Two successive convex corners turn a stream at \( M = 1.5 \) through \( 14.48 \) degrees and then \( 23.38 \) degrees. The final Mach number is
Answer: A
Turns simply add in the Prandtl-Meyer function, so \( \nu_3 = 11.90 + 14.48 + 23.38 = 49.76 \) degrees, giving Mach 3. Whether the turning is done in one corner or several makes no difference to the final state, since the whole process is isentropic.
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Assertion (A): An expansion in supersonic flow is spread through a fan rather than concentrated into a front.
Reason (R): A discontinuous expansion would require the entropy of the gas to decrease.
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Answer: B
Both are true and the second explains the first. The conservation equations formally permit an expansion discontinuity, but the second law forbids it, so nature spreads the expansion over an infinity of weak Mach waves instead. Compressions face no such prohibition, which is why they may coalesce into shocks.
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Assertion (A): Stagnation pressure is unchanged through a Prandtl-Meyer expansion.
Reason (R): Each wave in the fan is vanishingly weak, so the entropy rise through the fan is negligible.
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Answer: C
Both statements are true and the second is the reason for the first. Entropy rise scales with the cube of wave strength, so an infinity of infinitesimal waves still produces nothing measurable. This is the single sharpest contrast between an expansion and a shock.
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Assertion (A): An expansion fan opens out as the flow passes through it.
Reason (R): The Mach angle decreases as the flow accelerates.
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Answer: D
Both are true and the second explains the first. Each successive characteristic sits in faster flow and therefore lies at a smaller angle, so the family diverges. The compression case is the mirror image: characteristics converge and eventually coalesce into a shock.
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Assertion (A): An expansion fan reflects from a solid wall as a compression.
Reason (R): A solid wall requires the flow to remain tangent to it.
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Answer: A
The assertion is false and the reason is true. Tangency does have to be restored, but the incident expansion turned the flow away from the wall, so a further expansion is what turns it back. A solid wall reflects a wave in kind; it is a constant-pressure boundary that reverses its sense.
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Assertion (A): Shock diamonds form in the exhaust of a rocket whose nozzle is not perfectly expanded.
Reason (R): Waves reflecting from the constant-pressure plume boundary return with their sense reversed.
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Answer: B
Both statements are true and the second is the reason for the first. An expansion reflects from the free boundary as a compression and vice versa, so the plume oscillates between under- and over-pressure, producing the periodic bright regions. Viscous mixing eventually damps the pattern out.
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Assertion (A): The Prandtl-Meyer function makes successive turning angles simply additive.
Reason (R): The function is defined to be zero at Mach one.
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Answer: C
Both statements are true, but the second does not explain the first. Additivity follows from the differential relation between turning and Mach number that defines the function, and it would hold whatever constant of integration were chosen. Setting the value to zero at Mach one merely fixes a convenient datum.
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Assertion (A): The isentropic relations may be applied directly across a Prandtl-Meyer fan.
Reason (R): The expansion generates no significant entropy.
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Answer: D
Both are true and the second explains the first. With entropy constant, the ordinary stagnation-to-static relations hold on both sides of the fan and through it, so the downstream state follows from the downstream Mach number alone. Across a shock this would be quite wrong.
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Assertion (A): A sharp leading edge is used on a supersonic aerofoil.
Reason (R): A sharp edge allows the bow shock to remain attached and weak.
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Answer: A
Both statements are true and the second is the reason for the first. A small deflection can be turned by a weak attached oblique shock, whereas a blunt edge forces a detached bow wave with a strong nearly normal segment and a heavy wave drag penalty.
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Assertion (A): Turning a supersonic flow round a convex corner slows it down.
Reason (R): A convex corner turns the flow away from itself.
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Answer: B
The assertion is false and the reason is true. Turning away from itself is exactly what accelerates the flow: the corner produces an expansion, so the Mach number rises while pressure, temperature and density all fall. It is a concave corner that decelerates the flow, through a shock.
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