Consider the supersonic flow over an expansion comer, such...

Consider the supersonic flow over an expansion comer, such as given in Fig. 9.20. The deflection angle $\theta=23.38^{\circ}$. If the flow upstream of the comer is given by $M_1=2, p_1=0.7 \mathrm{~atm}, T_1=630^{\circ} \mathrm{R}$, calculate $M_2, p_2, T_2, P_2, p_{0.7}$, and $T_{n, 2}$ downstream of the comer. Also, obtain the angles the forward and rearward Mach lines make with respect to the upstream direction.

Key Concepts

Stagnation Properties

Stagnation properties are the total thermodynamic properties (pressure, temperature, density) that the flow would attain if brought to rest isentropically. They represent the maximum energy state of the fluid and remain constant in an isentropic process. These properties are used as a reference for comparing the state of the flow before and after processes like expansion, ensuring energy consistency in the analysis.

Flow Deflection Angle

The deflection angle is the angle through which the flow is turned, such as when encountering a corner. In supersonic flows, a positive deflection (turning away from the original flow direction) results in an expansion process. The magnitude of this turning angle directly influences the change in flow properties and is accounted for through the Prandtl-Meyer function in determining the new Mach number and thermodynamic states.

Expansion Fan

An expansion fan is a series of continuous, centered expansion waves that occur when a supersonic flow undergoes a smooth, gradual turn away from itself. This process results in an increase in the Mach number and a decrease in pressure and temperature. The expansion fan is analyzed using the Prandtl-Meyer function and isentropic relations, and it sets the basis for determining post-expansion flow properties.

Mach Angle

The Mach angle is defined as the angle between the flow direction and the characteristic line (or Mach wave) in a supersonic flow. It is given by the inverse sine of one over the Mach number. This concept is crucial for determining the orientation of the Mach lines, which are used to trace the propagation of disturbances in the flow, especially in regions where expansion waves or shock waves occur.

Isentropic Flow Relations

Isentropic flow relations describe the connection between various flow properties (such as pressure, temperature, and density) assuming the process is adiabatic and reversible (i.e., no entropy change). These relations are used to compute how static and stagnation properties change when the flow accelerates or decelerates, such as through an expansion process where the flow experiences a pressure and temperature drop.

Prandtl-Meyer Function

The Prandtl-Meyer function relates the change in the Mach number to the turning (or expansion) angle in supersonic isentropic flows. It is used to determine the final Mach number after a flow undergoes an expansion fan, and it incorporates the effects of the specific heat ratio of the gas. This function is fundamental for analyzing flows over expansion corners or ramps where the flow turns away from itself.

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