Category Theoretical and Applied Aerodynamics

14.7.2.1 Vortex Sheet Characteristics

Explain why the perturbation velocity components u and w are continuous across the vortex sheet behind a finite wing, i. e. < u >=< w >= 0.

How do you explain the existence of induced drag for a wing in incompressible, inviscid flow?

14.7.2.2 Circulation Representation

The circulation is represented by a Fourier series

■ Г[y(t)] = 2Ub Z“i An sin nt

y(t) = —§ cos t, 0 < t < n

Sketch the first three modes with unit coefficients A5 = A2 = A3 = 1.

14.7.2.3 Efficiency Factor

Give the definition of the Efficiency Factor e in terms of the Fourier coefficients.

The AMAT09 wing has been designed with a rectangular planform, b = 3.1m, c = 0.55 m, and is equipped with the SS1707-0723 double-element airfoil with high lift (Cl)max = 2.7. Prandtl Lifting Line theory indicates that the Efficiency Factor is e = 0.95 for all phases of flight (take-off, top speed, power-off descent). Find all the coefficients A1, A2,…, An (not to be confused with their 2-D counterparts) for the following three phases of flight (symmetrical)

• take-off: a = 18°, Cl = 2.5

• top speed: a = -3°, Cl = 1.0

• power-off descent: a = 4°, Cl = 1.6.

Here we assume that the rectangular wing is the “simplest” wing with e = 0.95, i. e., with the least number of non-zero coefficients.

14.7.1.1 Incompressible Flow (Mo = 0)

Cambered Plate Geometry

Consider a thin cambered plate of chord c of equation

d(x) = AcX (1 – 2X) (1 – X)

Calculate d'(x) to help you with the graph and make a plot of the plate (Hint: expand d (x) before taking the derivative, A is not determined at this point). Verify your result as much of the rest depends on it.

Fourier Coefficients

Find all the Fourier coefficients Ao, A1; A2,An for this thin cambered plate and give the incidence of adaptation aadapt. Use the identity cos21 = 2 (1 + cos 2t).

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), Cm, o (a) and Cm, a.c.

Static Equilibrium About an Axis

If an axis is located at the mid-chord, f = 5, find the equilibrium angle aeq if there are no forces other than the aerodynamic forces. Is the equilibrium stable, unstable, neutral?

14.7.1.2 Supersonic Flow (M0 > 1, в = JM( — 1)

The same cambered plate equips the fins of a missile 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).

Static Equilibrium About an Axis

If an axis is located at the mid-chord, f = 2, find the equilibrium angle aeq if there are no forces other than the aerodynamic forces. Is the equilibrium stable, unstable, neutral? (Hint: calculate Cm 1).

m, 2

The best design for the AMF IV has a span bm = 2.54 m and a constant chord cxm = 0.279 m. The corresponding tail is defined as bt = 1.0 m with constant chord cxt = 0.254 m. The equilibrium code provides the take-off velocity Vto = 12.5 m/s and a maximum take-off mass M = 19 kg.

14.6.3.1 Airplane Lift and Moment Curves

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

Cl (a, tt) = 4.391a + 0.757tt + 0.928
См, о(а, tt) = -1.733a – 0.686tt – 0.267

where a is the geometric incidence (in radians, measured from the fuselage axis) and tt is the tail setting angle (in radians). См, о 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.

Give the definition of the aerodynamic center.

Find the location of the aerodynamic center xac in m given that lref = 1.6 m. The center of gravity is located at xcg = 0.503 m. Find the static margin SM in

% of lref.

14.6.3.2 Take-Off Conditions

Write the longitudinal equilibrium equation for the moment. How do you interpret this equation?

The main wing lift coefficient is given by

CLm = 4.927a + 1.388

At take-off the lift coefficient for the main wing is (CLm)t-o = 2.1. Find the take-off incidence (a)t-o.

Find the tail setting angle at take-off.

Find the location of the center of pressure in m at take-off.

Is the lift force on the tail up or down at take-off.

14.6 Problem 7

14.6.2.1 Downwash Evolution

For a large aspect ratio (AR) wing without sweep (straight lifting line) how does the downwash vary with distance from the lifting line from upstream infinity to downstream infinity near the x-axis and in particular what is the relationship between ww(y) and wT(y), — b < У < 2, where the former is the downwash at the lifting line and the latter the downwash in the Trefftz plane, far downstream. Explain with a simple argument.

14.6.2.2 Wing with Ideal Loading

Consider an aircraft wing with ideal loading (minimum induced drag) at a cruise lift coefficient (Cl)cruise.

Calculate the maximum value of the circulation (root circulation) Г0 and the downwash at the lifting line ww(y).

14.6.2.3 Induced Downwash in Manoeuvre

The aircraft needs to roll to initiate a left turn. Given that the rolling moment about the aircraft axis is given in terms of the Fourier coefficients by

n

Cm,0x = —44 AR A2

sketch the circulation distribution and corresponding downwash that will allow such a manoeuvre. Hint: use the simplest Fourier distribution that needs to be added to the ideal loading.

14.6.2.4 Induced Drag in Manoeuvre

Calculate the induced drag and the gain/loss compared to the elliptic loading, assum­ing that the (Cl)cruise remains unchanged and that A2 = – A1/10.

14.6.1.1 Incompressible Flow (Mo = 0)

Airfoil Design

Consider a thin airfoil of chord c with parabolic camberline. The design requirements are the following:

• the take-off lift coefficient is (Ci)t-o = 1.8

• the location of the center of pressure at take-off is {xC-)t_o = 0.4

The airfoil geometry will be completely determined by the airfoil Fourier coeffi­cients.

First, write Cl in terms of the Fourier coefficients.

Second, express the center of pressure location in terms of the Fourier coefficients. Write as a system and solve for A0 and A1. Check your solution carefully.

Find the take-off incidence.

Find the profile relative camber.

Can you think of a profile that can fulfill these requirements?

Lift Curve

Give the value of the lift slope ^0..

Give the value of the Cl0 corresponding to a = 0. Sketch the lift curve on a graph Cl (a) = ^0.a + Cl0.

Equilibrium with Tail

If the center of gravity of the wing+tail configuration is located at Xa. = 0.5 at take-off, what sign do you expect the tail lift coefficient to be, Cl >=< 0?

Sketch this situation.

14.6.1.2 Supersonic Flow (M0 > 1, в = JM^ — 1)

A flat plate equips the fins of a missile cruising at Mach number M0 > 1ina uniform atmosphere. The chord of the airfoil is c.

Pressure Distribution and Global Coefficients

Plot —C + and —C – versus x for this airfoil at a > 0.

Write the formula for the lift coefficient C;(a), moment coefficient Cm, o(a) and drag coefficient Cd (a) according to inviscid, linearized supersonic flow theory.

Maximum Finess and Flow Features

Given a viscous drag coefficient Cd0 independent of incidence, form the expression of the inverse of the finess, 1/f = Cd / Ci where Cd now includes the viscous drag and find the value of a that maximizes f (minimizes 1/f).

Sketch the profile at incidence and indicate on your drawing the remarkable waves (shocks, expansions).

The center of gravity is located at xcg/ lref = 0.21.

Find the static margin in % of lref.

Derive the formula for the equilibrium incidence a(tt). Check your result carefully as other results depend upon it.

14.5.3.2 Take-Off Conditions

The take-off lift coefficient for the complete configuration is CL, to = 2.0. Find the tail setting angle and the incidence at take-off.

The tail lift curve is given by

CLt = 2.49a + 2.61tt – 0.36

Find the tail lift coefficient at take-off.

Find the lift force (in N) on the tail at take-off. Take p = 1.225 kg/m3. Is it up or down?

14.5.3.3 Extra Credit

Sketch qualitatively the forces (main wing and tail lifts, weight) and forces locations at take-off with respect to the aerodynamic centers of the main wing (xacm/ lref = 0.21), of the tail (xact /lref = 0.85), of the complete configuration (xac/lref) and with respect to the center of gravity.

14.5 Problem 6

The AMF III has a span bm = 2.1m and a constant chord cxm = 0.263 m. The corresponding tail is defined as bt = 0.527 m with constant chord cxt = 0.176 m, using the rapid prototyping code, which also provides the take-off velocity Vto =

13.7 m/s and a maximum take-off mass M = 15.5 kg.

14.5.3.1 Airplane Lift and Moment Curves

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

CL (a, tt) = 4.47a + 0.37tt + 1.11
CM, o(a, tt) = -1.30a – 0.33tt – 0.28

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).

What is the effective aspect ratio AR for the complete configuration? Compare it with the main wing aspect ratio ARm. Explain the difference. (Hint: look at how the total lift coefficient is calculated).

Find the location of the aerodynamic center xac given that lref = 2.062 m.

14.5.2.1 Origin of Induced Drag

Explain how an inviscid flow (no friction) past a finite wing, can produce a drag. (Hint: you can discuss an energy balance).

14.5.2.2 Maximum Lift of a Wing

The maximum lift of a wing would ideally correspond to a constant circulation along the span, Г(y) = rmax, where rmax is the maximum circulation that a profile can sustain, given the flow conditions (Reynolds number, etc.). Such a distribution is given by:

where y(t) = —f cos t, 0 < t < n. It corresponds to a single, finite strength horse­shoe vortex trailing the wing from the wing tips. The problem with this model, as found out by Prandtl, is that the induced drag is infinite. Since this is not feasible, we are going to consider only a few terms of the infinite series.

Consider modes 1 and 3 only (p = 0, 1).

Give the values of A1 and A3.

Sketch the corresponding distribution of circulation and downwash (Use sin 3t = sin t (3 — 4sin21)).

Show that there is no upwash.

14.5.2.3 Induced Drag

Calculate the induced drag and the gain/loss compared to the elliptic loading.
Show that the induced drag due to two finite strength vortices is infinite.

14.5.1.1 Incompressible Flow (Mo = 0)

Definition

In 2-D, define the following aerodynamic coefficients C;, Cm, o, and Cd in terms of the forces and moment per unit span (L’, M’o, D’), the density (p), incoming flow velocity (U) and chord (c). Give the expressions for Ci(a), Cm, o(a) and Cd for a symmetric profile.

Suction Force

Consider a symmetric profile (d(x) = 0). The lifting problem corresponds to the flow past a flat plate at incidence. Sketch the flat plate, the force due to pressure integration and the suction force.

Calculate the suction force coefficient Cs = F’s/( 1 pU2c), where F’s is the suction force per unit span, from the Ci expression.

In what sense does the thickness distribution contribute to recuperating the suction force? Explain.

Center of Pressure

Give the definition of the center of pressure.

Find the location of the center of pressure for a symmetric profile.

Calculate Cm, ac for a symmetric profile.

Verify your result for Cm, o(a) using the change of moment formula, from the aerodynamic center to the profile leading edge.

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

A thin double wedge airfoil equips the fins of a missile cruising at Mach number M0 > 1 in a uniform atmosphere. The chord of the airfoil is c. The profile camber and thickness are:

d (x) = 0

20x, 0 < x < c/2
29(c – x), c/2 < x < c

Fig. 14.2 Double wedge in г

supersonic flow at zero Mo

incidence *■

with z±(x) = d(x) ± e(x)/2 + a(c — x). See Fig. 14.2.

Pressure Distributions

Plot —C + and —C— versus x for this airfoil at a = 0.

Lift Coefficient

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

Drag Coefficient

Calculate (Cd)a=0.

Give the value of the coefficient Cd for the general case a = 0.

Moment Coefficient

Calculate the zero incidence moment coefficient (Cm, o) 0.

Give the values of the coefficient Cm, o for the general case a = 0.

Maximum Finess

Form the expression of the inverse of the finess, 1/f = Cd /Ci and find the value of a that maximizes f (minimizes 1 /f).

Sketch the profile at the maximum finess incidence and indicate on your drawing the remarkable waves (shocks, expansions).

Using the same linearized lift curve, find the take-off incidence, {aeq)t_o in deg corresponding to CLm = 1.9.

Show that at take-off, the tail has positive lift (Hint: calculate the tail setting angle (tt )t-o at take-off).

14.4.3.4 Extra Credit

Sketch the forces and forces locations at take-off with respect to the aerodynamic centers of the main wing (xacm/lref = 0.27), of the tail (xact /lref = 0.83), of the complete configuration (xac/lref) and with respect to the center of gravity, given that there is an 8 % static margin.

14.4 Problem 5