A • TRANSITION FROM LAMINAR TO TURBULENT FLOW 



of the roughness element based on free stream velocity and nominal 

 roughness height, it appears that Uk/v must be less than about 100 + 50 

 if the surface is to be considered aerodynamically smooth. 



Feindt [9] studied the effect of distributed roughness on a hollow 

 cylinder, using sand paper. The sand paper cylinder was extended 12 mm 

 beyond the sharpened leading edge of the supporting cyhnder. The sand 

 paper was smoothed with shellac for 3 mm from the leading edge, and the 

 leading edge of the sand paper sharpened. Roughness for which JJk/v was 

 less than 60 to 100 based on nominal sand grain size had little effect on 

 transition, the exact value depending somewhat on the pressure gradient, 

 the lower value applying to an adverse pressure gradient. This value is 

 approximately the same as that found in Holstein's measurements, al- 

 though Feindt's measurements show less scatter. 



The decrease in Re^ with increasing Uk/v is very rapid. For the case 

 of zero pressure gradient an increase of Uk/v from 100 to 150 reduces 

 Re^ to one half its initial value. 



Feindt proposed the use of the Gottingen equivalent sand grain rough- 

 ness, defined as that value of roughness in certain Gottingen measure- 

 ments of friction coefficients in roughened tubes which gave the same 

 friction coefficient as his samples. Based on this Gottingen equivalent 

 roughness height, the limiting value of Uk/v is from 100 to 170. 



The applicabihty of these hmiting values of Uk/v to smoother sur- 

 faces for which k hes outside the range of the experimental values or to 

 higher Reynolds numbers is somewhat uncertain. Since transition is be- 

 lieved to arise as a result of eddy production by the roughness elements 

 in the case of three-dimensional distributed roughness, it appears more 

 logical to base any extrapolation on Ukk/v where Uk is the velocity in the 

 boundary layer at the transition position at the height of the roughness 

 element. A value of Ukk/v of 30 to 50 was found for sand grain roughness 

 in air streams of relatively high turbulence. For distributed small spheres 

 in a flow of very low turbulence without pressure gradient, the observed 

 value was about 300. Since critical Reynolds numbers are usually depend- 

 ent on air stream turbulence as well as on shape, the difference between 

 30 to 50 and 300 may be due in large part to the different air stream 

 turbulence levels (see Art. 17). 



In summary, single roughness elements should be limited to heights 

 less than 0.2 times the displacement thickness of the boundary layer at the 

 element under the prevaihng conditions of speed, viscosity, and element 

 location to avoid reduction of transition Reynolds number. Distributed 

 roughness should be limited to heights for which the Reynolds number 

 based on nominal roughness height and the velocity in the boundary layer 

 at the transition position at the height of the element is less than a value 

 of the order of 30 to 300. The former value applies to sand roughness in a 

 stream of turbulence of 1.0 per cent; the latter to small spheres in a stream 



< 18 ) 



