Problem 36 - Aerodynamics for Engineers

Question

Calculate the force and the overturning moment exerted by a 45 -mph wind on a cylindrical flagpole that has a diameter of 6 in. and a height of \(15 \mathrm{ft}\). Neglect variations in the velocity of the wind over the height of the flagpole. The temperature of the air is \(85^{\circ} \mathrm{F} ;\) its pressure is \(14.4\) psi. What is the Reynolds number of this flow?

Step-by-Step Solution

Answer

Convert units and calculate force and moments using dynamic pressure. Compute Reynolds number with air properties.

Step 1: Convert Units

First, convert the wind speed from mph to feet per second (fps). There are 1.467 fps in 1 mph, so the wind speed is \(45 \text{ mph} \times 1.467 \text{ fps/mph} = 66.315 \text{ fps}\). Also, convert diameter from inches to feet (1 inch = 0.0833 feet) to get 0.5 feet.

Step 2: Calculate Dynamic Pressure

Use the dynamic pressure formula for the wind, which is \(q = 0.5 \times \text{air density} \times \text{wind speed}^2\). First, determine the air density. Use the equation for density, \( \rho = \frac{P}{R \cdot T} \), where \( P = 14.4 \) psi (convert to psf: 1 psi = 144 psf), \( R = 1716 \text{ ft}^2/\text{s}^2/(\text{lbf}\cdot\text{°R}) \), and \( T = 85 + 459.67 = 544.67 \text{ °R} \). This gives \( \rho \approx 0.0715 \text{ slug/ft}^3 \). Substitute this value into \(q\).

Step 3: Calculate the Force

The force exerted by the wind can be calculated using \( F = q \times A \), where \( A \) is the frontal area. The frontal area for a cylindrical column is \( A = \text{height} \times \text{diameter} = 15 \text{ ft} \times 0.5 \text{ ft} = 7.5 \text{ ft}^2 \). Substitute \(q\) and \(A\) to find the force \(F\).

Step 4: Calculate Overturning Moment

The overturning moment is given by \( M = F \times \frac{h}{2} \), where \( h \) is the height of the flagpole. With a height of 15 feet, substitute \(F\) (found in the previous step) and \(h\) into the formula to find the moment \(M\).

Step 5: Determine Reynolds Number

Reynolds number \(Re\) is calculated using the formula \(Re = \frac{\rho \cdot V \cdot D}{\mu}\). From previous steps, \(\rho \) is air density, \(V\) is wind speed, and \(D\) is diameter. Air viscosity \(\mu\) for 85°F is approximately 3.737x10^-7 lb/ft.s. Substitute these values to calculate \(Re\).

Key Concepts

Wind Force Calculation

When calculating the wind force acting on a structure, like a flagpole, it's crucial to determine the dynamic pressure first. This is essentially the pressure exerted by the wind as it moves. The dynamic pressure, denoted as \( q \), is calculated using the formula:

• \( q = 0.5 \times \text{air density} \times \text{wind speed}^2 \)

Air density is found by the equation \( \rho = \frac{P}{R \cdot T} \). Here, \( P \) is the air pressure, \( R \) is the specific gas constant for air, and \( T \) is the temperature in Rankine.
Once the dynamic pressure is known, the next step is to apply it to the structure's frontal area. In the case of a cylindrical flagpole, the frontal area \( A \) is the height multiplied by the diameter. The wind force \( F \) can then be calculated as:

• \( F = q \times A \)

This force is what pushes against the flagpole, and understanding how to find it helps anticipate the structural demands on it.

Overturning Moment

The overturning moment refers to the rotational force that can make a structure like a flagpole tip over. It results from the wind force applying a torque on the structure. The formula to compute the overturning moment \( M \) is:

• \( M = F \times \frac{h}{2} \)

Here, \( F \) represents the wind force, and \( h \) is the height of the pole.
The point of application of the force, which is at the pole's midpoint height-wise, is essential to calculate this moment. As the wind applies a force at this height, it creates a lever arm. Understanding the overturning moment helps assess the stability of tall structures and is critical in designing their foundations to resist toppling.

Reynolds Number

Reynolds number \(Re\) is a vital dimensionless quantity in fluid mechanics that indicates whether the flow will be laminar or turbulent. It's determined by the formula:

• \(Re = \frac{\rho \cdot V \cdot D}{\mu}\)

Here, \( \rho \) is the air density, \( V \) is the velocity or wind speed, \( D \) is the diameter of the structure, and \( \mu \) is the air viscosity.
Knowing the Reynolds number helps predict the nature of the flow around the flagpole. For high Reynolds numbers, the flow tends to be turbulent, which leads to increased drag force. On the other hand, low Reynolds numbers suggest laminar flow, meaning less resistance. This concept is crucial for engineers to design structures that efficiently handle different flow regimes.

Dynamic Pressure

Dynamic pressure is a specific pressure exerted by fluid motion. It's an essential part of calculating the forces acting on any object within a fluid flow, such as air around a flagpole. Dynamic pressure is extracted from the Bernoulli equation specific for moving fluids:

• \( q = 0.5 \times \text{air density} \times \text{wind speed}^2 \)

This pressure is distinguished from static pressure, which is the pressure exerted by a fluid at rest.
Determining the dynamic pressure accurately allows us to find how much force the moving air can exert on a structure. It's vital for identifying how various aerodynamic elements behave and interact with structures to ensure safety and efficiency.

Fluid Mechanics

Fluid mechanics is the branch of physics concerned with how fluids (liquids and gases) move and the forces they exert. It provides the foundation for understanding aerodynamics, including wind force, pressure, and flow characteristics.
Some major areas in fluid mechanics include:

• Fluid statics: The study of fluids at rest.
• Fluid dynamics: The study of fluids in motion, focusing on forces and energy conservation.
• Aerodynamics: A subset of fluid dynamics focusing on the interaction of air with solid surfaces.

In the context of wind on a flagpole, fluid mechanics helps determine how wind speed, density, and flow types (laminar or turbulent) impact the applied forces and moments. It serves as the underlying science that guides engineers in predicting and optimizing the behavior of structures in different environments.