Here’s the "Word" from the previous years when the course has been taught by n.k. There were 2 midterms each quarter then. One 6031 test is also attached because the courses have evolved.

1. The Pathfinder aircraft is described in Aviation Week & Space Technology, Sep.18, 1995, p. 67, in sufficient detail for you to analyze its aerodynamics.

a) Assume that the wing is rectangular and use lifting-line analysis to estimate the following:

1. Aircraft lift coefficient vs. angle of attack, assuming ideal lift curve slope for the airfoil.

2. Wing induced drag coefficient vs. angle of attack.

3. The drag (and thus the thrust required) to climb at 300fpm at 30,000 feet ISA, assuming an airspeed of 16 keas.

b) Try to design the wing shape to give the best lift/drag ratio at the 30,000 ft. flight condition. What climb rate can you get at this condition, assuming that the available power is the same as before?

c) With the aircraft of Part (a) flying at 20keas at 1,000 feet ISA, it encounters an updraft, which is in the shape of a rectangular vertical jet of air (same temperature as ISA 1,000 feet). The jet velocity varies sinusoidally from 0 at the edge to 10 m/s at its centerline, which is 500 meters from the edge.

    c-1. Calculate and plot the magnitude and direction of the acceleration vector experienced by the aircraft as a function of time, crossing the jet, if the aircraft controls are set.

   c- 2. Calculate CL as a function of time if the aircraft adjusts itself in real time to fly level through this jet.

1. a) A well-designed (optimized for long-endurance low-speed cruise) flying wing has an aspect ratio of 7, and a lift per unit span of 50N/m at midspan. It is cruising at 50 m/s at an altitude where air density is 1.0 kg/ m3. Estimate the lift per unit span at mid-semi-span. Also estimate the strength per meter of the vortex sheet directly downstream of mid-semi-span.

b) The velocity directly above a 2-D vortex sheet is 5 m/s, and below it is 3 m/s. After some time, an element of this sheet, 0.5m wide, rolled up into a line vortex. Find the strength of this vortex.

2. a) A flat-plate wing is flying at 3 deg. angle of attack, at 50 m/s. What is the normal velocity induced by its bound vortex system and wake at a point 75% chord from the leading edge?

b) Show that the elliptical spanwise lift distribution implies that downwash is constant along the span.

3. Plot the chordwise distribution of pressure coefficient along the upper and lower surfaces of an airfoil at small angle of attack in steady incompressible flow. Identify differences between a "viscous-flow" answer, a "full-potential" answer, and a "linearized potential" answer to this.

4. a) Prove that the integral on the r.h.s. of Kelvin's theorem vanishes if the flow is barotropic.

b) Show how to reduce the momentum equation in differential form to the Euler equation.

5. "The Laplace equation describes both steady and unsteady potential flow". Is this true in general, or in particular cases (which, if any?), or generally untrue? Please clarify: If you say it is true, please explain how this can be so: why bother studying unsteady flow specially? If it is untrue, then please explain why, because I swear I heard someone say this on the MARTA train just before they got off at Lindbergh (and I stayed on because I thought it was a Doraville train until it reached Dunwoody).

1. An NACA0008 airfoil of chord 1m is at a nominal angle of attack of 5 degrees to the steady freestream. The airfoil is executing a simple harmonic vertical translation with an amplitude of 0.05 times the chord, and a reduced frequency of 0.5. Find the lift coefficient and the pitching moment coefficient about the midchord and the phase lag between them. (35)

2. a) A wing of rectangular planform and aspect ratio 10 is operating in steady flow. The airfoil section is uniform from the root to mid-semi-span, and the same section continues from 3/4 semi-span to the tips. However, in the intervening distance, ailerons are deflected down on the right side, and up on the left side, by roughly the same amount. The ailerons are of rectangular planform, so that the circulation distribution on the wing changes abruptly at each spanwise edge of the aileron. Sketch the vortex system of this wing, and the wing circulation distribution. Make sure that the sense of rotation of each discrete vortex and vortex sheet is clearly identified. (15)

b) If you were trying to construct an unsteady-flow panel method for this configuration, and the ailerons were to be able to operate (rotate) rapidly, with a frequency f (cycles/sec.) and amplitude e degrees, show using diagram and mathematical formulae as appropriate, how you would construct the coordinate system, and write the boundary condition at the 3/4-chord point of the aileron, at the midspan of one aileron. (15)

3. Given the relation

where  = 0 at , and is singular at 

Write down the solution of the integral equation for  as a function of  ,  and . (15)

4. A symmetric airfoil of chord 1m is oscillating in pitch at a frequency of 30Hz with an amplitude of 2 degrees about a mean angle of attack of 5 degrees, in a freestream of 50m/s. Find the following:
(i) the pressure difference between points A and B. (ii) if the circulation on the airfoil is increasing by 50 m/s2 at the instant shown, find the value of vortex sheet strength at the trailing edge. (20)

1. A thin symmetric airfoil of 1m. chord undergoes simple harmonic pitch oscillations of 2-degree amplitude at 100Hz, about the midchord. The freestream speed is 100m/s and the mean angle of attack is 5 degrees. Find the lift coefficient as a function of time, and the time

delay between the airfoil motion and the airfoil response. (30)

2. For the airfoil of Problem 1, find the normal velocity at 3/4 chord if the pivot point is moved to quarter-chord. (20)

3. Given that you are dealing with a barotropic fluid, show that the Euler equation reduces to

(Hand-written equation: couldn’t find MSEquation Editor then) (20)

4. A thin 2-D flat plate of chord 0.5m is executing sinusoidal pitch oscillations in still air, with an amplitude of 5 degrees, pivoted about the quarterchord, and a frequency of 40 Hz. Find the additional torque required (also show its direction) because the motion is in air as opposed to vacuum (as calculated by linearized potential flow theory) and the phase relation of this torque with the resulting motion. Sketch (qualitatively) the torque and the motion history as functions of time on the same plot. (20)

5. Write (or derive) an expression for the normalwash in the frequency domain, for a surface executing small-amplitude simple harmonic motion. (10)

1. Convert the linearized potential equation for unsteady compressible subsonic flow into a form:

(Hand-written equation: couldn’t find MSEquation Editor then)

State the transformation to use, and other steps involved. What is in the parentheses on the right hand side? Separate this equation into two equations. What are these?

2. Write the general form of the integral equation which is to be solved in the process of computing the surface pressure distribution and lift of a thin finite wing at small angle of attack. State in words what the equation says. What is general about it?

3. A triangular thin flat-plate wing has 70-degree leading edge sweep and a root chord of 0.5m. It is placed at 4 degrees angle of attack in air at standard sea level moving at 25m/s. Estimate the total lift, the total induced drag, and the slopes of the lift curve and the induced drag curve at 4 degrees angle of attack.

If the airfoil is cambered, what will be the effect on the lift curve slope?

Estimate the pressure coefficient at mid-semi-span at the 40% chord from the apex, at 4 deg. angle of attack.

4. a) Prove that the elliptic spanwise distribution of circulation gives the best ratio of lift to induced drag on an unswept wing of high aspect ratio.

b) Prove that the elliptic distribution leads to a constant downwash distribution.

1. Shown below are the vortices trailing from a missile being tested in a wind tunnel. The tunnel roof is also shown. Locate the positions of these vortices a few feet downstream in the tunnel. Explain your answer.

2. You are asked to formulate and analyze two problems: The goal is to find the lift and its distribution along the chord in each case.

a) An 8% thick airfoil of chord 0.1m, whose zero-lift angle of attack is -2 deg., is placed at 5 deg. angle of attack in a steady incompressible air flow (density = 1.2kg/m³) at 30 m/s.

b) The same airfoil is shown at the same orientation in the same flowfield; however, it is shown at an instant, captured during the downward cycle of a rapid (100Hz) sinusoidal oscillation of 7 deg. amplitude.

Describe how each problem is to be analyzed. Discuss expected answers, and expected magnitude of the lift per unit span, and the shape of the lift distribution.

3. The point P is located at (0.7, 0.05). The surface at P is inclined at 3 deg. to the airfoil chord. The angle of attack is 2 deg. The freestream velocity 30m/s.

a) Write the boundary condition at P.

b) Write the boundary condition at P using the planar wing approximation.

4. The angle of attack history of a thin airfoil in incompressible flow is shown below. Plot the lift coefficient as a function of time. Explain.

1. Compare two delta wings, each of 60 deg. sweep. One has a sharp edge, while the other has a rounded edge. Sketch the difference in the cross-flow patterns at midchord, for

a) a = 5 degrees, M = 0.2

b) a = 10 degrees, M = 0.6

c) a = 45 deg., M = 3.

2. Why does vortex flow over an aircraft become asymmetric at high angles of attack?

3. Why does forebody vortex asymmetry cause rolling moment?

4. Why do simultaneous traces of the nosetip position and the rolling moment look uncorrelated?

5. A 70-deg. delta wing of 1m. chord is at 30 deg. angle of attack in a 30m/s air flow at sea-level standard conditions. Find the lift and drag using Betz cross-flow theory.

6. The 70-deg. delta wing mentioned above were taken to 5 deg. angle attack. Compare the lift and drag computed from slender body theory against that from Betz cross-flow theory.