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Inclusion of no-slip boundary conditions in the MEEVC scheme. (English) Zbl 1416.76104

Summary: This work presents three methods for enforcing tangential velocity boundary conditions for the MEEVC scheme, which was shown to be mass, enstrophy, energy and vorticity conserving scheme in the case of inviscid flow [the last two authors, ibid. 328, 200–220 (2017; Zbl 1406.76064)]. While the normal velocity component can be strongly imposed in a div-conforming formulation for the velocity field, inclusion of the tangential velocity needs to be set through an appropriate choice of vorticity boundary conditions. Three methods to impose the tangential velocity boundary condition will be discussed: the kinematic Dirichlet formulation, the kinematic Neumann formulation and the dynamic Neumann formulation. The conservation properties of each of the resulting schemes are analyzed and numerical results are shown for the Taylor-Green vortex and for the dipole collision test cases. These confirm that kinematic Neumann vorticity boundary conditions perform best.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids

Citations:

Zbl 1406.76064
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Full Text: DOI

References:

[1] Palha, A.; Gerritsma, M. I., A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations, J. Comput. Phys., 328, 200-220 (2017) · Zbl 1406.76064
[2] Budd, C. J.; Piggott, M. D., Geometric integration and its applications, Handb. Numer. Anal., 11, 35-139 (2003) · Zbl 1062.65134
[3] Christiansen, S. H.; Munthe-Kaas, H. Z.; Owren, B., Topics in structure-preserving discretization, Acta Numer., 20, 1-119 (2011) · Zbl 1233.65087
[4] Perot, J. B., Discrete conservation properties of unstructured mesh schemes, Annu. Rev. Fluid Mech., 43, 229-318 (2011) · Zbl 1299.76127
[5] Verstappen, R. W.C. P.; Veldman, A. E.P., Symmetry-preserving discretization of turbulent flow, J. Comput. Phys., 187, 343-368 (2003) · Zbl 1062.76542
[6] Rebholz, L. G., Conservation laws of turbulence models, J. Math. Anal. Appl., 326, 33-45 (2007) · Zbl 1110.76028
[7] Bochev, P.; Hyman, M., Principles of Mimetic Discretizations, IMA, vol. 142, 89-119 (2006), Springer Verlag · Zbl 1110.65103
[8] Palha, A.; Rebelo, P. P.; Hiemstra, R.; Kreeft, J.; Gerritsma, M. I., Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms, J. Comput. Phys., 257, 1394-1422 (2014) · Zbl 1352.65629
[9] Arnold, D. N.; Falk, R. S.; Winther, R., Finite element exterior calculus, homological techniques and applications, Acta Numer., 15, 1-155 (2006) · Zbl 1185.65204
[10] Arnold, D. N.; Falk, R. S.; Winther, R., Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Am. Math. Soc., 42, 281-354 (2010) · Zbl 1207.65134
[11] Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods, vol. 15, Springer Series in Computational Mathematics, vol. 44 (1991), Springer
[12] Gresho, P. M., Incompressible fluid dynamics: some fundamental formulation issues, Annu. Rev. Fluid Mech., 23, 413-453 (1991) · Zbl 0717.76006
[13] Gatski, T. B., Review of incompressible fluid flow computations using the vorticity-velocity formulation, Appl. Numer. Math., 7, 227-239 (1991) · Zbl 0714.76033
[14] Wu, J. Z.; Wu, J. M., Vorticity dynamics on boundaries, Adv. Appl. Mech., 32, 119-275 (1996) · Zbl 0870.76068
[15] Lighthill, M. J., Introduction: boundary layer theory. Chapter II, (Rosenhead, L., Laminar Boundary Layers (1963), Oxford at the Clarendon Press)
[16] Chorin, A. J., Vortex sheet approximation of boundary layers, J. Comput. Phys., 27, 428-442 (1978) · Zbl 0387.76040
[17] Wu, J. C., Fundamental solutions and numerical methods for flow problems, Int. J. Numer. Methods Fluids, 4, 185-201 (1984) · Zbl 0538.76007
[18] Morton, B. R., The generation and decay of vorticity, Geophys. Astrophys. Fluid Dyn., 28, 277-308 (1984) · Zbl 0551.76019
[19] Wu, J. Z.; Ma, H. Y.; Zhou, M. D., Vorticity and Vortex Dynamics (2006), Springer: Springer Berlin
[20] Wu, J. Z.; Wu, X. H.; Ma, H. Y.; Wu, J. M., Dynamic vorticity condition: theoretical analysis and numerical implementation, Int. J. Numer. Methods Fluids, 19, 905-938 (1994) · Zbl 0839.76061
[21] Taylor, C.; Hood, P., A numerical solution of the Navier-Stokes equations using the finite element technique, Comput. Fluids, 1, 73-100 (1973) · Zbl 0328.76020
[22] Guevremont, G.; Habashi, W.; Hafez, M., Finite element solution of the Navier-Stokes equations by a velocity-vorticity method, Int. J. Numer. Methods Fluids, 11, 611-675 (1990) · Zbl 0711.76021
[23] Guevremont, G.; Habashi, W.; Kotiuga, P.; Hafez, M., Finite element solution of the 3D compressible Navier-Stokes equations by a velocity-vorticity method, J. Comput. Phys., 107, 176-187 (1993) · Zbl 0776.76051
[24] Olshanskii, M. A.; Rebholz, L. G., Velocity-vorticity-helicity formulation and a solver for the Navier-Stokes equations, J. Comput. Phys., 229, 4291-4303 (2010) · Zbl 1334.76083
[25] Lee, H. K.; Olshanskii, M. A.; Rebholz, L. G., On error analysis for the 3D Navier-Stokes equations in velocity-vorticity-helicity form, SIAM J. Numer. Anal., 49, 711-732 (2011) · Zbl 1222.35143
[26] Benzi, M.; Olshanskii, M. A.; Rebholz, L. G.; Wang, Z., Assessment of a vorticity based solver for the Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 247-248, 216-225 (2012) · Zbl 1352.76040
[27] Olshanskii, M. A.; Heister, T.; Rebholz, L. G.; Galvin, K. J., Natural vorticity boundary conditions on solid walls, Comput. Methods Appl. Mech. Eng., 297, 18-37 (2015) · Zbl 1423.76100
[28] Heister, T.; Olshanskii, M. A.; Rebholz, L. G., Unconditional long-time stability of a velocity-vorticity method for the 2D Navier-Stokes equations, Numer. Math., 135, 143-167 (2017) · Zbl 1387.76052
[29] Trujillo, J.; Karniadakis, G., A penalty method for the velocity-vorticity formulation, J. Comput. Phys., 149, 32-58 (1999) · Zbl 0931.76072
[30] Charnyi, S.; Heister, T.; Olshanskii, M. A.; Rebholz, L. G., On conservation laws of Navier-Stokes Galerkin discretizations, J. Comput. Phys., 337, 289-308 (2017) · Zbl 1415.65222
[31] Brenner, S.; Scott, R., The Mathematical Theory of Finite Element Methods (2007)
[32] Kirby, C.; Logg, A.; Rognes, M. E.; Terrel, A. R., Common and unusual finite elements, (Automated Solution of Differential Equations by the Finite Element Method. Automated Solution of Differential Equations by the Finite Element Method, Lecture Notes in Computational Science and Engineering, vol. 84 (2012), Springer), 95-119
[33] Arnold, D. N., Spaces of finite element differential forms, (Gianazza, U.; Brezzi, F.; Franzone, P. C.; Gilardi, G., Analysis and Numerics of Partial Differential Equations (2013), Springer), 117-140 · Zbl 1290.58003
[34] Clercx, H. J.H.; Bruneau, C. H., The normal and oblique collision of a dipole with a no-slip boundary, Comput. Fluids, 35, 245-279 (2006) · Zbl 1160.76328
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