Stagnation pressure and temperature probes are located on...

Stagnation pressure and temperature probes are located on the nose of a supersonic aircraft. At $35,000 \mathrm{ft}$ altitude a normal shock stands in front of the probes. The temperature probe indicates $\bar{T}_{0}=420^{\circ} \mathrm{F}$ behind the shock. Calculate the Mach number and air speed of the plane. Find the static and stagnation pressures behind the shock. Show the process and the static and stagnation state points on a $T s$ diagram.

Key Concepts

Stagnation Conditions

Stagnation (or total) conditions refer to the state variables—pressure, temperature, and density—that a fluid would have if it were brought to rest isentropically. These values are useful in compressible flow analysis because they remain constant along a streamline in the absence of shocks or other dissipative processes.

Normal Shock Relations

Normal shock relations describe the abrupt changes in flow properties (such as Mach number, pressure, temperature, and density) that occur when a supersonic flow passes through a shock wave that is perpendicular to the flow direction. They provide equations to relate the upstream (before the shock) and downstream (after the shock) conditions, accounting for entropy increase and loss of mechanical energy.

Mach Number and Supersonic Flow

The Mach number, defined as the ratio of the flow velocity to the local speed of sound, is critical in characterizing supersonic flows. It helps determine how compressibility effects, such as shock waves, will influence the flow. In supersonic flight, the Mach number influences aerodynamic heating, wave drag, and overall aerodynamic performance.

T-s Diagram in Compressible Flow

A Temperature-Entropy (T-s) diagram is a graphical tool used in thermodynamics to illustrate the state changes of a flow, including processes like expansion and compression. In the context of shock waves, it visualizes the irreversible increase in entropy and the corresponding changes in temperature from upstream to downstream states, providing insight into energy losses and flow behavior.

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