A new Eulerian Monte Carlo method for the joint velocity-scalar PDF equations in turbulent flows

A new Eulerian Monte Carlo method for the joint velocity-scalar PDF equations in turbulent flows

Soulard, O.
Sabel'nikov, V.

2006-09-08

In the field of turbulent combustion, Lagrangian Monte Carlo (LMC) methods (Pope, 85) have become an essential component of the probability density function (PDF) approach. LMC methods are based on stochastic particles, which evolve from prescribed stochastic ordinary differential equations (SODEs). They are used to compute the one-point statistics of the quantities describing the state of a turbulent reactive flow: namely, the velocity field and the reactive scalars (species mass fractions and temperature). Numerous publications document the convergence and accuracy of LMC methods. They have been used in many complex calculations (including LES), and for several years now, they have been implemented in commercial CFD codes. Nonetheless, the development of a new Eulerian Monte Carlo (EMC) method is useful and stimulating, since the competition between LMC and EMC methods could push both approaches forward. EMC methods have already been proposed by Sabel'nikov and Soulard (2006) in order to compute the one-point PDF of turbulent reactive scalars. EMC methods are based on stochastic Eulerian fields, which evolve from prescribed stochastic partial differential equations (SPDE) statistically equivalent to the PDF equation. The extension of EMC methods to include velocity still remains to be done. Thus, the purpose of this article is to derive SPDEs allowing to compute a modeled one-point joint velocity-scalar PDF. To achieve this objective, we start from existing Lagrangian stochastic models. The latter are described by SODEs, which can be considered as modeled Navier-Stokes equations written in Lagrangian variables. Then, the idea is to transform these Lagrangian SODEs into Eulerian SPDEs, in the same way one transforms the Lagrangian Navier-Stokes equations into Eulerian equations, in classical hydrodynamics. However, our case differs from the classical one. Indeed, the stochastic velocity does not respect an instantaneous continuity constraint, but only a mean one. To account for this difference between the stochastic and the physical system, one must introduce a stochastic density, different from the physical density. As a result of this procedure, we eventually obtain hyperbolic conservative SPDEs giving the evolution of a stochastic velocity, of stochastic scalars, and of a stochastic density. In addition to the main result, an alternative EMC method for the scalar PDF is also derived as a special case of the full velocity-scalar method.

turbulence
combustion
probability density functions
stochastic partial differential equations
Eulerian Monte Carlo Method

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Conference proceedings

conference paper

(c) 2006 Soulard, O.; Sabel'nikov, V.