 H °o 0 n ^ c".n xv -j— * J U /\

0 ^ n v v>O V > r-, A

, V LJ

o ^ no v n

0 O xv

° o ° 0 D^No

0

0 -o

- -

- -

1 , 1 , 1 , 1 , 1 , 1 , 1 ,

0,0 0,4 0.8 1.2 1.,

Y / Y nnxFIGURE lib. RADIAL TURBULENT SHEAR STRESS "uv/v*2(9).

Re = 22.6 x 103; p/d = 1.107

40

oo

u u

0,8

0,6

0,,

0,2

0,0

-0,2

-0.6

-0,8

._1 n

I ' I ' I I ' I • I

RNGLE

- 20

- 15- + M

* & * ^ $ * * -10X ± ¥ W *s\ &• -r W pjjk c

o&x + + * n

O Q X + N ^ u

OQ A x " w c(•~S — 1 >^\ y\ "^ _I( T A J xv £-^ — '

rfJX^^ + |Q

^^ x/ s/* X

° ^^0 x ^ 15

cP^O X

o JP O° D

OO

-

1 , 1 , 1 , 1 , 1 , 1 , 1

-

x

X

D

0

O-

M

-

-

-

-

1

0.0 0,8Y / Y nnx

1,2 1,6

FIGURE 12a RADIAL TURBULENT SHEAR STRESS uv/v*2(8).Re = 207.6 x 103; p/d = 1.107

41

'- .- *-*

0,8

0,6

0,4

0,2CM-K-:>\ 0,0>

-0,2

-0,4

-0,6

-0,8

-1,00,

i ' 1 ' ! ' ! ' ! ' ! ' !

-

42

C\J

1,6

1,2 -

0,8

OA

0.0

-0.it

-0,8

-1.2

-1,60,0

flNGLE

* * * * * * *

*

x x x x x x x x x x x x

n _LJ n ̂ n nO

o

o0

O A -v '

10

20

OA 0,8V / V !̂

*1

j

,v

-

~

J

jI

FIGURE 13a AZIMUTHAL TURBULENT SHEAR STRESS uw/v*2(9).Re = 22.6 x 10 3; p/d = 1.107

43

C\j

I - 'J

0,8

0,6

OA

0,2

0.0

-0,2

-0.it

-0,6

-0.8

_1 n

i | . | . | • . | , | i | , |

flNGLE

25

30

35^0

^D

50

A -s -s /s 0 ° ° °" '" ^ O o O "" ^

o n a ° ° o ° °X hx n

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u

« + x

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X V V X

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^ + + + + +

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1.2 1,6

FIGURE 13b AZIMUTHAL TURBULENT SHEAR STRESS uw/v*2(0).Re = 22.6 x 103; p/d = 1.107

44

CNJ-x-

2,0

1,5 -

1.0

0,5

0,0

-0.5

-1.0

-1.5

-2,00,0

1 ' 1 ' 1 ' 1

w

1 ' 1

RNGLE

NN M -

N15

10

010

15

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D

D

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X

X X

+

X

n

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1 , 1 1 , 1

1,20,^. 0,8

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FIGURE 14a AZIMUTHAL TURBULENT SHEAR STRESS uw/v*2(6).Re = 207.6 x 103; p/d = 1.107

1.6

45

C\J-K-

!,0

1,5 -

1,0

0,5

0,0

-0,5

-1,0

-1,5

-2,0

• • '

RNGLE

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+ + +

** +

x

+

0 nX n n 0

n o o O o 0 o

, 1 , 1 , 1 , 1

25 +30 x35 D4 J Q

50

0,0 0,8

Y / Y HRX

1.2

O _

1,6

*2,FIGURE 14b. AZIMUTHAL TURBULENT SHEAR STRESS uw/v*z(9).Re = 207.6 x 10 3; p/d = 1.107

46

fvl

i . Q

0,8

0.6

OA

9.2

0,0

-0,2

-0.4

-0,6

-0,8

-1 -00.0

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X X

PCPx

Q

47

f\j

0,8

0,6

n ? r-

0,0 -

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; ,

\/ \j/j— Q— i """T^ ^v

, 1 , 1

.0 OA

1 ' 1 ' 1 ' 1 ' 1 '

L J X -

30 +35 x^0 r-

^5 o

50 0

C\J

.1 , 'J

0,8

0.6

O.^t

0,2

0,0

-0,2

-0.

u u

n ou,o

0,6

0,,

0,2C\J•X

^>

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H

-0,2

-0,6

-0,8

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i ' i ' i ' i ' i ' i ' i '

RNGLE20 *

25 +30 x "

* "3^X J-J D

+ * * + * Lf]* *+ + x +

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- ^0^+^ o^* x^^^o^O^O^

o »nx x n n

n

- -

, i , i i i i i i i . I , i

0 0,tf 0,8 1,2 1.

Y / Y nnxFIGURE 16b TRANSVERSE TURBULENT SHEAR STRESS vw/v*2(9).

Re = 207.6 x 103; p/d = 1.107

50

ori

CHCD

LLJHOZI

o:

a:

n r-o, j r

SYI1BOLS

7 Q — ri >K Radial turbulent shear stress

b -|- pressure con.ponent

C v radial integral of azimuthal5,0 r

;.o -

3,0

2 = 0

1,0

0.0

-1,0

ngradient of uw

balance term

xX

xX

\ /

X\..

X

Y RX

1 !

1.0

FIGURE 17a AXIAL MOMENTUM BALANCE. Re = 22.6 x 103;

ANGLE THETA = 0°; p/d = 1.107

51

U., 'J 1

i

7 , 0

!

6,0

•• i ' i • ' i ' i ' i ' i • -

: SYMBOLS :

- R ^radial turbulent shear stress ;

f- LJ -j- pressure component ,"*" ~-J F ~ . n *' :l— ' L- Q L_ L x radial integral of azimuthal ^ _1Q: t n gradient of uw ,

+ ^— d P C balance term *- :

m f .- ' :

r '̂ ' Tx t ' "1^." y- 7i\~^^ \L^-- |̂̂ . — <

; . | ^ . ' \ _^ "J

£ - .n t- /•" ^ \ 4' ! — " h- \,/ Vi/ -^1

cr^ 2, 0CC

1 . 0

0, 0

_ 1 n

f- -f- ffl 7\. _

/ \ ~

- ^K -: :

; -

: &-a :^ Nn. :

N u"1 ~ys_ G ^^^

; " ̂ X-x- x- x-x^ ̂ ^7^ :

: 1 , 1 , 1 , 1 , 1 , 1 , :

0U i 0,8

Y / Y MRX

1,2

FIGURE 17b AXIAL MOMENTUM BALANCE. Re = 22.6 xANGLE THETA = 15°; p/d = 1.107

1.6

52

7 . 0

LiJLJZ"or

IIccCD^i5h—i , ti_LJ£o

_jor,

a:

6,0

5.0

it, 0

3=0

2,0

1 = 0

0,0

-1.0

-2,0

_Q n

•1: 1

E SYMBOLS -]: R v radial turbulent shear stress :1

?!t *1- -1

I D + pressure component ,+ "]

— Q radial integral of azimuthal .^ -3gradient of uw , " Jn + 1: — I D balance term , ' 3

— . •' — j

: - * '\~- ->>-' -i? - 3E. '̂ _:;- • "j•t- -.. 7 • ^- :»̂ — w_))ê r̂ )j/ J

- ' Nv 1

;; ?r )it,__ ;i"̂•ii' -<

: ^ J~ "̂̂ v̂̂ "1U^K 1- ''H — -H -i

"̂ -i-i '1LJ" — Q. "]

: "' g- -ĝ _ _ .]Nj5 - -^r— G 3

- X^ x ^: ""X ^: x-^ -!r "x'"-x 1

""̂ v iA , -1X- 1

'*-t '"

" ' ^

~~- , I , ! , I , I , I , i , i , '\

0,0 0.^ 0,8

Y / *. MHX

FIGURE I7c AXIAL MOMENTUM BALANCE. Re = 22.6 x 10 ;

ANGLE THETA = 25°; p/d = 1.107

53

8 - 0!J F

CE_Ja:

LU

orxor

?. fi b

0,0

-1 .0

-3,00. 0

SYHBOLS

H x radial turbulent shear stress

I- —

^ D

(_•LJ 4. pressure component

Q radial integral of azimuthalgradient of uw

G balance term

-+'

+'

— ov7^ •— .̂xj/1

*" -- -

0,8

Y / Y nnx1,2

i

FIGURE 17d AXIAL MOMENTUM BALANCE. Re = 22.6 x 103;ANGLE THETA = 45°; p/d = 1.107

54

10,0

LJ•z.a:_iCECD

CTI 1

Xor

8,0

6,0

-0,0

-2,0

0,0

SYMBOLSfl ̂ radial turbulent shear stress

D 4. pressure component X

C x radial integral of azimuthal Xn gradient of uw v^ balance term ' -+

1,20,^f 0,8Y / Y nnx

FIGURE 18a AXIAL MOMENTUM BALANCE. Re = 207.6 x l()3.ANGLE THETA = 0°; p/d = 1.107

55

LJ

ororCD

O

CEi—iXcr

7,0 E

6,0 -

5,0

i i i

3,0

2,0

1,0

0,0

-1,0

2,0

SYMBOLS

H & radial turbulent shear stress

D 4. pressure component

C v radial integral of azimuthalgradient of uw

^ D balance term

0,0

N,

O.*t .1,20,8v / Y NRX

FIGURE 18b AXIAL MOMENTUM BALANCE. Re = 207.6 x 1Q3;ANGLE THETA = 15°; p/d = 1.107

1,6

56

LJ

CE_Jcren

IDi—z:LU

On

_Ja:i—ixcr

8,0

6.0 -

2,0

-0,0

-2,0

-4.0

-6,00,0

SYMBOLSp) * radial turbulent shear stress

g + pressure component

Q X radial integral of azimuthalgradient of uw

[] Q balance term 4.'

-*'

,.',,

3 - -B—e ^

x-

x.

1,2O.it 0,8Y / Y MRX

FIGURE 18c AXIAL MOMENTUM BALANCE. Re = 207.6 x 1Q3;ANGLE THETA = 25°; p/d = 1.107

1-6

57

LULJ

cc11

crDD

z:IDh-LUnozz_lori ^

Xcr

o, u

7,0

6,0

5,0

^0

3,0

2,0

1.0

O n3 U

-1,0

-2,0

_q 0

. 1 1 | 1 | ! | 1 | 1 1 | , J

j SYMBOLS i~ H ;K radial turbulent shear stress ~I D :z D 4. pressure component /•*" ~^_ C v radial integral of azimuthal ^ j:- p. ' gradient of uw ^ -- n balance term ^ ~-

'-- / -i

': ^ '-

~r ,'*' ~

'--_ ^ _

\ "^^^^ \

\ *'* ^"^HS^ =| ^^^^^^^^^^Ss^^ E''- ^^xX^B^^]^"s-H

1-' ^^"x ->v :: ^X :

: , 1 , 1 , 1 , 1 , 1 , 1 , 1 , :

0,0 (U 1.-20,8Y / Y nnx

FIGURE 18d AXIAL MOMENTUM BALANCE. Re = 207.6 x 103;ANGLE THETA = 45°; p/d = 1.107

1,6

58

5,0

UJ

\L\

i —y i nIT - - J

1 —

LlJ

m 2 = 0ccIDI —

uo

n n

: REYNOLDS NO- ?26Gn P/D

" r...l, LflUN

- \

- \: \

\~ s

: \\

— \

: \: x

\

i , i , i i. i ,

H7

r> o0, u

"-'

FIGURE 19a TURBULENT KINETIC ENERGY.Re = 22.6 x 1Q3; p/d = 1.107

X

59

1

jl

hL" 0 t--' J 1-

[- tv 1-

_,

:JD i •-. L01 "-t . '-> L: ; i i_

•"̂ J t

- tLL

•— ; L2 3, 0V"

t—21

~

--~

!t! -— ' ° !~!en ^ = JCC

,'h"

i , 0

r> T".u, u

-_----~

-

-

0, 0

-

RNGLE THETfl-r. j. i RUN :

30 + 135 x :

P- "*4 J Q

\ so o 1\ J\\ :

\\\

"\N :o "$ \X . \ ., "

O "^^vy \ v,- « ^ 5K ^" J X ^ r « V ) K ' ^ ' ^ . ^ + + +

rn^\y"r "•" ">" + -

o ^ \ X x x x ,0 \ x x xG N X

0 0 rX -

o ^v n :o o x .̂ G n

° o C^^~7\ A ^o ~/~*l *S j' >̂ J v/ ~"

o o o u -

1 i 1 1 1 1 1 1 1 1 1 1 1 1 1

OA 0,8 1.2 1,

Y / Y HRX,

FIGURE 19b TURBULENT KINETIC ENERGY.Re = 22.6 x i ( )3 ; p/d = 1.107

ou

6,0

LDCCUJ

LJ

LU

onor

5.0

3,0

2,0

C.J. LRUN

1,0

flNGLE THEFR;- 20 * :- 15 + :- 10- 5

10

X

D

0 o5 o

w

0.0

J.O 0,8

Y / Y MflX,

1,2

FIGURE 20a TURBULENT KINETIC ENERGY.Re = 207.6 x 1Q 3 ; p/d = 1.107

1.6

6,0 i i r

CDCC

LU

Oi—iI—LU-z.•^

t—ZLiJ_lID(TlCH

5,0

3,0

2,0

,Ja LRUNRNGLE TMFTR-

20 * :

25 + :30 x :35 D ;

^5 o50 ^

0,00,0 0,8

Y / Y nnx,1,2

FIGURE 20b TURBULENT KINETIC ENERGY.Re = 207.6 x 103. p/d = 1.107

1.6

C\JCE

O

CT01

^ n -^', -J

2 ,0

1-5 -

1.0

0,5

0,00,0

r ,'

0

N0

W

wD Dn n D °

Xx x

X

vX

Z J0

*

v/ t

X X X

I , I

OA 0,8Y / Y MRX

FIGURE 21a REYNOLDS SHEAR STRESS RATIO A2.Re = 22.6 x 103; p/d = 1.107

< x xi - 5 r "' x n a

1.0

0,5

0,00,

^ ^ f-j LJ LJ ps^ _J_ \s i-J 1 — '

- ^ x ^ x n D a- X"\CJ E.J 01 uJ ^v ^^ ^^^ t-J

*^ "̂in~ £± V

° -s ^ ° OV. > /li

: % ^ - °O ^

1 , 1 , 1 , 1 ,"" 1 , 1

0 OA 0,8 1.2

j/

-

-

—

-

—

-

—

-

—

-

-

-

L

Y / Y HRX

FIGURE 21b REYNOLDS SHEAR STRESS RATIO A2.Re = 22.6 x io3; p/d = 1.107

64

-i^,0*10

C\Ja:o

crcc

3,5 -

3,0

2., 5

2,0

1,5

1,0

0,5

0.00,0

X

ft x4-.

*

oo o

RNGLE- 20- 15- 10- 5

0

10

X

noO

4.t%

20

yx

n0n n n

o oo

° °1 , nl

0,8Y / Y MRX

1,2

FIGURE 22a REYNOLDS SHEAR STRESS RATIO A2.Re = 207.6 x 103; p/d = 1.107

1,6

65

4,0*10

CSJen

o

crcc

3,5

3,0

2,5

2,0

1,5

0,5

0,00,0

RNGLE20 *25 +30 x35 D

45

5055

o

* * ^_ +* +

X X

'+ x xX

X D

D

x ( < > D - © - 0°

cO o

O,H 0,8Y / Y nnx

1,2

FIGURE 22b REYNOLDS SHEAR STRESS RATIO A2.Re = 207.6 x lo3; p/d = 1.107

1.6