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Buffl

TUM

by Peter P.

Sketch the typical relation of the change in relative thickness with Mach number!

What is meant with the term „Zero twist“ ?

A twist angle distribution along the span of the wing, that causes the lift and downforce to cancel out

Sketch the typical relation of the change in wing sweep with Mach number!

Which aerodynamic considerations determine mainly this characteristic course?

The effective velocity which hits normal to the wing, decreases with the higher sweep. Therefore higher Ma numbers are reachable

Sketch the typical relation of the change in aspect ratio with Mach number!

Sketch for a rectangular wing (parabolic thickness contour) the maximum induced longitudinal velocities relative to the 2D case, u max/u max,2D, as function of aspect ratio Λ!

From which aspect ratio on a markable reduction in the induced velocities is present for the 3D case?

How is this reduction reasoned?

For Aspect ratio < 1.5, a noticeable drop in umax appears

The wing gets less “2D“ and therefore the maximum velocity in the middle of the wing decreases

Name the 3D kinematic flow condition

How big are approximately the maximum induced longitudinal velocities on a rectangular wing (parabolic thickness contour) of aspect ratio Λ = 10 relative to the 2D case?

Hence, which statement can be derived for the design of the thickness contour for the wing of finite aspect ratio?

(umax/Uinf)_2D/(umax/Uinf)_3D = 0.9983; for aspect ratio = 10

-> High aspect ratio wings lead to higher longitudinal overspeeds. At an AR of 10 the wing is almost “2D”

-> The thicker the airfoil, the higher the longitudinal overspeeds

-> Highest overspeeds and cp reached in the wing center

Explain via a sketch the change in the thickness related pressure loading on an oblique (swept) wing (Λ = ∞)!

Which relation is then derived for the pressure coefficient associated with an oblique (swept) wing?

Cp(phi) = Cp(phi=0)*cos(phi)

Sketch the cp distribution across the the cord with different sweep angles

Which numerical approximation methods are applied for the solution of the lift problem integral equation for the 3D case in context of potential theory?

Which method provides the most accurate solution (reasoning)?

Lifting Surface Method (Most precise method, by using a pressure Ansatz function, which is based on physical conditions)

Panel lattice method

Vortex lattice method

Outline the main points in the course of the solution for the lifting surface method?

(A) Appropriate Ansatz for pressure distribution Δ𝐶𝑝(𝑥, 𝑦) in x-direction and y-direction has to be applied.

(B) The integration via the product of the Kernel function and the pressure distribution function must be carried out. Necessary to identify the singularity functions of the integrand.

(D) This leads to a system of linear equations.

How does an enlargement of the wing aspect ratio (on the one hand) and a reduction of the wing sweep (on the other hand) influence the wing root bending moment?

The higher the Aspect Ratio, the higher the root bending moment (increased lever arm)

The lower the wing sweep, the lower the root bending moment (inwards shift of the lift)

What can be done to shift the lift distribution inwards?

Reducing the wing sweep

Reducing the taper ratio

Increasing twist and camber inward

Name the coefficients of longitudinal motion

Name their respective limits

Pressure distribution 𝜟𝑪𝒑:

-> Limited due to maximum Ma number (shocks)

Load 𝑐𝐿(𝑦) ⋅ 𝑙(𝑦) and Loading 𝑐𝐿(𝑦):

-> CL is limited due to flow separation (CL_max)

Bending moment 𝑪𝒃:

-> Limited due to maximum root bending moment (structure)

Torsional moment C𝑡:

-> Limited due to maximum torsional moment (divergence, rudder reversal)

What has to be the case to risk divergence?

Elastic axis behind the center of pressure

Explain the mechanism of „rudder reversal“!

Which means are used to counteract this phenomenon?

It is an aeroelastic phenomenon, that appears when the increased lift due to the deflection of the rudder is cancelled out by the torsional deformation of the airfoil.

Deflection of the rudder -> increased camber -> increased lift in the back -> rotation around the elastic axis -> decreased angle of attack -> decreased lift

Means to counteract:

-> Higher stiffness (weight)

-> High and low speed ailerons (High speed next to the wing root, low speed outward)

Sketch for a rectangular wing with aspect ratio Λ1 and Λ2 = 2Λ1 the course of the lift distribution (cLl/(4s))α and the lift coefficient (cL/CL)α over the wing half span?

How is then the influence of the aspect ratio Λ on cL und cL*l?

CL gets shifted outward for higher Λ

CL*l is more evenly distributed for higher Λ (gets higher for higher Λ)

(Higher Λ decrease of 3D Effects)

How is the influence of the taper ratio λ and sweep φ on (cL*l/(4s))α and (cL/CL)α

With decreasing taper ratio:

cL*l increases inboard -> Bending moment decreases

cL increases strongly in the tip region -> risk of flow separation

Which qualitative course is associated with CLα as a function of the wing aspect ratio Λ?

Which value of CLα can be obtained for the limit case Λ → 0?

CLα = 2Pi for AR->inf

Which aerodynamic coefficients are particularly related to the pitching motion?

How can they be calculated on basis of the lifting surface method?

How can they be calculated on basis of (known) steady derivatives?

For which lift distribution minimal induced drag is present (reasoning)?

For which wing planform this lift and circulation distribution, respectively, is present?

Can such a lift distribution be realized for a wing planform with a certain sweep and taper ratio (reasoning)?

Is this possible for different angles of attack?

Elliptical distribution, constant downwash across span

Reason:

For minimal Cdi, only A1 should exist.This appears for an elliptical distribution!

This distribution can be achieved or a given sweep and taper ratio, since the camber and twist distribution can be set accordingly (only for a given desing point).

Only for an elliptical planform (with the quarter line being unswept) the distribution can be kept elliptical for different AoAs.

Explain via a sketch the development of the induced rolling moment Clβ for a wing featuring positive wing sweep at sidewind conditions?

How does the aspect ratio influence the sideslip induced rolling moment?

Which conditions are relevant for the sideslip induced yawing moment Cnβ?

Which wing planform is associated with directional stable/unstable behaviour (reasoning)?

Higher Aspect ratios lead to higher induced rolling moments (bigger leverarm)

Which Conditions???

Backward swept wing: Cnβ >0 -> stable

Forward swept wing: Cnβ <0 -> unstable

For a backward swept wing, the effective sweep of the windward side decreases, this increases its lift. The induced drag of this side increases aswell, whereas the induced drag at the leeward side decreases. Therefore a yawing moment appears that turns the aircraft “into the wind“. This leads to a lower sidewind conditions therefore moment -> stable

Forward swept wing: vice-versa

How does a dihedral influence the sideslip induced rolling and yawing moment?

In general, for small dihedral angles, there is only a slight dependency of 𝐶𝑛𝛽 on 𝜈

How can the induced rolling moment at sidewind conditions be alleviated (compensated) for a high-wing configuration?

What about a Low-Wing?

Explain with a Sketch

A high wing configuration, should have an anhedral to compensate for the fuselage effect.

Low-wing -> dihedral

Which conditions are associated with the loss of (aerodynamic) roll damping?

The loss of roll damping appears for a one-sided flow separation.

Explain the development of the roll induced yawing moment Cnp at a roll motion?

Which typical trend is present?

Due to a rolling motion the effective angle of attack increases on the downward moving side. This leads to a higher lift on this side -> higher induced drag -> vice versa on the upward moving side -> yawing moment

Which is the main consideration motivating the Prandtl-Glauert transformation?

Which transformation quantity is related to the pressure coefficient (2nd form)?

Solving the incrompressible flow problem (potential functions) and then transforming it to the compressible case.

Incompressible:

Compressible 2st form:

A wing planform is given with Λ = 8, λ = 0.28, φ = 30° at a free stream Mach number of M∞ = 0.8.

Which planform parameters are attributed to the reference wing applying the Prandtl-Glauert transformation?

Which lift coefficient CL is then obtained for M∞ = 0.8 based on the value of the reference case CLi?

β = sqrt(1-M∞^2) = 0.6

Λ = Λi*β = 4.8

λ = λi = 0.28

φ = atan(tan(φi)/β) = 43.9°

CL = CLi/β

What is meant with „critical Mach number“ and „critical pressure coefficient“?

The Critical Mach number 𝑀∞* is the free stream mach number that results in a LOCAL Mach number M = 1 on a given airfoil

The Critical pressure coefficient cp* is the pressure coefficient that is present on the location at which M = 1 on a given airfoil

Which flow physical effects occur when exceeding the critical Mach number?

How are drag and lift coefficient affected?

Explain in this context design criteria for sub- and super-critical airfoils!

For higher mach numbers than the critical mach number, supersonic regions on the airfoil appear. When the supersonic flow decelerates, they may cause:

Shocks
Pressure increase
Tendency for flow separation
Drag increase
Decrease of lift force

A sub-critical airfoil tries to avoid any supersonic regions on the airfoil. This can be achieved when the thickness/camber distribution is changed in a way that the critical cp is not reached.

A super-critical airfoil doesnt avoid cp values lower than cp critical, therefore more lift can be achieved. This however has to be done carefully since a strong shock can cause flow separation and a decrease in performance.

Which critical pressure coefficient is obtained applying linear theory for M∞ = 0.8?

Sketch qualitatively the course of the critical pressure coefficient as function of Mach number!

cp*=-0.5

By which means the critical Mach number can be changed to higher values (reasoning)?

Which relation is then obtained for the critical pressure coefficient based on linear theory?

Lower relative thickness -> Cp ~ (d/l) -> Cp can be decreased at the same mach number

Higher sweep -> Cp can be decreased at the same mach number

Which characteristics are associated with the disturbance propagation in supersonic flow?

What is meant in this context by the hyperbolic radius?

Disturbances are emitted within the “downstream cone”
Disturbances are received within the “upstream cone”

It defines the surface area of a hyperboloid

Sketch the stream line pattern at an inclined flat plate for a free stream Mach number of M∞ = sqrt(2)!

Which values can be determined for the induced longitudinal velocities (u) with respect to the free stream velocity U∞ and the induced vertical velocity (w)?

Determine the lift coefficient!

Mach-Cone Angle (Formula)

Delta cp for cambered airfoil and inclined flat plate for supersonic flow

Cambered airfoil:

Inclined flat plate:

Explain the main differences in the aerodynamic forces and moments for an inclined flat plate comparing subsonic and supersonic flow!

Subsonic:

Experiences a suction peak in the front -> Nose suction

No drag -> d’Alembert paradox

Kutta-Condition holds true

Supersonic:

No nose suction

Wave drag appears

Subsonic: CLalpha = 2Pi

Supersonic: CLalpha = 4/beta

Which values are obtained for the lift coefficient at zero angle of attack, for the angle of attack at zero lift and the lift slope for an inclined parabolic camber line at supersonic flow (reasoning)?

Where is the aerodynamic center located (reasoning)?

In supersonic flow, lift cannot be produced by airfoil camber, only by AoA -> No lift of the parabolic shape without AoA

The aerodynamic center is always located at 50% of the chord (moves back compared to subsonic flow)

How does the flap length influence the lift coeficient in supersonic flow?

Why does the so called wave drag occur in supersonic flow?

Which variables determine the wave drag coefficient CDw (reason)?

Which features of the airfoil geometry result from the requirement for the lowest possible wave drag?

Since the flow is supersonic, the upper and lower sides of the airfoil are disconnected. This leads (for a flat plate) to a constant cp distribution along the airfoil. Since the lift is perpendicular to the incoming free stream, it also has component parallel to the free stream -> wave drag! “Waves can no longer propagate freely in space as for 𝑀∞ < 1, but that the body “drags” the system of compression and expansion waves”

Wave drag C𝐷𝑤 is defined as the pressure force in the direction of 𝑀∞

Alpha
Thickness Distribution
Camber Distribution

Lowest Wave Drag:

Low relative camber (No camber)
Low relative thickness

Which reference Mach number is related to the Goethert-Transformation?

How does the lift coefficient referred to the lift coefficient of the reference case CL/CL1 change with increasing Mach number M∞?

𝑀∞ = sqrt(2)

Considered is a wing with Λ = 4, λ = 0.25, φ = 50° at a free stream Mach number of M∞ = 1.1.

Which planform parameters are attributed to the reference wing applying the Goethert transformation?

Which lift coefficient CL is then obtained for M∞ = 1.1 based on the value of the reference case CL1?

beta = 0.4583

Λ_new = 1.833

λ_new = 0.25

φ_new = 68.97°

CL = CL1 * 2.182

Which pressure coefficient is obtained for an oblique wing (φ = 50°) of infinte span at M∞ = 2?

Evaluate this result in comparison to the characteristic in subsonic flow!

Cp = 1.591*alpha

Subsonic:

Which lift slope is present for a delta wing featuring supersonic edges only (reasoning)?

Which value for CLα is obtained for a delta wing with Λ = 3 at M∞ → 1?

Purely supersonic edges lead to no flow around the edges -> The sweep doesnt have an effect and therefore the solution is the same as for the 2D case

Explain the course of the calculation methodology for wings featuring subsonic and supersonic edges?

Sketch the lift distribution over the wing half span for a delta wing with

i) subsonic

ii) supersonic leading-edge (for a straight trailing-edge)

Subsonic: m<1

Supersonic: m>1

-> Lift gets shifted more towards the inner parts of the wing under supersonic conditions

Draw the lift slope of a wing with leading-edge sweep of φV = 45° and trailing-edge sweep of φH = –45° at supersonic flow relative to the 2D case, CLα/(CLα)2D, as a function of the Mach number M∞!

For which planform geometry a larger lift slope (lift gradient) can be achieved with respect to the 2D case?

High sweep and low taper ratio

Which tasks are typically dedicated to the tails of a transport aircraft?

Tasks of tail: Directional stability, control & trim

-> Directional stability: Cmalpha < 0, Cnbeta > 0

Which geometric and aerodynamic parameters determine mainly the horizontal tail efficiency?

How is this efficiency influenced by the roll-up process of the wing trailing vortex sheet?

Efficiency coefficient:

Mainly influence by:

Geometric and aerodynamic data of the wing
position of the horizontal stabilizer relative to the wing

The Efficiency coefficient generally increases due to the roll-up process

Determine the downwash angle αW for a wing with elliptical circulation distribution (Λ = 10; CL = 1.0) at a position of one wing half span (x/s = 1) downstream of the wing (i.e. here the lifting line)!

For large distances: