Problem 44 - Aerodynamics for Engineers

Question

There were early attempts in the development of the airplane to use rotating cylinders as airfoils, Consider such a cylinder having a diameter of \(1 \mathrm{~m}\) and a length of \(10 \mathrm{~m}\). If this cylinder is rotated at \(100 \mathrm{rpm}\) while the plane moves at a speed of \(100 \mathrm{~km} / \mathrm{h}\) through the air at \(2 \mathrm{~km}\) standard atmosphere, estimate the maximum lift that could be developed, disregarding end effects.

Step-by-Step Solution

Answer

The estimated maximum lift is approximately 4591.89 N.

Step 1: Convert Units

First, convert the plane's speed from kilometers per hour to meters per second. The speed is given as 100 km/h. Using the conversion factor of 1 km/h = 0.27778 m/s, we have\[100 \text{ km/h} \times 0.27778 \text{ m/s per km/h} = 27.778 \text{ m/s}\]

Step 2: Calculate Circumference

Since lift on a rotating cylinder is related to the rotational velocity, convert the rpm to a linear velocity at the surface of the cylinder. The circumference \( C \) of the cylinder is calculated by:\[ C = \pi \times \text{diameter} = \pi \times 1 \text{ m} = \pi \text{ m} \]

Step 3: Calculate Rotational Surface Velocity

With a diameter of 1 m, the surface at 100 rpm has a velocity \( V_c \) calculated by:\[V_c = \frac{100 \text{ revolutions}}{1 \text{ minute}} \times \frac{1 \text{ minute}}{60 \text{ seconds}} \times \pi \text{ m} = \frac{100}{60} \pi \text{ m/s} \approx 5.236 \text{ m/s }\]

Step 4: Use the Kutta-Joukowski Theorem

The Kutta-Joukowski theorem gives lift per unit length for a rotating cylinder as:\[L' = \rho V \Gamma\]where \( \rho \) is the air density, \( V \) is the velocity of the free stream, and \( \Gamma \) is the circulation, given by \( \Gamma = 2 \pi r V_c \). Assume standard atmosphere at 2 km altitude where \( \rho \approx 1.006 \text{ kg/m}^3 \).

Step 5: Calculate Circulation

For the rotation, the circulation \( \Gamma \) can be calculated by:\[\Gamma = 2 \pi \times 0.5 \times 5.236 \text{ m/s} \approx 16.451 \text{ m}^2\text{/s}\] where 0.5 m is the radius of the cylinder.

Step 6: Calculate Lift Per Unit Length

Substitute \( \Gamma \) and \( V \) into the lift per unit length equation:\[L' = 1.006 \text{ kg/m}^3 \times 27.778 \text{ m/s} \times 16.451 \text{ m}^2\text{/s} \approx 459.189 \text{ N/m}\]

Step 7: Calculate Total Lift

Finally, calculate the total lift by multiplying the lift per unit length by the length of the cylinder:\[L = L' \times 10 \text{ m} = 459.189 \text{ N/m} \times 10 \text{ m} = 4591.89 \text{ N}\]

Key Concepts

Rotating cylinders as airfoils

The concept of using rotating cylinders as airfoils comes from an interesting application in aerodynamics. Traditional wings generate lift by moving air faster over one surface than the other, thanks to their shape. But with a rotating cylinder, the idea is to produce lift by spinning the cylinder to manipulate the air flow around it. This phenomenon is somewhat similar to the Magnus effect seen in spinning balls, where a rotating object drags air with it, creating pressure differences. By controlling rotation speed and direction, a cylinder can generate significant lift. Historically, this was explored in the early days of aviation, although the practicality of such systems was limited by engineering constraints of the time. Advancement in materials and design could make this concept more viable in the future.

Kutta-Joukowski theorem

The Kutta-Joukowski theorem is a fundamental principle in aerodynamics that relates circulation around a wing or cylinder to the lift produced. It states that the lift per unit length of a cylinder or wing is proportional to the fluid density, the velocity of the approaching flow, and the circulation around the object. The formula is:

• \( L' = \rho V \Gamma \)

Here, \( \rho \) represents air density, \( V \) is the velocity of the oncoming fluid, and \( \Gamma \) is the circulation, which essentially measures how much the air is swirling around the body. Understanding this theorem helps in visualizing how air movement (and not just speed) can lead to lift. This principle is crucial for moving beyond static wing designs to dynamic models like rotating cylinders.

Lift calculation

Lift calculation is the process of determining the upward force that keeps an aircraft aloft. For rotating cylinders, we start with calculating the circulation, which depends on the cylinder's rotational speed and size. The formula for circulation \( \Gamma \) is:

• \( \Gamma = 2 \pi r V_c \)

Where \( r \) is the radius of the cylinder and \( V_c \) is the velocity at the cylinder's surface from its rotation. Once \( \Gamma \) is found, the Kutta-Joukowski theorem helps calculate lift per unit length. We multiply this by the cylinder's length to find total lift:

• \( L = L' \times \text{length of the cylinder} \)

This gives a concrete number to the theoretical lift, helping engineers understand the effectiveness of different designs.

Unit conversion

Unit conversion is crucial in solving aerodynamics problems as it ensures all measurements are in compatible units. In the given problem, we start by converting the plane's speed from kilometers per hour to meters per second, which is standard for calculating forces in physics. Using the conversion factor:

• 1 km/h = 0.27778 m/s

we find the velocity in meters per second. Similarly, rotational speed given in revolutions per minute (rpm) must be converted to meters per second to relate to linear speed. This ensures consistency in calculations, allowing you to correctly apply aerodynamic principles such as the Kutta-Joukowski theorem without errors.