Publication Cover
Numerical Heat Transfer, Part B: Fundamentals
An International Journal of Computation and Methodology
Volume 72, 2017 - Issue 3
145
Views
0
CrossRef citations to date
0
Altmetric
Original Articles

Optimum decomposition of the anisotropic diffusion term

, &
Pages 191-210 | Received 10 May 2017, Accepted 23 Aug 2017, Published online: 25 Sep 2017

References

  • S. V. Patankar, “A calculation procedure for two dimensional elliptic situations,” Numer. Heat Transfer, vol. 4, no. 4, pp. 409–425, 1981. DOI:10.1080/10407798108547025
  • S. V. Patankar, Numerical Heat Transfer and Fluid Flow. New York, NY: Hemisphere Publishing, 1980.
  • S. V. Patankar and D. B. Spalding, “A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows,” Int. J. Heat Mass Transfer, vol. 15, no. 10, pp. 1787–1806, 1972. DOI:10.1016/b978-0-08-030937-8.50013-1
  • F. Moukalled, L. Mangani, and M. Darwish. The Finite Volume Method in Computational Fluid Dynamics: An Advanced Introduction with OpenFOAM® and Matlab®. Springer, Switzerland, 2015.
  • G. D. Raithby and G. E. Schneider, “Numerical solution of problems in incompressible fluid flow: treatment of the velocity-pressure coupling,” Numer. Heat Transfer, vol. 2, no. 4, pp. 417–440, 1979. DOI:10.1080/10407797908547089
  • S. V. Patankar, “Numerical prediction of three-dimensional flow,” In Studies in Convection: Theory, Measurement, and Application, vol. 1, B. E. Launder Ed. New York: Academic, 1975, pp. 1–9.
  • C. M. Rhie and W. L. Chow, “Numerical study of the turbulent flow past an airfoil with trailing edge separation,” AIAA J., vol. 21, pp. 1525–1532, 1983. DOI:10.2514/3.8284
  • B. P. Leonard, “A stable and accurate convective modelling procedure based on quadratic upstream interpolation,” Comput. Methods Appl. Mech. Eng., vol. 19, pp. 59–98, 1979. DOI:10.1016/0045-7825(79)90034-3
  • P. H. Gaskell and A. K. C. Lau, “Curvature compensated convective transport: SMART, a new boundedness preserving transport algorithm,” Int. J. Numer. Methods Fluids, vol. 8, no. 6, pp. 617–641, 1988. DOI:10.1002/fld.1650080602
  • E. A. Fromm, “A method for reducing dispersion in convective difference schemes,” J. Comput. Phys., vol. 3, pp. 176–189, 1968. DOI:10.1016/0021-9991(68)90015-6
  • B. P. Leonard and H. S. Niknaffs, “Sharp monotonic resolution of discontinuities without clipping of narrow extrema,” Comput. Fluids, vol. 19, pp. 141–154, 1991. DOI:10.1016/0045-7930(91)90011-6
  • B. Van Leer, “Towards the ultimate conservation difference scheme V. A second order sequel to Godunov’s method,” J. Comput. Phys., vol. 23, pp. 101–136, 1977. DOI:10.1016/0021-9991(79)90145-1
  • J. Zhu and W. Rodi, “A low-dispersion and bounded convection scheme,” Comput. Methods Appl. Mechan. Eng., vol. 92, pp. 87–96, 1991. DOI:10.1016/0045-7825(91)90199-g
  • H. C. Yee, R. F. Warming, and A. Harten, Implicit total variation diminishing (TVD) schemes for steady-state calculations, NASA Technical Memorandum 84832, 1983.
  • P. K. Sweby, “High resolution schemes using flux limiters for hyperbolic conservation Laws,” SIAM J. Numer. Anal., vol. 21, no. 5, pp. 995–1011, 1984. DOI:10.1137/0721062
  • B. P. Leonard, “Simple high-accuracy resolution program for convective modelling of discontinuities,” Int. J. Numer. Methods Eng., vol. 8, pp. 1291–1318, 1988. DOI:10.1002/fld.1650081013
  • M. Darwish, “A new high resolution scheme based on the normalized variable formulation,” Numer. Heat Transfer, Part B, vol. 24, pp. 353–371, 1993. DOI:10.1080/10407799308955898
  • M. Darwish and F. Moukalled, “Normalized variable and space formulation methodology for high-resolution schemes,” Numer. Heat Transfer, Part B, vol. 26, no. 1, pp. 79–96, 1994. DOI:10.1080/10407799408914918
  • F. Moukalled and M. Darwish, “New family of adaptive very high resolution schemes,” Numer. Heat Transfer, Part B, vol. 34, pp. 215–239, 1998. DOI:10.1080/10407799808915055
  • M. Darwish and F. Moukalled, “The schemes: a new consistent high-resolution formulation based on the normalized variable methodology,” Comput. Methods Appl. Mechan. Eng., vol. 192, pp. 1711–1730, 2003. DOI:10.1016/s0045-7825(03)00199-3
  • M. Darwish and F. Moukalled, “TVD schemes for unstructured grids,” Int. J. Heat Mass Transfer, vol. 46, no. 4, pp. 599–611, 2003. DOI:10.1016/s0017-9310(02)00330-7
  • L. Mangani, L. Buchmayr, and M. Darwish, “Development of a novel fully coupled solver in openfoam: steady-state incompressible turbulent flows,” Numer. Heat Transfer, Part B, vol. 66, no. 1, pp. 1–20, 2014. DOI:10.1080/10407790.2014.894448
  • S.-K. Choi, S.-O. Kim, T.-H., Lee, and D. H. Dohee, “Computation of the natural convection of nanofluid in a square cavity with homogeneous and nonhomogeneous models,” Numer. Heat Transfer, Part A, vol. 65, no. 4, pp. 287–301, 2014. DOI:10.1080/10407782.2013.831695
  • S.-J. Liang, Y.-J. Jan, and C.-A. Huang, “A quasi-implicit time-advancing scheme for flow in a three-dimensional curved duct,” Numer. Heat Transfer, Part B, vol. 64, no. 4, pp. 306–325, 2013. DOI:10.1080/10407790.2013.806203
  • S. Vakilipour and S. J. Ormiston, “A coupled pressure-based co-located finite-volume solution method for natural-convection flows,” Numer. Heat Transfer, Part B, vol. 61, no. 2, pp. 91–115, 2012. DOI:10.1080/10407790.2012.642281
  • S. Das, S. R. Mathur, and J. Y. Murthy, “Finite-volume method for creep analysis of thin RF MEMS devices using the theory of plates,” Numer. Heat Transfer, Part B, vol. 61, no. 2, pp. 71–90, 2012. DOI:10.1080/10407790.2012.646170
  • S. Das, S. R. Mathur, and J. Y. Murthy, “Finite-volume method for structural analysis of RF MEMS devices using the theory of plates,” Numer. Heat Transfer, Part B, vol. 61, no. 1, pp. 1–21, 2012. DOI:10.1080/10407790.2011.630949
  • S.-K. Choi and C.-L. Lin, “A simple finite-volume formulation of the lattice Boltzmann method for laminar and turbulent flows,” Numer. Heat Transfer, Part B, vol. 58, no. 4, pp. 242–261, 2010. DOI:10.1080/10407790.2010.511965
  • H. M. Matos, M. A. Alves, and P. J. Oliveira, “New formulation for stress calculation: Application to viscoelastic flow in a T-junction,” Numer. Heat Transfer, Part B, vol. 56, no. 5, pp. 351–371, 2009. DOI:10.1080/10407790903507972
  • A. Dalal, V. Eswaran, and G. Biswas, “A finite-volume method for Navier-stokes equations on unstructured meshes,” Numer. Heat Transfer, Part B, vol. 54, no. 3, pp. 238–259, 2008. DOI:10.1080/10407790802182653
  • V. C. Mariani and A. T. Prata, “A Eulerian-Lagrangian method applied to fluid flow in lid-driven cavities with irregular bottom walls,” Numer. Heat Transfer, Part B, vol. 53, no. 3, pp. 206–233, 2008. DOI:10.1080/10407790701632568
  • C. D. Perez-Segarra, C. Farre, J. Cadafalch, and A. Oliva, “Analysis of different numerical schemes for the resolution of convection-diffusion equations using finite-volume methods on three-dimensional unstructured grids. Part I: discretization schemes,” Numer. Heat Transfer, Part B, vol. 49, no. 4, pp. 333–350, 2006. DOI:10.1080/10407790500314947
  • W. J. Ferguson and I. W. Turner. “Study of two-dimensional cell-centered and vertex-centered control-volume schemes applied to high-temperature timber drying,” Numer. Heat Transfer, Part B, vol. 27, no. 4, pp. 393–415, 1995. DOI:10.1080/10407799508914964
  • I. Demirdzic, “A finite volume method for computation of fluid flow in complex geometries,” Ph.D. thesis, University of London, London, UK, 1982.
  • S. Muzaferija, “Adaptive finite volume method for flow predictions using unstructured meshes and multigrid approach,” Ph.D. thesis, University of London, London, UK, 1994.
  • I. Demirdzic and S. Muzaferija, “Numerical method for coupled fluid flow, heat transfer and stress analysis using unstructured moving meshes with cells of arbitrary topology,” Comput. Methods Appl. Mech. Eng., vol. 125, pp. 235–255, 1995.
  • S. R. Mathur and J. Y. Murthy, “A pressure-based method for unstructured meshes,” Numer. Heat Transfer, Part B, vol. 31, pp. 195–215, 1997. DOI:10.1080/10407799708915105
  • H. Jasak, “Error analysis and estimation for the finite volume method with applications to fluid flow,” Ph.D. thesis, Imperial College, London, 1996.
  • I. Demirdzic, “On the discretization of the diffusion term in finite-volume continuum mechanics,” Numer. Heat Transfer, Part B, vol. 68, no. 1, pp. 1–10, 2015.
  • M. Darwish and F. Moukalled, “A compact procedure for discretization of the anisotropic diffusion operator,” Numer. Heat Transfer, Part B, vol. 55, no. 5, pp. 339–360, 2009. DOI:10.1080/10407790902816747
  • S. G. Rubin and P. K. Khosla, “Polynomial interpolation method for viscous flow calculations,” J. Comput. Phys., vol. 27, pp. 153–168, 1982. DOI:10.1016/0021-9991(77)90036-5
  • I. Aavatsmark, “An introduction to multipoint flux approximations for quadrilateral grids,” Comput. Geosci., vol. 6, pp. 405–432, 2002.
  • I. Aavatsmark, G. Eigestad, B. Mallison, and J. Nordbotten, “A compact multipoint flux approximation method with improved robustness,” Numer. Methods Partial Differ. Equations, vol. 24, no. 5, pp. 1329–1360, 2008. DOI:10.1002/num.20320
  • C. LePotier, “Schema volumes finis monotone pour des operateurs de diffusion fortement anisotropes sur des maillages de triangle non structures,” Comptes Rendus Mathématiques, vol. 341, pp. 787–792, 2005.
  • F. Brezzi, K. Lipnikov, M. Shashkov, and V. Simoncini, “A new discretization methodology for diffusion problems on generalized polyhedral meshes,” Comput. Methods Appl. Mech. Eng., vol. 196, pp. 3682–3692, 2007. DOI:10.1016/j.cma.2006.10.028
  • K. Lipnikov, D. Svyatskiy, M. Shashkov, and Y. Vassilevski, “Monotone Finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes,” J. Comput. Phys., vol. 227, pp. 492–512, 2007. DOI:10.1016/j.jcp.2007.08.008
  • K. Lipnikov, D. Svyatskiy, and Y. Vassilevski, “Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes,” J. Comput. Phys., vol. 228, pp. 703–716, 2009. DOI:10.1016/j.jcp.2008.09.031
  • K. Nikitin and Y. Vassilevski, “A monotone nonlinear finite volume method for advection–diffusion equations on unstructured polyhedral meshes in 3D,” Russ. J. Numer. Anal. Math. Modell., vol. 25, no. 4, pp. 335–358, 2010. DOI:10.1515/rjnamm.2010.022
  • G. Manzini and M. Putti, “Mesh locking effects in the finite volume solution of 2-D anisotropic diffusion equations,” J. Comput. Phys., vol. 220, pp. 751–771, 2007. DOI:10.1016/j.jcp.2006.05.026
  • I. Kapyrin, “A family of monotone methods for the numerical solution of three-dimensional diffusion problems on unstructured tetrahedral meshes,” Doklady Math., vol. 76, no. 2, pp. 734–738, 2007. DOI:10.1134/s1064562407050249
  • M. Darwish, L. Mangani, and F. Moukalled, “General fully implicit discretization of the diffusion term for the finite volume method,” Numer. Heat Transfer, Part B, vol. 71, no. 6, pp. 506–532, 2017. DOI:10.1080/10407790.2017.1330060
  • K. Lipnikov, D. Svyatskiy, and Y. Vassilevski, “A monotone finite volume method for advection-diffusion equations on unstructured polygonal meshes,” J. Comput. Phys., vol. 229, pp. 4017–4032, 2010. DOI:10.1016/j.jcp.2010.01.035
  • V. S. Borisov and S. Sorek, “On the monotonicity of difference schemes for computational physics,” SIAM J. Sci. Comput., vol. 25, pp. 1557–1584, 2004.
  • M. Keyhani and R. A. Polehn, “Finite difference modeling of anisotropic flows,” ASME J. Heat Transfer, vol. 117, pp. 458–464, 1995. DOI:10.1115/1.2822544
  • A. I. Shestakov, D. S. Kershaw, and G. B. Zimmerman, “Test problems in radiative transfer calculations,” Nucl. Sci. Eng., vol. 105, no. 1, pp. 88–104, 1990. DOI:10.13182/nse90-a19215

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.