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

Published online by Cambridge University Press:  18 March 2019

A. Cimarelli Open the ORCID record for A. Cimarelli [Opens in a new window] ,
A. Abbà  and
M. Germano Open the ORCID record for M. Germano [Opens in a new window]
A. Cimarelli*
Affiliation:
School of Engineering, Cardiff University, Cardiff CF24 3AA, UK
A. Abbà
Affiliation:
Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, 20156 Milano, Italy
M. Germano
Affiliation:
Department Civil and Environmental Engineering, Duke University, Durham, NC 27708, USA
*
†Email address for correspondence: CimarelliA@cardiff.ac.uk
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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.

JFM classification

Turbulent Flows: Turbulence modelling Turbulent Flows: Turbulence theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 866 , 10 May 2019 , pp. 865 - 896
Copyright
© 2019 Cambridge University Press 

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References

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, 209–222.10.1016/j.compfluid.2015.08.021Google Scholar
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), 589–610.10.1080/14685248.2017.1312001Google Scholar
Abbà, A., Cercignani, C. & Valdettaro, L. 2003 Analysis of subgrid scale models. Comput. Maths. Appl. 46, 521–535.10.1016/S0898-1221(03)90014-9Google Scholar
Bardina, J., Ferziger, J. & Reynolds, W.1980 Improved subgrid scale models for large eddy simulation. AIAA Paper 801357.Google Scholar
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.Google Scholar
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.Google Scholar
Borue, V. & Orszag, S. A. 1998 Local energy flux and subgrid-scale statistics in three-dimensional turbulence. J. Fluid Mech. 366, 1–31.10.1017/S0022112097008306Google Scholar
Carati, D. & Cabot, W. 1996 Anisotropic eddy viscosity models. Proceedings of Summer School Program, Center for Turbulence Research, pp. 249–259.Google Scholar
Cerutti, S. & Meneveau, C. 1998 Intermittency and relative scaling of subgrid scale energy dissipation in isotropic turbulence. Phys. Fluids 10, 928–937.10.1063/1.869615Google Scholar
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.96.084502Google Scholar
Cimarelli, A. & De Angelis, E. 2012 Anisotropic dynamics and sub-grid energy transfer in wall-turbulence. Phys. Fluids 24, 015102.10.1063/1.3675626Google Scholar
Cimarelli, A. & De Angelis, E. 2014 The physics of energy transfer toward improved subgrid-scale models. Phys. Fluids 26, 055103.10.1063/1.4871902Google Scholar
Cimarelli, A., De Angelis, E. & Casciola, C. M. 2013 Paths of energy in turbulent channel flows. J. Fluid Mech. 715, 436–451.10.1017/jfm.2012.528Google Scholar
Cimarelli, A., De Angelis, E., Jiménez, J. & Casciola, C. M. 2016 Cascades and wall-normal fluxes in turbulent channel flows. J. Fluid Mech. 796, 417–436.10.1017/jfm.2016.275Google Scholar
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, 407–423.10.1017/jfm.2015.182Google Scholar
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, 1–16.10.1017/S002211207900001XGoogle Scholar
Colosqui, C. & Oberai, A. 2008 Generalized Smagorinsky model in physical space. Comput. Fluids 37, 207–217.10.1016/j.compfluid.2007.09.002Google Scholar
Domaradzki, J. A., Teaca, B. & Carati, D. 2009 Locality properties of the energy flux in turbulence. Phys. Fluids 21 (2), 025106.10.1063/1.3081558Google Scholar
Domaradzki, J. A., Liu, W., Härtel, C. & Kleiser, L. 1994 Energy transfer in numerically simulated wall-bounded turbulent flows. Phys. Fluids 6, 1583–1599.10.1063/1.868272Google Scholar
Eyink, G. L. 2006 Multi-scale gradient expansion of the turbulent stress tensor. J. Fluid Mech. 549, 159–190.10.1017/S0022112005007895Google Scholar
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, 1667–1691.10.1016/j.cma.2005.05.045Google Scholar
Germano, M 1986 A proposal for a redefinition of the turbulent stresses in the filtered Navier–Stokes equations. Phys. Fluids 29 (7), 2323–2324.10.1063/1.865568Google Scholar
Germano, M. 1992 Turbulence: the filtering approach. J. Fluid Mech. 238, 325–336.10.1017/S0022112092001733Google Scholar
Germano, M. 2007 A direct relation between the filtered subgrid stress and the second order structure function. Phys. Fluids 19, 038102.10.1063/1.2714078Google Scholar
Germano, M. 2012 The simplest decomposition of a turbulent field. Physica D 241 (3), 284–287.Google Scholar
Härtel, C., Kleiser, L., Unger, F. & Friedrich, R. 1994 Subgrid-scale energy transfer in the near-wall region of turbulent flows. Phys. Fluids 6, 3130–3143.10.1063/1.868137Google Scholar
Horiuti, K. 1993 A proper velocity scale for modeling subgrid-scale eddy viscosities in large eddy simulation. Phys. Fluids A 5 (1), 146–157.10.1063/1.858800Google Scholar
John, V. & Kindl, A. 2010 Numerical studies of finite element variational multiscale methods for turbulent flow simulations. Comput. Meth. Appl. Mech. Engng 199, 841–852.10.1016/j.cma.2009.01.010Google Scholar
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), 197–208.10.1063/1.868827Google Scholar
Knight, D., Zhou, G., Okong’o, N. & Shukla, V.1998 Compressible large eddy simulation using unstructured grids. AIAA Paper 980535.Google Scholar
Leonard, A. 1974 Energy cascade in large-eddy simulations of turbulent fluid flows. Adv. Geophys. 18, 237–248.10.1016/S0065-2687(08)60464-1Google Scholar
Lu, H. & Porté-Agel, F. 2010 A modulated gradient model for large-eddy simulation: application to a neutral atmospheric boundary layer. Phys. Fluids 22, 015109.10.1063/1.3291073Google Scholar
Ni, R., Voth, G. A. & Ouellette, N. T. 2014 Extracting turbulent spectral transfer from under-resolved velocity fields. Phys. Fluids 26 (10), 105107.10.1063/1.4898866Google Scholar
Piomelli, U., Cabot, W. H., Moin, P. & Lee, S. 1991 Subgrid-scale backscatter in turbulent and transitional flows. Phys. Fluids A 3 (7), 1766–1771.10.1063/1.857956Google Scholar
Piomelli, U., Rohui, A. & Geurts, B. 2015 A grid-independent length scale for large-eddy simulations. J. Fluid Mech. 766, 499–527.10.1017/jfm.2015.29Google Scholar
Piomelli, U., Yu, Y. & Adrian, R. J. 1996 Subgrid-scale energy transfer and near-wall turbulence structure. Phys. Fluids 8, 215–224.10.1063/1.868829Google Scholar
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.90.104502Google Scholar
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.1.044401Google Scholar
Sagaut, P. 2001 Large-Eddy Simulation for Incompressible Flows: An Introduction. Springer.10.1007/978-3-662-04416-2Google Scholar
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), 115109.10.1063/1.5012546Google Scholar
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, 025114.10.1063/1.4941781Google Scholar
Vreman, B., Guerts, B. & Kuerten, H. 1996 Large eddy simulation of the temporal mixing layer using the Clark model. Theor. Comput. Fluid Dyn. 8, 309–324.10.1007/BF00639698Google Scholar
Vreman, B., Guerts, B. & Kuerten, H. 1997 Large-eddy simulation of the turbulent mixing layer. J. Fluid Mech. 339, 357–390.10.1017/S0022112097005429Google Scholar
Wang, J., Wan, M., Chen, S. & Chen, S. 2018 Kinetic energy transfer in compressible isotropic turbulence. J. Fluid Mech. 841, 581–613.10.1017/jfm.2018.23Google Scholar
Zhou, Y. 1993 Interacting scales and energy transfer in isotropic turbulence. Phys. Fluids A 5 (10), 2511–2524.10.1063/1.858764Google Scholar