Friction Factor Comparison Between Mixing Length Theory and Empirical Correlation

Abstract

Equivalent sand grain roughness is required for estimating friction factor for engineering applications from empirical relation via Haalands equation. The real surfaces are different from the sand grain profile. The correlations for friction factor were derived from the use of discrete roughness elements with regular shapes such as cones and bars. The purpose of the paper is to derive the analytical expression of friction factor for a two-dimensional (2D) semicylindrical roughness (not exactly a three-dimensional (3D) sand grain but for the circular profile of cross-section) using Navier–Stokes equation and mixing length theory. This is compared with the modified series mathematical representation of Haalands equation for friction factor in terms of equivalent sand grain roughness. The comparison is valid for a high Reynolds number where the velocity profile is almost flat beyond the boundary layer and approximately linear all throughout the boundary layer. The high Reynolds number approximation for Haalands equation is derived and the series form of the friction factor compares approximately with the series form derived from first principles, where in the exponents of the series expansion are close.