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

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.

Colebrook’s Flow Friction Explicit Approximations Based on Fixed-Point Iterative Cycles and Symbolic Regression

“…The simplified method is based on the fixed-point method, a variant of the Newton-Raphson method in which the first derivative of the unknown function is equalized to 1 [30]. In the worst case, the demonstrated iterative procedures need up to seven iterations to reach the final accurate solution [10], whereas in this study the explicit approximations derived from the iterative procedure will have relative errors of up to 1.81% and 0.317% in the case of one internal and two internal iterative steps, respectively. Moreover, when the novel simple rational function p 0 of Equation (2) is used as a starting point, the relative errors are significantly reduced to 0.156% and 0.0259%, respectively.…”

“…. Now, the new value log 10 (y 1 ) = ζ + 0.8686·p 01 should be used for the calculation x 1 = −2·ζ + 0.8686·p 01 where 0.8686 ≈ 2 ln (10) . To solve the Colebrook equation using a numerical fixed-point iterative procedure, the steps shown in Figure 1 should be followed.…”

“…should be chosen from the domain of applicability of the Colebrook equation, which is 0 < ε < 0.05 and 4000 < Re < 10 8 . Every initial starting point chosen from this domain will be suitable, but to reduce the required number of iterations, different strategies are offered [10]. We will demonstrate the approach with the fixed-value initial starting point, but also the approach with the starting point given by the rational function: Equation (2) was found using the symbolic regression tool Eureqa [31,32] following the approach of Gholamy and Kreinovich [33], in which existing absolute error minimizing software was used to minimize the relative error (various artificial intelligence tools have been used recently for the Colebrook equation, such as symbolic regression [34], novel artificial bee colony programming (ABCP) methods [35], artificial neural networks [36], genetic algorithms [37], etc.).…”

“…The initial starting point x = should be chosen from the domain of applicability of the Colebrook equation, which is 0 < ε < 0.05 and 4000 < Re < 10 . Every initial starting point chosen from this domain will be suitable, but to reduce the required number of iterations, different strategies are offered [10]. We will demonstrate the approach with the fixed-value initial starting point, but also the approach with the starting point given by the rational function: p = x = 1 f = 2600 · Re 657.7 · Re + 214600 · Re · ε + 12970000 − 13.58 · ε…”

“…The Colebrook equation, Equation (1) [1], is implicitly given in respect to the unknown flow friction factor f , which is captured implicitly in a logarithmic expression in a way that can be expressed explicitly only using special functions such as the Lambert W function or its cognate Wright Ω [2][3][4][5][6][7][8][9]. In the latter instance, the Colebrook equation can be solved only numerically using iterative procedures [10] or using specially developed explicit approximations [11][12][13][14][15].…”