A generalization of the Smagorinsky model. (English) Zbl 07833983
Summary: Direct computation in numerical simulations of turbulent flow are often unfeasible. Large eddy simulations (LES) have been shown to provide efficient alternative. We investigate a generalization of an LES model, the so-called Smagorinsky model, that attempts to fix the over dissipation which is the well known drawback of the Smagorinsky model. We study finite element analysis and numerical computations based on benchmark problems in 2-D and 3-D.
MSC:
| 76F65 | Direct numerical and large eddy simulation of turbulence |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
References:
| [1] | Borggaard, Jeff; Iliescu, Traian; Lee, Hyesuk; Roop, John Paul; Son, Hyunjin, A two-level discretization method for the Smagorinsky model, Multiscale Model. Simul., 7, 2, 599-621 (2008) · Zbl 1201.76086 |
| [2] | Brenner, S.; Scott, L. R., The Mathematical Theory of Finite Element Methods (1994), Springer-Verlag · Zbl 0804.65101 |
| [3] | Cao, Yu; Giorgini, Andrea; Jolly, Michael; Pakzad, Ali, Continuous data assimilation for the 3d Ladyzhenskaya model: analysis and computations, Nonlinear Anal., Real World Appl., 68, Article 103659 pp. (2022) · Zbl 1504.35300 |
| [4] | Ciarlet, Philippe G., The Finite Element Method for Elliptic Problems (2002), SIAM · Zbl 0999.65129 |
| [5] | Du, Qiang; Gunzburger, Max D., Finite-element approximations of a Ladyzhenskaya model for stationary incompressible viscous flow, SIAM J. Numer. Anal., 27, 1, 1-19 (1990) · Zbl 0697.76046 |
| [6] | Du, Qiang; Gunzburger, Max D., Analysis of a Ladyzhenskaya model for incompressible viscous flow, J. Math. Anal. Appl., 155, 1, 21-45 (1991) · Zbl 0712.76039 |
| [7] | Ervin, Vincent J.; Layton, William; Neda, Monika, Numerical analysis of a higher order time relaxation model of fluids, Int. J. Numer. Anal. Model., 4, 3, 648-670 (2007) · Zbl 1242.76061 |
| [8] | Ethier, C. Ross; Steinman, D. A., Exact fully 3d Navier-Stokes solutions for benchmarking, Int. J. Numer. Methods Fluids, 19, 5, 369-375 (1994) · Zbl 0814.76031 |
| [9] | Gunzburger, Max D., Finite Element Methods for Viscous Incompressible Flows: a Guide to Theory, Practice, and Algorithms (1989), Academic Press: Academic Press Boston · Zbl 0697.76031 |
| [10] | Gunzburger, Max D., Finite Element Methods for Viscous Incompressible Flows: a Guide to Theory, Practice, and Algorithms (2012), Elsevier |
| [11] | Heywood, John G.; Rannacher, Rolf, Finite-element approximation of the nonstationary Navier-Stokes problem. part iv: error analysis for second-order time discretization, SIAM J. Numer. Anal., 27, 2, 353-384 (1990) · Zbl 0694.76014 |
| [12] | Iliescu, T.; John, V.; Layton, W. J.; Matthies, G.; Tobiska, L., A numerical study of a class of LES models, Int. J. Comput. Fluid Dyn., 171, 75-85 (2003) · Zbl 1148.76327 |
| [13] | Iliescu, Traian; Volker, John; Layton, William J., Convergence of finite element approximations of large eddy motion, Numer. Methods Partial Differ. Equ., 18, 6, 689-710 (2002) · Zbl 1021.76019 |
| [14] | Volker, John, Large Eddy Simulation of Turbulent Incompressible Flows: Analytical and Numerical Results for a Class of LES Models, 34 (2003), Springer Science & Business Media |
| [15] | Labovsky, Alexandr; Layton, William J.; Manica, Carolina C.; Neda, Monika; Rebholz, Leo G., The stabilized extrapolated trapezoidal finite-element method for the Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 198, 9-12, 958-974 (2009) · Zbl 1229.76051 |
| [16] | Ladyzhenskaya, Olga Aleksandrovna, New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems, Tr. Mat. Inst. Steklova, 102, 85-104 (1967) · Zbl 0202.37802 |
| [17] | Ladyzhenskaya, Olga Aleksandrovna, On modifications of Navier-Stokes equations for large gradients of velocities, Zap. Nauč. Semin. POMI, 7, 126-154 (1968) · Zbl 0195.10602 |
| [18] | Layton, William, Introduction to the Numerical Analysis of Incompressible Viscous Flows, 6 (2008), Siam · Zbl 1153.76002 |
| [19] | Layton, William J.; Rebholz, Leo G., Approximate Deconvolution Models of Turbulence: Analysis, Phenomenology and Numerical Analysis, 2042 (2012), Springer Science & Business Media · Zbl 1241.76002 |
| [20] | Lions, Jacques Louis, Quelques méthodes de résolution des problemes aux limites non linéaires (1969) · Zbl 0189.40603 |
| [21] | Minty, George J., Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29, 3, 341-346 (1962) · Zbl 0111.31202 |
| [22] | Neda, Monika; Sun, Xudong; Yu, Lanxuan, Increasing accuracy and efficiency for regularized Navier-Stokes equations, Acta Appl. Math., 118, 1, 57-79 (2012) · Zbl 1427.76140 |
| [23] | Pakzad, Ali, Damping functions correct over-dissipation of the Smagorinsky model, Math. Methods Appl. Sci., 40, 16, 5933-5945 (2017) · Zbl 1382.76117 |
| [24] | Sagaut, Pierre, Large Eddy Simulation for Incompressible Flows: an Introduction (2006), Springer Science & Business Media · Zbl 1091.76001 |
| [25] | Siddiqua, Farjana; Xie, Xihui, Numerical analysis of a corrected Smagorinsky model, (Numerical Methods for Partial Differential Equations (2022)) · Zbl 1537.65140 |
| [26] | Taylor, G. I., Lxxv. on the decay of vortices in a viscous fluid, Lond. Edinb. Dublin Philos. Mag. J. Sci., 46, 274, 671-674 (1923) · JFM 49.0607.02 |
| [27] | Zang, Yan; Street, Robert L., A dynamic mixed subgrid-scale model and its application to turbulent recirculating flows, Phys. Fluids A, Fluid Dyn., 5, 12, 3186-3196 (1993) · Zbl 0925.76242 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
