Is aspect ratio efficiency related to the momentum added to the air?

Question

Many years ago I was taught that for generating lift or propulsion, lower losses were associated with imparting less momentum change to a larger volume of air, and higher losses were seen when imparting larger momentum changes to a smaller volume of air - and that the losses were due to friction (i.e., the air's viscosity).

The propulsive example of this is the commonly-cited turbojet-versus-high bypass turbofan engine comparison, and I had been under the impression that the airfoil example was high aspect ratio versus low aspect ratio wing designs. However, having just learned that this is in fact incorrect (it has to do with induced drag instead), is there in fact any connection at all between the imparted momentum change/volume of air argument and airfoil aspect ratio?

Answer 1

You're almost there. The difference lies in the kinetic energy required to move a certain mass of air, not in the air's viscosity. Copying freely from one of my earlier answers:

The force to keep the object aloft is $$F=m_{object}g$$ The force generated by downwards momentum transfer is $$F=\dot{m}v$$ with $\dot{m}$ indicating mass flow (kilogram per second) of the air (not the mass of the object). The energy flow (power) required to impart this momentum on the airflow is $$ P=\frac{1}{2}\dot{m}v^2$$ Here we can draw an important conclusion. The power requirement is arbitrarily small, by increasing the mass flow and decreasing the downwards velocity.

The power requirement expresses itself in the form of induced drag. By affecting a large mass of air (a long wingspan), you reduce the power requirement, which manifests itself in the form of lower induced drag.