《空气动力学》（双语版）chapter 12 The compressibilty correction rule for thin wing

* The compressibilty correction rule for thin wing The effect of compressibility in 3-D flows is somewhat less dramatic than with 2-D flows, but many of the same effects become important. Many of the same techniques for predicting linear compressibility effects work in 3-D too. For example we can transform the 3-D Prandtl-Glauert equation into the 3-D Laplace equation for incompressible flow by changing variables Just as in 2-D

*The compressibilty correction rule for thin wing The effect of compressibility in 3-D flows is somewhat less dramatic than with 2-D flows, but many of the same effects become important.Many of the same techniques for predicting linear compressibility effects work in 3-D too. For example, we can transform the 3-D Prandtl-Glauert equation into the 3-D Laplace equation for incompressible flow by changing variables just as in 2-D

a9 a. ao +~,2 0 2 Defining the geometry of a finite wing: y=f(x, z) So the boundary condition is: dB

0 ˆ ˆ ˆ 2 2 2 2 2 2 2 =   +   +   x y z     Defining the geometry of a finite wing: y=f(x,z) So the boundary condition is: x y V y   =     ˆ

Transform the(x,y,z)and 9 in the following way 人.x 1 =1 B203b,203b,0203b 0 2 O22A7 2元:a42

0 2 2 ˆ 2 2 2 ˆ 2 2 2 ˆ 2 2 =   +   +                   x y z Transform the (x,y,z) and in the following way:  ˆ           ˆ = ˆ = = = z y x z y x

Bn 2 If +0+00=0 Derive the boundary ao a, ao condition on 012,05 ox Ox n ,as If O an 2 Then 05

If           ˆ 2 ˆ 2 ˆ 2 2 x y z = = 0 2 2 2 2 2 2 =   +   +                     =     =    y x y x y V y ˆ Derive the boundary ˆ condition: x y V y   =     ˆ           =   V y x 2 ˆ 1 2 ˆ = y x    If Then       =   V

B222 57 77 βyz 0=B2 0=B

          ˆ 2 ˆ 2 ˆ 2 2 x y z = = 1 2 ˆ = y x            2 ˆ = = = = z y x       ˆ 1 = = = = z y x

The Geometry relation 0=B Bt 6=B6 6=B6 1=元 1=元 Ab= BA A0=B4 tan xo=o tan x tan no =o tan x

The Geometry relation:                tan 1 tan 0 0 0 0 0 0 = = = = = = A A                tan 1 tan 0 0 0 0 0 0 = = = = = = A A

The relation of Aerodynamic coefficients 210C 2210p Ba B25 P B 2p,0 C(M,t, 0,a, A, tan x,n)= PC,(M=0, Br,BB, Ba, BA,otan x, a) This correction rule is Goethert Rule

The relation of Aerodynamic coefficients: 2 ,0 2 1 2 1 ˆ 2 p x p C V V C      =   = − = −   tan , ) 1 ( 0, , , , , 1 ( , , , , ,tan , ) 2                 C M A C M A p p = = 2 ,0 2 1 2 1 ˆ 2 p x p C V V C      =   = − = −   This correction rule is Goethert Rule

Derivation of the 3-D Prandtl-Glauert correction rule from Goethert rule C(M,t, 0,a, A, tan x, n) C (M=0, T, 0, a, BA,tan x, n B Cn(M=0, Br, BB, Ba, BA,tan x, n) C(M=0,T, 6,a, BA,tan x, n

                              tan , 1 ( 0, , , , , tan , ) 1 ( 0, , , , , 1 tan , 1 ( 0, , , , , ( , , , , ,tan , ) 2 C M A C M A C M A C M A p p p p = = = = Derivation of the 3-D Prandtl-Glauert correction rule from Goethert rule:

Cn(M, T, 0,a, A, tan x, n) C(M=0, t, 0, a, BA,tan x, n) B This is the Prandt -Glauert correction rule

tan , ) 1 ( 0, , , , , 1 ( , , , , ,tan , )              C M A C M A p p = = This is the Prandtl-Glauert correction Rule

Nonetheless, changing the lift curve slope just by the Prandtl-Glauert factor does not do too badl 2几凸 阝(A+2 A somewhat better approximation is obtained by applying the Prandtl-Glauert correction to the 2-D lift curve slope, then applying the downwash correction from lift line theory 2n(-e)2t 2亿凸 A+2

Nonetheless, changing the lift curve slope just by the Prandtl-Glauert factor does not do too badly: A somewhat better approximation is obtained by applying the Prandtl-Glauert correction to the 2-D lift curve slope, then applying the downwash correction from lift line theory