Friction factor correlations for laminar, transition and turbulent flow in smooth pipes

Frition fator orrelations for laminar, transition and turbulent flow in smooth pipes By D.D. JOSEPH AND B.H. YANG Department of Aerospae Engineering and Mehanis, University of Minnesota, Minneapolis, 5555 Department of Mehanial and Aerospae Engineering, University of California, Irvine, 967 July, 008 In this paper we derive an aurate omposite frition fator vs. Reynolds number orrelation formula for laminar, transition and turbulent flow in smooth pipes. The orrelation is given as a rational fration of rational frations of power laws whih is systematially generated by smoothly onneting linear splines in log-log oordinates with a logisti dose urve algorithm. This kind of orrelation seeks the most aurate representation of the data independent of any input from theories arising from the researhers ideas about the underlying fluid mehanis. As suh, these orrelations provide an objetive metri against whih observations and other theoretial orrelations may be applied. Our orrelation is as aurate, or more aurate, than other orrelations in the range of Reynolds numbers in whih the orrelations overlap. However, our formula is not restrited to ertain ranges of Reynolds number but instead applies uniformly to all smooth pipe flow data for whih data is available. The properties of the lassial logisti dose response urve are reviewed and extended to problems desribed by multiple branhes of power laws. This extended method of fitting whih leads to rational frations of power laws is applied to data Marusi and Perry 995 for the veloity profile in a boundary layer on a flat plate with an adverse pressure gradient, to data of Nikuradse 9 and MKeon et al. 00 on frition fators for flow in smooth pipes and to the data of Nikuradse 9 for effetively smooth pipes.. Introdution The goal of this paper is to extrat analyti formulas relating the frition fator = d( dp / dx) /( ρu / ) to the Reynolds number Re = Ud / ν (U being the average bulk veloity) from proessing of data for flow in smooth pipes. No pipe an be perfetly smooth but when the roughness is small enough the flow depends only on the Reynolds number = f ( Re) (and not on roughness) and the pipe is said to be effetively smooth. The effets of roughness on a nominally smooth wall are well desribed in the following personal ommuniation with Ivan Marusi: The way I like to onsider the effet of roughness vs. a smooth wall is by onsidering the range of sales in the flow. For a pipe of radius a, the frition Reynolds number, Re τ = Uτ a / ν, is proportional to the ratio of the largest and smallest length sales in the flow that ontribute to the frition fator. The largest length sale is proportional to a, and the smallest attahed eddies sales with the ν / U τ (say 00ν / U and this is towards the wall). (There is also the Kolmogorov sale, τ but those motions do not ontribute to the energy-ontaining motions in a signifiant way, espeially if the Reynolds number is high enough). On a smooth wall the range of sales ontinues to inrease unabated as Re inreases, and so frition fator is free τ to hange likewise. On a rough wall the smallest sale is restrited by the roughness sale k. That is, as Re inreases the smallest sale τ is fixed by k and will no longer depend on the Reynolds number. For this reason the frition fator levels off and beomes independent of Re. Different roughness sales will ause the frition fator to level off at different values and this leveling off will our at lower Re for larger values of k / a. τ An expanded data set for flow in smooth pipes is reated by extrating results for effetively smooth pipes from data of Nikuradse 9 for flows spanning laminar, transition and turbulent flow in rough pipes. New results are obtained by diret omparison of data for smooth pipes with data for effetively smooth

pipes. We also present results omparing formulas for turbulent flow in smooth pipes based on modifiations of lassial log laws and the formula anhored in the theory of inomplete similarity (a power law for = f ( Re) with a pre-fator and exponent that also depend on Re) with eah other and with the expanded data set. The prinipal results of this paper arise from the introdution of a new and systemati method for proessing experimental data whih an be desribed fitting data pieewise by linear splines (whih are power laws in log-log oordinates). The data in the transition regions between the splines is proessed by fitting five points with a logisti dose algorithm. This method of fitting leads to rational frations of power laws and rational frations of rational frations of power laws. It is a fundamentally different than other methods of proessing data. Other methods are motivated by flow fundamentals modulated by the researhers imagination. Our goal is to get formulas of the highest auray and greatest range; ideas about fluid mehanis and turbulene do not enter at any stage. We will show that our orrelation is as aurate, or more aurate, than other orrelations in the range of Reynolds numbers in whih the orrelations overlap. However, our formula is not restrited to ertain ranges of Reynolds number but instead applies uniformly to all smooth pipe flow data for whih data is available. Other methods for desribing data on turbulent flow in pipes and boundary layers are rooted in hypotheses about the ontrolling fluid mehanis priniples under sway. It an be said that the implementation of the fluid mehanis ideas require a ertain number of operational hypotheses leading to funtional forms involving unknown quantities whih must finally be seleted to fit the data. In this sense, all these fluid mehanis motivated fitting methods are semi-empirial. An exellent desription of the most popular of these methods has appeared in the review paper of Barenblatt et al. 997 ited below. During the more than 60 years of ative researh into turbulent pipe flow, two ontrasting laws for the veloity distribution in the intermediate region have oexisted in the literature (see, eg. Shlihting 968): the first is the power or saling law, α φ = Cη (.) where the C and α are onstants (i.e. parameters independent of η) believed to depend weakly on Re. Laws suh as (.) were in partiular proposed by engineers in the early years of turbulene researh. The seond law found in the literature is the universal, Reynolds number independent logarithmi law, φ = ( ln η) / κ B (.) where κ (von Kármán s onstant) and B are assumed to be universal, i.e. Re-independent, onstants. In more reent deades, the logarithmi law (.) has been emphasized over the power law (.), sometimes even to the exlusion of the latter. The reasons have been mainly theoretial: it was not reognized that the power law has an equally valid theoretial derivation and satisfies the approximate self-onsisteny (overlap) ondition. This theoretial bias has been allowed to obsure the fat that the experimental data unequivoally militate in favor of the power law (.) It is generally thought that the universal logarithmi law (.) is in satisfatory agreement with the experimental data both in pipes and in boundary layers.however, the saling law (.) has also found experimental support, provided the dependene of the quantities C and α on the Reynolds number was properly taken into aount. Indeed, Shlihting, following Nikuradse, showed that the experimental data agree with the saling law over pratially the whole ross-setion of a pipe An important onlusion has been reahed: The power law (.) and the logarithmi law (.) an be derived with equal rigor but from different assumptions. The universal logarithmi law is obtained from the assumption of omplete similarity in both parameters η and Re; physially, this assumption means that neither the moleular visosity ν nor the pipe diameter d influenes the flow in the intermediate region. The saling law (.) is obtained from an assumption of inomplete similarity in η and no similarity in Re; this assumption means that the effets of both ν and d are pereptible in the intermediate region.. Experimental data The data we use omes from three soures: () the experiments on the flow of water of in smooth pipes of Nikuradse 9, () the experiments of Nikuradse 9 on water flow in rough pipes from whih we have extrated data for effetively smooth pipes. Figure shows how we have seleted effetively smooth pipe

data from the rough pipe data. Figure shows that the smooth pipe data and effetively smooth pipe data are in good agreement. The third soure of data is presented in the paper by MKeon, Swanson, Zagarolla, Donnelly and Smits in 00 (hereafter MSZDS). Their data for the flow of gases in smooth pipes is presented graphially in figure and in tabular form in table of MSZDS 00. They note that The Prineton (Zagarola & Smits 998; MKeon et al. 00a; MKeon, Zagarola & Smits 00b) and Oregon (Swanson et al. 00) researh groups have reently examined fully developed pipe flow using very different apparatus. Compressed air is used in the Prineton Superpipe, whereas the Oregon devie uses several room temperature gases: helium, oxygen, nitrogen, arbon dioxide and sulphur hexafluoride are used for relatively small Reynolds numbers, and normal liquid helium (helium I) is used for highest Reynolds numbers. The differene in of the two devies is dramati: for example, the Superpipe weighs about 5 tons, whereas the Oregon tube weighs about oune. Prior to the Prineton and Oregon experiments, the experiments performed by Nikuradse 9 overed the largest range of Reynolds numbers. Most other experiments span less than an order of magnitude in Reynolds number (see Zagarola 996, Table.). The data of Nikuradse 9, 9 and MSZDS 00 is relatively easy to proess and ompare with empirial formulas beause it is presented in tabular form. 0. Nikuradse Rough Pipe Data Nikuradse Effetively Smooth Pipe Data FIGURE : Frition fator ( ) vs. Reynolds number ( Re ) in rough pipes (Nikuradse 9). The dark points on the bottom envelope of urves whih depend on roughness do not depend on roughness = f ( Re) an be said to be effetively smooth. 0.0 00 000 0000 00000 000000 0. 0.0 Nikuradse Smooth Pipe Data in Turbulent Regime Nikuradse Effetively Smooth Pipe Data FIGURE. vs. Re (. 0 6 < Re <. 0 ) in smooth pipes (open red irles) from Nikuradse 9 (open blue squares) ompared with vs. Re ( 5. 0 < Re < 8.7 0 ) in effetively smooth rough pipes (Nikuradse 9, see figure ). Turbulent data for effetively smooth pipes oinides with data from smooth pipes. 0.00 00 000 0000 00000 000000 0000000

0 0. 0.0 Prineton Data FIGURE. vs. Re from experiments reported in MSZDS 00. The Prineton data is in good agreement with the smooth pipe data of Nikuradse 9 (see figure 6.5 in the thesis of MKeon 00). Figure shows that the Oregon data is not in agreement with the 9 data of Nikuradse for effetively smooth pipes. 0.00 0 00 000 0000 00000 000000 0000000 00000000 0. Nikuradse Effetively Smooth Pipe Data 0.0 000 0000 00000 FIGURE. Comparison of data from Nikuradse 9 for effetively smooth pipes with Oregon data (table ) for flow in smooth pipes. The Oregon data is greatly different than Nikuradse data in the transition region. Data in the transition is assoiated with the instability of laminar flow whih depends on parameters like the pipe length whih do not strongly influene laminar or turbulent flow. The data in the transition region forms a loud rather than a urve.. Comparison of formulas of Barenblatt 00 and MKeon 00, 005 with the data and eah other The fitting urve proposed by MKeon et al. 00 is ( ) 0. 57 =.90log Re ; (.) it fits Prineton data with a perentage error less than.5% for 6 00 0 < Re <.6 0. 6 0 < Re < 5 0 and 0.5% for The urve proposed by MKeon et al. 005 inludes a orretion for the visous deviation from the log law at the wall and is given by ( Re ) 7.0 =.90log 0.75. (.) ( Re ) 0. 55 This equation predits the frition fator to within.% of the Prineton data (0.6% at high Reynolds 6 number, 0 0 < Re < 0 0 ) and within.0% of the Blasius relation at low Reynolds numbers ( 0 0 < Re < 90 0 ). Barenblatt s 00 saling law (8.9) is derived from an extended theory (inomplete similarity) of

similarity (power law in the Reynolds number) to a form in whih the prefator and exponent of the power law also depend on the Reynolds number; the funtional form of the prefator and exponent are derived from theoretial assumptions. It is given by / ( 5α ) where e ψ = α α ( α )( α ) and α =. ln 8 / ψ = ( Re) ( α ), (.) 0.0 0.0 0.0 Prineton Data Correlation in equation (.) Correlation in equation (.) FIGURE 5. Comparison of the power law formula (.) and the modified log formula (.) with the data of MSZDS. The differenes between (.) and (.) are about % for large Re (see figure 0). 0.0 0 0000 00000 000000 0000000 00000000. Logisti dose urves for data sets with multiple power law regions Prior to the relatively reent work of Joseph and his oworkers, it was not known that the lassial logisti dose response urve ould be used to fit data arising from real and numerial experiments in fluid mehanis and other hard sienes. In this setion we will briefly review the properties of the lassial logisti dose response urve and extend this method of fitting for orrelations with multiple branhes of power laws. This extended method of fitting whih leads to rational frations of power laws and to rational frations of rational frations of power laws is applied here to data on the frition fator vs. Reynolds number for laminar, transition and turbulent flow in smooth pipes (MKeon et al. 00); this method of fitting leads to a omposite orrelation whih desribes all the available data unrestrited by Reynolds number. The formula follows from diret proessing of the data and does not depend on any assumption or orrelation from fluid mehanis... Classial logisti dose response urve The logisti funtion is one of the oldest growth funtions and a best andidate for fitting sigmoidal (also known as logisti ) urves. Espeially the logisti dose response funtion is a robust fitting funtion for transition phenomena. The -parameter, -parameter and 5-parameter logisti dose response urves are widely used in non-linear data fitting in life sienes; pharmay and agronomy (see Balakrishnan N. 99). A typial 5-parameter logisti dose response urve is given by 5

( b a) y = a m x t n, (.) where x and y are the independent and the dependent variables. It is widely used in the literature to desribe the orrelations between x and y featured by two plateau regions and a transition region. Two typial examples are shown in figure 6. 000 bakward S-shaped logisti dose response urve forward S-shaped logisti dose response urve y 00 0 00 00 y = x 50 00 y = x 50 FIGURE 6. Classial forward and bakward logisti dose response urves. 0.0 0. 0 00 000 0000 00000 000000 x To our knowledge the first studies using logisti dose urve fitting in fluid mehanis an be found in the study of sedimentation by Patankar et al. 00. This study was inspeted by R. Barree who explained this method of 5-parameter fitting in an appendix Fitting power-law data with transition regions by a ontinuous funtion: Appliation to the Rihardson-Zaki orrelation. We shall all the orrelation of data whih an be desribed by two power laws onneted by ontinuous data in the transition region a two power law orrelation (or bi-power orrelation). Two power law orrelations for sediment transport have been studied by Wang et al. (00) and for frition fators in turbulent gas-liquid flows by Garia et al (00, 005). Correlations of families of bi-power laws depending on a third parameter have been onstruted by Viana et al. (00) who orrelated data for the rise veloity of Taylor bubbles in round vertial pipes as a family of bi-power laws of the Froude number vs. Reynolds number indexed by the Eotvos number. A file of papers orrelating large data sets from real and numerial experiments an be found at (http://www.aem.umn.edu/people/faulty/joseph/pl-orrelations). The proessing of data overed by linear splines and onneted by the five point rule of the logisti dose urve (.) is an approximate method whose auray is judged by analysis of the error of the fit. Moreover, analytial funtions like (.) whih vary on some interval annot assume a onstant value at any finite x beause disontinuous derivatives are required at suh points. However, funtions with disontinuous derivatives an be losely modeled by analyti funtions as in the ase in the two logisti dose funtions shown in figure 6. The dose funtion (.) is atually a rather ompliated nonlinear funtion and the mathematial problem of approximation of experimental data with suh funtions deserves further study... Modified logisti dose response urve The lassial 5-parameter logisti dose response urve (.) is modeled to equation (.), in whih the two onstants a and b in equation (.) are replaed by two ontinuous funtions f L, respetively. x, m and n are onstants. x is an important onstant in determining the onvergene trend of f when the oeffiient m is negative and n is a small positive number. (Hereafter x is alled the threshold value of x ) 6

y = f = f L f R f L G = f L f R f x x L m n. (.) Here we show how to onstrut logisti dose urves (see equation (.)) for ompliated data whih an be desribed by pieewise ontinuous multiple power laws or multiple rational frations of power laws, where the rational frations of power laws themselves are also logisti dose funtions. There are three key steps in determining a modified logisti dose urve; these are () the seletion of two appropriate assembly members f L, () the estimate of the threshold value x whih identifies the point of intersetion of f L, and () the five-point sharpness ontrol for fitting the transitional part between the two assembly members. G in equation (.) to move towards or rapidly on different sides of the threshold value x one the independent variable x deviates from x, so that the logisti dose funtion an approah f L asymptotially on one side on the other side of x. The two assembly members f L and f R an be onstants (in this ase, it will redue to the lassial logisti dose response urve), power laws, rational frations of power laws, rational frations of rational frations of power laws, or any other type of ontinuous funtions. The splitting trends of f at x = x is the basis of the sequential onstrution of a rational fration of multiple segments of power laws. A main idea in the modified logisti dose urve fitting is to fore the denominator funtion.. Logisti dose urve fitting for mixed power laws and/or rational frations of power laws The logisti dose funtion for a two power law orrelation (.) is a rational fration of two power laws. Here we go one step further and replae one of the two member funtions f L in equation (.) with a rational fration of two power laws. Then we an use the modified logisti dose urve to reate a three power law orrelation. If both f L are replaed by rational frations of two power laws, the modified logisti dose urve an even fit data whih ontains four subsetions of power laws. Sine f L an be updated with new rational frations again and again by adding more and more power laws, we may expet to use the modified logisti dose response urve to fit a orrelation omposed of five or even more branhes of power laws. Suh orrelations lead inevitably to rational frations of rational frations of power laws... Threshold values x of the independent variable x We have found that the independent variable x is sensitive to the threshold value x in equation (.); x an be used to loate the point of intersetion of the two assembly members f L. That is to say, x = x when f L = f R (. When x < x, the denominator funtion G in equation (.) will approah and therefore f f L sine m is negative. When x > x, G approahes f x f x. asymptotially; therefore ( ) ( ) R.. Five-point sharpness ontrol for the transition region between f L Data whih is very losed to the two assembly members f L and f R at the point x of intersetion is said to have sharp transition. The onstants m and n an ontrol the smoothness of the transitional part between f L. On the transitional part, the smoothness of data inreases as the magnitude of m dereases. We use a five-point rule to judge the auray of the fitting urve. Five data points on or near the transition segment of the two assembly members are seleted; the errors between the five sample points and the fitting urve are alulated and evaluated by the R-square value. The goal of the 7

five-point rule is to make the fitting urve for the five seleted points as smooth as possible, whih an be seen in the zoom-in view of transition region... Constrution of a logisti dose funtion for data sets identified by two power laws and a smooth transition segment In this setion we illustrate the proedure for onstruting a logisti dose orrelation for two power laws and a smooth transition segment for data of Marusi and Perry (995) on the variation the dimensionless veloity φ = u / u vs the dimensionless distane from the wall η = u y / ν in boundary layers with an adverse pressure gradient. An example of the seletion of two power laws and the alulation of sharpness ontrol parameters m and n based on raw data is given here. The tabulated results are shown in table for 0 APG mean flow at Re = 9. 0. 0 APG MEAN FLOW (Re = 9.0) No. φ η No. φ η No. φ η No. φ η.9779 9.86 7.8959 0.6.985 96. 6.875 75.0.96.877 8.679 8.600 5.909 0.06 5 7.095 9.5.8 7.00 8.78 77.065 5 6.0605 57.57 6 7.60 7.0.9865 5.5069 5 9.99 0. 6 7.0885 9.67 7 7.877 6.55 5 5.586 6.586 6 9.59897 67.9655 7 8.867 6.568 8 7.9908 579.705 6 5.68 76.898 7 0.0 5.7588 8 9.56 590.75 9 8.075 795.95 7 6.089 9.966 8 0.8898 8.005 9 0.0 75.78 0 8.059 988.087 8 6.905 09.65 9.57 55.06 0.588 97.8 8.068 0. 9 6.869 9.5 0.995 67.657.7956 0.85 8.087 6.0 0 7.657 5.86.796 7.0556.898 0.65 8.0868 55.57 7.569 75.757.0 80.7 5.056 50.969 8.06908 75.9 TABLE. Experimental data (0APG mean flow at Re = 9. 0 ) for boundary layer flows with adverse pressure gradient [Reprodued after Marusi and Perry 995]. Figure 7 illustrates the typial proedure of the onstrution of a logisti dose orrelation for two power laws onneted by a smooth transition. Panel (a) shows all the data from table. Five data segments an be identified from the graph: data segments,, 5 are three power laws segments; segments and are transition segments between power laws. In this example, we do not onsider segments and 5. Panel (b) shows two power laws P and P whih are identified on segments and, respetively. The whole plane is divided into four regions by P and P (i.e.,, and in panels (b) and ()). Panel (b) learly shows that the transition segment is loated in region. We shall show that logisti dose funtion of P and P annot pass through any data points in region. After P and P are determined, we an easily obtain the rossing point η = η by letting P P = 0. The two parameters m and n are unknown, and they are obtained by substituting two data points into the following logisti funtion of P and P. P ( ) ( η) P ( η) φ = P η. (.) η / η m We tentatively hoose the point A (,φ ) figure 7). It follows that [ ( ) ] n η whih is loated right on the power law P (see panel (b) in 8

η = φ It follows that ( ) ( η ) ( η) P ( η) m n ( η / η ) [ ] ( η) ( η / η ) P P φ φ = P = φ. (.) [ ] m n P and P = P. Only the point of intersetion an satisfy this ondition but it is not near the transition segment. Similarly, if we hoose point B ( η,φ ) whih is loated on the power law P ( ) ( η ) P ( η ) φ P ( η ) φ P η = φ m n [ ( η / η ) ] ( ) m η / η Therefore P ( η ) = φ giving rise to the same problem of point A ( η,φ ). If we hoose a point C ( η,φ ) in region but at the left side of η = η P ( ) ( η ) P ( η ) φ = P η. ( η / η ) m It follows that [ ] n P, we have =. (.5) [ ] n [ ( / m n P ) ] ( η ) P ( η ) η η = φ P ( η ), we obtain. (.6) However, the LHS of equation (.6) is always positive, but the RHS of (.6) is negative beause P ( η ) < P ( η ) and φ > P ( η ). Parameters m and n annot be found to resolve this ontradition at η,φ. ( ) If we hoose a point D (,φ ) Therefore η in region but at the right side of ( η ) Sine φ > P ( ), we have < ( η / ) ( η ) P ( η ) ( η / η ) m P φ = P. [ ] n [ ( / m n P ) ] ( η ) P ( η ) η η = φ P ( η ) η = η, we obtain. (.7) m n [ ] η 0 η <. In this ase, (.) an be satisfied only if the exponent n is negative. However, if n is negative, φ in equation (.) annot approah the two power laws P and P on different sides of η = η, as expeted. We have shown that the logisti funtion for power laws P and P has no solution if m and n are hosen to math any points in region or on the two power laws. When transition segment is sharp (i.e. m is large and the radius of urvature of the data segment is small), the logisti funtion (.) follows the two power laws losely and the deviation of logisti funtion from power laws is not even appreiable. In these ases, we do not modify the power laws in data proessing beause the logisti funtion an easily satisfy the requirement of fitting error. However, if the radius of urvature of transition segment is large, we want the fitting urve to pass through the data points in a smooth transition. A most onvenient way to realize this is to modify the prefators of the two power laws P and P. Then we an obtain two new power laws P ' and P ' (see panel ()). The purpose of the modifiation is to move all the data points on transition segment out of region. In this ase, equation (.) an be proessed to fit points on the transition segment. This proedure is not sensitive to small hanges in the prefators of P ' and P '. After P ' and P ' are determined, we hoose two points on or nearby the transition segment and substitute into equation (.) to solve for m and n by iterations. In this example, 9

0.9 0.76 0.9 0.76 P, P, P ' and P ' are φ = 8.η, φ =.89η, φ = 8.76η and φ =.0η, respetively. The threshold value η an be determined by letting P P = 0 and then η = 70.95. If we hoose points No.5 and No.8 (see table ) to solve for m and n, then we an get m = 0. 95 and n =.070. Therefore, we have the omposite logisti funtion expressed as φ = P ' P ' η η.0η 8.76η η 70.95 0.76 0.9 0.9 P ' = 8.76η m n 0.95..070 (.8) Equation (.8) is the rational fration formula for the data in table without onsidering the last 0 points. The orrelation in equation (.8) is shown in panel (d) of figure 7, whih passes through the data points in segments, and perfetly. 00 DATA 00 PROCESSING Segment 5 Segment P : φ =. 89η 0.76 Segment φ Segment φ D B Segment P : φ = 8. η 0.9 C A 0 0 00 000 0000 η (a) 0 0 00 000 0000 η (b) 00 PROCESSING 00 FINAL P ': φ =. 0η 0.76 φ φ P ': φ = 8. 76η 0.9 0 0 00 000 0000 η () 0 0 00 000 0000 η FIGURE 7. Dimensionless veloity profile vs. dimensionless distane from wall for 0 APG mean flow at Re = 9.0. The onstrution of the logisti dose funtion as a rational fration of two power laws is shown in the graphs. The data are from the adverse pressure gradient experiments of Marusi and Perry (995). The four panels are: (a) the raw data are identified by three (d) 0

power law segments and two transition segments, (b) for segments and, two power laws P and P are identified, () P and P are modified to P ' and P ' so that equation (.) an be proessed to fit the points on or nearby the transition segment, (d) the logisti dose funtion is onstruted as a rational fration of P ' and P ' after m and n are determined (the fitting errors for segments, and are less than.9% or 0.9% if the first three points are not onsidered). Turbulent data suitable for proessing as two power law orrelations an be found in figures 8.5 and 8.8 in the saling book of Barenblatt 00. The orrelation in figure 7 orresponds to in figure 8.8 in the Barenblatt book in whih the data is desribed by two broken power laws but no desription of the smooth transition between them is given. Keller 00 introdued a method for onstruting a smooth transition between power laws. A differential equation for this problem is proposed and solved. It leads to a power law for the veloity profile followed by a smooth transition to a different power law. Two formulas giving different results are derived; they give different results. Comparisons to data are not given. 5. Proessing the data of MSZDS 00 for rational fration orrelations between and Re for laminar, transition and turbulent flow in smooth pipes More ompliated data whih requires the use of more than two power laws may require that the exponents as well as the prefators need to be modified. The seletion of these modifiations of the power laws depend on the details of data distribution and annot be speified a priori. The seletion of data points for proessing even on apparently smooth transition segments are also unertain in log-log plots where apparently small errors are atually rather large. If improper data points are hosen, the logisti dose funtion may not have a solution or may give rise to an inaurate solution. The onstrution of aurate logisti dose urve solutions is something of an art. A rule of thumb proedure is to estimate the value of m from the urvature and distribution of data, and then hoose only one point from the transition segment to solve for n. The proessing of multi-power law data is arried out in the appendix, from whih we obtain three frition fator orrelations, ' and " (see equations (A.5), (A.7) and (A.9) in appendix). and ' are generated by 6 onneting four segments of power laws = 6Re,. 0 0.55 = Re, = 0.5Re and 0.65 = 0.8Re from different sequenes and are in minute agreement with eah other. " is onstruted 0.8 with the addition of the fifth power law = 9Re due to the fat that the Oregon data does not agree with the laminar solution = 6Re for 950 < Re < 900. The omparison of " with the data of MSZDS (00) is shown in figure 8. 0 0. 0.0 Prineton Data '' FIGURE 8. The omparison of logisti fitting urve " (see equation (A.9) in appendix) with the data of MSZDS (00). " is a rational fration of five power 0.8 laws = 6Re, = 9Re, 6. 0 0.55 = Re, = 0.5Re 0.65 and = 0.8Re. The R-square value of " is 0.9966 for the full range of Prineton and Oregon data. 0.00 0 00 000 0000 00000 000000 0000000 00000000

6. Roughness An aurate formula for the full range data of MKeon et al 00 for pipe flow was derived as a rational fration of five power laws onneted smoothly by the logisti dose urve algorithm. The turbulent flow data is represented by a omposition of two power laws with errors less than % for Re < 0 6 but the error inreases rapidly thereafter. This rather sudden inrease of data an be interpreted as a manifestation of the effet of roughness in an effetively smooth pipe with honed roughness. It is of onsiderable interest that this ritial value was also obtained independently MKeon et al 00 as a lower bound for the manifestation of roughness by studying the error of their best log formula. 0.0 0.0 Prineton Data '' ~ Correlation in equation (A.9) Correlation in equation (.) Correlation in equation (.) FIGURE 9. Comparison of MSZDS s data with three fitting urves in equations (.), (.) and equation (A.9) in the appendix. 0.0 0.0 0 0000 00000 000000 0000000 00000000 Fitting Error ε (%) 0 8 6 Perentage errors between Prineton data and fitting urves ( 0 < < 6 0 6 ) Fitting error by our orrelation '' in (A.9) Fitting error by the orrelation of MKeon et al 005 in (.) Fitting error by the orrelation of Barenblatt 997 in (.) Fitting error by Blasius law FIGURE 0. Relative fitting error ε vs. Re between four ~ Re orrelations and the Prineton data in turbulent regime 6 ( 0 < Re < 6 0 ). 0 0000 00000 000000 0000000 00000000 We might hope that the quality of our method of logisti dose urve fitting an be improved to arbitrary auray, modulo experimental satter, by hoosing more and more branhes of power laws and adjusting the sharpness ontrol parameters m and n in equation (A.). However, as a pratial matter, the implementation of fitting more and more power laws is more and more diffiult. Moreover, in priniple, the five point rule of the logisti dose urve algorithm whih is at the enter of our fitting method annot possibly give a perfetly aurate representation of the ontinuous data onneting power laws.

7. Frition fator orrelation of Nikuradse s data (9, 9) for smooth and effetively smooth pipes We onlude our analysis with a omparison of our rational fration of power law orrelation (A.9) with Nikurdses data for smooth and effetively smooth pipes. Equation (A.9) was derived from data of MSZDS 00 but is in satisfatory agreement with the Nikuradse data. The reader will reall that the full range data given by MSKDS is not in good agreement with the full range data Nikuradse 9. A new full range orrelation whih oinides with (A.9) in the overlap region is derived in a ompanion paper by Yang & Joseph 008 on flow in rough pipes is shown figure. '' ~ orrelation in equation (A.9) Nikuradse Smooth Pipe Data in Turbulent Regime 0. FIGURE. Comparison of Nikuradse s 9 data for turbulent flow in smooth pipes with the orrelation " (A.9) in appendix. 0.0 0.00 00 000 0000 00000 000000 0000000 0. 0.0 0.00 Nikuradse Smooth Pipe Data in Turbulent Regime Nikuradse Effetively Smooth Pipe Data Another fitting urve s taken from 'Virtual Nikuradse' of Yang & Joseph (008) 00 000 0000 00000 000000 0000000 FIGURE. Comparison of Nikuradse s 9 and 9 data for smooth and effetively smooth pipes with the logisti dose urve s for effetively smooth pipes derived by Yang & Joseph 008. s is onstruted from Nikuradse s 9 data by a proedure similar to the one desribed in this paper; s is a rational fration of five power laws: = 6/ Re, 0.75 = 0.00008Re, 0.85 = 0.57Re and 0.5 = 0.6Re, 0.6 = 0.075Re. The R-square value of s is 0.995 for the full range of Nikuradse s data for smooth and effetively smooth pipes. 8. Conlusion and disussion An effetively smooth pipe is one for whih the frition fator depends only on the Reynolds number = f ( Re) and not on roughness. We showed that the bottom envelope of Nikuradse s 9 data for rough pipes are effetively smooth and in fat oinide with his 9 data for smooth pipes and with superpipe data presented by MKeon et al. 00 in the interval. 0 < Re < 8.7 0. The MKeon data is full range over all Reynolds numbers less than 0 6 ombining Oregon data for laminar, transition and turbulent flow with Prineton superpipe data for turbulent flow. The Oregon data for transition and near transition

flows of various gases differ strongly with Nikuradse s 9 data for the flow of water. We introdued a new method for proessing data. The orrelation is given as a rational fration of rational frations of power laws whih is systematially generated by smoothly onneting linear splines in log-log oordinates with a logisti dose urve algorithm. This kind of orrelation seeks the most aurate representation of the data independent of any input from theories arising from the researhers ideas about the underlying fluid mehanis. As suh, these orrelations provide an objetive metri against whih observations and other theoretial orrelations may be applied. Our orrelation is as aurate, or more aurate, than best log formula of MKeon et al 00 and inomplete similarity power law formula of Barenblatt et al 996 in the range of Reynolds numbers in whih the orrelations overlap. Moreover, our formula is not restrited to ertain ranges of Reynolds number but instead applies uniformly to all smooth pipe flow data for whih data is available. Our method was applied to data for the mean flow profile in a boundary layer on a flat plate with adverse pressure gradient. We obtained a orrelation formula of good auray as a rational fration of two power laws. A highly aurate formula for the full range data of MKeon et al 00 for pipe flow was derived as a rational fration of five power laws onneted smoothly by the logisti dose urve algorithm. The turbulent flow data is represented by a omposition of two power laws with errors less than % for Re <.6 0 6 but the error inreases rapidly thereafter. This rather sudden inrease of data an be interpreted as a manifestation of the effet of roughness in an effetively smooth pipe with honed roughness. It is of onsiderable interest, though possibly for tuitis, that this ritial value was also obtained independently MKeon et al 00 as a lower bound for the manifestation of roughness by studying the error of their best log formula. The ideas and methods presented here may be used to orrelate ompliated engineering data from real and numerial experiments diretly into expliit formulas whih ould never be obtained by mathematial analysis. Aknowledgement. This work was supported by the NSF/CTS under grant 007668. We have profited from onversations with G. I. Barenblatt, Ivan Marusi, Beverley MKeon and Alex Smits about topis disussed in this paper.

APPENDIX Fitting proedure for onstruting multiple power law orrelations between and Re for smooth pipes (I). Data for the orrelation between and Re for turbulent flow in smooth pipes Now we will fit data for the orrelations between the frition fator and the Reynolds number Re. The modified logisti dose response urve an be expressed as ( Re) = f ( Re) L ( Re) f L( Re) ( Re / Re ) m f = f, (A.) R [ ] n where Re is the ritial Reynolds number (i.e. threshold value of Reynolds number), two assembly members, m and n are onstants. f L are the Experimental data of MSZDS 00 for frition fators in turbulent flow in smooth pipes is presented in figure ; the figure shows that the data ontains four subsetions of straight lines that are distributed in the interval(0, 900), (900, 050), (050, 0000) and (0000, ), respetively. The four straight lines represent four branhes of power laws in the log-log plot shown in figure A, and they are f a = 6Re, f b 6 =. 0 Re, 0.55 f = 0.5Re and 0.65 f d = 0.8Re. (A.) 0 0. 0.0 f f d Prineton Data fa fb f fd FIGURE A. Data of MSZDS 00 for frition fators in laminar, transition and turbulent flow in smooth pipes. Four branhes of power laws are identified in the plot, and they are f, a f, b f and f d. f b f a 0.00 0 00 000 0000 00000 000000 0000000 00000000 (II). Modified logisti dose urve fitting for two power laws Substituting f L = fa and f R = fb into equation (A.), we obtain a rational fration for the two subsetions f a and f b in figure A; identified in figure A by a thik solid line in green olor, giving rise to the expression where the ritial Reynolds number ( f f ) F = f, (A.) a Re is 900. b a 50 [ ( Re / Re ) ] 0. 5 5

0 Prineton Data fa fb F FIGURE A. The omposite logisti dose funtion F is a rational fration of two 0. power laws (A.)) f and a f b (see equation 0.0 0.00 0 0 00 000 0000 00000 000000 0000000 00000000 0. Prineton Data f fd F FIGURE A. The omposite logisti dose funtion F is a rational fration of two power laws f and f d (see equation (A.)) 0.0 0.00 0 0 00 000 0000 00000 000000 0000000 00000000 0. 0.0 Prineton Data F F FIGURE A. The omposite logisti dose funtion is a rational fration of two rational frations F and F, whih are logisti dose funtions of two power laws (see equations (A.) and (A.)). is a logisti fitting urve based on four power laws f, a f, b f and f d. 0.00 0 00 000 0000 00000 000000 0000000 00000000 Similarly, another logisti dose funtion for the two subsetions on the right side is given by ( f f ) F = f, (A.) d ' [ ( Re / Re ) ] 0. 5 6

where the ritial Reynolds number ' Re is 0000 (see figure A). (III). Logisti dose urve fitting for two rational frations of power laws To desribe all of the data of MSZDS 00 for frition fators in laminar, transition and turbulent flow in smooth pipes onneting the four branhes of power laws, we first replae the two assembly members f L with F and F whih are defined in setion (II). This leads to the following rational fration of power laws: where the ritial Reynolds number ( F F ) = F, (A.5) " 50 [ ( Re / Re ) ] 0. 5 " Re is 050 (see figure A). Figure A shows that the fitting urve desribed in equation (A.5) an fit the data of MSZDS 00 nearly perfetly in the full range of Reynolds numbers. 0 Prineton Data F f F FIGURE A5. The omposite logisti dose funtion F is a rational fration of a two power law rational fration F and a single 0. power law f (see equation (A.6)). 0.0 0.00 0 00 000 0000 00000 000000 0000000 00000000 0 0. 0.0 Prineton Data F fd ' FIGURE A6. The omposite logisti dose funtion ' is a rational fration of a rational frations F and a power law F is a logisti dose funtions of three f. d power laws (see equation (A.6)). ' is a logisti fitting urve based on four power laws f, a f, b f and f d. The R-square value of ' is 0.996 for the full range of Prineton and Oregon data. 0.00 0 00 000 0000 00000 000000 0000000 00000000 Another proedure for fitting two rational frations of power laws is to onstrut a logisti dose funtion, first for a rational fration of three power laws f a, f b and f fit to a fourth power law f d. This proedure is exhibited in figures A5 and A6. Figure A5 shows the logisti dose urve fitting of three power laws by 7

replaing f L in equation (A.) with F and f ; this urve is expressed as ( f F ) F = F, (A.6) ''' 50 Re / Re 0. where the threshold value of Reynolds number [ ( ) ] 5 ''' Re = 050. ( - ' ) / (%) 5 0 0 00 000 0000 00000 000000 0000000 00000000 FIGURE A7. Comparison of logisti dose fitting urves and '. The logisti dose funtion F in equation (A.6) represents a fitting urve of three power laws. We an ombine F with the fourth power law f d to obtain the fitting urve for the data of MSZDS 00 for the full range of Reynolds number (see figure A6). The fitting urve is given by where the threshold value of Reynolds number ( f F ) ' = F, (A.7) d "" [ ( Re / Re ) ] 0. 5 ''" Re = 0000. Figures A and A6 show that both proedures lead to fitting urves whih agree well with the experimental data. Moreover, the two fitting urves in equations (A.5) and (A.7) are in a good agreement with one another. We have ompared the differene between the two urves in equations (A.5) and (A.7), and the results are shown in figure A7 whih shows that the relative error between the two urves an be ontrolled within ±%. The large errors mainly our in the range of small Re beause Oregon data does not agree well with the theoretial solution = 6 / Re for laminar flow, whih is used in the onstrutions of orrelations (A.5) and (A.7). Another reason is that the power law f b is too steep in the log-log plot and logisti funtion may not handle it perfetly. (IV). Analysis of fitting error ε Figures A and A6 show that the data does not math the fitting urves perfetly in a small interval around Re = 000. The deviation is due to the fat that the data of MSZDS 00 does not math the laminar orrelation f = 6/ Re. This deviation of the four power law orrelations (A.5) and (A.7) from the experimental data an be redued by representing the data with five rather than four power laws. The measure of the deviation used here is Predited value by fitting urve - Observed value in experiment ε =. (A.8) Observed value in experiment The five power law onstrution is arried out as follows: We first split the laminar orrelation into two parts and replae it with a rational fration f a = 6 / Re 8

~ f = f a a ( f f a a 0 [ ( Re / 950) ] 0. 5 0.8 of two power laws fa = fa = 6 / Re and f a = 9Re. Then the fitting urve (A.7) an be updated to ~ ~ ( f F ) " = F, (A.9) where ~ ~ ( f F ~ ) F = F and ''' 50 [ ( Re / Re ) ] 0. 5 d "" [ ( Re / Re ) ] 0. 5 ~ ~ ~ ( f f ) F = f. a ) b a 50 [ ( Re / Re ) ] 0. 5 The fitting urve " in equation (A.9) is shown in figure 8. A zoom-in plot of our fitting urve " for Re ranging from 000 to 000 is shown in figure A8; the fitting urve " passes through the data smoothly, the relative errors of some points, due to satter, annot be redued to less than 5-0%. 0.05 0.0 0.0 0.0 '' 000 500 000 500 000 FIGURE A8. The zoom-in view of the modified logisti dose fitting urves ' and " for 000 Re 000 shows that the logisti dose fitting urves an be improved by hoosing more power laws. The two fitting urves ' and " are based on four and five power laws, respetively. The fitting errors on some points annot be redued to less than 5-0% due to the satter of experimental data. The big irles show the five points whih are hosen to fit the data in transition regions. Fitting Error ε (%) 50 0 0 0 0 Fitting errors of the orrelations in equations (A.9) and (.) for Prineton and Oregon data in full range of (. 0 < < 6 0 6 ) Fitting error by '' of Joseph & Yang in (A.9) MKeon et al (005) in (.) 0 0 00 000 0000 00000 000000 0000000 00000000 FIGURE A9. Relative fitting error vs. Re between our logisti dose urve " (A.9), the best formula (.) of MKeon et al 005 and the data of MSZDS 00. MKeon et al. fit their data with an empirial formula based on logarithms rather than powers; their 6 formula fits the data well in a limited range of Reynolds numbers Re ( 0 Re 5 0 ) in the turbulent regime. The relative error of their formula (.) in representing the data is also shown in this figure. Our equation (A.9) applies to the full range of Reynolds numbers given by MSZDS 00 rather than a limited one. 9

REFERENCES BALAKRISHNAN N. 99 Handbook of the logisti distribution. New York: Marel Dekker. BARENBLATT, G. I. 00 Saling. Cambridge University Press. BARENBLATT, G. I. & CHORIN, A. J. 997 Saling laws for fully developed turbulent flow in pipes. Appl. Meh. Rev., 50(7), -9. BARENBLATT, G. I., CHORIN, A. J., & PROSTOKISHIN, V. M. 997 Saling laws for fully developed flows in pipes: Disussion of experimental data. Pro. Natl. Aad. Si., 9(), pp. 77-776. GARCIA, F., GARCIA, R., PADRINO, J. C., MATTA, C., TRALLERO, J. & JOSEPH, D. D. 00 Power law and omposite power law frition fator orrelations for laminar and turbulent gas-liquid flow in horizontal pipelines. Int. J. Multiphase Flow, 9, 605-6. GARCIA, F., GARCIA, R. Z. & JOSEPH, D. D. 005 Composite power law holdup orrelations in horizontal pipes, Int. J. Multiphase Flow, (), 76-0. MARUŠIC IVAN and PERRY, A. E. 995 A wall-wake model for the turbulene struture of boundary layers. Part. Further experimental support. J. Fluid Meh., 98, 89-07. MCKEON, B. J. 00 High Reynolds number turbulent pipe flow. Ph.D. thesis, Prineton University MCKEON, B. J., SWANSON, C. J., ZARAGOLA, M. V., DONNELLY, R. J. & SMITS, A. J. 00 Frition fators for smooth pipe flow, J. Fluid Meh., 5, -. MCKEON, B. J., ZARAGOLA, M. V. & SMITS, A. J. 005 A new frition fator relationship for fully developed pipe flow, J. Fluid Meh., 58, 9-. NIKURADSE, J. 9 Laws of turbulent flow in smooth pipes (English translation). NASA TT F-0: 59 (966). NIKURADSE, J. 9 Stromungsgesetz in rauhren rohren, vdi Forshungshefte 6. (English translation: Laws of flow in rough pipes). Tehnial report, NACA Tehnial Memorandum 9. National Advisory Commission for Aeronautis (950), Washington, DC. PATANKAR, N. A., JOSEPH, D. D., WANG, J., BARREE, R., CONWAY, M. & ASADI, M. 00 Power law orrelations for sediment transport in pressure driven hannel flows, Int. J. Multiphase Flow, 8(8), 69-9. VIANA, F., PARDO, R., YÄNEZ, R., TRALLERO, J. L. & JOSEPH, D. D. 00 Universal orrelation for the rise veloity of long gas bubbles in round pipes, J. Fluid Meh., 9, 79-98. WANG, J., JOSEPH, D. D., PATANKAR, N. A., CONWAY, M. & BARREE, B., 00 Bi-power law orrelations for sedimentation transport in pressure driven hannel flows, Int. J. Multiphase Flow, 9(), 75-9. YANG, B. H. & JOSEPH, D. D. 008 Virtual Nikuradse, J. Fluid Meh., submitted. ZAGAROLA, M.V. 996 Mean flow saling in turbulent pipe flow. PhD thesis, Prineton University 0