11 



FIGURE 2. Integration domain for plane 

 Poiseuille flow. 



there are basically two different approaches which 

 lead to entirely different conceptions of the trans- 

 ition simulation: 



1) Use of periodicity conditions at the upstream 

 (A-D) and downstream (B-C) boundary, i.e. 

 corresponding disturbance quantities are 

 equal at the two boundaries for all times . 



Here it is assumed that flow phenomena are spatially 

 periodic in downstream direction where the integra- 

 tion domain X contains integer multiples of the 

 spatial wavelength. When the spatial development 

 is forced to be periodic, the flow responds with a 

 temporal development. Thus, with this arrangement 

 the temporal reaction of the flow to an initial 

 disturbance (at t=0) of the flow field can be 

 studied. This case corresponds in linear stability 

 theory to an eigenvalue problem with wave number 

 a real and frequency B complex (6=3 +iB'), i.e. 

 amplification in time. Figure 3, for example, shows 

 a typical result of a finite-difference calculation 

 based on such an approach for a plane Poiseuille 

 flow [Bestek and Fasel (1977)]. Plotted here is a 

 time signal for a case which is unstable according 

 to linear stability theory. The flow is only dis- 

 turbed once at t=0. After a certain time span, 

 where considerable reorganization of the disturbance 

 flow takes place, the disturbances assume a periodic 

 character with a slight amplification in time- 

 direction. 



The Navier-Stokes calculation for this approach 

 may be conceived as a means of solving the eigen- 

 value problem as in linear stability theory, with 

 a and Reynolds number given and obtaining the fre- 

 quency Qj-, amplification rate £>i, and the amplitude 

 distribution of the distrubance flow. Of course 

 these answers could be obtained with considerably 

 less effort from linear stability analysis. The 

 advantage of this present approach is, however, 

 that it can be easily extended to investigations 



FIGURE 3. Temporal development of u' -disturbance at 

 y/Ay = 3 for initially disturbed flow (small ampli- 

 tude) ; spatially periodic case (plane Poiseuille flow) . 



of certain nonlinear effects by merely increasing 

 the amplitude level of the initial disturbances 

 [see, for example, George and Heliums (1972)]. An 

 equivalent study of nonlinear effects formulated 

 as an eigenvalue problem in a stability theory 

 analysis would, on the other hand, become consider- 

 ably more involved. 



A major drawback of this first approach is, how- 

 ever, that it is pratically only applicable for 

 basic flows that do not vary in downstream direction 

 (parallel flows) , because only then is the period- 

 icity assumption for the disturbance flow a real- 

 istic one. Thus, strictly speaking, boundary-layer 

 flows could not be treated in this manner since they 

 are basically (although very mildly) non-parallel. 

 It has been shown that non-parallel effects can 

 have a strong influence on the stability character- 

 istics of this flow [caster (1974) , Saric et al. 

 (1977)]. 



A second, perhaps even more serious disadvantage 

 of this model is that the disturbance development 

 in downstream direction cannot be investigated. As 

 observed in numerous laboratory experiements the 

 phenomena of transition are not periodic in space 

 but rather are inherently space dependent. The 

 disturbance flow may vary rapidly in downstream 

 direction. This space dependency of the transition 

 process does not only occur for flows where the 

 basic flow is already dependent on the downstream 

 location. It also occurs when the basic flow does 

 not vary in downstream direction, as was impress- 

 ively demonstrated experimentally by Nishioka et 

 al. (1975) for the parabolic profiles of plane 

 Poiseuille flow between parallel plates. Thus, 

 this model is not suitable for realistic studies 

 of transition phenomena. 



However, finite difference simulations based on 

 this approach become considerably less involved 

 and are less costly in practical execution than for 

 the second approach discussed subsequently. The 

 former approach is therefore applicable for funda- 

 mental investigations of various unresolved ques- 

 tions in hydrodynamic stability (such as certain 

 nonlinear effects) or for preliminary studies of 

 flow simulations based on the approach discussed 

 below. 



2) At the upstream boundary, time-dependent 



disturbances are introduced. Use of bound- 

 ary conditions at the downstream boimdary 

 which allow downstream propagation of the 

 spatial disturbance waves . 



This second approach differs entirely in concept 

 from the first one. Here, the reaction of the 

 flow field to the disturbances introduced at the 

 upstream boundary is of interest, particularly the 

 spatial developments of the ensuing disturbance 

 waves. In contrast to the previous approach, this 

 case corresponds in stability theory to an eigen- 

 value problem with a complex (a=aj.+ici^) and 6 real. 

 A typical result for a boundary-layer flow of a 

 calculation based on this concept is shown in Fig- 

 ure 4. Plotted is the disturbance variable u' 

 (velocity component in x-direction) versus the down- 

 stream coordinate x. The downstream development 

 of the disturbance (in this case amplification) may 

 be clearly observed. Thus, this approach enables 

 the calculation of the spatial reaction of the flow 

 to upstream disturbances, and therefore realistic 

 simulations of space -dependent transition phenomena 



