64 



from sixth to fourth order, but the velocity pro- 

 files and wave parameters are not transformed. 

 This approach is equally valid for the temporal 

 and spatial theories, but for the latter a growth 

 direction must be assigned before eigenvalues can 

 be computed. In M77 this direction was taken 

 equal to the direction of the real part of the 

 group velocity and numerical results were obtained 

 for two-dimensional incompressible and compressible 

 flat-plate boundary layers and for the rotating 

 disk boundary layer. 



In the present paper, a theoretical presentation 

 is given in Section 2 to justify the use of a 

 spatial mode whose direction of growth is determined 

 by the complex group velocity. In Section 3, some 

 results concerning three-dimensional spatial waves 

 in the Blasius boundary layer are given as an 

 example . In Section 4 , we adapt the family of 

 yawed-wedge three-dimensional boundary layers 

 [ Cooke (1950) ] for use in stability calculations. 

 In Section 5 , under Boundary Layers with Small 

 Crossflow, we consider the effect of the flow angle 

 (the angle between the local potential-flow direc- 

 tion and the direction of the pressure gradient) on 

 the maximum amplification rate for small pressure 

 gradients. Next, in Section 5 we take up cross- 

 flow instability and determine the critical Rey- 

 nolds number for several combinations of pressure 

 gradient and flow angle. We then obtain the max- 

 imum amplification rate and instability boundaries 

 of all unstable frequencies as a function of the 

 wavenumber vector for a favorable pressure-gradient 

 boundary layer which is unstable at low Reynolds 

 numbers only because of crossflow instability. 

 Finally, in the last part of Section 5, we repeat 

 the latter calculation for an adverse pressure- 

 gradient boundary layer with crossflow instability 

 at a Reynolds number where the boundary layer is 

 unstable even without crossflow instability. In 

 all of the examples, only the amplification rate 

 is calculated, and on the basis of locally uniform 

 flow. No results concerning wave amplitude are 

 given, although in Section 2, we make use of a 

 simple wave amplitude equation in order to properly 

 define the spatial amplification rate. 



^3 = 



iZj 



R ^ 



i (aU + 6W 



0)) + 



^3' 



for the determination of the eigenvalues. The 

 primes refer to differentiation with respect to 

 y, and the dependent variables are 



Zj(y) = af(y) + 6h(y), Z3 (y) = 4. (y) , 



Zij(y) = TT(y) . 



There are two additional uncoupled equations for 

 h(y) . In Eqs. (2) , a and 6 are the complex wave- 

 number components in the x and z directions, o) is 

 the complex frequency, U and W are the mean velo- 

 city components in the x and z directions , and R 

 is the Reynolds number UpL /v*, where the velocity 

 scale Up is the potential velocity and L* is a 

 suitable length scale. Asterisks refer to dimen- 

 sional quantities. The modes in Eq. (1) can be 

 termed plane waves in the x,z plane because of the 

 phase function, even though there is a modal struc- 

 ture in the y direction. 



The boundary conditions are 



Zj(0) = 



Zi(y) 







Z3(0) = 



Z3(y) * 



(3) 



as y 



If we choose x to be the direction of the local 

 potential flow, then z is the crossflow direction 

 and 



U(y) 



W(y) 







as y 



Thus U(y) is the mainflow velocity profile; W(y) is 

 the crossflow profile. 



In the temporal stability theory, a and g are 

 real, and Eqs. (2) can be reduced to two-dimension- 

 al form in two different ways. The first transfor- 

 mation is 



2 2 1^ _ g 

 a=(a+e) ,U=U+— w. 



SR = aR , S/S = uj/a. 



(4) 



THREE DIMENSIONAL STABILITY THEORY 



Formulation and Transformations 



The linearized, incompressible, parallel-flow, 

 dimensionless Navier-Stokes equations for the 

 elementary modes 



'u(x,y,z,t) 

 v(x,y,z,t) 

 w(x,y,z,t) 



\p(x,y,z,t) 



exp[i(ax + Bz - ut) ] , 



(1) 



where u,v,w are the velocity fluctuations and p 

 is the pressure fluctuation, can be reduced to 

 (M77) 



Zi = Zy, 



2 2 

 Z^ = [a +6 + iR(aU + 6W - u))]Zi 



2 2 

 + (aU' + 6W')RZ3 + i(a + 6 )RZi,, (2) 



When W = 0, this is the transformation of Squire 

 (1933). It relates the eigenvalues of a three- 

 dimensional wave of frequency oi in a velocity pro- 

 file (U,W) at Reynolds number R to the eigenvalues 

 of a two-dimensional wave of frequency a)/cosi|' in 

 a velocity profile U + W tanijj at Reynolds number 

 R cosij), where 



ijj = tan 



-1 



(e/a) 



is the phase orientation angle. 

 The second transformation. 



a = (a 



R = R 



2 h 

 3 )^ 



aU 



aU + BW, 



is that of Stuart [Gregory et al. (1955) ]. It 

 relates the eigenvalues of a three-dimensional 

 wave of frequency 10 in a velocity profile (U,W) at 

 Reynolds number R to the eigenvalues of a two- 

 dimensional wave of the same frequency in a veloc- 

 ity profile U cosi|) + W sinijj at the same Reynolds 



