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Why does turbulence make pipe flow increase less rapidly as pressure increases?

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differential manometer, Torricellian barometer, and siphon — fluid mechanics Figure 1: Schematic representations of (A) a differential manometer, (B) a Torricellian barometer, and (C) a siphon.

In fluid mechanics, the transition from laminar flow to turbulent flow fundamentally changes the relationship between pressure and discharge rate.

Under laminar conditions, Poiseuille’s law describes a linear relationship where the flow rate is directly proportional to the pressure gradient. However, once the Reynolds number exceeds a critical threshold, the flow becomes turbulent. In this state, the fluid motion is characterized by complex, unsteady fluctuations and eddies that dissipate energy more efficiently than the smooth, layered motion of laminar flow.12

As a result of this increased energy dissipation, the rate of discharge increases much less rapidly with pressure than predicted by Poiseuille’s equation. Instead of a linear relationship, the flow rate in a turbulent regime typically increases roughly as the square root of the pressure gradient, or slightly faster depending on the smoothness of the pipe's internal surface.1