Abstract
In this chapter pipe flows are studied for laminar Hagen-Poiseuille flows as well as turbulent flows; this situation culminates via a dimensional analysis in the well known Moody diagram based on the pioneering work of Johann Nikuradse’s hydraulically smooth and rough pipes. Plane turbulent flow is mathematically modeled by Prandtl and von Kármán on the basis of Prandtl’s mixing length parameterization of the turbulent shear stress. Wall parallel flows can computationally be determined by two additional assumptions, (1) that the mixing length grows linearly with distance from the wall and (2) the shear stress velocity is constant across the turbulent boundary layer. The result is the well known logarithmic velocity profile. It culminates in Prandtl’s and von Kármán’s universal representation of the coefficient of resistance, \(\lambda \). This universality property, assumed to hold since the 1930s, has been challenged by Barenblatt and associates in the 1990s and later on. These authors negate the logarithmic law and show that the laminar sub-layer influences the outer turbulent boundary layer.
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Notes
- 1.
Actually, the soap film analogy was established by Prandtl between membranes under transverse deflection and Saint Venant torsion of shafts. Correspondence between the BVPs of these two problems exists as well.
- 2.
After Evangelista Torricelli (1608–1647), who published this formula in 1644. He was physicist and mathematician in Italy of Galilei’s time , whom he worked for and whose successor he became after Galilei’s death. His research on hydrodynamics was extremely well received in the 17th century of Europe. For a portrait and short biography, see Fig. 3.30.
- 3.
For a short biography of Henry Philibert Gaspard Darcy (1803–1858) , see Fig. 10.9 .
- 4.
For a short biography of Henri-Émile Bazin (1829–1917) , see Fig. 10.10 .
- 5.
\(\log \) is the logarithm to the basis 10
- 6.
For a short biography of Johann Nikuradse (1894–1979), see Fig. 10.13 .
- 7.
In the German hydrodynamic literature (10.60) is known as Prandtl -Colebrook formula, in the English literature it is often referred to as Colebrook -White formula.
- 8.
This assumption is plausible, because very close to the wall \(\text{ d } u/\text{ d } y\) is approximately constant, so that Boussinesq’s formula \(\tau = \varepsilon \text{ d } u/\text{ d } y\) also suggests \(\tau = \text{ constant }\)
- 9.
This subsection illustrates how pressure losses in a sequential arrangement of pipe segments are technically performed in the context of the Prandtl-von Kármán turbulence model. Conceptual thoughts on its theoretical limitation are given in the subsequent subsection.
- 10.
We follow in this subsection closely Chap. 30 in Physics of Lakes, Vol. 3 [9].
- 11.
We have anticipated here consequences of the Buckingham theorem , in details explained in Chap. 20 of Volume 2.
- 12.
For a short biography of Grigory Isaakovich Barenblatt see Fig. 10.20 .
- 13.
For brief biographies of Prony, Eytelwein and du Buat, see [7].
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Hutter, K., Wang, Y. (2016). Pipe Flows. In: Fluid and Thermodynamics. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-33633-6_10
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