Consider a steady, isentropic, supersonic flow (\(M > 1\)) entering a convergent-divergent (CD) duct as shown. Which option correctly describes the flow at the throat?
Consider a steady, isentropic, supersonic flow (\(M > 1\)) entering a convergent-divergent (CD) duct as shown. Which option correctly describes the flow at the throat?
Show Hint
In any isentropic nozzle/duct, the minimum area is the \(A^*\) station and is always sonic. Supersonic or subsonic states occur only on sections with \(A > A^*\).
- Can only be supersonic
- Can only be sonic
- Can either be sonic or supersonic
- Can only be subsonic
The Correct Option is B
Solution and Explanation
Step 1: Area-Mach relation for isentropic quasi-1D flow.
\[ \frac{dA}{A}=(M^2-1)\frac{du}{u}. \] The minimum geometric area corresponds to the station where \(dA=0\), which requires \(M=1\) for an isentropic solution (otherwise \(du=0\) at a minimum area would be inconsistent with upstream and downstream states). Thus the station of minimum area is the sonic area \(A^{*}\).
Step 2: Implication for a CD nozzle with supersonic inlet.
A steady, isentropic solution passing through a minimum area must have \(M=1\) there, no matter whether the upstream is subsonic or supersonic. Hence at the throat the flow is necessarily sonic.
\[\boxed{\text{At the throat } M=1 \text{ (sonic only).}}\]
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- An aircraft is flying at an altitude of 4500 m above sea level, where the ambient pressure, temperature, and density are 57 kPa, 259 K, and 0.777 kg/m\(^3\), respectively. The speed of the aircraft \( V \) is 230 m/s. Gas constant \( R = 287 \, {J/kg/K} \), and specific heat ratio \( \gamma = 1.4 \). If the stagnation pressure is \( p_0 \), and static pressure is \( p \), the value of \[ \frac{p_0 - p}{\frac{1}{2} \rho V^2} \] is \_\_\_\_\_\_\_\_ (rounded off to two decimal places).
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