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4 Eulerian (Field) Monte Carlo Methods for Solving PDF Transport Equations in Turbulent Reacting Flows

Part 3. Gaseous and Liquid Fuels

  1. Vladimir Sabel'nikov1,
  2. Olivier Soulard2

Published Online: 15 JUL 2010

DOI: 10.1002/9783527628148.hoc043

Handbook of Combustion

Handbook of Combustion

How to Cite

Sabel'nikov, V. and Soulard, O. 2010. Eulerian (Field) Monte Carlo Methods for Solving PDF Transport Equations in Turbulent Reacting Flows. Handbook of Combustion. 3:4:75–119.

Author Information

  1. 1

    ONERA/DEFA, Palaiseau, France

  2. 2

    CEA/DAM/DIF, Arpajon, France

Publication History

  1. Published Online: 15 JUL 2010


This chapter reviews some recent progress made in the field of Eulerian (field) Monte Carlo (EMC) methods, used for solving probability density function (PDF) transport equations in turbulent reacting flows. The framework of this review includes both Reynolds-averaged Navier–Stokes (RANS) and large eddy simulation (LES) approaches, and both composition and velocity–composition PDFs. Instead of stochastic particles, as in Lagrangian Monte Carlo (LMC) methods, stochastic Eulerian fields are used to represent the PDF in EMC methods. These fields evolve according to stochastic partial differential equations (SPDEs), stochastically equivalent to the PDF equation. These SPDEs are derived from Lagrangian stochastic ordinary differential equations (SODEs), in the same way as in classical hydrodynamics one transforms the Navier–Stokes equations written in Lagrangian variables into Eulerian coordinates. Boundary conditions for the stochastic fields are discussed. From a numerical point of view, recent publications give evidence that EMC methods converge more rapidly than LMC methods. Further, EMC methods offer an additional advantage over LMC methods: they allow typical Eulerian transport solvers to be employed, thus facilitating the coupling in hybrid EMC–RANS or EMC–LES algorithms. The novel hybrid EMC–RANS is developed and then applied to a premixed methane–air flame stabilized by a backward-facing step. Other examples of recent applications are reviewed, including the simulation of the auto-ignition of a hydrogen jet issuing into a co-flow of vitiated hot air with a scalar EMC–LES method and the simulation of an NO plume in the atmosphere with a scalar EMC–RANS method. Finally, potential future developments and areas of EMC methods are highlighted.


  • Eulerian (field) Monte Carlo;
  • probability density function;
  • turbulent reacting flows;
  • Reynolds-averaged Navier–Stokes approach;
  • large eddy simulation;
  • stochastics