Suitability for coding of the Colebrook’s flow friction relation expressed through the Wright ω-function
DOI:
https://doi.org/10.31181/rme200101174pKeywords:
Flow friction, Colebrook equation, Wright ω-function, Artificial intelligence, Symbolic regression, Computational speedAbstract
This article analyses a form of the empirical Colebrook’s pipe flow friction equation given originally by the Lambert W-function and recently also by the Wright ω-function. These special functions are used to explicitly express the unknown flow friction factor of the Colebrook equation, which is in its classical formulation given implicitly. Explicit approximations of the Colebrook equation based on approximations of the Wright ω-function given by an asymptotic expansion and symbolic regression were analyzed in respect of speed and accuracy. Numerical experiments on 8 million Sobol’s quasi-Monte points clearly show that also both approaches lead to the approximately same complexity in terms of speed of execution in computers. However, the relative error of the developed symbolic regression-based approximations is reduced significantly, in comparison with the classical basic asymptotic expansion. These numerical results indicate promising results of artificial intelligence (symbolic regression) for developing fast and accurate explicit approximations.
References
Alfaro-Guerra, M., Guerra-Rojas, R., & Olivares-Gallardo, A. (2020). Evaluación de la profundidad de recursión de la solución analítica de la ecuación de Colebrook-White en la exactitud de la predicción del factor de fricción. Ingeniería, investigación y tecnología, 21(4), https://doi.org/10.22201/fi.25940732e.2020.21.4.036
Assuncao, G. S. C., Marcelin, D., von Hohendorff Filho, J. C., Schiozer, D. J., de Castro, M. S. (2020). Friction factor equations accuracy for single and two-phase flows. ASME 2020 39th International Conference on Ocean, Offshore and Arctic Engineering (OMAE2020), Virtual Online, August 3 – 7, 2020. Available from: https://www.researchgate.net/publication/346503366 (accessed on 06 December 2020)
Barry, D. A., Parlange, J. Y., Li, L., Prommer, H., Cunningham, C. J., & Stagnitti, F. (2000). Analytical approximations for real values of the Lambert W-function. Mathematics and Computers in Simulation, 53(1-2), 95-103. https://doi.org/10.1016/S0378-4754(00)00172-5
Biberg, D. (2017). Fast and accurate approximations for the Colebrook equation. Journal of Fluids Engineering, 139(3), 031401, https://doi.org/10.1115/1.4034950
Brkić, D. (2011a). Review of explicit approximations to the Colebrook relation for flow friction. Journal of Petroleum Science and Engineering, 77(1), 34-48. https://doi.org/10.1016/j.petrol.2011.02.006
Brkić, D. (2011b). Determining friction factors in turbulent pipe flow. Chemical Engineering (New York), 119(3), 34-39. Available from: https://www.chemengonline.com/determining-friction-factors-in-turbulent-pipe-flow/?printmode=1 (accessed on 06 December 2020)
Brkić, D. (2011b). W solutions of the CW equation for flow friction. Applied Mathematics Letters, 24(8), 1379-1383. https://doi.org/10.1016/j.aml.2011.03.014
Brkić, D. (2011c). An explicit approximation of Colebrook's equation for fluid flow friction factor. Petroleum Science and Technology, 29(15), 1596-1602. https://doi.org/10.1080/10916461003620453
Brkić, D. (2012a). Can pipes be actually really that smooth? International Journal of Refrigeration, 35(1), 209-215. https://doi.org/10.1016/j.ijrefrig.2011.09.012
Brkić, D. (2012c). Lambert W function in hydraulic problems. Mathematica Balkanica (New Series), 26(3-4), 285-292. Available from: http://www.math.bas.bg/infres/MathBalk/MB-26/MB-26-285-292.pdf (accesed on 06 December 2020)
Brkić, D. (2012d). Comparison of the Lambert W‐function based solutions to the Colebrook equation. Engineering Computations, 29(6), 617-630. https://doi.org/10.1108/02644401211246337
Brkić, D., & Ćojbašić, Ž. (2017). Evolutionary optimization of Colebrook’s turbulent flow friction approximations. Fluids, 2(2), 15. https://doi.org/10.3390/fluids2020015
Brkić, D., & Praks, P. (2019). Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function. Mathematics, 7(1), 34. https://doi.org/10.3390/math7010034
Colebrook, C. F., & White, C. M. (1937). Experiments with fluid friction in roughened pipes. Proceedings of the Royal Society of London. Series A-Mathematical and Physical Sciences, 161(906), 367-381. https://doi.org/10.1098/rspa.1937.0150
Colebrook, C.F. (1939). Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws. Journal of the Institution of Civil Engineers, 11(4), 133–156. https://doi.org/10.1680/ijoti.1939.14509
Corless, R. M., Gonnet, G. H., Hare, D. E., Jeffrey, D. J., & Knuth, D. E. (1996). On the LambertW function. Advances in Computational Mathematics, 5(1), 329-359. https://doi.org/10.1007/BF02124750
Corless, R.M., & Jeffrey, D.J. (2002). Wright ω Function. In: Calmet, J., Benhamou, B., Caprotti, O., Henocque, L., & Sorge, V. (eds.) Artificial Intelligence, Automated Reasoning, and Symbolic Computation; AISC 2002, Calculemus 2002, Lecture Notes in Computer Science, vol. 2385, pp. 76–89. Berlin/Springer, Heidelberg. https://doi.org/10.1007/3-540-45470-5_10
Dubčáková, R. (2011). Eureqa: software review. Genetic Programming and Evolvable Machines, 12(2), 173-178. https://doi.org/10.1007/s10710-010-9124-z
Fukushima, T. (2020a). Precise and fast computation of Lambert W function by piecewise minimax rational function approximation with variable transformation. Available online: https://doi.org/10.13140/RG.2.2.30264.37128 (accessed on 05 December 2020)
Fukushima, T. (2020b). Fast computation of Wright ω function by piecewise minimax rational function approximation. Available online: https://doi.org/10.13140/RG.2.2.27086.28481 (accessed on 06 December 2020)
Guo, X., Wang, T., Yang, K., Fu, H., Guo, Y., & Li, J. (2020). Estimation of equivalent sand–grain roughness for coated water supply pipes. Journal of Pipeline Systems Engineering and Practice 11(1), 04019054. https://doi.org/10.1061/(ASCE)PS.1949-1204.0000433
Horchler, A.D. (2017). Complex double-precision evaluation of the Wright Omega function. Available online: https://github.com/horchler/wrightOmegaq (accessed on 28 December 2019)
Mikata, Y., & Walczak, W.S. (2016). Exact analytical solutions of the Colebrook-White equation. Journal of Hydraulic Engineering, 142(2), 04015050. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001074
Moody, L.F. (1944) Friction factors for pipe flow. Transactions of the American Society of Mechanical Engineers, 66(8), 671–684.
Praks, P., & Brkić, D. (2020). Review of new flow friction equations: Constructing Colebrook explicit correlations accurately. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, 36(3), 41, https://doi.org/10.23967/j.rimni.2020.09.001
Rollmann, P., & Spindler, K. (2015). Explicit representation of the implicit Colebrook–White equation. Case Studies in Thermal Engineering, 5, 41-47. https://doi.org/10.1016/j.csite.2014.12.001
Sobol, I. M., Turchaninov, V. I., Levitan, Y. L., & Shukhman, B. V. (1992). Quasi-random sequence generators; Distributed by OECD/NEA Data Bank; Keldysh Institute of Applied Mathematics; Russian Academy of Sciences: Moscow, Russia. 1992. https://ec.europa.eu/jrc/sites/jrcsh/files/LPTAU51.rar (accessed on 28 December 2019)
Sonnad, J. R., & Goudar, C. T. (2004). Constraints for using Lambert W function-based explicit Colebrook–White equation. Journal of Hydraulic Engineering, 130(9), 929-931. https://doi.org/10.1061/(ASCE)0733-9429(2004)130:9(929)
Vatankhah, A.R. (2018). Approximate analytical solutions for the Colebrook equation. Journal of Hydraulic Engineering, 144(5), 06018007. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001454
Viccione, G., & Tibullo, V. (2012). An effective approach for designing circular pipes with the Colebrook-White formula. AIP Conference Proceedings, 1479, 205-208. https://doi.org/10.1063/1.4756098
