北京航空航天大学：《工程流体力学》（双语版） Chapter 5 Boundary Layer

Chapter 5 Boundary Layer (BL) 5.1 Introduction D'Alembert Paradox(佯谬,疑题): A body immersed in a frictionless fluid has zero drag au a ou au l-+1 p ax 2 Ox- Oy

Chapter 5 Boundary Layer (BL) 5.1 Introduction * D’Alembert Paradox（佯谬，疑题）: A body immersed in a frictionless fluid has zero drag.           +   +   = −   +   2 2 2 2 1 y u x u x p X y u v x u u   

+一=X ~,2 p、ox2oy Inviscid 0 or Ox- ay 0 2

2  = 0           +   +   = −   +   2 2 2 2 1 y u x u x p X y u v x u u    Inviscid x p X y u v x u u   = −   +    1 0 2 2 2 2 =   +   y u x u or = 0   = y u  

Boundary layer(边界层) Although frictional effects in slightly viscous fluid are indeed present, they are confined to a thin layer near the surface of the body, and the rest of the flow can be considered inviscid (1904 PrandtL) 第三届国际数学大会 德国海登堡 (论性很小的流体运动) Ludwig Prandtl(1875-1953)

3 Boundary layer (边界层) Although frictional effects in slightly viscous fluid are indeed present, they are confined to a thin layer near the surface of the body, and the rest of the flow can be considered inviscid. (1904, Prandtl) Ludwig Prandtl(1875-1953) 第三届国际数学大会 德国海登堡 （论粘性很小的流体运动）

5.2 Three Thicknesses of a Boundary Layer bL Thickness d(名义厚度,厚度) The locus of points where the velocity u parallel to the plate reaches 99 percent of the external velocity U 2 Displacement Thickness 8*(排移厚度,移厚度) U

5.2 Three Thicknesses of a Boundary Layer 2. Displacement Thickness  * (排移厚度，位移厚度） d U y u x 1. BL Thickness d (名义厚度，厚度） The locus of points where the velocity u parallel to the plate reaches 99 percent of the external velocity U

U real ud ldeal s Udy=pU △m= !- pU=pr(-Mb=DU「(1-p rea PU8-8

5 u dy d *    0 udy d y U  =   0 udy    0 Udy = U   −     0 0 Udy udy ( )  = −   0 U u dy * U        = −   0 1 dy U u U        = −   0 * 1 dy U u m  real = m  ideal = m  = ( ) * m  real = U  −

3. Momentum thicknessδ(O)(动量厚度,动量损失厚度) ,= pudy. u y pua △M=0w-v △M=pUO" p8U2=p u(u-uky 0 ="= ou

3. Momentum Thickness   ** ( ) （动量厚度，动量损失厚度） y d U u ( )   = −   0 2 M Uu u dy  udy u    0  udy U   0 M U U **  =    U u(U u)dy  = −    0 ** 2 dy U u U u        = = −    0 ** 1 M  real = M  ideal =

5.3 Momentum Integral relation for flat-plate BL (平板边界层动量积分方程) streamline P O= const p= const△z=b U Steady incompressible dddΦe1 +-ldqpout-(ddp)in] RTT CV dt dt dmv dmv [(厘m-()m(,)m=pAU·U=pbhU anm 6 out PdAuu=l pbu?dy

5.3 Momentum Integral Relation for flat-plate BL (平板边界层动量积分方程) U=const p=const Steady & incompressible 1 [( ) ( ) ] s cv out in d d d d dt dt dt   = +  −  [( ) ( ) ] out in dmV dmV dmV dt dt dt = − RTT CV x y h U P d streamline o z = b 2 ( ) AU U bhU dt dmV i n =   =     =  =     0 2 0 ( ) dAu u bu dy dt dmV out 

dmy df )in] obu dy-pbhU/2 ann streamline P Momentum equation U ∑F=「m?- phu2° CV ∑F=D(g)→D=mbU2-mbbh Continuity:phu= pbudy=>pbhU2= pbuUdy D=pbl(U-u)hy速度积分法求阻力 0

8 Momentum equation: F D drag = − ( ) Continuity: 0 D b u U u dy ( )  = −   [( ) ( ) ] out in dmV dmV dmV dt dt dt = − 2 0 2 bu dy bhU  = −  2 0 2 F bu dy bhU  = −      = −    0 2 2 D bhU bu dy  =    0 bhU budy   =    0 2 bhU buUdy CV x y h U P d streamline o 速度积分法求阻力

D=pblu(U-u)dy=pbU2d1-)dy =pbu-8 D(x)=TA=bT(x)dx t, (xdx=puB 2d0 Von Ka'rma'n 1921 (1881-1963)

0 D b u U u dy ( )  = −   2 =   bU 2 w d U dx    = Von Ka rma n   1921 (1881-1963)  = −   0 2 (1 )dy U u U u bU  = x w x dx U 0 2  ( )   D(x) = w A  = x b w x dx 0  ( )

0 U ud(u/l δa(y/δ)0 u=a+by+cy BC:y=0=0 Q0 y=8u≈U u。0 a=0.b= C 2yy0≤y≤6(x) Uδδ

10 0 (1 ) u u dy U U   = −  y  ( ) x u(y) x 2 u a by cy = + + 2 2 0, , - U U a b c   = = = 2 2 u y y 2 U   = − 0 ( )  y x  y u = = 0 0 0 u y    y u U =   BC: 0 ( / ) ( / ) =   = y y U u U    =0   w = y y u  