Algorithmic Aspects of the LES-PBE-PDF Method for Modeling Soot Particle Size Distributions in Turbulent Flames

ABSTRACT

In recent times, the LES-PBE-PDF framework has been developed to couple large eddy simulation (LES) and population balance models (PBE) for the description of soot formation in turbulent hydrocarbon flames. This approach is based on a modeled evolution equation for the LES-filtered probability density function (pdf) associated with the instantaneous gas composition and soot particle size distribution. Here, the interaction of turbulence with chemical reactions and soot formation can be represented without approximations on part of the chemical and soot formation kinetics, while effects due to turbulent transport and molecular diffusion require closure. In view of an efficient numerical solution scheme, we previously proposed to combine a statistically equivalent reformulation of the joint scalar-number density pdf based on Eulerian stochastic fields with a time-explicit adaptive grid discretization in particle size space and a fractional time stepping scheme. In this article, we present algorithmic aspects and relay implementational details for a consistent semi-discrete formulation of the PBE fractional step as well as an effective dynamic load balancing scheme for both the chemical reaction and PBE fractional steps. Considering soot formation in the Delft III turbulent diffusion flame as a test case, we show that the persisting load imbalance is almost negligible on average and give evidence of linear strong scaling on a modern high performance computer for moderate numbers of compute nodes.

KEYWORDS:

• LES–PDF method
• population balance
• stochastic fields
• grid adaptivity
• dynamic load balancing

Acknowledgments

This work used the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk).

Notes

¹ By convention, we use uppercase letters (PDF) to refer to PDF approaches in general and italic lowercase letters (pdf) when discussing a particular probability density function.

² With a slight abuse of notation, we write, for brevity, $G(l,\mathit{\theta })$ and $\dot{s}(l,\mathit{\theta })$ instead of $G(l,{\theta }_1(t;\mathbf{x}),\dots ,{\theta }_{n_s}(t;\mathbf{x}))$ and $\dot{s}(l,{\theta }_1(t;\mathbf{x}),\dots ,{\theta }_{n_s}(t;\mathbf{x}),\hat{\rho }(\mathit{\theta }){\theta }_{N_{\rho }}(t;\cdot ,\mathbf{x}))$, respectively, and similarly for $\hat{\rho }(\mathit{\theta })$, $\dot{\omega }(\mathit{\theta })$ and $\mathbf{m}(\mathbf{x},t,\mathit{\theta })$.

³ For clarity, we employ the short overbar $\bar{\cdot }$ to distinguish between the coordinate transformation ($\bar{l}$ and $\bar{\tau }$) and particular values of physical or transformed size ($l$ and $\tau$). The long overbar $\bar{\cdot }$, on the other hand, indicates the LES-operator.

Additional information

Funding

This work was supported by the UK Consortium on Turbulent Reacting Flows [EP/K026801/1].