Evolutionary Optimization of Colebrook’s Turbulent Flow Friction Approximations

Abstract

This paper presents evolutionary optimization of explicit approximations of the empirical Colebrook's equation that is used for the calculation of the turbulent friction factor (λ), i.e., for the calculation of turbulent hydraulic resistance in hydraulically smooth and rough pipes including the transient zone between them. The empirical Colebrook's equation relates the unknown flow friction factor (λ) with the known Reynolds number (R) and the known relative roughness of the inner pipe surface (ε/D). It is implicit in the unknown friction factor (λ). The implicit Colebrook's equation cannot be rearranged to derive the friction factor (λ) directly, and therefore, it can be solved only iteratively [λ = f(λ, R, ε/D)] or using its explicit approximations [λ ≈ f(R, ε/D)], which introduce certain error compared with the iterative solution. The optimization of explicit approximations of Colebrook's equation is performed with the aim to improve their accuracy, and the proposed optimization strategy is demonstrated on a large number of explicit approximations published up to date where numerical values of the parameters in various existing approximations are changed (optimized) using genetic algorithms to reduce maximal relative error. After that improvement, the computational burden stays unchanged while the accuracy of approximations increases in some of the cases very significantly.

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

“…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.…”

“…The Serghides approximation [29] is based on Steffensen iterative scheme [4], and the shown version, Eq. ( 5) is improved by genetic algorithms [35,45,46]. It is given in Eq.…”

Turbulence: Numerical Analysis, Modeling, and Simulation

“…Brkić andĆojbašić [7] present evolutionary optimization for approximations of the Colebrook's equation for the turbulent friction factor. This calculation is used for the calculation of turbulent hydraulic resistance in hydraulically smooth and rough pipes including the transient zone.…”