《空气动力学》（双语版）chapter11 SUBSONIC COMPRESSIBLE FLOW OVER AIRFOILS：LINEAR THEORY（2/2）

CHAPTER 11 SUBSONIC COMPRESSIBLE FLOW OVER AIRFOILS LINEAR THEORY 11. 4 PRANDTL-GLAUERT COMPRESSIBILITY CORRECTION The methods that approximately take into account of the effects of compressibility by correct the incompressible flow results is called compressible corrections

CHAPTER 11 SUBSONIC COMPRESSIBLE FLOW OVER AIRFOILS: LINEAR THEORY 11.4 PRANDTL-GLAUERT COMPRESSIBILITY CORRECTION The methods that approximately take into account of the effects of compressibility by correct the incompressible flow results is called compressible corrections

We will derive the most widely known correction of Prandtl-Glauert compressibility correction in this section Since the prandtl-glauert method is based on the linearized perturbation velocity potential equation (1-M2) 3×∞2 0 So it has restrictions: thin airfoil at small angle of attack purely subsonIC give inappropriate results at M。≥0.7

We will derive the most widely known correction of Prandtl-Glauert compressibility correction in this section. Since the Prandtl-Glauert method is based on the linearized perturbation velocity potential equation: 0 ˆ ˆ (1 ) 2 2 2 2 2 =   +   −  x y M   So it has restrictions: thin airfoil at small angle of attack; purely subsonic; give inappropriate results at M  0.7

B2=(1-M2) 02a2 0 7= (5,)=B0(x,y)

0 ˆ ˆ 2 2 2 2 2 =   +   x y    (1 ) 2 2  = − M  = x  = y ( , ) ˆ  ( ,) =  x y

ax as ax an ax Ba5 oOo on n ao ay as ay an Oy 00(1a)05_102 Ox2 a5( ox Ba5 00(00ma0 Oy2an(n丿ay =P0m1

            =     +     =   1 ˆ ˆ ˆ x x x            =     +     =   y y y ˆ ˆ ˆ 2 2 2 2 1 1 ˆ            =               =   x x 2 2 2 2 ˆ           =               =   y y

B B 00+P0m O 0o Ba5 0 0202=0

0 1 ˆ ˆ 2 2 2 2 2 2 2 2 2 2 =   +   =   +             x y 0 2 2 2 2 =   +      

Boundary Condition In(x,y) space df a 1 a ao dx ay B an an In transformed space dg ao dx an So This equation implies that the shape of the airfoil in the transformed space is the same as the physical space. Hence, the above tranform tion relates the compressible flow over an airfoil in(x, y) space to the in (5, n) space over the same airfoil

Boundary Condition :         =   =    = ˆ 1 ˆ dx y df V      = dx dq V In (x,y) space: In transformed space: dx df dx dq So = This Equation implies that the shape of the airfoil in the transformed space is the same as the physical space. Hence, the above tranforma￾tion relates the compressible flow over an airfoil in (x,y) space to the in space over the ( ,) same airfoil

2 2 a 2 1 ao v ax B ax 12 B 0p=1 as

) 2 ( 1 ˆ 2 1 ˆ 2 ˆ 2         = −   = −   = − = −     V V V x V x u Cp = u     ) 2 ( 1  = − V u Cp 

p,0 B 0 P 1-M 11.51) l.0 M m.0 2

 p,0 p C C = 2 ,0 1−  = M C C p p 2 ,0 2 ,0 1 1   − = − = M c c M c c m m l l (11.51)

11.5 IMPROVED COMPRESSIBILITY CORRECTIONS -1.4r Laitone Karman-Tsien Prandtl. glauert e Experiment o.20.40.6O.8 C=1=M2+M+=M2)2(1 p,0 P (11.55) M2+[M2/1+2M2/2V1-M。2)C P

11.5 IMPROVED COMPRESSIBILITY CORRECTIONS 1 [ /(1 1 )] ,0 / 2 2 2 2 ,0 p p p M M M C C C −  +  + −  = ,0 2 2 2 2 ,0 / 2 1 )] 2 1 1 [ /(1 p p p M M M M C C C    −  − − + + =  (11.54) (11.55)

11.6 CRITICAL MACH NUMBER In this section we deal with several aspects of transonic flow from a qualitative point of view Local M =0.435 What is the definition of M=0.3 Critical Mach Number? The critical Mach number Local M4 =0.77 Ma=0.5 is that free stream Mach number at which sonic flow Local Ma =1.0 0.61 is first achieved on the airfoil surface. Sonic line where M I 0.65>M

11.6 CRITICAL MACH NUMBER In this section we deal with several aspects of transonic flow from a qualitative point of view. What is the definition of Critical Mach Number? The critical Mach number is that free stream Mach number at which sonic flow is first achieved on the airfoil surface