Question

Using EES (or other) software, calculate and plot the entropy change of air across the normal shock for upstream Mach numbers between 0.5 and 1.5 in increments of \(0.1 .\) Explain why normal shock waves can occur only for upstream Mach numbers greater than \(\mathrm{Ma}=1\).

Short answer

Normal shock waves only occur for upstream Mach numbers greater than 1 because the flow is supersonic and "choked". This means it cannot respond to downstream pressure changes by accelerating or decelerating without the need for a normal shock. In contrast, when the upstream Mach number is less than 1, the flow is not choked and can respond to pressure changes without requiring a normal shock wave. A normal shock wave leads to an increase in entropy and a decrease in total pressure, slowing down the supersonic flow to subsonic speeds.

Step-by-step solution

Calculate the entropy change for each Mach number

We will be using the following isentropic flow relations in combination with the normal shock relations to calculate the entropy change: \( \frac{P_2}{P_1} = \frac{(\gamma + 1) \mathrm{Ma}^2}{2+(\gamma - 1) \mathrm{Ma}^2} \) \( \frac{T_2}{T_1} = \frac{2(\gamma - 1)\mathrm{Ma}^2+3-\gamma}{(\gamma + 1)^2 \cdot \mathrm{Ma}^2} \) \( \frac{\rho_2}{\rho_1} = \frac{(\gamma + 1)\mathrm{Ma}^2}{2+(\gamma - 1) \mathrm{Ma}^2} \) where P is pressure, T is temperature, ρ is density, γ is the specific heat ratio (γ = 1.4 for air), and Ma is the Mach number. The subscripts 1 and 2 denote the upstream and downstream conditions, respectively. Now, we can use these relations to calculate the entropy change for each Mach number by increments of 0.1: \( \Delta S = C_p \ln{\frac{T_2}{T_1}} - R \ln{\frac{P_2}{P_1}} \) where C_p is the specific heat at constant pressure and R is the specific gas constant.

Plot the entropy change

Once the entropy change has been calculated for each Mach number between 0.5 and 1.5 in increments of 0.1, plot the results with the x-axis being the Mach number and the y-axis being the entropy change. You should notice a trend in how the entropy change behaves as the Mach number increases.

Explain why normal shock waves can only occur for upstream Mach numbers greater than 1

Normal shock waves occur when a supersonic flow (Mach number greater than 1) encounters a sudden increase in pressure, causing the flow to slow down and become subsonic (Mach number less than 1). The normal shock wave is a result of the conservation of mass, momentum, and energy across the shock. The normal shock wave cannot occur for upstream Mach numbers less than 1 because the flow is already subsonic and cannot be further slowed down through a normal shock. The reason for this lies in the fact that when the upstream Mach number is less than 1, the flow is not "choked," meaning that it can still respond to pressure changes downstream by accelerating or decelerating without the need for a normal shock. When the upstream Mach number is greater than 1, the flow is "choked," and the only way to slow down the flow to subsonic speeds is through a normal shock wave, accompanied by an increase in entropy and a decrease in total pressure.

Key concepts

Entropy Change in Fluid Dynamics

In fluid dynamics, entropy is a measure of disorder within a system. It's a concept borrowed from thermodynamics and plays a crucial role in understanding the behavior of fluids, such as air, when they undergo various processes.

A particularly interesting scenario arises when dealing with normal shock waves. As the air transitions from supersonic to subsonic speeds through the shock wave, there is an irreversible process resulting in an increase in entropy. This change, denoted as \( \Delta S \), can be computed using specific heat at constant pressure (\( C_p \) and specific gas constant (\( R \)) given the upstream and downstream conditions of pressure (\( P \) and temperature (\( T \)).

The formula for entropy change is expressed as:
\[ \Delta S = C_p \ln{\frac{T_2}{T_1}} - R \ln{\frac{P_2}{P_1}} \]
Here, \( T_2/T_1 \) and \( P_2/P_1 \) can be derived from isentropic flow relations coupled with normal shock relations for specific Mach numbers. When the airflow changes from a high-speed supersonic condition to a slower subsonic state, energy dissipation occurs, leading to a higher level of chaos or entropy within the system.

Mach Number

The Mach number (\( \mathrm{Ma} \)) is a dimensionless quantity in fluid dynamics that represents the ratio of the speed of an object moving through a fluid to the speed of sound in that fluid. It is a critical parameter in the study of supersonic and subsonic flows, as it helps characterize the nature of the flow around various objects, such as aircraft wings or through channels.

When the Mach number is less than 1, the flow is subsonic, and pressure disturbances can propagate upstream. However, for Mach numbers greater than 1, the flow becomes supersonic, and information about pressure changes cannot travel upstream against the flow. This leads to the fundamental criterion for the formation of normal shock waves: they can only occur when the upstream Mach number is greater than one, as explained in the exercise solution.

Therefore, in the context of normal shock waves, the Mach number serves as the dividing line between predictable, steady subsonic flow and the unpredictable, often turbulent supersonic flow that necessitates a shock to restore continuity and balance in the system.

Isentropic Flow Relations

Isentropic flow relations are mathematical expressions that describe the behavior of a perfect, or ideal, gas undergoing a reversible (no entropy change) and adiabatic (no heat transfer) process. These relations are fundamental to the study of fluid flow, especially when dealing with high-speed aerodynamics where changes in pressure, temperature, and density are significant.

The exercise provided uses these relations to calculate the changes across a normal shock wave. For a given Mach number, the pressure, temperature, and density ratios between the downstream and upstream states of the air can be determined using the isentropic flow equations:
\[ \frac{P_2}{P_1} = \frac{(\gamma + 1) \mathrm{Ma}^2}{2+(\gamma - 1) \mathrm{Ma}^2} \]
\[ \frac{T_2}{T_1} = \frac{2(\gamma - 1)\mathrm{Ma}^2+3-\gamma}{(\gamma + 1)^2 \cdot \mathrm{Ma}^2} \]
\[ \frac{\rho_2}{\rho_1} = \frac{(\gamma + 1)\mathrm{Ma}^2}{2+(\gamma - 1) \mathrm{Ma}^2} \]
These relations highlight the direct link between the Mach number and the thermodynamic properties of the flow. The ability to determine these changes is crucial for designing and analyzing applications in aerodynamics, including engines, aircraft performance, and wind tunnels.