What is turbulence? (And why are helicopters never quiet?)

Fluid mechanics is very often left out of the undergraduate physics curriculum.  This is a shame, as it's very interesting and directly relevant to many broad topics (atmospheric science, climate, plasma physics, parts of astrophysics).  Fluid mechanics is a great example of how it is possible to have comparatively simple underlying equations and absurdly complex solutions, and that's probably part of the issue.  The space of solutions can be mapped out using dimensionless ratios, and two of the most important are the Mach number (\(\mathrm{Ma} \equiv u/c_{s}\), where \(u\) is the speed of some flow or object, and \(c_{s}\) is the speed of sound) and the Reynolds number (\(\mathrm{Re} \equiv \rho u d/\mu\), where \(\rho\) is the fluid's mass density, \(d\) is some length scale, and \(\mu\) is the viscosity of the fluid). 

From Laurence Kedward, wikimedia commons

There is a nice physical interpretation of the Reynolds number.  It can be rewritten as \(\mathrm{Re} = (\rho u^{2})/(\mu u/d)\).  The numerator is the "dynamic pressure" of a fluid, the force per unit area that would be transferred to some object if a fluid of density \(\rho\) moving at speed \(u\) ran into the object and was brought to a halt.  This is in a sense the consequence of the inertia of the moving fluid, so this is sometimes called an inertial force.  The denominator, the viscosity multiplied by a velocity gradient, is the viscous shear stress (force per unit area) caused by the frictional drag of the fluid.  So, the Reynolds number is a ratio of inertial forces to viscous forces.  

When \(\mathrm{Re}\ll 1\), viscous forces dominate.  That means that viscous friction between adjacent layers of fluid tend to smooth out velocity gradients, and the velocity field \(\mathbf{u}(\mathbf{r},t) \) tends to be simple and often analytically solvable.  This regime is called laminar flow.  Since \(d\) is just some characteristic size scale, for reasonable values of density and viscosity for, say, water, microfluidic devices tend to live in the laminar regime.  

When \(\mathrm{Re}\gg 1\), frictional effects are comparatively unimportant, and the fluid "pushes" its way along.  The result is a situation where the velocity field is unstable to small perturbations, and there is a transition to turbulent flow.  The local velocity field has big, chaotic variations as a function of space and time.  While the microscopic details of \(\mathbf{u}(\mathbf{r},t)\) are often not predictable, on a statistical level we can get pretty far since mass conservation and momentum conservation can be applied to a region of space (the control volume or Eulerian approach).

Turbulent flow involves a cascade of energy flow down through eddies at length scales all the way down eventually to the mean free path of the fluid molecules.   This right here is why helicopters are never quiet.  Even if you started with a completely uniform downward flow of air below the rotor (enough of a momentum flux to support the weight of the helicopter), the air would quickly transition to turbulence, and there would be pressure fluctuations over a huge range of timescales that would translate into acoustic noise.  You might not be able to hear the turbine engine directly from a thousand feet away, but you can hear the resulting sound from the turbulent airflow.  

If you're interested in fluid mechanics, this site is fantastic, and their links page has some great stuff.