A normal shock wave forms in front of a two-dimensional blunt-nosed obstacle in a supersonic air stream. The pressure at the nose of the obstacle is three times the static pressure upstream of the shock wave. Determine the upstream Mach number, the density ratio across the shock and the velocity immediately after the shock if the upstream static temperature is \(10^{\circ} \mathrm{C}\). If the air were subsequently expanded isentropically to its original pressure what would its temperature then be?

Supersonic flow refers to the condition where an object's velocity is greater than the speed of sound in the surrounding medium, typically air. During supersonic flight, speeds exceed Mach 1. Here are some key characteristics:

• Formation of shock waves: These are thin regions where the flow properties like pressure, temperature, and density change drastically.
• Compressibility effects: The air behaves differently at high speeds as it's compressed, contrasted to the nearly incompressible behavior at low speeds.
• Loss of subsonic assumptions: Simplified models for subsonic flow don’t apply; factors like pressure waves and shock interactions become crucial.

The problem deals with a supersonic air stream encountering a blunt object, generating a normal shock wave. This shock wave divides the flow into upstream and downstream regions with distinct properties, analyzed using specific shock relations.
Understanding supersonic flow involves considering how changes in flow properties affect Mach number, pressure, temperature, and density. Solving for the upstream Mach number and downstream characteristics highlights these shifts in properties due to the shock wave.

Isentropic processes are thermodynamic processes where entropy remains constant, representing idealized scenarios with no heat transfer or irreversible losses. These principles are absolutely vital in high-speed aerodynamics.
For instance:

• Isentropic relations: These involve pressure, temperature, and density changes in a reversible adiabatic process.
• Flow acceleration and deceleration: In supersonic conditions, flow can be accelerated or decelerated isentropically, affecting Mach number and other properties.
• Application to shock waves: After a normal shock, if the flow is isentropically expanded back to its initial pressure, changes in temperature are evaluated using isentropic relations.

In the given problem, post-shock, the air is expanded isentropically back to the initial pressure. The temperature at this state can be found through isentropic relations, establishing the link between pressure and temperature changes under constant entropy.
Understanding these processes helps highlight differences between idealized and real flows and is pivotal for solving high-speed aerodynamic problems like those involving shock waves.

High-speed aerodynamics studies the behavior of air and its interaction with objects moving at speeds comparable to or exceeding the speed of sound. These studies involve:

• Shock waves: Sudden changes in flow properties occur, categorized as normal, oblique, or bow shocks based on their formation location and orientation.
• Mach number: A critical dimensionless quantity representing the ratio of the object's speed to the local speed of sound, dictating the flow regime as subsonic, transonic, supersonic, or hypersonic.
• Flow properties: High-speed conditions alter basic assumptions, making compressibility and non-linear effects prominent.

In our problem, high-speed aerodynamics principles help calculate the upstream Mach number, density ratio, and post-shock velocity by leveraging relationships unique to supersonic and shock-induced flows.
High-speed aerodynamics also encompasses the effects of heat and friction, wave drag, and techniques to manage or control shock waves, crucial for designing efficient supersonic vehicles.