《热力学》课程教学资源（PPT课件）Chapter 11 SUBSONIC COMPRESSIBLE FLOW OVER AIRFOILS：LINEAR THEORY（1/2）

CHAPTER 11 SUBSONIC COMPRESSIBLE FLOW OVER AIRFOILS LINEAR THEORY 11. 1 Introduction This chapter mainly deal with the properties of two-dimensional airfoils at mach number above 0.3 but below 1, where the compressibil ity must be considered

CHAPTER 11 SUBSONIC COMPRESSIBLE FLOW OVER AIRFOILS: LINEAR THEORY 11.1 Introduction This chapter mainly deal with the properties of two- dimensional airfoils at Mach number above 0.3 but below 1, where the compressibility must be considered

Velocity potential equation Prandtl-Glauet Compressibilty correction Linearized velocity potential equation Improved compressibilty Correction Critical mach number The area rule for transonic flow F gure 11 Supercritical airfoils Road Map for Chap 1

Velocity potential equation Linearized velocity potential equation Prandtl-Glauet Compressibilty correction Improved compressibilty Correction Critical Mach number The area rule for transonic flow Supercritical airfoils Figure 11.1 Road Map for Chap.11

11.2 The Velocity Potential Equation For two-dimensional, steady irrotational, isentropic flow, a veloc potential o=o(x, y) can be defined such that V=Vo The introduction of velocity potential can greatly simplify the governing equations, we can derive the velocity potential quation from continuity, momentum, energy equations

11.2 The Velocity Potential Equation For two-dimensional , steady, irrotational , isentropic flow, a velocity potential can be defined such that :  = (x, y) V =  The introduction of velocity potential can greatly simplify the governing equations, we can derive the velocity potential equation from continuity, momentum, energy equations:

The continuity equation for steady, two-dimensional flow S 无法显示该图片 pm)+(py)=0 ( 0 or ptu+vtp Substituting u=,v= Into it, we ge 0 oo op oo op p(2+2)+ ox

The continuity equation for steady,two-dimensional flow is : 0 ( ) ( ) =   +   y v x u  = 0   +   +   +   y v y v x u x u     or Substituting into it, we get y v x u   =   =   , ( ) 0 2 2 2 2 =     +     +   +   x y x x y y       

To eliminate p from above equation, we consider the momentum equation dp=-prdr dp=-ovdv-pdv2=pd(211,2 pJ00、2,0 Since the flow we are considering is isentropic, so dp o a O

To eliminate from above equation, we consider the momentum equation :  dp = −VdV ( ) 2 2 2 2 2 dp = − VdV = − dV = − d u + v            +   = − 2 2 ( ) ( ) 2 x y dp d    Since the flow we are considering is isentropic, so 2 a p d dp s =           =  

dp 02,0 d(x)+( 2a SO 0、2,0 2 or p2 a or 2a ay oX 0y、,0op,0 2人x

        +   = − 2 2 2 ( ) ( ) 2 x y d a d             +     = −   2 2 2 ( ) ( ) x 2a x x y     so         +     = −   2 2 2 ( ) ( ) y 2a y x y     ( ) 0 2 2 2 2 =     +     +   +   x y x x y y       

We get the velocity potential equation +1 100、2|0 a ox 03=0 (11.12) In this equation the speed of sound is also the function of o y-1ab2,0 +() or

( )( ) 0 2 ( ) 1 ( ) 1 1 1 2 2 2 2 2 2 2 2 2 2 =        −           + −           − a x y x y a x x a y y        We get the velocity potential equation: In this equation , the speed of sound is also the function of :          +  −  = − 2 2 2 0 2 ( ) ( ) 2 1 x y a a    (11.12)

For subsonic flow, Eq. 11 12 is an elliptic partial differential equation. For supersonic flow, Eq 11 12 is a hyperbolic partial differential equation. For transonic flow, Eq 11 12 is mixed type equation

For subsonic flow, Eq. 11.12 is an elliptic partial differential equation. For supersonic flow, Eq.11.12 is a hyperbolic partial differential equation. For transonic flow, Eq.11.12 is mixed type equation

Eq. 11.12 represents a combination of continuity, momentum, energy equations. In principle, it can be solved to obtain o for the flow field around any two-dimensional flow The infinite boundary condition is ao=va 0 The wall boundary condition is 0p=0 an

Eq. 11.12 represents a combination of continuity, momentum, energy equations. In principle, it can be solved to obtain for the flow field around any two-dimensional flow. The infinite boundary condition is =    = V x u  = 0   = y v  The wall boundary condition is = 0   n  

Once is known, all the other value flow variables are directly obtained as follows p Calculate u and y. u= ant 2. Calculate a 2y-1002,02 a=a +( 3. Calculate M: M_V-Vu'+v 4. Calculate Tp, p: T=To(1+y-IM)-1 2 B=P0(1+2-1 M2) 尸=P0(1+ M2)

Once is known, all the other value flow variables are directly  obtained as follows: 1. Calculate u and v: and x u   =  y v   =  2.Calculate a:         +  −  = − 2 2 2 0 ( ) ( ) 2 1 x y a a    3.Calculate M: a u v a V M 2 2 + = = 4. Calculate T,p,  : 1 1 2 0 2 1 0 2 1 0 ) 2 1 (1 ) 2 1 (1 ) 2 1 (1 − − − − − − = + − = + − = +         M p p M T T M