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Boundary treatment for fourth-order staggered mesh discretizations of the incompressible Navier-Stokes equations. (English) Zbl 1351.76182
Summary: Harlow and Welch [Phys. Fluids 8 (1965) 2182-2189] introduced a discretization method for the incompressible Navier-Stokes equations conserving the secondary quantities kinetic energy and vorticity, besides the primary quantities mass and momentum. This method was extended to fourth order accuracy by several researchers [25,14,21]. In this paper we propose a new consistent boundary treatment for this method, which is such that continuous integration-by-parts identities (including boundary contributions) are mimicked in a discrete sense. In this way kinetic energy is exactly conserved even in case of non-zero tangential boundary conditions. We show that requiring energy conservation at the boundary conflicts with order of accuracy conditions, and that the global accuracy of the fourth order method is limited to second order in the presence of boundaries. We indicate how non-uniform grids can be employed to obtain full fourth order accuracy.

##### MSC:
 76M20 Finite difference methods applied to problems in fluid mechanics 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids
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##### References:
 [1] Axelsson, O.; Kolotilina, L., Monotonicity and discretization error estimates, SIAM J. Numer. Anal., 27, 6, 1591-1611, (1990) · Zbl 0719.65036 [2] Botella, O.; Peyret, R., Benchmark spectral results on the lid-driven cavity flow, Comput. Fluids, 27, 4, 421-433, (1998) · Zbl 0964.76066 [3] Desjardins, O.; Blanquart, G.; Balarac, G.; Pitsch, H., High order conservative finite difference scheme for variable density low Mach number turbulent flows, J. Comput. Phys., 227, 7125-7159, (2008) · Zbl 1201.76139 [4] Gresho, P. M.; Sani, R. L., Incompressible flow and the finite element method. vol. 2: isothermal laminar flow, (2000), Wiley · Zbl 0988.76005 [5] Ham, F. E.; Lien, F. S.; Strong, A. B., A fully conservative second-order finite difference scheme for incompressible flow on nonuniform grids, J. Comput. Phys., 177, 117-133, (2002) · Zbl 1066.76044 [6] Harlow, F. H.; Welch, J. E., Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids, 8, 2182-2189, (1965) · Zbl 1180.76043 [7] Hundsdorfer, W.; Verwer, J., Numerical solution of time-dependent advection-diffusion-reaction equations, (2007), Springer [8] Lorenz, J., Zur inversmonotonie diskreter probleme, Numer. Math., 27, 227-238, (1977) · Zbl 0325.65014 [9] Mahesh, K.; Constantinescu, G.; Moin, P., A numerical method for large-eddy simulation in complex geometries, J. Comput. Phys., 197, 215-240, (2004) · Zbl 1059.76033 [10] Manteuffel, T. A.; White, A. B., The numerical solution of second-order boundary value problems on non-uniform meshes, Math. Comput., 47, 176, 511-535, (1986) · Zbl 0635.65092 [11] Mattheij, R. M.M.; Rienstra, S. W.; ten Thije Boonkkamp, J. H.M., Partial differential equations: modeling, analysis, computation, (2005), SIAM · Zbl 1090.35001 [12] Mattson, K.; Nordstrøm, J., Summation by parts operators for finite difference approximations of second derivatives, J. Comput. Phys., 199, 503-540, (2004) · Zbl 1071.65025 [13] Mittal, R.; Moin, P., Suitability of upwind-biased finite difference schemes for large-eddy simulation of turbulent flows, AIAA J., 35, 8, (1997) · Zbl 0900.76336 [14] Morinishi, Y.; Lund, T. S.; Vasilyev, O. V.; Moin, P., Fully conservative higher order finite difference schemes for incompressible flows, J. Comput. Phys., 143, 90-124, (1998) · Zbl 0932.76054 [15] Morton, K. W.; Mayers, D. F., Numerical solution of partial differential equations, (2005), Cambridge University Press · Zbl 1126.65077 [16] Moser, R. D.; Kim, J.; Mansour, N. N., Direct numerical simulation of turbulent channel flow up to $$\mathit{Re}_\tau = 590$$, Phys. Fluids, 11, 943-945, (1999) · Zbl 1147.76463 [17] Perot, B., Conservation properties of unstructured staggered mesh schemes, J. Comput. Phys., 159, 58-89, (2000) · Zbl 0972.76068 [18] Sanderse, B., Energy-conserving Runge-Kutta methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 233, 100-131, (2012) · Zbl 1286.76034 [19] Sani, R. L.; Shen, J.; Pironneau, O.; Gresho, P. M., Pressure boundary condition for the time-dependent incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 50, 673-682, (2006) · Zbl 1155.76322 [20] Strand, B., Summation by parts for finite difference approximations for $$\operatorname{d} / \operatorname{dx}$$, J. Comput. Phys., 110, 47-67, (1994) · Zbl 0792.65011 [21] Vasilyev, O. V., High order finite difference schemes on non-uniform meshes with good conservation properties, J. Comput. Phys., 157, 746-761, (2000) · Zbl 0959.76063 [22] Veldman, A. E.P., “missing” boundary conditions? discretize first, substitute next, and combine later, SIAM J. Sci. Stat. Comput., 11, 82-91, (1990) · Zbl 0683.76028 [23] Veldman, A. E.P., High-order symmetry-preserving discretization of convection-diffusion equations on strongly stretched grids, (Lube, G.; Rapin, G., Proceedings of the Int. Conf. on Boundary and Interior Layers, (2006)) [24] Verstappen, R. W.C. P., When does eddy viscosity damp subfilter scales sufficiently?, J. Sci. Comput., 49, 94-110, (2011) · Zbl 1432.76129 [25] Verstappen, R. W.C. P.; Veldman, A. E.P., Direct numerical simulation of turbulence at lower costs, J. Eng. Math., 32, 143-159, (1997) · Zbl 0911.76072 [26] 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 [27] Wesseling, P., Principles of computational fluid dynamics, (2001), Springer
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