### Abstract

Eulerian-Lagrangian simulations often require a two-way coupling, where the discrete phase affects the continuous phase and vice versa. The continuous phase in a CFD-simulation is typically represented on a computational mesh, which consists of a number of discrete cells. As the discrete phase is not directly associated with this mesh, a mapping from the discrete phase to the continuous phase is required in order to couple the two phases. This will distribute the exchange rates from the discrete to the continuous phase, where the exchanged properties can be thermal energy, momentum, species etc.

The mapping can be done in several ways, and different methods exist. Some of these include a smoothening step, which ensures that the exchange fields do not contain large spikes between any two neighbouring cells, which typically increases the convergence rate of the CFD simulation. Gaussian smoothening is often applied, where the discrete phase exchange rates are spread to the surrounding cells by a normal distribution with a specified standard deviation. This paper proposes a new weight for distributing the discrete phase exchange rates to the continuous phase cells. This is done by analytically integrating any distribution over the entirety of the particle trajectory, and only requires the probability- and cumulative density functions to be specified.

The proposed weight is used in combination with three different smoothening methods: node-based, kernel-based, and cell-based. Common existing weight distributions are also investigated, which are used in combination with the different methods. Each method and weight combination is compared with the analytical ground-truth exchange field, where the distributions are accurately integrated over the cell volumes.

The results show that the proposed weight has an average error of ≈ 2 % in the exchange field for the test case used. This is about three times more accurate compared to existing weights, where the exchange field is not integrated along the trajectory of the particle. Alternative distributions were investigated to decrease the computational requirements. It was found that a fourth-order polynomial distribution was about 20 times faster to evaluate, whilst the error induced by switching to this distribution was ≈ 4.5 % compared with the previous ≈ 2 %.

The mapping can be done in several ways, and different methods exist. Some of these include a smoothening step, which ensures that the exchange fields do not contain large spikes between any two neighbouring cells, which typically increases the convergence rate of the CFD simulation. Gaussian smoothening is often applied, where the discrete phase exchange rates are spread to the surrounding cells by a normal distribution with a specified standard deviation. This paper proposes a new weight for distributing the discrete phase exchange rates to the continuous phase cells. This is done by analytically integrating any distribution over the entirety of the particle trajectory, and only requires the probability- and cumulative density functions to be specified.

The proposed weight is used in combination with three different smoothening methods: node-based, kernel-based, and cell-based. Common existing weight distributions are also investigated, which are used in combination with the different methods. Each method and weight combination is compared with the analytical ground-truth exchange field, where the distributions are accurately integrated over the cell volumes.

The results show that the proposed weight has an average error of ≈ 2 % in the exchange field for the test case used. This is about three times more accurate compared to existing weights, where the exchange field is not integrated along the trajectory of the particle. Alternative distributions were investigated to decrease the computational requirements. It was found that a fourth-order polynomial distribution was about 20 times faster to evaluate, whilst the error induced by switching to this distribution was ≈ 4.5 % compared with the previous ≈ 2 %.

Original language | English |
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Journal | Computers & Fluids |

Volume | 186 |

Pages (from-to) | 15-23 |

Number of pages | 9 |

ISSN | 0045-7930 |

DOIs | |

Publication status | Published - 30 May 2019 |

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### Keywords

- Gaussian
- Mapping
- Smoothening
- Weighting

### Cite this

Simonsen, A. S., Deen, N. G., Sørensen, K., Condra, T., & Rosendahl, L. (2019). Trajectory integrated smoothening of exchange fields for discrete phase simulations.

*Computers & Fluids*,*186*, 15-23. https://doi.org/10.1016/j.compfluid.2019.03.015