Talk:Darcy friction factor formulae

A comment on notation
The Colebrook Equation and the Haaland's solution are written for 1/sqrt(f) rather than for 'f' and this is IMHO the right thing to do. However the Swamee-Jain and Serghide's solutions are written for 'f' instead of 1/sqrt(f) which makes them inconsistent with the first two equations and makes them more difficult to understand. BTW Swamee-Jain is very similar to Haaland, but that's hard to see it because of the way Swamee-Jain is written. Consider a change ?


 * I favor giving the most visibility to the versions introduced by the creators of the various formulae. Having shown the creator's version, then presentation of other, published forms seems useful to me.  I suggest being bold if you have seen other forms and want them here.


 * If Swamee-Jain, etc., is usually written in terms of 'f' instead of '1/sqrt(f)' then it might cause confusion to _replace_ the usually-expected form with a revised form.


 * While the '1/sqrt(f)' form is traditional for the Colebrook equation, the other equations which calculate 'f' seem more user-friendly to me in that they don't require the user to unpack '1/sqrt(f)'.


 * About the similarity noted between Haaland and Swamee-Jain:
 * Interesting observation!
 * Playing fast and loose with the exponent, the constant in the Re term is 6.96 ( = 5.74^1.11 -- rounding 1/0.9 to 1.11) when Swamee-Jain is put into the Haaland form.


 * That is, the transformation from Swamee-Jain to almost-Haaland seems to involve this operation:


 * (a + b)^c = (a^c + b^c)


 * As one cannot simply move the exponent onto each term of a polynomial and have the result be _exactly_ the same, I would caution against showing the almost-Haaland form of Swamee-Jain in the article.


 * I realize that Colebrook may have done something comparable in deriving his formula, but his formula has historical acceptance. Putting the almost-Haaland form of Swamee-Jain in the article would show an approximation which is probably in the realm of original research.
 * --Ac44ck 00:28, 28 October 2007 (UTC)

Origin and approximations of Colebrook's equation
There is an intriguing statement in this document: http://www.psig.org/papers/2000/0112.pdf
 * "the Colebrook-White equation ... is nothing more than a combination of" the Nikuradse equation and the Karman-Prandtl equation.

If one pursues that notion, a more precise form of Colebrook's equation seems to be:


 * 1/sqrt(f) = -2 * log [e/D / 3.71535... + 2.51188... / (Re * sqrt(f))]

Here are the details:

Nikuradse -- rough pipe law (infinite Reynolds number)


 * f approaches zero as e/D approaches zero.

1/sqrt(f) = 2*log(D/e) + 1.14
 * = 1.14 - 2*log(e/D)
 * = 2*log(3.71535...) - 2*log(e/D)
 * = -2*[log(e/D) - log(3.71535...)]
 * = -2*log(e/D / 3.71535...)


 * where:
 * 3.71535... = 10^(1.14/2)

Karman and Prandtl -- smooth pipe law (e/D = 0)


 * f approaches zero as Re approaches infinity.

1/sqrt(f) = 2*log(Re * sqrt(f)) - 0.8
 * = 2*log(Re * sqrt(f)) - 2 * log(2.51188...)
 * = 2*log(Re * sqrt(f) / 2.51188...)
 * = -2*log[2.51188... / (Re * sqrt(f))]


 * where:
 * 2.51188... = 10^(0.8/2)

Combining terms (which are mutually-exclusive at the limits) within the log by simple superposition:


 * 1/sqrt(f) = -2 * log [e/D / 3.71535... + 2.51188... / (Re * sqrt(f))]

Which is the Colebrook equation -- with more "available" figures.

How "significant" those extra "available" figures are is a question which might be explored elsewhere.

Regardless of how significant those extra digits are in the end, it seems that attempts to create explicit formulae may not be aiming at quite the right target. They are trying to get as close as possible to the results of a formula which was derived from two others -- and _already_ contains rounding errors.

It seems that any curve-fitting to the Colebrook equation should be done with several more "available" figures in the Colebrook equation's constants.

The time to worry about significant digits should be _after_ the "best" constants have been found in the explicit formula. That is, after the results of the explicit formula provide the best fit to a precise form of Colebrook's equation.

Perhaps that has been happening, but 3.7 and 2.51 may have been treated as sacred in the development of most of the explicit formulae which are intended to approximate the Colebrook equation.

Exactly those constants certainly loom large in Serghide's solution. And exactly 3.7 appears in the other approximations listed in this article so far.

Perhaps this is the stuff of a Wikipedia article of its own. --Ac44ck 00:18, 21 October 2007 (UTC)

Log10?
Numerically solving the Colebrook-White equation as given in this article does not reproduce the Moody chart. Try this in Mathematica:

The correct result is obtained, however, when replacing the base-10 logarithms in _all_ the formulas in this article with the natural logarithm. All approximative formulas in this article should be written with the natural logarithm also.

--sb —Preceding unsigned comment added by 128.138.42.170 (talk) 15:36, 22 August 2008 (UTC)


 * Numbers, please.


 * I believe that Log10 is correct.


 * Let's pick some numbers.


 * Say:
 * Reynold's number = 1e5
 * e/D = 0.001
 * f = 0.022175


 * 1/sqrt(0.022175) =  6.71534277873
 * -2*log(0.001/3.7 + 2.51/1e5/sqrt(0.022175)) =  6.71541653669


 * The calculations above were performed with Calcute, a handy freeware program that is available here:
 * http://calcute.com
 * The left hand sides of the calculations above can be pasted directly into Calcute to yield the values on the right hand sides.


 * f = 0.022175 seems pretty close to the convergence value using log10.


 * The calculator here
 * http://www.calctool.org/MOBILE/eng/civil/friction_factor
 * says 0.0221745.


 * Let's look at the Moody diagram here:
 * http://biosystems.okstate.edu/darcy/DarcyWeisbach/MoodyDiagram.htm
 * which is linked here:
 * http://en.wikipedia.org/wiki/Darcy_friction_factor


 * f = 0.022 seems like a credible value from the graph when e/D and Re have the values given above.


 * The convergence value is closer to 0.005 using Ln (natural log) instead of Log10.


 * I don't use Mathematica. I didn't try to parse the Mathematica instructions posted above.  I suspect that something is amiss in how Mathematica is being used. Ac44ck (talk) 03:43, 27 August 2008 (UTC)

You are correct, there is a typo in the Mathematica code above. Fixing that reproduces the chart qualitatively and quantitatively. Thanks for your help and sorry for the confusion. Best, sb.


 * The natural/common log ambiguity can make things messy. Ditto degrees/radians for trig functions.


 * As 1) the Mathematica code isn't paragraphs long and 2) you have identified the typo, might you post the code which you found produces a Moody diagram for the benefit of others who are using Mathematica?


 * Thanks. -Ac44ck (talk) 05:36, 8 September 2008 (UTC)

Here is some (working :-) code that plots the friction factor for relative roughness 1e-6, 1e-5, 1e-4, 1e-3 and 1e-2.

--sb —Preceding unsigned comment added by 128.138.42.230 (talk) 07:43, 7 October 2008 (UTC)

The redirect from Darcy friction factor‎
The previous redirect from Darcy friction factor to Darcy–Weisbach equation seemed more appropriate to me than the recent redirect which points to this Darcy friction factor formulae article.

The purpose of this article was to be a repository for various formulae for the friction factor. It provides little or no explanation of what the friction factor is or how it is used.

The Darcy–Weisbach equation article contains information which may be of more interest to one who is seeking general information about the Darcy friction factor. I favor reverting to the previous redirection of Darcy friction factor to Darcy–Weisbach equation. -Ac44ck (talk) 21:34, 27 October 2008 (UTC)


 * I changed the redirects, as well as the wording of the lead sections of this article and Darcy–Weisbach equation. Thanks for pointing this out. -- Crowsnest (talk) 08:06, 28 October 2008 (UTC)

Accuracy of the Swamee-Jain equation
I removed this note and commentary (which belongs on a talk page) from the article:
 * The equation was found to match the Colebrook-White equation within 1.0% for 10-6 < ε/D < 10-2 and 5000 < Re < 108. This claim appears false. Swamee-Jain is 2 percent high at ε/D = 0.0064 and Re = 10,000.

Some numbers (which were missing from the commentary):
 * Value for Colebrook equation according to
 * http://www.ajdesigner.com/php_colebrook/colebrook_equation.php
 * absolute roughness (ε) = .0064 meter
 * diameter (d) = 1 meter
 * Reynolds number (Re) = 10000
 * Darcy friction factor (f) = 0.0392618990804


 * That is consistent with result from http://www.calctool.org/CALC/eng/civil/friction_factor using Serghide's solution:
 * f = 0.0392619


 * Value for Swamee-Jain equation from a spreadsheet:
 * f = 0.040041

0.040041 / 0.0392619 =  1.0198

Or, 2 percent high.

Good catch, 64.9.235.10. - Ac44ck (talk) 02:51, 16 December 2008 (UTC)

Shanks transformation
The section on Serghides' solution mentions that it is the result of a Shanks transformation. Page 6 of this document http://www.lib.umd.edu/drum/bitstream/1903/5060/1/TR_91-12.pdf suggests how that may have been done. It seems that the initial guess for f was around 0.044 ( = (2.51/12)^2).

This document http://anale.steconomice.evonet.ro/arhiva/2007/statistics-and-economic-informatics/5.pdf mentions the Aitken's delta-squared process.

I am unfamiliar with this area. The latter two links suggest to me that the label in the first link may be incorrect. Is Serghides' solution actually the result of a Aitken's delta-squared process rather than a Shanks transformation?- Ac44ck (talk) 18:12, 24 December 2008 (UTC)
 * The References section on this page http://www.cheresources.com/colebrook3.shtml suggests Serghides' solution was obtained using Steffensen's method‎. It is not clear to me that the Colebrook equation satisfies the "converging to a fixed point" criterion mentioned in the Aitken's delta-squared process article. -Ac44ck (talk) 05:02, 26 December 2008 (UTC)

The original Serghides paper from 1984 says the following: "Eq. 2 is derived by applying Steffenson's accelerated-convergence technique to an iterative, numerical solution of Eq. 1. The constants A, B and C are approximations of Eq. 1 obtained by three iterations of the direct-substitution method. Eq. 2 is the approximation obtained by combining those constants according to Steffenson's formula." Eq. 1 is the Colebrook equation, while Eq. 2 is Serghides' solution.

His paper also references a previous paper which "discusses the conditions for convergence in solving the Colebrook equation by direct substitution, and explains Steffenson's method" (Serghides, T.K., Iterative solution by direct substitution, Chem. Eng., Sept. 6, 1982). I only have the 1984 paper. -Kybber (talk) 10:10, 6 January 2009 (UTC)

Reducing fractions involving e/D
The article contains a mixture of fractional forms involving relative roughness (e/D).

Consistency seems desirable. Which form is preferred?
 * Complex Fraction: (e/D) / 3.7
 * Reduced fraction: e / (3.7 * D)

The latter form may allow the spreadsheet calculation to proceed more quickly. Someone referring to the article would not need to reduce the fraction themselves, possibly introducing an error in the process.

To me, the notion of "relative roughness" is somewhat obscured by the reduced form "e / (3.7 * D)." On the other hand, the complex fraction may be an anachronism. Anyone using a Moody diagram doesn't need the formula. Someone using the formula may be hindered by the complex fraction.

Shall we strive for uniformity in this article? If so, shall we reduce fractions where possible? Would that be WP:OR if the sources use complex fractions? - Ac44ck (talk) 19:12, 29 February 2012 (UTC)


 * I think it's better to keep the complex fraction ε/D (relative roughness), as that is the dimensionless group which is actually important, not the individual values of ε and D, so that makes more sense on physical grounds. Also, charts displaying the Fanning friction factor plot f as a function of relative roughness and Reynolds number. Ryan ChemE (talk) 20:25, 11 March 2015 (UTC)

New formula
The general form of the formula is:

$$ \frac{1} = - c \cdot \ln \left( {a + b\frac{1}} \right) $$

where

$$ a = \frac{1} \frac{\epsilon }{D} $$

$$ b = \frac $$

$$ c = \frac{2} $$

Applying the transformation

$$ \frac{1} = \left( {c \cdot y - \frac{a}{b}} \right) $$

$$ x = \frac{1}{c}\left( {\frac{1} + \frac{a}{b}} \right) $$

we get

$$ \left( {\frac{a} - \ln \left( {b \cdot c} \right)} \right) = y + \ln \left( y \right) $$

so we have to solve the equation

$$ x = \left( {y + \ln \left( y \right)} \right) $$

so

$$F(x)$$ is the revert of $$y+ln(y)$$

$$y = F(x)$$

with

$$x=F(x)+ln(F(x))$$

an ascending function

$$ {\text{x}} \in \left( { - \infty {\text{,}}\infty } \right),{\text{y}} \in \left( {0{\text{,}}\infty } \right) $$

The Newton's method is

$${y_{n + 1}} = {y_n}\left( {1 - \frac} \right)$$

$${y_{n + 1}} = \frac\left( {1 - \ln \left( \right) + x} \right)$$

the solution is

$$ f = \frac{1}\frac{1} = \frac $$

where

$$ x = \left( {\frac{a} - \ln \left( {b \cdot c} \right)} \right) = \left( {\frac{\epsilon }\frac{2}\frac + \ln \left( {\frac{2}\frac} \right)} \right) $$

$$ x = \left( {0.1239681863 \cdot \frac{\epsilon }{D}{N_R } + \ln \left( {0.4586822894 \cdot {N_R }} \right)} \right) $$

Because $$N_R>4000 \Rightarrow x>7.5$$

a good approximation is

$$y = F\left( x \right) \approx x\left( {1 - \frac{1}\ln \left( x \right)} \right)$$

$$x \in [0.6,\infty )$$ with $$error{\text{ }} < {\text{ }}1.5\%$$

Sorry it the same solution with the Brkić solution i haven't notice before!!!!

Agepap (talk) 15:32, 10 October 2013 (UTC).

I added the reference for the Churchill equation 1977 - see http://www.scribd.com/doc/200078357/Churchill-S-W-Friction-factor-equation-spans-all-fluid-flow-regimes?secret_password=xqwq9ri6m7t4lsjtivt if you want to check data (I will delete it in a few weeks)Turner chris1 (talk) 10:04, 16 January 2014 (UTC) 91.137.130.129 (talk) 12:16, 3 July 2018 (UTC)I think, there are some problem with these (Churchill 1977 and Fang 2011, and Tsal 1989) forms: i got mainly different f values with these forms, such as other 23 forms or with goal seek solve!91.137.130.129 (talk) 12:16, 3 July 2018 (UTC)

Churchill equation ref
I added reference to the Churchill equation - hope I did it right - you can see paper at  http://www.scribd.com/doc/200078357/Churchill-S-W-Friction-factor-equation-spans-all-fluid-flow-regimes?secret_password=xqwq9ri6m7t4lsjtivt to check the data. I will delete the file in a few days (copyright), after that email me if you want to check it. Turner chris1 (talk) 10:21, 16 January 2014 (UTC)

I think the Churchill equation quoted here is wrong - gives about x 8 the factor in the turbulent region.

The correct one (that is to say, one which gives sensible answers when you plumb some numbers in !) is quoted in Mustafa Asker, Oguz Emrah Turgut, Mustafa Turhan Coban's 2014 review. — Preceding unsigned comment added by 217.37.222.89 (talk) 16:25, 17 September 2015 (UTC)


 * For anyone reviewing the 1977 reference, eq. 18 (the final equation for f in the paper) differs from the one in this article by a factor of 8. But after eq. 5 it says, "So, all equations in this article can be converted to the Fanning or Darcy friction factors, by multiplying the nonsubscripted f by 2 or 8, respectively."
 * So, the one in the article is correct for the Darcy friction factor. 38.18.112.72 (talk) 13:53, 6 April 2023 (UTC)

Availability of Rugosity values
Information about rugosity is needed for the recent addition under Blasius correlations.

The description in the wikilink above is not helpful. It brings to mind the question, "How Long Is the Coast of Britain?," because the "real surface area" depends on how it is measured.

How can one use the given formula which includes a parameter for rugosity? Where are such values tabulated or how may they be calculated from information that is tabulated, such as absolute roughness? - Ac44ck (talk) 19:32, 18 January 2014 (UTC)


 * Fixed by 128.178.23.109. - Ac44ck (talk) 22:30, 21 January 2014 (UTC)

Removing redundancies
Several approximations are listed both in their own subsections of the article and in the table of approximations at the end. Might we preserve points of interest from the subsections with notes to the table, and eliminate most of the dedicated subsections? A sample for the Haaland equation is inserted in the article. The footnote is long, but it could eliminate an entire subsection while preserving the explanatory material that is currently in the subsections. Any thoughts on implementing this? - Ac44ck (talk) 02:02, 27 June 2014 (UTC)

Colebrook-White equation
This type of Colebrook-White equation can be computed accurately. Also there are about 6 different Colebrook-White equations that all cane be solved accurately. If a number is in a Log with other numbers, and without numbers on the other side, it can be solved easily. My training years ago what use X=Log(200+X*2). You can guess an X and compute the right side, and use the solution for the next X. Because the Log makes the compute much smaller, it makes the correct solution quick. In the Colebrook-White equation can have the 1/Sqrt(f) changed to X, and since the right side has a number divided by sqrt(f) it can be changed to  *X. So the equation will be X=-2*Log(e/(3.7*Dh)+(2.51/Re)*X). To make the solution easier, compute the e/(3.7*Dh) as A... and compute the 2.51/Re as B. Then the easy solution will be  X=-2*Log(A+B*X). Usually in about 5 loops the X will stop changing will about 15 decimals not changing. But some times it may take 10 loops to get 15 decimals. The the f will be the X ... 1/X/X (one divided by the X twice). There are a few approximations that are right to 15 decimals. But if can compute to 50 digits you will see that all the approximation are not right. To check the solution, both the left and right sides will both be the X you computed. See the web site ...  keisan.casio.com/exec/system/1381988562 for all six different Colebrook equations, That place is shearing the whole world with this solutions. (There you will see who I am).

PS: Someone always removes my Easy and True solution, because it would show only the correct solution shows "Goudar–Sonnad" is about right 99% percent for 15 decimals and "Serghides" is about right 95% of the time for 15 decimals for only the main Colebrook equation. But very few are right for about 8 decimals. — Preceding unsigned comment added by Harrell Geron (talk • contribs) 3 January 2015


 * Note: I have moved the above contribution from the article page to here Noyster  (talk),  11:35, 3 January 2015 (UTC)

Do we need Excel walkthroughs?
I'm wondering if it is appropriate to spend so much time walking through how to numerically solve the Colebrook equation. This is nothing more than a nonlinear equation, and any engineer could think of multiple ways to solve it, e.g. Goal Seek in Excel, Solver in Excel, or implementing Newton's method. To me it seems kind of distracting to spend so much time explaining how to solve the equation, and this information is completely unnecessary for anyone reading this article. It probably makes sense to mention explicit approximations to the equation, as that's actually new material and facilitates the solution of complex piping problems using Newton-Raphson. But why don't we just say "this equation could be solved by any nonlinear equation solver" and be done with it? Ryan ChemE (talk) 19:30, 11 March 2015 (UTC)

Haaland equation
According to my 7th edition of Massey the RHS of the Haaland equation is =-3.6log10... not -1.8

Magicchouffe (talk) 14:54, 9 October 2015 (UTC)

Suggested edits to improve this article
This article uses inconsistent notation:
 * Darcy friction factor: both f and &lambda; are used for the purpose.
 * Hydraulic diameter is given as both Dh and D.
 * Reynolds number: both roman (Re) and italic (Re) appear.
 * Natural logarithm erroneously appears italic (ln).

Some suggested alterations:
 * It seems worthwhile to explain the notation once and for all near the beginning of the article, then use consistent notation thereafter.
 * The roughness height ε invariably appears scaled to the hydraulic diameter D in the ratio ε / D (roughness ratio), so the various formulas ought to keep this dimensionless quantity intact in so far possible.
 * Prefer hydraulic diameter D to hydraulic radius R; they invariably stand in a 2:1 ratio.
 * Left-hand side of equation $$\frac{1}{\sqrt{\lambda}}$$ for all formulae (instead of &lambda;).

Finally, this article is entitled "Darcy friction factor formulae", but treats only the Colebrook–White relation, which is by no means the only formula that is used to fit the data (see Darcy–Weisbach equation for several formulae). Modern experiments have extended the range of the data greatly and have shed light on a number of failings of Colebrook–White. While this article is not the place to hash out the controversy, we owe it to the discerning reader to avoid conflating Colebrook–White with Darcy–Weisbach.

ArthurOgawa (talk) 04:37, 1 July 2016 (UTC)

Haaland equation
The formula for Haaland in the section "Approximations of the Colebrook equation" is not the same as the formula for Haaland presented in the section "Table of approximations" at the end of the article

$$ \frac{1}{\sqrt {f}} = -1.8 \log_{10} \left[ \left( \frac{\varepsilon/D}{3.7} \right)^{1.11} + \frac{6.9}{\mathrm{Re}} \right] $$

vs.

$$ \frac{1}{\sqrt{\lambda}} = -1.8 \log \left[\left(\frac{\varepsilon}{3.7D}\right)^{1.11} + \frac{6.9}{Re}\right] $$

Note the difference in the base of log. — Preceding unsigned comment added by 134.132.52.221 (talk) 19:26, 19 July 2016 (UTC)

I tried to help things along a bit. ArthurOgawa (talk) 22:26, 19 July 2016 (UTC)

Brkić solution
I read the papers referred to, Brkić(2011) and Brkić(2012), and the formula given in this section does not faithfully portray the formula in Brkić(2012), Equation 4.3. The equation for this section would be changed to:
 * $$ \frac{1}{\sqrt {f}} = -2\log \left(\frac{\varepsilon/D}{3.71} + \frac{2.18}{\mathrm{Re}}W\left(\frac{\mathrm{Re}}{2.18}\right) \right) $$

where I have reduced Brkić's numerical value $$ 5.02/\ln{10} $$ to 2.18.

Unless I hear objections, I propose to make this correction on 1 December 2017 ArthurOgawa (talk) 17:55, 15 November 2017 (UTC)

Error in formular of "Bellos, Nalbantis, Tsakiris 2018"
I implemented the formular 33-35 of "Bellos, Nalbantis, Tsakiris" from https://ascelibrary.org/doi/pdf/10.1061/%28ASCE%29HY.1943-7900.0001540 for a libary I am building and could not recreate fig. 6. of the paper. I think they have a minor mistake in the formular in the paper.

In my opinion there are some problems with formular 33-35 and fig. 6 of the paper: - the lables in fig. 6 are wrong. With growing D/ks, the factor f the curve converges to should be lower. -> my assuption is that they lables are inverted. The curve lables (D/ks=1014 and D/Ks=30) are switched, the ones (D/ks=504 and D/Ks=61.2) and the ones (D/ks=504 and D/Ks=120). - Now if you implement formula 33-35, and try to recreate the plot, you get different curves. Comparing the proposed formula to the one of Cheng (https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9429%282008%29134%3A9%281357%29) we also reference, the mayor difference is that cheng uses log10 and Bellos at. al are using ln. This explains the differences in the factors infront of the logs. The other mayor difference is the factor in the second log of eg. 33. (cheng is using 3.7 and bellos is using 6.82). My assuption here is that, Bellos at al. are actually using radius instead of diameter in the second ln, because it explains the difference to chengs formular and if you use the radius here, the formular creates fig. 6 exactly. So either a factor of 3.41=6.82/2 sould be correct or the use of R instead of D.

My suggestion is to correct the formular in the table or at least mention the mistake. — Preceding unsigned comment added by SRSonix (talk • contribs) 08:30, 3 September 2020 (UTC)


 * I have copies of the two cited papers (Bellos, Nalbantis, Tsakiris, 2018 and Cheng, Nian-Sheng, 2008) and am trying to understand what SRSonix is talking about. ArthurOgawa (talk) 16:45, 6 September 2020 (UTC)


 * There's a erratum for that paper: Bellos, V., Nalbantis, I., & Tsakiris, G. (2020). Erratum for “Friction Modeling of Flood Flow Simulations” by Vasilis Bellos, Ioannis Nalbantis, and George Tsakiris. Journal of Hydraulic Engineering, 146(10), 08220005. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001802. This erratum probably fixes the errors. IvesC (talk) 21:05, 27 August 2021 (UTC)

Free-surface flow? Open-channel flow?
I know in Bellos et al., 2018, they use the term "free surface flow", but basically it's talking about wide open-channel flow. A flood plain is a special case of open-channel flow: a channel with an infinite width. I think free-surface flow has a broader meaning than open-channel flow in other areas/applications of fluid dynamics. Should we use open-channel flow regardless of what Bellos et al. use? Moreover, I feel like it's overstating that the equation from Bellos et al., 2018 is the only one formula, among others in this wiki article, works for open-channel flow. The most naive way to work with open-channel flow is to use the hydraulic diameter of an open channel in formulas of closed pipe flows. As no validation (i.e., against experimental data) is available, no one can really say the most naive approach is not working, In this sense, it may be too much to say Bellos et al., 2018 is the only one working with open channels. IvesC (talk) 22:30, 27 August 2021 (UTC)