GATE Aerospace Engineering Aerodynamics Flashcards

24 sample cards from the Aerodynamics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. In thin airfoil theory, give the moment coefficient about the quarter chord for a cambered airfoil.

   $$c_{m,c/4} = \frac{\pi}{4}(A_2 - A_1)$$ It is independent of angle of attack, confirming the quarter chord as the aerodynamic center. For a symmetric airfoil $A_1=A_2=0$, so $c_{m,c/4}=0$.

2. Write the fundamental equation of thin airfoil theory (the camber-line boundary condition integral).

   $$\frac{1}{2\pi}\int_0^{c}\frac{\gamma(\xi)\,d\xi}{x-\xi} = V_\infty\left(\alpha - \frac{dz}{dx}\right)$$ where $\gamma$ is the vortex sheet strength along the chord and $\dfrac{dz}{dx}$ is the camber-line slope.

3. Why does a finite (3-D) wing experience induced drag while a 2-D airfoil does not?

   On a finite wing the pressure difference drives flow around the wingtips, creating trailing vortices that induce a downwash. The downwash tilts the local lift vector rearward, producing a streamwise force component called induced drag (drag due to lift).

4. How does downwash $w$ change the effective angle of attack of a wing section?

   The downwash reduces the angle: $$\alpha_{eff} = \alpha - \alpha_i$$ where the induced angle is $\alpha_i = \tan^{-1}\!\left(\dfrac{w}{V_\infty}\right)\approx \dfrac{w}{V_\infty}$ for small angles.

5. State the formula for the induced drag coefficient of a finite wing and define the span efficiency factor.

   $$C_{D,i} = \frac{C_L^{2}}{\pi e\, AR}$$ where $AR$ is the aspect ratio and $e$ ($\leq 1$) is the span (Oswald) efficiency factor. $e=1$ for an elliptical lift distribution (minimum induced drag).

6. Define aspect ratio and state how increasing it affects induced drag.

   $$AR = \frac{b^{2}}{S}$$ ( $b$ = span, $S$ = planform area). Higher aspect ratio reduces induced drag since $C_{D,i}\propto 1/AR$.

7. Write Prandtl's lifting-line (fundamental monoplane) equation.

   $$\alpha(y_0) = \frac{\Gamma(y_0)}{\pi V_\infty c(y_0)} + \alpha_{L=0}(y_0) + \frac{1}{4\pi V_\infty}\int_{-b/2}^{b/2}\frac{(d\Gamma/dy)\,dy}{y_0 - y}$$ It relates geometric angle of attack to the spanwise circulation distribution $\Gamma(y)$.

8. What lift distribution minimizes induced drag for a given lift and span, and what is its planform consequence?

   An elliptical lift (circulation) distribution gives the minimum induced drag, with $e=1$ and constant downwash across the span. It is produced by an elliptical planform (untwisted).

9. For an elliptical lift distribution, give the induced drag coefficient and constant downwash.

   Downwash is constant: $w = \dfrac{\Gamma_0}{2b}$, induced angle $\alpha_i = \dfrac{C_L}{\pi AR}$, and $$C_{D,i} = \frac{C_L^{2}}{\pi AR}$$ (i.e. $e=1$).

10. Define the critical Mach number $M_{cr}$.

    $M_{cr}$ is the freestream Mach number at which the local flow speed first reaches sonic ($M=1$) somewhere on the body (typically the point of maximum surface velocity / minimum pressure). Below it the flow is everywhere subsonic.

11. Define the drag-divergence Mach number and describe what happens there.

    The drag-divergence Mach number $M_{drag\text{-}divergence}$ (slightly above $M_{cr}$) is where the drag coefficient rises sharply due to the formation of shock waves and shock-induced flow separation (wave drag). It marks the onset of the transonic drag rise.

12. State the Prandtl–Glauert compressibility correction for the pressure coefficient in subsonic flow.

    $$C_p = \frac{C_{p,0}}{\sqrt{1 - M_\infty^{2}}}$$ where $C_{p,0}$ is the incompressible value. Likewise $c_l = \dfrac{c_{l,0}}{\sqrt{1-M_\infty^2}}$. Valid for thin bodies at small angles in subsonic flow.

13. What is the definition of the speed of sound and its formula for a calorically perfect gas?

    $$a = \sqrt{\left(\frac{\partial p}{\partial \rho}\right)_s} = \sqrt{\gamma R T}$$ It is the speed at which small pressure disturbances propagate through the gas; for air $\gamma=1.4$, $R=287\ \text{J/(kg·K)}$.

14. When is a flow generally treated as incompressible versus compressible, in terms of Mach number?

    Flow is usually treated as incompressible for $M < 0.3$ (density changes < ~5%). For $M \geq 0.3$ compressibility effects must be included. Regimes: subsonic, transonic ($M\approx 0.8\text{–}1.2$), supersonic ($M>1$), hypersonic ($M>5$).

15. Write the energy equation for adiabatic 1-D flow relating temperature to the stagnation temperature.

    $$c_p T + \frac{V^{2}}{2} = c_p T_0 = \text{const}$$ Equivalently $$\frac{T_0}{T} = 1 + \frac{\gamma-1}{2}M^{2}$$

16. Give the isentropic relations linking stagnation to static pressure and density in terms of Mach number.

    $$\frac{p_0}{p} = \left(1 + \frac{\gamma-1}{2}M^{2}\right)^{\frac{\gamma}{\gamma-1}}, \qquad \frac{\rho_0}{\rho} = \left(1 + \frac{\gamma-1}{2}M^{2}\right)^{\frac{1}{\gamma-1}}$$

17. State the isentropic process relations among $p$, $\rho$, and $T$.

    $$\frac{p_2}{p_1} = \left(\frac{\rho_2}{\rho_1}\right)^{\gamma} = \left(\frac{T_2}{T_1}\right)^{\frac{\gamma}{\gamma-1}}$$ i.e. $p/\rho^{\gamma} = \text{const}$ for an isentropic (adiabatic, reversible) process.

18. What are the stagnation-to-critical (sonic) property ratios for air ($\gamma=1.4$) at $M=1$?

    At $M=1$: $$\frac{T^{*}}{T_0}=\frac{2}{\gamma+1}=0.833,\quad \frac{p^{*}}{p_0}=\left(\frac{2}{\gamma+1}\right)^{\frac{\gamma}{\gamma-1}}=0.528,\quad \frac{\rho^{*}}{\rho_0}=\left(\frac{2}{\gamma+1}\right)^{\frac{1}{\gamma-1}}=0.634$$

19. Write the area–Mach number relation (area ratio) for isentropic 1-D flow in a variable-area duct.

    $$\left(\frac{A}{A^{*}}\right)^{2} = \frac{1}{M^{2}}\left[\frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^{2}\right)\right]^{\frac{\gamma+1}{\gamma-1}}$$ where $A^*$ is the sonic throat area.

20. In a converging–diverging (Laval) nozzle, how must area change to accelerate the flow in subsonic vs supersonic regimes?

    From $\dfrac{dA}{A} = (M^{2}-1)\dfrac{dV}{V}$: subsonic ($M<1$) needs a converging area to accelerate; supersonic ($M>1$) needs a diverging area to accelerate. Sonic flow ($M=1$) can only occur at the throat (minimum area).

21. Define Fanno flow and state its key assumptions.

    Fanno flow is steady, 1-D, adiabatic flow with friction in a constant-area duct, no work and no heat transfer. Key feature: stagnation temperature $T_0$ is constant; entropy increases due to friction.

22. In Fanno flow, in which direction does friction drive the Mach number for subsonic and supersonic inlets?

    Friction drives the flow toward $M=1$ (Fanno limit): subsonic flow accelerates ($M\uparrow$ toward 1), supersonic flow decelerates ($M\downarrow$ toward 1). $M=1$ cannot be crossed; choking occurs at the duct exit.

23. What happens to entropy and stagnation pressure along a Fanno line, and at what point is entropy maximum?

    Entropy increases in the flow direction (friction is irreversible) and stagnation pressure $p_0$ decreases. Entropy is maximum at the sonic point $M=1$, which is the tip of the Fanno curve on the T–s diagram.

24. Write the Fanno-flow friction-length relation $\dfrac{4\bar{f}L^{*}}{D}$ in terms of Mach number.

    $$\frac{4\bar f L^{*}}{D} = \frac{1-M^{2}}{\gamma M^{2}} + \frac{\gamma+1}{2\gamma}\ln\!\left[\frac{(\gamma+1)M^{2}}{2\left(1+\frac{\gamma-1}{2}M^{2}\right)}\right]$$ where $L^{*}$ is the duct length needed to reach $M=1$.

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