Assessment of subgrid-scale models for the incompressible Navier-Stokes equations. (English) Zbl 1273.76082
Summary: We assess two kinds of subgrid finite element methods for the two-dimensional (2D) incompressible Naver-Stokes equations (NSEs). These methods introduce subgrid-scale (SGS) eddy viscosity terms which do not act on the large flow structures. The eddy viscous terms consist of the fluid flow fluctuation strain rate stress tensors. The fluctuation tensor can be calculated by a elliptic projection or a simple \(L^{2}\) projection (projective filter) in finite element spaces. The finite element pair \(P_{2}/P_{1}\) is adopted to numerically implement analysis and computation. We give a complete error analysis based on the assumptions of some regularity conditions. On the part of numerical tests, the numerical computations for the stationary flows show that the numerical results agree with some benchmark solutions and theoretical analysis very well. Furthermore, the given SGS models are applied to the non-stationary fluid flows.
MSC:
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
Navier-Stokes equations; subgrid model; eddy viscosity; pressure stabilization method; error analysis; numerical testsReferences:
| [1] | John, V., Large Eddy Simulation of Turbulent Incompressible Flows (2004), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 1035.76001 |
| [2] | Frisch, U.; Orszag, S. A., Turbulence: Challenges for theory and experiment, Phys. Today, 24-32 (1990) |
| [3] | Iliescu, T.; Layton, W. J., Approximating the larger eddies in fluid motion. III. The Boussinesq model for turbulent fluctuations, An. Ştiinţ. Univ. Al. I. Cuza Iaşi., Mat. (N.S.), 44, 245-261 (1998) · Zbl 1078.76553 |
| [4] | Smagorinsky, J., General circulation experiments with the primitive equation, I: The basic experiment, Mon. Weather Rev., 91, 99-164 (1963) |
| [5] | Guermond, J.-L.; Marra, A.; Quartapelle, L., Subgrid stabilized projection method for 2D unsteady flows at high Reynolds numbers, Comput. Methods Appl. Mech. Engrg., 195, 5857-5876 (2006) · Zbl 1121.76036 |
| [6] | Kaya, S.; Layton, W.; Brivière, B., Subgrid stabilized defect correction methods for the Navier-Stokes equations, SIAM J. Numer. Aanal., 44, 4, 1639-1654 (2006) · Zbl 1124.76029 |
| [7] | John, V.; Kaya, S., Finite element error analysis of a variational multiscale method for the Navier-Stokes equations, Adv. Comput. Math., 28, 43-61 (2008) · Zbl 1126.76030 |
| [8] | Hughes, T. J.R.; Mazzei, L.; Jansen, K. E., Large eddy simulation and the variational multiscale method, Comput. Visual. Sci., 3, 47-59 (2000) · Zbl 0998.76040 |
| [9] | He, Yinnian; Wang, Aiwen, A simplified two-level method for the steady Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 197, 1568-1576 (2008) · Zbl 1194.76120 |
| [10] | Temam, R., Navier-Stokes Equations, Theory and Numerical Analysis (1983), NorthHolland: NorthHolland Amsterdam · Zbl 0555.76030 |
| [11] | Girault, V.; Raviat, P. A., (Finite Element Methods for Navier-Stokes. Finite Element Methods for Navier-Stokes, Springer Series in Computational Mathematics, vol. 5 (1986), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0585.65077 |
| [12] | Quarteroni, A.; Valli, A., Numerical Approximation of Partial Differential Equations (1998), Springer-Verlag, World Publishing Co. |
| [13] | Layton, W., A connection between subgrid-scale eddy viscosity and mixed methods, Appl. Math. Comput., 133, 1, 147-157 (2002) · Zbl 1024.76026 |
| [14] | Kaya, S.; Riviére, B., A two-grid stabilization method for solving the steady-state Navier-Stokes equations, Inc. Numer. Methods Partial Differential Equations, 22, 728-743 (2005) · Zbl 1089.76034 |
| [15] | Ghia, U.; Ghia, K. N.; Shin, C. T., High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys., 48, 387-411 (1982) · Zbl 0511.76031 |
| [16] | He, Yinnian; Zhang, Yan, A subgrid model for the time-dependent Navier-Stokes equations, Adv. Numer. Anal. (2009) · Zbl 1410.76205 |
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