Question

Are the isentropic relations of ideal gases applicable for flows across \((a)\) normal shock waves, \((b)\) oblique shock waves, and ( \(c\) ) Prandtl-Meyer expansion waves?

Short answer

Answer: The isentropic relations of ideal gases are not applicable for flows across normal shock waves and oblique shock waves, but they are applicable for flows across Prandtl-Meyer expansion waves.

Step-by-step solution

Understanding isentropic relations of ideal gases

Isentropic relations are applicable for an ideal gas when the process is reversible and adiabatic, meaning that there is no heat transfer and no entropy generation during the process. With this understanding, we can now investigate whether these conditions hold for the specified cases.

(a) Normal shock waves

Normal shock waves are formed when a supersonic flow comes across a sudden change in the area (e.g., a sudden contraction or expansion of the channel). In this situation, there is a rapid deceleration of the flow, leading to a sharp increase in both pressure and temperature. This process is neither reversible nor adiabatic, as there are significant entropy generation and energy dissipation. Therefore, isentropic relations of ideal gases are not applicable for flows across normal shock waves.

(b) Oblique shock waves

Oblique shock waves are generated when a supersonic flow encounters a sharp corner, such as a wedge or a cone. Similar to normal shock waves, oblique shock waves also cause a sudden increase in pressure and temperature due to the rapid deceleration of the flow. This process is also irreversible and non-adiabatic, with considerable entropy generation and energy dissipation. Consequently, isentropic relations of ideal gases are not applicable for flows across oblique shock waves.

(c) Prandtl-Meyer expansion waves

Prandtl-Meyer expansion waves occur when a supersonic flow encounters an expansion corner (e.g., the exterior corner of two diverging plates). In this case, the flow accelerates, and both pressure and temperature decrease. Unlike the cases of normal and oblique shock waves, the flow in Prandtl-Meyer expansion waves is isentropic, which means it is reversible and adiabatic. Therefore, the isentropic relations of ideal gases are applicable to Prandtl-Meyer expansion waves. In conclusion, the isentropic relations of ideal gases are: - Not applicable for flows across (a) normal shock waves - Not applicable for flows across (b) oblique shock waves - Applicable for flows across (c) Prandtl-Meyer expansion waves