21 



7. CONCLUDING REMARKS 



The objective of the present review was to discuss 

 possible approaches to numerical simulations of sta- 

 bility and transition based on numerical solutions of 

 the Navier-Stokes equations using finite-difference 

 methods. The approach, allowing investigations of 

 spatially propagating disturbance waves, mainly 

 elaborated upon in this paper, appears most promis- 

 ing for realistic numerical investigations of physi- 

 cal phenomena occurring in transition. The immense 

 amount of reproducible data obtained from such cal- 

 culations allows detailed information of any part 

 of the flow field which may be helpful to gain in- 

 sight into essential mechanisms occurring in tran- 

 sition. 



The restriction of the numerical model to two- 

 dimensional flows has also a positive side. With 

 this model truly two-dimensional numerical experi- 

 ments can be performed while in laboratory experi- 

 ments it is always difficult to completely exclude 

 unwanted three-dimensional effects. Of course the 

 later stages of transition are inherently three- 

 dimensional in nature and therefore for a study of 

 these later developments a three-dimensional model 

 would be desirable . 



The main difficulties and limitations of such 

 simulations result from the large gradients which 

 occur in the transition process. For adequate 

 resolution of the large gradients which become even 

 larger for more complicated phenomena, increasing 

 numbers of grid points are required which may lead 

 to excessive requirements of computer storage and 

 computation time . 



In spite of these difficulties the number of 

 numerical simulations of transition, similar to 

 the approach discussed in this paper, is likely to 

 increase due to the enormous potential inherent in 

 such investigations. Emphasis will probably be on 

 the development of difference methods with higher 

 accuracy which are applicable for such studies. 

 Additionally, increasing use of numerical methods 

 other than finite-difference methods is likely, 

 such as spectral methods or finite-element methods. 

 Finally, with continuing progress in the develop- 

 ment to high-speed digital computers, detailed 

 quantitative investigations of three-dimensional 

 transition phenomena will probably become feasible 

 in the near future . 



This research is supported by the Deutsche 

 Forschungsgemeinschaft, Bonn-Bad Godesberg, con- 

 tract Ep 5/7 . 



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