Exact Analytical Solutions of the Colebrook-White Equation

Abstract

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,...