16.121 ANALYTICAL SUBSONIC AERODYNAMICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY Similarity Rules Consider the linearized equation for the perturbation equation Φ1 (x, y) in plane, steady ﬂow: 2 (1 − M ∞ ) ∂2 Φ1 ∂2 Φ1 + =0 ∂y 2 ∂x 2 Now, consider two separate ﬂows with their mach numbers M ∞1 and M ∞2 . 0 < M ∞1 < 1 0 < M ∞2 < 1 M ∞1 6= M ∞2 Also: V∞1 = Free stream velocity c = Airfoil chord T1 = Thickness ratio = t 1 /c ¡ ¢ ¡ ¢ y = t 1 f 1 xc = T1 c f 1 xc y = Airfoil shape/boundary or ¡x¢ y = T 1 f1 c c The boundary condition on the airfoil surface may be written ³ ∂Φ ´ 1 ∂y y=0 = U∞1 ³d y ´ dx B od y = U∞1 T1 f 10 ³x ´ c From lecture notes, the linearized pressure coefﬁcient may be written C p1 = − 2 ³ ∂Φ1 ´ U∞1 ∂x y=0 1 Now consider a second ﬂow: Φ2 = Φ2 (ξ, η) Assume: U∞ Φ1 (x, y) = A U∞1 Φ2 (ξ, η) 2 ³ 1−M 2 ´1/2 U∞1 ∞ Φ1 (x, y) = A U∞ Φ2 (x, 1−M 2 1 y) ∞2 2 Where A = constant, to be determined And 2= ξ=x ³ 1−M 2 ´1/2 ∞1 2 1−M ∞ 2 y Hence, equation governing Φ2 (ξ, η) becomes 2 (1 − M ∞ ) 2 ∂2 Φ2 ∂2 Φ2 + =0 ∂η2 ∂ξ2 Recall: Φ1 (x, y) is a solution corresponding to M ∞1 Φ2 (ξ, η) is a solution corresponding to M ∞2 The boundary condition along the airfoil proﬁle may be written: ¡ ∂Φ1 ¢ ∂x U∞ 1 y=0 = A U∞ ³ 1−M 2 ´1/2 ¡ ∞1 2 2 1−M ∞ ¡2 ¢ 0 x = U∞1 T1 f 1 c ∂Φ2 ¢ ∂η η=0 Now let: v u 2 u 1 − M∞ 1 T1 = A t T2 2 1 − M∞ 2 2 And let f 1 = f 2 (f is the same in both ﬂows) which means both airfoils are of the same family. The boundary condition along the airfoil in (ξ, η): µ ∂Φ2 ∂η ¶ η=0 = U∞2 T2 f 20 µ ¶ ξ c The pressure coefﬁcient may be written µ ¶ µ ¶ 2 ∂Φ1 2 ∂Φ2 C p1 = − =− A U∞2 ∂ξ η=0 U∞1 ∂x y=0 Likewise: C p2 = − µ ¶ 2 ∂Φ2 U∞2 ∂ξ η=0 Therefore: C p 1 = AC p 2 A= A1 A2 Two airfoils of the same family of shapes characterized by the thickness ratios T1 and T2 have pressure distributions given by coefﬁcients C p 1 and C p 2 . If the mach numbers of the ﬂows are M ∞1 and M ∞2 , respectively, then C p 1 = C p 2 provided: v u 2 u 1 − M∞ 1 T1 = A t T2 2 1 − M∞ 2 A= A1 A2 Or, formally: Cp A µ T =fn A q ¶ 2 1 − M∞ Recall A is a constant. 3 4 5

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