Stability and Transition Investigations 

 Using the Navier-Stokes Equations 



Hermann F. Fasel 



Universitat Stuttgart 



Stuttgart, Federal Republic of Germany 



SUMMARY 



With this paper an attempt is made to review the 

 stability and transition simulations, performed at 

 the University of Stuttgart, which are based on 

 finite-difference solutions of the Navier-Stokes 

 equations. Research in this area has demonstrated 

 that implicit finite-difference methods for the 

 solution of the complete Navier-Stokes equations 

 for unsteady, two-dimensional, incompressible flows 

 can be successfully applied to investigations of 

 hydrodynamic stability and to certain aspects of 

 transition. This approach of numerically solving 

 the partial differential equations describing the 

 underlying flow mechanisms promises to be a valuable 

 aid in transition research. In particular, this 

 concept may prove to be especially rewarding for 

 investigations of aspects of stability and transition 

 which as yet are not feasible with other theoretical 

 models. 



There are two main reasons for the attractiveness 

 of this approach: Firstly, no assumptions whatsoever 

 have to be made concerning the basic flow field under 

 investigation. Thus, for example, all possible 

 effects resulting from the growth of a boundary layer 

 in downstream direction can be included in such 

 investigations. Even strongly converging or diverg- 

 ing flows, or flows with separation and/or reattach- 

 ment can be studied. Secondly, no restrictions 

 have to be made concerning amplitude and form of 

 the disturbances which are injected into the flow. 

 Therefore, using larger disturbance amplitudes 

 certain nonlinear effects of the amplification 

 process can be readily investigated. 



The major aspects of this approach will be dis- 

 cussed in this paper. Emphasis will be placed not 

 only on conveying the advantages of such investi- 

 gations but also on elaborating the difficulties 

 and shortcomings of such numerical simulations. 



Finally, a conjecture concerning the course of 



future developments will be attempted. 



1. INTRODUCTION 



The phenomena occurring in transition from laminar 

 to turbulent flow have been the subject of inten- 

 sive research ever since the discovery that these 

 two entirely different states of flow exist. 

 From all the research efforts basically only one 

 universally-accepted theoretical concept evolved, 

 namely, linear stability theory, verified experi- 

 mentally by the famous experiments of Schubauer 

 and Skramstad (1943) . 



However, experimental evidence has also shown 

 that linear stability theory is only applicable 

 for one 'special' transition process, namely, 

 transition initiated by the presence of very small 

 disturbances in the flow. In this case a substan- 

 tial portion of the entire transition process is 

 indeed well described by this theory, i.e. the 

 amplification of two-dimensional disturbance waves 

 (the so-called Tollmien-Schlichting waves) can be 

 predicted adequately. But even for this special 

 transition process, triggered by small disturbances, 

 linear stability theory is inadequate in the 

 description and investigation of the mechanisms 

 that follow the growth of Tollmien-Schlichting 

 waves, and which finally cause the breakdown to 

 fully turbulent flow. Nevertheless, due to the 

 relative success of the linear stability theory 

 and its impressive experimental verification, the 

 vast majority of theoretical transition investi- 

 gations were, and still are, based on stability 

 theory concepts, thus constantly improving and 

 perfecting this theory. 



The inherent shortcomings of this concept 

 nontheless (such as being applicable only when 

 transition is initiated by small disturbances, or 

 that certain assumptions concering the basic and 

 disturbance flow have to be made to keep the 

 resulting equations tractible) led to a search for 

 other means to investigate transition. One of the 

 more promising concepts that has emerged in recent 



