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Processes 2018, 6(8), 130; https://doi.org/10.3390/pr6080130

Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation

1,2,†,* and 1,†,*
1
European Commission, Joint Research Centre (JRC), Directorate C - Energy, Transport and Climate, Unit C3: Energy Security, Distribution and Markets, Via Enrico Fermi 2749, 21027 Ispra (VA), Italy
2
IT4Innovations National Supercomputing Center, VŠB—Technical University of Ostrava, 17. listopadu 2172/15, 708 00 Ostrava, Czech Republic
Both authors contributed equally to this study.
*
Authors to whom correspondence should be addressed.
Received: 24 July 2018 / Revised: 10 August 2018 / Accepted: 14 August 2018 / Published: 16 August 2018
(This article belongs to the Section Computational Methods)
Full-Text   |   PDF [369 KB, uploaded 21 August 2018]

Abstract

The Colebrook equation is implicitly given in respect to the unknown flow friction factor λ; λ = ζ ( R e , ε * , λ ) which cannot be expressed explicitly in exact way without simplifications and use of approximate calculus. A common approach to solve it is through the Newton–Raphson iterative procedure or through the fixed-point iterative procedure. Both require in some cases, up to seven iterations. On the other hand, numerous more powerful iterative methods such as three- or two-point methods, etc. are available. The purpose is to choose optimal iterative method in order to solve the implicit Colebrook equation for flow friction accurately using the least possible number of iterations. The methods are thoroughly tested and those which require the least possible number of iterations to reach the accurate solution are identified. The most powerful three-point methods require, in the worst case, only two iterations to reach the final solution. The recommended representatives are Sharma–Guha–Gupta, Sharma–Sharma, Sharma–Arora, Džunić–Petković–Petković; Bi–Ren–Wu, Chun–Neta based on Kung–Traub, Neta, and the Jain method based on the Steffensen scheme. The recommended iterative methods can reach the final accurate solution with the least possible number of iterations. The approach is hybrid between the iterative procedure and one-step explicit approximations and can be used in engineering design for initial rough, but also for final fine calculations. View Full-Text
Keywords: Colebrook equation; Colebrook–White; iterative methods; three-point methods; turbulent flow; hydraulic resistances; pipes; explicit approximations Colebrook equation; Colebrook–White; iterative methods; three-point methods; turbulent flow; hydraulic resistances; pipes; explicit approximations
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Praks, P.; Brkić, D. Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation. Processes 2018, 6, 130.

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