Understanding Airfoils: An In-Depth Guide

Posted on Oct 13, 2024 in Design and Engineering

Airfoils

General Values:

NACA 2412

Conservation of Mass: ṁ1 = ṁ2 (ρ1*V1*A1 = ρ2*V2*A2)

#1: Chamber (0.02 * cord length)

Bernoulli’s Principle:

#2: Chamber Location (0.4 * cord length)

Dynamic Pressure (q): q = (1/2) * ρ * V^2

#3+4: Max Thickness (0.12 * cord length)

Ideal Gas Law: P = ρ / (R * T)

Lift Coefficient (cL): cL = L / (q * span * cord)

Gas Constant (R): R = (8.314 / M), where M is the molar mass of the gas. For air, R = 287 (m^2/(s^2K))

Drag Coefficient (cD): cD = D / (q * span * cord)

Total Pressure (Po): Po = P (static) + P (dynamic) = Ps + (1/2) * ρ * V^2

Lift and Drag per Unit Span (D’, L’): D’, L’ = cL, cD * q * c

Dynamic Viscosity (μo): μo = 1.71 x 10^-5 kg/(m*s)

Reynolds Number (Re): Re = ρ * V * c / μ

Standard Temperature (To): To = 273K

Solution Steps:

1. Find Re, pitch, and dynamic pressure (q).
2. Find the lift coefficient (cL).
3. Solve for lift per unit span (L’) and drag per unit span (D’).

Mean Cord:

Compressibility

Forces

Mach Number (M): M = V / a

Speed of Sound (a): a = sqrt(KRT), where K = 1.4 and R = 287

Normal Force and Axial Force Coefficients (Cn and An): Cn and An = A, C / (q * s)

Steady Flow: d/dt = 0

Mass Flux (mdot): mdot = cρ

2D Flow: d/dz = 0

Scaling (Similarity)

Geometric Scaling: Lm / Lp = constant

Kinematic Scaling: Vm / Vp = constant, cDm = cDp

Dynamic Scaling: Re, M are constant

Navier-Stokes (Incompressible)

Wind Tunnels

x-direction:

Ideal Gas: Find ρ

Constant Mass: Relate V1 and V2

Bernoulli’s Principle: Solve for V2

Isentropic Flow: Use V1, M, and P0

y-direction:

Total Pressure (Po): Po = pitot reading

z-direction:

Isentropic Flow (for more accurate M): , where γ = 1.4 and M is the Mach number.

Average Pressure (P): P = (Px + Py + Pz) / 3

Bernoulli’s Principle

P + (1/2) * ρ * V^2 = constant

Sutherland’s Law: (Reference Values)

Viscous Shear:

1. Inviscid Flow: τ = 0

2. Incompressible Flow: ρ = constant (M∇ * V) + 2μ(du/dx)

3. Steady Flow: d/dt = 0

Shear Stress (τyy): τ(yy) = λ * (∇ * V) + 2μ(dv/dy)

4. Flow Along Streamline or Irrotational Flow: (Plug points into streamline and see if they are the same, ∇ x V = 0)

Shear Stress (τzz): τ(zz) = λ * (∇ * V) + 2μ(dw/dz)

Flow Patterns

Couette Flow “Sliding Surface”

Incompressible: ρ = constant

Steady: d/dt = 0

2D: d/dz = 0

Fully Developed: Flow velocity is symmetric

Viscous: V(y=0) = 0; V(y=h) = Vinf

-∇P = 0