1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Archived Iterative method to solving the Colebrook-White equation

  1. Mar 17, 2013 #1
    1. The problem statement, all variables and given/known data
    In our fluid mechanics class we were taught that we could use the following equation to solve for the Darcy friction factor f:


    To do this by hand:
    1. Guess a value for 1/sqrt(F), guess 3
    2. Get the right hand side result of the equation using 3
    3. Use that result for the next value of 1/sqrt(F)
    4. Continue using the result for the next value.
    5. To find F, just divide one by that value squared.

    This iterative approach works but I am not too sure why. Can anyone explain why it works? I'm guessing it requires some knowledge of mathematical proofs?
  2. jcsd
  3. Feb 5, 2016 #2
    You are trying to solve an equation of the form x = f(x) using successive substitution. The successive substitution scheme is $$x^{n+1}=f(x^n)$$ where n signifies the n'th iteration. If we also consider the previous iteration, we have $$x^n=f(x^{n-1})$$. If we subtract the two equations, we have:
    If we expand the rhs in a taylor series about xn, we obtain:
    In order for the scheme to converge, the magnitude of the changes in x from one iteration to the next must be getting smaller. If x is in the close vicinity of the solution, this means the, in order for the scheme to converge, $$|f'(x)|<1$$
    That is, the absolute value of the derivative of the function f must be less than 1 for the scheme to converge.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Discussions: Iterative method to solving the Colebrook-White equation
  1. Equation solving (Replies: 5)