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General formalism for a reduced description and modelling of momentum and energy transfer in turbulence

  • A. Cimarelli (a1), A. Abbà (a2) and M. Germano (a3)

Abstract

Based on hierarchies of filter lengths, the large eddy decomposition and the related subgrid stresses are recognized to represent generalized central moments for the study and modelling of the different modes composing turbulence. In particular, the subgrid stresses and the subgrid dissipation are shown to be alternative observables for quantitatively assessing the scale-dependent properties of momentum flux (subgrid stresses) and the energy exchange between the large and small scales (subgrid dissipation). In this work we present a theoretical framework for the study of the subgrid stress and dissipation. Starting from an alternative decomposition of the turbulent stresses, a new formalism for their approximation and understanding is proposed which is based on a tensorial turbulent viscosity. The derived formalism highlights that every decomposition of the turbulent stresses is naturally approximated by a general form of turbulent viscosity tensor based on velocity increments which is then recognized to be a peculiar property of small-scale stresses in turbulence. The analysis in a turbulent channel shows the rich physics of the small-scale stresses which is unveiled by the tensorial formalism and usually missed in scalar approaches. To further exploit the formalism, we also show how it can be used to derive new modelling approaches. The proposed models are based on the second- and third-order inertial properties of the grid element. The basic idea is that the structure of the integration volume for filtering (either implicit or explicit) impacts the anisotropy and inhomogeneity of the filtered-out motions and, hence, this information could be leveraged to improve the prediction of the main unknown features of small-scale turbulence. The formalism provides also a rigorous definition of characteristic lengths for the turbulent stresses, which can be computed in every type of computational elements, thus overcoming the rather elusive definition of filter length commonly employed in more classical models. A preliminary analysis in a turbulent channel shows reasonable results. In order to solve numerical stability issues, a tensorial dynamic procedure for the evolution of the model constants is also developed. The generality of the procedure is such that it can be employed also in more conventional closures.

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Corresponding author

Email address for correspondence: CimarelliA@cardiff.ac.uk

References

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Abbà, A., Bonaventura, L., Nini, M. & Restelli, M. 2015 Dynamic models for large eddy simulation of compressible flows with a high order DG method. Comput. Fluids 122, 209222.10.1016/j.compfluid.2015.08.021
Abbà, A., Campaniello, D. & Nini, M. 2017 Filter size definition in anisotropic subgrid models for large eddy simulation on irregular grids. J. Turbul. 18 (6), 589610.10.1080/14685248.2017.1312001
Abbà, A., Cercignani, C. & Valdettaro, L. 2003 Analysis of subgrid scale models. Comput. Maths. Appl. 46, 521535.10.1016/S0898-1221(03)90014-9
Bardina, J., Ferziger, J. & Reynolds, W.1980 Improved subgrid scale models for large eddy simulation. AIAA Paper 801357.
Bardina, J., Ferziger, J. & Reynolds, W.1983a Improved turbulence models based on large eddy simulation of homogeneous, incompressible, turbulent flows. Tech. Rep. NASA NCC 2-15.
Bardina, J., Ferziger, J. H. & Reynolds, W. C.1983b Improved turbulence models based on LES of homogeneous incompressible turbulent flows. Tech. Rep. TF-19. Thermosciences Division, Department of Mechanical Engineering, Stanford University.
Borue, V. & Orszag, S. A. 1998 Local energy flux and subgrid-scale statistics in three-dimensional turbulence. J. Fluid Mech. 366, 131.10.1017/S0022112097008306
Carati, D. & Cabot, W. 1996 Anisotropic eddy viscosity models. Proceedings of Summer School Program, Center for Turbulence Research, pp. 249259.
Cerutti, S. & Meneveau, C. 1998 Intermittency and relative scaling of subgrid scale energy dissipation in isotropic turbulence. Phys. Fluids 10, 928937.10.1063/1.869615
Chen, S., Ecke, R. E., Eyink, G. L., Rivera, M., Wan, M. & Xiao, Z. 2006 Physical mechanism of the two-dimensional inverse energy cascade. Phys. Rev. Lett. 96 (8), 084502.10.1103/PhysRevLett.96.084502
Cimarelli, A. & De Angelis, E. 2012 Anisotropic dynamics and sub-grid energy transfer in wall-turbulence. Phys. Fluids 24, 015102.10.1063/1.3675626
Cimarelli, A. & De Angelis, E. 2014 The physics of energy transfer toward improved subgrid-scale models. Phys. Fluids 26, 055103.10.1063/1.4871902
Cimarelli, A., De Angelis, E. & Casciola, C. M. 2013 Paths of energy in turbulent channel flows. J. Fluid Mech. 715, 436451.10.1017/jfm.2012.528
Cimarelli, A., De Angelis, E., Jiménez, J. & Casciola, C. M. 2016 Cascades and wall-normal fluxes in turbulent channel flows. J. Fluid Mech. 796, 417436.10.1017/jfm.2016.275
Cimarelli, A., De Angelis, E., Schlatter, P., Brethouwer, G., Talamelli, A. & Casciola, C. M. 2015 Sources and fluxes of scale energy in the overlap layer of wall turbulence. J. Fluid Mech. 771, 407423.10.1017/jfm.2015.182
Clark, R. A., Ferziger, J. H. & Reynolds, W. C. 1979 Evaluation of subgrid-scale models using an accurately simulated turbulent flow. J. Fluid Mech. 91, 116.10.1017/S002211207900001X
Colosqui, C. & Oberai, A. 2008 Generalized Smagorinsky model in physical space. Comput. Fluids 37, 207217.10.1016/j.compfluid.2007.09.002
Domaradzki, J. A., Teaca, B. & Carati, D. 2009 Locality properties of the energy flux in turbulence. Phys. Fluids 21 (2), 025106.10.1063/1.3081558
Domaradzki, J. A., Liu, W., Härtel, C. & Kleiser, L. 1994 Energy transfer in numerically simulated wall-bounded turbulent flows. Phys. Fluids 6, 15831599.10.1063/1.868272
Eyink, G. L. 2006 Multi-scale gradient expansion of the turbulent stress tensor. J. Fluid Mech. 549, 159190.10.1017/S0022112005007895
Farhat, C., Rajasekharan, A. & Koobus, B. 2006 A dynamic variational multiscale method for large eddy simulations on unstructured meshes. Comput. Meth. Appl. Mech. Engng 195, 16671691.10.1016/j.cma.2005.05.045
Germano, M 1986 A proposal for a redefinition of the turbulent stresses in the filtered Navier–Stokes equations. Phys. Fluids 29 (7), 23232324.10.1063/1.865568
Germano, M. 1992 Turbulence: the filtering approach. J. Fluid Mech. 238, 325336.10.1017/S0022112092001733
Germano, M. 2007 A direct relation between the filtered subgrid stress and the second order structure function. Phys. Fluids 19, 038102.10.1063/1.2714078
Germano, M. 2012 The simplest decomposition of a turbulent field. Physica D 241 (3), 284287.
Härtel, C., Kleiser, L., Unger, F. & Friedrich, R. 1994 Subgrid-scale energy transfer in the near-wall region of turbulent flows. Phys. Fluids 6, 31303143.10.1063/1.868137
Horiuti, K. 1993 A proper velocity scale for modeling subgrid-scale eddy viscosities in large eddy simulation. Phys. Fluids A 5 (1), 146157.10.1063/1.858800
John, V. & Kindl, A. 2010 Numerical studies of finite element variational multiscale methods for turbulent flow simulations. Comput. Meth. Appl. Mech. Engng 199, 841852.10.1016/j.cma.2009.01.010
Kerr, R. M., Domaradzki, J. A. & Barbier, G. 1996 Small-scale properties of nonlinear interactions and subgrid-scale energy transfer in isotropic turbulence. Phys. Fluids 8 (1), 197208.10.1063/1.868827
Knight, D., Zhou, G., Okong’o, N. & Shukla, V.1998 Compressible large eddy simulation using unstructured grids. AIAA Paper 980535.
Leonard, A. 1974 Energy cascade in large-eddy simulations of turbulent fluid flows. Adv. Geophys. 18, 237248.10.1016/S0065-2687(08)60464-1
Lu, H. & Porté-Agel, F. 2010 A modulated gradient model for large-eddy simulation: application to a neutral atmospheric boundary layer. Phys. Fluids 22, 015109.10.1063/1.3291073
Ni, R., Voth, G. A. & Ouellette, N. T. 2014 Extracting turbulent spectral transfer from under-resolved velocity fields. Phys. Fluids 26 (10), 105107.10.1063/1.4898866
Piomelli, U., Cabot, W. H., Moin, P. & Lee, S. 1991 Subgrid-scale backscatter in turbulent and transitional flows. Phys. Fluids A 3 (7), 17661771.10.1063/1.857956
Piomelli, U., Rohui, A. & Geurts, B. 2015 A grid-independent length scale for large-eddy simulations. J. Fluid Mech. 766, 499527.10.1017/jfm.2015.29
Piomelli, U., Yu, Y. & Adrian, R. J. 1996 Subgrid-scale energy transfer and near-wall turbulence structure. Phys. Fluids 8, 215224.10.1063/1.868829
Rivera, M. K., Daniel, W. B., Chen, S. Y. & Ecke, R. E. 2003 Energy and enstrophy transfer in decaying two-dimensional turbulence. Phys. Rev. Lett. 90 (10), 104502.10.1103/PhysRevLett.90.104502
Rouhi, A., Piomelli, U. & Geurts, B. 2016 Dynamic subfilter-scale stress model for large-eddy simulations. Phys. Rev. Fluids 1 (4), 044401.10.1103/PhysRevFluids.1.044401
Sagaut, P. 2001 Large-Eddy Simulation for Incompressible Flows: An Introduction. Springer.10.1007/978-3-662-04416-2
Trias, F. X., Gorobets, A., Silvis, M. H., Verstappen, R. W. C. P. & Oliva, A. 2017 A new subgrid characteristic length for turbulence simulations on anisotropic grids. Phys. Fluids 29 (11), 115109.10.1063/1.5012546
Vollant, A., Balarac, G. & Corre, C. 2016 A dynamic regularized gradient model of the subgrid-scale stress tensor for large-eddy simulation. Phys. Fluids 28, 025114.10.1063/1.4941781
Vreman, B., Guerts, B. & Kuerten, H. 1996 Large eddy simulation of the temporal mixing layer using the Clark model. Theor. Comput. Fluid Dyn. 8, 309324.10.1007/BF00639698
Vreman, B., Guerts, B. & Kuerten, H. 1997 Large-eddy simulation of the turbulent mixing layer. J. Fluid Mech. 339, 357390.10.1017/S0022112097005429
Wang, J., Wan, M., Chen, S. & Chen, S. 2018 Kinetic energy transfer in compressible isotropic turbulence. J. Fluid Mech. 841, 581613.10.1017/jfm.2018.23
Zhou, Y. 1993 Interacting scales and energy transfer in isotropic turbulence. Phys. Fluids A 5 (10), 25112524.10.1063/1.858764
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