Self Evaluation

• negative dCm/dCL and positive pitching moment at zero Lift

• negative dCm/dCL and positive pitching moment at zero Lift

• evaluation task: known: geometry, wanted: pressure and lift distribution, unique solution.

• design task: known: target lift and pressure distribution, wanted: geometry.

• Design task does not lead to an unique solution, because different geometries can lead to the same lift and pressure distribution

C_L = (2*m*g) / (rho*V^2*S)

• increases with area loading ((m*g) / S)

• increases with flight altitude (1 / rho)

• decreases quadratically with the velocity (1 / V^2)

• stability: horizontal and vertical tail

• control: horizontal and vertical rudders, ailerons

• high lift: flaps (trailing edge) and slats (leading edge)

• center of pressure: where all acting forces can be summed up (lift, drag) to a resultant force without any moment / Defined as the intersection point of the line of action of the resulting aerodynamic force composed of lift and drag with the airfoil chord

• neutral point: where the pitching moment coeff. does not vary with a changing lift coeff. (i.e. AoA) / Defined as the intersection point of the line of action of the additional force based on a change of AoA with the airfoil chord

• center of pressure is influenced by AoA, because C_L is proportional to AoA

• location of center of pressure changes forward to the neutral point, because AoA increases

• to fix the center of pressure we can set C_m0 to null (or to a very small number); if camber line is negative in the front and positive in the rear

• symmetrical, anti symmetrical and asymmetrical motions

• longitudinal motion (phugoid)

  • forces and moment coeffs: lift, drag and pitching moment coefficients.

  • degrees of freedom: Ma, AoA and pitch angular velocity

• lateral motion (dutch roll)

  • forces and moment coeffs.: yawing moment coeff., rolling moment coeff. and side force coeff.

  • degrees of freedom: yawing and rolling angular velocity, sideslip angle

• coupled motion at anti symmetrical configs due to flow separation (spin)

  • forces and moment coeffs.: longitudinal force, side force, lateral force, normal force

  • degrees of freedom: AoA, sideslip angle, pitch-, roll- and yaw angular velocity

• laplace equation

• under the assumption of inviscid and incompressible flow it is an exact solution

• Ma = free stream velocity / speed of sound, compressibility and sonic regime 

• Re = (free stream velocity * length) / kinematic viscosity, inertial forces vs friction force

• the angle of the stabilizer decreases from CL1 to CL2

• irrotational flow

• inviscid flow

• incompressible flow

• stationary flow

• CG is in front of the neutral point (aerodynamic center)

• C_m0 is > 0

• (dC_m / dC_L) < 0

• C_L = n*C_W

  
  

• drag coeff. at optimum glide number = 2 * zero drag coeff.

• induced drag coeff. = zero drag coeff.

  

• the flow around a body can be divided into two regions

• a thin viscous boundary layer with velocity -> 0 at airfoil surface (rotational, viscous)

• and a region with an inviscid flow outside the boundary layer (irrotational, inviscid)

-> we only have to calculate with the irrotational flow and can neglect everything happening in the boundary layer since the pressure in the outer area is nearly the same as on the surface in the boundary layer (constant in boundary layer, no pressure discontinuity between b.l. and outer domain.)

Source

Vortex

• mass continuity and momentum equation (energy conservation equation not required)

• incompressible flow: Ma = 0=>a = sound velocity = infinite,

• inviscid flow: viscosity is zero

• Obtain velocity potential equation and Bernoulli equation by integrating on a streamline

• thickness problem: source distribution

• lift/camber line problem: vortex distribution

• Using velocity potential, Kinematic Boundary condition applied, attached flow (flow follows contourline/camberline for example)

• kinematic boundary condition: the flow is attached/follows the contour of the airfoil. velocity vector*surface normal vector=0, vectors normal to each other.

• Linearization: w/U=df/dx = contour gradient. Formula to be known. 

• no normal forces (no lift) due to equal pressures above and below the airfoil

• no forces in free stream direction (no drag) -> d’Alembert paradox

• AoA and contour gradient.

u << U_infinite and dS/dx small -> neglegtable

U and AoA

• Delta Cp -> infinite at root bc of root singularity

• Delta Cp -> 0 at trailing edge bc of KAB

• Circulation follows bc Delta Cp = (2*gamma)/U_infinite

• trailing edge: delta Cp = 0 and circulation is 0

• Kutta summuary

  • For a given AoA the value of the circulation around the airfoil is such that the flow leaves the trailing edge smoothly

  • trailing edge angle finite -> trailing edge = stagnation point

  • trailing edge cusped -> velocitys at bottom and top are finite and equal in direction and magnitude

• trailing edge: delta Cp = 0 and circulation is 0

• Kutta summuary

  • For a given AoA the value of the circulation around the airfoil is such that the flow leaves the trailing edge smoothly

  • trailing edge angle finite -> trailing edge = stagnation point

  • trailing edge cusped -> velocitys at bottom and top are finite and equal in direction and magnitude

• normal force Fy = rho*U_infinity*Gamma

• axial Force Fx = -rho*U_infinity*Gamma*AoA -> minus leads to suction force (nose thrust)

• due to d’Alembert paradoxon no drag

• lift coefficient gradient 2*pi

• locations of center of pressure and aerodynamic center both at 1/4

• Lift Coeff: A0 and A1

• Pitching moment coefficient: A1 and A2

• for thin airfoils separation starts at leading edge -> leading edge stall

• for thick airfoils separation starts at trailing edge and progresses upstream -> trailing edge stall

• thin airfoil stall = flow separates at leading edge but reattaches downstream -> separation bubble in 2D and reattachment in turbulent area

• thicker airofils lead to an increasing max. lift coeff. but if its too thick (>12%) the max lift coeff. decreases -> optimum around 12%

• Also with an increase in thickness can increase radius at Leading edge causing suction

• trailing edge flap: affect Kutta cond., positive deflection -> increase in CL0 and CLmax

• leading-edge flap: CLmax increases at higher AoA, higher Aoa possible, allows for the CLmax to be reached at higher alphas

• if both are active you can superposition them

• increasing camber -> increasing CL0 -> increasing CL

• increasing surface

• energisation of the top layer via through flow

Flap deflection is more effective than a slat, because max. camber position is shifted dowstream and increases the zero lift coefficient.

Horseshoe vortex model

• trailing edge flap: affect Kutta cond., positive deflection -> increase in CL0 and CLmax

• leading-edge flap: CLmax increases at higher AoA, higher Aoa possible, allows for the CLmax to be reached at higher alphas

• if both are active you can superposition them

difference in pressure between upper and lower side; naturally p wants to equalize: high pressure from bottom curls up

Induction of tip vortices: induced downward velocity, thus reduction in effective AoA from nominal AoA

For elliptical lift distribution: b’ = (pi/4) b

• system of infinitesimal horseshoe vortices whose “heads” (bound/root vortex) are placed on a common line, the lifting line

• Ansatz: Fourier sine series

• Lift: only Gamma 1

• Induced Drag: Gamma n (for elliptical distribution only Gamma 1)

• C_Di min, constant downwash velocity

• alpha ~ CL and C_Di ~ C_L^2

• Elliptical wing shape, quarter line is unswept

• AR up -> C_Di down

• Lambda down -> C_Di down (not a big influence), taper ratio influences circulation distribution

• Gamma 2 -> antimetrical; Gamma2 sin(2*theta) is antimetrical. So one side of wing has up force, other side down force -> rolling moment

Mx =

with:

• Point where downwash of the bounded vortex corresponds to the downwash according to the camber theory x= 3/4 * l

• kinematic b.c. is applied there

Each lattice element corresponds to a small wing on which Prandtls theory is applied. The circulation distribution of each lattice element is concentrated on a bound vortex located at 1/4 line. Each bound vortex is continued by two trailing vortices. The circulatoin of each of those horseshoe vortices is determined based in the condition that the kinematic b.c. is fulfilled at the 3/4 point of each lattice element

• NACA 0012 (see appendix)

• pressure minimum ahead of max thickness

• increase of AoA: suction peak moves upstream and magnitude inreases

• Transition Re<10^6 downstream of pressure minimum, Re>10^6 upstream of pressure minimum

• high adverse pressure gradient -> danger of separation

• slats to energize flow

• no harsh contour gradients

• thicker airfoil

• direct and inverse design

• direct: geometry given (calculation task), modifications to airfoil geometry to improve performance

• inverse: optimization of airfoil geometry based on objective function

Constant flow acceleration, pressure gradient, rear placed max camber and thickness allowing a smooth acceleration and negative pressure gradient. This allows laminar flow and an indicaiton of flow separation

With an increasing angle of attack, the flow starts to separate on the upper side of the airfoil - first in the rear section → ”End of the linear range”, drop in the lift!