Skip to main content
SpringerLink
Log in

Viscosity, Fast Flows and the Science of Flight

  • Chapter
  • First Online:
Book cover Continuum Mechanics through the Ages - From the Renaissance to the Twentieth Century

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 223))

Abstract

In line with our previous perusing of the early developments of hydraulics and the foundation works on porous media, the present contribution pays more due attention to the effects of viscosity, to increasing levels of velocity of the fluid flows, and to the inception of the mechanics of vortices. The main characters at work are essentially Helmholtz (his vortex filaments) and Reynolds (his celebrated number). The application side materializes in the groundbreaking ideas of Lanchester and the more mathematical approach of Prandtl that were going to bring a true efficient revolution in the just born science of flight. The role played by non-dimensional numbers, dimensional analysis, and new methods such as good asymptotic expansions is rightly emphasized, leading us to the successes of the mathematics of aeronautics and astronautics in the twentieth century. Again, the offered approach is more discursive and historical than mathematically detailed and rigorous.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The reader can check by himself that Eqs. (7), (6), and (5) are none other than Eqs. (1, p. 44), (3, p. 49) and the unnumbered equation in p. 52 in the English translation of the original German text; cf. Helmholtz (1858). In particular Eq. (3, p. 49) is none other than the modern equation.

    $$ \frac{{\partial \underline{\omega } }}{\partial t} - \left( {\underline{\omega } \cdot \nabla } \right){\mathbf{v}} = {\mathbf{0}}. $$

    For the notion of connectedness Helmholtz refers to recent work by Riemann in the previous volume of the same journal.

  2. 2.

    Of course Prandtl was a very serious man who took all matters at heart but also a bit naïve so that we cannot resist the temptation to report the gossip told by Anderson (1997, p. 259): Having reached the age of 34 he decided he should get married. To that purpose he went to visit his master August Föppl and his wife who had two daughters, asking if he could marry one of them, without further precision. The Föppls made a family decision by selecting the eldest daughter. Prandtl and Gertrude Föppl married, lived happy and themselves had two daughters. The story does not tell if one of them or even the two, married some of Prandtl’s students, but this is quite possible if the tradition had to be respected. This almost happened to the present writer.

  3. 3.

    This is a theory in which the elastic potential energy is quadratic in the finite strain measure. The resulting equations surprisingly have a pure geometric meaning.

  4. 4.

    I personally think that the corresponding French expression “allée des rouleaux (ou tourbillons) de von Kármán” sounds less vulgar and more “chic” (but this is a snob’s opinion).

  5. 5.

    See the snapshots of the atomic explosion in pp. 236–238 in the French edition (1977) of Sedov (1944) [These are in fact reproduced from Taylor (1950a, b)].

  6. 6.

    This chapter exceptionally bears the print of the initial formation of the author as an aeronautical engineer and simultaneously a university graduate student in fluid mechanics in the 1960s.

References

  • Anderson JD Jr (1997) A history of aerodynamics. C.U.P, Cambridge, UK

    Book  Google Scholar 

  • Barenblatt GI (1979) Scaling, self similarity and intermediate asymptotics (translation from the Russian, 1st edn). Consultant Bureau, New York (Reprint, Springer, New York, 2012)

    Google Scholar 

  • Barenblatt GI (1987) Dimensional analysis (English translation by P. Makerin). Gordon and Breach, New York

    Google Scholar 

  • Barré de Saint-Venant A (1843) Note à joindre au mémoire sur la dynamique des fluides. C R Acad Sci Paris 17:1240

    Google Scholar 

  • Bertrand J (1878) Sur l’homogénéité dans les formules de physique. C R Acad Sci Paris 86:916–920

    MATH  Google Scholar 

  • Blasius H (1912) Das Aehnlichkeitsgesetz bei Reibungsvorgängen. VDI-Z 56:639–643

    Google Scholar 

  • Born M, von Kármán T (1912) On fluctuations in spatial grids. Zeit Physik 13:297–309 [Also: On the distribution of natural vibrations of spatial lattices. Zeit Phys 14: 65–71, 1913]

    Google Scholar 

  • Bridgman PW (1931) Dimensional analysis. Yale University Press, Connecticut

    Google Scholar 

  • Buckingham E (1914) On physically similar systems. Illustrations of the use of dimensional equations. Phys Rev 4:345–376

    Article  Google Scholar 

  • Eckhardt B (2009) Introduction. Turbulence transition in pipe flow: 125th anniversary of Reynolds paper. Phil Trans Roy Soc Lond A367:449–455

    Article  MathSciNet  Google Scholar 

  • Favre A, Kovasnay LG, Dumas R, Caviglio J (1979) La turbulence en mécanique des Fluides. Gauthier-Villars, Paris

    Google Scholar 

  • Flamant A (1900) Hydraulique, 2nd edn. Béranger, Paris (First edition, 1891)

    Google Scholar 

  • Germain P (1977) Méthodes asymptotiques en mécanique des fluides. In: Peube JL (ed) Fluid dynamics, (Lectures at Les Houches, 1973, France). Gordon and Breach, London, pp 1–147

    Google Scholar 

  • Hagen G (1869) Über die Bewegung des Wassers in cylindrischen nahe horizontalen Leitungen. Math Abh Akad Wiss, Berlin, pp 1–29

    Google Scholar 

  • Lagerstrom PA (1988) Matched asymptotic expansions. Springer, New York

    Book  MATH  Google Scholar 

  • Lamb H (1912–1913) Osborne Reynolds. Proc Roy Soc Lond 88A:XV–XXI

    Google Scholar 

  • Lumley JL (1979) Toward a turbulent constitutive equation. J Fluid Mech 41:413–434

    Article  Google Scholar 

  • Maugin GA (2014a) Continuum mechanics through the eighteenth and nineteenth Centuries. Springer, Dordrecht

    Book  MATH  Google Scholar 

  • Maugin GA (2014b) Hydraulics, Preprint UPMC-Paris 6 (see Chapter 2 in this book)

    Google Scholar 

  • McDowell DM, Jackson JD (eds) (1970) Osborn Reynolds and engineering science today. Manchester University Press, UK [papers presented at the Osborne Reynolds Centenary Symposium, Manchester, Sept 1968]

    Google Scholar 

  • Moffatt K (2008) Vortex dynamics: the legacy of Helmholtz and Kelvin. In: Borisov AV et al (ed) Proceedings of IUTAM Symposium on Hamiltonian dynamics. Springer, Heidelberg, pp 1–10

    Google Scholar 

  • Navier C LMH (1822) Sur les lois du mouvement des fluides. Bull Soc Philomatique 75–79 (also: Mémoire sur les lois du mouvement des fluides. Mémoires Acad Roy Sciences (Paris), 6:389–416, 1823)

    Google Scholar 

  • Poiseuille JLM (1846) Recherches expérimentales sur le mouvement des liquides dans les tubes de très petit diamètre. Mém Acad Sci 9:433–564

    Google Scholar 

  • Prandtl L (1905) Über Flüssigkeitbewegung bei sehr kleiner Reibung. In: Proceedins of 3rd international mathematical congress, Heidelberg 1904. Leipzig, pp 484–492

    Google Scholar 

  • Prandtl L (1910) Eine Beziehung zwischen Wärmeaustausch und Strömungswiderstand der Flüssigkeiten. Phys Zeit 11:1072–1078

    MATH  Google Scholar 

  • Prandt L (1931) Führer durch die Strömungslehre. Vieweg Verlag, Braunschweig (reprint of 12th edition in 2008)

    Google Scholar 

  • Prandtl L, Tietjens O (1931) Hydro- und Aeromechanik. Julius Springer, Berlin

    MATH  Google Scholar 

  • Rayleigh L (1892) On the question of the stability of the flow of fluids. Phil Mag 34:177–180 (also, in Collected Scientific Papers, 3:575–584, CUP, 1920)

    Google Scholar 

  • Rayleigh L (1913) Sur la résistance des sphères dans l’air en écoulement. C R Acad Sci Paris 156:109–113 (also in Collected Scientific Papers, 6:136, CUP, 1920)

    Google Scholar 

  • Reynolds O (1874) Proceedings of Manchester Literary and Philosophical Society, 14, p 7 on, (not seen by the author)

    Google Scholar 

  • Reynolds O (1883) An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and the law of resistance in parallel channels. Phil Trans 174:935–982

    Article  MATH  Google Scholar 

  • Reynolds O (1885) On the dilatancy of media composed of rigid particles in contact. Phil Mag, December issue (see Reynolds collected papers II:203–216)

    Google Scholar 

  • Reynolds O (1886) On the theory of lubrification and the experimental determination of the viscosity of olive oil. Phil Trans Roy Soc, Part I. (see Collected papers, 1900–1901 II:203–216)

    Google Scholar 

  • Reynolds O (1894) On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Phil Trans Roy Soc 186:123–164

    Article  Google Scholar 

  • Reynolds O (1900–1901) Papers on mechanical and physical subjects, 2 vols. Cambridge University Press, UK

    Google Scholar 

  • Riabouchinsky D (1911). Méthode des variables de dimension zéro et son application en Aérodynamique. L’Aérophile 407–408

    Google Scholar 

  • Rott N (1990) Note on the history of the Reynolds number. Ann Rev Fluid Mechanics 22:1–11

    Article  MathSciNet  Google Scholar 

  • Rotta J (1951) Statistischer Theorie nichthomogener Turbulenz. Zeit Physik 129:547–572

    Article  MathSciNet  MATH  Google Scholar 

  • Rouse H, Ince S (1957) History of hydraulics. Iowa City, State Univ. Iowa (Reprint by Dover, New York, 1963)

    Google Scholar 

  • Saffman PG (1995) Vortex dynamics. Cambridge University Press, UK

    Google Scholar 

  • Saph AV, Schoder EW (1903) An experimental study of the resistance to the flow of water in pipes. Trans Amer Soc Civil Eng 51:253–330

    Google Scholar 

  • Schlichting H, Gersten K, Krause E, Oertel H, Mayes C (2004) Boundary layer theory, 8th edn. Springer, Berlin

    Google Scholar 

  • Sedov LI (1944) Similarity and dimensional methods in mechanics (original Russian edition), Moscow [English translation of 4th edn, Infosearch Ltd, London, 1959; French translation of the sixth Russian edition of 1967, Editions MIR, Moscow, 1977]

    Google Scholar 

  • Sommerfeld A (1908) Ein Betrag zur hydrodynamischen Erklärung der turbulenten Flüssigkeitsbewegung, In: Proceedings of the 4th international congress Math., Rome, vol 3, pp 116–124

    Google Scholar 

  • Speziale CG (1981) On turbulent Reynolds stress closure and modern continuum Mechanics. Int J Non-Linear Mech 16:387–393

    Article  MathSciNet  MATH  Google Scholar 

  • Speziale CG, Eringen AC (1981) Nonlocal fluid mechanics description of wall turbulence. Com Meth Applications 7:321–343

    MathSciNet  Google Scholar 

  • Stokes GG (1845) On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Trans Cambridge Phil Soc 8:287–319

    Google Scholar 

  • Taylor GI (1950a) The formation of a blast wave by a very intense explosion, I. Theoretical discussion, Proc Roy Soc Lond A vol 201, pp 159–174

    Google Scholar 

  • Taylor GI (1950b) The formation of a blast wave by a very intense explosion, II. The atomic explosion, ibid, A vol 201, pp 175–186

    Google Scholar 

  • Tytell ED, Standen EM, Lauder GV (2008) Escaping Flatland: three-dimensional kinematics and hydrodynamics of median fins in fishes. J Exp Biol 211:187–195

    Google Scholar 

  • Van Dyke M (1982) An album of fluid motion. Parabolic, Stanford

    Google Scholar 

  • Vaschy A (1892) Sur les lois de similitude en physique. Annales Télégraphiques 19:25–28

    Google Scholar 

  • von Helmholtz H (1858) Über Integrale der Hydrodynamischen Gleichungen, welche denWirbelbewegungen entspreche. J reine angew Math 55:25–55 [English translation available on the web at: www.21stcenturysciencetech.com/Articles_2009/Helmholtz.pdf; pp 41–68]

    Google Scholar 

  • von Kármán Th (1910) Festigkeit im Maschinenbau. Encycl Math Wiss 4:311

    Google Scholar 

  • von Kármán Th (1911) Über die Turbulenareibung verschierdener Flüssigkeiten. Phys Zeit 12:283–284

    MATH  Google Scholar 

  • von Kármán Th (1961) From low-speed aerodynamics to astronautics. Pergamon Press, Oxford, UK

    Google Scholar 

  • Zeytounian RKH (2002) Asymptotic modelling of fluid flow phenomena. Kluwer, Dordrecht

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Jean Le Rond d’Alembert, Université Pierre et Marie Curie, Paris Cedex 05, France

    Gérard A. Maugin

Authors
  1. Gérard A. Maugin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gérard A. Maugin .

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Maugin, G.A. (2016). Viscosity, Fast Flows and the Science of Flight. In: Continuum Mechanics through the Ages - From the Renaissance to the Twentieth Century. Solid Mechanics and Its Applications, vol 223. Springer, Cham. https://doi.org/10.1007/978-3-319-26593-3_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-26593-3_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-26591-9

  • Online ISBN: 978-3-319-26593-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation