Abstract
The unknown friction factor from the implicit Colebrook equation cannot be expressed explicitly in an analytical way, and therefore to simplify the calculation, many explicit approximations can be used instead. The accuracy of such approximations should be evaluated only throughout the domain of interest in engineering practice where the number of test points can be chosen in many different ways, using uniform, quasi-uniform, random, and quasi-random patterns. To avoid picking points with undetected errors, a sufficient minimal number of such points should be chosen, and they should be distributed using proper patterns. A properly chosen pattern can minimize the required number of testing points that are sufficient to detect maximums of the error. The ability of the Sobol quasi-random vs. random distribution of testing points to capture the maximal relative error using a sufficiently small number of samples is evaluated. Sobol testing points that are quasi-randomly distributed can cover the domain of interest more evenly, avoiding large gaps. Sobol sequences are quasi-random and are always the same, which allows the exact repetition of scientific results.
Highlights
The Colebrook equation, formally established in 1939 for calculating the flow friction through pipes, is empirical but widely accepted in engineering practice as an informal standard [1]
An unfortunate circumstance is that the Colebrook equation is expressed in an implicitly given logarithmic form with respect to the unknown
This section describes the calculation of the relative error δ% of the chosen approximation of the Colebrook equation and evaluates different quantities of testing points and related patterns, i.e., their distribution over the domain of its applicability in engineering practice
Summary
The Colebrook equation, formally established in 1939 for calculating the flow friction through pipes, is empirical but widely accepted in engineering practice as an informal standard [1]. The Colebrook equation is used in engineering practice for the Reynolds number Re between 4000 and 108 , and for the relative roughness of an inner pipe surface ε between 0 and 0.05, i.e., for a turbulent condition of flow. The distribution of the relative error δ% over the domain of applicability of the Colebrook equation is uneven and is different for every new approximation [17,18] This error should be evaluated in a sufficient number of points dispersed over the domain of applicability in engineering practice, and the points should be uniformly, randomly or quasi-randomly distributed. This section describes the calculation of the relative error δ% of the chosen approximation of the Colebrook equation and evaluates different quantities of testing points and related patterns, i.e., their distribution over the domain of its applicability in engineering practice
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