Analytical calculation of the drag force near drag crisis of a falling sphere

TL;DR

This paper analytically derives the quadratic velocity dependence of drag force near the drag crisis for a falling sphere, accurately predicting the drag coefficient by considering turbulent boundary layer effects using Navier-Stokes equations.

Contribution

It provides an analytical calculation of drag force behavior near the drag crisis, incorporating turbulence effects with Navier-Stokes equations, aligning well with experimental data.

Findings

Derived $v^2$ dependence of drag force near drag crisis

Calculated drag coefficient at the drag crisis accurately

Validated results with experimental observations

Abstract

We obtain analitically the $v^2$ dependence of the drag force on a falling sphere close to the drag crisis, as well as the drag coefficient at the drag crisis, with excellent agreement with experiment. We take into account the effects of viscosity in creating a turbulent boundary layer and perform the calculations using the Navier-Stokes equation.