A numerically-stable method for enforcing numerical conservation in transported probability density function models: Application to MMC-IEM with one reference variable

A numerically-stable method for enforcing numerical conservation in transported probability density function models: Application to MMC-IEM
with one reference variable

Turbulent combustion closure models – for Transported Probability Density Function (TPDF) models – where more than one other stochastic particle influences the mixing process for any given particle may not be mathematically conservative. A technique is proposed here which makes the mixing process conservative within numerical precision by correcting the matrix governing the contribution each particle makes to each particle’s new value. In each iteration, the matrix is normalised to become conservative, then renormalised to become well-mixed so that it complies with numerical stability requirements. The technique inherently converges until the error in conservation is lower than the desired tolerance. This technique is tested on the Multiple Mapping Conditioning (MMC) model using the Interaction-by-Exchange-with-the-Mean (IEM) model for the turbulent micro-mixing closure (MMC-IEM) for the approach where the two closest particles in reference space are used to compute the conditional mean towards which a particle relaxes. For a single reference variable, these two particles are the immediate neighbour in either direction in reference space. While this method is implicitly conservative if all particles have identical weights, it is inherently unconservative otherwise. This is a challenge for applying this method to standard inhomogeneous combustion codes, where varying particle weights is used to manage computational load and accuracy by removing stochastic particles if there are too many in a region and creating stochastic particles if there are insufficient. The technique is tested using an inhomogeneous lifted-flame open flow, with the introduction of numerical conservation having an insignificant impact on the mean results, so the essence of MMC-IEM is preserved. The conservation error in the scalars is 3 orders of magnitude lower than the specified tolerance for the matrix. However, because the chemical kinetics scheme only used 48 species – of which, 28 were steady-state – it is expected that the application of this technique to practical cases with more species and at higher pressures will have a more significant impact.