Problem 222 - Calculus Volume 1

Question

Using \(\mu=2 \sin ^{-1}\left(\frac{1}{M}\right)\), find the Mach number \(M\) for the following angles. a. \(\quad \mu=\frac{\pi}{6}\) b. \(\quad \mu=\frac{2 \pi}{7}\) c. \(\quad \mu=\frac{3 \pi}{8}\)

Step-by-Step Solution

Answer

a) M ≈ 3.87, b) M ≈ 2.30, c) M ≈ 1.80.

Step 1: Understand the Formula

The given formula is \( \mu = 2 \sin^{-1}\left(\frac{1}{M}\right) \). This is the Prandtl-Meyer function, where \( \mu \) is the Mach angle and \( M \) is the Mach number. The formula needs to be rearranged to solve for \( M \).

Step 2: Rearrange the Formula

To solve for \( M \), rearrange the formula: \( \mu = 2 \sin^{-1}\left(\frac{1}{M}\right) \) implies \( \sin^{-1}\left(\frac{1}{M}\right) = \frac{\mu}{2} \). Then, \( \frac{1}{M} = \sin\left(\frac{\mu}{2}\right) \) and finally, \( M = \frac{1}{\sin\left(\frac{\mu}{2}\right)} \).

Step 3: Solve for Mach Number \( M \) for \( \mu = \frac{\pi}{6} \)

First, substitute \( \mu = \frac{\pi}{6} \) into the rearranged formula: \( M = \frac{1}{\sin\left(\frac{\pi}{12}\right)} \). Calculate \( \sin\left(\frac{\pi}{12}\right) \), knowing \( \frac{\pi}{12} \approx 15^\circ \), it's approximately 0.2588. Thus, \( M \approx \frac{1}{0.2588} \approx 3.87 \).

Step 4: Solve for Mach Number \( M \) for \( \mu = \frac{2\pi}{7} \)

Substitute \( \mu = \frac{2\pi}{7} \) into the formula: \( M = \frac{1}{\sin\left(\frac{\pi}{7}\right)} \). Calculate \( \sin\left(\frac{\pi}{7}\right) \approx 0.4339 \). Therefore, \( M \approx \frac{1}{0.4339} \approx 2.30 \).

Step 5: Solve for Mach Number \( M \) for \( \mu = \frac{3\pi}{8} \)

Substitute \( \mu = \frac{3\pi}{8} \) into the formula: \( M = \frac{1}{\sin\left(\frac{3\pi}{16}\right)} \). Calculate \( \sin\left(\frac{3\pi}{16}\right) \approx 0.5556 \). Therefore, \( M \approx \frac{1}{0.5556} \approx 1.80 \).

Key Concepts

Prandtl-Meyer function

The Prandtl-Meyer function is a crucial concept in the study of supersonic flow, particularly relating to expansion waves. It describes the relation between the Mach number, denoted as \( M \), and the angle of expansion, which is termed as the Prandtl-Meyer angle or \( u \).
In the context of the Prandtl-Meyer function, the Mach number is an essential variable if you are analyzing the supersonic expansion where changes in the flow direction occur in a smooth and gradual manner. This function is calculated as an inverse trigonometric equation that helps to find the Mach number based on a given angle, like in the exercise provided.
Simply put, if you need to find how fast the flow is moving past a point in terms of sound speed fractions (i.e., the Mach number), understanding the expansion angle through this function aids in determining that. Notably, the angle \( \mu \) in this context is crucial as it is a measure directly related to flow deflection and the Mach number in supersonic speeds.

Mach angle

Mach angle is a pivotal aspect when dealing with the concepts of supersonic flow. In simple terms, the Mach angle \( \mu \) represents the angle between the direction of the Mach wave and the direction of the flow in a supersonic flow field. It is directly linked to the Mach number by the equation \( \mu = \sin^{-1}\left(\frac{1}{M}\right) \).
This angle helps visualize how shock waves or expansion waves are oriented in the flow. When the Mach number is higher, the Mach angle becomes smaller, indicating that the flow is more compressed and faster relative to the speed of sound.

• The Mach angle helps determine the wave patterns in the flow.
• It allows engineers to design better aerodynamic structures by understanding flow characteristics.
• By altering the Mach number, one can see direct changes in the Mach angle, influencing how waves like shock waves propagate through the air.

Understanding the Mach angle grants us insight into the transition from subsonic to supersonic flows, a concept that's instrumental in aerospace engineering.

Trigonometric function

Trigonometric functions are essential mathematical tools used to relate angles to the sides of a triangle, and they frequently appear in equations that describe wave motion, periodic phenomena, and in this case, supersonic flow through Mach numbers and angles.
In the given exercise, the function \( \sin^{-1} \) or arcsine is used to find angles from known sine values, reflecting the inverse property of the sine function.
Let's explore some reasons why trigonometric functions are essential here:

• Trigonometric functions help translate the geometric representation of the flow into mathematical terms, allowing us to solve for the Mach number using angles.
• They provide a universal language to describe oscillations and repetitive movements.
• In dynamics, they simplify complex wave motions into manageable calculations.

By utilizing trigonometric functions, you can harness the power of these mathematical relationships to intuitively and effectively solve for unknown variables – such as angles or Mach numbers – in fluid dynamics and beyond. This transformation of geometric data into numerical solutions underscores their importance.