Henry Darcy and His Law
Darcy on Velocity Distributions and the Boundary Layer
Glenn Brown
Oklahoma State University
1/29/00
revised 9/7/01
Henry Darcy was the first researcher to systematically examine turbulent velocity distributions. In his investigation of pipe flow he quantified not only total flow and average velocity, but also made detailed measurements of the velocity distribution within the pipe (Darcy, 1857). These measurements were made possible by his improvements to the Pitot tube. His careful measurements and insightful analysis allowed him to develop a generalized relationship for the velocity distribution.
In modern terms it is,
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where R is the pipe radius, y is the distance from the wall, U is the mean velocity, u is the velocity at y, and v* is the friction velocity given by,
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where r is the fluid density, and to is the wall shear stress. Schlichting, (1968; page 370) compared Darcy's experimental results to Prandtl's and von Kármán's theoretical relations that are valid for large Reynolds Number flows, (Re > 106). Prandtl's relation based on Stanton's universal velocity-distribution law and smooth pipe velocity distribution is
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Von Kárman's similarity law predicts
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where c is an empirical mixing length constant normally
assumed to equal 0.4. Figure 1 plots Equations 1, 3 and 4, (however
the value of c
used in Equation 4 is 0.36.) If measured data were plotted, it
would fall close to the line for von Kármán's relation.
Download
this data
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Figure
1. Universal velocity-distribution law for smooth and rough pipes
(after Schlichting, 1986). Note Re > 106 and c = 0.36.
As can be seen, Darcy's formula is in agreement with the latter relationships for all y/R > 0.2. Considering the difficulty of using Pitot tubes next to the wall, the error in his equation is understandable. Darcy's efforts are even more significant when it is considered that Prandtl did not present his relation until 1925 and von Kármán published in 1930; 73 and 78 years after Darcy respectively.
Due to the limitation of Darcy's equipment and technique, he was unable to gain quantitative data at the boundary. However, comparison of flow in smooth and rough pipes led him to suspect a "fluid layer at the boundary". Rouse and Ince (1954; page 170) provide the following translation. In it, Darcy addresses the transition from rough to smooth pipes and its impact on the coefficient in the head loss relationship.
"If one uses very smooth pipes, of lead, recovered with glazed bitumen, etc, the coefficient of V2 decreases continuously as the degree of polish increases.
But the reduction nevertheless is far from appearing proportional to the degree of polish obtained. In vain one would say that the influence of asperites inappreciable to the eye persists for the fluid molecules; that explanation would not seem at all satisfactory.
In effect, the term in V2 does not appear to correspond only to the resistance caused by the asperities, but also to that produced by the fluid layer next to the boundary."
It is clear that Darcy suspected correctly that the fluid boundary layer was at the root of the variation between smooth pipe and fully rough flows. In the first, resistance is a function of velocity to the 1.75 power, while in the latter resistance is a function of velocity squared. This diversity was the cause of the confusion and debate that lead to the eventual adoption of what we call the "Darcy-Weisbach" equation over the prevailing Prony type relationships that were popular at the time.
References
Darcy, H. 1857. Recherches Experimentales Relatives au Mouvement de L'Eau dans les Tuyaux, 2 volumes, Mallet-Bachelier, Paris. 268 pages and atlas. ("Experimental Research Relating to the Movement of Water in Pipes")
Rouse, H. and S. Ince. 1957. History of Hydraulics. Iowa State Univ., Ames, Iowa. 269 pages.
Schlichting, H. 1968. Boundary-Layer Theory, sixth edition. McGraw-Hill Book Co. New York. 748 pages. (This is a classic text on boundary-layer theory.)