mathematical theory will be given in the complete publication. Let us now go on to 

 make some remarks on the problem of transition. 



5. Some remarks on transition 



I shall not attempt to give a general survey of the transition problem but wish 

 merely to discuss a few points about the relation between the stability theory and the 

 transition phenomenon. 



First, we know that the minimum critical Reynolds number obtained from 

 stability calculations is usually much lower than the transition Reynolds number. Still, 

 there are two useful applications of the stability theory. First, the parameters occurring 

 in the instability theory usually turn out to be the right ones for correlating transition 

 data. For example, the Gortler parameter 



R e (d/r)i (18) 



for the boundary layer over a concave surface. Another example is the form parameter 



H = 8*/d (19) 



which is found to be useful for correlating stability calculations of the boundary layer 

 with suction or pressure gradient or both. A third example is the surface temperature. 

 In all the three cases just cited, the trend of variation of the critical Reynolds number 

 with the parameter in question is the same for stability theory and transition experiment. 



Some results obtained by Klebanoff, Schubauer, and Tidstrom [21] give an even 

 more direct correlation between theory and experiment. They found that below the 

 minimum critical Reynolds number of the stability theory, it is practically impossible 

 to induce transition either by roughness or by a spark. The turbulence spot produced 

 by a spark does not spread until it reaches beyond the location defined by the minimum 

 critical Reynolds number. However, if one would consider the situation in a pipe, it 

 is clear that we cannot yet consider such a situation as a general rule for transition from 

 laminar to turbulence. On the other hand, if further experiments can show its validity 

 even only for boundary layers, the practical value of the stability theory would be 

 greatly enhanced. 



I wish now to make a few remarks about factors which influence transition but 

 which lie outside the scope of the usual stability theory. Two such important factors are 



a. the level of turbulence in the free stream, 



b. roughness of the surface. 



The influence of the level of turbulence is described by the well-known parameter 



u' ( x V 



Al) 



u 



(20) 



given by Taylor [1]. I wish now to turn my attention to the effect of roughness. 



Dryden [22] analyzed a series of experiments (especially those of Tani and 

 Hama) where the transitions is induced by a single two-dimensional roughness element 

 and arrived at the correlation parameter 



k/8 k *, (21) 



where S&* is the displacement thickness of the boundary layer at the roughness elements 

 of height k. In the case of distributed roughness, Klebanoff, Schubauer and Tidstrom 

 [21] found that such a parameter was not good for correlating experimental data 

 (Fig. 2). They instead proposed the parameter 



u k k/v, (22) 



where u h is the velocity of the flow at the edge of the roughness element. The data still 

 show some scatter, but much less (Fig. 3). Now from purely dimensional considera- 

 tions, it appears that both parameters must be used, since the location of the roughness 



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