Determine, for the airflow through the frictionless and...

Determine, for the airflow through the frictionless and adiabatic converging-diverging duct of Example $11.8,$ the ratio of duct exit pressure to duct inlet stagnation pressure that will result in a standing normal shock at: (a) $x=+0.1 \mathrm{m},$ (b) $x=+0.2 \mathrm{m}$ (c) $x=+0.4 \mathrm{m} .$ How large is the stagnation pressure loss in each case?

Determine, for the airflow through the frictionless and adiabatic converging-diverging duct of Example $11.8,$ the ratio of duct exit pressure to duct inlet stagnation pressure that will result in a standing normal shock at: (a) $x=+0.1 \mathrm{m},$ (b) $x=+0.2 \mathrm{m}$
(c) $x=+0.4 \mathrm{m} .$ How large is the stagnation pressure loss in each case?

Key Concepts

Converging?Diverging Nozzle Geometry

This concept involves a duct or nozzle whose cross-sectional area first decreases (converges) and then increases (diverges). It is fundamental in compressible flow applications such as supersonic wind tunnels and rockets. The geometry influences the acceleration and deceleration of the flow, enabling the formation and position of shock waves in regions where the duct shape changes the Mach number distribution.

Adiabatic and Frictionless Flow

Adiabatic flow indicates that no heat is added to or removed from the fluid as it flows, while frictionless flow means that viscous effects (and thus energy losses due to friction) are neglected. These idealized assumptions simplify the analysis by ensuring that energy is conserved along streamlines (except across shocks) and that variations in properties are governed primarily by changes in geometry and flow conditions.

Normal Shock Waves

A normal shock wave is a thin region in a supersonic flow where the flow abruptly transitions from supersonic to subsonic speeds. Across a normal shock, the fluid experiences significant increases in pressure, temperature, and density while the Mach number drops sharply. These shocks are highly non-isentropic, meaning they generate entropy and cause irreversible losses in total pressure.

Stagnation Pressure

Stagnation pressure is the pressure a fluid would attain if it were isentropically brought to rest from its flow state. It represents the maximum pressure in the flow and is a crucial parameter in compressible flow analysis, particularly in determining the energy available in a flow. Losses in stagnation pressure, especially across shock waves, indicate irreversible energy dissipation.

Isentropic Flow Relations

In portions of the flow where no shocks or losses occur, the flow can be assumed to be isentropic, meaning the process is both adiabatic and reversible. This allows the use of specific mathematical relations to connect flow properties such as pressure, density, temperature, and Mach number with the duct’s geometry. These relations are vital for predicting where shocks might occur when the flow adjusts to external conditions.

Shock?Induced Stagnation Pressure Loss

When a normal shock occurs within the flow, the process is no longer isentropic, and there is a distinct loss in stagnation pressure. This loss quantifies the irreversible energy dissipation that occurs due to the shock wave. Understanding and calculating the stagnation pressure loss across shocks are important for designing ducted systems where maintaining performance and control over pressure ratios is critical.

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