A platform for research: civil engineering, architecture and urbanism
A platform for research: civil engineering, architecture and urbanism
Exact Analytical Solutions of the Colebrook-White Equation
AbstractThe primary results of this paper are two exact analytical solutions of the Colebrook-White equation (1939), one by an infinite recursion and the other by an integral. These solutions are the first exact analytical solutions of the Colebrook-White equation that do not use a special transcendental function such as the Lambert W function. An explicit approximate formula for the Colebrook-White equation, called the nth formula, is also developed on the basis of one of the exact solutions by truncation. The key result in this development is the closed-form expression for an associated function of the Lambert W function, called the Y function, expressed by an infinite recursion. Once the Y function is obtained in a closed-form using a recursive function, it can be applied to the W function and to the Colebrook-White equation. It is shown numerically that the absolute error of the Darcy friction factor decreases geometrically toward zero as the recursion depth of the nth formula increases. Additionally, the recursion depth of the nth formula may be preferentially chosen to yield a solution that is consistently larger or smaller than the exact solution.
Exact Analytical Solutions of the Colebrook-White Equation
AbstractThe primary results of this paper are two exact analytical solutions of the Colebrook-White equation (1939), one by an infinite recursion and the other by an integral. These solutions are the first exact analytical solutions of the Colebrook-White equation that do not use a special transcendental function such as the Lambert W function. An explicit approximate formula for the Colebrook-White equation, called the nth formula, is also developed on the basis of one of the exact solutions by truncation. The key result in this development is the closed-form expression for an associated function of the Lambert W function, called the Y function, expressed by an infinite recursion. Once the Y function is obtained in a closed-form using a recursive function, it can be applied to the W function and to the Colebrook-White equation. It is shown numerically that the absolute error of the Darcy friction factor decreases geometrically toward zero as the recursion depth of the nth formula increases. Additionally, the recursion depth of the nth formula may be preferentially chosen to yield a solution that is consistently larger or smaller than the exact solution.
Exact Analytical Solutions of the Colebrook-White Equation
Mikata, Yozo (author) / Walczak, Walter S
2016
Article (Journal)
English