University:
Massachusetts Institute of Technology
Course:
16.100 | Aerodynamics
Academic year:

2023
Views:

311

Pages:

2
Author:

sperrtrab7z

Subsonic Small Disturbance Potential Flow v v v 1. V = (V∞ + uˆ )i + vˆj where | uˆ 2 + vˆ 2 | << 1 V∞2 small disturbances are assumed v v 2. uˆi + vˆj = ∇φˆ ← φˆ ≡ perturbation potential ∂φˆ ∂φˆ ⇒ uˆ = vˆ = ∂x ∂y 3. small-disturbance (?) and bc’s= ∂ 2φˆ ∂ 2φˆ (1 − M ∞2 ) 2 + 2 = 0 ∂y ∂x dy c ( x) where yc = camber line dx Note: this eqn and bc are valid for subsonic and supersonic flow. p − pu 2uˆ Also note, C p = =− q∞ V∞ BC: vˆ( x,0) = V∞ 4. For subsonic flow, we utilize the following mathematical transformation: Define: ∂ 2φ ∂ 2φ ξ=x + =0 ∂ξ 2 ∂η 2 2 β = 1− M∞ ⇒ η = βy dy ∂φ (ξ ,0) = V∞ c (ξ ) φ = β φˆ ∂η dx The implications are that the subsonic compressible flow around an airfoil can be related to the incompressible ( M ∞ = 0) flow about the airfoil. M∞ = 0 Same eqn ∂ 2φˆ ∂ 2φˆ =0 + ∂x 2 ∂y 2 Same bc ) dy ∂φ ( x,0) = V∞ c ( x) ∂y dx 0 < M∞ <1 ∂ 2φ ∂ 2φ + =0 ∂ξ 2 ∂η 2 dy ∂φ = V∞ c (ξ ) ∂η dx Subsonic Small Disturbance Potential Flow Implications: uˆ ( x, y, M ∞ ) = ∂φˆ 1 ∂φ 1 = = u 0 (ξ ,η ) ∂x β ∂ξ β M∞ = 0 flow M∞ ≠ 0 flow C p ( x, y , M ∞ ) = − 2uˆ 1 2u 0 =− V∞ β V∞ M∞ = 0 ⇒ C p ( x, y , M ∞ ) = 1 β C p0 (ξ ,η ) What about the forces? 16.100 2002 2