Research Article
Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction
Table 10
Calculation of W(y) where y = Re/2.18 using the Newton–Raphson, the Halley, and the Schröder iterative methods.
| | | Re = 5·106, y = Re/2.18 = 2293411.45 | Re = 3·104, y = Re/2.18 = 13760.47 | | Newton–Raphson | Halley | Schröder | Newton–Raphson | Halley | Schröder |
| | Iteration 0 | z0 = 15 | z0 = 15 | z0 = 15 | z0 = 15 | z0 = 15 | z0 = 15 | | Iteration 1 | 14.10634749 | 13.29860556 | 13.68208338 | 14.06276308 | 13.1333396 | 13.59610616 | | Iteration 2 | 13.28604947 | 12.2757343 | 12.62556802 | 13.12986539 | 11.29171838 | 12.20340124 | | Iteration 3 | 12.62863905 | 12.14855784 | 12.17738057 | 12.20257063 | 9.520829163 | 10.83006437 | | Iteration 4 | 12.25343232 | z = 12.14835704 | 12.14836628 | 11.28354302 | 8.068323472 | 9.505729616 | | Iteration 5 | 12.15407754 | 12.14835704 | z = 12.14835704 | 10.37904335 | 7.530266826 | 8.341483562 | | Iteration 6 | 12.14837461 | — | 12.14835704 | 9.504505014 | 7.512930233 | 7.637280252 | | Iteration 7 | z = 12.14835704 | — | — | 8.697314341 | z = 7.512929679 | 7.513654122 | | Iteration 8 | 12.14835704 | — | — | 8.037456295 | 7.512929679 | z = 7.512929679 | | Iteration 9 | — | — | — | 7.640105762 | — | 7.512929679 | | Iteration 10 | — | — | — | 7.52154464 | — | — | | Iteration 11 | — | — | — | 7.512971011 | — | — | | Iteration 12 | — | — | — | 7.51292968 | — | — | | Iteration 13 | — | — | — | z = 7.512929679 | — | — | | Iteration 14 | — | — | — | 7.512929679 | — | — |
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