Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

Abstract

The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 < Re < 108) through pipes with roughness between negligible relative roughness (ε/D ⟶ 0) to very rough (up to ε/D = 0.05). The Colebrook equation includes the flow friction factor λ in an implicit logarithmic form, λ being a function of the Reynolds number Re and the relative roughness of inner pipe surface ε/D: λ = f(λ, Re, ε/D). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, λ ≈ f(Re, ε/D), it is necessary to determinate the value of the friction factor λ from the Colebrook equation as accurately as possible. The most accurate way to achieve that is by using some kind of the iterative method. The most used iterative approach is the simple fixed-point method, which requires up to 10 iterations to achieve a good level of accuracy. The simple fixed-point method does not require derivatives of the Colebrook function, while the most of the other presented methods in this paper do require. The methods based on the accelerated Householder’s approach (3rd order, 2nd order: Halley’s and Schröder’s method, and 1st order: Newton–Raphson) require few iterations less, while the three-point iterative methods require only 1 to 3 iterations to achieve the same level of accuracy. The paper also discusses strategies for finding the derivatives of the Colebrook function in symbolic form, for avoiding the use of the derivatives (secant method), and for choosing an optimal starting point for the iterative procedure. The Householder approach to the Colebrook’ equations expressed through the Lambert W-function is also analyzed. Finally, it is presented one approximation to the Colebrook equation with an error of no more than 0.0617%.

Nonrecursive Equivalent of the Conjugate Gradient Method without the Need to Restart

“…Therefore, IRM-CG can be successfully applied to nonlinear problems (including optimisation), where conjugacy is not even defined. This issue is vitally important, as iterative methods are used exclusively in these cases [12].…”

Short Overview of Early Developments of the Hardy Cross Type Methods for Computation of Flow Distribution in Pipe Networks

“…The here described multipoint method can substitute the Newton-Raphson iterative procedure used in all the above described methods. Recently, we successfully used the here presented multipoint method for acceleration of the iterative solution of the Colebrook equation for flow friction modelling [16,17]. On the contrary, for the gas network example of Figure 1, the multipoint method requires the same number of iterations as the original Newton-Raphson procedure.…”

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

“…The approximation with one internal iterative cycle will be shown in Section 3.1, while the approximation with two internal iterative cycles will be presented in Section 3.2. The iterative procedures are sensitive, depending on the chosen initial starting point [38], and therefore carefully conducted numerical tests are provided separately for the approximation with one and two internal iterative cycles. In both cases, fixed numerical values for the initial starting points are chosen in such a way as to reduce the final maximal relative error over the domain of applicability of the Colebrook equation.…”