# Symbolic Regression-Based Genetic Approximations of the Colebrook Equation for Flow Friction

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods Used, Preparation of Data and Software Tool, Results, Structure of Approximations, Accuracy and Comparative Analysis

^{8}(whole turbulent flow covered) and for the relative roughness of inner pipe surface, ε/D up to 0.05 (pipe covered from practically smooth to the very rough) [20] with a mesh which consists of 90 thousand intersection points {Re, ε/D} for which we calculated very accurately the flow friction factor, λ using the Colebrook equation; Equation (1); Figure 1a,b.

_{i}, {λ}

_{i}(see Supplementary Material attached to this paper) hoping that it will connect {Re, ε/D}

_{i}→{λ}

_{i}accurately. The input sample was generated according to the uniform density function of each input variable. The low-discrepancy Sobol sequences were employed [21]. These so-called quasirandom sequences have useful properties. In contrary to random numbers, quasirandom numbers cover the space more quickly and evenly. Thus, they leave very few holes. We used [computer software] Eureqa by Nutonian, Inc., Boston, MA, as a genetic programming tool [22,23]. The symbolic regression approach adopted herein [24,25,26,27,28,29] is based upon genetic programming wherein a population of functions is allowed to breed and mutate with the genetic propagation into subsequent generations based on survival-of-the-fittest criteria [30]. The main goal of this study is to make accurate and computationally cheap explicit approximations of the Colebrook equation, where computationally cheap means to contain the least possible number of logarithmic functions and non-integer powers [31,32,33,34,35,36].

#### 2.1. Input Parameters in Their Raw Form

_{i}→{λ}

_{i}, Eureqa, the used genetic programming tool gives a set of approximations in polynomial forms [12]. Knowing that logarithmic expressions and non-integer powers are expensive for computation, we hoped that we have fully accomplished our task. Unfortunately, Eureqa gives a number of not very accurate solutions and here we show Equation (2) with the relative error of λ

_{0}even up to 16.56% in respect to the accurate λ, where the relative error [5,26] is defined as (|λ

_{accurate}− λ|/λ

_{accurate})·100%, where λ

_{accurate}is calculated in an iterative procedure using the original implicitly given Colebrook equation [6,9]; Equation (1), while λ is obtained through the presented approximations; Equations (2)–(6). In Equation (2), “↔” means related but not sufficiently accurate:

_{1}increases up to 0.98%.

_{0}→16.56%, λ

_{1}→0.98%, λ

_{2}→0.13%, etc. (Figure 2). Thus, using only two logarithmic forms, high accuracy of λ

_{2}→0.13% is reached. Results are in the form {λ}

_{0}↔{Re, ε/D}

_{0}, {λ}

_{1}≈ {log

_{10}(λ

_{0})}

_{1}, {λ}

_{2}≈{log

_{10}(log

_{10}(λ

_{0}))}

_{2}, etc., where “↔” means related but not sufficiently accurate, while “≈” is reasonably accurate enough. This approach with acceleration is widely used in development of approximations of the Colebrook equation [37,38,39,40,41,42]. The error can be further reduced by using one more accelerating step as shown, or using genetic algorithms [25,29,36], Excel fitting tool [27] or the Monte Carlo method [43,44].

#### 2.2. Normalized Input Parameters

_{10}(Re), b = −log

_{10}(ε/D), in order to avoid discrepancy in the scale which are in raw form 1000 < Re < 10

^{8}and ε/D << 1 and after normalization 3.5 < a < 8 and 1.3 < b < 6.5 (Eureqa, software used a as genetic programming tool also suggested to us a data normalization process) [34,35,36]. The normalization gives relatively good results, and the genetic programming tool generated more accurate results without knowing that the logarithmic form of the Colebrook equation was originally used but only knowing the predicted input and output datasets; Figure 3:

_{10}(Re), b = −log

_{10}(ε/D), the genetic programming tool generated a dozen equations with different levels of accuracy and complexity, but fortunately none of them contain logarithms or non-integer power terms. Here, we present the four most successful explicit approximations; Equations (3)–(6). Adding one additional logarithmic form for acceleration using one additional fixed-point iterative step [9,45]; Equation (2a), the accuracy of the approximations increases significantly (about 10 times); Equations (3a)–(6a). Results are in the form {λ}

_{0}↔{a = log

_{10}(Re), b = −log

_{10}(ε/D)}

_{0}, {λ}

_{1}≈{log

_{10}(λ

_{0})}

_{1}, {λ}

_{2}≈{log

_{10}(log

_{10}(λ

_{0}))}

_{2}, etc., where “↔” means related but not sufficiently accurate, while “≈” is reasonably accurate enough.

#### 2.3. Discussion—Comparative Analysis

_{2}→0.13% compared with the approximately same error of Equation (6a) with three logarithmic forms (two for normalization and one for acceleration); Equation (6a)—λ

_{1}→0.17%. Also, the expression for λ

_{0}in case of Equation (2) is a polynomial while Equations (4)–(6) contain sinus trigonometric function [46]. After this careful analysis it can be concluded that it is better to use computationally expensive logarithmic functions for acceleration through Equation (2a) and not for normalization.

- Highly accurate: Compared with the similar approximations to the Colebrook equation, accelerated Equation (5a); with the relative error up to 0.28% and accelerated Equations (2a) and (6a) with the relative error up to 0.13% and 0.17%, respectively, are accurate as approximations by Bar [47] (0.2%), Chen [38] (0.36–0.18%), Zigrang and Sylvester [41] (0.14–0.08%, simpler: 1–0.775%), Fang et al. [48] (0.61–0.56%), Serghides [38] (0.14–0.0026%, simpler 0.35–0.27%), Buzzelli [49] (0.14–0.08%), Sonad and Goudar [50] (0.8–improved by Vatankhah and Kouchakzadeh [51]: 0.15%) and Romeo et al. [52] (0.14–0.008%); where the higher reported accuracy is achieved through genetic optimization [25,29]. These approximations are among the most accurate available to date [3,4,5,6,7,8], but at the same time in many cases much more complex compared to the approximations presented in our paper [4,11,12,13]. For example; approximations by Barr [47] and by Chen [38] contain two logarithmic expressions and two non-integer powers; by Romeo et al. [52], three logarithmic expressions and two non-integer powers, etc. which means that they introduce a higher computational burden to achieve the same accuracy. In this case, our Equation (2a), which after two steps of acceleration, contains only two logarithmic forms.
- Moderately accurate: Our Equation (6) with the relative error up to 2% does contain only two logarithmic expressions used for normalization and no non-integer power, and its accuracy can be compared with approximations by Swamee and Jain [53] (2.18–1.75%), Manadili [54] (2–1.5%), Brkić [42,55,56,57] (2–1.3%), Haland [58] (1.4–1.1%), etc., all with the same or higher complexity as Equation (6). Equation (2a) after the first step of acceleration with only one logarithmic function and with the relative error of up to 2.6% is even more efficient.
- Low accuracy: Our accelerated Equation (3a) is very simple with the relative error up to 5.35% but with only one peak of high error (otherwise up to 3% as can be seen from Figure 4); it is more accurate compared with approximations by Round [59] (10.9–5.5%), Eck [60] (8.2–5.7%) and Avci and Karagoz [61] (4.8–3.1%), Wood [62] (23.7–16.6%), Moody [63] (21.5–18.1%), etc.

## 3. Possible Simplifications

^{2})/(60 + 3x

^{2}) can potentially overwhelm the problem [64]. The Padé approximation x·(60 − 7x

^{2})/(60 + 3x

^{2}) within the domain of interest; −0.08821 < x < 1.18456; compared with sin(x) can introduce the relative error up to 0.068%. Within the same domain, Eureqa gives a number of approximations for sin(x), but we have chosen to present here one accurate but relatively simple; sin(x) ≈ x − x

^{2}/5350.6747 − x

^{3}/6.0171 + x

^{5}/127.4678 with the relative error up to 0.003%. This error of approximations for sin(x) also has impact on the final error of our approximations; Equations (4)–(6) and their accelerated pairs; Equations (4a)–(6a).

_{10}(ε/D) and to use the Padé polynomial in the form ln (1 − θ), where θ is given by Equation (8). In Equation (7), both 2·log

_{10}(3.71) ≈ 1.1387478 and 2/ln(10) ≈ 0.8686 are constant, while b = log

_{10}(ε/D) is recycled as already evaluated during the normalization of input parameters.

^{−9}. In practice [20], pipes are never that smooth to reach the value ε/D = 0.

_{10}(ε/D) and λ

_{0}from Equations (2)–(6).

^{−6}within the practical domain of the Colebrook equation; 4000 < Re < 10

^{8}and 0 < ε/D < 0.05.

_{10}(Re), we hoped also to use the fact that the practical domain of the Reynolds number Re, is from 4000 to 10

^{8}which has as a consequence log

_{10}(Re) ≈ log

_{10}(1 + Re), where for ln(1 + y) numerous approximations are available, but mostly for the argument z around 0. This is because some computer algebra systems and programming languages provide a special natural logarithm plus 1 function alternatively named to give more accurate results for values of x close to zero compared to using ln(1 + y) directly. In our case this is not of interest, knowing that for y = Re; y >> 1. Of course, to use only numbers between 1 and 10, we can use the rule ln(Re) = ln(z10

^{n}) = ln(z) + nln(10) where n = len(int(z)) and z = Re/10

^{n}; len is a function which calculates number of digits in a number while int is a function which gives a number down to the nearest integer; ln(10) = 2.30258509. A similar method can be used for the normalized parameter b = −log

_{10}(ε/D) where ε/D is between 0 and 0.05.

^{−10}%. Thus, Equation (10) is very accurate when the iterative process defined by Equation (2a) is initiated by a good starting point (A comparison of iterative methods for solving the Colebrook equation is given in [65]). Consequently, the argument z of Equation (9) is very close to one in all cases within the domain of interest of the Colebrook equation [10].

## 4. Conclusions

## Supplementary Materials

_{i}, {λ}

_{i}used to feed the genetic programming tool are available online at https://www.mdpi.com/2073-4441/10/9/1175/s1. We used: Eureqa [computer software], Nutonian, Inc., Boston, MA, USA to generate the presented approximations. Data used to feed the software are based on the Colebrook equation; Equation (1) of this paper. The dataset of 200 combinations of {Re, ε/D}→{λ} was chosen using Sobol quasirandom sequences among 90 thousand generated triplets.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) An example of Eureqa [computer software] interface: Best approximate solutions of the Colebrook equation with various complexity, which were automatically found by Eureqa. (

**b**) An example of Eureqa [computer software] interface: The residual error plot of a selected analytical model for 200 pairs together with an accuracy vs complexity plot of solutions.

**Figure 2.**Distribution of the relative error of λ over the domain of applicability of the Colebrook equation introduced by polynomial Equation (2)—(

**up**), Equation (2a) after first step of fixed-point acceleration—(

**middle**), and the second step of acceleration—(

**down**); relative error up to 16.56%, up to 0.98% and up to 0.13% respectively.

**Figure 3.**Genetic programming tool makes λ ≈ f (a,b) without knowing the physical law that connects a and b.

**Figure 4.**Distribution of the relative error of λ over the domain of applicability of the Colebrook equation introduced by Equation (3)—(

**a**), and accelerated Equation (3a)—(

**b**); relative error up to 20% and up to 5.35%, respectively.

**Figure 5.**Distribution of the relative error of λ over the domain of applicability of the Colebrook equation introduced by Equation (4)—(

**a**), and accelerated Equation (4a)—(

**b**); relative error up to 60% and up to 6.29%, respectively.

**Figure 6.**Distribution of the relative error of λ over the domain of applicability of the Colebrook equation introduced by Equation (5)—(

**a**), and accelerated Equation (5a)—(

**b**); relative error up to 6% and up to 0.28%, respectively.

**Figure 7.**Distribution of the relative error of λ over the domain of applicability of the Colebrook equation introduced by Equation (6)—(

**a**), and accelerated Equation (6a)—(

**b**); relative error up to 2% and up to 0.17%, respectively.

Complexity: Number of log Functions | |||
---|---|---|---|

Accuracy | 1 | 2 | 3 |

High | Equation (2a)—λ_{2}→0.13% | Equation (6a)—λ_{1}→0.17%Equation (5a)—λ _{1}→0.28% | |

Moderate | Equation (2a)—λ_{1}→0.98% | Equation (6)—λ_{0}→2% | |

Low | Equation (5)—λ_{0}→6% | Equation (4a)—λ_{1}→6.29%Equation (3a)—λ _{1}→5.35% |

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Praks, P.; Brkić, D. Symbolic Regression-Based Genetic Approximations of the Colebrook Equation for Flow Friction. *Water* **2018**, *10*, 1175.
https://doi.org/10.3390/w10091175

**AMA Style**

Praks P, Brkić D. Symbolic Regression-Based Genetic Approximations of the Colebrook Equation for Flow Friction. *Water*. 2018; 10(9):1175.
https://doi.org/10.3390/w10091175

**Chicago/Turabian Style**

Praks, Pavel, and Dejan Brkić. 2018. "Symbolic Regression-Based Genetic Approximations of the Colebrook Equation for Flow Friction" *Water* 10, no. 9: 1175.
https://doi.org/10.3390/w10091175