18 



in order to be capable at all of undertaking inves- 

 tigations of this nature with the computers available 

 today. 



Of the requirements discussed here, numerical 

 stability is the most stringent one and hence has 

 to be given most consideration. For this reason 

 only implicit methods are suitable. Implicit meth- 

 ods are generally much more stable than their im- 

 plicit counterparts. For the adequate resolution 

 of the large gradients, resulting from the strongly 

 time-dependent flow fields to be investigated, rel- 

 atively small spatial intervals Ax and Ay are re- 

 quired. Using explicit methods this could lead to 

 excessively small time-steps required to maintain 

 numerical stability. For example, using an explicit 

 counterpart to the present implicit method, the 

 time-step, according to a linearized stability anal- 

 ysis, would have to be more than 100 times smaller 

 for a practical calculation than when using the 

 corresponding implicit scheme. To satisfy require- 

 ment (iv) attention has to be given to making the 

 implicit difference method extremely efficient and 

 also to meeting the other requirements discussed 

 previously. 



Experimentation with various implicit difference 

 schemes suggested that 'fully' implicit schemes are 

 the most promising for transition studies. 'Fully' 

 implicit means that all difference approximations 

 and nodal values for the approximation of governing 

 equations and boundary conditions are taken at the 

 most recent time-level. For our fully implicit 

 method three time-levels are employed to obtain a 

 truncation error of second order for the time de- 

 rivative 3u/3t in Eq. (1) . 



For all space derivatives, central difference 

 approximations with second-order truncation error 

 are employed. The implementation of the boundary 

 conditions into the numerical scheme requires 

 special care so that overall second-order accuracy 

 can be maintained. 



This implicit scheme leads to two systems of 

 equations for the a),i|) formulation and to three 

 systems of equations for the a),u,v formulation. 

 These systems of equations can be solved by itera- 

 tion. Because of the retention of full implicity 

 the equation system resulting from the vorticity- 

 transport equation is coupled with the Poisson 

 equation systems via the nonlinear convection terms. 

 It is additionally coupled with the systems result- 

 ing from the Poisson equations via the calculation 

 of the wall vorticity from Eq. (17) for the a>,ij) 

 formulation and from either Eqs. (18) or (19) for 

 the a),u,v formulation. 



A very effective solution algorithm based on 

 line-iteration has been developed for our method 

 for this coupled system. It is discussed elsewhere 

 in more detail [Fasel (1978) ] . This solution algo- 

 rithm has shown to be equally effective when the 

 basic equations are transformed to allow for a vari- 

 able mesh in the physical plane such as, for exam- 

 ple, to concentrate grid points close to walls where 

 high gradients are expected. Overrelaxation to 

 accelerate convergence can be easily implemented as 

 has been done for several calculations [Fasel et al. 

 (1977)]. Tinother advantage is that the solution 

 algorithm is readily exchangeable to be applied for 

 both the governing equations in tiitip and ii),u,v formu- 

 lation. This has been successfully exploited in the 

 investigations of the effects of a backward-facing 

 step on transition. In this study both formulations 

 were used in the integration domain; the uj,i)j formu- 



lation was used in the region containing the corners 

 of the step which can be treated more conveniently 

 with this formulation. For the domain bounded by 

 the downstream boundary the io,u,v formulation was 

 applied, because it allows use of less restrictive 

 boundary conditions as discussed in Section 4. 



The effectiveness of this solution algorithm can 

 be best judged by presenting a typical computation 

 time for a practical calculation. For a periodi- 

 cally disturbed flow with small disturbance ampli- 

 tudes, using a 35 x 41 grid and calculating 260 

 time-steps , the required CPU time on a CDC 6600 

 is about five minutes, including the calculation 

 of the steady flow. This is relatively little, 

 considering that the flow is disturbed at every- 

 time level and that full implicity is retained in 

 the numerical method. 



6. NUMERICAL RESULTS 



The implicity difference method which we have devel- 

 oped has been subjected to crucial test calcula- 

 tions to verify its applicability to investigations 

 of stability and transition. First, the reaction 

 of the boundary- layer on a flat plate to periodic 

 disturbances of small amplitudes was investigated 

 in detail . It was demonstrated that the spatial 

 propagation of Tollmien-Schlichting waves could be 

 simulated where comparison of the numerical calcu- 

 lations with results of linear stability theory and 

 laboratory measurements showed good agreement. Re- 

 sults of such calculations for the numerical method 

 based on the a),u,v formulation are presented and 

 discussed elsewhere [Fasel (1976)]. 



The usefulness of the numerical simulations for 

 the investigation of two-dimensional, nonlinear 

 effects was demonstrated by calculating the reaction 

 of a boundary- layer flow to periodic disturbances of 

 larger amplitudes . Investigating the propagation 

 of spatially growing or decaying disturbance waves 

 in a plane Poiseuille flow (both in the linear and 

 nonlinear regime) verified that the numerical method 

 is not limited to boundary-layer flows but rather 

 that it is equally applicable to other flows of 

 importance. Finally, numerical investigations of 

 of transition phenomena in the presence of a two- 

 dimensional roughness element (backward-facing step) 

 showed that simulations with this numerical model 

 allow insight into processes which may possibly be 

 important for understanding certain transition mech- 

 anisms . Results of this investigation and of the 

 investigations mentioned before are discussed in 

 another paper [Fasel et al . (1977)]. 



Because the purpose of this paper is to review 

 main aspects of numerical transition simulations, 

 emphasis here is not on conveying new results or 

 details of numerical calculations. Rather, results 

 presented here are intended to be of exemplary 

 nature and were selected in order to clearly demon- 

 strate essential aspects of such simulations and 

 to show what can be expected from such numerical 

 calculations. 



The drawings' in Figures 7 and 8 should facilitate 

 an evaluation of the potential of such numerical 

 simulations, and, of course, also point out possible 

 disadvantages and limitations. Figures 7 and 8 

 show results for a boundary-layer flow on a flat 

 plate, disturbed at the upstream boundary with small 

 periodic disturbances. This case is particularly 

 suitable for demonstration purposes. The ensuing 



