The summary of the results as reported by Manohar are shown in Figure 1* 

 The explanation given by these two investigators to the rather singular 

 behavior at the rough region was that the transition from laminar to 

 turbulent is mainly due to the instability of the flow along the wake 

 formed between the individual particles. The characteristic length then 

 is the size of the wake which, although a function of the roughness, 

 diameter, is not necessarily proportional to it. This explanation, 

 although reasonable, is not quite convincing. The writer's reservations 

 were confirmed by some observations made during a more recent phase of 

 the investigation. These observations were made while measuring the phase 

 shift on the vertical in the same experimental flume. The method used 

 involved again the use of dye which this time was injected from above 

 through a thin brass tube. It was observed that the flow was definitely 



unstable at a value of _2 equal to 1.3x10 for a two-dimensional rough- 



v 

 ness with e = .0625'. Moreover unstable flow conditions were observed 

 while experimenting with three-dimensional roughness of e = .017' when 



u o 4 



the value of — was about equal to 10 . These points when plotted in 



Figure 1 demonstrate that the constant value of the critical Reynolds 

 number as defined for the transition regime may well be extended to cover 

 the rough case too. This implies that in Li's and Manohar 's experiments 

 the flow in this region was already unstable before it could be established 

 as such from observations. There is a possibility that the crystals and 

 the high-density dye solution were confined within the layer of the dead 

 water under the theoretical bed. The fluid in this layer oscillated back 

 and forth with the plate and as long as the velocities were small it never 

 had a chance to spill over the crest of the roughness elements and mix 

 with the flow above. So even if the flow were unstable there was no way 

 of detecting it. On the other hand, one may argue that the more recent 

 observations do not describe the behavior of the real model either , be- 

 cause the observed premature transition might have been triggered by the 

 tube itself. We want to make clear, however, that the tip of the tube 

 was maintained at a level several millimeters above the bed at a distance 

 where the amplitude of the fluid velocity was negligible while the first 

 indication of instability was observed near the bed. In concluding this 

 Appendix we would like to point out that this phase of the problem needs 

 further investigation so that more reliable information will become 

 available . 



* Figure number refers to figures following main text, 



A-4 



