×

zbMATH — the first resource for mathematics

Weak and strong wall boundary procedures and convergence to steady-state of the Navier-Stokes equations. (English) Zbl 1245.76020
Summary: We study the influence of different implementations of no-slip solid wall boundary conditions on the convergence to steady-state of the Navier-Stokes equations. The various approaches are investigated using the energy method and an eigenvalue analysis. It is shown that the weak implementation is superior and enhances the convergence to steady-state for coarse meshes. It is also demonstrated that all the stable approaches produce the same convergence rate as the mesh size goes to zero. The numerical results obtained by using a fully nonlinear finite volume solver support the theoretical findings from the linear analysis.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
Software:
EDGE
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abarbanel, S.; Gottlieb, D., Optimal time splitting for two- and three-dimensional navier – stokes equations with mixed derivatives, Journal of computational physics, 41, 1-43, (1981) · Zbl 0467.76062
[2] Abbas, Q.; Nordström, J., Weak versus strong no-slip boundary conditions for the navier – stokes equations, Engineering applications of computational fluid mechanics, 4, 1, 29-38, (2010)
[3] Berg, J.; Nordström, J., Stable Robin solid wall boundary conditions for the navier – stokes equations, Journal of computational physics, 230, 7519-7532, (2011) · Zbl 1280.76023
[4] Carpenter, M.H.; Nordström, J.; Gottlieb, D., A stable and conservative interface treatment of arbitrary spatial accuracy, Journal of computational physics, 148, 341-365, (1999) · Zbl 0921.65059
[5] Costa, Bruno; Don, Wai Sun; Gottlieb, David; Sendersky, Radislav, Two-dimensional multi-domain hybrid spectral-WENO methods for conservation laws, Communications in computational physics, 1, 3, 548-574, (2006) · Zbl 1114.76050
[6] S.C. Dias, D.W. Zingg, A high-order parallel Newton-Krylov flow solver for the Euler equations, in: 19th AIAA Computational Fluid Dynamics Conference, 2009.
[7] Don, Wai-Sun; Gottlieb, David; Jung, Jae-Hun, A weighted multi-domain spectral penalty method with inhomogeneous grid for supersonic injective cavity flows, Communications in computational physics, 5, 5, 986-1011, (2009) · Zbl 1364.76146
[8] Efraimsson, G.; Gong, J.; Svärd, M.; Nordström, J., An investigation of the performance of a high-order accurate navier – stokes code, (), 11
[9] P. Eliasson, Edge, a Navier-Stokes solver for unstructured grids, in: Proceedings to Finite Volumes for Complex Applications III, 2002, pp. 527-534. · Zbl 1177.76285
[10] P. Eliasson, S. Eriksson, J. Nordström, The influence of weak and strong solid wall boundary conditions on the convergence to steady state of the Navier-Stokes equations, AIAA Paper 2009-3551, 2009.
[11] P. Eliasson, P. Weinerfelt, Recent applications of the flow solver Edge. in: Proceedings to 7th Asian CFD Conference, 2007. · Zbl 1390.76426
[12] Engquist, B.; Gustafsson, B., Steady state computations for wave propagation problems, Mathematics of computations, 49, 39-64, (1987) · Zbl 0632.65099
[13] Eriksson, S.; Abbas, Q.; Nordström, J., A stable and conservative method for locally adapting the design order of finite difference schemes, Journal of computational physics, 230, 11, 4216-4231, (2011), (Cited By (since 1996): 1) · Zbl 1220.65112
[14] Eriksson, S.; Nordström, J., Analysis of the order of accuracy for node-centered finite volume schemes, Applied numerical mathematics, (2009) · Zbl 1173.65055
[15] Gong, J.; Nordström, J., A stable and efficient hybrid scheme for viscous problems in complex geometries, Journal of computational physics, 226, 1291-1309, (2007) · Zbl 1121.76041
[16] Gottlieb, D.; Hesthaven, J.S., Spectral methods for hyperbolic problems, Journal of computational and applied mathematics, 128, 1-2, 83-131, (2001), (special issue SI) · Zbl 0974.65093
[17] Gustafsson, B.; Kreiss, H.-O.; Oliger, J., Time dependent problems and difference methods, (1995), John Wiley & Sons, Inc.
[18] Hellsten, A., New advanced κ−ω turbulence model for high lift aerodynamics, AIAA journal, 43, 9, 1857-1869, (2005)
[19] Hesthaven, J.S.; Gottlieb, D., A stable penalty method for the compressible navier – stokes equations. 1. open boundary conditions, SIAM journal on scientific computing, 17, 3, 579-612, (1996) · Zbl 0853.76061
[20] Hesthaven, J.S.; Warburton, T., Nodal high-order methods on unstructured grids - I. time-domain solution of maxwell’s equations, Journal of computational physics, 181, 1, 186-221, (2002) · Zbl 1014.78016
[21] Hicken, J.E.; Zingg, D.W., Superconvergent functional estimates from summation-by-parts finite-difference discretizations, SIAM journal on scientific computing, 33, 2, 893-922, (2011) · Zbl 1227.65102
[22] Horn, R.A.; Johnson, C.R., Topics in matrix analysis, (1991), Cambridge University Press · Zbl 0729.15001
[23] X. Huan, J.E. Hicken, D.W. Zingg, Interface and boundary schemes for high-order methods, in: 19th AIAA Computational Fluid Dynamics Conference, 2009.
[24] A. Jameson, W. Schmidt, E. Turkel, Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes, AIAA Paper 81-1259, 1981.
[25] Kozdon, J.E.; Dunham, E.M.; Nordström, J., Interaction of waves with frictional interfaces using summation-by-parts difference operators: weak enforcement of nonlinear boundary conditions, Journal of scientific computing, 50, 2, 341-367, (2012) · Zbl 1325.74161
[26] Kreiss, H.-O.; Scherer, G., Finite element and finite difference methods for hyperbolic partial differential equations, () · Zbl 0355.65085
[27] Mattsson, K., Boundary procedures for summation-by-parts operators, Journal of scientific computing, 18, 1, 133-153, (2003) · Zbl 1024.76031
[28] Mattsson, K.; Nordström, J., Summation by parts operators for finite difference approximations of second derivatives, Journal of computational physics, 199, 2, 503-540, (2004) · Zbl 1071.65025
[29] Mavriplis, D.J., Accurate multigrid solution of the Euler equations on unstructured and adaptive meshes, AIAA journal, 28, 2, (1990)
[30] Nordström, J., The influence of open boundary conditions on the convergence to steady state for the navier – stokes equations, Journal of computational physics, 85, 210-244, (1989) · Zbl 0679.76039
[31] Nordström, J.; Carpenter, M.H., Boundary and interface conditions for high order finite difference methods applied to the Euler and navier – stokes equations, Journal of computational physics, 148, 621-645, (1999) · Zbl 0921.76111
[32] Nordström, J.; Carpenter, M.H., High-order finite difference methods, multidimensional linear problems and curvilinear coordinates, Journal of computational physics, 173, 149-174, (2001) · Zbl 0987.65081
[33] Nordström, J.; Forsberg, K.; Adamsson, C.; Eliasson, P., Finite volume methods, unstructured meshes and strict stability, Applied numerical mathematics, 45, 453-473, (2003) · Zbl 1019.65066
[34] Nordström, J.; Gong, J., A stable and efficient hybrid method for aeroacoustic sound generation and propagation, Comptes rendus mecanique, 333, 713-718, (2005) · Zbl 1102.76043
[35] Nordström, J.; Gong, J., A stable hybrid method for hyperbolic problems, Journal of computational physics, 212, 436-453, (2006) · Zbl 1083.65085
[36] Nordström, J.; Gong, J.; van der Weide, E.; Svärd, M., A stable and conservative high order multi-block method for the compressible navier – stokes equations, Journal of computational physics, 228, 24, 9020-9035, (2009) · Zbl 1375.76036
[37] Nordström, J.; Gustafsson, R., High order finite difference approximations of electromagnetic wave propagation close to material discontinuities, Journal of scientific computing, 18, 2, 215-234, (2003) · Zbl 1029.78012
[38] Nordström, J.; Ham, F.; Shoeybi, M.; van der Weide, E.; Svärd, M.; Mattsson, K.; Iaccarino, G.; Gong, J., A hybrid method for unsteady fluid flow, Computers and fluids, 38, 875-882, (2009) · Zbl 1242.76181
[39] Nordström, J.; Svärd, M., Well-posed boundary conditions for the navier – stokes equations, SIAM journal on numerical analysis, 43, 3, 1231-1255, (2005) · Zbl 1319.35163
[40] M. Osusky, J.E. Hicken, D.W. Zingg, A parallel Newton-Krylov-Schur flow solver for the Navier-Stokes equations using the sbp-sat approach. in: 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, 2010.
[41] Schiesser, W.E., The numerical method of lines: integration of partial differential equations, (1991), Academic Press · Zbl 0763.65076
[42] Shoeybi, M.; Svrd, M.; Ham, F.E.; Moin, P., An adaptive implicit – explicit scheme for the DNS and LES of compressible flows on unstructured grids, Journal of computational physics, 229, 17, 5944-5965, (2010) · Zbl 1425.76108
[43] Strand, B., Summation by parts for finite difference approximation for d/dx, Journal of computational physics, 110, 1, 47-67, (1994) · Zbl 0792.65011
[44] Strikwerda, J.C., Initial boundary value problems for incompletely parabolic systems, Communications on pure and applied mathematics, 9, 3, 797-822, (1977) · Zbl 0351.35051
[45] Svärd, M.; Carpenter, M.H.; Nordström, J., A stable high-order finite difference scheme for the compressible navier – stokes equations: far-field boundary conditions, Journal of computational physics, 225, 1, 1020-1038, (2007) · Zbl 1118.76047
[46] Svärd, M.; Gong, J.; Nordström, J., Stable artificial dissipation operators for finite volume schemes on unstructured grids, Applied numerical mathematics, 56, 12, 1481-1490, (2006) · Zbl 1103.65096
[47] Svärd, M.; Gong, J.; Nordström, J., An accuracy evaluation of unstructured node-centred finite volume methods, Applied numerical mathematics, 58, 1142-1158, (2008) · Zbl 1139.76040
[48] Svärd, M.; Nordström, J., Stability of finite volume approximations for the Laplacian operator on quadrilateral and triangular grids, Applied numerical mathematics, 51, 101-125, (2004) · Zbl 1065.65111
[49] Svärd, M.; Nordström, J., A stable high-order finite difference scheme for the compressible navier – stokes equations: no-slip wall boundary conditions, Journal of computational physics, 227, 10, 4805-4824, (2008) · Zbl 1260.76021
[50] Wallin, S.; Johansson, A.V., An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows, Journal of fluid mechanics, 403, 89-132, (2000) · Zbl 0966.76032
[51] Zhou, L.; Walker, H.F., Residual smoothing techniques for iterative methods, SIAM journal on scientific computing, 15, 297-312, (1994) · Zbl 0802.65041
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.