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On a Proper Tensor-Diffusivity Model for Large-Eddy Simulation of Buoyancy-Driven Turbulence

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Abstract

In this work, we aim to shed light to the following research question: can we find a nonlinear tensorial subgrid-scale (SGS) heat flux model with good physical and numerical properties, such that we can obtain satisfactory predictions for buoyancy-driven turbulent flows? This is motivated by our findings showing that the classical (linear) eddy-diffusivity assumption, \(\varvec{q}^{eddy} \propto \nabla \overline{T}\), fails to provide a reasonable approximation for the actual SGS heat flux, \(\varvec{q}= \overline{\varvec{u}T} - \overline{\varvec{u}} \overline{T}\): namely, a priori analysis for air-filled Rayleigh-Bénard convection (RBC) clearly shows a strong misalignment. In the quest for more accurate models, we firstly study and confirm the suitability of the eddy-viscosity assumption for RBC carrying out a posteriori tests for different models at very low Prandtl numbers (liquid sodium, \(Pr=0.005\)) where no heat flux SGS activity is expected. Then, different (nonlinear) tensor-diffusivity SGS heat flux models are studied a priori using DNS data of air-filled (\(Pr=0.7\)) RBC at Rayleigh numbers up to \(10^{11}\). Apart from having good alignment trends with the actual SGS heat flux, we also restrict ourselves to models that are numerically stable per se and have the proper cubic near-wall behavior. This analysis leads to a new family of SGS heat flux models based on the symmetric positive semi-definite tensor \(\mathsf {G}\mathsf {G}^{T}\) where \(\mathsf {G}\equiv \nabla \overline{\varvec{u}}\), i.e. \(\varvec{q}\propto \mathsf {G}\mathsf {G}^{T}\nabla \overline{T}\), and the invariants of the \(\mathsf {G}\mathsf {G}^{T}\) tensor.

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Acknowledgements

F.X.T. and F.D. are supported by the Ministerio de Economía y Competitividad, Spain (ENE2017-88697-R). F.X.T. and C.O. are supported by the Generalitat de Catalunya RIS3CAT-FEDER, FusionCAT project (001-P-001722). F.X.T. was financially supported by a Ramón y Cajal postdoctoral contract (RYC-2012-11996). F.X.T. and A.G. are supported by the Research project 20-02-01 of the Department of the Moscow Center for Fundamental and Applied Mathematics at the Keldysh Institute of Applied Mathematics of RAS. F.D. is supported by the Austrian Federal Ministry for Digital and Economic Affairs, the National Foundation for Research, Technology and Development, and the K1MET center for metallurgical research in Austria (www.k1-met.com). Calculations have been performed on the MareNostrum 4 supercomputer at the BSC (PRACE 15th Call, Ref. 2016163972, “Exploring new frontiers in Rayleigh-Bénard convection”; and RES project FI-2019-1-0040 “Exploring nonlinear subgrid-scale heat flux models for buoyancy driven flows”). Preliminary simulations were carried out using computational resources of MCC NRC “Kurchatov Institute”, http://computing.nrcki.ru/. The authors thankfully acknowledge these institutions.

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Authors and Affiliations

  1. Heat and Mass Transfer Technological Center (CTTC), Technical University of Catalonia, C/Colom 11, 08222, Terrassa, Barcelona, Spain

    F. X. Trias, F. Dabbagh & C. Oliet

  2. Christian Doppler Laboratory for Multi-Scale Modeling of Multiphase Processes, Johannes Kepler University, Altenbergerstraße 69, 4040, Linz, Austria

    F. Dabbagh

  3. Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, 4A, Miusskaya Sq., Moscow, Russia, 125047

    A. Gorobets

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  1. F. X. Trias
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Trias, F.X., Dabbagh, F., Gorobets, A. et al. On a Proper Tensor-Diffusivity Model for Large-Eddy Simulation of Buoyancy-Driven Turbulence. Flow Turbulence Combust 105, 393–414 (2020). https://doi.org/10.1007/s10494-020-00123-3

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