Aerodynamics: Force, Atmosphere and Pressure

• Aerodynamic force is the force due to air flow.
• Thrust is in the positive X direction, drag is in the negative X direction, and weight (mg) acts downwards.
• Aerodynamic efficiency = Lift/Drag, which is approximately 10 for normal aircraft.
• For equilibrium flight conditions, lift (L) is 100g and if L/D = 10, then drag (D) is 10g.
• An engine requires a thrust of at least 10g to power the aircraft.
• Gliders possess a larger Lift/Drag ratio, indicating very high efficiency.

Atmosphere

• Ideal Atmosphere: Winds, clouds, and other objects are neglected.
• Most aircraft fly in the mid-range of the troposphere and stratosphere.
• Tropopause ranges from approximately 11km to 20km.
• Stratosphere ranges from approximately 20km to 47km.

Pressure Variation

• Pressure varies with altitude, and can be modeled using the equation dP = -ρ * g * dh.

Lapse Rate

• Lapse Rate: the rate of change of temperature with respect to height.
• L = dT/dh.
• Temperature varies in the atmosphere because greenhouse gases trap certain regions/wavelengths from electromagnetic radiations.
• Temperature increases in Stratosphere due to ozone presence.
• Formula to calculate dp/P

• dp/P = (-g)/(R*T) * dTdh
  

• Formula to calculate pressure at a specific altitude

• T(h) = To + a*h
  

• a decreases where T decreases
• a increases where T increases

Aircraft Performance

• Max Velocity
• Pressure Altitude: Altitude associated to ISA model altitude by knowing pressure.
• Not true altitude.
• Radar or GPS systems can be used for some flights.
• Radar or GPS systems might not work at altitudes above 18,000 ft.
• Nose hole measures pressure.
• Pressure is distorted around a craft.
• Distortion means it might not measure the correct ISA height.
• Tail wind and head wind may need to be considered

Air Speed, Tail Wind and Head Wind

• Ground Speed (GS) = Air Speed (AS) + Tail Wind Speed (TWS)
• Ground Speed = Air Speed – Head Wind Speed (HWS).
• Jet streams happen during the winter season.
• Jet streams affect the flow of air and wind.
• The altitude of jet streams is approximately 9 km.
• Jet streams are strong during winter in polar regions.
• Jet streams are slower in subtropical regions.
• All commercial flights have velocities exceeding 800 mph.

Aerodynamics

• It is the study of relative motion between air and bodies.
• Relative motion is caused by "wind" or "motion of body"
• In aircraft's reference, wind only moves.
• In aircraft's reference, relative velocity is affected, but pressure, temperature, and density remain constant according to the local atmospheric surrounding.
• Pressure distribution matters though Bernoulli Equation:

• P + 1/2 * ρ * v^2 = constant
  

• For a rigid body, F = m * (d^2r/dt^2).

Continuum Mechanics

• Continuum mechanics properties:
  • Viscous force
  • Streamlines

Steady/Unsteady Flow

• Steady Flow: Same element property value at all times.
  • But may not have same value writ position
  • So not uniform throughout
• Unsteady Flow: Element property changes

Momentum Equation

• Properties: (P, T, S, v) can be written as function of position, but mass and volume can't be written for a point
• Pressure is calculated as force/area
  • Can't be defined for a point
  • Relates to Eulers / Naviers Stokes

Newton's Law of Viscosity

• Viscous stress (τ) = μ * dv/dy

• In Euler's equation, assume fluids to be INVISCID

  • Viscous effects are neglected
  • Pressure and gravity effects are considered

• Force acts along a certain direction and depends on acceleration along that direction.

• 2D flow assumption

• The assumption of steady flow: Fs = m âs.

• dP/dS = ∫(dA ds dv)/dt
  

• Stream line: a path a mass particle would take when carried in a stream.

• Local velocity is tangent to a stream line.

• There can be no mass flow across a stream line.

  • Means fluid has to be inviscid in this situation

• For steady flow, one assumption that is followed is:

• d/ds * (P + 1/2*v^2) = 0
  

• On a streamline

• Constant relationships apply

• Free Stream Flow: constant pressure and velocity.

• Depending on the magnitude of the radius of curvature of the streamlines, the value of dP/dn varies.

• For points A and beyond, the radius trends to infinite meaning dP trends to 0.

• With Bernoulli's equation, the velocity is minimum when P is highest.

• Curvature reduces from B to infinite, which means that pressure trends to P infinity.

• The curvature in C increases but the pressure reduces, i.e., Pc < P∞

• Pressure reduces from B -> C

• In a free stream flow

• Can apply this equation: P + 1/2 * ρ * v^2 = C

• When v is 0 at stagnation point can find max pressure

Coefficient of Pressure

• More/Higher the curvature, more is the change in pressure values.
• Coefficient of pressure

• Cp = (P-Poo) / (pw * vo^2)
  Cp_max  = 1
   --> P = Poo +  (pw * vo^2)