Physical/numerical duality of explicit/implicit subgrid-scale modelling

Abstract

The development of implicit approaches has prompted debate on the actual usefulness of any explicit subgrid-scale modelling in large-eddy simulation. This question is addressed here by considering two generic turbulent flows: (i) the Taylor-Green vortex problem; (ii) the pipe flow. For both flow configurations, implicit modelling is found to overtake the very popular Smagorinsky model. To understand this robust observation, an analysis in the Fourier space is presented for the Taylor-Green vortex problem. The concept of spectral eddy viscosity, widely used in the pioneer work of Marcel Lesieur in two-point closure and subgrid-scale modelling, is revisited in a general framework based on explicit/implicit subgrid-scale modelling. In particular, the essentially anisotropic nature of implicit modelling is exhibited, as a favourable feature in terms of consistency with the computational mesh. Smagorinsky's model, considered as a generic explicit subgrid-scale model in the framework of Boussinesq's hypothesis, is found to be highly sensitive to numerical errors. Removing the latter is easy but makes computationally inefficient this type of explicit modelling. Comparisons between a priori and a posteriori spectral eddy viscosities show that neither Smagorinsky's model nor implicit modelling can mimic the expected spectral behaviour. Smagorinsky's model is observed to be weakly scale-selective with a poor ability to actually filter the solution. The feature of scale-selectivity is well replicated by implicit modelling which exhibits excellent capabilities for filtering. However, its lack of influence at the largest scales is against the expected behaviour for the spectral eddy viscosity at low wavenumber through the establishment of a non-zero plateau value. This lack of consistency of implicit LES could be overcome thanks to an extra explicit modelling but the attempt to mix Smagorinsky's model and implicit LES is not successful in this study. The potential of implicit large-eddy simulation is also exhibited for the accurate computation of near-wall turbulence inside a pipe flow despite the use of a regular Cartesian mesh with an immersed boundary method. Interestingly, the resulting coarse wall-normal resolution in the near-wall direction does not prevent the reliable prediction of statistical profiles up to the capture of subgrid-scale details. It is suggested that the regularisation associated with implicit modelling is a necessary condition to reach numerical accuracy. However, to faithfully represent the large-scale dynamics, present results confirm that non-local triad interactions must be taken into account as widely discussed in the inspiring textbook Lesieur [Turbulence in fluids. 4th ed. Springer; 2008] of Marcel Lesieur.

Keywords:

• Large eddy simulation
• turbulence modelling: subgrid-scale
• implicit LES
• spectral eddy viscosity

Acknowledgments

This study is dedicated to Marcel Lesieur (1945–2022) in memory of his passion for turbulence and his unique talent for sharing it with students and young researchers with generosity and encouragement.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The pressure contribution is not expressed in the present incompressible framework.

2 This average procedure is meaningful only for conditions not too far from isotropic turbulence, at least for the range of scales considered.

3 The mesh refinement in one direction, enabled by Incompact3d, is not used here.

4 In this section, computational mesh cells are assumed to be cubic leading to the same cutoff wavenumber in the three directions.

5 This strong assumption will be assessed in the next section.

6 Present immersed boundary method is not kinetic-energy conserving, we only refer here to the basic structure of the numerical differentiation.

Additional information

Funding

This work was granted access to the HPC resources of TGCC/CINES under the allocations A0052A07624, A0072A07624, A0092A07624, and A0112A07624 made by Grand Equipement National de Calcul Intensif.