Limits of the Model and Turbulent Regime

Bernoulli’s theorem holds for an ideal fluid: incompressible, non-viscous, in steady motion along a flow line. Real fluids, however, have viscosity: layers of fluid in relative motion hinder each other through internal friction, dissipating energy. To establish when viscosity dominates, or when instead it is negligible, we introduce a pure number, the Reynolds number:

$\ev{\text{Re} = \frac{\rho\,v\,L}{\eta}}$

where $L$ is a characteristic scale of the problem (the pipe diameter, the wing chord) and $\eta$ is the dynamic viscosity of the fluid. The Reynolds number compares the importance of inertial forces with that of viscous forces: when it is small the latter dominate, when it is large the former dominate. Depending on its value, three flow regimes are distinguished.

• Laminar ($\text{Re} \lesssim 2000$): the fluid flows in well-ordered parallel layers and Bernoulli’s theorem works very well.
• Transition ($2000 \lesssim \text{Re} \lesssim 4000$): the motion becomes irregular, with intermittent vortices appearing and disappearing.
• Turbulent ($\text{Re} \gtrsim 4000$): vortices form at all scales, kinetic energy is dissipated as heat and Bernoulli fails.

The table below shows how much the Reynolds number varies across very different situations, from the microscopic swimming of a sperm cell to the flight of an airliner.

Situation	Typical $\text{Re}$
Swimming sperm cell	$\sim 10^{-2}$
Blood in arterioles	$\sim 1$
Blood in the aorta	$\sim 10^3$
Air in a room	$\sim 10^4$
Boeing 747 wing	$\sim 10^8$

Summary

Continuity: $A\,v = \text{cost.}$ — Bernoulli: $P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{cost.}$ — small Reynolds: viscous fluid, layered flow; large Reynolds: turbulent flow. Venturi effect: where the fluid is faster, the pressure is lower.

Links

Topics: Fluidostatica e fluidodinamica

Related exercises: Esercizio svolto — Pressa per la frutta · Esercizio svolto — Pressione del cuore · Esercizio svolto — Un pezzo di legno in acqua con benzina