Review of new flow friction equations: Constructing Colebrook’s explicit correlations accurately

Abstract

Using only a limited number of computationally expensive functions, we show a way how to construct accurate and computationally efficient approximations of the Colebrook equation for flow friction.The presented approximations are based on the asymptotic series expansion of the Wright ω-function and symbolic regression. The results are verified with 8 million of Quasi-Monte Carlo points covering the domain of interest for engineers. In comparison with the built-in "wrightOmega" feature of Matlab R2016a, the herein introduced related approximations of the Wright ω-function significantly accelerate the computation. With only two logarithms and several basic arithmetic operations used, the presented approximations are not only computationally efficient but also extremely accurate. The maximal relative error of the most promising approximation which is given in the form suitable for engineers' use is limited to 0.0012%, while for a little bit more complex variant is limited to 0.000024%.

Suitability for coding of the Colebrook’s flow friction relation expressed by symbolic regression approximations of the Wright-ω function

“…On the other hand, although accurate, build-in Matlab function for the Wright ω-function is extremely slow. Numerical experiments in Table 1 also show that the alternative open-source implementation given by the library "wrightOmegaq" is much faster than the Matlab build-in implementation (Horchler 2017 As reported in Praks and Brkić (2020), as extension to the results from Table 1, approximation y=lnx./(x -0.5564*lnx + 1.207). -lnx, is to date the most accurate.…”

“…Number of explicit approximations of the Colebrook equation exists (Assuncao et al 2020, Brkić 2011ab, Brkić and Ćojbašić 2017, and here we will offer few very accurate and computationally efficient which are based on the Wright ω-function, a cognate of the Lambert W-function (Corless et al 1996). Thanks to these special functions, it is capable to transform expressions from their implicit in an explicit form, which is suitable for further processing (Brkić 2011bc, Brkić 2012c, Brkić and Praks 2019, Praks and Brkić 2020.…”

Excel VBA-Based User Defined Functions for Highly Precise Colebrook’s Pipe Flow Friction Approximations: A Comparative Overview

“…After thorough examination of the approximations of the Colebrook equation from available literature [6][7][8][9][10][11], nine most accurate explicit approximations [23][24][25][26][27][28][29][30] were selected for analysis and for comparisons performed in this review paper. The examined approximations are ranked in Table 1 in terms of 1) accuracy, and 2) time taken for execution:…”

“…The results are [44]: Addition-23.40sec, Subtraction-27.50sec, Multiplication-36.20sec, Division-31.70sec, Squared-51.10sec, Square root-53.70sec, Fractional exponential-77.60sec, Napierian natural logarithm-63.00sec, and Briggsian decimal logarithm to base 10-78.80sec. Accuracy is checked using 2 Million quasi-random and as well 90 thousand and 740 uniformly distributed samples, as in [9,23,35,36], which covers the whole domain of the Reynolds number, Re and of the relative roughness of inner pipe surface, ε, which are commonly used in engineering practice; 2320<Re<10 8 and 0<ε<0.05.…”

Oil pipeline hydraulic resistance coefficient identification

“…It has been shown that most of the available approximations of the Colebrook formula (including the Altshul's formula) are accurate with a difference of a few percent. Brkić and Praks (2019), (Praks & Brkić, 2020) showed a way to construct accurate and efficient approximations of the Colebrook equation. The use of the shifted Lambert W function, also known as the Wright omega function, made it possible to speed up the calculations and obtain the data with a relative error of 0.0096% to 0.000024%.…”