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Chapter 5: Problem 17

Consider a uniform flow with a Mach number of 2 . What angle does a Mach wave make with respect to the flow direction?

Short Answer

Expert verified
The Mach wave makes a 30掳 angle with the flow direction.

Step by step solution

01

Understanding Mach Waves

A Mach wave is a pressure wave that forms due to supersonic speeds (when the Mach number is greater than 1). It occurs at an angle to the direction of flow.
02

Formula for Mach Angle

The angle of a Mach wave, also known as the Mach angle, is given by \( \mu = \arcsin \left( \frac{1}{M} \right) \), where \( M \) is the Mach number.
03

Substitute the Mach Number

Given \( M = 2 \), substitute this into the formula: \[ \mu = \arcsin \left( \frac{1}{2} \right) \].
04

Calculate the Mach Angle

Compute \( \mu = \arcsin \left( 0.5 \right) \), which means \( \mu = 30^\circ \).

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Key Concepts

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

Mach Number
The Mach number is a fundamental concept in aerodynamics, representing the ratio of an object's speed to the speed of sound in the surrounding medium. When the Mach number is less than 1, the flow is subsonic. When it equals 1, the flow is sonic. For values greater than 1, the flow is supersonic.
Understanding the Mach number helps predict how an object will interact with the air or fluid around it.
  • Subsonic: Mach number < 1
  • Sonic: Mach number = 1
  • Supersonic: Mach number > 1
In the given exercise, the Mach number is 2, indicating that the flow is supersonic. This plays a crucial role in phenomena like shock waves and Mach waves, which we'll explore further.
Supersonic Flow
Supersonic flow occurs when an object moves faster than the speed of sound in the medium. This results in unique aerodynamic forces and wave patterns that differ significantly from subsonic flow.
When an object travels at supersonic speeds, it creates a compressible flow, where density changes are significant. One prominent feature of supersonic flow is the formation of shock waves, which are abrupt changes in pressure and density.
  • Compressible flow behavior
  • Significant density changes
  • Formation of shock waves
The understanding of supersonic flow is essential for designing aircraft and spacecraft that can efficiently travel at high speeds.
Pressure Wave
A pressure wave is a disturbance that travels through a medium and involves compressions and rarefactions of the particles. In the context of high-speed aerodynamics, when the flow becomes supersonic, these pressure waves converge into Mach waves.Mach waves are a type of pressure wave associated with objects moving at supersonic speeds. The angle at which these waves form with the flow direction is known as the Mach angle, \(\mu\), calculated using the formula \(\mu = \arcsin \left( \frac{1}{M} \right)\).Understanding pressure waves is crucial to comprehend how they influence the lift and drag on supersonic vehicles and play a role in the design of aerodynamics at these speeds.
Uniform Flow
Uniform flow indicates that the velocity of the fluid or air is consistent in magnitude and direction. This simplification helps in analyzing and solving aerodynamic problems effectively. In uniform flow:
  • Velocity is constant throughout the flow field.
  • No acceleration is present within the flow.
  • Analysis becomes straightforward due to the consistency.
The given exercise assumes a uniform flow, which means that changes in the Mach angle can be directly attributed to variations in the Mach number without complicating factors like velocity gradients.

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Most popular questions from this chapter

The ratio of lift to drag \(L D\) for a wing or airfoil is an important aerodynamic parameter, indeed, it is a direct measure of the aerodynamic efficiency of the wing. If a wing is pitched through a range of angle of attack, \(L D\) first increases, then goes through a maximum, and then decreases. Consider an infinite wing with an NACA 2412 airfoil. Estimate the maximum value of \(L / D\). Assume that the Reynolds number is \(9 \times 10^{6}\).

Consider a wing in a high-speed wind tunnel. At a point on the wing, the velocity is \(850 \mathrm{ft} / \mathrm{s}\). If the test-section flow is at a velocity of \(780 \mathrm{ft} / \mathrm{s}\), with a pressure and temperature of \(1 \mathrm{~atm}\) and \(505^{\circ} \mathrm{R}\), respectively, calculate the pressure coefficient at the point.

Consider a finite wing with an aspect of ratio of 7; the airfoil section of the wing is a symmetric airfoil with an infinite-wing lift slope of \(0.11\) per degree. The lift-todrag ratio for this wing is 29 when the lift coefficient is equal to \(0.35\). If the angle of attack remains the same and the aspect ratio is simply increased to 10 by adding extensions to the span of the wing, what is the new value of the lift-to-drag ratio? Assume that the span efficiency factors \(e=e_{1}=0.9\) for both cases.

During the \(1920 \mathrm{~s}\) and early \(1930 \mathrm{~s}\), the NACA obtained wind tunnel data on different airfoils by testing finite wings with an aspect ratio of 6 . These data were then "corrected" to obtain infinite-wing airfoil characteristics. Consider such a finite wing with an area and aspect ratio of \(1.5 \mathrm{ft}^{2}\) and 6 , respectively, mounted in a wind tunnel where the test-section flow velocity is \(100 \mathrm{ft} / \mathrm{s}\) at standard sea- level conditions. When the wing is pitched to \(\alpha=-2^{\circ}\), no lift is measured. When the wing is pitched to \(\alpha=10^{\circ}\), a lift of \(17.9 \mathrm{lb}\) is measured. Calculate the lift slope for the airfoil (the infinite wing) if the span effectiveness factor is \(0.95\).

By the method of dimensional analysis, derive the expression \(M=q_{\infty} S c c_{w}\) for the aerodynamic moment on an airfoil, where \(c\) is the chord and \(c_{m}\) is the moment coefficient.
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