Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function: Reply to Discussion

Abstract

This reply gives two corrections of typographical errors in respect to the commented article, and then provides few comments in respect to the discussion and one improved version of the approximation of the Colebrook equation for flow friction, based on the Wright ω-function. Finally, this reply gives an exact explicit version of the Colebrook equation expressed through the Wright ω-function, which does not introduce any additional errors in respect to the original equation. All mentioned approximations are computationally efficient and also very accurate. Results are verified using more than 2 million of Quasi Monte-Carlo samples.

Rational Approximation for Solving an Implicitly Given Colebrook Flow Friction Equation

“…Our rational approximation approach is based on Padé approximants [20,21], symbolic regression [30,31] and although not used directly, it is inspired by the Wright ω-function, a cognate of the Lambert W-function [32]. To avoid detailed explanations about the Lambert W-function [33], here it should be noted that in this context it is used to transform the Colebrook equation from the shape implicitly given in respect to the unknown flow friction factor to the explicit form [23,24,[27][28][29]34,35].…”

“…Based on that approach, few very accurate and efficient explicit approximations suitable for coding and for engineering practice have been constructed [22]. In addition, the same authors developed few approximations of the Colebrook equation based on the Wright ω-function, which are among the most accurate to date [23][24][25]. On the other hand, they contain one or two logarithmic functions, depending on the chosen version [23,24] (these procedures are based on the previous efforts by Praks and Brkić for symbolic regression [26] and by Brkić with Lambert W-function [27][28][29]).…”

What Can Students Learn While Solving Colebrook’s Flow Friction Equation?

“…However, the Colebrook equation can be rearranged in the explicit form using the Lambert W function, where further, this special function can be evaluated only approximately [5]. Some numerical constraints in using the Lambert W function, which can occur in our case, will be detected [37] and a way to mitigate this inconvenience will be shown [27][28][29].…”

“…On the contrary, basic mathematical operations (addition, subtraction, multiplication, and division) are very fast on computers, because they are executed directly. Students can learn how to avoid using these time-consuming functions [27][28][29][30]. This simplification process can have a large effect on the students' future jobs, regarding the design of complex pipe networks, which involve extensive computations [31][32][33][34][35][36].…”

“…For this reason, the next step involves the series expansion of the Lambert W function, and its cognate, the Wright ω function [39]. In this step, students will learn how to make accurate, but at the same time, efficient approximations of the Colebrook equation using the Wright ω function [27][28][29]40,41]. Last, but not least, students can use raw numerical data that can efficiently simulate the Colebrook equation and, in that way, learn principles of using artificial intelligence in everyday engineering practice [12,[42][43][44][45].…”