0.0000 



0.05 



0,00 



0.0015 



0000 



-0 0015 



19 



0015 



0000 



0015 



FIGURE 7. Disturbance variables versus x/Ax and y/Ay (perspective representation) at t/At = 80; a) u' 

 b) v', c) u', d) (j' (different view). 



Tollmien-Schlichting waves that can be studied from 

 such calculations are thoroughly investigated, 

 experimentally as well as theoretically, and the 

 results of these calculations are therefore more 

 intelligible than those of more complicated phe- 

 nomena of transition. 



For these calculations, based on the a),u,v for- 

 mulation, the Reynolds number at the upstream bound- 

 ary is Re*=630. For the periodic disturbance input, 

 for which perturbation profiles of linear stability 

 theory are used, the frequency parameter (defined 

 as F=10 Bv/U , with disturbance frequency g) is 

 F=1.3. In this case the flow is unstable according 

 to linear stability theory (the location of the left 

 boundary corresponds to a point on the neutral curve) 

 and therefore the disturbances should become ampli- 

 fied in downstream direction. For the calculations 

 an equi-distant grid with 35 points in y-direction 

 and 41 points in x-direction was used. 



In Figures 7 and 8 the function values of the 

 disturbance flow (obtained by subtracting the 

 quantities of the basic flow from those of the total 

 flow) are plotted for all three fields of variables 

 u',v',a)', for which the total flow variables are 

 directly obtained from the numerical calculations. 

 To allow simultaneous representation of the func- 

 . tion values at all grid points a perspective rep- 

 resentation was chosen where the function values 

 are plotted versus the downstream coordinate x/Ax 



and the coordinate normal to the wall y/Ay. These 

 perspective representations allow the best possible 

 qualitative survey of the large amount of data ob- 

 tained from such calculations . 



In Figure 7 the disturbance variables u' ,v' ,0)' 

 are plotted for a time instance of t=80At, which 

 corresponds to a time of two time periods after 

 initiation of the disturbances at the upstream 

 boundary. In Figures 7a, 7b, and 7c the view is in 

 the direction away from the wall, looking slightly 

 in upstream direction. In Figure 7d the view 

 is also in the direction away from the wall, look- 

 ing now, however, downstream. From these figures 

 the propagation of the disturbance waves into the 

 undisturbed flow field can be clearly observed. 



Figure 8 shows the corresponding drawings for 

 the three variables u' ,v' ,u' at a time instance of 

 t=250At, that is, more than two time periods after 

 the disturbance wave reached the downstream bound- 

 ary. These plots demonstrate that the downstream 

 botmdary conditions work properly. Obviously, the 

 waves can smoothly pass through this boundary, 

 causing no noticeable reflections. Even after hun- 

 dreds of time-steps the flow at and near this 

 boundary maintains its time-periodic character and 

 therefore the state of the disturbance flow as rep- 

 resented in Figure 8 would repeat itself periodi- 

 cally if the calculations were continued for further 

 time-steps. 



