68 



and in the x direction 



0.1073, -a^ 



3.085 



10 



-3 



The eigenvalues computed with (6) differ from these 

 values in the fourth decimal place, which means 

 that a. has an unacceptable error of 16.5%, an 

 error which can also be calculated directly from 

 (15b) . Consequently, this example reiterates that 

 (6) , or the real Squire and Stuart transformations, 

 can only be used if i|) = (or i|j = ij'gj.) • 



For the check of the transformation of an ele- 

 mentary spatial mode with growth direction x to the 

 'physical' mode with growth direction Xg^-, we start 

 by calculating the eigenvalues as a function of t() 

 for < i|) < 95° and the same F, R and i|) as in the 

 previous example. In addition, we calculate the 

 complex group velocity by evaluating the complex 

 derivatives of (2) from central differences for 

 increments in a,-, 6j- of ±0.001 about the calculated 

 Oj-, Pr ^t each ii . The real and imaginary parts of 

 the complex angle ijjg are listed in columns 2 and 3 

 of Table 1. The angle iig-j- of the growth direction 

 x' , as obtained from (l9b) , is listed in Column 

 4. Eigenvalues were computed as a function of ijj 

 with ii = 50° by integrating (2) , starting at y/L = 

 8.0, with a fourth-order Runge-Kutta integration 

 and 80 equal integration steps. The results are 

 listed in columns 5 and 6 . 



If (13) with ipgr = tan"l (C^j^/Cj^r' ^^ applied to 

 the a (i)) given in column 6, a nearly constant a^^ 

 is obtained out to about t = 60°. For ^ > 60°, Ogj. 

 decreases steadily, and at i|; = 95° it is 21% lower 

 than the Ogj- for t|; = iigj-- Columns 7 and 8 give the 

 angle i|) and wavenumber k for ii = ij^gr as calculated 

 from the phase-shift formula (22) of the transform- 

 ation for complex group velocity. The corresponding 

 amplification rate, as calculated from {21a), is 

 listed in column 9. Comparisons of directly com- 

 puted eigenvalues with these k and a are provided 

 in the last two columns. Column 10 lists the eigen- 

 value k computed for the i/jgj. of coliram 2 and the ijj 

 of column 7. Column 11 lists the amplification 

 rate a obtained from the eigenvalue a„j- accompany- 

 ing k and from (21b) . 



We see that the transformation formulas work 

 quite well out to ip = 60°, where the difference 



between columns 9 and 11 is 0.13%. The change of 

 the a in column 9 with ijj is only about half of the 

 change given by the transformation with real group 



velocity and the correct ij) 



gr 



given by C^jj. and C^^-- 



In this particular example, at least, the smallest 

 change of a with ijj is found if (13) is used with 

 ij; also computed from (13) on the basis of two 

 neighboring values of a(ijj) obtained with the fre- 

 quency held constant and Ogj. assumed to be indepen- 

 dent of ii . The conclusion to be drawn is that in 

 order to obtain the desired spatial amplification 

 rate a as defined by (21b), a(ii/)may be computed at 

 some convenient ij) which can differ from the correct 

 i|j by as much as 40° or 50°, but should be as close 

 as possible^ Only later, after ijjgj- and ii^^ are 

 know, is a(ii)) converted to o by the transformation 

 formulas. Almost any of the methods discussed above 

 for applying the transformations gives acceptable 

 numerical accuracy. 



Effect of Obliqueness Angle on Instability 



The frequency F = 0.2225 x 10""* used in the examples 

 of the previous Section is the most unstable fre- 

 quency at R = 1600, and the maximiim amplification 

 rate for this frequency occurs for ^ = 0° . The 

 distribution of a with i); is shown in Figure 2 for 

 this frequency and F x lo"* = 0.280, 0.1490 and 

 0.1008. The latter two frequencies are the most 

 unstable for ip = 60° and 75°, respectively. They 

 have their peak amplification rates, not for i|i = 0°, 

 but for ^ = 34.4° and 61.8°, respectively. These 

 results demonstrate that although the maximum am- 

 plification rate at a given Reynolds number with 

 respect to both frequency and orientation occurs 

 for a two-dimensional wave, the maximum amplification 

 rate with respect to orientation of given frequency 

 occurs for. a three-dimensional wave if the frequency 

 is less than the most unstable frequency. 



The envelope curve formed by the individual 

 frequency curves is also shown in Figure 2 . This 

 curve gives cij^^jj, the maximum amplification rate 

 with respect to frequency, as a function of iC- The 

 envelope curve emphasizes the wide range of unstable 

 orientations in a two-dimensional boiandary layer. 

 It can be seen that Om^x ^^ "°*- reduced to one-half 



TABLE 1 Numerical check of spatial-mode transformation for complex group 

 velocity. R = 1600, F = 0.2225 x 10""*, ^ = 50°. 



gr 



^i^ 



gr 



k a(iij) X 10- 



gi (19b) eig. , ii = 50" 



t k 

 (22) 



a X 10- 



a X iqS 



(21a) ^^5-' <!, = 4, 



i|j of C7 

 gr 



10 



11 



9.23 



9.21 9.21 



30.0 9.16 



-4.03 9.18 

 -4.02 9.16 

 -4.02 9.11 



60.0 9.08 -4.00 9.04 

 90.0 8.67 -3.88 8.63 

 95.0 8.36 -3.63 8.33 



0.1668 3.171 49.99 0.1669 3.145 



0.1669 3.127 50.00 0.1669 3.143 



0.1670 3.340 50.02 0.1669 3.137 



0.1672 4.928 50.06 0.1670 3.122 



0.1687 19.66 50.30 0.1677 2.977 



0.1710 46.04 50.66 0.1688 2.710 



0.1669 3.146 



0.1669 3.143 



0.1669 3.138 



0.1670 3.126 



0.1676 3.060 



0.1686 2.957 



