16.121 ANALYTICAL SUBSONIC AERODYNAMICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY Linearized Subsonic Flow We desire to solve (1 − M 2 )φxx + φ y y + φzz = 0 (1.1) 0 < M∞ < 1 (1.2) (1 − M 2 ) ≥ 0 (1.3) (1 − M 2 )φxx + φ y y = 0 (1.4) ∞ ∞ In two dimensions, we have ∞ v 0 = U∞ ³ ∂y ´ ∂x (1.5) BODY u 0 → 0, x, y → ∞ (1.6) v 0 → 0, x, y → ∞ (1.7) Now let β= q 2 1 − M∞ (1.8) and transform independent coordinates as follows: ξ=x (1.9) η = βy (1.10) And likewise, the dependent perturbation velocity potential e(ξ, η) = βφ(x, y) φ (1.11) This series of transformations lead to the following ∂ξ =1 ∂x φx = ∂ξ =0 ∂y ∂η =0 ∂x ∂η =β ∂y (1.12) e 1 h ∂φ e ∂ξ ∂φ e ∂η i 1 ∂φ e 1 ∂φ 1 ∂φ eξ = = + = = φ ∂x β ∂x β ∂ξ ∂x ∂η ∂x β ∂ξ β (1.13) 1 eξξ φ β (1.14) φxx = 1 φy = e 1 h ∂φ e ∂ξ ∂φ e ∂η i ∂φ e ∂φ 1 ∂φ eη = = + = =φ ∂y β ∂y β ∂ξ ∂y ∂η ∂y ∂η eηη φ y y = βφ (1.15) (1.16) Our transformed governing equation becomes β2 ³1 β ´ eξξ + βφ eηη = 0 φ (1.17) or eξξ + φ eηη = 0 φ (1.18) Our analysis drives us to the following question: H OW CAN WE EXPLOIT INCOMPRESSIBLE RESULTS TO ACCOUNT FOR COMPRESSIBILITY EFFECTS ? Compare the forms φxx + φ y y = 0, M∞ = 0 (1.19) β>0 (1.20) eξξ + φ eηη = 0, φ Consider the boundary condition on the airfoil surface. y = f (x), airfoil shape in x, y (1.21) η = q(ξ), airfoil shape in ξ, η (1.22) Our boundary condition may be expressed as follows: U∞ e ∂φ e df ∂φ 1 ∂φ = = = d x ∂y β ∂y ∂η (1.23) (x, y) space Similarly in (ξ, η) space U∞ e d q ∂φ = d ξ ∂η (1.24) (ξ, η) space Therefore, U∞ or df dq = U∞ dx dξ (1.25) df dq = dx dξ (1.26) Conclusions (a) The shape of the airfoil in x, y space is the same in ξ, η space. e, ξ, η implies that the compressible ﬂow over an airfoil in x, y space is related to an incom(b) φ pressible ﬂow in ξ, η space over the same airfoil. Now, let’s return to the pressure coefﬁcient: 0 u c p = −2 U∞ 1 ∂φ = −2 U∞ ∂x e 2 1 ∂φ =− U∞ β ∂ξ (1.27) 2 Let the incompressible pressure coefﬁcient be 0 c p 0 ≡ −2 e ue 1 ∂φ = −2 U∞ ∂ξ U∞ (1.28) Therefore, substituting 1 cp β 0 cp0 cp = cp = q (1.29) (1.30) 2 1 − M∞ This is the Prandtl-Glauert rule. It is a similarity rule that relates incompressible ﬂow over a given two-dimensional proﬁle to subsonic compressible ﬂow over the same proﬁle. From above results, it can be shown that cL0 cL = q (1.31) 2 1 − M∞ C M0 CM = q (1.32) 2 1 − M∞ 0 ue u =q (1.33) 2 1 − M∞ What does it mean? 3

Linearized Subsonic Flow

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