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A FIC-based stabilized finite element formulation for turbulent flows. (English) Zbl 1439.65111

Summary: We present a new stabilized finite element (FEM) formulation for incompressible flows based on the Finite Increment Calculus (FIC) framework [the third author, ibid. 151, No. 1–2, 233–265 (1998; Zbl 0916.76060)]. In comparison to existing FIC approaches for fluids, this formulation involves a new term in the momentum equation, which introduces non-isotropic dissipation in the direction of velocity gradients. We also follow a new approach to the derivation of the stabilized mass equation, inspired by recent developments for quasi-incompressible flows [the third author, A. Franci and J. M. Carbonell, “Lagrangian formulation for finite element analysis of quasi-incompressible fluids with reduced mass losses”, Internat. J. Numer. Methods Fluids 74, No. 10, 699–731 (2014; doi:10.1002/fld.3870)]. The presented FIC-FEM formulation is used to simulate turbulent flows, using the dissipation introduced by the method to account for turbulent dissipation in the style of implicit large eddy simulation.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76Fxx Turbulence
76M10 Finite element methods applied to problems in fluid mechanics

Citations:

Zbl 0916.76060
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References:

[1] Pope, S. B., Turbulent Flows (2000), Cambridge University Press · Zbl 0966.76002
[2] Tejada-Martínez, A. E.; Jansen, K. E., On the interaction between dynamic model dissipation and numerical dissipation due to streamline upwind/Petrov-Galerkin stabilization, Comput. Methods Appl. Mech. Engrg., 194, 9-11, 1225-1248 (2005) · Zbl 1091.76027
[3] Trofimova, A. V.; Tejada-Martínez, A. E.; Jansen, K. E.; Lahey, Jr., R. T., Direct numerical simulation of turbulent channel flows using a stabilized finite element method, Comput. & Fluids, 38, 4, 924-938 (2009) · Zbl 1242.76141
[4] Gravemeier, V.; Gee, M. W.; Kronbichler, M.; Wall, W. A., An algebraic variational multiscale-multigrid method for large eddy simulation of turbulent flow, Comput. Methods Appl. Mech. Engrg., 199, 13-16, 853-864 (2010), Turbulence modeling for large eddy simulations · Zbl 1406.76027
[5] Hughes, T. J.R., Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg., 127, 1-4, 387-401 (1995) · Zbl 0866.76044
[6] Hughes, T. J.R.; Feijóo, G. R.; Mazzei, L.; Quincy, J.-B., The variational multiscale method-a paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg., 166, 1-2, 3-24 (1998), Advances in stabilized methods in computational mechanics · Zbl 1017.65525
[7] Hughes, T. J.R.; Mazzei, L.; Jansen, K. E., Large eddy simulation and the variational multiscale method, Comput. Vis. Sci., 3, 47-59 (2000) · Zbl 0998.76040
[8] Codina, R.; Príncipe, J.; Guasch, O.; Badia, S., Time dependent subscales in the stabilized finite element approximation of incompressible flow problems, Comput. Methods Appl. Mech. Engrg., 196, 21-24, 2413-2430 (2007) · Zbl 1173.76335
[9] Guasch, O.; Codina, R., Statistical behavior of the orthogonal subgrid scale stabilization terms in the finite element large eddy simulation of turbulent flows, Comput. Methods Appl. Mech. Engrg., 261-262, 0, 154-166 (2013) · Zbl 1286.76066
[10] Boris, J. P.; Grinstein, F. F.; Oran, E. S.; Kolbe, R. L., New insights into large eddy simulation, Fluid Dyn. Res., 10, 4-6, 199 (1992)
[11] Akkerman, I.; Bazilevs, Y.; Calo, V.; Hughes, T.; Hulshoff, S., The role of continuity in residual-based variational multiscale modeling of turbulence, Comput. Mech., 41, 371-378 (2008) · Zbl 1162.76355
[12] Bazilevs, Y.; Calo, V.; Cottrell, J.; Hughes, T.; Reali, A.; Scovazzi, G., Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows, Comput. Methods Appl. Mech. Engrg., 197, 1-4, 173-201 (2007) · Zbl 1169.76352
[13] Avila, M.; Codina, R.; Principe, J., Large eddy simulation of low Mach number flows using dynamic and orthogonal subgrid scales, Comput. & Fluids, 99, 44-66 (2014) · Zbl 1391.76196
[14] Colomés, O.; Badia, S.; Codina, R.; Príncipe, J., Assessment of variational multiscale models for the large eddy simulation of turbulent incompressible flows, Comput. Methods Appl. Mech. Engrg., 285, 0, 32-63 (2015) · Zbl 1423.76152
[15] Oñate, E., Derivation of stabilized equations for numerical solution of advective-diffusive transport and fluid flow problems, Comput. Methods Appl. Mech. Engrg., 151, 12, 233-265 (1998), Containing papers presented at the Symposium on Advances in Computational Mechanics · Zbl 0916.76060
[16] Oñate, E., A stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation, Comput. Methods Appl. Mech. Engrg., 182, 3-4, 355-370 (2000) · Zbl 0977.76050
[17] Oñate, E., Possibilities of finite calculus in computational mechanics, Internat. J. Numer. Methods Engrg., 60, 1, 255-281 (2004) · Zbl 1060.76576
[18] Oñate, E.; Valls, A.; García, J., Computation of turbulent flows using a finite calculus-finite element formulation, Internat. J. Numer. Methods Fluids, 54, 6-8, 609-637 (2007) · Zbl 1128.76038
[19] Oñate, E.; Valls, A.; García, J., Modeling incompressible flows at low and high Reynolds numbers via a finite calculus-finite element approach, J. Comput. Phys., 224, 1, 332-351 (2007), Special Issue Dedicated to Professor Piet Wesseling on the occasion of his retirement from Delft University of Technology http://dx.doi.org/10.1016/j.jcp.2007.02.026 · Zbl 1116.76056
[20] Oñate, E.; Nadukandi, P.; Idelsohn, S. R.; García, J.; Felippa, C., A family of residual-based stabilized finite element methods for Stokes flows, Internat. J. Numer. Methods Fluids, 65, 1-3, 106-134 (2011) · Zbl 1427.76056
[21] Oñate, E.; Idelsohn, S. R.; Felippa, C. A., Consistent pressure Laplacian stabilization for incompressible continua via higher-order finite calculus, Internat. J. Numer. Methods Engrg., 87, 1-5, 171-195 (2011) · Zbl 1242.76135
[22] Oñate, E.; Franci, A.; Carbonell, J. M., Lagrangian formulation for finite element analysis of quasi-incompressible fluids with reduced mass losses, Internat. J. Numer. Methods Fluids, 74, 10, 699-731 (2014) · Zbl 1455.76091
[23] Codina, R.; Principe, J.; Ávila, M., Finite element approximation of turbulent thermally coupled incompressible flows with numerical sub-grid scale modeling, Internat. J. Numer. Methods Heat Fluid Flow, 20, 492-516 (2010) · Zbl 1231.76140
[24] Temam, R., Navier-Stokes Equations: Theory and Numerical Analysis (2001), American Mathematical Soc. · Zbl 0981.35001
[25] Hughes, T. J.R.; Mallet, M., A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg., 58, 3, 305-328 (1986) · Zbl 0622.76075
[26] Codina, R., A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation, Comput. Methods Appl. Mech. Engrg., 110, 3-4, 325-342 (1993) · Zbl 0844.76048
[27] Hughes, T. J.R.; Franca, L. P.; Hulbert, G. M., A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations, Comput. Methods Appl. Mech. Engrg., 73, 2, 173-189 (1989) · Zbl 0697.76100
[28] Hughes, T. J.R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (2000), Dover Publications, Inc. · Zbl 1191.74002
[29] Dadvand, P.; Rossi, R.; Gil, M.; Martorell, X.; Cotela-Dalmau, J.; Juanpere, E.; Idelsohn, S. R.; Oñate, E., Migration of a generic multi-physics framework to HPC environments, Comput. & Fluids, 80, 301-309 (2013) · Zbl 1426.76644
[30] Tennekes, H.; Lumley, J. L., A First Course in Turbulence (1972), MIT Press · Zbl 0285.76018
[31] Moser, R. D.; Kim, J.; Mansour, N. N., Direct numerical simulation of turbulent channel flow up to \(\text{Re}_\tau = 590\), Phys. Fluids, 11, 4, 943-945 (1999) · Zbl 1147.76463
[32] Piomelli, U.; Balaras, E., Wall-layer models for large-eddy simulations, Annu. Rev. Fluid Mech., 34, 349-374 (2002) · Zbl 1006.76041
[33] Bou-Zeid, E.; Meneveau, C.; Parlange, M., A scale-dependent Lagrangian dynamic model for large eddy simulation of complex turbulent flows, Phys. Fluids (1994-present), 17, 2, Article 025105 pp. (2005) · Zbl 1187.76065
[34] Ham, F. E.; Lien, F.; Strong, A. B., A fully conservative second-order finite difference scheme for incompressible flow on nonuniform grids, J. Comput. Phys., 177, 1, 117-133 (2002) · Zbl 1066.76044
[35] Norberg, C., Fluctuating lift on a circular cylinder: review and new measurements, J. Fluids Struct., 17, 1, 57-96 (2003)
[36] Lienhard, J. H., Synopsis of Lift, Drag, and Vortex Frequency Data For Rigid Circular Cylinders, Tech. Rep. (1966), Washington State University
[37] Ong, L.; Wallace, J., The velocity field of the turbulent very near wake of a circular cylinder, Exp. Fluids, 20, 441-453 (1996)
[38] Kravchenko, A. G.; Moin, P., Numerical studies of flow over a circular cylinder at \(R e_D = 3900\), Phys. Fluids, 12, 2, 403-417 (2000) · Zbl 1149.76441
[39] Beaudan, P.; Moin, P., Numerical Experiments on the Flow Past a Circular Cylinder at Sub-critiecal Reynolds Number, Tech. Rep. TF-62 (1994), Department of Mechanical Engineering, Stanford University
[40] Clark, R. A.; Ferziger, J. H.; Reynolds, W. C., Evaluation of subgrid-scale models using an accurately simulated turbulent flow, J. Fluid Mech., 91, 1-16 (1979) · Zbl 0394.76052
[41] Lu, H.; Porté-Agel, F., A modulated gradient model for large-eddy simulation: Application to a neutral atmospheric boundary layer, Phys. Fluids, 22, 1 (2010) · Zbl 1183.76328
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