appear, their velocity downwards being about three times the mean flow velocity, and 

 it is in fact, quite straight forward to measure all features of the transition. It should 

 be noted that this transition is far removed from the development of turbulence, unlike 

 that in the more familiar cases of instability of laminar flow. The film cannot be de- 

 scribed as fully turbulent until the Reynolds number is increased to values in the 

 hundreds, at least. 



The influence of surface tension must be recognized in any theory which is to 

 be of practical use for water or other common liquids. Surface tension is the only 

 restoring force when the plate is vertical, although, of course, gravity exerts a restoring 

 force when the plate is inclined. 



During his talk on Monday, Professor Lighthill remarked that the existing theory 

 of roll waves, as for instance developed by R. F. Dressier, is applicable to flow in thin 

 films. Since this theory has been developed by Dressier for a generalized law of fric- 

 tional resistance, one may use his results simply by substituting the appropriate resistance 

 of a laminar film. In this case, of course, the resistance is directly proportional to flow 

 speed, and inversely to the film thickness. 



In fact, a lengthy treatment having much in common with this approach, was 

 published a number of years ago by the Russian scientist Kapitza. However, a more 

 exact treatment of this problem is possible. The problem may be formulated as a 

 characteristic-value problem in the familiar manner of stability theory for the boundary 

 layer. The only differences are that special boundary conditions are needed to account 

 for the free surface, and that, in the range of interest, the Reynolds number is small. 

 Thus one may resort to asymptotic expansions in ascending powers of Reynolds number, 

 instead of descending powers as in the usual case. 



Dr. Yih has already published a description of this method of approach, but 

 restricted his treatment to the case of zero surface tension, and to terms of only the 

 first power of aR. Actually the Reynolds number is about two or three in the range of 

 greatest interest, but alpha is kept fairly small due to the result of surface tension. aR 

 is considerably less than one, but couldn't be regarded as a small quantity in the usual 

 sense. 



In our work at Cambridge, we have considered expansions up to the third power 

 of aR, which has required a power series expansion of the stream function in terms of 

 the coordinate perpendicular to the flow as far as the 15th power. It remains to prove 

 the convergence of this approximation, but the physical aspects are sufficiently reassuring. 



The result of a good deal of heavy algebra is a series of curves of neutral stability, 

 with R and alpha as coordinates in the familiar fashion of stability theory, and with 

 Weber number, representing surface tension, and the slope of the plane as parameters. 



A good agreement with experiment has been found, and we have also made 

 some progress with the more difficult problem where the film is stressed by a turbulent 

 gas stream confined with an additional fixed boundary. 



In conclusion, I would very much like to comment the film problem for further 

 study by workers in this field. It has a number of important practical aspects. I hope, 

 for instance, that it will attract the great talents of Professor Lin. 



P. S. Klebanoff 



The comments I have to make consist of some of the recent results obtained 

 from a continuing experimental investigation of boundary layer transition which is 

 being conducted at the National Bureau of Standards under the sponsorship of the 

 National Advisory Committee for Aeronautics. They bear on Dr. Lin's paper in that 

 they deal with the region of finite amplitude with which we are all concerned. One 

 objective of the investigation is to observe by hot-wire techniques the growth and 

 evolution of a wave from its source to transition. It was possible to carry out this 

 objective by using the vibrating ribbon technique used so successfully by Schubauer 

 and Skramstad in their experimental confirmation of the Tollmien-Schlicting stability 



366 



