estimation of friction function

E-proceedings of the 38th IAHR World Congress

September 1-6, 2019, Panama City, Panama

doi:10.3850/38WC092019-0188

4309

Estimation of Friction Function

BALKRISHNA SHANKAR CHAVAN(1), (1) Reservoir and Appurtenant Structures,

Central Water and Power Research Station,Pune-411046, Maharashtra, India

[email protected]

ABSTRACT Estimating Friction function Bs is complex because it is sensitive to changes in size, shape, spacing

and spacing pattern that is It is sensitive to conveyance and shape factors. Many researchers such as

Colebrook- White, Afzal, Yalin and Garcia improved its estimation. Various parameters of fluid flow and

geometrical section are combined to describe the friction function. Estimation of friction loss in a flow is

important step in finalizing discharge passing through a water conductor system. surface roughness is

present, there is no universality of scaling of the friction factor with respect to the traditional Reynolds

number Re, and different expressions are evolved. Friction factor important role in the computation of

discharge, deciding geometry of the flow section. In the past attempts were made to express Friction

function Bs as a function of Reynolds number. Re can be estimated with empirical equation. Because

of its complexity so far published literature is not enough to arrive at generalized equation for the

computation of friction function accurately. There is need to analyze and develop empirical equations

for friction function. In this paper attempt has been made to analyze 878 published data collected from

various sources such as various project sites globally(123 observations). and from laboratory (755

observations). Data collected include geometrical parameters of flow i.e. width, depth and longitudinal

slope. Other parameters are grain size, roughness, shear velocity, discharge and fluid characteristics

such as density, viscosity and specific weight. Present works include; the analysis of data for the

computation of Reynolds number for the estimation of friction function with respect to depth of flow and

grain roughness. Mathematical simple equations are proposed obtained from plots were based on the

analysis of data for depth of flow and roughness.

Keywords: Empirical, Friction function, Friction factor Roughness Reynolds number,

1 INTRODUCTION

Friction function Bs term originates in a law of logarithmic distribution for velocity first given by Karman

Prandtl. Friction function Bs is complex because it is sensitive tohydraulic parameters and bed-material

particles, channel gradient in natural stable channels, depth of flow, suspended material, and the bed

level. Naturally occurring flow and its interaction within fluid and with flow boundaries result in two

unavoidable problems. One is that there are a great variety of semi-empirical and semi- theoretical or

full-empirical friction formulae based on the same, similar or different theories and data. Comparisons

of these formulae through theoretical analysis, and with laboratory and field data leads to interesting

conclusions and information. Methods for finding the friction factor f was using a diagram, such as

the Moody chart, or solving implicit equations such as the Colebrook–White equation upon which the

Moody chart is based, or explicit equations proposed by various investigators as given in Table 1 in

Annexure A.

The ASCE Task Force on Friction Factors in Open Channels (1963) highlighted usefulness of Darcy Weisbach formulation of friction factor in computing resistance to flow in open channels. Task Force noted formulation as fundamental and mentioned that the Manning equation could be used for fully rough conditions. Task Force presented figure for variation of resistance with Reynolds number, which showed with Mannings equation, there is continual decay of resistance with Reynolds number, even in the limit of large values. From this, one could deduce that Mannings equation is fundamentally flawed. The recommendations of the Task Force have almost entirely been ignored, and the Gauckler-Manning-Strickler formulation continued to dominate, even though, with the exception of the Strickler formula,

mailto:balkrishna0@

https://en.wikipedia.org/wiki/Moody_chart

https://en.wikipedia.org/wiki/Colebrook%E2%80%93White_equation



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there are few general research results available and use of tables/graphs continued. Even though the Task Force presented a number of experimental and analytical results, there was no simple solution path to follow for friction factor problems.

2. REVIEW OF LITERATURE:

Law of logarithmic distribution for velocity first given by Karman - Prandtl - velocity distribution.𝑢

𝑢∗=

1

𝜅ln (𝐴𝑠

𝑧

𝑘𝑠) + 𝐵𝑠 (1)

In a typical engineering applications, there will be a set of given or known quantities. The acceleration of gravity g and the dynamic/kinematic viscosity of the fluid μ ,ν are known, as are the hydraulic radius of the R and its roughness height ε.

1

√𝑓= 𝐴𝑙𝑜𝑔

𝑅

𝑦′ + 𝐵𝑠

(2) The constant in resistance equation is due to the numerical integration, and is a function of shape of the cross-section. For circular section A = 2.0, B = -0.62 For rectangular section: A = 2, B = -0.79 (for large ratio of width/depth.

The Moody diagram proposed in 1940 is still relevant especially at remote sites where limited computing facility is available. A simplification and a rough approximation: the conditions it describes (fully developed, isothermal, incompressible, dissipative, pseudo-steady-state flow) are never quite attained in practice. Conditions within pipelines are inherently non-uniform over the flow cross section and over a given length. Fluids are more or less compressible, rather than incompressible, and have finite thermal conductivity. The flowing fluid within pipelines is not isothermal in the radial direction because of the very dissipation that is quantified by the Darcy-Weisbach equation. Frictional dissipation and finite thermal conductivity together give rise to differences in temperature—and density and viscosity—between different parts of the flow. Inevitably too, any length of pipe for which the fully-developed flow requirement is approximated must be connected to entry and exit systems within which this is not the case. When the flow is turbulent, vortices form and collapse relentlessly over time. In the flow of liquids, vapour pressure may play a role. Nevertheless, the diagram is a very useful design tool. For computational purposes, it is easily represented as a data set of discrete points, or by the equations that define it along with appropriate solution algorithms. The same four zones that were labeled by Moody (laminar, critical, transition and complete turbulence) are shown.

Figure 1 Moody Diagram



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Yalin (1992) suggested equation

1

√𝑓= 𝐴𝑙𝑜𝑔

𝑅

𝑦′ + 𝐵𝑠 (3)

It follows that As and Bs are related by

𝐴𝑠 = 𝑒𝜅𝐵𝑠 (4)

If as well the head loss per unit length S is a known quantity, then the friction factor f can be calculated directly from the chosen fitting function. Solving the Darcy–Weisbach equation:

Head loss due to channel friction, calculated with the help of Darcy Weisbach

Head loss = ℎ𝑓 = 𝑓𝐿𝑉2

2𝑔𝐷

(5)

3.Semi-Empirical Equations

3.1Colebrook and White

The smooth, rough and transition laws for Nikuradse's sand roughness in which the grains are of uniform size and closely packed together do not resemble with roughness of real pipes. Experimental results of Nikuradse considerably differ in the region of the transition curve for a pipe having a roughness

composed of isolated particles. Semi-Empirical Equation suggested by Colebrook and White overcome

variations in transition region is: 1

√𝑓= 1.74 − 2𝑙𝑜𝑔 (

𝑘

𝑟0+

18.7

𝑅𝑒√𝑓) (6)

when Re is small, it is consistent with smooth pipe flow, when Re is large, it is consistent with rough pipe flow. However its performance in the transitional domain overestimates the friction factor by a substantial margin.[12] Colebrook acknowledges the discrepancy with Nikuradze's data but argues that his relation is consistent with the measurements on commercial pipes. Indeed, such pipes are very different from those carefully prepared by Nikuradse: their surfaces are characterized by many different roughness heights and random spatial distribution of roughness points, while those of Nikuradse have surfaces with uniform roughness height, with the points extremely closely packed.

Estimation of friction factor Implicit equation for estimation of friction factor was given by Colbrook in 1939

1

√ 𝑓= 𝑙𝑜𝑔 [

𝑒

𝐷

3.7+

2.51

𝑅𝑒√𝑓] (7)

𝑅𝑒 =𝜌𝐷𝑉

𝜇 (8)

Where 𝑓 = Friction factor

Re = Reynolds Number given by

μ = Coefficient of Dynamic viscosity

Alternative equations to Equation 7 for the friction factor of smooth pipes have been proposed and there is some evidence to suggest that the smooth pipe friction factor curve should be a little higher at high Reynolds numbers. However, corroborating data are still scarce. Also, where friction factor has been measured at very high Reynolds numbers the fluid has been a gas. Equation (7) is implicit in nature and needs time consuming iterative steps for its solution.

3.2Two power explicit equation given by the author Chavan (2017) is simple and convenient:

https://en.wikipedia.org/wiki/Darcy%E2%80%93Weisbach_equation#cite_note-Shockling2006-15



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𝑓 =0.0075(

(4000)

𝑅𝑒

0.05)

0.25+(4000

𝑅𝑒)

5 +20

𝑅𝑒 (9)

Many investigators (some of them are listed in Annexure A) proposed approximate solution of the Colbrook-White equation in a form of explicit equation. Plots given by various researchers is shown in Figure 2.

Figure 2 –Friction factor plots given by various researchers

3.3 Garcia Friction function Bs given by Garcia as a function of Roughnes (friction) Reynolds number Re* as

following empirical equation:

𝐵𝑠 = 8.5 + [2.5ln (𝑅𝑒∗) − 3]𝑒−0.121[ln(𝑅𝑒∗)]2.42

(10) Where Roughness Reynold Number Re* is

Re∗ =u∗ ks

ν (11)

In above expression friction function is expressed in only in terms of roughness Reynolds number, no

effect or consideration for 𝑒

𝐷 for higher values of Reynolds number which is logical. Plot is shown in

Figure 3



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Figure 3 Plot of Friction function Bs and Re* edited by Garcia

above graph exhibit following features

when ks = 0, then Re* is identically zero: flow is in the smooth regime. The data for thesepoints lie to the left extreme of the abscissa and are not within the frame of the graph.

When Re* < 5, the data lie on the line Bs = Re*; flow is in the smooth regime.

When Re* > 100, the data asymptotically approach a horizontal line; they are independentof Re, f and ks/D

While the Colebrook–White relation is, in the general case, an iterative method needs more computing time. In the filed it is very convenient to use two power explicit equation

Roughness Reynolds Number R* is defined by

R∗ =1

√8. 𝑅𝑒 √𝑓

∈

𝐷 (12)

Friction(Roughness) Function Bs is defined by Afzal as

𝐵𝑠 =1

9.3√𝑓+ 𝐿𝑜𝑔 (

1.9

√8

∈

𝐷) (13)



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Figure-4. Plot of Roughness Reynolds Number and Friction Function given by Afzal.

From the above graph exhibit following features:

When ε = 0, then Re* is identically zero: flow is always in the

smooth pipe regime. The data for these points lie to the left extreme of the abscissa and arenot within the frame of the graph.

When Re* < 5, the data lie on the line B(R) = Re*; flow is in the smooth pipe regime.

When Re* > 100, the data asymptotically approach a horizontal line; they are independentof Re, f and ε/D

The intermediate range of 5 < Re* < 100 constitutes a transition from one behavior to theother. The data depart from the line B(R) = R very slowly, reach a maximum near Re* = 10,then fall to a constant value.

A fit to these data in the transition from smooth to rough flow employs an exponential expression in Re* that ensures proper behavior for 1 < Re* < 63 (the transition from the smooth regime to the rough regime) function shares the same values for its term in common with the Kármán–Prandtl resistance equation, plus one parameter 0.34 to fit the asymptotic behavior for Re* → ∞ along with one further parameter, 11, to govern the transition from smooth to rough flow. It is exhibited in Figure 3.

4. Data Collection:878 runs of basic flow parameters such as width, slope, discharge, depth of flow and sediment size offlumes, channel and rivers collected by different investigators has been compiled and analysed in thispaper. Experimental data from laboratory flumes, irrigation channels and Natural Rivers were collectedfrom publication of following sources.

a. Laboratory Flumes : USGS(1966), Bharat Singh(1971), J. R. Barton & P. N. Lin(1955), P. K.

Pande(1960), G. P. Williams(1970), E. M. Laursen(1957), E. O. J. Plate (1957) and H. P. (1996)

b. Irrigation Channels : U S channels (1970), Japanese channels (1978) and Pakistan channels

(1931)

c. Natural Rivers: Amezon, Rio Orinoco, Luznic, Ishikari, Skive-Karup and Tigus rivers.

Above data is analyzed for velocity, friction factor and roughness Reynolds, is plotted to obtain

proposed equation is shown in Figure 5



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Figure-5. Plot of Roughness Reynolds Number and Friction Factor.

.5.0 Equatons obtained by analysis of data for flume data and field data are :

5.1 For Flume data 1

√𝑓= −0.57𝑙𝑛(𝑅𝑒) + 9.379 (14)

5.2 For Field data

1

√𝑓= −0.59𝑙𝑛(𝑅𝑒) + 8.011 (15)

uncertainty of collected data (Chien and Wan, 1999) (although laboratory-collected data has a relatively high accuracy, the scale problem has limited its extensive use). For example, in order to collect field data (in natural river course) to validate hydraulic parameter formulae, flow and sediment transport conditions are assumed to satisfy “uniform flow and equilibrium sediment transport”, which is rarely the case in practice. Another inevitable problem for the collected data is “it is one dimensionality”. When these formulae are used in two-dimensionally, numerically modeling of sediment transport and bed deformation in natural rivers, the results are seldom expected.

Thus, it is important to examine these formulae’s performance when Data mentioned above was plotted

for Roughness Reynolds Number Vs Friction factor Bs as shown in Figure 6.

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

10.00

1.00 10.00 100.00 1000.00 10000.00 100000.00

Roughness Re*

Variation of friction with Roughness Reynolds no

Re1/Sqrtffield

Re1/sqrtfflume

Log. (Re1/Sqrtffield)

Log. (Re1/sqrtfflume)

1

√𝑓= −0.57𝑙𝑛(𝑅𝑒) + 9.379

1

√𝑓= −0.59𝑙𝑛(𝑅𝑒) + 8.011



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Figure 6 Plot of Roughness Reynolds Number and Friction Function for data in Figure 5

Plot of Friction function Bs vs. roughness Reynolds number Re, the data fall on a single

trajectory when plotted.. The regime Re < 1 is effectively that of smooth pipe flow. For large Re,

the friction function Bs approaches a constant value. of 8.5

5. 3 Proposed equation for Bs

𝐵𝑠 = 8.5 + [2.6ln (𝑅𝑒∗) − 3]𝑒−0.12[ln(𝑅𝑒∗)]2.4 (16)

From the above plot exhibit following features:

When ε = 0, then Re* is identically zero: flow is always in the smooth pipe regime. The datafor these points lie to the left extreme of the abscissa and are not within the frame of thegraph.

When Re* < 7, the data lie on the line Bs = Re*; flow is in the smooth pipe regime.

When Re* > 30, the data asymptotically approach a horizontal line; they are independentof Re, f and ε/D

The intermediate range of 7 < Re* < 30 constitutes a transition from one behavior to the other.The data depart from the line Bs = Re*; very slowly, reach a maximum near Re* = 11, thenfall to a constant value.

Limitations of studies- As naturally flowing water has limited range of hydraulic parameters,

data collected has limited range of Reynolds number. There is need of additional data at

higher Roughness Reynolds number to refine the result of studies

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

10.00

11.00

12.00

1.00 10.00 100.00 1000.00 10000.00 100000.00

Bs

Roughness Reynolds number=Re* = u*ks/v

Bs Curve

ObservedValue

HPFLUME



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Figure-7. Plot of Roughness Reynolds Number Calculated and Observed.

6. CONCLUSIONS:

1 The proposed equation (16) is logarithmic exponential equation, explicit in nature. Friction function varies directly with log Re∗ and inversely with exponent of Re∗Roughness Reynolds Number.

2. The proposed equation (14) and (15) are logarithmic equations, simple explicit in nature.

Friction factor varies inversely with square of Roughness Reynolds Number. Equation (15) and

(16) would be of help in simulation of complicated water conductor networks in predicting flow

parameters.

3. Equation (14) give values of friction factor for laboratory flume lower than Field values

predicted by equation (15)

4. Figure 7 shows 25% and 50% envelop curve for Roughness Reynolds Number. It is seen that

99.66% of data is covered in 50% envelop, while 72 % data is covered in 25 % envelop.

5. Data collected has limited range. There is need of additional data at higher Roughness

Reynolds number to refine the result of studies.

6. Further scope to improve with additional parameters as pointed out by Hunter Rouse

e.g.Froude Number, porous bed & banks which affects friction factor after getting additional

data

7. REFERENCES

Afzal Noor 2007 Friction Factor Directly From Transitional Roughness in a Turbulent Pipe Flow Journal of Fluids Transactions of the ASME Vol. 129, October 2007. Pp 1255-1267.

ASCE Task Force Report, 1963 Jour. Hydraulics Division, Amer. Soc. of Civ. Engs., vol. 89, no HY2, March,pp 97-143.

Bawdy, D.R., 1979, Flood frequency estimates on alluvial fans: American Society of Civil Engineers, Journal of Hydraulics Division, HY11, v. 105, p. 1407-1413.

Bobbie, C.H., and Wolf, P.O., 1953, The Lynmouth flood of August 1952:London, Proceedings of the Institution of Civil Engineers, v. 2, Part 3,Dec. 1953, pp. 522-588.

0

5000

10000

15000

20000

25000

30000

0 5000 10000 15000 20000 25000 30000

Calculated Vs Observed

50% Lower envelop

50% Upper envelop

25% envelop

25% Envelop

Series6

ReFrlaboratory



4117

Colebrook, C. F. (1939) Turbulent Flow in Pipes with particular reference to Transition Regions between Smooth and Rough Pipe Laws, Journal of Institution of Civil Engineers, Paper No. 5204 pp 133-156 + tables

Chavan Balkrishna Shankar (2017), Unified Equation for Estimation of Friction Factor, Proceedings of 37th IAHR World Congress, Aug 13-18, held at Kuala Lumpur, Malaysia Pages 5885-5893

Chow, V.T., 1959, Open Channel Hydraulics, New York, McGraw-Hill, 680 p.

Costa, J.E., and Jarrett, R.D., 1981, Debris flows in small mountain stream channels of Colorado and their hydrologicimplications: Bulletin of the Association of Engineering Geologists, v. 18, no. 3, p. 309-322.

Cowan, W.L.,1956, Estimating hydraulic roughness coefficients: Agricultural Engineering, v. 37, no. 7, p. 473-475.Davidian, Jacob, 1984, Computation of water-surface profiles in open channels: U.S. Geological Survey Techniques of Water-Resources Investigations, Book 3, Chapter A15, 48 p.

Gol'din, B.M., and Lyubashevskiy, L.S., 1966, Computation of velocity of mudflows for Crimean Rivers: Soviet Hydrology, v. 5, no. 2, p. 179-181.

Golubtsov, V.V., 1969, Hydraulic resistance and formula for computing the average flow velocity of mountain rivers: Soviet Hydrology, v. 8, no. 5, p. 500-510.

Hejl, H.R., Jr., 1977, A method for adjusting values of Manning's roughness coefficients for flooded urban areas: U.S. Geological Survey, Journal of Research, v. 5, no. 5, p. 541-545.

Herbich, J.B., and Shulits, Sam, 1964, Large-scale roughness in open-channel flow: Journal of the Hydraulics Division, American Society of Civil Engineers, v. 90, HY6, p. 203-230.

Jarrett, R. D. (1985) Determination of Roughness Coefficients for Streams in Colorado Lakewood, Colorado

Moody LF. 1944. Friction factors for pipe flow. Trans. A.S.M.E 66: p671-684.

Yalin M. S. (1992) River Mechanics, Pergamon Press, Newyork



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ANNEXURE-A

TABLE-1 Application of explicit Friction Factor Formulae

No Investigator Year Formula Validity Range R2

1 Chavan 2017 ff =20

Re+ [

0.0075 (4000

Re)

0.05

0.25 + (4000

Re)

5 ]

Valid for wide range of Reynolds number

0.9777

2 Morrison 2013 f =16

Re+ [

0.0076 (3170

Re)

0.165

1 + (3170

Re)

7 ]

Inaccuracy crept in for higher values of Reynolds numbers

3 Barenblatt 2005 f =8

ψ2(1+α) Re< 13×106

4 Mckeon 2005

1

√f= 1.920 log (Re√f − 0.475

−7.04

(Re√f)0.55

5 Yen 1991 f =

1

4[−log (

e

12R) +

5.2

(4Re)0.9]

−2 30000<Re , e

D< 0.05

6 Barr 1981 f =1

4[−log (

e

14.8R) +

5.2

(4Re)0.89]

−2

7 Chen 1979 1

√f= −2log [

1

3.7605[

e

D] −

5.0452A

Re]

4000<Re<108

0.000001< e

D

0.9148

8 Churchil 1977 f = 8 [ (8/Re)12 + 1

(A + B)1.5]

112

0.9066

9 Haaland 1983 f =

0.308642

[log ⟨(e

3.7D)

1.11

+ |6.9Re

⟩]2 0.8509

10 Colebrook 1938-39

1

√f= −2log [

eD

3.7+

2.51

Re√f]

0.90416

11 Serghides 1984 1

√f= ⌊A −

(B − A)

C − 2B + A

2

⌋ Re>2100 and any

e/D

12 Swami-Jain 1976 1

√f= 1.14 − 2log [[

e

D] +

21.25

Re0.9]

5000<Re<108

0.000001˂e

D˂0.05

13 Barr 1972 f =1

4[log (

e

14.8R) +

5.76

(4Re)0.9]

−2

0.9051

14 Zigrang & Sylvester

1982 1

√f= −2log [

1

3.7[

e

D] −

5.02A

Re]

4000<Re<108 and

0˂e

D˂0.04

0.9041

15 Wood 1966 f = 0.094 (

e

D)

0.223

+ 0.53 (e

D)

+ 88 (e

D)

0.44

(Re)−ψ

4000˂Re˂5*108

0.00001˂e

D˂0.04 0.7075

16 Moody 1947 f = 0.0053 [1 + (2 ∗ 104ε

D+

106

Re)

13

]

4000˂Re˂5*108

0˂ε

D˂0.01

estimation of friction function

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