{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,12]],"date-time":"2025-12-12T13:35:20Z","timestamp":1765546520277,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2019,9,3]],"date-time":"2019-09-03T00:00:00Z","timestamp":1567468800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Centre for Energy of the Czech Republic","award":["TN01000007"],"award-info":[{"award-number":["TN01000007"]}]},{"name":"Technology Agency of the Czech Republic, Energy System for Grids","award":["TK02030039"],"award-info":[{"award-number":["TK02030039"]}]},{"name":"Ministry of Education, Youth, and Sports of the Czech Republic through the National Programme of Sustainability (NPS II) project \u201cIT4Innovations excellence in science\u201d","award":["LQ1602"],"award-info":[{"award-number":["LQ1602"]}]},{"name":"Ministry of Education, Science, and Technological Development of the Republic of Serbia through the project \u201cDevelopment of new information and communication technologies, based on advanced mathematical methods, with applications in medicine, telecommunic","award":["iii44006"],"award-info":[{"award-number":["iii44006"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>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 \u03a9 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 Pad\u00e9 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.<\/jats:p>","DOI":"10.3390\/computation7030048","type":"journal-article","created":{"date-parts":[[2019,9,4]],"date-time":"2019-09-04T08:28:13Z","timestamp":1567585693000},"page":"48","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Colebrook\u2019s Flow Friction Explicit Approximations Based on Fixed-Point Iterative Cycles and Symbolic Regression"],"prefix":"10.3390","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2502-0601","authenticated-orcid":false,"given":"Dejan","family":"Brki\u0107","sequence":"first","affiliation":[{"name":"IT4Innovations, V\u0160B\u2014Technical University of Ostrava, 708 00 Ostrava, Czech Republic"},{"name":"Research and Development Center \u201cAlfatec\u201d, 18000 Ni\u0161, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3913-7800","authenticated-orcid":false,"given":"Pavel","family":"Praks","sequence":"additional","affiliation":[{"name":"IT4Innovations, V\u0160B\u2014Technical University of Ostrava, 708 00 Ostrava, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,9,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1680\/ijoti.1939.13150","article-title":"Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws","volume":"11","author":"Colebrook","year":"1939","journal-title":"J. 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