Abstract
It is well known that the overwhelming majority of both natural and man-made flows of fluids do not vary smoothly in space and time but fluctuate in a quite disordered manner, exhibiting sudden and irregular (but still continuous) space- and time-variations. Such irregular flows are called “turbulent”. A very large amount of information is required to describe the whole of a field of turbulence in space and time, but as a rule only statistical properties, such as time averages are useful to scientists and engineers. Various semi-empirical, approximate methods have been devised to calculate the simpler statistical averages directly.
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Notes
- 1.
As usual the word ‘fluid’ denotes here any liquid or gaseous medium.
- 2.
The word turbulence (Latin turbulentia) originally refers to the disorderly motion of a crowd (turba). It was first used (in a sense close to that accepted today) around the year 1500 by the famous painter (and, as it was discovered much later, also a remarkable inventor and scientist) Leonardo da Vinci (who used the Italian spelling la turbolenza; see Frisch (1995, p. 112). However Leonardo did not write his scientific notes for publication and, being left-handed, his writings must be read with the help of a mirror. Leonardo’s notes remained little known up to the modern times, and by this reason the word ‘turbulence’ in its scientific meaning is often attributed to Kelvin; see, e.g., Lamb (1932), Sect. 366.
- 3.
All material presented in this chapter can be found in a great number of textbooks and monographs on fluid mechanics. Therefore the few references presented here must be considered as only some examples of numerous books containing the stated results.
- 4.
This use of the thermodynamic quantities and equations may raise some doubts, since a fluid flow with nonzero gradients of velocity and temperature does not constitute a system in thermodynamic equilibrium. However in all the books cited here it is explained that in the case of the moderate gradients encountered in real fluid flows, the fundamental thermodynamic quantities may be defined in such a way that all their ordinary properties, and the corresponding equations will be valid (for more details see, e.g., Sect. 49 of the book by Landau and Lifshitz (1987) or one of the more special publications, such, as, e.g., the paper by Tolman and Fine (1948) and the books by Chapman and Cowling (1952) and Hirschfelder et al. (1954)).
- 5.
It seems natural to expect that the thickness δ will decrease with the increase of flow viscosity (a proof of this statement will be presented somewhat later). Note also that the value of δ cannot be determined uniquely since the boundary layer has no strictly defined upper edge and the velocity u = u(x, z) of flow in this layer tends with increase of z to the free-stream velocity U only asymptotically as z→∞. Hence δ must be defined somewhat artificially. In practice it is often taken to be equal to the distance from the plate at z = 0 up to the level z at which u attains a given, sufficiently great fraction of U, e.g., 0.99 U (in this case δ is μ sometimes denoted by δ 99). Some other possible definitions of the thickness δ will be indicated later.
- 6.
- 7.
This remarkable figure was included in the first German edition of the well-known book by Schlichting (1951) and then it was reproduced in all revised re-publications of this book and in many other books and survey papers dealing with flat-plate boundary layers.
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Yaglom, A., Frisch, U. (2012). The Equations of Fluid Dynamics and Some of Their Consequences. In: Frisch, U. (eds) Hydrodynamic Instability and Transition to Turbulence. Fluid Mechanics and Its Applications, vol 100. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4237-6_1
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