16 



This would be impractical due to the excessive 

 amounts of grid points required. On the other hand, 

 the conditions given below allow a relatively small 

 integration domain in y-direction. They only postu- 

 late that the disturbances decay asymtotically in 

 y-direction. For the co,i() formulation such a condi- 

 tion is 



3 i|)' 

 3y 



ail' 



(25) 



and for the (»),u,v formulation 

 " 8 u' 



3y 



3 V 

 3y 



au' 



av' 



(26) 



where a is the local wave number of the resulting 

 disturbance waves. Test calculations have shown 

 that with the conditions (25) or (26) , together 

 with the Dirichlet-type vorticity condition dis- 

 cussed previously, physically meaningful results 

 can be obtained when the integration domain in y- 

 direction includes only two to three boundary-layer 

 thicknesses. 



Selection and implementation of the boundary 

 conditions at the downstream boundary B-C represents 

 a very difficult task. These boundary conditions 

 have to enable propagation of disturbances right 

 through this boundary, where any effects causing 

 even the slightest wave reflection have to be 

 avoided. The conditions found most satisfactory 

 in this respect are for the u,i() formulation 



3^u)' 

 3x2 



3x2 



2 

 a (jj' 



(27) 



~'i>' 



and for the a),u,v formulation 



(28) 



3x2 



Numerical experiments with conditions (27) and (28) 

 have shown that physically reasonable results are 

 already possible when, for periodic upstream dis- 

 turbance input, the length of the integration domain 

 includes only three to four wavelengths. 



For the calculation of the steady flow (for the 

 boundary-layer flow, for example) boundary condi- 

 tions which are compatible with those of the unsteady 

 calculations are for the a),i|) system 



3x2 



3^ 

 3x2 



(29) 



and for the (u,u,v system 



3^n 



3x2 



= 



3^0 



3x^ 



= 



3fV 

 3x2 



The boundary conditions Eqs. (27) or Eqs. (28) for 

 the downstream boundary [also Eqs. (25) and (26) 

 for the free stream boundary] can be derived assum- 

 ing neutral, periodic behaviour of the disturbance 

 flow. However, extensive test calculations have 

 shown that use of such conditions does not enforce 

 a strict periodic behaviour of the disturbance flow 

 near these boundaries. Rather, these conditions 

 allow damping or amplification of the disturbances 

 even on these boundaries themselves. These con- 

 ditions have also proven to be applicable for cal- 

 culations with periodic disturbance input of large 

 amplitudes as well as for non-periodic disturbance 

 input (random disturbances, for example) [see Fasel 

 et al. (1977) ]. 



For cases where a is not known a priori it can 

 be determined interatively . Starting with an ini- 

 tial guess ao(x) (a is generally a function of x, 

 of course, although for the derivation of the 

 boundary conditions it was assumed constant to 

 arrive at simple relationships) an improved a(x) 

 can be determined from the resulting disturbance 

 waves developing in the integration domain. Even 

 with relatively crude initial guesses ag (x) (for 

 example ao=0) this interation loop converges 

 rapidly, and for practical purposes two or three 

 iterations are sufficient. 



There is no formal difference between the bound- 

 ary conditions (27) and (28) used for the u,i|) and 

 ii),u,v formulation, respectively. Both sets of con- 

 ditions specify relationships for the second deriva- 

 tives in the disturbance variables. Nevertheless a 

 siibtle difference does exist. Condition (27) for 

 i|i' implies that (due to the definition of ii , Eq. 4b) 

 for v' a relationship involving the first derivative 

 is prescribed 



3v' 

 3x 



- a^^' 



(31) 



This is obviously more restrictive than condition 

 (28c) where for v' a second derivative is prescribed. 

 For small periodic disturbances the two sets of 

 conditions lead to practically the same results, 

 although the results with the u),\J; system, together 

 with conditions (27) , exhibit subtle irregularities 

 near the downstream boundary for the waves propa- 

 gating through this boundary. The ii),u,v system, 

 together with conditions (28) , however become su- 

 perior to the (i),i|) system with conditions (27) when 

 larger disturbance amplitudes are involved. In this 

 case, reflection-type phenomena can be observed in 

 increasing manner at the downstream boundary for the 

 uii'ii system. For the investigation of the effects 

 of a backward- facing step on transition [Fasel et 

 al. (1977)] the small vortices traveling downstream 

 are caught at the downstream boundary when the to,i(; 

 system and conditions (27) are used, rendering the 

 numerical results worthless. Using conditions (28) 

 with the u,u,v system, on the other hand, allows 

 smooth passage of these vortices through that bound- 

 ary. 



For these reasons conditions (28) , in connection 

 with the u,u,v system, have proven to be the best 

 choice so far in properly treating the downstream 

 boundary. The relatively small upstream influence 

 of these conditions can be best demonstrated with 

 typical results from test calculations. Figure 5 

 for example, shows a comparison of the disturbance 

 variable u' for calculations with small periodic 

 disturbances where first in Eqs. (28) an adequate 

 value for a (a=35.6, obtained from linear stability 



