There is an entire cottage industry producing explicit approximations to the Colebrook-White Equation (CW):
It appears to be a buyer's market.
In 2014, I analyzed 25 such explicit approximations to determine their relative accuracy compared to CW. I am not the first person to do such an analysis, but I may be the first person to use Mathcad for it. You can find the documents here:
For readers without Mathcad Prime 3.0, the .zip file includes Adobe Acrobat .pdf files. In my research, I came across many published papers and articles, but I don't recall finding Churchill's paper. I may have it, I just don't recall seeing it.
For my data set, I used 21 relative roughness curves (typical for most of the Moody Diagrams I own) and 45 Reynolds Numbers on each curve. This generated a matrix of 945 data points for comparison with CW. I calculated the relative error (RE) compared to CW for each of the 945 data points and reported the minimum and maximum values. I also calculated the mean squared error for all 945 data points, then added them up to create a single value (SumMSE) for each equation that I used to rank them for overall accuracy.
Churchill's 1973 equation has REs ranging from -0.61% to +3.34%, while Churchill's 1977 equation has REs ranging from -0.64% to +3.10%. The SumMSEs are 6.3*10^-5 and 6.1*10^-5, respectively. These aren't bad, considering the error bars on the original data used to create CW (IIRC, it was ±5%).
However, I found 15 explicit approximations to CW that are more accurate than Churchill's two equations. At least two are as simple as Churchill 1973 and more accurate than Churchill 1977, including Jain 1976 (SumMSE = 5.4*10^-5) and Haaland 1983 (SumMSE = 3.0*10^-5). Zigrang & Sylvester #1 and #2 (both 1982) are much more accurate and still less complex than Churchill 1977 (SumMSE = 1.2*10^-5 and 1.4*10^-7, respectively. By far the most accurate and by far the most complex of the explicit approximations is Goudar & Sonnad #2 (2008), which nowhere deviates from CW by more than about 10^-10%. Seriously. The SumMSE for Goudar & Sonnad #2 is a miniscule 3.6*10^-26.
This is a very long way of saying that there are many explicit approximations to CW that are better than either of Churchill's equations.
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"Is it the only lesson of history that mankind is unteachable?"
--Winston S. Churchill
