A supersonic aircraft cruises at M=2.2 at 12 km altitude. A pitot tube is used to sense pressure

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A supersonic aircraft cruises at M=2.2 at 12 km altitude. A...

A supersonic aircraft cruises at $M=2.2$ at $12 \mathrm{km}$ altitude. A pitot tube is used to sense pressure for calculating air speed. A normal shock stands in front of the tube. Evaluate the local isentropic stagnation conditions in front of the shock. Estimate the stagnation pressure sensed by the pitot tube. Show static and stagnation state points and the process path on a $T s$ diagram.

Key Concepts

Supersonic Flow

Supersonic flow refers to the motion of a fluid (typically air) in which the flow velocity exceeds the local speed of sound. This regime is characterized by phenomena such as shock waves, expansion fans, and significant compressibility effects. The Mach number, which is the ratio of the flow speed to the local speed of sound, is a key parameter that defines whether the flow is supersonic and determines the associated aerodynamic behavior.

Mach Number

The Mach number is the dimensionless quantity defined as the ratio of the flow velocity to the local speed of sound in the medium. It plays a crucial role in compressible flows, dictating the nature of the flow field – whether it is subsonic, transonic, supersonic, or hypersonic – and influencing shock wave formation and the thermodynamic changes in the fluid.

Normal Shock

A normal shock is a type of shock wave that is perpendicular to the incoming supersonic flow. It causes an abrupt and non-isentropic change in the flow properties, such as a sudden increase in pressure, temperature, and density, accompanied by a drop in Mach number. Normal shock relations are used to quantify these changes and are essential for analyzing flow conditions upstream and downstream of the shock.

Stagnation Pressure

Stagnation pressure is the pressure a flowing fluid attains when it is brought to rest isentropically. In compressible flow, especially when shock waves are present, the stagnation pressure is affected by the non-isentropic processes. It is a critical parameter in aerodynamic measurements and is used to infer the true dynamic pressure and speed of the aircraft, particularly in applications involving pitot tubes.

Isentropic Flow

Isentropic flow is a flow process that is both adiabatic (no heat transfer) and reversible, meaning that entropy remains constant throughout. In compressible aerodynamics, isentropic relations are used to connect local (static) properties with stagnation properties under ideal conditions, serving as a baseline for comparison with real, non-isentropic processes such as those occurring across shock waves.

Pitot Tube

A pitot tube is an aerodynamic instrument used to measure the stagnation pressure of a fluid by bringing the flow to rest. In high-speed flows where shock waves occur, the pitot tube must be correctly interpreted to account for the pressure losses and discontinuities introduced by the shock. This device is fundamental in determining the speed and other flow characteristics in supersonic aircraft.

T-s Diagram

A Temperature-Entropy (T-s) diagram is a thermodynamic graphical representation that plots temperature versus entropy. It is used to visualize the processes undergone by a fluid, including isentropic (constant entropy) and non-isentropic changes. In the context of shock waves and stagnation processes, the T-s diagram helps illustrate the changes in thermodynamic states and the effects of irreversibilities during the flow deceleration and pressure recovery stages.

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