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

Abstract

Widely used in hydraulics, the Colebrook equation for flow friction relates implicitly to the input parameters; the Reynolds number, Re and the relative roughness of inner pipe surface, ε/D with the output unknown parameter; the flow friction factor, λ; λ=f(λ, Re, ε/D). In this paper, a few explicit approximations to the Colebrook equation; λ≈f(Re, ε/D), are generated using the ability of artificial intelligence to make inner patterns to connect input and output parameters in explicit way not knowing their nature or the physical law that connects them, but only knowing raw numbers, {Re, ε/D}→{λ}. The fact that the used genetic programming tool does not know the structure of the Colebrook equation which is based on computationally expensive logarithmic law, is used to obtain better structure of the approximations which is less demanding for calculation but also enough accurate. All generated approximations are with low computational cost because they contain a limited number of logarithmic forms used although for normalization of input parameters or for acceleration, but they are also sufficiently accurate. The relative error regarding the friction factor λ, in best case is up to 0.13% with only two logarithmic forms used. As the second logarithm can be accurately approximated by the Padé approximation, practically the same error is obtained also using only one logarithm.

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.…”

Unified Friction Formulation from Laminar to Fully Rough Turbulent Flow

“…The unified flow friction approach presented here is flexible, as proposed equations for a certain hydraulic flow regime can be easily altered using interchangeable formulas for laminar, smooth turbulent, and rough turbulent flow. Although our previous experiences with artificial intelligence [38][39][40] have shown that an encapsulation of all flow friction regimes into one coherent model is not a straightforward task, the form proposed here is simple. Thus, the unified approach presented here can be easily implemented with software codes.…”

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

“…Every initial starting point chosen from this domain will be suitable, but to reduce the required number of iterations, different strategies are offered [10]. We will demonstrate the approach with the fixed-value initial starting point, but also the approach with the starting point given by the rational function: Equation (2) was found using the symbolic regression tool Eureqa [31,32] following the approach of Gholamy and Kreinovich [33], in which existing absolute error minimizing software was used to minimize the relative error (various artificial intelligence tools have been used recently for the Colebrook equation, such as symbolic regression [34], novel artificial bee colony programming (ABCP) methods [35], artificial neural networks [36], genetic algorithms [37], etc.). The novel starting point given by Equation (2) has a maximal relative error of only 6.7%, whereas the previous version of the starting point [34] had a maximal relative error of 40%.…”