Publication Cover
Numerical Heat Transfer, Part B: Fundamentals
An International Journal of Computation and Methodology
Volume 65, 2014 - Issue 6
203
Views
15
CrossRef citations to date
0
Altmetric
Original Articles

A High-Accurate Fourier-Galerkin Solution for Buoyancy-Driven Flow in a Square Cavity

&
Pages 495-517 | Received 29 Oct 2013, Accepted 01 Dec 2013, Published online: 09 May 2014

REFERENCES

  • F. Arpino , N. Massarotti , and A. Mauro , High Rayleigh Number Laminar-Free Convection in Cavities: New Benchmark Solutions , Numer. Heat Transfer B , vol. 58 , pp. 73 – 97 , 2010 .
  • G. de Vahl Davis , Natural Convection of Air in a Square Cavity: A Bench Mark Numerical Solution, Int. J. Numer. Meth. Fluids , vol. 3, pp. 227–248, 1983.
  • M. E. Newell and F. W. Schmidt , Heat Transfer by Laminar Natural Convection within Rectangular Enclosures , ASME J. Heat Transfer , vol. 92 , pp. 159 – 168 , 1970 .
  • Y. Guo and K. J. Bathe , A Numerical Study of a Natural Convection Flow in a Cavity , Int. J. Numer. Meth. Fluids , vol. 40 , pp. 1045 – 1057 , 2002 .
  • W. Wu and C. Y. Ching , The Effect of the Top Wall Temperature on the Laminar Natural Convection in Rectangular Cavities with Different Aspect Ratios , J. Heat Transfer , vol. 131 , pp. 1 – 11 , 2009 .
  • A. Campo , J. Y. Chang , and E. H. Ridouane , Heat Transfer Comparison between a Vertical Rectangular Cavity and Isosceles Right-Angled Triangular Cavity of Equal Cross Sectional Area , Thermal Sci. , vol. 15 , pp. 357 – 365 , 2011 .
  • M. A. H. Mamun , T. R. Tanim , M. M. Rahman , R. Saidur , and S. Nagata , Analysis of Mixed Convection in a Lid Driven Trapezoidal Cavity , in A. Ahsan (ed.), Convection and Conduction Heat Transfer , chap. 3, InTech Open Access Publisher, Rijeka, Croatia , 2011 .
  • M. Hasnaoui , P. Vasseur , and E. Bilgen , Natural Convection Heat Transfer in Inclined Tall Cavities Bounded by Porous Layer , Int. J. Numer. Meth. Fluids , vol. 3 , pp. 91 – 105 , 1993 .
  • O. Polat and E. Bilgen , Laminar Natural Convection in Inclined Open Shallow Cavities , Int. J. Thermal. Sci , vol. 41 , pp. 360 – 368 , 2002 .
  • K. Sundaravadivelu and P. Kandaswamy , Natural Convection of Water in an Inclined Cavity with Heat Generation , J. Appl. Math. Comput. , vol. 12 , pp. 281 – 289 , 2003 .
  • O. Polat and E. Bilgen , Conjugate Heat Transfer in Inclined Open Shallow Cavities , Int. J. Heat Mass Transfer , vol. 46 , pp. 1563 – 1573 , 2003 .
  • E. Bilgen and H. Oztop , Natural Convection Heat Transfer in Partially Open Inclined Square Cavities , Int. J. Heat Mass Transfer , vol. 48 , pp. 1470 – 1479 , 2005 .
  • E. Tuliszka-Sznitko and W. Majchrowski , LES and DNS of the Flow with Heat Transfer in Rotating Cavity , Comput. Meth. Sci. Tech. , vol. 16 , pp. 105 – 114 , 2010 .
  • S. Biringen and G. Danabasoglu , Oscillatory Flow with Heat Transfer in a Square Cavity , Phys. Fluids A vol. 1 , pp. 1796 – 1812 , 1989 .
  • K. Vafai and J. Ettefagh , Thermal and Fluid Flow Instabilities in Buoyancy-Driven Flows in Open-Ended Cavities , Int. J. Heat Mass Transfer , vol. 33 , pp. 2329 – 2344 , 1990 .
  • Z. Alloui , P. Vasseur , and M. Reggio , Natural Convection of Nanofluids in a Shallow Cavity Heated from Below , Int. J. Thermal Sci. , vol. 50 , pp. 385 – 393 , 2011 .
  • S. Saha , G. Saha , and N. Hasan , Mixed Convection in a Lid-Driven Cavity with Internal Heat Source, Proc. 13th Annual Paper Meet, Dhaka , 2010 .
  • A. Namprai and S. Witayangkurn , Fluid Flow and Heat Transfer in Square Cavities with Discrete Two Source-Sink Pairs , Adv. Stud. Theor. Phys. , vol. 6 , pp. 743 – 753 , 2012 .
  • H. Ambarita , K. Kishinami , M. Daimaruya , T. Saitoh , H. Takahashi , and J. Suzuki , Laminar Natural Convection Heat Transfer in Air Filled Square Cavity with Two Insulated Baffles Attached to Its Horizontal Walls , Thermal Sci. Eng. , vol. 14 , pp. 35 – 46 , 2006 .
  • R. F. Brito , H. S. Alencar , L. O. Rodrigues , G. J. Menon , and M. A. R. Ascimento , Numerical Simulation of Fluid Flow in a Cubic Cavity with a Four-Finned Dissipator Placed on the Bottom Surface , Thermal Eng. , vol. 6 , pp. 54 – 61 , 2007 .
  • P. Kandaswamy , J. Lee , and A. K. Abdul Hakeem , Natural Convection in a Square Cavity in the Presence of Heated Plate , Nonlinear Anal. Model. Control , vol. 12 , pp. 203 – 212 , 2007 .
  • P. Wang and P. Li , Parallel Computation and Visualization of 3D, Time-Dependent, Thermal Convective Flows, Proc. 4th Applied Parallel Computing, Large Scale Scientific and Industrial Problems, PARA, Umea, Sweden , 1998 .
  • P. Rauwoens , J. Vierendeels , and B. Merci , Numerical Study of the Flow in a Three-Dimensional Thermally Driven Cavity, J. Comput. Appl. Math. , vol. 215, pp. 538–546, 2008.
  • P. Le Quéré and T. Alziary de Roquefort , Computation of Natural Convection in Two-Dimensional Cavities with Chebyshev Polynomials , J. Comput. Phys. , vol. 57 , pp. 210 – 228 , 1985 .
  • P. Le Quéré , Accurate Solutions to the Square Thermally Driven Cavity at High Rayleigh Number , Comput. Fluids , vol. 20 , pp. 29 – 42 , 1991 .
  • R. J. A. Janssen and R. A. W. M. Henkes , Accuracy of Finite-Volume Discretizations for the Bifurcating Natural-Convection Flow in a Square Cavity , Numer. Heat Transfer B , vol. 24 , pp. 191 – 207 , 1993 .
  • P. D. Minev , F. N. van de Vosse , L. J. P. Timmermans , and A. A. van Steenhoven , A Second Order Splitting Algorithm for Thermally-Driven Flow Problems , Int. J. Numer. Meth. Heat Fluid Flow , vol. 6 , pp. 51 – 60 , 1995 .
  • D. A. Mayne , A. S. Usmani , and M. Crapper , h-Adaptive Finite Element Solution of High Rayleigh Number Thermally Driven Cavity Problem , Int. J. Numer. Meth. Heat Fluid Flow , vol. 10 , pp. 598 – 615 , 2000 .
  • D. C. Wan , B. S. V. Patnaik , and G. W. Wei , A New Benchmark Quality Solution for the Buoyancy-Driven Cavity by Discrete Singular Convolution , Numer. Heat Transfer B , vol. 40 , pp. 199 – 228 , 2001 .
  • D. A. Mayne , A. S. Usmani , and M. Crapper , h-Adaptive Finite Element Solution of Unsteady Thermally Driven Cavity Problem , Int. J. Numer. Meth. Heat Fluid Flow , vol. 11 , pp. 172 – 194 , 2001 .
  • J. Vierendeels , B. Merci , and E. Dick , Benchmark Solutions for the Natural Convective Heat Transfer Problem in a Square Cavity with Large Horizontal Temperature Differences , Int. J. Numer. Meth. Heat Fluid Flow , vol. 13 , pp. 1057 – 1078 , 2003 .
  • H. N. Dixit and V. Babu , Simulation of High Rayleigh Number Natural Convection in a Square Cavity Using the Lattice Boltzmann Method , Int. J. Heat Mass Transfer , vol. 49 , pp. 727 – 739 , 2006 .
  • N. B. Cheikh , B. B. Beya , and T. Lili , Benchmark Solution for Time-Dependent Natural Convection Flows with an Accelerated Full-Multigrid Method , Numer. Heat Transfer B , vol. 52 , pp. 131 – 151 , 2007 .
  • A. Younes , A. Makradi , A. Zidane , Q. Shao , and L. Bouhala , A Combination of Crouzeix-Raviart, Discontinuous Galerkin and MPFA Methods for Buoyancy-Driven Flows , Int. J. Numer. Meth. Heat Fluid Flow , in press , 2013 .
  • H. R. Henry , Effects of Dispersion on Salt Encroachment in Coastal Aquifers, in Sea Water in Coastal Aquifers , U. S. Geol. Surv. Supply Papers , vol. 1613 , pp. 70 – 84 , 1964 .
  • G. Segol , Classic Groundwater Simulations Proving and Improving Numerical Models , Prentice-Hall , Old Tappan , N.J. , 1994 .
  • M. Christou and C. I. Christov , Fourier–Galerkin Method for Localized Solutions of Equations with Cubic Nonlinearity , J. Comput. Anal. Appl. , vol. 4 , pp. 63 – 77 , 2002 .
  • M. A. Christou and C. I. Christov , Fourier–Galerkin Method for Interacting Localized Waves , J. Neural Parallel Sci. Comput. , vol. 4 , pp. 63 – 77 , 2002 .
  • M. A. Christou and C. I. Christov , Localized Waves for the Regularized Long Wave Equation via a Galerkin Spectral Method , J. Math. Comp. Simul. , vol. 69 , pp. 257 – 268 , 2005 .
  • M. Christou and C. I. Christovb , Fourier–Galerkin Method for 2D Solutions of Boussinesq Equation , Math. Comput. Simul. , vol. 74 , pp. 82 – 92 , 2007 .
  • A. Zidane , A. Younes , P. Huggenberger , and E. Zechner , The Henry Semi-analytical Solution for Saltwater Intrusion with Reduced Dispersion , Water Resources Res. , vol. 48 , pp. 1 – 10 , 2012 .
  • C. Canuto , M. Hussaini , A. Quarteroni , and T. Zang , Spectral Methods: Fundamentals in Single Domains , chap. 3, Springer-Verlag , New York , 2006.
  • B. Costa , Spectral Methods for Partial Differential Equations , Cubo Math. J. , vol. 6 , pp. 1 – 32 , 2004 .
  • D. A. Kopriva , Implementing Spectral Methods for Partial Differential Equations , chap. 4 , Springer Science , New York , 2009 .
  • D. Durran , Numerical Methods for Wave Equations in Geophysical Fluid Dynamics , chap. 4 , Springer-Verlag , New York , 1999 .
  • M. A. Christou and C. I. Christov , Interacting Localized Waves for the Regularized Long Wave Equation via a Galerkin Spectral Method , Math. Comput. Simul. , vol. 69 , pp. 257 – 268 , 2005 .
  • M. J. D. Powell , An Efficient Method for Finding the Minimum of a Function of Several Variables without Calculating Derivatives , Comput. J. , vol. 7 , pp. 155 – 162 , 1964 .
  • M. J. D. Powell , On the Calculations of Orthogonal Vectors , Comput. J. , vol. 1 , pp. 302 – 304 , 1968 .
  • W. H. Press , S. A. Teukolsky , W. T. Vetterling , and B. P. Flannery , Numerical Recipes: The Art of Scientific Computing, , 3rd ed. , chaps. 9–10 , Cambridge University Press , New York , 2007 .
  • I. J. Rao and K. R. Rajagopai , The Effect of the Slip Boundary Condition on the Flow of Fluids in a Channel , Acta Mechanica , vol. 135 , pp. 113 – 126 , 1999 .
  • M. Behr , On the Application of Slip Boundary Condition on Curved Boundaries , Int. J. Numer. Meth. Fluids , vol. 45 , pp. 43 – 51 , 2004 .
  • Y. D. Shikhmurzaev , Singularities at the Moving Contact Line: Mathematical, Physical and Computational Aspects . Physica D , vol. 217 , pp. 121 – 133 , 2006 .
  • L. Bocquet and J. L. Barrat , Flow Boundary Conditions from Nano- to Micro-Scales , Soft Matter , vol. 3 , pp. 685 – 693 , 2007 .
  • H. Qiaolin and W. Xiao-Ping , Numerical Study of the Effect of Navier Slip on the Driven Cavity Flow , ZAMM J. Appl. Math. Mech. , vol. 89 , pp. 857 – 868 , 2009 .
  • Y. Stokes and G. Carey , On Generalized Penalty Approaches for Slip, Free Surface and Related Boundary Conditions in Viscous Flow Simulation , Int. J. Numer. Meth. Heat Fluid Flow , vol. 21 , pp. 668 – 702 , 2011 .
  • L. Wang , Z. Xin , and A. Zang , Vanishing Viscous Limits for 3D Navier–Stokes Equations with a Navier-Slip Boundary Condition , J. Math. Fluid Mech. , vol. 14 , pp. 791 – 825 , 2012 .
  • C. G. Broyden , A Class of Methods for Solving Nonlinear Simultaneous Equations , Math. Comput. , vol. 19 , pp. 577 – 593 , 1965 .
  • M. Crouzeix and P. Raviart , Conforming and Nonconforming Finite Element Methods for Solving the Stationary Stokes Equations , RAIRO Sér. Rouge , vol. 7 , pp. 33 – 75 , 1973 .
  • V. Girault and P. A. Raviart , Finite Element Methods for Navier-Stokes Equations , chap. 4 , Springer-Verlag , Berlin , 1986 .
  • F. Brezzi and M. Fortin , Mixed and Hybrid Finite Element Methods , chap. 3 , Springer-Verlag , Berlin , 1991 .
  • P. M. Gresho and R. L. Sani , Incompressible Flow and the Finite Element Method , vol. 2 , chap. 3 , John Wiley , New York , 1998 .
  • D. N. Arnold , On Nonconforming Linear-Constant Elements for Some Variants of the Stokes Equations , Istit. Lombardo Accad. Sci. Lett. Rend. A , pp. 83 – 93 , 1993 .
  • F. Schieweck and L. Tobiska , A Nonconforming Finite Element Method of Upstream Type Applied to the Stationary Navier-Stokes Equation , Modél. Math. Anal. Numér. , vol. 23 , pp. 627 – 647 , 1989 .
  • K. Djadel and S. Nicaise , A Non-conforming Finite Volume Element Method of Weighted Upstream Type for the Two-Dimensional Stationary Navier–Stokes System, Appl. Numer. Math. , vol. 58, pp. 615–634, 2008.
  • B. Cockburn , S. Y. Lin , and C. W. Shu , TVB Runge Kutta Local Projection Discontinuous Galerkin Finite Element Method for Conservative Laws III: One Dimensional Systems , J. Comput. Phys. , vol. 84 , pp. 90 – 113 , 1989 .
  • D. N. Arnold , F. Brezzi , B. Cockburn , and L. D. Marini , Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , SIAM J. Numer. Anal. , vol. 39 , pp. 1749 – 1779 , 2002 .
  • R. Kirby , A Posteriori Error Estimates and Local Time-Stepping for Flow and Transport Problems in Porous Media, Ph.D. thesis, The University of Texas at Austin , 2000 .
  • A. Younes , M. Fahs , and P. Ackerer , An Efficient Geometric Approach to Solve the Slope Limiting Problem with the Discontinuous Galerkin Method on Unstructured Triangles . Int. J. Numer. Meth. Bio. Eng. , vol. 26 , pp. 1824 – 1835 , 2010 .
  • I. Aavatsmark , T. Barkve , Ø. Bøe , and T. Mannseth , Discretization on Non-orthogonal, Quadrilateral Grids for Inhomogeneous, Anisotropic Media , J. Comput. Phys. , vol. 127 , pp. 2 – 14 , 1996 .
  • A. Younes and V. Fontaine , Efficiency of Mixed Hybrid Finite Element and Multi Point Flux Approximation Methods on Quadrangular Grids and Highly Anisotropic Media , Int. J. Numer. Meth. Eng. , vol. 76 , pp. 314 – 336 , 2008 .
  • A. Younes and P. Ackerer , Solving the Advection-Dispersion Equation with Discontinuous Galerkin and Multipoint Flux Approximation Methods on Unstructured Meshes , Int. J. Numer. Meth. Fluids , vol. 58 , pp. 687 – 708 , 2008 .
  • A. Younes and P. Ackerer , Empirical versus Time Stepping with Embedded Error Control for Density-Driven Flow in Porous Media , Water Resources Res. , vol. 46 , pp. 1 – 8 , 2010 .

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.