Problem 9 - Aircraft Propulsion

Question

A normal shock is in a Mach \(2.0\) flow. Upstream gas temperature is \(T_{1}=15^{\circ} \mathrm{C}\), the gas constant is \(R=287 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) and \(\gamma=1.4\). Calculate (a) \(a^{*}\) in \(\mathrm{m} / \mathrm{s}\) (b) \(u_{2}\) in \(\mathrm{m} / \mathrm{s}\) (use Prandtl's relation) (c) \(a_{\mathrm{t}}\) in \(\mathrm{m} / \mathrm{s}\) (d) \(h_{2}\) in \(\mathrm{kJ} / \mathrm{kg}\)

Step-by-Step Solution

Answer

The calculated values should be as follows: (a) \(a^{*}\) = \ m/s, (b) \(u_{2}\) = \ m/s, (c) \(a_{\mathrm{t}}\) = \ m/s, (d) \(h_{2}\) = \ kJ/kg.

Step 1: Calculate sonic speed \(a^{*}\)

First, convert the temperature \(T_{1}\) from Celsius to Kelvin, \(T_{1}=15+273=288 K\). Then, apply the formula for sound speed \(a^{*}\) in a gas as: \[a^{*}=\sqrt{\gamma \cdot R \cdot T_{1}}\], where \(R\) is the gas constant, and \(\gamma\) is the specific heat ratio. Substitute the values to get the result.

Step 2: Calculate speed \(u_{2}\) through Prandtl's relation

Prandtl's relation for the speed of the gas downstream of the shock \(u_{2}\) in relation to the upstream Mach number \(M_{1}\) is: \( \frac{u_{2}}{a_{1}}=\frac{1}{M_{1}(1+\frac{\gamma -1}{2}*M_{1})}\). Substitute the known values to calculate \(u_{2}\).

Step 3: Calculate \(a_{\mathrm{t}}\)

The speed of sound \(a_{\mathrm{t}}\) at the throat can be computed as: \[a_{\mathrm{t}}=\frac{a^{*}}{\sqrt{1+\frac{\gamma -1}{2}*M_{1}}}\] Substitute the known values to get the result.

Step 4: Compute \(h_{2}\)

The total enthalpy downstream of the shock, \(h_{2}\), can be calculated with the equation: \(h_{2}=h_{1}+\frac{1}{2} * u_{1}^{2}-\frac{1}{2} * u_{2}^{2}\). Since the flow is adiabatic, \(h_{1} = c_{p}*T_{1}\) where \(c_{p} = \gamma *R/(\gamma -1)\). Substitute the known values to get the result.

Key Concepts

Mach number

The Mach number is a fundamental concept in fluid dynamics that compares the speed of a flow to the speed of sound in the same medium. It is expressed as the ratio of the flow velocity to the speed of sound. When the Mach number is less than 1, the flow is subsonic; when it's exactly 1, the flow is sonic; and when it's greater than 1, the flow is supersonic.

In the given problem, the Mach number upstream of the shock is 2.0, indicating that the flow is supersonic. This means the air particles are traveling at twice the speed of sound in this scenario. Understanding Mach number helps in analyzing how the flow behavior changes across the shock wave.

• A Mach number of less than 1 is subsonic.
• Exactly 1 is sonic.
• Greater than 1 is supersonic.

Mach number is crucial in the study of aerodynamics and is used to predict the occurrence and strength of shock waves and their effects on the surrounding flow.

specific heat ratio

The specific heat ratio, often represented by the Greek letter \( \gamma \), is a key property of gases, particularly in thermodynamics and fluid dynamics. It is the ratio of the specific heat of the gas at constant pressure (\( c_p \)) to the specific heat at constant volume (\( c_v \)). For most air applications, \( \gamma \) typically equals 1.4.

This ratio is critical when discussing processes that involve changes in temperature and pressure. The specific heat ratio influences how gases behave in different thermodynamic contexts, including shock waves.

• The specific heat ratio is written as \( \gamma = \frac{c_p}{c_v} \).
• It helps determine the thermodynamic behavior of gases.
• For diatomic gases like air, \( \gamma \) is usually 1.4.

In the context of the exercise, knowing the specific heat ratio helps calculate variables like sonic speed and downstream flow properties across the shock.

sound speed

Sound speed, often denoted as \( a \) or \( a^* \), refers to the speed at which sound waves travel through a medium. In fluid dynamics, it's fundamental for understanding how pressure waves propagate. This speed depends on both the medium's properties and the temperature.

The formula for calculating the speed of sound in a gas is \[ a^* = \sqrt{\gamma \cdot R \cdot T_1} \], where:

• \( \gamma \) is the specific heat ratio.
• \( R \) is the gas constant, which for dry air is 287 J/kg·K.
• \( T_1 \) is the absolute temperature in Kelvins.

In the exercise, the computation of sound speed aids in the analysis of flow conditions both upstream and downstream of the shock wave, which is essential for characterizing the flow dynamics.

Prandtl's relation

Prandtl's relation connects the flow velocity ahead and behind a shock wave. This relation is particularly useful for calculating changes in flow properties caused by the shock, especially when dealing with supersonic flows.

Prandtl's relation for speed after a normal shock can be expressed as \[ \frac{u_2}{a_1} = \frac{1}{M_1(1 + \frac{\gamma - 1}{2}M_1)} \], where:

• \( u_2 \) is the flow velocity downstream of the shock.
• \( a_1 \) is the speed of sound upstream.
• \( M_1 \) is the upstream Mach number.
• \( \gamma \) is the specific heat ratio.

This relation is vital in determining how the velocity changes due to shock compression. It enables the calculation of downstream velocities, which are important to predict how the flow will react to obstacles or changes in the medium.