×

A hybrid incremental projection method for thermal-hydraulics applications. (English) Zbl 1349.76190

Summary: A new second-order accurate, hybrid, incremental projection method for time-dependent incompressible viscous flow is introduced in this paper. The hybrid finite-element/finite-volume discretization circumvents the well-known Ladyzhenskaya-Babuška-Brezzi conditions for stability, and does not require special treatment to filter pressure modes by either Rhie-Chow interpolation or by using a Petrov-Galerkin finite element formulation. The use of a co-velocity with a high-resolution advection method and a linearly consistent edge-based treatment of viscous/diffusive terms yields a robust algorithm for a broad spectrum of incompressible flows. The high-resolution advection method is shown to deliver second-order spatial convergence on mixed element topology meshes, and the implicit advective treatment significantly increases the stable time-step size. The algorithm is robust and extensible, permitting the incorporation of features such as porous media flow, RANS and LES turbulence models, and semi-/fully-implicit time stepping. A series of verification and validation problems are used to illustrate the convergence properties of the algorithm. The temporal stability properties are demonstrated on a range of problems with \(2 \leq C F L \leq 100\). The new flow solver is built using the Hydra multiphysics toolkit. The Hydra toolkit is written in C++ and provides a rich suite of extensible and fully-parallel components that permit rapid application development, supports multiple discretization techniques, provides I/O interfaces, dynamic run-time load balancing and data migration, and interfaces to scalable popular linear solvers, e. g., in open-source packages such as HYPRE, PETSc, and Trilinos.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
76M12 Finite volume methods applied to problems in fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kan, J. V., A second-order accurate pressure-correction scheme for viscous incompressible flow, SIAM J. Sci. Statist. Comput., 7, 870-891 (1986) · Zbl 0594.76023
[2] Bell, J. B.; Colella, P.; Glaz, H. M., A second-order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys., 85, 257-283 (1989) · Zbl 0681.76030
[3] Gresho, P. M., On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 1: theory, Int. J. Numer. Methods Fluids, 11, 587-620 (1990) · Zbl 0712.76035
[4] Gresho, P. M.; Chan, S. T., On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 2: implementation, Int. J. Numer. Methods Fluids, 11, 621-659 (1990) · Zbl 0712.76036
[5] Gresho, P. M.; Chan, S. T.; Christon, M. A.; Hindmarsh, A. C., A little more on stabilized \(q 1 q 1\) for transient viscous incompressible flow, Int. J. Numer. Methods Fluids, 21, 837-856 (1995) · Zbl 0862.76034
[6] Gresho, P. M.; Chan, S. T., Projection 2 goes turbulent- and fully implicit, Int. J. Comput. Fluid Dyn., 9, 3-4, 249-272 (1998) · Zbl 0917.76036
[7] Almgren, A. S.; Bell, J. B.; Colella, P.; Howell, L. H., An adaptive projection method for the incompressible Euler equations, (Eleventh AIAA Computational Fluid Dynamics Conference (1993), AIAA), 530-539
[8] Almgren, A. S.; Bell, J. B.; Szymcyzk, W. G., A numerical method for the incompressible Navier-Stokes equations based on an approximate projection, SIAM J. Sci. Comput., 17, 2, 358-369 (1996) · Zbl 0845.76055
[9] Almgren, A. S.; Bell, J. B.; Crutchfield, W. Y., Approximate projection methods: Part I. Inviscid analysis, SIAM J. Sci. Comput., 22, 4, 1139-1159 (2000) · Zbl 0995.76059
[10] Rider, W. J., The robust formulation of approximate projection methods for incompressible flows (1994), Los Alamos National Laboratory, Tech. rep. LA-UR-3015
[11] Rider, W. J., Filtering nonsolenoidal modes in numerical solutions of incompressible flows (September 1994), Los Alamos National Laboratory: Los Alamos National Laboratory Los Alamos, New Mexico, Tech. rep. LA-UR-3014
[12] Rider, W. J.; Kothe, D. B.; Mosso, S. J.; Cerutti, J. H.; Hochstein, J. I., Accurate solution algorithms for incompressible multiphase flows (January 1995), AIAA: AIAA Reno, Nevada, Tech. rep. AIAA-95-0699
[13] Rider, W. J., Approximate projection methods for incompressible flow: implementation, variants and robustness (July 1995), Los Alamos National Laboratory: Los Alamos National Laboratory Los Alamos, New Mexico, Tech. rep. LA-UR-2000
[14] Minion, M. L., A projection method for locally refined grids, J. Comput. Phys., 127, 158-178 (1996) · Zbl 0859.76047
[15] Guermond, J.-L.; Quartapelle, L., Calculation of incompressible viscous flow by an unconditionally stable projection fem, J. Comput. Phys., 132, 12-23 (1997) · Zbl 0879.76050
[16] Puckett, E. G.; Almgren, A. S.; Bell, J. B.; Marcus, D. L.; Rider, W. J., A high-order projection method for tracking fluid interfaces in variable density incompressible flows, J. Comput. Phys., 130, 269-282 (1997) · Zbl 0872.76065
[17] Sussman, M.; Almgren, A. S.; Bell, J. B.; Colella, P.; Howell, L. H.; Welcome, M. L., An adaptive level set approach for two-phase flows, J. Comput. Phys., 148, 81-124 (1999) · Zbl 0930.76068
[18] Knio, O. M.; Najm, H. N.; Wyckoff, P. S., A semi-implicit numerical scheme for reacting flow. II. Stiff operator-split formulation, J. Comput. Phys., 154, 2, 428-467 (1999) · Zbl 0958.76061
[19] Christon, M. A.; Patil, R. S., A finite element projection method for low-Mach number reacting flows, (Bathe, K. J., Third MIT Conference on Computational Fluid and Solid Mechanics (2005), Elsevier: Elsevier New York), 617-622
[20] Schofield, S. P.; Christon, M. A.; Dyadechko, V.; Garimella, R. V.; Lowrie, R. B.; Swartz, B. K., Multi-material incompressible flow simulation using the moment-of-fluid method, Int. J. Numer. Methods Fluids, 63, 931-952 (2010) · Zbl 1406.76052
[21] Brown, D. L.; Minion, M. L., Performance of under-resolved two-dimensional incompressible flow simulations, J. Comput. Phys., 122, 165-183 (1995) · Zbl 0849.76043
[22] Minion, M. L.; Brown, D. L., Performance of under-resolved two-dimensional incompressible flow simulations, II, J. Comput. Phys., 138, 734-765 (1997) · Zbl 0914.76063
[23] Wetton, B. B., Error analysis of pressure increment schemes, SIAM J. Numer. Anal., 38, 1, 160-169 (1998) · Zbl 0973.76070
[24] Guermond, J.-L., Some implementations of projection methods for Navier-Stokes equations, Math. Model. Numer. Anal., 30, 5, 637-667 (1996) · Zbl 0861.76065
[25] Guermond, J.-L., A convergence result for the approximation of the Navier-Stokes equations by an incremental projection method, C. R. Acad. Sci. Paris, 325, 1329-1332 (1997) · Zbl 0899.76271
[26] Guermond, J.-L.; Quartapelle, L., On the approximation of the unsteady Navier-Stokes equations by finite element projection methods, Numer. Math., 80, 207-238 (1998) · Zbl 0914.76051
[27] Guermond, J.-L.; Quartapelle, L., On stability and convergence of projection methods based on pressure Poisson equation, Int. J. Numer. Methods Fluids, 26, 1039-1053 (1998) · Zbl 0912.76054
[28] Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow (1969), Gordon and Breach Science Publishers, Inc. · Zbl 0184.52603
[29] Babuška, I., Error-bounds for the finite element method, Numer. Math., 16, 322-333 (1971) · Zbl 0214.42001
[30] Babuška, I., The finite element method with Lagrangian multipliers, Numer. Math., 20, 179-192 (1973) · Zbl 0258.65108
[31] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Revue Francais d’Automatique, Informatique, et Recherce Opérationnelle (R.A.I.R.O.), R-2, 129-151 (1974) · Zbl 0338.90047
[32] Gresho, P. M.; Sani, R. L., Incompressible Flow and the Finite Element Method, Advection-Diffusion and Isothermal Laminar Flow (1998), John Wiley & Sons: John Wiley & Sons Chicester, England · Zbl 0941.76002
[33] Tu, S.; Aliabadi, S., Development of a hybrid finite volume/element solver for incompressible flows, Int. J. Numer. Methods Fluids, 20, 177-203 (2007) · Zbl 1205.76173
[34] Tu, S.; Aliabadi, S.; Patel, R.; Watts, M., An implementation of the Spalart-Allmaras DES model in an implicit unstructured hybrid finite volume/element solver for incompressible turbulent flow, Int. J. Numer. Methods Fluids, 30, 1051-1062 (2009) · Zbl 1259.76013
[35] Wan, T.; Aliabadi, S.; Bigler, C., A hybrid scheme based on finite element/volume methods for two immiscible fluid flows, Int. J. Numer. Methods Fluids, 61, 930-944 (2009) · Zbl 1252.76049
[36] Flanagan, D. P.; Belytschko, T., A uniform strain hexahedron and quadrilateral with orthogonal hourglass control, Int. J. Numer. Methods Eng., 17, 679-706 (1981) · Zbl 0478.73049
[37] Christon, M. A., Hydra-TH theory manual (September 2011), Los Alamos National Laboratory, Tech. rep. LA-UR 11-05387
[38] Christon, M. A.; Martinez, M. J.; Voth, T. E., Generalized Fourier analysis of the advection-diffusion equation - part I: one-dimensional domains, Int. J. Numer. Methods Fluids, 45, 839-887 (2004) · Zbl 1085.76054
[39] Voth, T. E.; Martinez, M. J.; Christon, M. A., Generalized Fourier analysis of the advection-diffusion equation - part II: two-dimensional domains, Int. J. Numer. Methods Fluids, 45, 889-920 (2004) · Zbl 1085.76059
[40] Christon, M. A.; Ketcheson, D. I.; Robinson, A. C., An assessment of semi-discrete central schemes for hyperbolic conservation laws, SAND2003-3238 (May 2003), Sandia National Laboratories: Sandia National Laboratories Albuquerque, New Mexico
[41] Baptista, A. M.; Adams, E. E.; Gresho, P., Benchmarks for the transport equation: the convection-diffusion equation and beyond, (Quantitative Skill Assessment for Coastal Ocean Models, vol. 47 (1995)), 241-268
[42] Yidong, X.; Wang, C.; Luo, H.; Christon, M.; Bakosi, J., Assessment of a hybrid finite element and finite volume code for turbulent incompressible flows, J. Comput. Phys., 307, 653-669 (2016) · Zbl 1351.76087
[43] Christon, M. A.; Lu, R.; Bakosi, J.; Nadiga, B.; Karoutas, Z.; Berndt, M., Large-eddy simulation, fuel rod vibration and grid-to-rod fretting, LA-UR 14-28497 (October 2014), Los Alamos National Laboratory: Los Alamos National Laboratory Los Alamos, New Mexico
[44] Erturk, E.; Dursun, B., Numerical solutions of 2-d steady incompressible flow in a driven skewed cavity, Z. Angew. Math. Mech., 87, 377-392 (2007) · Zbl 1115.76055
[45] de Vahl Davis, G.; Jones, I. P., Natural convection in a square cavity: a comparison exercise, Int. J. Numer. Methods Fluids, 3, 227-248 (1983) · Zbl 0538.76076
[46] de Vahl Davis, G., Natural convection of air in a square cavity: a bench mark numerical solution, Int. J. Numer. Methods Fluids, 3, 249-264 (1983) · Zbl 0538.76075
[47] Beavers, G. S.; Joseph, D. D., Boundary conditions at a naturally permeable wall, J. Fluid Mech., 1, 197-207 (1967)
[48] Brinkman, H. C., A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res., 1, 1, 27-34 (1949) · Zbl 0041.54204
[49] Khaled, A. R.A.; Vafai, K., The role of porous media in modeling flow and heat transfer in biological tissues, Int. J. Heat Mass Transf., 46, 26, 4989-5003 (2003) · Zbl 1121.76521
[50] Nithiarasu, P.; Sujatha, K. S.; Ravindran, K.; Sundararajan, T.; Seetharamu, K. N., Non-Darcy natural convection in a hydrodynamically and thermally anisotropic porous medium, Comput. Methods Appl. Mech. Eng., 188, 1-3, 413-430 (2000) · Zbl 0980.76047
[51] Vafai, K.; Kim, S., Fluid mechanics of the interface region between a porous medium and a fluid layer - an exact solution, Int. J. Heat Fluid Flow, 11, 3, 254-256 (1990)
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.