At a speed of \(6 \mathrm{~m} \cdot \mathrm{s}^{-1}\) the resistance to motion of a rotor-ship is \(80 \mathrm{kN}\). It is propelled by two vertical cylindrical rotors, each \(3 \mathrm{~m}\) diameter and \(9 \mathrm{~m}\) high. If the actual circulation generated by the rotors is \(50 \%\) of that calculated when viscosity and end effects are ignored, determine the magnitude and direction of the rotational speed of the rotor necessary when the ship travels steadily south-east at \(6 \mathrm{~m} \cdot \mathrm{s}^{-1}\) in a \(14 \mathrm{~m} \cdot \mathrm{s}^{-1}\) north-east wind. For these conditions use inviscid flow theory to determine the positions of the stagnation points and the difference between the maximum and minimum pressures. (Assume an air density of \(1.225 \mathrm{~kg} \cdot \mathrm{m}^{-3}\).)

Rotor-ship resistance refers to the forces that oppose the motion of the ship. This resistance comes due to water friction against the hull and aerodynamic factors such as drag. In this exercise, the resistance is given as 80 kN when the ship is traveling at a speed of 6 m/s. To overcome this resistance, the rotors need to generate enough lift and propulsion.
When considering resistance, it is essential to understand how much force is needed to move the ship through the water and air. In practical scenarios, engineers account for various factors such as water friction, hull shape, and even weather conditions.
Lower resistance means that the ship requires less force to move, making it more fuel-efficient. Therefore, reducing resistance is a key goal in ship design. Bulbous bows, streamlined hulls, and advanced materials are commonly used to achieve this.

Effective wind velocity is the combined effect of the wind's velocity and the ship's velocity. It determines the relative wind the rotors experience and affects their performance.
In this exercise, the ship is traveling southeast at 6 m/s, and the wind is blowing northeast at 14 m/s. To find the effective wind velocity, you subtract the ship's velocity vector from the wind's velocity vector. By representing the southeast and northeast directions as vectors, you get a more accurate understanding.
The effective wind velocity concept is crucial because it impacts the force that the rotor needs to generate. Accurate calculation helps to optimize rotor speed and direction, ensuring efficient propulsion.

The lift coefficient, denoted as \(C_L\), is a dimensionless number that describes a rotor's ability to generate lift. It essentially tells you how effectively the rotor converts wind into lift force.
Lift is generated when air flows over the rotor, creating a lower pressure on one side and a higher pressure on the other. The higher the lift coefficient, the more lift the rotor can produce for a given wind speed.
In this exercise, you use formulas that ignore viscosity and end effects to calculate \(C_L\). These simplifications make the calculations easier but less accurate. Understanding \(C_L\) is vital for designing efficient rotors and achieving the desired propulsion.

Rotor circulation efficiency refers to how effectively the rotor converts rotational speed into useful lift force. In real-world scenarios, various factors like drag, viscosity, and end effects reduce efficiency.
In the exercise, the actual circulation generated by the rotors is only 50% of the theoretical value when ignoring viscosity and end effects. This reduction means the theoretical calculations need to be adjusted.
Understanding circulation efficiency is crucial for optimizing rotor design and speed. Efficient rotors require less energy to produce the same amount of lift, making the ship more fuel-efficient and reducing operating costs.

Inviscid flow theory assumes that the fluid (in this case, air) has no viscosity. It simplifies calculations by ignoring factors like friction and turbulence.
This theory is often used in initial design stages to get a rough estimate of performance. However, it lacks the precision needed for real-world applications. Using inviscid flow theory makes it easier to calculate the positions of stagnation points and pressure differences.
Inviscid flow theory is simpler but less accurate than viscous flow models. Engineers often start with inviscid calculations and later refine them with more complex models. This approach allows them to balance simplicity and accuracy in their designs.