# RESULTS

## Task 1: Improvement of the LES approach in scalar simulations

- A dynamic regularized gradient model of the subgrid-scale scalar flux for large eddy simulations, G. Balarac, J. Le Sommer, X. Meunier and A. Vollant, *Phys. Fluids*, in press, 2013

Abstract: Accurate predictions of scalar fields advected by a turbulent flow is needed for various industrial and geophysical applications. In the framework of large-eddy simulation (LES), a subgrid-scale (SGS) model for the subgrid-scale scalar flux has to be used. The gradient model, which is derived from a Taylor series expansions of the filtering operation is a well-known approach to model SGS scalar fluxes. This model is known to lead to high correlation level with the SGS scalar flux. However, this type of model can not be used in practical LES because it does not lead to enough global scalar variance transfer from the large to the small scales. In this work, a regularization of the gradient model is proposed based on a physical interpretation of this model. The impact of the resolved velocity field on the resolved scalar gradient is decomposed into compressional, stretching and rotational effects. It is shown that rotational effect is not associated with transfers of variance across scales. Conversely, the compressional effect is shown to lead to forward transfer, whereas the stretching effect leads to back-scatter of scalar variance. The proposed regularization is to neglect the stretching effect in the model formulation. The accuracy of this regularized gradient model is tested in comparison with direct numerical simulations (DNS) and compared with other classic SGS models. The accuracy of the regularized gradient model is evaluated in term of structural and functional performances, i.e. the model ability to locally approximate the SGS unknown term and to reproduce its global effect on tracer variance, respectively. It is found that the regularized gradient model associated with a dynamic procedure exhibits good performances in comparison with the standard dynamic eddy diffusivity model and the standard gradient model. In particular, the dynamic regularized gradient model provides a better prediction of scalar variance transfers than the standard gradient model. The dynamic regularized gradient model is then evaluated in a series of large-eddy simulations. This shows a substantial improvement for various scalar statistics predictions.

- Development of a new dynamic procedure for the Clark model of the subgrid-scale scalar flux using the concept of optimal estimator, Y. Fabre and G. Balarac, *Phys. Fluids*, 23, 115103, 2011

Abstract: Accurate prediction of a scalar advected by a turbulent flow is needed for various applications. In the framework of large-eddy simulation (LES), an accurate subgrid-scale (SGS) model for the subgrid-scale scalar flux has to be used. In this work, the performance of various dynamic SGS models is first evaluated by a priori tests through the concept of optimal estimator. Direct numerical simulation (DNS) in homogeneous isotropic turbulence is performed on 512^{3} grid points. Filtered quantities are extracted from the DNS data using a box or a spectral cut-off filter. The models’ accuracy is then evaluated in term of structural and functional performances, i.e., the model capacity to locally approximate the SGS unknown term and to reproduce its energetic action, respectively. It is shown that the Clark model has the best set of parameters to describe the SGS scalar flux. However, the classic dynamic procedure usually applied to compute the model coefficient leads to a large error. A new dynamic procedure is thus proposed to reduce this error. The results show that the new dynamic model leads to a good accuracy, which is not expectable from a model based only on the parameters of the classic dynamic Smagorinsky model. To better evaluate the improvement of the new dynamic procedure, a posteriori (large-eddy simulation) tests are performed for three different Schmidt numbers. It is shown that the new model allows to improve substantially the prediction of various scalar statistics.

- Subgrid-scale modeling of velocity and passive scalar for the Large-Eddy Simulation of non homogeneous turbulent flows, Y. Favre, H. Touil, G. Balarac and E. Lévêque, *13 European Turbulence Conference*, 2011

Abstract: Subgrid-scale modeling that relies on the separation between the mean and the fluctuating part of a turbulent field is introduced in the context of large-eddy simulations of non-homogeneous turbulent flows. Subgrid-scale models are proposed explicitly for both the velocity and passive scalar fields. Performances are examined, and comparisons are made with the classic and dynamic Smagorinsky models for the simulation of a temporal turbulent jet.

- Passive scalar LES using an optimal estimator as SGS model, A. Vollant, G. Balarac and C. Corre, *9th European Fluid Mechanics Conference*, 2012

- A dynamic regularized gradient model of the subgrid-scale scalar flux, G. Balarac, J. Le Sommer and A. Vollant, *Proceeding of the CTR summer program (Stanford Univ.)*, 2012

Abstract: Accurate prediction of a scalar advected by a turbulent flow is needed for various applications. In the framework of large-eddy simulation (LES), a subgrid-scale (SGS) model for the subgrid-scale scalar flux has to be used. A gradient model derived from Taylor series expansions of the filtering operation is a well-known approach to model SGS quantities. This model is known to lead to high correlation levels between the unknown term and the model. However, this type of model can not be used in practical LES because it does not lead to sufficient global energy transfer from the large to the small scales. In this work, we propose a regularization of the gradient model based on a physical interpretation of this model. We compare the regularized gradient model with classic models through a priori tests. It is found that the new proposed model associated with a dynamic procedure exhibits very good performances in comparison with the standard dynamic eddy diffusivity gradient models. To better evaluate the new dynamic procedure, we perform a posteriori (large-eddy simulation) tests, showing a substantial improvement for various scalar statistics predictions.

## Task 2: Particle methods and their coupling with spectral and finite volume flow solvers

- DNS of turbulent transport by a particle/spectral method, J.B. Lagaert, G. Balarac and G.-H. Cottet, *9th European Fluid Mechanics Conference*, 2012

- Particle method: an efficient tool for direct numerical simulations of a high Schmidt number passive scalar in turbulent flow, J.B. Lagaert, G. Balarac, G.-H. Cottet and P. Bégou, *Proceeding of the CTR summer program (Stanford Univ.)*, 2012

Abstract: In this work, an efficient way to predict the dynamics of a scalar at high Schmidt numbers, advected by a turbulent flow, is presented. For high Schmidt numbers, the spatial resolution required for the scalar field has to be finer than the resolution required for the velocity field, leading to a significant computational cost due to the Courant‚ Friedrichs‚ Lewy (CFL) constraint. We propose here a remeshed particle method coupled to a spectral flow solver to overcome this computational cost limitation. This allows us to perform a systematic analysis of flows over a wide range of Reynolds and Schmidt numbers. For high enough Reynolds and Schmidt numbers, the results presented here recover the spectral behavior predicted by theory. First, the classic law (where is the wave number) is found for the inertial-convective range. At intermediate scales, the viscous-convective range exhibits a law for Schmidt numbers higher than unity. Finally, the numerical results indicate that the dissipation range agree well with the Kraichnan model for high Schmidt numbers.