\(\mathcal{M}_1(N)\): Mappings for Velocity–Scalar and Position–Scalar Pdfs

  • Wolfgang KollmannEmail author


The extension of mapping methods to velocity–scalar (spatial description) and position–scalar (material description) Pdfs requires some preparations. First, a useful property of the Pdf equation is derived and its consequences for mapping methods are considered.


  1. 1.
    Courant, R.: Differential & Integral Calculus. vol. II, Blackie & Sons Ltd, London (1962)Google Scholar
  2. 2.
    Pope, S.B.: Mapping closures for turbulent mixing and reaction. Theoret. Computat. Fluid Dyn. 2, 255–270 (1991)ADSCrossRefGoogle Scholar
  3. 3.
    Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge, U.K. (2001)zbMATHGoogle Scholar
  4. 4.
    Poinsot, T., Veynante, D.: Theoretical and Numerical Combustion. R.T. Edwards, Philadelphia, PA (2001)Google Scholar
  5. 5.
    Kollmann, W.: Fluid Mechanics in Spatial and Material Description. University Readers, San Diego (2011)Google Scholar
  6. 6.
    Wu, J.-Z., Ma, H.-Y., Zhou, M.-D.: Vorticity and Vortex Dynamics. Springer, Berlin (2006)CrossRefGoogle Scholar
  7. 7.
    Duffy, D.G.: Greenś Functions with Applications. Chapman & Hall/CRC, Boca Raton, Florida (2001)CrossRefGoogle Scholar
  8. 8.
    Chen, H., Chen, S., Kraichnan, R.H.: Probability distribution of a stochastically advected scalar field. Phys. Rev. Lett. 62, 2657 (1989)ADSCrossRefGoogle Scholar
  9. 9.
    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
  10. 10.
    Constantin, P.: Lagrangian-Eulerian methods for uniqueness in hydrodynamic systems. SIAM J. Math. Anal. 49, 1041–1047 (2015)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of California DavisDavisUSA

Personalised recommendations