《热力学》课程教学资源（PPT课件）Chapter 11 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-Ma)2+ 0 2 or 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) 209+0=0 or X n= By 0(,)=B0(x,y)

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

05 doan 1 ao as Ox an ax B as do doas ao an do i5_10 B5丿axBa 00(0)0naO2b ay2 On(an ay

            =     +     =   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

100+p0m1 βa2 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 ao 1 ao ag dx ay Ban an In transformed space dq ao dx an df So dx d 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

2221p P ax y 120p B v as 12 B V

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

M 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 Laitone Karman· Tsien Prandtl-Glauert e Experment 0.4 0.2O p,0 1-M2+M2/l+1-M2)Cn0/2(154) p,0 (11.55) +[M2(1+2,M2/2V1-M)Cp0

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 =0.3 Critical Mach Number? The critical mach number Local M=0.77- M。=0.5 is that free stream Mach (6 number at which sonic flow Local M,=1.0 0.61 is first achieved on the (c) arton surface Sonic line where M I M=0.65>Mc

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