J. Nikuradse.

It is evident from Nikuradse's paper[1] that, between 1914, when Stanton and Pannell published their work, and 1930, there had been a great deal of uncoordinated experimenting on pipes of all sorts. It had become evident that the same pipe could behave at one velocity of flow as if it were smooth and at another velocity as if it were rough. These two conditions came to be called hydraulically smooth and hydraulically rough. Nikuradse thought that it was time to attempt to find out whether there was some underlying order in all the results.

 

Stanton and Pannell had found that one pipe could not be tested over the "whole range"[2] of Reynolds number and so they used pipes of several diameters. By chance, it did not matter in the work of Stanton and Pannell that the pipes were not geometrically similar between the surface finish and the diameter because they only worked with hydraulically smooth flow. Nikuradse now proposed to find out the relationship between the friction loss in a pipe and the roughness for pipes that were geometrically similar over the whole range of Reynolds number. As this involves the use of pipes of small and large diameter this meant that they required pipes of different diameter but equal roughness. Such pipes would have to be specially made for Nikuradse.

 

Nikuradse argued that a pipe may be artificially roughened by coating it internally with graded sand and that the roughness may be quantified as  where  is the radius of the pipe and  the maximum size of the holes in the sieve used to grade the sand. This involved finding a way to stick the sand to the inside of the pipe. Nikuradse used Japanese lacquer, a runny compound made from a natural resin from trees found in Japan and China using turpentine as a solvent. The sand used was "builders" sand that has irregular rounded grains free from the sharp corners of sharp sand. The sand was sieved and typically the  mm size was obtained by sieving between a mesh size of  mm and one of  mm. Pipes were abraded, to improve the bond between the pipe and the lacquer, filled with lacquer and then allowed to drain. After half an hour, when the lacquer was tacky, the pipe was filled with sand. It was then heated for a period of 2-3 weeks to drive off the solvent and to set the resin.

 

Unfortunately this did not give an adequate bond and the sand quickly washed out. In order to improve the bond the pipes, with the sand coating in place, were again filled with lacquer thinned with more turpentine, drained and allowed to dry for another 4 weeks. The result was a pipe covered internally with sand with a thin coating of lacquer over the sand. This solved the problem of bonding the sand to the pipe at the expense of a change of the surface from one that is familiar to most engineers and physicists to one that would not be familiar at all. I suppose that one might varnish pieces of sand paper of different grades to see what the inner surfaces might have looked like. By using suitable combinations of  and  Nikuradse made six ranges of geometrically similar pipes of different diameters with all the pipes in one range having the same ratio of . If there was any order in pipe friction each range of pipes, when tested, should give a single line on the  versus  plot. Nikuradse made pipes having  ratios of 15, , 60, 126, 252, and 507. These pipes were tested over the "whole range"[3] of Reynolds number. The details of the pipes together with the ranges of Re for which Nikuradse gives results are given in table 7-1.

Text Box: Nikuradse’s valueof r/k Nominal value of k/d d in mm k in mm Approximate range of Re 507 0.001 99.4 0.1 104 to 106 252 0.002 49.4 99.2 0.1 0.2 104 to 106 104.7 to 106 126 0.004 24.74 99.2 0.2 0.4 103.6 to 105.2 105 to 106 60 0.008 24.34 98.0 0.2 0.08 103.6 to 105.2 105 to 106 30.6 0.016 24.34 48.7 96.4 0.4 0.4 0.8 103.6 to 105.2 104.4 to 105.6 105 to 106 15 0.033 24.12 48.2 0.8 1.6 103.8 to 105.3 104.4 to 106 Table 7-1

Text Box: Graph 7-2 Nikuradse presented his results on a graph of  against  where  was equal to  in the expression . This plot does not now fit in with the Moody diagram so, in graph 7-2, I re-plotted Nikuradse's results on the same logarithmic scales as have been used by Moody so that Nikuradse's curves can be directly compared with Moody. In addition, as it is easier to think of relative roughness as the ratio of the size of the sand divided by the diameter, Nikuradse's r/k has been replaced with k/d.

 

The most important outcome of Nikuradse's work is that pipes for which there is geometrical similarity between the surfaces and the diameter have a single relationship between  and . This means that there is an underlying order to friction loss in pipes.

 

Nikuradse had also shown that, for values of  that would be relevant in engineering, the curves first followed the smooth pipe curve[4] and then, after a transition range was passed, the value of  becomes independent of Re. In accepting this outcome it should be realised that  for a new steel pipe of 50 mm diameter is of the order of  and this corresponds to the least rough of Nikuradse's pipes.

 


[1]“ Stromungsgesetze in rauhen Rohren”, VDI-Forschungsheft 361 (1933)

[2] There cannot be a limit to the range of Reynolds number. For water the value of Re is primarily dependent on the diameter because the practical upper limit of velocity is about 3 m/s. There is no practical limit to the diameter and therefore no limit to Re. For any really large pipe of, say, 10 m diameter the flow would be hydraulically smooth and then an extrapolation of the Stanton and Pannell curve could be used to predict friction loss.

[3] In fact Nikuradse tested up to log Re = 6. The Moody diagram goes to log Re equals 8.

[4] This is, in some ways, an unfortunate choice of name. It would be acceptable if it were not for the fact that the name is often interpreted literally. It is the curve for hydraulically smooth flow in any practical pipe. All such pipes will behave as if they are rough at high enough values of Reynolds number.