17 



40 x/Ak 



a in eq (28) from linear stability theory '^ 

 a= in eq (28) y 



FIGURE 5. Downstream development of u' -disturbance at 

 y/Ay = 3 for different boundary conditions at the 

 downstream boundary (boundary layer on a flat plate) . 



theory) was used while for the other calculation a 

 was simply set zero. It is obvious that even with 

 the poor value for a the upstream influence is re- 

 stricted to a region of approximately one wavelength, 

 while the disturbance further upstream is practi- 

 cally unaffected. This relatively minor upstream 

 influence can also be observed in Figure 6 where 

 the amplification curves (for the maximum of u') 

 are compared for the two cases. The disturbance 

 amplification further than one wavelength upstream 

 is practically unaffected by the value used for a 

 in Eqs. (28) . 



5. NUMERICAL METHOD 



A numerical method for transition studies has to 

 generally allow for numerical solutions of a 

 boundary-value problem for the calculation of the 

 steady flow, i.e. solution of Eqs. (1) and (2) or 

 Eqs. (1) and (7) (without 3u/3t in Eq. 1) with ap- 

 propriate boundary conditions discussed in Section 

 4. Further the solution of a mixed initial-boundary- 

 value problem for the calculation of the unsteady 

 flow is required, i.e. solution of Eqs. (1) and (2) 

 or Eqs. (1) and (7) with the boundary conditions for 

 the unsteady, disturbed flow and initial conditions 

 discussed in Section 4. The partial differential 

 equations are of fourth order for the u,i|j formula- 

 tion and of even higher order for the a),u, v-system. 

 For both formulations the governing equations are 

 elliptic for the calculation of the steady flow and 

 parabolic for the unsteady flow. In this paper the 

 discussion is restricted to application of finite- 

 difference methods for the solution of the mathe- 

 matical problems posed. 



A difference method for investigations of hydro- 

 dynamic stability and transition phenomena has to 

 meet a number of requirements in order to ensure 



15 



a. in eq (28) from 

 linear stability theory. 



1.0 



t 



10 



20 



30 



40 x/Ax 



FIGURE 6. Amplification curves for maximum of u' for 

 different boundary conditions at the downstream bound- 

 ary (boundary layer on flat plate) . 



success. Some of the requirements deemed most 

 important in this context are as follows : 



(i) Stability , convergence 



Rigorous mathematical proofs of (numerical) stabil- 

 ity and convergence for nonlinear problems as dif- 

 ficult as the one at hand have not been accomplished 

 as yet. For the present investigation, however, 

 stability of the numerical method is of fundamental 

 importance. Numerical instability is frequently 

 exhibited in form of oscillations which would be 

 hardly discernible from the physically meaningful 

 oscillations caused by introduced forced perturba- 

 tions . Hence , a prospective difference method has 

 to be highly stable, even for relatively large 

 Reynolds numbers . 



In general, for transition studies of the kind 

 considered in this paper convergence is also quite 

 serious. Convergence is not necessarily guaranteed 

 if for a properly posed problem the numerical scheme 

 is stable and consistent as is the case for linear 

 partial differential equations of second order 

 [Lax's equivalence theorem, see Richtmyer and 

 Morton (1967)]. However, experimenting first with 

 small periodic disturbances one can at least empir- 

 ically check the convergence behaviour of the nu- 

 merical method by comparing calculations for various 

 grid sizes with linear-stability-theory results and 

 experimental measurements. Then for other dis- 

 turbance inputs, such as large amplitude periodic 

 disturbances, one hopes that the convergence char- 

 acteristics do not change significantly. 



(li) Accuracy of second order 



For these investigations at least second-order ac- 

 curacy of the numerical method (i.e. the truncation 

 error of the difference analogue to the governing 

 equations, initial and boundary conditions at least 

 of second order) is required to exclude or minimize 

 undesirable non-physical effects, such as artificial 

 viscosity, when mesh intervals of practical sizes 

 are used. 



(Hi) Realistic resolution of the transient char- 

 acter of unsteady flow fields 



Transition phenomena are of highly unsteady nature, 

 with the time-dependent behaviour of the flow being 

 of special interest. Thus, the difference method 

 has to be such that realistic resolution of the 

 transient character of such flow fields is possible. 

 Therefore truly second-order accuracy is also de- 

 sirable for the time derivative. 



(iv) Efficiency with respect to computational 

 speed and required fast-access storage capacity 



Numerical solutions of the complete Navier-Stokes 

 equations for unsteady flows at high Reynolds 

 numbers require niamerous time-consuming numerical 

 operations. Therefore computers with large, fast- 

 access computer storage capacity, reaching even the 

 limits of modern computer systems, are necessary. 

 A prospective difference method for transition 

 simulations has to be extremely efficient, i.e. 

 maximizing computational speed and minimizing re- 

 quired computer storage capacity as much as possible. 



