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16th Australasian Fluid Mechanics ConferenceCrown Plaza, Gold Coast, Australia2-7 December 2007

The development of turbulent pipe ow

J. Doherty, P.Ngan, J. Monty, M. Chong

Department of Mechanical Engineering, The University of Melbourne, Parkville, VIC 3010, Australia

Abstract

Whilst turbulence still remains one of the great mysteries of classical physics, its reputation for a chaotic lack of structure isunder intense scrutiny. Research now points to the existence of highly organized large-scale structures within turbulent ows.Much of this research have almost exclusively been carried outwithin a so called fully-developed region of ow. This regionis downstream of a point beyond which the ow elds behav-ior is invariant. However, it has not been conclusively proventhat a fully developed region actually exists. A literature survey

found that most facilities are cited as being sufciently long,yet no formal denition or documentary evidence of the fullydeveloped condition is widely accepted. The aim of the studyis to produce a detailed analysis of the ow from the uniforminlet conditions through to the fully-developed turbulent state,along with an accurate denition of what constitutes fully de-veloped ow. This investigation concerns the affect of both thegrowth of large-scale structures and their role in the evolution of the ow to the fully-developed condition. With the increasingacceptance of the turbulent large-scale structures model, previ-ous research has not yet shown how large-scale structures af-fect this development length. These aims were achieved by thedesign, development and deployment of a carriage to transporthot-wires within the pipe allowing measurements at any streamwise point within the working section

Introduction

The study of development length of duct ows has been ap-proached many ways over the years, however recent develop-ments in the understanding of the nature of turbulent ows in-cluding the existence of highly ordered eddy trains or large-scale structures [4] within the ow have led to the authors re-newed interest in development length work. Previous investi-gations of internal ows such as pipes, have almost exclusivelybeen carried out within a so-called fully-developed region of ow. This region of ow is believed to be streamwise invariant.However, it has not been conclusively proven that a fully devel-oped region actually exists, or how far downstream if it does.A literature survey found that most facilities are cited as being

sufciently long, yet no formal denition or documentary ev-idence of the fully developed condition is widely accepted, thisis however not a product of inactivity on the part of researchers.As data acquisition and analysis methods have advanced, moreandmore stringentconditionshavebeen placed on thedenitionof fully developed ow. Each subsequent work on the matterhas indeed shown the existence of a fully developed ow satis-factory to their own conditions often adding a subsequent mea-sure to the work preceding theirs. Despite this extensive timeand effort spent on pipe ow development research, the lack thereof is mainly attributed to the difculty in gaining accessto sufcient stream-wise positions within the working sectionfor data collection. Taking, for example, the pipe ow appara-tus used in the current study measuring stations were originallyonly available at 60 D intervals, where D is the pipe diamter.

Early judgements of development length took only mean veloc-ity into consideration. For example, [7] suggested fully devel-oped mean ow was achieved in his experiment at 40 D. More

detailed work by [2] showed a mean velocity overshoot withinthe inlet region of the pipe before the centerline velocity settledto its nal fully developed value. This overshoot phenomenonwas shown to be subtle and of short streamwise duration. Thelatter observation would imply that any detailed study of devel-oping pipe ow will likely miss important ow changes due tothe aforementioned difculty of having good streamwise reso-lution of measurement stations.

In later work by Perry & Abell [8] at higher Reynolds numbersup to 175 103 , a development length of 71 .9 D was suggestedfor the ow to become fully developed (Reynolds number is de-ned as Re = U b D/ , where U b is the bulk velocity and iskinematic viscosity). Their denition of fully developed owrequired that the mean velocity and the turbulence intensity, u 2proles were the same for two consecutive measurement sta-tions. Even more stringent conditions were imposed by Za-garola & Smits [9] who suggested an empirical model based onthe characteristic time scale for the development of large scalestructures. This empirical form was calibrated using measure-ments by Dean & Bradshaw [3], again acquired at only twopoints.

Recent investigations into the large-scale features of turbulence(e.g. Adrian [1], Hutchins & Marusic [4] and Monty et al. [6])have revealed coherent structures up to 20 R [6], where R is thepipe radius. With such large stream-wise lengths, it seems rea-sonable to assume that their development to an invariant statemay be critical in the development length problem [9].

Clearly then, the development length required to ensure fullydeveloped turbulent ow in a pipe is neither certain nor is thedevelopment process well documented. Therefore, the aim of the study is to produce a highly detailed analysis of the owfrom the uniform inlet conditions through to the fully-developedturbulent state, along with an accurate denition of what consti-tutes fully developed ow.

Experimental apparatus

Pipe ow facility

The pipe ow facility is located in the Walter Bassett Aerody-namics Laboratory at the University of Melbourne. A radial fanlocated at the exit sucks air through a honeycomb screen intoa settling chamber interspersed with 5 mesh screens at the en-trance to the pipe. Immediately following the settling chamberis a 5.2:1 circular contraction. These provisions are intendedto introduce a uniform, low turbulence intensity ow into thepipe. The entrance to the 0.0988 m diameter, drawn brass work-ing section is lined with a 150 mm long, 60 grit sand paper trip.Each section is nominally 6 metres in length and the overallpipe length is approximately 400 D. Five pressure tappings lo-cated at approximately six meter intervals enable the pressuregradient to be measured accurately. Results were obtained at Re 100 103 and Re 200 103 .

Carriage

In order to obtain the measurements required to improve our un-derstanding of turbulent pipe ow development, it was required

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that a probe be built capable of any streamwise movementwithin the pipe working section whilst minimizing the ow dis-ruption. Furthermore it was required that the positioning of the probes was accurate such that a gure for the fully devel-oped length could be determined with minimal error. Following

much discussion, the authors realised that the desired measure-ments could only be obtained with an internal carriage. The -nal result is illustrated in Figure 1. The slender design featuresa narrow body and two sets of three legs spaced equally aboutthe centerline (i.e., 120 ). The upper legs are equipped withtwo magnets which are attracted to magnets housed in an exter-nal guide apparatus, through the brass wall of the pipe. Mag-netic coupling between the two components allows the probeassembly to be moved within the working section without theneed to stop the ow or disassemble the working section. Fivehot-wire anemometers are arranged vertically, each attached to15cm tubes protruding from a vertical bar mounted on the frontsection of the carriage. Measurements were taken simultane-ously at 0 . 3 R, 0 . 5 R, 0 . 7 R and 0 . 8 R from the pipe wall with thefth hot-wire mounted at the centerline R. To enable in situ cal-ibration of the hot-wire anemometers, a custom built pitot-statictube was mounted at 0 . 2 R. A seven-meter-long, custom-madeumbilical cable attached the carriage to an AA Labs AN1003anemometry system and MKS 680 pressure transducer. A Mi-crostar Laboratories 4000a data acquisition board allowed si-multaneous sampling on all six channels (ve hotwires and onepressure transducer) at 20 kHz for two minutes. This total mea-surement time corresponds approximately to the time taken for36000 and 72000 pipe radii to pass the hot-wire probes for thelow and high Re respectively.

Figure 1: Hot-wire probe carriage illustrated within the work-ing section of the pipe. The external magnetic guide used toposition the probes during measurement is also shown.

Experimental Procedure and Data Analysis Methods

Measurement procedure

Utilizing the aforementioned pitot static setup, calibrationwas performed before each length of pipe was measured.Each of the hot-wires was calibrated in situ at the end of afully-developed channel ow. The resulting calibration datawere subsequently tted with a fourth order polynomial,allowing the velocity of the data sampled from the pipe tobe calculated. The inherent delicacy of the hot-wire arrayand the time required to traverse each six metre section of pipe required that only one 6 metre section could be sampledbefore re-calibration. For each section of the pipe, velocity

measurements were taken at intervals of 2.5 D, spanning fromthe pipe entrance to a distance 228 D downstream. The pressuredrop in the streamwise direction, the air temperature andpressure were measured before and after the data collection foreach 6-metre section.

Non-dimensionalised quantities

The inability to traverse the entire 200 D length of pipe in onesitting led to slight mismatching of Reynolds number betweeneach 60 D experiment. This led to the need to scale data fromseparate runs to produce consistency in the statistical and spec-tral analysis contained in later sections. For this reason, thefriction velocity, derived from the the wall shear stress, w wasrequired:

U 2

= w = D4

d pdx

. (1)

The pressure gradient is reported to become constant veryearly in the development of the ow [2], meaning that pressuretappings located more than 100 D from the inlet should providean accurate value of U .

Statistical analysis

As discussed in the introduction, previous research has focussedon the convergence of ow statistical quantities as an indicationof the development length. Thus, for each measurement loca-tion these same statistical quantities were calculated for eachhot-wire from the data collected over two minutes of samplingtime. Included in the analysis are the local mean velocity, vari-ance, u 2 , skewness, u 3 and atness, u 4 .

With a renewed importance being placed upon the role of large-

scale structures, the main1

analysis performed was that of spec-tral analysis via Fourier transforms. The benet of this tech-nique is that it identies the wavenumbers of structures con-tributing most to the kinetic energy of the ow. As such, the de-velopment of the spectra was analysed, paying particular atten-tion to the energy associated with the low wavenumber (large-scale) structures. Fourier spectral analysis is a common tech-nique employed to detect these structures, however it should benoted that it has the limitation of assuming the structures havesinusoidal velocity signatures. For this comparative study, thelimitations of the technique are not expected to impact the con-clusions drawn.

Results and Discussions

Azimuthal invarianceThe rotational mobility of the probe enabled ow measurementsto be taken at radial positions 90 degrees apart. This enabledin situ measurement of the ow at different azimuthal orien-tation with the same experimental rig which permitted a mea-sure of the symmetry of the ow. The data recorded at vari-ous azimuthal positions indicate axis-symmetry to within 2%.These measurements were taken to help establish the quality of the ow and ensure all conclusions were due to axis-symmetricow development only.

Velocity proles

Figure 2 shows the mean velocities at the centreline, U CL withdashed lines marking 1% from the nal value. All velocitieswere scaled with the average U CL , except over the rst 45 D,where U CL was taken to be the mean of the last three data points

1 in terms of CPU hours required

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Figure 2: Centreline mean velocity along the streamwise direc-tion for x / D = 0 to 230 for Re 100 103 and 200 103 . Datapoints for Re 200 103 vertically shifted 0 .2 units for clarity.

Figure 3: Centreline mean velocity scaled with U and the cor-responding turbulence intensity u 2 plotted against streamwisedistance, x / D .

of the section (due to the lack of development). These resultsagree reasonably well with the ndings of Abell and Perry [8]who suggested that the mean velocity was invariant after 71 .9 D .In fact, as this study provided measurements taken at very smallstreamwise intervals, the mean velocity appears to be invariantat a streamwise distance of x / D 60 for Re 100 103 . Therequired length for invariant mean velocity was extremely sim-ilar for Re 200 103 , although no conclusions can be drawnabout Reynolds number effects since the two Re are not signi-cantly different in magnitude.

The two velocity variations shown in gure 2 support the veloc-ity peak observations of [2]. Centerline velocity distributionsfor both Reynolds numbers indicate a distinct overshoot beforethe mean velocity becomes invariant. It should be remarked thatthis overshoot occurs over a relatively short streamwise distanceof approximately 10 D . As previously discussed, measurementsperformed at larger intervals by past workers may well havemissed the peak all together. A plausible physical argument forthe existence of the velocity peak may be: as the ow entersthe pipe and slows due to shear at the wall, the central coreof the ow speeds up to maintain a constant mass efux. Thecentreline velocity must then reach a peak value as the core di-minishes, before settling to a lower mean velocity. Comparisonof the variance with the position of the mean velocity peak is

Figure 4: Variance scaled with U 2 , skewness, and kurtosis for Re 100 103 , in the streamwise direction.

Figure 5: Variance scaled with the fully develop U 2 , skewness,and kurtosis for Re 100 103 , along the streamwise directionfor x / D = 0 230.

shown in gure 3 and suggests that the peak velocity is dueto the reduction in cross-sectional area of the laminar core asthe axially symmetric boundary layer encroaches on the centerof the pipe; the peak occurring just before the boundary layerreaches the pipe centerline and the transition to a fully turbu-lent mean velocity value is complete. The lagging peak of theintensity statistic is indicative of the peak centreline velocity oc-curring within the intermittently turbulent core region with thecentreline velocity decreasing as the transition to fully turbulentow is made.

Higher-order statistics

Figures 4 and 5 display development of the higher-order tur-bulence quantities (variance, skewness, kurtosis). Referring tokurtosis in particular, peaks are associated with sudden largeuctuations about the mean, which is indicative of the transi-tion from ow with intermittent turbulence to fully turbulent.Shortly after the peak, u 4 becomes invariant for the remain-ing length of the pipe. The skewness displays much the samebehaviour. For the variance, the higher Reynolds number datashow that this quantity reaches its fully developed value beforethe higher-order statistics which might be expected. However,

for the lower Re the variance slowly drops to its nal value at 110 D . However, since this appears to be an anomalous result,further investigation will be required to conrm it is accurate.Setting this result aside, it would appear that the highest-order

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statistics are fully developed by 80 D as suggested by Perry &Abell [8].

Spectral development

Plots of pre-multiplied spectra k x uu versus k x R for increasingvalues of x/ D yields a more complete picture of the structurelengths present within the early stages of the transition to fullydeveloped ow. Due to the unprecedented amount of data beinggenerated and the investigation focussing on how the ow phys-ically develops, a more intuitive method was required for thepresentation of data. Conventional methods of plotting k x uuversus k x on two-dimensional log-normal axes did not allowfor the streamwise development to be clearly visualised. Assuch, the maps presented in gures 6 and 7 below attempt toconvey the desired information. A top down view of the con-tour map of the pre-multiplied energy spectra enables the rapididentication of important ow features. The horizontal axis isthe streamwise position within the pipe working section withthe vertical axis denoting wave number, k x; red colouring indi-

cates high energy regions and blue the low energy regions. Bothgures illustrate the growth of the turbulence from the laminarinlet conditions. With laminar conditions existing at the inletthe plots show extremely low energy for all wavenumbers, asexpected. The region of the ow where the core becomes in-termittently turbulent is indicated by the growth of the energypeak between k x R = 80 and 100. It is thought that this methodfor the presentation of large amounts of spectral data enablesbetter isolation of trends within the development of turbulentow than would have otherwise been possible.

Figure 8: Pre-multiplied spectra k x uu at y/ R = 0.5 for x/ D =57 and 107. Note the shift in the position of the secondary, lowwavenumber peak.

As x/D increases past 50, the bulk shape of the pre-multipliedspectra begins to converge to a nal shape. This initial transitionis well dened and easily identied, implying that the turbulentow quickly reaches a fully developed condition with respectto the energy spectrum. However, a closer look in gure 8 il-lustrates the divergence towards lower wave numbers of the lowwave number energetic peak centred roughly around k x R = 5.As the ow develops, the eddy packet hypothesis [5] suggeststhat larger numbers of eddies, termed hairpin vortices are fallinginto line. These large-scale structures continue to entrain ed-dies as the ow develops, thus shifting energy further into thesmaller wavenumbers. As these eddies are merely becomingmore aligned, the contribution made to the overall ow energy

is not substantially increased (as evidenced by constant varianceshown in gure 5). This therefore represents a redistribution of the energy of the turbulent motions of the ow, with the prevail-ing tendency towards development of the longer structures.

Figure 9: Illustration of the collapse of ten individual EnergySpectra lines k x uu versus k x R from between x/ D = 184 and210.

If the fully developed condition is to actually exist it is impor-tant that the movement of this peak be traced to an invariant po-sition in line with the denition of fully developed ow. Figure9 illustrates the collapse of ten separate energy spectra curvestaken at a wall normal distance of 0 .5 R. Note that the plotscome from between x/ D = 184 and 210 to ensure that fully de-veloped conditions were prevailing. The invariance of the po-sition of the low wavenumber energetic peak, indicative of thelarge-scale structures, would suggest that the long eddy trainshave reached a critical length beyond which they do not grow.This entrainment process would then present the nal hurdle inthe process of the ow becoming fully developed.

Conclusion and further research

To bridge the 60 D gap between measurement stations, the suc-cessful design and construction of a novel internal carriagehousing ve hot-wire anemometry probes has enabled the rstpipe ow measurements to be made with high streamwise reso-lution within the working section. The results have helped pro-vide a much better understanding of how the turbulent boundarylayer develops from the trip; in particular, the growth of large-scale structures and their role in the evolution of the ow to thefully-developed condition.

This preliminary study shows the gradual development of thecentreline velocity, with indication of the location where theboundary layers may have merged, and the interaction between

them. Noting that the mean velocity prole requires develop-ment over approximately 50 D to become invariant, statisticalanalyses reveal that when consideration is made of the higher-order statistics, a longer length of approximately 80 D is re-quired. However, this length was further increased by the ad-dition of the development of the large-scale ow structures asa criteria for fully developed ow. It was shown these struc-tures grow relatively slow, even though their presence was notedvery early in the working section. It was found that large scalestructures were continuing to grow in size past the points earlierthought to be fully developed. As the large-scale structures re-quire a longer development length, it is proposed that the maincriteria for a fully-developed pipe ow should focus on the de-velopment of these structures, rather than on the mean velocityprole or higher order statistics.

References

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Figure 6: Pre-multiplied pipe spectra k x uu for y/ R = 1, plotted against distance from inlet x/ D and k x R for Re = 200 103

Figure 7: Pre-multiplied pipe spectra k x uu for y/ R = 0.5, plotted against distance from inlet x/D and k x R for Re = 200 103

[1] Adrian, R. J., Hairpin vortex organization in wall turbu-lence, Physics of Fluids, 19 , 2007, 041301.

[2] Barbin, A. R. and Jones, J. B., Turbulent ow in the inletregion of a smooth pipe, Transactions of the ASME, 85 ,1963, 29 to 34.

[3] Dean, R. B. and Bradshaw, P., Measurements of interactingturbulent shear layers in a duct, Journal of Fluid MechanicsDigital Archive, 78 , 2006, 641676.

[4] Hutchins, N. and Marusic, I., Evidence of very long me-andering features in the logarithmic region of turbulentboundary layers, Journal of Fluid Mechanics, 579 , 2007,128.

[5] Kim, K. C. and Adrian, R. J., Very large-scale motion in theouter layer, Physics of Fluids, 11 , 1999, 417422.

[6] Monty, J. P., Stewart, J. A., Williams, R. C. and Chong,M. S., Large-scale features in turbulent pipe and channelows, Journal of Fluid Mechanics Digital Archive, 589 ,2007, 147156.

[7] Nikuradse, J., Gesetzmassigkeiten der turbulenten stro-mung in glatten rohren, ,, Forschung auf dem Gebiet desIngenieurwesens, Translated in NASA TT F-10, 359, 1966,

3 , 1932, 136.

[8] Perry, A. E. and Abell, C. J., Scaling laws for pipe-owturbulence, Journal of Fluid Mechanics, 67 , 1978, 257271.

[9] Zagarola, M. V. and Smits, A. J., Mean-ow scaling of tur-bulent pipe ow., Journal of Fluid Mechanics, 373 , 1998,3379.

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