Incompressible Flow over Airfoils Questions and Answers

Answer: D

The airfoil is NACA 2512.

In the NACA four-digit series, the first digit is the maximum camber as a percentage of chord, the second digit is the position of maximum camber in tenths of chord from the leading edge, and the last two digits are the thickness as a percentage of chord.

NACA 2512 has its maximum camber of 2% at 5/10 = 0.5 of the chord, which is half chord, and is 12% thick. NACA 5212, 1225 and 2215 all have 2 as their second digit, so their maximum camber is at 0.2 of the chord, not at half chord.

Answer: B

The angular locations are ±30° from the front stagnation point.

Inviscid flow over a circular cylinder gives a surface speed \(2U\sin\theta\), measured from the front stagnation point, and so a pressure coefficient

\[ C_{p} = 1 - 4\sin^{2}\theta \]

The static pressure equals the free-stream pressure where \(C_{p} = 0\): \(\sin\theta = 1/2\), so \(\theta = \pm 30^{\circ}\) on the front of the cylinder (and \(\pm 150^{\circ}\) at the rear, which the options do not offer).

At 0° the pressure is the stagnation value, \(C_{p} = 1\). At ±60°, \(C_{p} = 1 - 4(0.75) = -2\), well below free-stream pressure. ±38° does not satisfy the inviscid result that the key follows.

Answer: C

The pressure coefficient will be about -1.5.

The Prandtl-Glauert rule scales a thin airfoil's pressure coefficient with Mach number: \(C_{p} = C_{p,0}/\sqrt{1 - M^{2}}\), where \(C_{p,0}\) is the incompressible value. First recover \(C_{p,0}\) from the reading at M = 0.1, then scale it to M = 0.6:

\[ \begin{aligned} C_{p,0} &= -1.2\sqrt{1 - 0.01} \approx -1.194 \\ C_{p} &= \frac{-1.194}{\sqrt{1 - 0.36}} \\ &= \frac{-1.194}{0.8} \\ &\approx -1.49 \end{aligned} \]

At M = 0.1 the correction is negligible, so the answer is essentially -1.2 divided by 0.8. -0.75 multiplies by a factor less than 1 instead, as if compressibility weakened the suction; it strengthens it. -2.9 and -3.5 are far larger than any compressibility correction gives at M = 0.6 (even the Karman-Tsien rule gives only about -1.75).

Answer: D

A clockwise circulation, \(\Gamma \lt 0\), must be added.

In the sketch, the flow from below the airfoil wraps around the sharp trailing edge and the rear stagnation point sits on the upper surface ahead of it. This is the zero-circulation potential flow, which predicts no lift and infinite velocity at the trailing edge. The Kutta condition requires the flow to leave the sharp trailing edge smoothly, with the rear stagnation point at the trailing edge itself.

With the free stream from left to right, a clockwise vortex speeds up the flow over the upper surface and slows it underneath, moving the rear stagnation point back along the upper surface until it reaches the trailing edge. That clockwise circulation is what produces positive lift by the Kutta-Joukowski theorem.

A counter-clockwise circulation would move the stagnation point the wrong way and give negative lift. A source or sink adds or removes mass, changing the thickness of the body the flow sees, but it cannot fix the flow at the trailing edge or create lift.

Answer: C

The correct option is \(C_{P_{I}} = 1,\ C_{P_{II}} \lt 1\).

The Kutta condition requires the flow to leave the trailing edge smoothly. With a finite trailing-edge angle the upper and lower surfaces point in different directions there, so the only way both streams can leave with one velocity is for that velocity to be zero: the trailing edge is a stagnation point and

\[ C_{P} = 1 - \left(\frac{V}{U}\right)^{2} = 1 \]

At a cusped trailing edge the two surfaces are tangent, so the flow can leave with a common, finite, non-zero velocity, and \(C_{P} \lt 1\).

So the airfoils differ: the options giving both the same value, or reversing them, are wrong.

Answer: B

The maximum camber is at 0.40 of the chord.

In the NACA four-digit series the digits are read as M P XX: the first digit is the maximum camber in per cent of chord, the second is its position in tenths of chord from the leading edge, and the last two are the maximum thickness in per cent of chord. For NACA 4415: maximum camber 4% (0.04c) located at 0.4c, with 15% thickness.

0.04 is the camber itself, not its location, and 0.15 is the thickness ratio. 0.44 simply runs the first two digits together and has no meaning in the designation.