\(\mathcal{M}_1(1)\): Single Scalar in Homogeneous Turbulence

  • Wolfgang KollmannEmail author


The equations resulting from the treatment of marginal statistics are necessarily indeterminate, hence require additional information to obtain a solvable system of equations, a non-rigorous operation called closure. This holds also for the mapping equations, which are considered in the present section for homogeneous turbulence. The single scalar case is a convenient approach to introduce and explain mapping methods in elementary form [1], the velocity and multi-scalar, non-homogeneous cases are much more complex and discussed in the following section.


  1. 1.
    Pope, S.B.: Mapping closures for turbulent mixing and reaction. Theoret. Computat. Fluid Dyn. 2, 255–270 (1991)ADSCrossRefGoogle Scholar
  2. 2.
    Chen, H., Chen, S., Kraichnan, R.H.: Probability distribution of a stochastically advected scalar field. Phys. Rev. Lett. 62, 2657 (1989)ADSCrossRefGoogle Scholar
  3. 3.
    Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge, U.K. (2001)zbMATHGoogle Scholar
  4. 4.
    Adler, R.J., Taylor, J.E.: Random Fields and Geometry. Springer, New York (2007)zbMATHGoogle Scholar
  5. 5.
    Yadrenko, M.I.: Spectral Theory of Random Fields. Optimization Software Inc., Publication Division, New York (1983)zbMATHGoogle Scholar
  6. 6.
    Wilczek, M., Meneveau, C.: Pressure Hessian and viscous contributions to velocity gradient statistics based on Gaussian random fields. JFM 756, 191–225 (2014)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Olver, F.W., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, U.K. (2010)zbMATHGoogle Scholar
  8. 8.
    Courant, R.: Differential & Integral Calculus. vol. II, Blackie & Sons Ltd, London (1962)Google Scholar
  9. 9.
    Duffy, D.G.: Greenś Functions with Applications. Chapman & Hall/CRC, Boca Raton, Florida (2001)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of California DavisDavisUSA

Personalised recommendations