A • TRANSITION FROM LAMINAR TO TURBULENT FLOW 



pressure gradient, curvature, and surface roughness; that the critical 

 Reynolds number is the same as if all of these quantities were constant 

 and independent of the distance s along the surface. Attempts have been 

 made to check the adequacy of this assumption by Stephens and Haslam 

 [47] in actual flight tests. In the flight tests values of {Res*)oT were ob- 

 served ranging from 1460 to 3240, values of X at the point of transition 

 between +0.3 and —7.2, values of the curvature parameter 8*/r from 

 0.06 X 10-3 to 0.86 X 10-^ No correlation whatever of (i^esOer with X 

 and 8/r was observed. The values of X seem sufficiently far from the 

 Pohlhausen value —12 to preclude the presence of separation bubbles, 

 although in view of the shortcomings of the Pohlhausen method it may 

 not be safe to conclude that this phenomenon was absent in all cases. 

 No quantitative measures of surface roughness and waviness were given, 

 and it is now believed that the results obtained by Stephens and Haslam 

 were controlled by this parameter, whose effect overshadowed the influ- 

 ence of the other parameters. This conclusion is drawn since in later flight 

 tests where exceptional attention was given to the smoothness of the sur- 

 face and more particularly freedom from waviness, values of {Rei*)^^ in 

 the range of 6000 to 6500 were obtained by both British and United 

 States investigators as described in Art. 17. We conclude that the question 

 of the adequacy of theories based on local parameters is not settled by 

 these measurements. 



There is one further bit of evidence resulting from tests on a smooth 

 low drag airfoil in the NACA low turbulence wind tunnel over a wide 

 range of Reynolds numbers (from 14 to 58 million). In these tests [48] 

 the position of transition on both upper and lower surfaces varied from 

 the 25 per cent to the 50 per cent chord location. The observed values of 

 {Res*)cT were between 5150 and 6150. The assumption of a fixed value of 

 6150 gives computed transition positions agreeing with the observed po- 

 sitions with a maximum difference of 7 per cent of the chord. 



We have already seen that a roughness element may introduce a dis- 

 turbance which produces transition at some distance downstream from 

 the element. Hence it is clear that the disturbances originating upstream, 

 which are not indicated by instruments measuring average values, must 

 be considered in addition to the local values of 8*, X, k, and r. Whether a 

 theory based on local parameters is adequate or not depends on whether 

 all the important local parameters are included. So far as disturbances 

 from upstream roughness elements are concerned, we may consider in- 

 stead the direct influence of all upstream roughness elements on the 

 critical Reynolds number for transition. If any upstream value of k/8* 

 produces a value of {Rei*)„ greater than the value of Res* at the point 

 under study, its effect is negligible at that point. If, however, the local 

 Res- equals the {Res*)cT of any upstream roughness element, transition will 

 occur at that point due to roughness. By this device it may be possible to 



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