AN EXACT ANALYTICAL FRAMEWORK FOR IMPLICIT LOGARITHMIC RESISTANCE LAWS WITH AN ENERGY-BASED UNIFICATION OF FLOW REGIMES

Abstract

Since its publication in 1939, the Colebrook-White equation has remained the international reference for predicting the Darcy friction factor in fully developed turbulent pipe flow. Nevertheless, despite more than eight decades of research and the publication of hundreds of explicit correlations, an exact explicit analytical solution has not yet been reported. Consequently, the evaluation of the Darcy friction factor still relies either on iterative numerical procedures or on approximate explicit formulas that inevitably introduce approximation errors.

The present work addresses this long-standing problem through a fundamentally different approach. Rather than proposing another empirical approximation, the Colebrook-White equation is first reformulated exactly into a compact logarithmic representation by means of elementary algebraic transformations, without altering its mathematical content. This exact reformulation reveals a simple analytical structure that naturally lends itself to further mathematical treatment.

The reformulated equation is subsequently transformed into a fully explicit analytical expression by introducing the Lambert W function. A rigorous step-by-step derivation demonstrates that the proposed solution is mathematically equivalent to the original implicit Colebrook-White equation rather than an empirical approximation. Consequently, when the Lambert W function is evaluated with sufficient numerical precision, the proposed formulation reproduces exactly the friction factor obtained from the classical iterative solution of the Colebrook-White equation.

A detailed numerical example illustrates the complete computational procedure. The corresponding friction factors obtained from the proposed explicit formulation and from the original Colebrook-White equation coincide to all reported decimal places, yielding a relative deviation effectively equal to zero within the adopted numerical precision.

The applicability of the proposed W₀ approximation is further demonstrated through the derivation of an explicit Lambert W-based solution for the classical implicit Prandtl-von Kármán equation. Numerical comparisons show excellent agreement with the original implicit formulation, thereby confirming the generality, robustness, and practical value of the proposed approach.

Furthermore, the proposed approximation of the principal branch W₀ is generalized to encompass a broad class of implicit logarithmic friction laws, thereby extending its applicability well beyond the Colebrook-White and Prandtl-von Kármán equations.

Finally, an energetic branch formulation is proposed to provide a unified explicit framework for both laminar and turbulent flow regimes while preserving the exact analytical expressions governing the two asymptotic solutions and introducing no additional empirical parameters.