If an plane is in straight and level flight, is the rate of change of momentum of the surrounding air in the vertical direction equal to its weight?

Question

Assuming that there are no effects due to atmospheric stability, it seems evident that a plane would impart downward momentum to the surrounding air at a rate equal to its weight. Is this true? I think that the momentum and kinetic energy in the wake diffuses and eventually is hidden in the background noise of the atmosphere. However, theoretically, does this downward momentum persist all the way to the ground?

Answer 1

The net downward momentum flux of the air must balance the net weight (including the effects of buoyancy). This is a requirement based on drawing a control volume large enough that the pressure has returned to the static value. This downward momentum flux leads to the transfer of energy to the fluid. There is no requirement for the flux to be purely downward, though. The airfoil can impart other directions of momentum too, which leads to reduced efficiency of the airfoil.

Technically, the transfer of momentum to into the medium of the air must extend to the ground. Otherwise, there would be a net motion of the atmosphere, but the atmosphere is held to the earth. If the atmosphere is thought of as equal weight all around the planet (an ideal static atmosphere), then the plane flying creates a portion under itself where there is increased atmospheric pressure on the earth's surface. However, this increase in pressure does not "push the earth downward" except for the fact that it is very small, as you may be thinking. Rather, this increase in pressure is actually the reaction of the plane's gravitational pull on the earth.

In effect, when the plane is on the runway, the gravitational pulls are counteracted by the normals. When the plane is flying, the pull of the earth's gravity on the plane is counteracted by the downward transfer of momentum to the atmosphere by the airfoil, and the pull of the plane's gravity on the earth is counteracted by "stopping" this downward momentum that was induced into the atmosphere by the airfoil by the bluntearth surface.

For what it's worth, the drag on the plane, which imparts forward momentum to the atmosphere, is balanced by the thrust of the propulsion system, which imparts reverse momentum to the atmosphere. Thus, no net horizonal momentum is transferred to the atmosphere in steady state operation, and the downward momentum transferred by the plane is balanced by the upward momentum transferred by the earth surface. In all cases, thrust, drag, and the downwash of lift (which eventually impacts the blunt earth surface) will first lead to increased kinetic energy in the atmosphere, but eventually will become increased internal energy instead.