Answer:
Step-by-step explanation: To calculate the boundary layer thickness, Reynolds number, displacement thickness, and shear stress for a Boeing 787 flying at 600 mph at 30,000 ft, you can follow these steps.
Calculate the Reynolds Number (Re):
Reynolds number is given by the formula:
Re = (ρ * V * L) / μ
where:
ρ = air density at 30,000 ft
V = velocity (600 mph)
L = distance from the leading edge (150 ft)
μ = dynamic viscosity of air at 30,000 ft
First, convert the velocity from mph to ft/s:
V = 600 mph * 1.467 ft/s per mph = 880.2 ft/s
The dynamic viscosity of air at 30,000 ft can be looked up in a table, and for this calculation, let's assume μ = 3.49 x 10^-4 lb/(ft·s).
Calculate Re:
Re = (8.91 x 10^-4 slug/ft³ * 880.2 ft/s * 150 ft) / (3.49 x 10^-4 lb/(ft·s))
Re ≈ 8.97 x 10^7 (dimensionless)
Calculate the Boundary Layer Thickness (δ):
Use the empirical Blasius equation for a flat plate:
δ ≈ (5.0 * x) / √Re
where x is the distance from the leading edge (150 ft).
δ ≈ (5.0 * 150 ft) / √(8.97 x 10^7)
δ ≈ 0.004 ft or 0.048 inches
Calculate the Displacement Thickness (δ*):
δ* ≈ (1.72 * δ)
δ* ≈ 1.72 * 0.004 ft
δ* ≈ 0.00688 ft or 0.0825 inches
Calculate Shear Stress (τ):
τ = 0.332 * ρ * V²
where τ is the shear stress on the surface of the flat plate.
τ = 0.332 * 8.91 x 10^-4 slug/ft³ * (880.2 ft/s)²
τ ≈ 0.211 lb/ft²
Plot the Velocity Profile:
To plot the velocity profile, you can use the concept of boundary layer growth and assume that the velocity increases linearly within the boundary layer, reaching the free stream velocity outside the boundary layer. You'll have a boundary layer thickness of 0.004 ft (or 0.048 inches) at a distance of 150 ft from the leading edge.