f is equal to zero at a separation point. Boundary-Layer ... responsible for the growth of the boundary-layer. ⢠Continuity equation. ⢠Outer flow dy x u v y v x u y. â«.
Aerodynamics
Boundary-Layer Approximations Pressure gradient effect • Shape of the velocity profile
∂u 1 ∂p ∂u 1 dp ≅− ⇔ ≅− ∂s ρu ∂s ∂x ρu dx • Aplication of Bernoulli’s equation along a streamline in the interior of the boundary-layer neglecting friction dp • Pressure gradient effect increases with the decrease dx of u
Masters of Mechanical Engineering
Aerodynamics
Boundary-Layer Approximations Pressure gradient effect • Shape of the velocity profile
1 dp ∂ 2u =ν 2 ρ dx ∂y
∂ 2u =Λ 2 ∂η
δ 2 dU e u y , η= with u = and Λ = − Ue δ ν dx
• Second derivative at the wall is defined by the pressure gradient Masters of Mechanical Engineering
Aerodynamics
Boundary-Layer Approximations Pressure gradient effect • Shape of the velocity profile
∂ 2u =Λ 2 ∂η
η =0⇒u =0 η =1⇒ u =1
∂u =0 ∂η
∂ 2u =0 2 ∂η
Λ u = 2η − 2η + η − η − 3η 2 + 3η 3 − η 4 6 3
Masters of Mechanical Engineering
4
(
)
Aerodynamics
Boundary-Layer Approximations Pressure gradient effect • Shape of the velocity profile 1 0.9 0.8 0.7 0.6
u
0.5
Λ=-12 Λ=-6 Λ=0 Λ=12 →(τw=0) Λ=24
0.4 0.3 0.2 0.1 0 0
Masters of Mechanical Engineering
0.25
0.5
η
0.75
1
Aerodynamics
Boundary-Layer Approximations Pressure gradient effect • Shape of the velocity profile
