Pressure-driven flows in helical pipes: bounds on flow rate and friction factor

TL;DR

This paper derives rigorous bounds on flow rate and friction factor for pressure-driven flows in helical pipes using the background method, accounting for pipe geometry and flow conditions.

Contribution

It introduces a novel application of the background method to helical pipes, providing bounds that depend on curvature and torsion, and offers criteria for its applicability.

Findings

Derived lower bounds on volume flow rate at high Reynolds numbers.

Established upper bounds on the friction factor based on pipe geometry.

Extended bounds to toroidal and straight pipes as special cases.

Abstract

In this paper, we use the well-known background method to obtain a rigorous lower bound on the volume flow rate through a helical pipe driven by a pressure differential in the limit of large Reynolds number. As a consequence, we also obtain an equivalent upper bound on the friction factor. These bounds are also valid for toroidal and straight pipes as limiting cases. By considering a two-dimensional background flow with varying boundary layer thickness along the circumference of the pipe, we obtain these bounds as a function of the curvature and torsion of the pipe and therefore capture the geometrical aspects of the problem. In this paper, we also present a sufficient criterion to find out which pressure-driven flow and surface-velocity-driven flow problems can be tackled using the background method.