A number of two dimensional staggered unstructured discretisation schemes for the solution of fluid flow and heat transfer problems have been developed. All schemes store and solve velocity vector components at cell faces with scalar variables solved at cell centres. The velocity is resolved into face-normal and face-parallel components and the various schemes investigated differ in the treatment of the parallel component. Steady-state and time-dependent fluid flow and thermal energy equations are solved with the well known pressure correction scheme, SIMPLE, employed to couple continuity and momentum. The numerical methods developed are tested on well known benchmark cases: the Lid-Driven Cavity, Natural Convection in a Cavity and Melting of Gallium in a rectangular domain. The results obtained are shown to be comparable to benchmark, but with accuracy dependent on scheme selection.
CrossMPericleousKCroftTNMcBrideDLawrenceLAWilliamsA, Computational Modelling of Mold Filling and Related Free Surface Flows in Shape Casting, Metal. Mater. Trans. B, 37B: 879–885, 2006.
2.
FarihaS, A Computational Fluid Dynamics Study of Air Flow Through Human Nasal Cavity, MSc Thesis, University of Greenwich, London, 2007.
3.
HarlowFHWelchJE. Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface, The Physics of Fluids, 8: 2182–2189, 1965.
4.
PatankarSV, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, 1980.
5.
RhieCMChowWL. Numerical study of the turbulent flow past an airfoil with training edge separation, AIAA Journal, 21: 1525–1532, 1983.
CroftTN, Unstructured mesh Finite Volume Algorithms for Swirling, Turbulent, Reacting Flows, PhD Thesis, University of Greenwich, 1998.
8.
WittPJ, Development of a Numerical Model of the Flow Processes in a Fluidised Bed, PhD thesis, Swinburne University of Technology, 1997.
9.
PatankarSVSpaldingDB. A calculation procedure for heat and mass transfer in three-dimensional parabolic flows, Int. J Heat and Mass Transfer, 15: 1787–1806, 1972.
10.
Chow PMY.Control volume unstructured mesh procedure for Convection-Diffusion Solidification Process, PhD Thesis, University of Greenwich, London, 1993.
NicolaidesRA. Direct discretisation of planar div-curl problems, ICASE Report, 89–76, 1989.
13.
NicolaidesRA. The covolume approach to computing incompressible flow, Incompressible Computational Fluid Dynamics, GunzburgerM. D. & NicolaidesR. A., eds., Cambridge University Press, 295–234, 1993.
14.
NicolaidesRAWuX.Covolume solutions of three-dimensional div-curl equations, ICASE Report, 95–4, 1995.
15.
HallCAPetersonJSPorschingTASledgeFR. The dual variable method for finite element discretisations of Navierl/Stokes equations, Int. J. Num. Meth. Eng., 21: 883–898, 1985.
16.
HwangYH. Calculations of incompressible flow on a staggered triangular grid, part I: Mathematical formulation, Num. Heat Transfer B, 27: 323–336, 1995.
17.
HwangYH. Calculations of incompressible flow on a staggered triangular grid, part II: Mathematical formulation, Num. Heat Transfer B, 27: 337–353, 1995.
18.
KobayashiMHPereiraJMCPereiraJCF. A conservative finite-volume second-order accurate projection method on hybrid unstructured grids, J. Comp. Physics, 150: 40–75, 1999.
19.
RidaSMcKentyFMengFLReggioM.A staggered control volume scheme for unstructured triangular grids, Int. J. Num. Meth. Fluids, 25: 697–717, 1997.
20.
PerotJB. Conservation properties of unstructured staggered mesh schemes, J. Comp. Physics, 159: 58–89, 2000.
21.
ZhangXSchmidtDPerotJB. Accuracy and conservation properties of a three-dimensional unstructured staggered mesh scheme for Fluid Dynamics, J. Comp. Physics, 176: 1–28, 2002.
22.
WennekerISegalAWesselingP.Computation of compressible flows on unstructured staggered grids, Proceedings ECCOMAS 2000, Barcelona, 11–14 Sept. 2000.
23.
WennekerI.Computation of flows using unstructured staggered grids, PhD thesis, Delft University of Technology, the Netherlands, 2002.
24.
WennekerISegalA.WesselingP.Conservation properties of a new unstructured staggered scheme, Computers & Fluids, 32: 139–147, 2003.
van der VorstHA. Bi-CGSTAB: A fast and smoothly convergent variant of BiCG for solution of non-symmetric linear systems, SIAM J Sci. Stat. Comp., 13: 631–644, 1992.
28.
SaadYSchultzM.GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems, SIAM J. Sci. Statist. Comput., 7: 856–869, 1986.
29.
SleijpenGLGvan der VorstHA. Maintaining convergence properties of BiCGSTAB methods in finite precision arithmetic, Numerical Algorithms, 10: 203–223, 1995.
30.
GhiaUGhiaKNShinCT. High-Re Solutions for Incompressible Flow Using the Navier-Stokes Equations and a Multigrid Method, Journal of Computational Physics, 48: 387–411, 1982.
31.
JonesIP. A comparison Problem for Numerical Methods in Fluid Dynamics, The ‘Double-Glazing’ Problem, Numerical Methods in Thermal Problems, Proceedings of the First International Conference, 338–348, 1979.
32.
de Vahl DavisGJonesIP. Natural Convection in a Square Cavity: A Comparison Exercise, International Journal for Numerical Methods in Fluids, 3: 227–248, 1983.
33.
MarkatosNCPericleousKA, Laminar and Turbulent Natural Convection in an Enclosed Cavity, Int. J Heat and Mass Transfer, 27(5): 755–772, 1984.
34.
VollerVRCrossM.Application of Control Volume Enthalpy Methods in the Solution of Stefan Problem, Computational Techniques in Heat transfer, Editors: LewisR. W. and MorganK., pub. Pineridge Press, 245–276, 1985.
35.
VollerVRCrossMMarkatosNC. An Enthalpy Method for Convection/Diffusion Phase Change, International Journal of Numerical Methods in Engineering, 24: 271–284, 1987.
