Advanced iterative procedures for solving the implicit Colebrook equation for fluid flow friction

TL;DR

This paper compares iterative methods for solving the implicit Colebrook equation, demonstrating that accelerated Householder and three-point methods achieve high accuracy with fewer iterations, improving computational efficiency in fluid flow friction calculations.

Contribution

It introduces and analyzes accelerated iterative procedures, including Householder, Halley, Schroder, and Newton-Raphson methods, for solving the Colebrook equation more efficiently and accurately.

Findings

Accelerated methods reduce iterations needed for high accuracy.

Householder approach expressed via Lambert W-function analyzed.

A new approximation with less than 0.0617% error presented.

Abstract

Empirical Colebrook equation from 1939 is still accepted as an informal standard to calculate friction factor during the turbulent flow through pipes from smooth with almost negligible relative roughness to the very rough inner surface. The Colebrook equation contains flow friction factor in implicit logarithmic form where it is, aside of itself, a function of the Reynolds number Re and the relative roughness of inner pipe surface. To evaluate the error introduced by many available explicit approximations to the Colebrook equation, it is necessary to determinate value of the friction factor from the Colebrook equation as accurate as possible. The most accurate way to achieve that is using some kind of iterative methods. Usually classical approach also known as simple fixed point method requires up to 8 iterations to achieve the high level of accuracy, but does not require derivatives of the Colebrook function as here presented accelerated Householder approach (3rd order, 2nd order: Halley and Schroder method and 1st order: Newton-Raphson) which needs only 3 to 7 iteration and three-point iterative methods which needs only 1 to 4 iteration to achieve the same high level of accuracy. Strategies how to find derivatives of the Colebrook function in symbolic form, how to avoid use of the derivatives (Secant method) and how to choose optimal starting point for the iterative procedure are shown. Householder approach to the Colebrook equations expressed through the Lambert W-function is also analyzed. One approximation to the Colebrook equation based on the analysis from the paper with the error of no more than 0.0617% is shown.