Lumo has up a clever video of human powered flapping flight. It turns out to be a clever CGI fake, but I thought it might be fun to analyze the aerodynamics. Heavier than air fliers overcome gravity by accelerating air downward. That takes work against drag. It turns out that there are two components to that drag - so called induced drag that is a direct side effect of lift and frictional drag, often called form drag. Form drag is proportional to the square of velocity while induced drag is inversely proportional to the square of velocity - circumstances that combine to produce a velocity sweet spot of minimum drag that depends mostly on the wing loading, the ratio of mass to wing area.
That velocity sweet spot is given approximately by V = Sqrt[W/0.38A], where W is weight, A is area and the numerical factor (appropriate for MKS units) combines plausible values of air density and angle of attack. If we assume conservatively that the flier of the video plus equipment weighs at least 700 N (70 kg)and wing area is about 5 m^2 we get V = 19 m/s - an implausibly high velocity compared to what we see.
We can also compute the power needed to be expended against just induced drag at that speed. It's given approximately by P = W^2/(rho*V*b^2), where P is power, W is weight, rho is air density (1.2 kg/m^3), V is airspeed, and b is wingspan. If we figure b = 5 m for our aircraft, that works out to 860 Watts, more than 1 horsepower and a lot more than any athlete can sustain. There will also be a comparable contribution from form drag.
Real human powered aircraft feature 30 m wings with 30 m^2 of area, extensive streamlining, ultralight materials, and elite bicyclists retrained as pilot/power plants. They generate roughly 200 Watts and manage a speed of around 7 m/s. So the video is a fake, as CGI experts have independently established.
The equations and their derivations can be found in Henk Tennekes wonderful book: The Simple Science of Flight