Osborne Reynolds

How to use this course

Every engineering student meets Osborne Reynolds through one dimensionless number. That number is the smallest part of his legacy. Reynolds explained why hot fluids cool faster when they flow faster, why a film of oil can carry a railway axle, why wet sand dries around a footprint, and how the chaotic eddies of a turbulent flow exert a stress on the mean motion. He did this as a professor of engineering, with a workshop training and a Cambridge mathematical education, at a time when engineering was barely an academic subject.

Each technical lesson begins with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The order roughly follows Reynolds’s own: heat transfer and gases (1874–1879), pipe flow and similarity (1883–1887), lubrication (1886), granular matter (1885), and the averaged equations and energy method of 1895.

Timeline

Belfast, Dedham and the workshop, 1842–1863

Osborne Reynolds was born in Belfast on 23 August 1842. His father, also Osborne, was an Anglican clergyman and a Cambridge graduate of 1837 who had been a fellow of Queens’ College. At the time of his son’s birth he was headmaster of the Belfast Collegiate School. Three generations of the family had been rectors of Debach-with-Boulge in Suffolk. Soon after Osborne’s birth the family moved to Dedham in Essex, where the father became headmaster of the grammar school. A younger brother, Edward, became a priest.

The boy was taught first at Dedham and then privately, and above all by his father, a capable mathematician who also took out patents on agricultural machinery. Reynolds later wrote that he had enjoyed his father’s constant guidance in mechanics through his boyhood. The combination of mathematics and machines stayed with him for life.

The apprenticeship

Instead of going directly to university, Reynolds in 1861 became an apprentice in the workshop of Edward Hayes, a mechanical engineer at Stony Stratford in Buckinghamshire who built engines and small coastal steamers. There he learned how machines were made and how they failed. He came to believe that the practical questions he met there could only be answered by mathematics, and that conviction sent him to Cambridge.

Cambridge and the London sewers, 1863–1868

At Queens’ College, Cambridge, his father’s college, Reynolds read mathematics. The Mathematical Tripos of the 1860s was a hard competitive examination in mathematics and mathematical physics, taught through private coaches. John William Strutt, later Lord Rayleigh, was a year ahead of him, and they attended some classes together. In 1867 Reynolds graduated as seventh wrangler, that is, seventh in the first class, and was elected to a fellowship at Queens’.

He did not stay to teach. He spent a year as a practising civil engineer with the London firm of Lawson and Mansergh, working on the sewage system for Croydon. Moving large volumes of water through pipes and channels was the bread and butter of Victorian civil engineering. The resistance of pipes, the subject of his most famous paper, was in front of him from the start.

Engineering in the universities

In the 1860s engineering was learned by apprenticeship. Glasgow had had a chair of engineering since 1840, held from 1855 by W. J. M. Rankine, and University College London offered engineering courses. In England outside London there was nothing. When the industrialists of Manchester endowed a chair at Owens College in 1868, they created one of the first university posts in the country with the title Professor of Engineering.

The Manchester chair, 1868–1880

The post at Owens College was advertised in 1868 at £500 a year. Reynolds applied, writing that he had an irresistible liking for mechanics and the physical laws on which it rests. He was appointed at 25 and stayed for 37 years. Owens College became a constituent of the new federal Victoria University in 1880 and later the University of Manchester. Reynolds built its engineering school and, later, the Whitworth engineering laboratories.

In 1868 he married Charlotte Jemima Chadwick. She died in 1869 after the birth of their son Osborne, who died in childhood in 1879. In the early 1880s Reynolds married Annie Charlotte Wilkinson; they had three sons and a daughter.

From electricity to fluids

His first research, from 1868 to about 1873, was on electricity and magnetism and on electrical theories of the Sun and comets. Around 1873 he turned to mechanics, and for more than two decades fluid motion was his main subject. He published in the Memoirs and Proceedings of the Manchester Literary and Philosophical Society, which he joined in 1869 and served as secretary from 1874 to 1884, and increasingly in the journals of the Royal Society, which elected him a fellow in 1877.

His teaching was demanding. Students found his lectures hard to follow, because he worked out ideas at the board rather than delivering polished notes. J. J. Thomson, who studied at Owens College in the 1870s before going to Cambridge, later recalled him as a teacher in his memoirs. Horace Lamb, professor of mathematics at Owens from 1885 and Reynolds’s obituarist, remarked that he had no taste for advertisement.

Boilers, gases and the Reynolds analogy

For the general reader. Blowing on soup cools it faster. Reynolds explained why: the same swirling motions that drag a fast fluid against a wall also carry heat between the fluid and the wall. So if you know the friction, you know the heat transfer. This idea, from a paper on steam boilers, is still the first estimate engineers make.

The analogy (1874)

Reynolds’s 1874 paper on the heating surface of steam boilers argued that heat passes between a flowing fluid and a wall in two ways: by molecular conduction, and by the mixing motions of the fluid itself, which carry hot fluid to the wall and cold fluid away. The mixing part grows with speed. The same mixing transports momentum toward the wall and appears as friction.

Model both transports in a turbulent layer next to a wall by a common eddy diffusivity \(\varepsilon\). With \(y\) the distance from the wall, \(u(y)\) the mean velocity and \(T(y)\) the mean temperature,

If \(\tau\) and \(q\) vary across the layer in the same proportion, then \(q/\tau=-c_p\,dT/du\) is constant, and integrating from the wall (\(u=0,\ T=T_w\)) to the outer flow (\(u=U,\ T=T_\infty\)) gives \(q_w/\tau_w=c_p(T_w-T_\infty)/U\).

The hypothesis of equal diffusivities holds near a wall only when the molecular diffusivities are also equal, that is, when the Prandtl number \(\mathrm{Pr}=\nu/\kappa\) is close to 1, as for gases. For other fluids Thomas Chilton and Allan Colburn proposed in 1933 the empirical correction \(\mathrm{St}\,\mathrm{Pr}^{2/3}=\tfrac12C_f\), valid roughly for \(0.6<\mathrm{Pr}<60\). The analogy also fails when pressure gradients change the friction without changing the heat flux in the same way, for example in separated flow.

Thermal transpiration (1879)

The radiometer of William Crookes (1873), a vane that spins in a partial vacuum when lit, provoked a long argument about its mechanism. Reynolds traced the force to gas flowing along temperature gradients at surfaces, and in 1879 published a memoir on thermal transpiration: when two vessels at temperatures \(T_1\) and \(T_2\) are joined by a porous plug whose pores are narrow compared with the mean free path, gas creeps from the cold side to the hot side until a pressure difference builds up. James Clerk Maxwell refereed the memoir and wrote his own paper on rarefied gases in response. In the limit of very narrow pores, a steady state requires equal molecular fluxes through the pores in both directions, and the flux of molecules through a small hole is proportional to \(p/\sqrt{T}\). Hence

For wide pores the pressures equalize. Reynolds measured the transition between these regimes as the ratio of pore size to mean free path changed, an early systematic study of what is now called rarefied gas dynamics.

The two manners of motion

For the general reader. Turn on a tap gently and the water runs in a smooth glassy stream. Open it further and the stream becomes ragged. In 1883 Reynolds showed with a thread of dye in a glass pipe that water flows in two different ways, smoothly or chaotically, and that one number built from the speed, the pipe width and the viscosity decides which.

The experiment

The paper, submitted to the Royal Society in March 1883 and published in the Philosophical Transactions, has a long title: an experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Reynolds mounted glass tubes with flared, bell-mouthed entrances inside a tank of still water and drew water through them with a valve at the far end. A fine jet of coloured water was released at the centre of the entrance.

At low speeds the dye formed a straight thread along the whole tube. As the speed increased, at a definite point the thread suddenly mixed into the water and filled the tube with colour. Lit by an electric spark, the coloured cloud resolved into distinct eddies. Reynolds called the two regimes direct and sinuous; we say laminar and turbulent. He found that the critical speed was inversely proportional to the tube diameter and proportional to the kinematic viscosity, which he varied by changing the water temperature. The critical combination was therefore a pure number.

He found that the value at which turbulence appeared depended on how calm the water was in the tank, and could be raised far above its usual value with great care. On the other hand, a turbulent flow slowed down gradually always became laminar below a lower value, close to 2000 in modern units. Precise measurements in 2011 placed the threshold for sustained pipe turbulence at about 2040.

Dynamical similarity

Write the incompressible Navier–Stokes equations in terms of a velocity scale \(U\), a length \(L\), time \(L/U\) and pressure \(\rho U^2\). With \(\mathbf u=U\mathbf u^*\), \(\mathbf x=L\mathbf x^*\), and so on,

George Gabriel Stokes had identified this combination in his 1851 paper on pendulums. Reynolds showed experimentally that it controls the most important qualitative change a flow can undergo. Arnold Sommerfeld attached Reynolds’s name to it in 1908.

Laws of resistance

For a pipe of diameter \(D\), mean speed \(V\) and length \(\ell\), the pressure drop is written \(\Delta p=f\,\frac{\ell}{D}\,\frac12\rho V^2\), with \(f\) the Darcy friction factor. In the second half of his paper Reynolds measured the resistance of long pipes and found that it rises in proportion to the speed below the critical value, as the Poiseuille law requires, and roughly as a power of the speed close to 2 above it.

Dynamical similarity and models

For the general reader. Engineers test small model ships, aircraft and harbours before building the real thing. A model only tells the truth if the right dimensionless numbers match. Reynolds was among the first to use scale models systematically in research, building small working models of tidal estuaries to study how sand banks form.

Proof idea. Dimensions form a vector space over \(\mathbb Q\) with basis M, L, T. Monomials \(\prod q_i^{a_i}\) are dimensionless when \((a_i)\) lies in the kernel of the \(r\times n\) dimension matrix, a space of dimension \(n-r\). Rescaling the base units acts on the remaining \(r\) quantities freely, so a unit-independent relation can depend only on the invariants.

Froude against Reynolds

William Froude’s towing-tank experiments of the 1870s showed that wave-making resistance of ships scales with the Froude number \(\mathrm{Fr}=U/\sqrt{gL}\). Frictional resistance, which Reynolds’s work showed depends on \(\mathrm{Re}\), scales differently. Both cannot be matched in water, so model tests split the resistance into a wave part scaled by Froude and a friction part estimated from flat-plate data as a function of \(\mathrm{Re}\). Reynolds wrote on ship design and screw propulsion for the Institution of Naval Architects. His studies of the racing of screw propellers, when the blades lose their grip on the water, touched on what is now called cavitation, the formation of vapour cavities where the local pressure falls too low.

Estuary models

In the late 1880s Reynolds built small tidal models, with sand beds and a mechanism to raise and lower the sea level periodically, to reproduce the banks and channels of estuaries. He reported them in his 1887 address to the engineering section of the British Association. Such models cannot match \(\mathrm{Re}\), yet he found that they reproduced the main features of real estuaries. Hydraulic laboratories later used the same approach for rivers and harbours throughout the twentieth century.

The theory of lubrication

For the general reader. A spinning shaft in an oiled bearing does not touch the metal around it. It floats on a thin film of oil whose pressure can exceed many atmospheres, generated by the motion itself. Reynolds explained this in 1886 with an equation that is still the basis of bearing design and of the whole science of tribology.

Tower’s experiments

In 1883 Beauchamp Tower, working for the Institution of Mechanical Engineers, tested a railway axle bearing running in an oil bath. When he drilled a hole in the bearing cap for an oiler, oil was forced out of it; a pressure gauge on the hole showed a pressure far above the average load per unit area. The oil was supporting the load hydrodynamically. Reynolds’s paper On the theory of lubrication and its application to Mr. Beauchamp Tower’s experiments, in the Philosophical Transactions of 1886, explained this and also reported his measurements of the viscosity of olive oil as a function of temperature.

Thin-film equations

Let the film occupy \(0<z<h(x)\), with \(h\) small compared with the length \(L\) of the bearing, the lower surface sliding with speed \(U\) in the \(x\) direction and the upper surface fixed. If \(\mathrm{Re}\,(h/L)^2\ll1\), inertia is negligible and the Navier–Stokes equations reduce to

So \(p=p(x)\), and with \(u(0)=U,\ u(h)=0\),

The slider bearing

Take \(h\) decreasing linearly from \(h_1\) at \(x=0\) to \(h_2\) at \(x=L\), with \(p=0\) at both ends. Integrating once, \(dp/dx=6\mu U(h-h^*)/h^3\), where \(h^*\) is the gap at the pressure maximum. The boundary conditions give \(h^*=2h_1h_2/(h_1+h_2)\) and

With \(k=h_1/h_2\), the load per unit width is

The load grows as \(1/h_2^2\): a heavier load squeezes the film thinner, which raises the pressure until it balances. This self-regulation is why hydrodynamic bearings are stable. The bracket is maximal at \(k\approx2.2\), where \(W\approx0.16\,\mu UL^2/h_2^2\).

Dilatancy of granular media

For the general reader. Step on wet sand at the water’s edge and the sand around your foot turns dry and pale for a moment. Reynolds explained why in 1885. Tightly packed grains cannot be sheared without spreading apart, so the sand expands, sucks water in from the surface and looks dry. He called this property dilatancy.

The idea

In his 1885 paper on the dilatancy of media composed of rigid particles in contact, and in lectures at the Royal Institution in 1885–1886, Reynolds pointed out that a dense packing of rigid grains is locked. To deform it, grains must ride up over one another, and the total volume must increase. A loose packing, by contrast, compacts when sheared. He demonstrated the effect with a rubber bag filled with sand or shot and water: when squeezed, the bag became rigid and drew in water, rather than yielding.

The modern form of this idea is the critical state theory of soil mechanics developed at Cambridge in the 1950s and 1960s. Arthur Casagrande introduced the critical void ratio in 1936, the density at which a sand shears without volume change. Dilatancy explains why dense sands are strong and loose saturated sands can liquefy in earthquakes. Reynolds later tried to build a theory of the aether on the same principle (Lesson 11).

A geometric model

Consider layers of equal spheres stacked in two ways. In the simple cubic packing each sphere sits directly on the one below, with packing fraction \(\pi/6\approx0.524\). In the densest packing each sphere sits in a hollow of the layer below, with fraction \(\pi/(3\sqrt2)\approx0.740\). Shearing one dense layer over the next forces the upper spheres to climb out of the hollows, raising the layer.

Reynolds averaging and the turbulent stresses

For the general reader. A turbulent flow is a tangle of eddies that never repeats. Engineers usually care only about its average behaviour. In 1895 Reynolds showed how to write equations for the average flow. They contain new terms, the Reynolds stresses, which describe how the eddies carry momentum. Finding these terms without solving for every eddy is the closure problem of turbulence, still unsolved in general.

The decomposition

In On the dynamical theory of incompressible viscous fluids and the determination of the criterion, read in 1894 and published in the Philosophical Transactions in 1895, Reynolds split each velocity component into a mean and a fluctuation, \(u_i=U_i+u_i'\), with \(\overline{u_i'}=0\). He used local space averages; today one uses ensemble or time averages. The averaging operator is linear, commutes with derivatives and satisfies \(\overline{U_iu_j'}=0\).

Derivation. Write the convective term in conservation form \(\partial_j(u_iu_j)\), which is allowed because \(\partial_ju_j=0\). Averaging, \(\overline{u_iu_j}=U_iU_j+\overline{u_i'u_j'}\), because the cross terms average to zero. The extra term is moved to the right-hand side, where it acts like a stress. Its trace is \(-2\rho k\), with \(k=\tfrac12\overline{u_i'u_i'}\) the turbulent kinetic energy.

The closure problem

The averaged equations have more unknowns than equations. Equations for \(\overline{u_i'u_j'}\) can be derived, but they contain triple correlations, and so on without end. Joseph Boussinesq had proposed in 1877 to model turbulent stress by an eddy viscosity, \(-\overline{u_i'u_j'}=\nu_t(\partial_jU_i+\partial_iU_j)-\tfrac23k\delta_{ij}\). Ludwig Prandtl’s mixing length (1925) and the \(k\)–\(\varepsilon\) models of the 1970s are ways of choosing \(\nu_t\). Every Reynolds-averaged (RANS) code used in engineering today starts from Reynolds’s 1895 equation.

The energy method and the critical number

For the general reader. Why does turbulence appear at a Reynolds number near 2000 and not at 20 or 200 000? Reynolds tried to answer this from the equations themselves by tracking the energy of a disturbance: the mean flow feeds it, viscosity drains it. His bookkeeping gives a speed below which every disturbance must die. The gap between that bound and the observed transition is still a central theme of the subject.

The second half of Reynolds’s 1895 paper sought the critical number theoretically, completing the title’s promise of a criterion.

Reynolds’s energy equation

The second half of the paper sought the critical Reynolds number theoretically. Reynolds derived an equation for the kinetic energy of the fluctuations. In the form refined by William Orr in 1907, for a disturbance \(\mathbf u'\) of a steady flow \(\mathbf U\) in a bounded domain with no-slip walls,

The first term transfers energy from the mean flow to the disturbance; the second is viscous dissipation.

Proof sketch. In dimensionless form the energy equation reads \(\dot E=-\int u_i'u_j'\partial_jU_i\,dV-\mathrm{Re}^{-1}\int|\nabla\mathbf u'|^2dV\le-\big(\mathrm{Re}^{-1}-\mathrm{Re}_E^{-1}\big)\int|\nabla\mathbf u'|^2dV\). The Poincaré inequality \(\int|\nabla\mathbf u'|^2\ge\lambda_1\int|\mathbf u'|^2\) then gives \(\dot E\le-cE\) with \(c>0\). The maximization is a linear eigenvalue problem, the Euler–Lagrange equation of the ratio.

This gives a rigorous lower bound for the transition Reynolds number. For pipe flow, based on the mean speed and diameter, \(\mathrm{Re}_E\approx81\), far below the observed value of about 2000. Between the two, disturbances can grow for a while before they decay, a phenomenon now called transient growth, and turbulence can sustain itself only above about 2000. Orr corrected and sharpened Reynolds’s analysis in 1907, and the energy method in this form became a standard tool of stability theory.

Linear theory

A second approach linearizes the equations about the base flow and looks for exponentially growing modes. William Orr and Arnold Sommerfeld formulated the resulting Orr–Sommerfeld equation in 1907–1908. For plane Poiseuille flow it predicts instability above \(\mathrm{Re}\approx5772\) (based on centreline speed and half-width), a value computed by Steven Orszag in 1971. For pipe flow linear theory finds no instability at all. Pipe turbulence is therefore a finite-amplitude phenomenon, exactly as Reynolds’s observations of calm and disturbed inlets suggested.

Heat, Joule and the sub-mechanics of the universe, 1888–1903

By the late 1880s Reynolds was one of the leading engineering scientists in Britain. The Royal Society awarded him a Royal Medal in 1888, and he became president of the Manchester Literary and Philosophical Society. The new Whitworth laboratories at Owens College gave him space for large experiments.

The largest was a new determination of the mechanical equivalent of heat. Manchester was James Prescott Joule’s city, and after Joule’s death in 1889 Reynolds wrote a long memoir of him for the Lit & Phil, published in 1892. He then used a large steam engine and a hydraulic brake of his own design to measure the work needed to heat water from freezing to boiling point. The result, a mean specific heat of water expressed in mechanical units, was reported to the Royal Society in the late 1890s and is regarded as one of the classical determinations of a physical constant. He gave the Royal Society’s Bakerian Lecture in 1897. He also wrote on the refraction of sound by wind and temperature gradients, on condensation of steam, and on group velocity, where Lamb credited him with showing that the group velocity is also the speed at which wave energy travels.

Collected papers and the granular aether

Between 1900 and 1903 Cambridge University Press published his Papers on Mechanical and Physical Subjects in three volumes, gathering about seventy papers. The third volume was a single work, The Sub-Mechanics of the Universe (1903), developed from his 1902 Rede Lecture at Cambridge. In it Reynolds proposed that space is filled with a dense medium of tiny rigid grains, a dilatant aether like his rubber bag of shot, and tried to derive light, gravitation and electricity from its mechanics. In passing he stated the transport theorem for integrals over moving regions that now carries his name:

The book was hard to read and appeared just as relativity and quantum theory were making aether models obsolete. George Bryan reviewed it sympathetically in Nature, but it had little influence.

Decline, death and legacy, 1905–1912

In the early 1900s Reynolds’s health began to fail, and his mental powers declined with it. He retired from his chair in 1905, after 37 years. He received the Dalton Medal of the Manchester Lit & Phil in 1903. He lived his last years in Somerset and died of influenza at Watchet on 21 February 1912, aged 69. He is buried at St Decuman’s church in Watchet. Horace Lamb wrote his obituary for the Royal Society, and Arthur Gibson later wrote a book-length account of his work in hydraulics.

The school

Reynolds trained a generation of engineers at Manchester. Thomas Stanton studied in the Whitworth laboratory under him from 1888 and stayed as demonstrator until 1896. At the National Physical Laboratory, Stanton and J. R. Pannell confirmed in 1914 that pipe friction for air and water collapses onto a single curve of friction factor against Reynolds number, the decisive test of dynamical similarity. J. J. Thomson, who passed through Owens College in the 1870s, went on to the Cavendish chair and the discovery of the electron.

Legacy

• Fluid mechanics: the Reynolds number organizes all of fluid dynamics, from bacteria swimming at \(\mathrm{Re}\sim10^{-4}\) to atmospheric flows at \(10^{9}\) and more. The question of how laminar pipe flow becomes turbulent was studied intensively for more than a century after 1883.
• Turbulence: Reynolds averaging and the Reynolds stresses are the starting point of turbulence modelling and computational fluid dynamics.
• Tribology: the Reynolds equation is the foundation of hydrodynamic bearing design; Duncan Dowson counted him among the founding figures of the subject.
• Heat transfer and granular physics: the Reynolds analogy and the concept of dilatancy remain standard.
• Commemoration: the University of Manchester held a centenary symposium in 1968 for the founding of his chair and has held an annual Reynolds Day on fluid mechanics since 2001.

Connections

People linked to Osborne Reynolds across this collection. A name without a link has no page here yet. See Osborne Reynolds in the network of all 337 lives →

Students

• J. J. Thomson 1870-1876
• Thomas Stanton 1888-1896

Influenced by

• George Gabriel Stokes
• James Prescott Joule
• Joseph Boussinesq

Influenced

• Arnold Sommerfeld
• Ludwig Prandtl
• William McFadden Orr

Colleagues and collaborators

• James Clerk Maxwell 1879
• James Prescott Joule
• John William Strutt, Lord Rayleigh 1863-1867
• William Crookes 1879
• Horace Lamb 1885-1905

Also mentioned in this life

• Henry Cavendish
• James Chadwick
• John Dalton
• Leonhard Euler
• Joseph-Louis Lagrange
• Gottfried Wilhelm Leibniz
• Claude-Louis Navier
• Henri Poincaré

Fields

• fluid mechanics 6 lives
• engineering 6 lives
• turbulence 5 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Osborne Reynolds, MacTutor History of Mathematics.
2. Osborne Reynolds, Wikipedia.
3. J. Allen, Reynolds, Osborne, Dictionary of Scientific Biography; online at Encyclopedia.com.
4. H. Lamb, Osborne Reynolds (obituary), Proc. Roy. Soc. A 88 (1913).
5. A. H. Gibson, Osborne Reynolds and His Work in Hydraulics and Hydrodynamics, Longmans, 1946.
6. D. M. McDowell and J. D. Jackson (eds.), Osborne Reynolds and Engineering Science Today, Manchester University Press, 1970.
7. J. J. Thomson, Recollections and Reflections, Bell, 1936.
8. APS News, March 1883: Osborne Reynolds and the onset of turbulence.

Physics

1. O. Reynolds, On the extent and action of the heating surface of steam boilers, Proc. Manchester Lit. Phil. Soc. 14 (1874–75).
2. O. Reynolds, On certain dimensional properties of matter in the gaseous state, Phil. Trans. R. Soc. 170 (1879).
3. O. Reynolds, An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, Phil. Trans. R. Soc. 174 (1883) 935–982.
4. O. Reynolds, On the dilatancy of media composed of rigid particles in contact, Phil. Mag. 20 (1885).
5. O. Reynolds, On the theory of lubrication and its application to Mr. Beauchamp Tower’s experiments, Phil. Trans. R. Soc. 177 (1886) 157–234.
6. O. Reynolds, On the dynamical theory of incompressible viscous fluids and the determination of the criterion, Phil. Trans. R. Soc. A 186 (1895) 123–164.
7. O. Reynolds, Papers on Mechanical and Physical Subjects, 3 vols., Cambridge University Press, 1900–1903.
8. N. Rott, Note on the history of the Reynolds number, Annu. Rev. Fluid Mech. 22 (1990) 1–11.
9. K. Avila et al., The onset of turbulence in pipe flow, Science 333 (2011) 192–196.
10. S. B. Pope, Turbulent Flows, Cambridge University Press, 2000.
11. D. Dowson, History of Tribology, Longman, 1979.
12. Reynolds number and Reynolds transport theorem, Wikipedia.