• Wolfgang KollmannEmail author


Any fluid consists of elementary particles and molecular compounds. The first step in the analysis of flow phenomena is the decision on the range of physical conditions measured by temperature, pressure and chemical composition to be considered and then the fundamental approach for their mathematical description can be selected.


  1. 1.
    Van Kampen, N.G.: Stochastic Processes in Physics and Chemistry, 3rd edn. Elsevier, Amsterdam, The Netherlands (2007)zbMATHGoogle Scholar
  2. 2.
    Bogachev, V.I.: Measure Theory, vol. 1. Springer, New York (2006)Google Scholar
  3. 3.
    Berge, P., Pomeau, Y., Vidal, C.: Order within Chaos: Towards a Deterministic Approach to Turbulence. Wiley, New York (1984)zbMATHGoogle Scholar
  4. 4.
    Mandelbrot, B.B.: Fractals and Chaos: The Mandelbrot Set and Beyond. Springer, New York (2004)zbMATHCrossRefGoogle Scholar
  5. 5.
    Karniadakis, G., Beskok, A., Aluru, N.: Microflows and Nanoflows. Springer, New York (2000)zbMATHGoogle Scholar
  6. 6.
    Bradshaw, P., Ferriss, D.H., Johnson, R.F.: Turbulence in the noise-producing region of a circular jet. JFM 19, 591–624 (1964)ADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Oertel Sr., H.: Modern developments in shock tube research. Shock Tube Research Soc. Japan, 488–495 (1975)Google Scholar
  8. 8.
    Schmid, P.J., Henningson, D.S.: Stability and Transition in Shear Flows. Springer (2001)Google Scholar
  9. 9.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (2003)zbMATHGoogle Scholar
  10. 10.
    Kollmann, W.: Fluid Mechanics in Spatial and Material Description. University Readers, San Diego (2011)Google Scholar
  11. 11.
    Woyczynski, W.A.: Burgers-KPZ Turbulence: Goettingen Lectures. Lecture Notes in Math, vol. 1700. Springer, Berlin (1999)Google Scholar
  12. 12.
    Kraichnan, R.H.: Lagrangian history statistical theory for Burgers’ equation. Phys. Fluids II, 265–277 (1968)ADSzbMATHCrossRefGoogle Scholar
  13. 13.
    Onsager, L.: Statistical hydrodynamics. Nuovo Cimento 6(Suppl.), 279–287 (1949)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Lumley, J.L.: Stochastic tools in Turbulence. Academic Press (1970)Google Scholar
  15. 15.
    Frisch, U.: Turbulence: The Legacy of A.N. Kolmogorov. Cambridge University Press (1995)Google Scholar
  16. 16.
    Tsinober, A.: An Informal Conceptual Introduction to Turbulence, 2nd edn. Springer, Dordrecht (2009)zbMATHCrossRefGoogle Scholar
  17. 17.
    Vishik, M.J.: Analytic solutions of Hopf’s equation corresponding to quasilinear parabolic equations or to the Navier-Stokes system. Sel. Math. Sov. 5, 45–75 (1986)zbMATHGoogle Scholar
  18. 18.
    Vishik, M.J., Fursikov, A.V.: Mathematical Problems of Statistical Hydromechanics. Kluwer Academic Publication, Dordrecht (1988)zbMATHCrossRefGoogle Scholar
  19. 19.
    Lewis, R.M., Kraichnan, R.H.: A Space-time functional formalism for turbulence. Comm. Pure Applied Math. XV, 397–411 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Dopazo, C., O’Brien, E.E.: Functional formulation of nonisothermal turbulent reactive flows. Phys. Fluids 17, 1968–1975 (1974)ADSzbMATHCrossRefGoogle Scholar
  21. 21.
    Rogers, M.M., Moser, R.D.: Direct simulation of a self-similar turbulent mixing layer. Phys. Fluids 6, 903–923 (1994)ADSzbMATHCrossRefGoogle Scholar
  22. 22.
    Lee, M., Moser, R.D.: Direct numerical simulation of turbulent channel flow up to \(Re_{\tau }=5200\). JFM 774, 395–415 (2015)ADSCrossRefGoogle Scholar
  23. 23.
    Hoyas, S., Jimenez, J.: Scaling of velocity fluctuations in turbulent channels up to \(Re_\tau =2000\). Phys. Fluids 18, 011702 (2006)Google Scholar
  24. 24.
    Ostilla-Monico, R., Verzicco, R., Grossmann, S., Lohse, D.: The near-wall region of highly turbulent Taylor-Couette flow. JFM 788, 95–117 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Wu, X., Moin, P., Adrian, R.J., Baltzer, J.R.: Osborne Reynolds pipe flow: direct simulation from laminar through gradual transition to fully developed turbulence. In: Proceedings of the National Academy of Sciences of the United States of America, vol. 112 (26), pp. 7920–7924 (2015)ADSCrossRefGoogle Scholar
  26. 26.
    Barkley, D.: Theoretical perspective on the route to turbulence in a pipe. JFM 803, P1–1 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Sagaut, P., Cambron, C.: Homogeneous Turbulence Dynamics. Cambridge University Press, New York (2008)CrossRefGoogle Scholar
  28. 28.
    Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge, U.K. (2001)zbMATHGoogle Scholar
  29. 29.
    Poinsot, T., Veynante, D.: Theoretical and Numerical Combustion. R.T. Edwards, Philadelphia, PA (2001)Google Scholar
  30. 30.
    Oberlack, M., Busse, F.H.: Theories of Turbulence. Springer, New York (2002)zbMATHCrossRefGoogle Scholar
  31. 31.
    Fox, R.O.: Computational Models for Turbulent Reacting Flows. Cambridge University Press, U.K. (2003)CrossRefGoogle Scholar
  32. 32.
    Davidson, P.A.: Turbulence. Oxford University Press, Oxford, U.K. (2004)zbMATHGoogle Scholar
  33. 33.
    Garnier, E., Adams, N., Sagaut, P.: Large Eddy Simulation for Compressible Flows. Springer, New York (2009)zbMATHCrossRefGoogle Scholar
  34. 34.
    Leschziner, M.: Statistical Turbulence Modelling for Fluid Dynamics - Demystified. Imperial College Press, London (2015)zbMATHCrossRefGoogle Scholar
  35. 35.
    Jakirlic, S., Hanjalic, K., Tropea, C.: Modeling rotating and swirling turbulent flows: a perpetual challenge. AIAA J. 40, 1984–1996 (2002)ADSCrossRefGoogle Scholar
  36. 36.
    Novikov, E.A.: Functionals and the Random-force method in turbulence theory. Sov. Phys. JETP 20, 1290–1294 (1965)ADSMathSciNetGoogle Scholar
  37. 37.
    Lundgren, T.S.: Distribution functions in the statistical theory of turbulence. Phys. Fluids 10, 969–975 (1967)ADSCrossRefGoogle Scholar
  38. 38.
    Novikov, E.A.: Random-force method in turbulence theory. Sov. Phys. JETP 17, 1449–1454 (1963)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Kuo, H.-H.: Gaussian Measures in Banach Spaces. Lecture Notes in Mathematics, vol. 463. Springer, Berlin (1975)Google Scholar
  40. 40.
    Hosokawa, I.: A functional treatise on statistical hydromechanics with random force action. J. Phys. Soc. Jpn 25, 271–278 (1968)ADSCrossRefGoogle Scholar
  41. 41.
    Kraichnan, R.H.: Models for intermittency in hydrodynamic turbulence. Phys. Rev. Lett. 65, 575 (1990)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Pope, S.B.: Mapping closures for turbulent mixing and reaction. Theoret. Computat. Fluid Dyn. 2, 255–270 (1991)ADSzbMATHCrossRefGoogle Scholar
  43. 43.
    Pierce, C.D., Moin, P.: Progress-variable approach for large-eddy simulation of non-premixed turbulent combustion. JFM 504, 73–97 (2004)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Rosales, C., Meneveau, C.: A minimal multiscale Lagrangian map approach to synthesize non-Gaussian turbulent vector fields. Phys. Fluids 18, 075104 (2006)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Hopf, E.: Remarks on the functional-analytic approach to turbulence. Proc. Symp. Appl. Math. XIII, 157–163 (1962)Google Scholar
  46. 46.
    Batchelor, G.K.: The Theory of Homogeneous Turbulence. Cambridge University Press (1967)Google Scholar
  47. 47.
    Schumacher, J., Sreenivasan, K.R., Yakhot, V.: Asymptotic exponents from low-Reynolds-number flows. New J. Phys. 9, 1–19 (2007)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Wilczek, M., Daitche, A., Friedrich, R.: On the velocity distribution in homogenous isotropic turbulence: correlations and deviations from Gaussianity. JFM 676, 191–217 (2011)ADSzbMATHCrossRefGoogle Scholar
  49. 49.
  50. 50.
    Gelfand, I.M., Vilenkin, N.Y.: Generalized Functions, vol. 4. Academic Press, New York (1964)Google Scholar
  51. 51.
    Rogallo, R.S.: Numerical Experiments in Homogeneous Turbulence (1981). NASA-TM-81315Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of California DavisDavisUSA

Personalised recommendations