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FIGURE 8. Disturbance variables versus x/Ax and y/Ay (perspective representation) at t/At - 250; a) u', 

 b) v' , c) u', d) Lo' (different view). 



In Figures 7 and 8 the large gradients normal 

 to the wall of the u' and u' disturbances become 

 clearly visible (for u' this can be best observed 

 from Figures 7d and 8d) while v' changes more grad- 

 ually. The large gradients observable in these re- 

 sults indicate already the major difficulties and 

 limitations in numerical simulations of transition 

 phenomena. In a numerical solution method these 

 large gradients have to be adequately resolved to 

 obtain meaningful representation of essential physi- 

 cal phenomena. For nonlinear disturbance waves re- 

 sulting from disturbance input with larger amplitudes 

 [Fasel et al. (1977)] or for other more complicated 

 transition phenomena the gradients may become even 

 considerably larger. Using finite-difference meth- 

 ods of a given accuracy (for example, second order 

 as for the present method) better resolution can 

 only be achieved by using additional grid points. 

 This, however, leads to ever larger equation sys- 

 tems the sizes of which are limited by computer 

 storage capacity and computation time. 



Some help can be expected from employing vari- 

 able mesh systems allowing allocation of more grid 

 points closer to walls, where the gradients are 

 largest, and using fewer points further away where 

 gradients are small. This can be best achieved 

 using coordinate transformations for which test 

 calculations have shown that sizable savings in 

 the number of grid points, and also in computation 



time , are possible to achieve accuracy comparable 

 with calculations in an equidistant grid. Addi- 

 tional improvement may be expected from application 

 of higher-order accurate difference schemes (higher 

 than second order) which are presently in the state 

 of development and about to be used in our numerical 

 method. 



The results shown in Figures 7 and 8 also unveil 

 the considerable potential and advantages of such 

 numerical simulations. The finite-difference so- 

 lutions produce a bulk of data, i.e. the values of 

 the variables directly involved in the solution 

 procedure are obtained for all grid points and for 

 all time-levels that are calculated. The data can 

 be conveniently stored on mass storage devices, 

 such as magnetic tape (used for the present calcu- 

 lations, for example). The data stored can be 

 processed immediately or at any later data to ob- 

 tain any specific information desired, or to produce 

 additional data that might be deemed necessary for 

 a more detailed evaluation of particular flow phe- 

 nomena. For example, they can be used to obtain 

 frequency spectra, Reynolds stresses, energy bal- 

 ances, amplitude distributions, or to produce con- 

 tour plots (equivorticity lines, stream lines) etc. 

 Another positive side of such numerical simulations 

 is that if the data would be destroyed or lost, they 

 could be reproduced identically, which would be 

 hardly possible in comparable laboratory experiments . 



