I have an assingment that says:

How is the friction factor $f$ dependent of the roughness $\epsilon$ and Reynolds number $Re$?

The equaton for the friction factor is $\displaystyle\frac{1}{\sqrt{f}}=1.14-2 \log_{10}\left(\frac{\epsilon}{D_h}+\frac{9.35}{Re\sqrt{f}}\right)$

But how can I say something about this if $f$ is also dependent of it self?

Can anynone help?


1 Answer 1


It is an implicit function which you have to solve numerically. Typically you would use "fixed point iteration".

You can do this in Excel with some user defined functions where for a given geometry you start with $f=1$ and then use the above $\frac{1}{\sqrt{f}} = L(f)$ a few times until in converges to a value $$f \rightarrow \frac{1}{L^2(f)} $$

In my example with $\epsilon/D_h = 0.005$ and $R_e = 1000$ it converges to $f=0.06561561\ldots$ after 10 iterations.

So after you build a table with various $\epsilon/D_h$ and $R_e$ you can talk about how it behaves.

  • $\begingroup$ So was it decreasing or increasing in your case? $\endgroup$ Jun 17, 2015 at 18:50
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    $\begingroup$ I only did 1 point, so I cannot comment. It is up to you now. $\endgroup$ Jun 17, 2015 at 18:56
  • $\begingroup$ Alternatively you can try $L' = \frac{\partial L}{\partial f}f' + \frac{\partial L}{\partial \epsilon}\epsilon'+\frac{\partial L}{\partial R_e}R_e'$ but it looks to be very messy. $\endgroup$ Jun 17, 2015 at 19:02
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    $\begingroup$ Hmm I think I understand it. So you just pick some values for $\epsilon / D_h$ and $Re$. Then you start with $f=1$ and make iterations until it converges. Then you pick some new values for $\epsilon / D_h$ and $Re$ and make the whole process again. $\endgroup$ Jun 17, 2015 at 19:28
  • $\begingroup$ Yes, this is exactly how you can do it. $\endgroup$ Jun 18, 2015 at 12:10

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