A • TRANSITION FROM LAMINAR TO TURBULENT FLOW 



curvature obtaining changes in transition Reynolds number from 2.2 X 

 10« to 0.9 and 3.2 X lO^ for X = 0, -5.7, and +1.9, respectively. 



Feindt [9] studied the influence of pressure gradient on smooth and 

 roughened hollow cylinders with axes parallel to the air stream at a 

 stream turbulence of approximately 1.2 per cent, the turbulence level 

 being inferred from sphere measurements. (From the observed value of 

 Re^, for the smooth cylinder at zero pressure gradient, the turbulence level 

 derived from Fig. A,2b is 1.0 per cent.) For pressure gradient parameter 

 X = 0, —4.4, and 3.7, the observed transition Reynolds numbers for the 

 smooth cylinder were 0.66 X 10«, 0.36 X 10«, and 0.80 X 10«, respec- 

 tively. The effects of pressure gradient on roughened cylinders were also 

 large. Thus the influence of pressure gradient has been observed to be 

 large and qualitatively the same over a wide range of roughness and free 

 stream turbulence. 



A,4. Effect of Curvature of Surface on Transition of a Two- 

 Dimensional Boundary Layer. Liepmann made a systematic study 

 [10] of the effect of a uniform radius of curvature of the surface on the 

 transition of a two-dimensional laminar boundary layer. On convex sur- 

 faces up to values of displacement thickness b* equal to 0.0026 times the 

 radius of curvature r, the same Tollmien-Schlichting instability occurs as 

 for the flat plate and the effect of curvature is negligible. The effect of 

 turbulence is large as in the case of the flat plate. 



On a concave surface, the behavior is the same as for a flat plate, 

 provided the ratio 5*/r is less than 0.00013. If 6*/r exceeds 0.0013, the 

 laminar flow is dynamically unstable because of centrifugal forces pro- 

 ducing three-dimensional disturbances as studied theoretically by Gortler 

 [11]. Gortler used as a measure of the stability boundary the parameter 

 Ree \^Q/r, based on the momentum thickness 6 which is approximately 

 equal to 0.3865*. Liepmann found the Gortler parameter equal to 9.0 in 

 an air stream of the lowest turbulence available to him (turbulence in- 

 tensity 0.2 per cent as judged from his flat plate measurements, u' fii^ = 

 0.0006, v' and w' not measured), whereas at a higher turbulence level 

 {u' /u^ = 0.003, v'/ue and w'/ue not measured) the value was about 6.0. 

 For values of 8*/r between 0.00013 and 0.0013 there appears to be a 

 more or less continuous change from the Tollmien-Schlichting instability 

 to the Gortler instability. 



A, 5. Effect of Surface Roughness and Waviness on Transition of 

 a Two-Dimensional Boundary Layer. Surface roughness and wavi- 

 ness are known to influence transition presumably because of the disturb- 

 ances introduced by their presence. The general nature of the effect of a 

 single roughness element has been studied in detail by Liepmann [12], 

 the roughness element being a half cylinder with axis normal to the stream 



<8> 



