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Navier–Stokes Equations

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Navier-Stokes Turbulence

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Abstract

The dynamics of the Newtonian fluids considered here are determined by the laws of classical mechanics, a selection of references for the derivation of the fundamental pdes from these laws are Lamb [1], Landau and Lifshitz [2], Serrin [3], Majda and Bertozzi [4], Wu et al. [5]. The description and analysis of fully developed turbulent flows is the central theme. Turbulence in superfluids, i.e. liquid helium below the \(\lambda \)-temperature \(T=2.17\) K at atmospheric pressure, exhibits properties similar to turbulence in classical Newtonian fluids for vanishing viscosity, but are also subject to quantum-mechanical phenomena that do not have counterparts in classical fluid mechanics.

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References

  1. Lamb, H.: Hydrodynamics. 3rd ed. Cambridge University Press (1906)

    Google Scholar 

  2. Landau, L.D., Lifshitz, E.M.: Fluid Mechanics. Pergamon Press, Oxford, U.K. (1956)

    Google Scholar 

  3. Serrin, J.: The initial value problem for the Navier-Stokes equations. In: Langer, R. (ed.) Nonlinear Problems, Symp. Proc., University of Wisconsin Press, p. 69 (1963)

    Google Scholar 

  4. Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge, U.K. (2002)

    MATH  Google Scholar 

  5. Wu, J.-Z., Ma, H.-Y., Zhou, M.-D.: Vorticity and Vortex Dynamics. Springer, Berlin (2006)

    Book  Google Scholar 

  6. Guénault, A.M.: Basic Superfluids. Taylor & Francis, London, U.K. (2003)

    Book  MATH  Google Scholar 

  7. Vinen, W.F.: Superfluid turbulence: experiments, physical principles, and outstanding problems. In: Smits, A.J. (ed.) IUTAM Symposium on Reynolds Number Scaling in Turbulent Flow, pp. 101–108. Kluwer Academic Publishers, Dordrecht (2004)

    Chapter  Google Scholar 

  8. Donnelly, R.J.: Background on Superfluid turbulence. In: Smits, A.J. (ed.) IUTAM Symposium on Reynolds Number Scaling in Turbulent Flow, pp. 92–100. Kluwer Academic Publishers, Dordrecht (2004)

    Chapter  Google Scholar 

  9. Skrbek, L.: Quantum turbulence. J. Phys. Conf. Ser. 318, 012004 (2011)

    Article  Google Scholar 

  10. Kollmann, W.: Fluid Mechanics in Spatial and Material Description. University Readers, San Diego (2011)

    Google Scholar 

  11. Bennett, A.: Lagrangian Fluid Dynamics. Cambridge University Press, Cambridge, U.K. (2006)

    Book  MATH  Google Scholar 

  12. Cantwell, B.J.: Introduction to Symmetry Analysis. Cambridge University Press, U.K. (2002)

    MATH  Google Scholar 

  13. Gallavotti, G.: Foundations of Fluid Dynamics. Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  14. Tsinober, A.: An Informal Conceptual Introduction to Turbulence, 2nd edn. Springer, Dordrecht (2009)

    Book  MATH  Google Scholar 

  15. Poinsot, T., Veynante, D.: Theoretical and Numerical Combustion. R.T. Edwards, Philadelphia, PA (2001)

    Google Scholar 

  16. Frisch, U.: Turbulence: The Legacy of A.N. Kolmogorov. Cambridge University Press (1995)

    Google Scholar 

  17. Oberlack, M.: Symmetrie, Invarianz und Selbstähnlichkeit in der Turbulenz. Shaker, Aachen, Germany (2000)

    Google Scholar 

  18. Oberlack, M., Waclawczyk, M., Rosteck, A., Avsarkisov, V.: Symmetries and their importance for statistical turbulence theory. Bull. JSME 2, 1–72 (2015)

    Google Scholar 

  19. Duistermaat, J.J., Kolk, J.A.C.: Lie Groups. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  20. Von Wahl, W.: The Equations of Navier-Stokes and Abstract Parabolic Equations. Vieweg, Braunschweig (1985)

    Book  MATH  Google Scholar 

  21. Constantin, P., Foias, C.: Navier-Stokes Equations. University of Chicago Press (1988)

    Google Scholar 

  22. Kreiss, H.O., Lorentz, J.: Initial-Boundary Value Problems and the Navier-Stokes Equations. Academic Press (1989)

    Google Scholar 

  23. Sohr, H.S.: The Navier-Stokes Equations. Springer, Basel (2001)

    Book  MATH  Google Scholar 

  24. Constantin, P.: Euler Equations, Navier-Stokes equations and turbulence. In: Mathematical Foundation of Turbulent Viscous Flows. Lecture Notes in Mathematics, vol. 1871, pp. 1–43 (2006)

    Google Scholar 

  25. www.claymath.org/millenium/

  26. Eyink, G.L.: Energy dissipation without viscosity in ideal hydrodynamics: fourier analysis and local energy transfer. Physica D 78, 222–240 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Duchon, J., Robert, R.: Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations. Nonlinearity 13, 249–255 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Spalart, P.R.: Numerical Simulation of Boundary Layers. Part 1. Weak Formulation and Numerical Method, NASA TM-88222 (1986)

    Google Scholar 

  29. Seregin, G.: Lecture Notes on Regularity Theory for the Navier-Stokes Equations. World Scientific, New Jersey (2015)

    MATH  Google Scholar 

  30. Hopf, E.: Uber die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachrichten 4, 213–231 (1950)

    Article  MATH  Google Scholar 

  31. Kiselev, A.A., Ladyshenskaya, O.A.: On existence and uniqueness of the solution of the nonstationary problem for a viscous incompressible fluid. Izvestiya Akad. Nauk SSSR 21, 655 (1957)

    MathSciNet  Google Scholar 

  32. Kuksin, S., Shirikyan, A.: Mathematics of two-dimensional turbulence, Cambridge Tracts in Mathematics, vol. 194. Cambridge University Press, U.K. (2012)

    Book  MATH  Google Scholar 

  33. Serrin, J.: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Rat. Mech. Anal. 9, 187–196 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  34. Pileckas, K.: Navier-Stokes system in domains with cylindrical outlets to infinity. In: Friedlander, S., Serre, D. (eds.) Handbook of Mathematical Fluid Dynamics, vol. 4, North Holland, pp. 447–647 (2007)

    Google Scholar 

  35. Temam, R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988)

    Book  MATH  Google Scholar 

  36. Vishik, M.J., Fursikov, A.V.: Mathematical Problems of Statistical Hydromechanics. Kluwer Academic Publication, Dordrecht (1988)

    Book  MATH  Google Scholar 

  37. Vishik, M.J.: Asymptotic Behaviour of Solutions of Evolutionary Equations. Cambridge University Press, Lezioni Lincee (1992)

    MATH  Google Scholar 

  38. Morse, E.C.: Eigenfunctions of the curl in cylindrical geometry. J. Math. Phys. 46, 113511-1–113511-13 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Truesdell, C.A.: The Kinematics of Vorticity. Indiana University Press, Bloomington, Indiana (1954)

    MATH  Google Scholar 

  40. Brown, G.L., Roshko, A.: On density effects and large structure in turbulent mixing layers. JFM 64, 775–816 (1974)

    Article  ADS  MATH  Google Scholar 

  41. Constantin, P.: Analysis of Hydrodynamic Models. CBMS-NSF Regional Conference Series in Applied Mathematics (2017)

    Google Scholar 

  42. Ciarlet, P.G.: Mathematical Elasticity: Three-dimensional Elasticity, vol. 1. Elsevier Sci. Pub, Amsterdam, The Netherlands (1988)

    MATH  Google Scholar 

  43. Monin, A.S., Yaglom, A.M.: Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 1. The MIT Press, Cambridge Mass (1975)

    Google Scholar 

  44. Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge, U.K. (2001)

    MATH  Google Scholar 

  45. Casey, J., Naghdi, P.M.: On the lagrangian description of vorticity. Arch. Rational Mech. Anal. 115, 1–14 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. Vladimirov, V.A., Moffatt, H.K.: On general transformations and variational principles for the magnetohydrodynamics of ideal fluids. Part 1. Fundamental principles. JFM 283, 125–139 (2001)

    Article  ADS  MATH  Google Scholar 

  47. Milnor, J.: Morse Theory. Princeton University Press (1963)

    Google Scholar 

  48. Meneveau, C.: Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows. Annua. Rev. Fluid Mech. 43, 219–245 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Vieillefosse, P.: Local interaction between vorticity and shear in a perfect incompressible fluid. J. Phys. France 43, 837–842 (1982)

    Article  MathSciNet  Google Scholar 

  50. Vieillefosse, P.: Internal motion of a small element of fluid in an inviscid flow. Physica A 125, 150–162 (1984)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  51. Cantwell, B.J.: Exact solution of a restricted Euler equation for the velocity gradient tensor. Phys. Fluids A 4, 782–793 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. Gibbon, J.D., Galanti, B. and Kerr, R.M.: Stretching and compression of vorticity in the 3D euler equations. In: Hunt, J.C.R., Vassilicos, J.C. (eds.) Turbulence Structure and Vortex Dynamics. Cambridge University Press (2000)

    Google Scholar 

  53. Wilczek, M., Meneveau, C.: Pressure Hessian and viscous contributions to velocity gradient statistics based on Gaussian random fields. JFM 756, 191–225 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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  1. Department of Mechanical Engineering, University of California Davis, Davis, CA, USA

    Wolfgang Kollmann

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Kollmann, W. (2019). Navier–Stokes Equations. In: Navier-Stokes Turbulence. Springer, Cham. https://doi.org/10.1007/978-3-030-31869-7_2

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