Exact Analytical Solutions of the Colebrook-White Equation
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Volume 142, Issue 2
Abstract
The 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 function. An explicit approximate formula for the Colebrook-White equation, called the th 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 function, called the function, expressed by an infinite recursion. Once the function is obtained in a closed-form using a recursive function, it can be applied to the 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 th formula increases. Additionally, the recursion depth of the th formula may be preferentially chosen to yield a solution that is consistently larger or smaller than the exact solution.
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Journal of Hydraulic Engineering
Volume 142 • Issue 2 • February 2016
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© 2015 American Society of Civil Engineers.
History
Received: May 24, 2014
Accepted: Jun 27, 2015
Published online: Sep 9, 2015
Published in print: Feb 1, 2016
Discussion open until: Feb 9, 2016
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