Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 9: Problem 25

A rectangular aerofoil of \(100 \mathrm{~mm}\) chord and \(750 \mathrm{~mm}\) span is tested in a wind-tunnel. When the air velocity is \(30 \mathrm{~m} \cdot \mathrm{s}^{-1}\) and the angle of attack \(7^{\circ}\) the lift and drag are \(32.8 \mathrm{~N}\) and \(1.68 \mathrm{~N}\) respectively. Assuming an air density of \(1.23 \mathrm{~kg} \cdot \mathrm{m}^{-3}\) and an elliptical distribution of lift, calculate the coefficients of lift, drag and vortex drag, the corresponding angle of attack for an aerofoil of the same profile but aspect ratio \(5.0\), and the lift and drag coefficients at this aspect ratio.

Short Answer

Expert verified
Lift coefficient: 0.791. Drag coefficient: 0.0404. Vortex drag coefficient: 0.0266. New angle: 5.83°.

Step by step solution

01

- Convert dimensions to meters

Given the chord length is 100 mm and the span is 750 mm, first convert them into meters: \[ \text{Chord} = 100 \text{ mm} = 0.1 \text{ m} \] \[ \text{Span} = 750 \text{ mm} = 0.75 \text{ m} \]
02

- Calculate the area of the aerofoil

Calculate the area using the formula for the area of a rectangle: \[ \text{Area} = \text{Chord} \times \text{Span} = 0.1 \text{ m} \times 0.75 \text{ m} = 0.075 \text{ m}^2 \]
03

- Calculate the dynamic pressure

Dynamic pressure can be calculated using the formula: \[ q = \frac{1}{2} \rho V^2 \]Substitute \( \rho = 1.23 \mathrm{~kg} \, \mathrm{m}^{-3} \) and \( V = 30 \mathrm{~m} \, \mathrm{s}^{-1} \): \[ q = \frac{1}{2} \times 1.23 \times (30)^2 = 553.5 \, \mathrm{Pa} \]
04

- Calculate the lift coefficient \(C_L\)

Use the lift formula: \[ L = C_L q S \]Rearrange to find \(C_L\): \[ C_L = \frac{L}{q S} = \frac{32.8}{553.5 \times 0.075} = 0.791 \]
05

- Calculate the drag coefficient \(C_D\)

Use the drag formula: \[ D = C_D q S\]Rearrange to find \(C_D\): \[ C_D = \frac{D}{q S} = \frac{1.68}{553.5 \times 0.075} = 0.0404 \]
06

- Calculate the vortex drag coefficient \(C_{D_v}\)

Use the formula for vortex drag (induced drag), knowing the aspect ratio (AR): \[ C_{D_v} = \frac{C_L^2}{\pi e AR} \]Assuming elliptical distribution, efficiency factor \( e = 1 \): \[ AR = \frac{b^2}{S} = \frac{0.75^2}{0.075} = 7.5 \]So, \[ C_{D_v} = \frac{(0.791)^2}{\pi \times 7.5} = 0.0266 \]
07

- Find new corresponding angle of attack

Given the new aspect ratio (AR) of 5.0, use the approximation for small angle of attack: \[ \alpha' = \frac{\alpha}{1 + \frac{C_L}{\pi e AR}} \]Assuming \( e = 1 \) and \( \alpha = 7^{\circ} = 0.122 \text{ radians} \): \[ \alpha' = \frac{0.122}{1 + \frac{0.791}{\pi \times 5}} = 0.102 \, \text{radians} = 5.83^{\circ} \]
08

- Calculate the lift coefficient at new AR

For the lift coefficient at new AR, it remains the same if angle of attack doesn't change significantly: \[ C_L' = C_L = 0.791 \]
09

- Calculate the drag and vortex drag coefficients at new AR

Calculate total drag coefficient, knowing vortex drag decreases with aspect ratio: \[ C_{D_v}' = \frac{C_L^2}{\pi AR} = \frac{(0.791)^2}{\pi \times 5} = 0.0399 \]Assume profile drag part remains same, so total drag: \[ C_D' = C_{D_v}' = 0.0399 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Aerodynamics
Aerodynamics is the study of how air interacts with solid objects, like an aerofoil. When an object moves through the air, it affects and is affected by the airflow around it. This involves the forces of lift, drag, and moments. Lift is the force that acts perpendicular to the relative airflow. Drag is the force that opposes the object’s motion through the air. Good aerodynamic design minimizes adverse effects, such as high drag, to improve efficiency.
Deciphering Lift Coefficient
The lift coefficient (\( C_L \)) quantifies the lift produced by an aerofoil relative to the dynamic pressure and the area of the wing. It's calculated using the formula: \[ C_L = \frac{L}{q S} \]Here, \(L\) is the lift force, \(q\) is the dynamic pressure, and \(S\) is the surface area of the aerofoil. The lift coefficient helps in predicting how much lift an aerofoil generates at different angles of attack and at various speeds. A higher \( C_L \) indicates more lift.
Exploring Drag Coefficient
The drag coefficient (\( C_D \)) measures the resistance an object encounters as it moves through the air. It is derived using the formula: \[ C_D = \frac{D}{q S} \]where \(D\) represents the drag force. Like the lift coefficient, the drag coefficient aids in understanding how different shapes and surfaces influence the drag force experienced by an object. Minimizing \(C_D\) is crucial in designing efficient aircraft.
Unpacking Vortex Drag
Vortex drag, also known as induced drag, arises from the creation of vortices at the wingtips due to the pressure difference between the upper and lower wing surfaces. It is a component of the overall drag and can be calculated using: \[ C_{D_v} = \frac{C_{L}^2}{\pi e AR} \]where \(e\) is the span efficiency factor (usually close to 1 for elliptical lift distribution), and \(AR\) is the aspect ratio. Vortex drag decreases with higher aspect ratios, making aspect ratio an important design consideration.
Understanding Aspect Ratio
The aspect ratio (AR) of a wing is the ratio of its span to its chord length: \[ AR = \frac{b^2}{S} \]where \(b\) is the span and \(S\) is the wing area. A higher aspect ratio indicates a longer, narrower wing, which generally reduces vortex drag and increases aerodynamic efficiency. However, there are practical limits, as structures with excessively high aspect ratios can be less robust and harder to build. Optimizing aspect ratio is key to balancing performance and structural integrity in aircraft design.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A set of paddles of radius \(R\) is rotated with angular velocity \(\omega\) about a vertical axis in a liquid having an unlimited free surface. Assuming that the paddles are close to the free surface and that the fluid at radii greater than \(R\) moves as a free vortex, determine the difference in elevation between the surface at infinity and that at the axis of rotation.

Show that the two-dimensional flow described (in metre-second units) by the equation \(\psi=x+2 x^{2}-2 y^{2}\) is irrotational. What is the velocity potential of the flow? If the density of the fluid is \(1.12 \mathrm{~kg} \cdot \mathrm{m}^{-3}\) and the piezometric pressure at the point \((1,-2)\) is \(4.8 \mathrm{kPa}\), what is the piezometric pressure at the point \((9,6)\) ?

An enclosed square duct of side \(s\) has a horizontal axis and vertical sides. It runs full of water and at one position there is a curved right-angled bend where the axis of the duct has radius \(r\). If the flow in the bend is assumed frictionless so that the velocity distribution is that of a free vortex, show that the volume rate of flow is related to \(\Delta h\), the difference of static head between the inner and outer sides of the duct, by the expression $$ Q=\left(r^{2}-\frac{s^{2}}{4}\right)(s g \Delta h / r)^{1 / 2} \ln \left(\frac{2 r+s}{2 r-s}\right) $$

A kite may be regarded as equivalent to a rectangular aerofoil of \(900 \mathrm{~mm}\) chord and \(1.8 \mathrm{~m}\) span. When it faces a horizontal wind of \(13.5 \mathrm{~m} \cdot \mathrm{s}^{-1}\) at \(12^{\circ}\) to the horizontal the tension in the guide rope is \(102 \mathrm{~N}\) and the rope is at \(7^{\circ}\) to the vertical. Calculate the lift and drag coefficients, assuming an air density of \(1.23 \mathrm{~kg} \cdot \mathrm{m}^{-3}\)

To the two-dimensional infinite flow given by \(\psi=-U y\) are added two sources, each of strength \(m\), placed at \((0, a)\) and \((0,-a)\) respectively. If \(m>|2 \pi U a|\), determine the stream function of the combined flow and the position of any stagnation points. Sketch the resulting body contour and determine the velocity at the point where the contour cuts the \(y\)-axis.
See all solutions

Recommended explanations on Physics Textbooks

Further Mechanics and Thermal Physics

Read Explanation

Force

Read Explanation

Quantum Physics

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Thermodynamics

Read Explanation

Mechanics and Materials

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free