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.

Unified Friction Formulation from Laminar to Fully Rough Turbulent Flow

“…The here presented unified flow friction approach is flexible, as proposed equations for certain hydraulic flow regime can be easily changed. Although our previous experiences with artificial intelligence [38][39][40] showed that encapsulation of all flow friction regimes into a one coherent model is not a straightforward task, the here proposed form is simple. Thus, the here presented unified approach can be easily implemented in software codes.…”

Symbolic Regression-Based Genetic Approximations of the Colebrook Equation for Flow Friction

“…We used [computer software] Eureqa by Nutonian, Inc., Boston, MA, as a genetic programming tool [22,23]. The symbolic regression approach adopted herein [24][25][26][27][28][29] is based upon genetic programming wherein a population of functions is allowed to breed and mutate with the genetic propagation into subsequent generations based upon a survival-of-the-fittest criteria [30]. The main goal of this study is to make accurate and computationally cheap explicit approximations of the Colebrook equation, where computationally cheap means to contain the least possible number of logarithmic functions and non-integer powers [31][32][33][34][35][36].…”

One-Log Call Iterative Solution of the Colebrook Equation for Flow Friction Based on Padé Polynomials

“…Usually, with the initial guess that is close to the exact solution, the iterative procedure converges to it in five or fewer iterations. (Brkić 2017). Here the goal is to avoid use of logarithmic functions and therefore, this starting point is not suitable.…”