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

Abstract

The logarithmic Colebrook flow friction equation is implicitly given in respect to an unknown flow friction factor. Traditionally, an explicit approximation of the Colebrook equation requires evaluation of computationally demanding transcendental functions, such as logarithmic, exponential, non-integer power, Lambert W and Wright Ω functions. Conversely, we herein present several computationally cheap explicit approximations of the Colebrook equation that require only one logarithmic function in the initial stage, whilst for the remaining iterations the cheap Padé approximant of the first order is used instead. Moreover, symbolic regression was used for the development of a novel starting point, which significantly reduces the error of internal iterations compared with the fixed value staring point. Despite the starting point using a simple rational function, it reduces the relative error of the approximation with one internal cycle from 1.81% to 0.156% (i.e., by a factor of 11.6), whereas the relative error of the approximation with two internal cycles is reduced from 0.317% to 0.0259% (i.e., by a factor of 12.24). This error analysis uses a sample with 2 million quasi-Monte Carlo points and the Sobol sequence.

Hybrid Models for Solving the Colebrook–White Equation Using Artificial Neural Networks

“…Moreover, the study and development of the approximation of the Colebrook-White equation are still going on today. Some of the latest studies on this can be seen in several kinds of literature [18][19][20][21][22][23].…”

Precise Explicit Approximations of the Colebrook-White Equation for Engineering Systems

“…Friction losses in most cases are significant in the total pressure losses. Most flows in the systems are turbulent, and the friction losses correspond to the accurate but implicit Colebrook-White equation [1]:…”

“…At May 2020, the last entry corresponds to 2018 [12]. Dejan Brkić and Pavel Prax [1] obtained the most accurate modern approximations in 2019 by one and two steps of Padé approximation. It has 0.0259% of deviation, but it is too bulky.…”

Rational Approximation for Solving an Implicitly Given Colebrook Flow Friction Equation

“…Praks and Brkić [19] recently developed a one-log call iterative method, which uses only one computationally demanding function, and even then only in the first iteration (for all succeeding iterations, cheap Padé approximants are used [20,21]). 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].…”