The helicopter shown has a mass m and hovers in position by...

The helicopter shown has a mass $m$ and hovers in position by imparting downward momentum to a column of air defined by the slipstream boundary shown. Find the downward velocity $v$ given to the air by the rotor at a section in the stream below the rotor, where the pressure is atmospheric and the stream radius is $r$. Also find the power $P$ required of the engine. Neglect the rotational energy of the air, any temperature rise due to air friction, and any change in air density $\rho$.

Key Concepts

Momentum Conservation in Fluid Dynamics

This concept involves applying the conservation of momentum to the airflow manipulated by the helicopter rotor. In the context of rotorcraft, the momentum change of the air (as it is accelerated downward) is directly linked to the thrust produced by the rotor that counteracts gravity. The analysis typically involves setting up a control volume that encloses the rotor disk and applying the momentum equation to relate the downward velocity imparted to the air with the lift (or weight) generated by the rotor.

Mass Flow Rate

Mass flow rate is a key concept for determining the amount of air moved by the rotor per unit time. It is calculated using the product of the air density, the area of the rotor or slipstream (which, in the problem, is defined by a stream radius), and the induced velocity. This rate is crucial for linking the momentum imparted to the air with the force required to hover, as it provides a measure of how much mass is accelerated downward.

Energy and Power Considerations

The work done on the air by the rotor is associated with the kinetic energy imparted to the moving air. In determining the power required by the engine, one must assess the rate at which energy is transferred to the air (i.e., the change in kinetic energy per unit time). This involves computing the kinetic energy of the accelerated air mass and understanding that the power output must overcome this energy requirement while maintaining hover.

Actuator Disk (Momentum) Theory

This theoretical framework models the rotor as an idealized disk that imparts a uniform pressure jump and accelerates the air to a specific induced velocity. It provides a simplified yet powerful method for analyzing the lift and power of helicopters, allowing one to derive relationships between the induced velocity, mass flow rate, thrust, and power. The theory assumes ideal flow conditions, neglecting any energy loss mechanisms such as friction or rotational energy in the air.

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