Sampling the Darcy Friction Factor Using Halton, Hammersley, Sobol, and Korobov Sequences: Data Points from the Colebrook Relation

Abstract

When the Colebrook equation is used in its original implicit form, the unknown pipe flow friction factor can only be obtained through time-consuming and computationally demanding iterative calculations. The empirical Colebrook equation relates the unknown Darcy friction factor to a known Reynolds number and a known relative roughness of a pipe’s inner surface. It is widely used in engineering. To simplify computations, a variety of explicit approximations have been developed, the accuracy of which must be carefully evaluated. For this purpose, this Data Descriptor gives a sufficient number of pipe flow friction factor values that are computed using a highly accurate iterative algorithm to solve the implicit Colebrook equation. These values serve as reference data, spanning the range relevant to engineering applications, and provide benchmarks for evaluating the accuracy of the approximations. The sampling points within the datasets are distributed in a way that minimizes gaps in the data. In this study, a Python Version v1 script was used to generate quasi-random samples, including Halton, Hammersley, Sobol, and deterministic lattice-based Korobov samples, which produce smaller gaps than purely random samples generated for comparison purposes. Using these sequences, a total of 2²⁰ = 1,048,576 data points were generated, and the corresponding datasets are provided in in the zenodo repositoryWhen a smaller subset of points is needed, the required number of initial points from these sequences can be used directly.

Dataset: https://doi.org/10.5281/zenodo.17280142

Dataset License: CC-BY 4.0

1. Summary

The empirical Colebrook equation [1] is an implicit empirical relation for turbulent flow that relates the unknown Darcy friction factor λ to a known Reynolds number Re and a known relative roughness of a pipe’s inner surface ε/D. It is widely used in engineering. Its primary limitation is its implicit nature, which requires iterative numerical methods for solution rather than a direct analytical expression. To provide reliable reference values for engineering applications, comprehensive datasets based on the Colebrook equation were generated and presented in this Data Descriptor. They are intended to serve as benchmarks for assessing the accuracy of numerous explicit approximations of the Colebrook equation and can also serve as a computational substitute for the Moody chart. These datasets cover the practically relevant parameter space encountered in engineering, spanning Reynolds numbers (Re) from 4000 to 10⁸ and relative pipe roughness values (ε/D) from 0 to 0.05.

This Data Descriptor answers the following questions:

• How were the data points generated? To obtain accurate values, a highly precise iterative solver of the Colebrook equation was applied. In this way, the datasets were generated by computing the unknown Darcy friction factor λ from a known Reynolds number Re and a known roughness of a pipe’s inner surface ε/D—(Re, ε/D)→λ.
• How are the data points distributed? To minimize gaps in coverage, the data points were distributed using sampling methods. Specifically, Halton, Hammersley, Sobol, and Korobov sequences were employed. The Halton, Hammersley, and Sobol methods are quasi-random (low-discrepancy) techniques, while the Korobov method is a deterministic lattice-based approach. For each of these four sequences, 2²⁰ samples were generated, yielding 1,048,576 friction factor data points for each method. When a smaller subset of points is needed, the required number of initial points from these large sequences can be used directly.

Appendix A provides a Python script that can be used to generate Halton, Hammersley, Sobol, and Korobov samples normalized for the Colebrook equation—{S_j→Re, S_k→ε/D}→λ.

2. Data Description

2.1. Generation of the Data Points

The empirical Colebrook relation, given in Equation (1), dating from the 1930s [1], is widely accepted in engineering as an informal standard for calculating the flow friction factor in pipes:

$${\displaystyle \frac{1}{\sqrt{\mathsf{\lambda }}}}=-0.8686·\ln \left({\displaystyle \frac{2.51}{\mathrm{R}\mathrm{e}}}·{\displaystyle \frac{1}{\sqrt{\mathsf{\lambda }}}}+{\displaystyle \frac{1}{3.71}}·{\displaystyle \frac{\mathsf{\epsilon }}{\mathrm{D}}}\right),$$

(1)

where

λ is the Darcy friction factor (dimensionless);

Re is the Reynolds number (dimensionless);

D is the inner diameter of a pipe (meters);

ε/D is the relative roughness of a pipe’s inner surface (dimensionless);

ln stands for natural logarithm.

The Colebrook equation is based on an experiment performed by Colebrook and White [2] regarding the flow of a fluid through sets of pipes with different levels of roughness (starting from smooth).

Unfortunately, the Colebrook equation is an implicit equation for the unknown friction factor λ, with λ appearing on both sides of the equals sign in a logarithmic form from which it cannot be extracted or transformed in explicit form without approximations (an exception is the Lambert W-function [3,4,5,6,7,8]—further evaluation of the Lambert W-function is also approximate but highly accurate when using various mathematical approaches to evaluate this function). Many such approximations have been developed, and they can be compared with respect to their accuracy and complexity [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24].

Therefore, an iterative procedure, which can achieve practically any desired accuracy, is used in this Data Descriptor to compute the highly accurate values of the points employed in constructing the presented datasets available at https://doi.org/10.5281/zenodo.17280142. The iterative solution, using recommendations from this Data Descriptor, is treated as accurate, and therefore the Colebrook equation uses the sign ‘=’, while its approximations use ‘≈’.

To start iterations, the initial step is estimated by omitting the ${\displaystyle \frac{2.51}{\mathrm{R}\mathrm{e}}}·{\displaystyle \frac{1}{\sqrt{\mathsf{\lambda }}}}$ term from Equation (1), yielding a reasonable initial guess, λ₀, which is based only on the roughness of the inner surface of a pipe, ε, which is valid for a fully developed rough turbulent regime λ_∞, as given in Equation (2):

$${\displaystyle \frac{1}{\sqrt{{\mathsf{\lambda }}_0}}}={\displaystyle \frac{1}{\sqrt{{\mathsf{\lambda }}_{\mathrm{\infty }}}}}\approx -0.8686·\ln \left({\displaystyle \frac{1}{3.71}}·{\displaystyle \frac{\mathsf{\epsilon }}{\mathrm{D}}}\right),$$

(2)

The fixed-point iterative process [25] continues with the calculation of the values λ₁ (a function of λ₀), λ₂ (a function of λ₁), λ_i (a function of λ_i−1),$\cdots$, λ_n (a function of λ_n−1) using Equation (1), where n typically ranges from a few iterations up to 100 iterations to reach an accuracy of 10⁻⁹ (for highly accurate engineering applications, a precision of 10⁻⁶ is sufficient). However, given the empirical nature of the Colebrook equation, an accuracy of about two significant digits is adequate for most calculations, while additional digits beyond that are essentially numerical noise.

A limitation in accuracy can be noticed in cases where the roughness ε is very small (approaching zero) when the logarithm becomes problematic, leading to precision issues. Measurement and evaluation of pipe roughness also introduce a certain level of uncertainty [26,27].

The widely recognized Moody diagram [28,29,30,31,32] offers a graphical representation of the Colebrook relation. Instead of reading from the Moody diagram [33,34], datasets from https://doi.org/10.5281/zenodo.17280142 should be used. These data can also be used to redraw the Moody chart.

Datasets similar to those given here, but with a uniform distribution (1000-by-1000 mesh), are available [35].

2.2. Distributions of the Data Points

The Colebrook equation involves two input parameters: the Reynolds number (Re) and relative roughness of a pipe’s inner surface (ε/D). Therefore, a two-dimensional sequence of pairs (S_j, S_k) is required as an input. Using Equation (3) [36], pairs (S_j, S_k)_, each between 0 and 1, are transformed into the Reynolds numbers Re (4000 < Re < 10⁸) and relative roughness of a pipe’s inner surface, corresponding to 0 < ε/D < 0.05 (S_j→Re and S_k→ε/D).

$$\left.\begin{array}{c}{Re}_j={10}^{S_j·\left({\mathit{log}}_{10}\left({10}^8\right)-{\mathit{log}}_{10}\left(4000\right)\right)+{\mathit{log}}_{10}\left(4000\right)}\\ {\epsilon /D}_k={10}^{-\left(S_k·\left(6.5{-\mathit{log}}_{10}\left(1/0.05\right)\right)+{\mathit{log}}_{10}\left(1/0.05\right)\right)}\end{array}\right\},$$

(3)

where

log₁₀ denotes the base-10 (Briggs) logarithm.

Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5 show 2⁸ = 256 data points, while

Table A1. First 2² = 4 data points of the Halton quasi-random sequence for the Colebrook equation.

S1	S2	Re	ε/D	λ
0.125	0.444444444	14,184.12312	0.000244521	0.028646643
0.625	0.777777778	2,242,706.784	4.52 × 10−6	0.010356771
0.375	0.222222222	178,355.9058	0.003496579	0.027989349
0.875	0.555555556	28,200,544.83	6.47 × 10−5	0.011135324

,

Table A2. First 2² = 4 data points of the Hammersley quasi-random sequence for the Colebrook equation.

S1	S2	Re	ε/D	λ
0.75	0.003891051	7,952,707.288	0.047724408	0.069941163
0.125	0.007782101	14,184.12312	0.045552382	0.07014454
0.625	0.011673152	2,242,706.784	0.043479209	0.067049409
0.375	0.015564202	178,355.9058	0.041500391	0.065798735

,

Table A3. First 2² = 4 data points of the Sobol quasi-random sequence for the Colebrook equation.

S1	S2	Re	ε/D	λ
0.5	0.5	632,455.532	0.000125743	0.014350582
0.75	0.25	7,952,707.288	0.002507422	0.024895569
0.25	0.75	50,297.33719	6.31 × 10−6	0.020886224
0.375	0.375	178,355.9058	0.000561508	0.019315882

and

Table A4. First 2² = 4 data points of the Korobov deterministic lattice-based sequence for the Colebrook equation.

S1	S2	Re	ε/D	λ
0.003891051	0.249027237	4160.759355	0.002536792	0.041979229
0.007782101	0.498054475	4327.979604	0.000128706	0.039126016
0.011673152	0.747081712	4501.920406	6.53 × 10−6	0.038552156
0.015564202	0.996108949	4682.851862	3.31 × 10−7	0.038106614

of Appendix B show the first 2² = 4 data points. The full datasets with 2²⁰ = 1,048,576 samples are available at https://doi.org/10.5281/zenodo.17280142. If a smaller dataset is required, one can be obtained by randomly selecting the desired number of data points by uniformly omitting points (e.g., every nth point), or, in some cases, by selecting the first portion of the data points when appropriate—always ensuring that no gaps in data coverage occur. In general, the first subset of points (e.g., 2¹⁰ = 1024 from 2²⁰ = 1,048,576) generated by Halton, Hammersley, Sobol, and Korobov sequences can be used directly, as they already provide good spatial coverage; among them, the Sobol sequence typically yields the most uniform distribution, while the others exhibit only slightly lower uniformity.

Halton, Hammersley, Sobol, and Korobov sequences were specifically chosen to generate the datasets provided at https://doi.org/10.5281/zenodo.17280142. These sequences were employed to generate the points for the Colebrook dataset, ensuring thorough and reproducible coverage of the Reynolds number Re and relative roughness ε/D parameter space. Each method for sampling offers a different balance between uniformity, computational simplicity, and scalability. All are designed to generate well-distributed points in multidimensional space—being applied specifically to two-dimensional datasets for modeling purposes in this study. Unlike random distributions, quasi-random and deterministic lattice-based sequences provide coverage with minimized gaps and produce identical point arrangements across realizations, facilitating proper repetitions and reevaluations of computational experiments (in this Data Descriptor, the focus is on two-dimensional space—S_i→Re and S_k→ε/D; the distribution of dots in random distribution changes as in Section 2.2.1, while that in Section 2.2.2 remains identical each time in each realization). Their practical value lies in generating well-distributed sampling points in multidimensional spaces; theoretical details on their construction are beyond the scope of this Data Descriptor.

2.2.1. Random Distributions

The random distributions are inherently non-deterministic, resulting in different arrangements of data points across the spaces S₁ and S₂ in each realization (the distribution of dots from Figure 1 will be different in each realization). This can cause certain problems regarding repetitions and reevaluations of the results of computational experiments.

Figure 1. Random distribution.

Figure 1. Random distribution.

2.2.2. Quasi-Random Distributions

The Halton [37,38,39,40], Hammersley [40,41], and Sobol [42,43,44,45,46,47] methods are quasi-random (low-discrepancy) techniques designed to generate well-distributed points in multidimensional space—being applied specifically to two-dimensional datasets for modeling purposes in this case.

• Halton Quasi-Random Distribution (Figure 2)

The Halton sample is simple to implement; it provides good, uniform coverage in low-dimensional spaces such as those discussed here, where two dimensions are required (performance deteriorates beyond ~10 dimensions).

Figure 2. Halton quasi-random distributions.

Figure 2. Halton quasi-random distributions.

• Hammersley Quasi-Random Distribution (Figure 3)

The Hammersley sample shows behavior similar to that of the Halton distribution, especially when a finite number of samples is predefined.

Figure 3. Hammersley quasi-random distributions.

Figure 3. Hammersley quasi-random distributions.

• Sobol Quasi-Random Distribution (Figure 4)

Sobol sequences can be extended dynamically, point by point, without losing uniformity. This characteristic makes them particularly well-suited for selecting a smaller subset of points from a larger dataset simply by truncating the sequence.

Figure 4. Sobol quasi-random distributions.

Figure 4. Sobol quasi-random distributions.

2.2.3. Deterministic Lattice-Based Korobov Distribution

The Korobov sequence belongs to the broader family of lattice rules, which are based on modular arithmetic rather than randomization or discrepancy minimization in the quasi-Monte Carlo sense. The Korobov distribution generates structured, grid-like data distributions (which are less effective for non-periodic or irregular domains). While Korobov sequences can also produce evenly distributed points and are used for modeling similarly to quasi-random sequences, they are more accurately described as deterministic lattice-based or number-theoretic methods rather than quasi-random ones. They provide a compromise between uniformity and quasi-random sampling.

Figure 5. Korobov deterministic lattice-based distribution.

Figure 5. Korobov deterministic lattice-based distribution.

3. Methods: Evaluation of Accuracy Using the Datasets

The relative error ${\mathsf{\delta }}_{\%,\mathrm{j}\mathrm{k}}$ of the evaluated explicit approximation of the Colebrook equation can be calculated using Equation (4):

$${\mathsf{\delta }}_{\%,\mathrm{j}\mathrm{k}}={\displaystyle \frac{\left|{\mathsf{\lambda }}_{\mathrm{a}\mathrm{p}\mathrm{r},\mathrm{j}\mathrm{k}}-{\mathsf{\lambda }}_{\mathrm{j}\mathrm{k}}\right|}{{\mathsf{\lambda }}_{\mathrm{j}\mathrm{k}}}}\times 100\%,$$

(4)

where

${\mathsf{\delta }}_{\%,\mathrm{j}\mathrm{k}}$ is the relative error (dimensionless);

λ_apr is the Darcy friction factor (dimensionless) obtained from the observed approximation;

λ is the accurate Darcy friction factor (dimensionless) derived from the datasets (available at https://doi.org/10.5281/zenodo.17280142);

jk is the position of the accurate Darcy friction factor (dimensionless) with respect to the datasets from https://doi.org/10.5281/zenodo.17280142;

|| denotes the absolute value.

Maximal relative error across 2²⁰ = 1,048,576 quasi-random samples from the datasets from https://doi.org/10.5281/zenodo.17280142 should be identified accordingly.

Practical examples are given in Appendix C.