Question

Air enters a normal shock at \(26 \mathrm{kPa}, 230 \mathrm{K},\) and \(815 \mathrm{m} / \mathrm{s} .\) Calculate the stagnation pressure and Mach number upstream of the shock, as well as pressure, temperature, velocity, Mach number, and stagnation pressure downstream of the shock.

Short answer

#Short Answer# The properties of air before (upstream) and after (downstream) the normal shock wave are as follows: Upstream: Mach number (M1) = 2.407 Stagnation pressure (P0_1) = 153.27 kPa Downstream: Pressure (P2) = 85.07 kPa Temperature (T2) = 516.62 K Velocity (V2) = 346.47 m/s Mach number (M2) = 0.721 Stagnation pressure (P0_2) = 72.19 kPa

Step-by-step solution

Calculate the Mach number upstream of the shock (M1)

First, we need to calculate the Mach number (M1). We can use the formula for the speed of sound in an ideal gas and the given speed: \(M1 = \frac{V_1}{a_1}\) Where: \(V_1 = 815 \frac{m}{s}\) (velocity upstream of the shock) \(a_1 = \sqrt{\gamma \cdot R \cdot T_1}\) (speed of sound upstream of the shock) \(\gamma = 1.4\) (specific heat ratio for air) \(R = 287 \frac{J}{kg \cdot K}\) (ideal gas constant) \(T_1 = 230 K\) (temperature upstream of the shock) First, we calculate the speed of sound, and then we calculate the Mach number: \(a_1 = \sqrt{1.4 \cdot 287 \cdot 230}\) \(a_1 \approx 338.52 \frac{m}{s}\) \(M1 = \frac{815}{338.52} \approx 2.407\)

Calculate the stagnation pressure upstream of the shock (P0_1)

To find the stagnation pressure upstream of the shock, we use the isentropic flow relationship: \(P0_1 = P_1 \cdot \left(1 + \frac{\gamma - 1}{2} M1^2\right)^{\frac{\gamma}{\gamma-1}}\) Where: \(P_1 = 26 kPa\) (pressure upstream of the shock) Now, let's plug in the numbers: \(P0_1 = 26 \cdot \left(1 + \frac{1.4 - 1}{2} \cdot 2.407^2\right)^{\frac{1.4}{1.4-1}}\) \(P0_1 \approx 153.27 kPa\)

Use normal shock wave relations to calculate downstream properties (P2, T2, V2, M2, P0_2)

Now, we will use the normal shock wave relationships to calculate the downstream properties, using the upstream Mach number, M1. For pressure (P2): \(P2 = P_1(1 + 2\gamma\frac{(M1^2 - 1)}{1 + \gamma})\) \(P2 \approx 85.07 kPa\) For temperature (T2): \(T2 = T_1\frac{(2+\gamma(M1^2-1))(2\gamma(M1^2-1)+1)}{(\gamma+1)^2M1^2}\) \(T2 \approx 516.62 K\) For the downstream Mach number (M2): \(M2 = \sqrt{\frac{\frac{2}{\gamma - 1}(M1^2 - 1) + 1}{\gamma(M1^2 - 1) + 1}}\) \(M2 \approx 0.721\) For velocity (V2), we can use the conservation of mass: \(\frac{\rho_1 V_1}{\rho_2 V_2} = 1\) \(\rho_1 = \frac{P_1}{R T_1}\) \(\rho_2 = \frac{P_2}{R T_2}\) \(V_2 = V_1 \frac{\rho_1}{\rho_2}\) \(V_2 \approx 346.47 \frac{m}{s}\) Finally, for the downstream stagnation pressure (P0_2): \(P0_2 = P0_1 \cdot \frac{P_2}{P_1} \cdot \frac{(1 + \frac{\gamma - 1}{2} M2^2)^{\frac{\gamma}{\gamma-1}}}{(1 + \frac{\gamma - 1}{2} M1^2)^{\frac{\gamma}{\gamma-1}}}\) \(P0_2 \approx 72.19 kPa\) We have now calculated all needed properties: - Upstream Mach number (M1) = 2.407 - Upstream stagnation pressure (P0_1) = 153.27 kPa - Downstream pressure (P2) = 85.07 kPa - Downstream temperature (T2) = 516.62 K - Downstream velocity (V2) = 346.47 m/s - Downstream Mach number (M2) = 0.721 - Downstream stagnation pressure (P0_2) = 72.19 kPa

Key concepts

Understanding Mach Number

Mach number is essential in the study of aerodynamics and fluid flow. It’s a dimensionless quantity that represents the ratio of an object's speed to the speed of sound in the surrounding medium. The formula for Mach number is given by:
\( M = \frac{V}{a} \)
where \(M\) is the Mach number, \(V\) is the object's velocity, and \(a\) is the speed of sound in the fluid. Speeds below Mach 1 are considered subsonic, while those above Mach 1 are classified as supersonic.
In the given exercise, the upstream Mach number was calculated using the upstream velocity and the speed of sound in air at the given temperature. Understanding the concept of Mach number is crucial when analyzing the behavior of flows, such as the transformation across a normal shock wave where we see a sudden drop in Mach number from supersonic to subsonic.

Stagnation Pressure in Fluid Dynamics

Stagnation pressure is closely tied to the energy of a fluid. It represents the pressure a fluid would attain if brought to rest isentropically. In other words, it’s the pressure of the fluid when its kinetic energy is completely converted into pressure energy. This concept extends to aerodynamics and thermodynamics as a critical factor for understanding the changes in a fluid's state when passing through different processes, including shock waves.
Using the isentropic flow relations, the stagnation pressure ahead of the shock can be calculated. The exercise previously showed how the upstream stagnation pressure was obtained using the formula:
\(P_0 = P \cdot \left(1 + \frac{\gamma - 1}{2} M^2\right)^{\frac{\gamma}{\gamma-1}}\)
where \(P\) is the static pressure, \(\gamma\) is the specific heat ratio, and \(M\) is the Mach number. Knowing the stagnation pressure before and after a shock provides insights into the losses of total pressure due to shockwave interaction.

Isentropic Flow and its Significance

Isentropic flow refers to an idealized process in which a fluid flows at constant entropy. This assumption leads to several simplifications in analyzing compressible flow problems. Since the entropy remains constant, the flow is reversible, and no energy is dissipated as heat within the fluid, which is not the case in reality especially during shockwaves. However, this assumption helps engineers to get estimates and understand the maximum possible efficiency of fluid systems.
In isentropic conditions, relationships are derived for various flow properties, such as pressure, temperature, density, and velocity, as functions of the Mach number. The exercise leveraged these isentropic relations to calculate the upstream stagnation pressure. The use of isentropic flow relations provides an ideal benchmark, which is critical for understanding real fluid dynamics when factoring in inefficiencies and energy losses.