Consistency and convergence of Eulerian Monte Carlo field method for solving transported probability density function equation in turbulence modeling

Abstract

The transported probability density function (PDF) method is an attractive model for the closure of turbulent mixing and turbulent reactive flows. The transported PDF method suffers from the curse of dimensionality and an efficient numerical implementation of the method has always been a research topic of great importance. The Eulerian Monte Carlo field (EMCF) method (also termed as the stochastic field method) has been proposed as an efficient solution approach for solving the PDF transport equation for two decades. In this work, we revisit the EMCF method and examine its mathematical consistency analytically and numerically. It is found that the EMCF method is not mathematically consistent with the PDF transport equation that the method intends to solve. This creates a serious inconsistency issue and causes uncertainties in the yielded numerical solutions by EMCF. It is imperative to examine the effect of the inconsistency. We evaluate the effect of the inconsistency in a simplified turbulent mixing layer test case. Corrections to remedy the mathematical inconsistency are proposed and examined. The effectiveness of the corrections is demonstrated numerically through convergence testing. The effect of the Reynolds number on the inconsistency is explored. The impact of the inconsistency is also investigated in a thermal wake behind a line source in grid turbulence to assess the importance of the issue in real turbulence problems.

The Eulerian Stochastic Fields Method Applied to Large Eddy Simulations of a Piloted Flame with Inhomogeneous Inlet

“…In the present work, no correction terms have yet been considered due to various reasons. First, Wang et al (2018) clearly state that the first moments obtained with the ESF method are correct, which is eventually the quantity we are interested in. Second, Wang et al (2018) based their analysis on RANS where the second moments are more important and of higher magnitude as the cells are bigger.…”

“…First, Wang et al (2018) clearly state that the first moments obtained with the ESF method are correct, which is eventually the quantity we are interested in. Second, Wang et al (2018) based their analysis on RANS where the second moments are more important and of higher magnitude as the cells are bigger. Third, they (Wang et al 2018) state that the inconsistency does especially matter at small Reynolds numbers where turbulent diffusivity D t is considerably small, e.g.…”

“…Recently Wang et al (2018) reported on the mathematical inconsistency of the ESF method and provided correction terms for the RANS context for the case of a single scalar field. They (Wang et al 2018) argue that the second and any higher moments of scalars obtained from the ESF method are not consistent with the actual PDF equation ( 15). According to them, the original ESF formulation (Valiño 1998)…”

“…Second, Wang et al (2018) based their analysis on RANS where the second moments are more important and of higher magnitude as the cells are bigger. Third, they (Wang et al 2018) state that the inconsistency does especially matter at small Reynolds numbers where turbulent diffusivity D t is considerably small, e.g. Wang et al (2018) used test cases at Re = 20 .…”

Concentration Fluctuations from Localized Atmospheric Releases

“…Due to the high dimensionality, stochastic approaches are commonly the preferred methods to numerically solve the PDF transport equation. The approaches can be Eulerian stochastic field methods (Valiño 1998;Sabel'nikov and Soulard 2005;Garmory et al 2006;Cassiani et al 2010;Wang et al 2018) or more commonly the Lagrangian particle-mesh methods (Pope 2000;Fox 2003;Heinz 2003), which has been briefly outlined above for the special case of the calculation of the concentration PDF by extending atmospheric Lagrangian one-particle dispersion models.…”