Level 1 — RecognitionCompressible Flow & Aerodynamics

Compressible Flow & Aerodynamics

⏱ 20 minutes30 marksprintable — key stays hidden on paper

Time limit: 20 minutes
Total marks: 30
Instructions: Answer all questions. For True/False, a justification is required for full marks. Use $\gamma = 1.4$ for air unless stated.

Section A — Multiple Choice (1 mark each, 12 marks)

Q1. The speed of sound in a perfect gas is given by:

(a) $a = \sqrt{RT}$
(b) $a = \sqrt{\gamma R T}$
(c) $a = \gamma R T$
(d) $a = \sqrt{\gamma P \rho}$

Q2. A flow with Mach number $M = 6.5$ is classified as:

(a) subsonic
(b) transonic
(c) supersonic
(d) hypersonic

Q3. The area–velocity relation is $\dfrac{dA}{A} = (M^2 - 1)\dfrac{dV}{V}$ . For a supersonic flow to accelerate ( $dV>0$ ), the area must:

(a) decrease
(b) increase
(c) remain constant
(d) first increase then decrease

Q4. At the throat of a choked converging–diverging nozzle, the Mach number is:

(a) 0
(b) less than 1
(c) exactly 1
(d) exactly $\gamma$

Q5. Across a normal shock wave, which quantity remains constant?

(a) static pressure
(b) stagnation temperature
(c) stagnation pressure
(d) Mach number

Q6. The stagnation temperature ratio for isentropic flow is:

(a) $\dfrac{T_0}{T} = 1 + \dfrac{\gamma-1}{2}M^2$
(b) $\dfrac{T_0}{T} = 1 + \gamma M^2$
(c) $\dfrac{T_0}{T} = \left(1 + \dfrac{\gamma-1}{2}M^2\right)^{\gamma/(\gamma-1)}$
(d) $\dfrac{T_0}{T} = M^2 - 1$

Q7. Induced drag on a finite wing is most strongly reduced by:

(a) decreasing the aspect ratio
(b) increasing the aspect ratio
(c) increasing camber only
(d) decreasing chord only

Q8. Prandtl–Meyer expansion waves occur when a supersonic flow turns:

(a) into itself (compression corner)
(b) away from itself (expansion corner)
(c) at exactly Mach 1
(d) only in subsonic flow

Q9. The critical Mach number $M_{cr}$ corresponds to the free-stream Mach number at which:

(a) the flow first becomes hypersonic everywhere
(b) local flow first reaches $M = 1$ somewhere on the body
(c) a normal shock detaches
(d) lift becomes zero

Q10. Whitcomb's area rule is used to reduce:

(a) induced drag at low speed
(b) skin-friction drag
(c) transonic wave drag
(d) aerodynamic heating

Q11. For thin airfoil theory of a symmetric airfoil, the lift coefficient slope $dC_L/d\alpha$ is approximately:

(a) $\pi$ per radian
(b) $2\pi$ per radian
(c) $\pi/2$ per radian
(d) $4\pi$ per radian

Q12. A nozzle in which the exit pressure is below the ambient (back) pressure is called:

(a) perfectly expanded
(b) under-expanded
(c) over-expanded
(d) choked

Section B — Matching (6 marks)

Q13. Match each term in Column A with its correct description in Column B. (1 mark each)

Column A	Column B
(i) Detached bow shock	(P) angle between the shock and upstream flow direction
(ii) Shock (wave) angle $\beta$	(Q) forms ahead of a blunt body in supersonic flow
(iii) Deflection angle $\theta$	(R) governs area–Mach relation $A/A^*=f(M)$
(iv) Chord	(S) angle the flow is turned through by an oblique shock
(v) Isentropic relation	(T) straight line from airfoil leading edge to trailing edge
(vi) Recovery temperature	(U) temperature felt by a surface in high-speed viscous flow

Section C — True / False with Justification (2 marks each, 12 marks)

(1 mark correct T/F, 1 mark justification)

Q14. In a converging (only) nozzle fed from a reservoir, the exit flow can be accelerated to supersonic speeds. (T/F + justify)

Q15. Stagnation (total) temperature is conserved across a normal shock in a calorically perfect, adiabatic flow. (T/F + justify)

Q16. A Prandtl–Meyer expansion fan is an isentropic process. (T/F + justify)

Q17. Increasing the aspect ratio of a wing increases the induced drag. (T/F + justify)

Q18. For steady adiabatic flow through an open system with no shaft work and negligible potential energy, the stagnation enthalpy $h_0 = h + V^2/2$ is constant along the flow. (T/F + justify)

Q19. Downstream of a normal shock the flow is always supersonic. (T/F + justify)

Answer keyMark scheme & solutions

Section A (1 mark each)

Q1 — (b) $a=\sqrt{\gamma R T}$ .
Why: Sound propagation is an isentropic small disturbance; $a^2 = (\partial P/\partial \rho)_s = \gamma RT$ for a perfect gas.

Q2 — (d) hypersonic. $M = 6.5 > 5$ , high-temperature/real-gas effects dominate.

Q3 — (b) increase. With $M>1$ , $(M^2-1)>0$ , so $dV>0 \Rightarrow dA>0$ . This is why the diverging section of a de Laval nozzle accelerates supersonic flow.

Q4 — (c) exactly 1. Choking occurs when the throat reaches sonic conditions; maximum mass flow is fixed.

Q5 — (b) stagnation temperature. Adiabatic flow ⇒ $T_0$ constant; $P_0$ drops (entropy rise), static $P$ rises, $M$ decreases.

Q6 — (a). $T_0/T = 1 + \tfrac{\gamma-1}{2}M^2$ from $h_0 = h + V^2/2$ . Option (c) is $P_0/P$ .

Q7 — (b) increasing aspect ratio. Induced drag coefficient $C_{D,i} = C_L^2/(\pi e\,AR)$ ; larger $AR$ ⇒ smaller induced drag.

Q8 — (b) away from itself. Expansion corner turns flow away, expanding it isentropically (accelerating, dropping pressure).

Q9 — (b). $M_{cr}$ : free-stream Mach at which the peak local Mach first equals 1.

Q10 — (c) transonic wave drag. Smooth cross-sectional area distribution reduces shock-induced wave drag.

Q11 — (b) $2\pi$ per radian. Thin airfoil theory: $C_L = 2\pi\alpha$ .

Q12 — (b) under-expanded. Exit pressure > back pressure; flow expands further outside the nozzle.

Q13:
(i) → (Q) detached bow shock forms ahead of blunt body
(ii) → (P) shock angle between shock and upstream flow
(iii) → (S) deflection angle = flow turning by oblique shock
(iv) → (T) chord = LE to TE line
(v) → (R) isentropic relation gives $A/A^*=f(M)$
(vi) → (U) recovery temperature felt by surface

Section C (2 marks each: 1 T/F, 1 justification)

Q14 — FALSE. A converging nozzle can accelerate subsonic flow at most to $M=1$ at the exit (choked). Supersonic flow requires a diverging section (area–velocity relation). (1+1)

Q15 — TRUE. Energy equation for adiabatic flow gives $h_0$ constant; for calorically perfect gas $h_0 = c_p T_0$ , so $T_0$ is unchanged even though $P_0$ falls. (1+1)

Q16 — TRUE. Expansion occurs through infinitesimal Mach waves; entropy change per wave is second-order and sums to zero ⇒ isentropic. (1+1)

Q17 — FALSE. $C_{D,i}\propto 1/AR$ ; increasing $AR$ decreases induced drag. (1+1)

Q18 — TRUE. First law for a steady open system with $\dot Q=0$ , $\dot W_s=0$ , $\Delta PE\approx0$ gives $h + V^2/2 = h_0 =$ const. (1+1)

Q19 — FALSE. A normal shock always makes the downstream flow subsonic ( $M_2<1$ ). (1+1)

[
  {"claim":"Speed of sound in air at 288K is ~340 m/s (a=sqrt(gamma R T))","code":"import sympy as sp; g=sp.Rational(7,5); R=287; T=288; a=sp.sqrt(g*R*T); result = abs(float(a)-340.17) < 1.5"},
  {"claim":"T0/T at M=2 with gamma=1.4 equals 1.8","code":"g=sp.Rational(7,5); M=2; ratio=1+(g-1)/2*M**2; result = sp.simplify(ratio-sp.Rational(9,5))==0"},
  {"claim":"Thin airfoil symmetric CL slope is 2*pi per rad","code":"a=2*sp.pi; result = sp.simplify(a-2*sp.pi)==0"},
  {"claim":"Downstream Mach of normal shock at M1=2, gamma=1.4 is subsonic (~0.577)","code":"g=sp.Rational(7,5); M1=2; M2sq=(1+(g-1)/2*M1**2)/(g*M1**2-(g-1)/2); M2=sp.sqrt(M2sq); result = float(M2) < 1 and abs(float(M2)-0.5774)<0.001"}
]

Section A — Multiple Choice (1 mark each, 12 marks)

Section B — Matching (6 marks)

Section C — True / False with Justification (2 marks each, 12 marks)

Section A (1 mark each)

Section B (1 mark each)

Section C (2 marks each: 1 T/F, 1 justification)