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Velocity-Vorticity Formulation for 2D Natural Convection in an Inclined Cavity by the DQ Method

Published online by Cambridge University Press:  05 May 2011

D. C. Lo ,
T. Liao ,
D. L. Young  and
M. H. Gou
D. C. Lo*
Affiliation:
Institute of Navigation Science and Technology, National Kaohsiung Marine University, Kaohsiung, Taiwan 81157, R.O.C.
T. Liao*
Affiliation:
Department of Shipping Technology, National Kaohsiung Marine University, Kaohsiung, Taiwan 81157, R.O.C.
D. L. Young*
Affiliation:
Department of Civil Engineering and Hydrotech Research Institute, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
M. H. Gou*
Affiliation:
Department of Civil Engineering and Hydrotech Research Institute, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
*
*Assistant Professor
**Associated Professor
***Professor
****Ph.D. student
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Abstract

The aim of this paper attempts to apply the differential quadrature (DQ) method for solving two-dimensional natural convection in an inclined cavity. The velocity-vorticity formulation is used to represent the mass, momentum, and energy conservations of the fluid medium in an inclined cavity. We employ a coupled technique for four field variables involving two velocities, one vorticity and one temperature components. In this method, the velocity Poisson equation, continuity equation, vorticity transport equation and energy equation are all solved as a coupled system of equations so as to we are capable of predicting four field variables accurately. The main advantage of present approach is that coupling the velocity and the vorticity equations allows the determination of the boundary values implicitly without requiring the explicit specification of the vorticity values at the boundary walls. A natural convection in a cavity with different angle of inclinations for Rayleigh number equal to 103, 104, 105 and 106 and H/L aspect ratios varying from 1 to 3 is investigated. It is shown that with the use of the present algorithm the benchmark results for temperature and flow fields could be obtained using a coarse mesh grid.

Keywords

Differential quadrature (DQ)Velocity-vorticity formulationRayleigh number.
Type
Articles
Information
Journal of Mechanics , Volume 23 , Issue 3 , September 2007 , pp. 261 - 268
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2007

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References

1.Marshall, R., Heinrich, J. and Zienkiewicz, O., “Natural Convection in a Square Enclosure by a Finite-Element, Penalty Function Method using Primitive Fluid Variables,” Numerical Heat Transfer, 1, pp. 331–349 (1978).CrossRefGoogle Scholar
2.de Vahl Davis, G., “Natural Convection of Air in a Square Cavity, a Bench Mark Numerical Solution,” Int. J. Numerical Methods in Fluids, 3, pp. 249–264 (1983).CrossRefGoogle Scholar
3.Ismail, K. A. R. and Scalon, V. L., “A Finite Element Free Convection Model for the Side Wall Heated Cavity,” Int. J. Heat and Mass Transfer, 43, pp. 1373–1389 (2000).CrossRefGoogle Scholar
4.Fasel, H., “Investigation of the Stability of Boundary Layers by a Finite-Difference Model of the Navier-Stokes Equations,” J. Fluid Mechanics, 78, pp. 355–383 (1976).CrossRefGoogle Scholar
5.Daube, O., “Resolution of the 2D Navier-Stokes Equations in Velocity-Vorticity Form by Means of an Influence Matrix Technique,” J. Computational Physics, 103, pp. 402–414 (1992).CrossRefGoogle Scholar
6.Guj, G. and Stella, F., “A Vorticity-Velocity Method for the Numerical Solution of 3D Incompressible Flows,” J. Computational Physics, 106, pp. 286–298 (1993).CrossRefGoogle Scholar
7.Lo, D. C. and Young, D. L., “Two-Dimensional Incompressible Flows by Velocity-Vorticity Formulation and Finite Element Method,” The Chinese Journal of Mechanics—Series A, 17, pp. 13–20 (2001).Google Scholar
8.Lo, D. C. and Young, D. L., “Arbitrary Lagrangian-Eulerian Finite Element Analysis of Free Surface Flow Using a Velocity-Vorticity Formulation,” J. Computational Physics, 195, pp. 175–201 (2004).CrossRefGoogle Scholar
9.Eldho, T. I. and Young, D. L., “Two-Dimensional Incompressible Viscous Flow Simulation Using Velocity-Vorticity Dual Reciprocity Boundary Element Method,” Journal of Mechanics, 20, pp. 177–185 (2004).CrossRefGoogle Scholar
10.Lo, D. C. and Murugesan, K. and Young, D. L., “Numerical Solution of Three-Dimensional Velocity-Vorticity Navier-Stokes Equations by Finite Difference Method,” Int. J. Numerical Methods in Fluids, 47, pp. 1469–1487 (2005).CrossRefGoogle Scholar
11.Bellman, R. E., Kashef, B. G. and Casti, J., “Differential Quadrature: a Technique for the Rapid Solution of Nonlinear Partial Differential Equations,” J. Computational Physics, 10, pp. 40–52 (1972).CrossRefGoogle Scholar
12.Shu, C. and Richards, B. E., “Application of Generalized Differential Quadrature to Solve 2-Dimensional Incompressible Navier-Stokes Equations,” Int. J. Numerical Methods in Fluids, 15, pp. 791–798 (1992).CrossRefGoogle Scholar
13.Shu, C., Differential Quadrature and Its Application Engineering, Springer, London (2000)CrossRefGoogle Scholar
14.Lo, D. C., Young, D. L. and Murugesan, K., “GDQ Method for Natural Convection in a Square Cavity using a Velocity-Vorticity Formulation,” Numerical Heat Transfer B, 47, pp. 321–341 (2005).CrossRefGoogle Scholar
15.Lo, D. C., Young, D. L. and Murugesan, K., “GDQ Method for Natural Convection in a cubic Cavity using a Velocity-Vorticity Formulation,” Numerical Heat Transfer B, 48, pp. 363–386 (2005).CrossRefGoogle Scholar
16.Wang, L. W., Kung, Y. C., Wu, C. Y., Kang, M. F. and Wang, S. L., “Thermosolutal Convection in an Inclined Rectangular Enclosure with a Partition,” Journal of Mechanics, 20, pp. 233–239 (2004).CrossRefGoogle Scholar
17.Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P., Numerical recipes in Fortran 90, 2nd Ed., Cambridge University Press, New York (1996).Google Scholar