oerror, to which Patel (1965) responded with a new calibration,withcalibprovequa

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limits on the pressure gradient conditions in which theration was valid. Although others have suggested im-ements to Patels work [e.g., Poll (1983b)], the calibrationtions as given by Patel (1965) have since been widely

. Lect., Engrg. Sys. Dept., Royal Military Coll. of Sci., Cranfield, Shrivenham, Swindon SN6 8LA, U.K.aj., Singapore Armour, SAF Armour Ctr., Sungei Gedong Camp,pore 2471.te. Discussion open until November 1, 2000. To extend the closingne month, a written request must be filed with the ASCE Managerrnals. The manuscript for this paper was submitted for review andle publication on September 1, 1998. This paper is part of theal of Hydraulic Engineering, Vol. 126, No. 6, June 2000. qASCE,0733-9429/00/0006-04070415/$8.00 1 $.50 per page. Paper No..

measurements were carried out near his lower Reynolds num-ber limit, nor is it possible to deduce their influence upon thefinally presented calibration equations. However, it seemslikely that his measurements in lower Reynolds number flowscorresponded to the lower regions of x* = log[Dpd 2/(4rn 2)]and y* = log[t0d 2/(4rn 2)], and here the plotted data appear tobe relatively sparse with more scatter.

Poll (1983a) presented an analysis for a laminar boundarylayer with pressure gradient limitations, the outcome of whichwas a calibration equation valid in a limited range of Dpd 2/4rn 2. After applying the same equation to a turbulent boundarylayer and comparing it with Patels calibration in the rangeDpd 2/4rn 2 < 850, Poll concluded that it should be capable ofproviding an estimate of skin friction to within 10%, for lam-inar and turbulent boundary layers with 120 < Dpd 2/4rn 2

1,000. However, given the apparent sparseness of Patels datain this region, together with the absence of any measurementsof a Reynolds number below about 6.5 3 103, the physicalevidence for validating this or any calibration equation appliedto low Reynolds number turbulent flows seems far from ade-quate.

The work presented in this paper was carried out to redressthis situation by providing a database of measurements for lowReynolds number turbulent flows. Circular pipe flows werechosen so that the wall shear stress could be readily and ac-curately obtained by measuring the pressure gradient. Furtherwork on developing boundary layers under various pressuregradient conditions and on developed flows in rectangularducts is required to give a more comprehensive treatment.However, the present results show the scope and limitationsassociated with using the Preston tube in low Reynolds num-ber turbulent flows.

SIGNIFICANCE OF REYNOLDS NUMBER ANDPRESTON TUBE DIAMETER RATIO

The methodology adopted for the present study was to in-itially analyze the measurements as though the physics of thePreston tube could be fully represented by the near-wall var-iables x* and y*. Then, the other variables believed to be ofphysical significance, namely, Reynolds number and the ratiot of the Preston tube internal to external diameter, were takeninto account to improve the model. Here we briefly discussthe significance of the latter variables.

Reynolds NumberAs previously mentioned, Patel and Head (1969) reported

measurements in low Reynolds number pipe flows that clearlyshowed that the velocity profile in the logarithmic region ofthe law of the wall deviated significantly from the standardform for Reynolds numbers of less than about 9.2 3 103. Us-ing these results, Afzal and Yajnik (1973) demonstrated theReynolds number dependence of the additive constant B in thelogarithmic law.

More recently, Barenblatt (1993) controversially proposedthat a better model for the fully turbulent region was a powerlaw scheme in which both the power exponent and the mul-tiplicative factor were functions of the Reynolds number. Sup-port for this was adduced by Barenblatt and Prostokishin(1993) from the measurements of Nikuradze (1932).

However, on the basis of more accurate measurements, Za-garola et al. (1997) showed that the overlap between the wallregion and the outer region was better represented by a powerlaw (approximately the 1/7-power law) for 50 < y1 < 500 andfurther from the wall by a logarithmic law. This logarithmicsection ceased to exist for Reynolds numbers of

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FIG. 1. General Arrangement of Equipment

FIG. 2. Friction Coefficient-Reynolds Number Relationships forfor Larger Pipe Diameters Have Been Successively Shifted Upwar410 / JOURNAL OF HYDRAULIC ENGINEERING / JUNE 2000Four Pipes; Graph for D = 6.184 mm Is Plotted as Measured; Graphsd by log 2 for ClarityTABLE 1. Preston Tube Experimental Parameters

Pipe diameter(mm)(1)

R range(2)

Preston Tube Gauge Numbers20(3)

21(4)

22(5)

23(6)

24(7)

25(8)

26(9)

27(10)

29(11)

15.789 1,0008,500 * * * * * * * * *12.684 1,00010,000 * * * * * * * * *

9.483 1,00010,000 * * * * * * *6.184 1,0007,000 * * * *

Note: Combinations indicated by asterisk.

open downstream end of the selected pipe, in contact with thepipe inside wall, with the Preston tube open end about 1-mmupstream of the downstream static pressure tapping. Thus, theposition of the Preston tube unambiguously defined its block-age effect upon the static pressure measurement. The Prestontube dynamic pressure was measured by means of a secondelectrical pressure transducer with ranges of 01, 10, and 100N/m2, taking account of the blockage effect upon the staticpressure.

In every experiment the atmospheric pressure was recorded.The airflow temperature and relative humidity were measured

at the pipe exit, and the gauge pressure was measured at theoutlet from the flow meters. The indicated airflow rate wascorrected in each case for temperature and pressure, to give avalue corresponding to pipe exit conditions.

Experimentation proceeded in two stages. Stage 1 was de-signed to establish an experimental method that would deliverresults with an acceptable degree of accuracy. Stage 2 con-sisted of the Preston tube measurements and analysis.

In Stage 1, the pressure drop and flow rate in each pipewere measured over the full range of Reynolds numbers pro-posed for the Preston tube measurements (but with smaller

incremrelatioFromdata atine, ftime-aspondobtainwas pthan wcordedmethoestimain thethe flosure trline) vfor thewith sin theconjunA 24-the sm

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n Velocity with Reynolds Number in Four Pipesents in R), and the friction coefficient-Reynolds numbernship for each pipe was compared with standard results.the results of previous tests, a 30-s pause time beforecquisition was programmed into the measurement rou-ollowed by sampling at about 400 Hz for 62 s. Eachveraged digital signal in conjunction with the corre-ing pressure transducer calibration equation was used to

the time-averaged pressure. This order of calculationreferred as it gave a much faster rate of data acquisitionhen the time series of pressures was calculated and re-for final averaging. The error introduced by the first

d arising from a slight nonlinearity in the system wasted to be less than 60.01%. The instrumentation usedfirst test was the higher range pressure transducer andw meters. In a second test using the lower range pres-ansducer and the flow meters, the ratio of peak (center-elocity to cross-sectional mean velocity was obtainedsame range of Reynolds numbers and again compared

tandard results. Peak velocity was found by searchingvicinity of the pipe centerline with a pitot tube used inction with the downstream static pressure wall tapping.

gauge stainless steel tube was used for the pitot tube inallest pipe (D = 6.184 mm) and a 21-gauge tube in the

other three pipes. A similar measuring routine with pand data acquisition was employed.

In Stage 2, the pressure drop in the pipe test lPreston tube dynamic pressure were measured inpipes, with the nine Preston tubes, over the widesReynolds numbers practicable within the prescribed103 # R # 104. Constraints were imposed both bsitivity of the equipment and its range of measuremPatels rule of thumb that the Preston tube outsideshould be

FIG. 4. Complete Data Set (611 Points) for Laminar, Transit

412 / JOURNAL OF HYDRAULIC ENGINEERING / JUNE 2000ional, and Turbulent Flows, Compared with Patels Calibrationbeen shifted successively upward by log 2 for clarity, with theD = 6.184-mm graph being as measured.

An interesting feature is that the start of transition occurredat a significantly higher Reynolds number for the smallest di-ameter pipe (i.e., at R = 2,700 for D = 6.184 mm comparedwith R = 2,040 for D = 15.789 mm, 2,110 for D = 12.684mm, and 2,250 for D = 9.483 mm). Also, at the end of thetransition regime, the smallest diameter pipe gave fully tur-bulent flow at a significantly higher Reynolds number of R =3,250 (D = 6.184 mm) compared with R = 2,840, 2,770, and2,820 for D = 15.789, 12.684, and 9.483 mm, respectively.No further investigation was carried out on this phenomenon,but there is a plausible connection between the delay in tran-sition and the very large development length of 506D upstreamof the test length in the 6.184-mm-diameter pipe. It seemslikely that, in the 6.184-mm-diameter pipe for R < 2,700, inletdisturbances were damped out and turbulence was not self-sustaining (Sibulkin 1961).

Measuring the theoretical ratio U/Um = 0.5 for laminar flowin the pipes was difficult and was aggravated by the rathercrude traversing mechanism. However, as shown in Fig. 3,quite good results were obtained for D = 9.483, 12.684, and15.789 mm. At the higher Reynolds numbers, the data ap-proached the 1/7-power velocity distribution without quitereaching it. For D = 6.184 mm, midtransition occurred at ahigher Reynolds number than in the other pipes, which wasconsistent with the friction coefficient results.

Fig. 4 shows all of the data (611 points) acquired in Stage

2 of the experiments, plotted as y* against x* and split intotwo groups: (1) Fully turbulent flow (425 points); and (2) lam-inar and transitional flows (186 points). The criteria adoptedfor fully turbulent flow were R > 2,850 for D = 15.789,12.684, and 9.483 mm and R > 3,250 for D = 6.184 mm. Alsoshown are two of Patels three calibration equations appropri-ate to the present range of x*; that is

2 3y* = 0.8287 2 0.1381x* 1 0.1437x* 2 0.0060x* (3)y* = 0.037 1 0.5x* (4)

for 2.9 < x* < 5.6 and 0 < x* < 2.9, respectively. The datafollow a clearly discernible trend that roughly conforms toPatels calibration over much of the range, with the pointsscattered in a band that broadens with decreasing x*. The lam-inar and transitional flow data, which do not occur for x* >3.55, especially contribute to this broadening process andcause considerable scatter for x* < 3.

Following an analysis of laminar pipe flow in which thevelocity profile can be expressed as

u U y yt= 1 2 (5)S DU n Dt

a form of calibration equation in x*, y*, and d/D was consid-ered by New (1995), but no systematic variation in the datawith d/D could be detected. Given our primary interest in lowReynolds number turbulent flow, and our lack of success inexplaining the large scatter in the data for laminar and tran-

FIG. 5. (a) Third-Order Polynomial [Eq. (6)], Fitted for Fully TurbStress, Predicted by Fitted Curve, and 95% Prediction Intervalulent Flows with R # 10 3 103 (425 Points); (b) Errors in Wall Shearsitional flows, attention was consequently focused on flows inthe fully turbulent regime. In Fig. 5(a) only the fully turbulentflow measurements are plotted, and the data set is fitted by asingle third-order polynomial

2 3y* = 20.84534 1 1.3455x* 2 0.27758x* 1 0.032531x* (6)Although the scatter was considerably reduced by removing

the laminar and transitional flow data, the residuals were stillhigh. Fig. 5(b) shows the corresponding percentage differencebetween the measured shear stresses t0 and the shear stresses

predicted by (6). The 95% confidence limits of the predic-t0tion interval for y* vary from 64.37% at low values of x* to61.21% for high values of x*. The corresponding predictioninterval for t0 is shown in Fig. 5(b) and is asymmetric about

the mean: the positive interval varies in the range of 18.08 to17.59% and the negative interval in the range of 27.47 to27.06%, from low to high values of x*.

Fig. 5(a) also shows Patels calibration [(3) and (4)]. It canbe seen that in the high and low regions of x*, Patels equationoverpredicts y* and in the middle region it underpredicts y*.

To test for Reynolds number dependence, the y* residualsdy* obtained by fitting (6) were plotted against x* for fourReynolds numbers and a fixed Preston tube diameter ratio t.The ratio t was determined from the nominal internal and ex-ternal tube diameters provided by the manufacturer, and forthe test illustrated in Fig. 6, 24- and 26-gauge tubes were used,both with t 0.55. Fig. 6 shows the residuals scattered aboutthe zero line with no evidence of any systematic variation withJOURNAL OF HYDRAULIC ENGINEERING / JUNE 2000 / 413

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FIG. 7. y* Residuals from Eq. (6) Plotted for Two Values of Diameter Ratio t and Fixed Reynolds Number

eynolds number. The same negative inference was drawnrom similar analyses at other values of t. Therefore, contrary

expectation, it was concluded that no Reynolds number de-endence could be deduced from the present measurements.To test for dependence on the Preston tube diameter ratio t,e y* residuals were plotted against x* for two values of thereston tube diameter ratio at a fixed Reynolds number. Theatio t for all of the tubes employed varied in the approximateange 0.49 # t # 0.67, and for the test illustrated in Fig. 7,4- and 26-gauge data (t 0.55) were compared with 23- and5-gauge data (t 0.51 and 0.50, respectively), both at R =3 103. Fig. 7 shows the residuals scattered about the zero

ne, and the data dispersed with no evident systematic varia-on with t. Neither did any pattern emerge at other Reynoldsumbers or by comparing other values of t. Therefore, it wasoncluded that no dependence on t could be deduced from theresent set of data.

UMMARY AND CONCLUSIONS

Preston tube measurements were carried out in low Reyn-lds number pipe flows and 425 items of data for fully tur-

bulent flows, with Reynolds numbers up to R = 10 3 103were selected for subsequent analysis. When plotted as y*against x*, the data exhibited a degree of scatter such that anycalibration curve in these variables alone was bound to havea wide error band in predicting the wall shear stress. Howeverwe were unable to detect any systematic variation due toReynolds number or the Preston tube diameter ratio t. Therefore the data were analyzed in terms of y* as a function of x*with randomly distributed errors.

Patels calibration was found to be applicable, though thedata were better fitted by (6). Using the latter, the predictioninterval for t0, with 95% confidence limits, was found to varyin the range of 18.08 to 17.59% on the positive interval andin the range of 27.47 to 27.06% on the negative intervalfrom low to high values of x*.

Although no systematic dependence on R or t could be identified, it would be premature to conclude that such dependencedoes not exist. As previously discussed, experimental evidenceby other workers indicates that there should be a degree odependence on the Reynolds number and on the Preston tubediameter ratio. Why the present experiments did not identifythese dependencies is not known.FIG. 6. y* Residuals from Eq. (6) Plotted for Fo r Reynolds Numbers and Fixed Diameter Ratio t

The results reported here are strictly applicable to pipeflows, and (6) should not be used for wall shear stress mea-surements in low Reynolds number turbulent developingboundary layers or developed rectangular duct flows.

ACKNOWLEDGMENTSSome of the work reported here was carried out in the project phase

of the 30 ACSC Div. 1 MSc in Defence Technology, Royal MilitaryCollege of Science, Cranfield University, Shrivenham, U.K. It was partlyfunded by Grant No. GR/K75231 of the Engineering and Physical Sci-ences Research Council, U.K. Curve fitting and estimation of confidencelimits were carried out using software by the Numerical AlgorithmsGroup (NAG), Oxford, U.K. The writers are grateful to Dr. G. W. Morgan(NAG) for his advice.

APPENDIX I. REFERENCESAfzal, N., and Yajnik, K. (1973). Analysis of turbulent pipe and channel

flows at moderately large Reynolds number. J. Fluid Mech., Cam-bridge, U.K., 61(1), 2331.

8307, Coll. of Aeronautics, Cranfield Institute of Technology, Cranfield,Bedford, U.K.

Poll, D. I. A. (1983b). A note on the algebraic representation of thePreston tube calibration. Rep. No. 8308, Coll. of Aeronautics, Cran-field Institute of Technology, Cranfield, Bedford, U.K.

Preston, J. H. (1954). The determination of turbulent skin friction bymeans of Pitot tubes. J. Royal Aero. Soc., 58, 109121.

Sibulkin, M. (1962). Transition from turbulent to laminar pipe flow.Phys. Fluids, 5(3), 280284.

Spalding, D. B. (1961). A single formula for the law of the wall. J.Appl. Mech., 28, 455458.

Zagarola, M. V., Perry, A. E., and Smits, A. J. (1997). Log laws orpower laws: The scaling in the overlap region. Phys. Fluids, 9(7),20942100.

Zagarola, M. V., and Smits, A. J. (1997). Scaling of the mean velocityprofile for turbulent pipe flow. Phys. Rev. Lett., 78(2), 239242.

Zagarola, M. V., and Smits, A. J. (1998). Mean flow scaling of turbulentpipe flow. J. Fluid Mech., Cambridge, U.K., 373, 3379.

APPENDIX II. NOTATIONThe following symbols are used in this paper:Barenblatt, G. I. (1993). Scaling laws for fully developed turbulent shearflows. Part 1. Basic hypotheses and analysis. J. Fluid Mech., Cam-bridge, U.K., 248, 513520.

Barenblatt, G. I., and Prostokishin, V. M. (1993). Scaling laws for fullydeveloped turbulent shear flows. Part 2. Processing of experimentaldata. J. Fluid Mech., Cambridge, U.K., 248, 521529.

Chue, S. H. (1975). Pressure probes for fluid measurement. Prog.Aerosp. Sci., Great Britain, 16(2), 147223.

den Toonder, J. M. J., and Nieuwstadt, F. T. M. (1997). Reynolds num-ber effects in a turbulent pipe flow for low to moderate Re. Phys.Fluids, 9(11), 33983409.

Head, M. R., and Rechenberg, I. (1962). The Preston tube as a meansof measuring skin friction. J. Fluid Mech., Cambridge, U.K., 14, 117.

Hsu, E. Y. (1955). The measurement of local turbulent skin friction bymeans of surface pitot tubes. Rep. No. 957, David W. Taylor ModelBasin, U.S. Department of Navy.

MacMillan, F. A. (1954). Viscous effects on pitot tubes at low speeds.J. Royal Aero. Soc., 58, 570572.

National Physical Laboratory. (1958). On the measurement of local sur-face friction on a flat plate by means of Preston tubes. Rep. and Mem.No. 3185, Aerodynamics Div., U.K., 124.

New, A. P. (1995). Preston tube calibration in low Reynolds numberflow. 30 ACSC Div. 1 MSc (Defence Technol.), Royal Military Coll.of Sci., Cranfield University, Shrivenham, U.K.

Patel, V. C. (1965). Calibration of the Preston tube and limitations onits use in pressure gradients. J. Fluid Mech., Cambridge, U.K., 23(1),185208.

Patel, V. C., and Head, M. R. (1969). Some observations on skin frictionand velocity profiles in fully developed pipe and channel flows. J.Fluid Mech., Cambridge, U.K., 38(1), 181201.

Poll, D. I. A. (1983a). A note on the use of surface pitot tubes for themeasurement of skin friction in laminar boundary layers. Rep. No.Cf = friction coefficient, t0/(1/2)rU2;D = internal diameter of pipe;d = external diameter of Preston tube;F = local Froude number U/ gh;

f, F = function;g = acceleration due to gravity;h = water depth in open channel flow, semidepth in rectan-

gular closed duct flow;R = pipe flow Reynolds number, UD/n based on mean veloc-

ity;Rm = pipe flow Reynolds number, UmD/n based on maximum

velocity;t = ratio of internal to external diameter of Preston tube;

U = depth-averaged or cross-sectional mean velocity (see con-text);

Um = centerline (i.e., maximum) velocity in pipe flow or rectan-gular closed duct flow;

Ut = friction velocity, t /r;0u = local velocity;

x* = log[Dpd 2/(4rn2)];y = distance from wall;

y* = log[t0d 2/(4rn2)];Dp = Preston tube dynamic pressure;dy* = y* residual after fitting Eq. (6);

n = kinematic viscosity of fluid;r = density of fluid;t0 = wall shear stress in general, or measured wall shear stress

in particular (see context); andt0 = predicted wall shear stress.JOURNAL OF HYDRAULIC ENGINEERING / JUNE 2000 / 415