At a static temperature of \( 250 \) K, the flow speed corresponding to \( M = 0.3 \) is nearest to
Answer: C
First the speed of sound: \( \sqrt{1.4 \times 287 \times 250} = \sqrt{100\,450} = 316.9 \) m/s. Then \( 0.3 \times 316.9 = 95.1 \) m/s. The colder air gives a lower threshold speed than the 102 m/s that applies at sea level, so compressibility sets in at a lower true airspeed at altitude.
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The speed of sound may also be written in terms of the isentropic bulk modulus as
Answer: D
The general result for any medium is the square root of the isentropic bulk modulus over the density. For a perfect gas the bulk modulus is the specific heat ratio times the pressure, and substituting the equation of state recovers the familiar temperature-only form. The general expression is what applies to liquids and solids.
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The continuity equation for steady one-dimensional compressible flow states that the constant quantity is
Answer: A
Mass conservation fixes the product of density, velocity and cross-sectional area. In incompressible flow the density drops out and the familiar velocity-area product survives, but retaining the density is essential in compressible flow and is exactly what produces the area-velocity relation and its reversal at Mach one.
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A choked orifice is used to meter gas flow in industry chiefly because
Answer: B
Once the orifice is choked, the flow is fixed by the upstream stagnation state and the throat area, so fluctuations downstream cannot disturb it. That insensitivity makes it a stable and repeatable metering device, and the same principle governs rocket motor throats and gas turbine nozzle guide vanes.
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Assertion (A): A Mach wave may be treated as isentropic while a finite shock may not.
Reason (R): The property changes across a Mach wave are infinitesimal, so the entropy rise is of second order.
Select the correct answer from the options below.
Answer: C
Both statements are true and the second explains the first. Entropy rise across a wave scales with the cube of its strength, so for a vanishingly weak wave it is negligible. This is why expansion fans, built from Mach waves, are isentropic while compression shocks of finite strength are not.
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Assertion (A): A sonic boom is heard only at the moment an aircraft passes through Mach one.
Reason (R): An aircraft in steady supersonic flight trails a shock system continuously.
Select the correct answer from the options below.
Answer: D
The assertion is false and the reason is true, and the reason is what refutes it. The shock system persists for the whole of the supersonic flight, sweeping across the ground as a continuous carpet. Every observer along the track hears a boom as it passes, not just those beneath the point of acceleration.
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Assertion (A): An isentropic analysis is inappropriate for flow through a strong shock wave.
Reason (R): A shock wave raises the entropy of the gas passing through it.
Select the correct answer from the options below.
Answer: A
Both are true and the second explains the first. Isentropic relations assume constant entropy, which a shock plainly violates, so the Rankine-Hugoniot relations must be used across it instead. Isentropic relations remain perfectly good on either side of the shock, just not through it.
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Consider the following statements about the assumptions behind the isentropic relations:
1. The flow is adiabatic.
2. The flow is reversible, so no shocks or friction are present.
3. The gas is perfect, with constant specific heats.
Which of the statements given above is or are correct?
Answer: B
All three are required. Dropping the adiabatic condition or admitting irreversibility breaks the constant-entropy assumption, and allowing the specific heats to vary, as they do at high temperature, changes the exponents in the relations themselves. Real-gas effects at hypersonic speeds break exactly this last assumption.
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Which of the following flows may NOT be treated with the isentropic relations?
Answer: C
A shock raises entropy, so the constant-entropy assumption fails through it and the Rankine-Hugoniot relations are needed instead. The other three are all isentropic to a good approximation, an expansion fan notably so because it is built from vanishingly weak Mach waves.
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A supercritical aerofoil section delays drag divergence chiefly by
Answer: D
The relatively flat upper surface keeps the local supersonic flow at a modest Mach number over a long run, so the shock closing the pocket is weak and the wave drag and shock-induced separation are both reduced. It does not prevent supersonic flow; it manages it, and the aft camber recovers the lift lost from the flattened crest.
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