■0,01 



0,02 0,03 0.04 0,05 0,06 



Numerical Solutions 



bounded by a limiting curve of r versus D/h (Reynolds number versus ratio 

 of spacing D to mesh size; D/h is actually the number of meshes across the 

 channel in its upstream section). 



As a means of verifying the accuracy of the difference scheme used (which 

 was checked at each step by iterating until the discrepancies were reduced to a 

 prescribed value), and also as a means of testing the stability of the scheme be- 

 fore applying it to the long calculations for the unsteady problem under investi- 

 gation, the finite-difference system was applied to a disturbed uniform flow, 

 i.e., a flow within parallel straight boundaries for which the vorticity distribu- 

 tion had been initially prescribed to be quite different from the one for uniform 

 steady flow. The entire system of equations, including those for the boundary 



conditions, was used in calculating the 

 transient flow that should lead from the 

 disturbed flow to the steady uniform 

 flow. Were the scheme unstable, it was 

 reasoned, the disturbed flow would fail 

 to return to the original uniform flow; 

 were the scheme stable, but still con- 

 vergent to a different solution, this would 

 also be discovered. Figure 2 shows what 

 happens when a calculation for uniform 

 flow becomes unstable: Such rapidly ex- 

 panding oscillations with a wave length 

 directly related to the mesh size are 

 quite typical of numerical instability as 

 opposed to hydrodynamically originated 

 instability. Figure 3 shows the result of 

 a study of instability in the case of a 

 two-dimension expansion. The effects of iterating in different ways are also 

 shown (the influence of the paths of iteration is not great, but a less biased dis- 

 tribution of errors results from sweeping the field diagonally, N.W. to S.E., N.E. 

 to S.W., S.E. to N.W., and S.W. to N.E.). 



The exact position of the neutral line was not sought, because the process 

 is time-consuming. 



DISCUSSION 



In reply to an inquiry by A. M. O. Smith (Douglas Aircraft Co., Long Beach, 

 Calif.) whether the results obtained had been compared with other similar work, 

 a recent study by a graduate student at the University of Notre Dame was men- 

 tioned by S. Piacsek (University of Notre Dame, Ind.). In this investigation a 

 uniform stream parallel to a wall with a step was treated, and similar results 

 were found. 



Another comment was that symmetry of boundary geometry does not ensure 

 flow symmetry, as is assumed in the present work. A calculation at Reynolds 

 numbers from 100 to 200 of flow in an axisymmetric conduit which yielded non- 

 axisymmetric flow was cited. 



Figure Z 



1612 



