《热力学》课程教学资源（PPT课件）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

8 09 06 2+x.2+ 0 oX Defining the geometry of a finite wing: y=f(x, z) So the boundary condition is or

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 in the following way 0=2 B203b 2⌒2 d,200 0 0242n 2

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

2 B-7 2 0o 09. ao 0 2 02n2o5 Derive the boundary condition ay n an OO Oy n an or Oxn a o an If Then an

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

22 Bn 2 xX X 77= 77= βyz 0=B2y =B0

          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 to=Br o=Br 6o=B6 6o=B0 o=ba o=Ba 1= A= BA A=BA tan yo=tan x tan Xo=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 41210 20.210 V B2 OE B205 p,0 B 2p,0 C(M,I, 6,a, A, tan x,)= OC(M=0, Br, Be, Ba, BA, tan x, d) 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, 6,a, A, tan x, n) C,(M=0, t, 8, a, BA,tan x, n C,(M=0, Bt, 0,Ba, BA,tan x, n) 2 C,(M=0,, 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:

C,(M,t, 0,a, A, tan x, n) C(M=0, T, 0,a, BA,tan x, n) B This is the prandtl-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 badly 2I A 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(-8)2m(-n) 2 GA+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