Category Theoretical and Applied Aerodynamics

From the result derived in class

find the distribution of induced velocity.

Sketch the distribution ww [y(t)] in terms of y or t. Show that the wing experiences upwash.

14.10.2.3 Induced Drag

Calculate the induced drag of the wing without tip vortices and compare with the elliptic loading.

What is the percent change in induced drag?

What is the corresponding efficiency factor e?

14.10.2.1 Lifting Line Theory

In Lifting Line theory, explain the origin of downwash.

In terms of drag, what is the difference between downwash and upwash?

Is there a net drag benefit to design a wing with upwash?

14.10.2.2 Design of a Wing

The circulation distribution is represented by the Fourier series

Г[y(t)] = 2U&£~1 An sin nt

b

Design the simplest wing (meaning the wing with the least number of non-zero Fourier coefficients) with zero strength tip vortices, in order to minimize risks for airplanes caught in the wake of the wing. To achieve this goal, combine modes 1 and 3 in such a way that

at y = ±­7 2

Note: you can impose

d Г [y(t)] n _ n л = 0, at t = 0 and n dt

to find a relation between A1 and A3(verify your result as the rest depends on it), then verify that

Use the following identity: cos 3t = cos t (4 cos21 — 3).

Sketch the circulation distribution Г[y(t)] in terms of y or t. Use the following identity: sin3t = sin t (4 cos21 — 1).

14.10.1.1 Incompressible Flow (M0 = 0)

Profile Camber Estimation

A wing profile lift curve, calculated with a numerical method, gives a value of the lift coefficient to be (Ci)profiie = 2.1878 at a = 4°. Using the result of thin airfoil theory, find all the Fourier coefficients of a thin parabolic plate equivalent to this airfoil. Estimate the relative camber dm/c of the thin parabolic plate. Check your result as the rest depends on it.

Take-Off Incidence

For this equivalent thin parabolic plate, find the incidence in deg. for which the lift coefficient is Cl = 2.5.

Nose Pitching Moment Coefficient

Give the pitching moment coefficient Cm, o(a) for the parabolic plate. Predict the nose pitching moment coefficient of the wing profile (Cm, o)projile at a = 4°. Compare with the calculated result of -0.935.

Aerodynamic Center Pitching Moment Coefficient

Give the definition of the aerodynamic center. Give the pitching moment coefficient about the aerodynamic center Cm, a.c..

14.10.1.2 Supersonic Flow (M0 > 1, в = ^M(J — 1)

Consider the thin parabolic plate of equation

X

d (x ) = 4dm —

c

where the relative camber is given to be dm/c = 0.14.

Lift Coefficient

Give the lift coefficient C; (a) for this airfoil.

Drag Coefficient

Calculate the drag coefficient (Cd )a=0 and give the expression of Cd (a) for this airfoil.

Pitching Moment Coefficient

If the profile is allowed to rotate freely about an axis placed at the quarter-chord, find the equilibrium incidence aeq in deg. (Hint: Use the change of moment formula to evaluate Cmf/4). How would you qualify the equilibrium situation: stable, unstable, neutral?

The AMAT11 has a rectangular main wing with span bm = 2.1m and constant chord cxm = 0.3 m. The tail is also rectangular with span bt = 1.0 m and chord cxt = 0.3 m. The equilibrium code provides the aircraft aerodynamic characteristics and a maximum take-off mass M = 19 kg. The reference area is Aref = Am + At.

14.9.3.1

Aspect Ratios of Lifting Elements—Global Lift Slope

give the global lift slope dCL /da for the wing+tail configuration.

14.9.3.2 Airplane Center of Gravity

The aerodynamic center is located at xa. c./lref = 0.3.

Find the center of gravity xc. g./ lref, given a 6 % static margin (SM).

14.9.3.3 Equilibrium Condition and Static Stability

The equilibrium code calculates the linear model for lift and moment coefficients for the complete configuration, at low incidences, to be:

CL(a, tf) = 3.955a + 0.984tf + 0.712
CM, o(a, tf) = -1.188a – 0.907tf – 0.008

Derive the moment coefficient at the center of gravity, CM, c.g.(a, tf) and write the condition for equilibrium. Verify your result as the rest depends on it. Is the equilibrium stable?

Solve for aeq (tf).

14.9.3.4 Take-Off Conditions

The take-off speed of U = 13.11 m/s is obtained for tf = 9.2°. Find aeq at take-off.

Find the lift coefficient of the tail at take-off : the tail aerodynamic lift curve is given by

CLt = 2.696a + 2.705tf – 0.347

Calculate the force on the tail in (N), given that p = 1.2 kg/m3. Is the force up or down?

14.10 Problem 10

The WWII Spitfire was designed with an elliptic wing of span b = 11.2m and wing area S = 22.5 m2. It is equipped with a NACA2209.4 profile of 2 % relative camber (d/c = const = 0.02). The top speed in cruise is V = 170 m/s (378mph) with a take-off mass of M = 3,000 kg.

14.9.2.1 Vortex Sheet

Explain succinctly the key features of the vortex sheet (physically and mathemati­cally) and the effect it has on the flow past a large aspect ratio wing (lifting line).

14.9.2.2 Lift Coefficient

Given the above data, calculate the lift coefficient CL at top speed in cruise (take p = 1.2kg/m3 and g = 9.81 m/s2).

14.9.2.3 Elliptic Loading

Assuming an elliptic planform, a constant relative camber and zero twist, calculate the induced drag CDi and the first mode amplitude A i in the Fourier Series expansion of the circulation

Г[y(ff)] = 2Ub ~1 An sin nd
y(Q) = —§ cos в, 0 < в < п

Write the equation for the lift coefficient CL in terms of aspect ratio AR, geometric incidence a and relative camber d/c.

Find a, the geometric incidence in cruise.

14.9.2.4 Added Twist

The designer of the Spitfire added “washout” (—2.5° of twist between root and tip) so that the local lift coefficient is larger at the root than at the tip, to avoid tip stall. This corresponds to adding only the third mode with an amplitude A3 = —0.002. (A2 = A4 = A5 = … An = 0, n > 4).

Calculate the induced drag (CDi )waShout of the wing with twist.

What is the percent change in induced drag?

What is the corresponding Efficiency factor e?

14.9.1.1 Incompressible Flow (M0 = 0)

Aerodynamic Center

Give the definition of the aerodynamic center.

Where is the aerodynamic center located for thin airfoils at low speeds?

Second Mode only Airfoil

Consider a thin cambered plate such that the vorticity distribution is given by the second mode with A2 > 0 as

/ Г'[x(t)]=2U {A01+ft + A2 cos 2t}
x (t) = 2 (1 – cos t), 0 < t < n

Use the formula for d'[x(t)] to find A0 and find the incidence of adaptation for this airfoil (Hint: use the identity cos 2t = 2 cos21 – 1, and integrate in x: d'(x)dx from zero to c; or in t: d'[x(t)]dxdt from zero to n.)

Eliminate A0 and sketch the slope of the cambered plate along with the profile itself.

Aerodynamic Coefficients

Give the expression of C;(a), Cm, o(a), Cm, a.c. and Cd for this airfoil.

Moment About an Axis

Calculate the aerodynamic moment about the mid-chord, Cm, c/2 (Hint: use the change of moment formula.)

If the profile is allowed to rotate without friction about an axis located at mid­chord, find the equilibrium incidence, aeq, if only aerodynamic forces and moment are present.

Is the equilibrium stable (Answer by Yes or No)?

14.9.1.2 Supersonic Flow (M0 > 1, в = ^M^ — 1)

Consider the cubic plate of equation

4 x x x

d (x) = – Ac 1 – 2 1 – , A > 0

3 c V c c)

The slope is given by

. 4 ( x x2

d (x) = 3 A 1 — 6 —+ 6 —2

This plate equips the fins of a supersonic rocket.

Pressure Distribution and Flow Features

Calculate and plot — C + and —C— versus x for this airfoil at a = 0. Sketch the flow at a = 0 (shocks, characteristic lines, expansion shocks).

Moment Coefficient

At a = 0 , calculate the moment coefficient (Cm,0) 0 and give the expression of

Cm, o(a).

Use the change of moment formula to evaluate Cm, c/2 and discuss whether or not there is an equilibrium about a mid-chord axis and why. How would you qualify this situation: stable, unstable, neutral?

The AMAT10 has a rectangular main wing with span bm = 3.6 m and constant chord cxm = 0.35 m. The tail is also defined with a 33% moving flap. The equilibrium code provides the aircraft characteristics and a maximum take-off mass M = 24 kg.

14.8.3.1 Airplane Aerodynamic Center and Static Margin

The equilibrium code calculates the lift and moment coefficients for the complete configuration at low incidences to be:

CL(a, tf) = 4.479a + 0.808tf + 0.9314
Cm, o (a, tf) = -1.469a – 0.7479tf – 0.1565

where a is the geometric incidence (in radians, measured from the fuselage axis) and tf is the tail flap setting angle (in radians). Cm, o is the aerodynamic moment about the origin of the coordinate system (located at the nose O). We will use this linear model for simplicity.

The center of gravity is located at xcg/lref = 0.268.

Find the aerodynamic center and the static margin SM in % of lref.

14.8.3.2 Equilibrium Condition and Static Stability

Derive the moment coefficient at the center of gravity, CM, c.g.(a, tf) and write the condition for equilibrium. Verify your result as the rest depends on it. Is the equilib­rium stable?

Solve for a(tf).

14.8.3.3 Take-Off Conditions

The take-off speed of U = 11.49 m/s is obtained for tf = 7.13° .Find aeq at take-off.

Find the lift coefficient of the tail at take-off : the tail aerodynamic lift curve is given by

CLt = 3.032a + 2.886tf – 0.3259

Calculate the force on the tail in (N), given that the tail reference area is At = 0.49 m2, p = 1.2kg/m3.

Is the force up or down?

14.9 Problem 9

14.8.2.1 Circulation Representation

A paper airplane wing is made with manila folder by cutting an ellipse of large aspect ratio (b, c0). Then the wing is given some small camber such that the relative camber is constant (d/c = const) along the span, and there is no twist, t (y) = 0. The circulation for an arbitrary wing is represented by a Fourier series

Г[y(0)] = 2Ub “=1 An sin n0
y(0) = -—b cos 0, 0 < 0 < n

The chord distribution of the wing can be expressed as c[y(0)] = c0sin0 with the above change of variable.

Show that the circulation is given by the first mode only (Hint: use Prandtl Integro- differential equation to prove it).

What is the relationship between the root circulation Г0, the incidence a, the constant induced incidence a and the relative camber d/c?

14.8.2.2 Ideal Angle of Attack

From the previous result, calculate the lift coefficient, CL = Ci that corresponds to the geometric incidence, aideal, that will make the effective incidence zero, i. e. aeff = aideal + ai = 0. (Hint: first find Г0 in this case and use the relationship between Г0 and Cl to eliminate Г0).

Find the corresponding value of aideal in term of the wing aspect ratio, AR, and d/c, given that the lift of this ideal wing is

Make a sketch of the flow in a cross section of the wing (for example the wing root).

14.8.2.3 Including Twist

The paper wing is actually slightly warped with a linear twist t (y) = —2tx y/b, where tx is a small positive number that represents the tip twist.

Show that this linear twist is represented by the second mode in the Fourier series. Use Prandtl Integro-differential equation to find A2 in terms of tx. (Hint: use the identity sin(20) = 2sin0cos0).

14.8.2.4 Efficiency Factor

The warped wing loading is represented by the first two modes in the Fourier series.

Given that A2 = A1 /10, calculate the induced drag of the warped wing and compare it with that of the ideal (untwisted) wing.

What is the percentage of increase of the induced drag?

Calculate the efficiency factor e.

14.8.1.1 Incompressible Flow (M0 = 0)

Profile Geometry

Consider a half-double wedge profile of chord c of equation

where © is the wedge angle, see Fig. 14.3.

Calculate the distributions of camber d(x) and thickness e(x) for this profile. Check your result.

Fig. 14.3 Half double-wedge geometry

Fourier Coefficients

Give the expressions of the Fourier coefficients A0 and An in the expansion of the vorticity for an arbitrary profile.

Calculate the Fourier coefficients A0, A1 and A2 for this thin profile (Hint: you

c

need to split the integral into two pieces J02 + Jc).

Give the incidence of adaptation aadapt.

Sketch the flow at the incidence of adaptation, showing in particular the stream­lines near the leading and trailing edges.

Definition of Aerodynamic Center

Give the definition of the aerodynamic center.

Aerodynamic Coefficients

Give the aerodynamic coefficients C;(a) and Cm, o (a).

14.8.1.2 Supersonic Flow (M0 > 1, в = yjM( — 1)

The same profile equips the wing of an airplane cruising at Mach number M0 > 1 in a uniform atmosphere.

Pressure Distribution and Flow Features

Calculate and plot —C + and – C— versus x for this airfoil at a = 0. Sketch the flow at a = 0 (shocks, characteristic lines, expansion shocks).

Aerodynamic Coefficients

At a = 0 , calculate the drag and moment coefficients (Cd)a=0 and (Cm, o) 0

Static Equilibrium About an Axis

If an axis is located at the leading edge, x = 0, find the equilibrium angle aeq if there are no forces other than the aerodynamic forces. Is the equilibrium stable, unstable, neutral? (Hint: calculate Cm, o (a)).

The best design for the AMAT09 has a rectangular main wing with span bm = 3.1m and constant chord cxm = 0.55 m. The lifting tail is also defined with a 33% moving flap. The equilibrium code provides the aircraft characteristics and a maximum take­off mass M = 30 kg.

14.7.3.1 Airplane Aerodynamic Center and Static Margin

The equilibrium code calculates the lift and moment coefficients for the complete configuration at low incidences to be:

Cl (a, tt) = 3.912a + 0.626tt + 0.844
CM, o(a, tt) = -1.124a – 0.625tt – 0.138

where a is the geometric incidence (in radians, measured from the fuselage axis) and tt is the tail setting angle (in radians). CM, o is the aerodynamic moment about the origin of the coordinate system (located at the nose O). We will use this linear model, even for take-off conditions.

The center of gravity is located at xcg / lref = 0.227. Find the aerodynamic center and the static margin SM in % of lref.

14.7.3.2 Equilibrium Condition and Static Stability

Derive the moment coefficient at the center of gravity, CM, c.g.(a, tt) and write the condition for equilibrium. Verify your result as the rest depends on it. Is the equilib­rium stable?

Solve for a(tt).

14.7.3.3 Top Speed

The top speed is obtained for tt = 7.6°. Find aeq at top speed.

Find the top speed, given that Aref = 2.225 m2, p = 1.2kg/m3 (Hint: use the equilibrium equation for horizontal flight).

14.7 Problem 8