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Original Articles

Comparisons of different implementations of turbulence modelling in lattice Boltzmann method

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Pages 67-80 | Received 01 Apr 2014, Accepted 09 Aug 2014, Published online: 10 Oct 2014
 

Abstract

In this paper, we present an alternative approach for the turbulence modelling in the single-relaxation-time lattice Boltzmann method (LBM) framework by treating the turbulence term as an extra forcing term, in addition to the traditional approach of modifying the relaxation time. We compare these two different approaches and their mixture in large-eddy simulation (LES) of three-dimensional decaying isotropic homogenous turbulence using the Smagorinsky model and the mixed similarity model. When the LES was conducted using the Smagorinsky model, where the Boussinesq eddy-viscosity approximation is adopted, the results showed that these three different implementations are equivalent. However, when the mixed similarity model is adopted, which is beyond the Boussinesq eddy-viscosity approximation, our results showed that an equivalent eddy-viscosity will lead to errors, while the forcing approach is more straightforward and accurate. This provides an alternative and more general framework of simulation of turbulence with models in LBM, especially when the Boussinesq eddy-viscosity approximation is invalid.

Acknowledgements

We appreciate Lianping Wang and Zuoli Xiao for many useful discussions. Numerical simulations were finished at Dawn clusters in Peking University.

Notes

1. As pointed out by one of the referees that the word ‘general’ here is somewhat misleading. We would like to emphasise here that the word ‘general’ focuses on the capability to treat different types, within or beyond the Boussinesq approximation, of turbulence models. When LBM is used to simulate turbulence incorporated with modelling approaches, different perspectives exist. If a turbulence model (SGS or RANS) was derived through the Boltzmann equation based on the kinetic theory, then the notion of effective relaxation time is way more general [Citation28,Citation43]. From this perspective, a well-defined characteristic relaxation time scale could describe self-consistent dynamics of turbulent fluctuations better, and usually the notion of eddy-viscosity was adopted, as reviewed in this paper, to estimate the effective relaxation time in applications. However, if we viewed LBM as a tool to solve the Navier–Stokes (NS) equations, then we might use the extended LBE (Equation (Equation9)) to solve flow problems incorporated with turbulence models (RANS or LES models) instead. From this perspective, LBM is just a special solver, as the finite-volume method, to NS equations, and we could adopt any type of turbulence models, within or beyond eddy-viscosity assumption, as claimed in this work. If we further took the notion of eddy-viscosity, we could simply use the effective relaxation time to account for the turbulence models. At this point, these two different perspectives reach the same formulas.

Additional information

Funding

We acknowledge the financial support provided by National Natural Science Foundation of China [grant number 91130001], [grant number 11221061], [grant number 11302006]. Author Z. Xia wants to thank the support from National Science Foundation for Postdoctoral Scientists of China [grant number 2012M520109].

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