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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:

    86e8e7da627a4be30b61a4130a2a05eb.png

    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:
    $$x^{n+1}-x^n=f(x^n)-f(x^{n-1})$$
    If we expand the rhs in a taylor series about xn, we obtain:
    $$x^{n+1}-x^n=f'(x^n)(x^n-x^{n-1})$$
    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.
     
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