Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T07:24:11.988Z Has data issue: false hasContentIssue false
  • English
  • Français

Fully HOC Scheme for Mixed Convection Flow in a Lid-Driven Cavity Filled with a Nanofluid

Published online by Cambridge University Press:  03 June 2015

Dingfang Li ,
Xiaofeng Wang  and
Hui Feng
Dingfang Li*
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China
Xiaofeng Wang*
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China Department of Mathematics, Henan Institute of Science and Technology, Xinxiang 453003, Henan, China
Hui Feng*
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China
*
Email: [email protected]
*Corresponding author.Email: [email protected]
Email: [email protected]
Get access

Abstract

A fully higher-order compact (HOC) finite difference scheme on the 9-point two-dimensional (2D) stencil is formulated for solving the steady-state laminar mixed convection flow in a lid-driven inclined square enclosure filled with water-Al2O3 nanofluid. Two cases are considered depending on the direction of temperature gradient imposed (Case I, top and bottom; Case II, left and right). The developed equations are given in terms of the stream function-vorticity formulation and are non-dimensionalized and then solved numerically by a fourth-order accurate compact finite difference method. Unlike other compact solution procedure in literature for this physical configuration, the present method is fully compact and fully higher-order accurate. The fluid flow, heat transfer and heat transport characteristics were illustrated by streamlines, isotherms and averaged Nusselt number. Comparisons with previously published work are performed and found to be in excellent agreement. A parametric study is conducted and a set of graphical results is presented and discussed to elucidate that significant heat transfer enhancement can be obtained due to the presence of nanoparticles and that this is accentuated by inclination of the enclosure at moderate and large Richardson numbers.

Keywords

35K5765N0676T20HOC difference schemepseudo-time derivativemixed convectionADI method
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Issue 1 , February 2013 , pp. 55 - 77
Copyright
Copyright © Global-Science Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Choi, S.U.S., Enhancing thermal conductivity of fluids with nanoparticle, ASME FED, 231 (1995), pp. 99–105.Google Scholar
[2]Eastman, J. A., Choi, S. U. S., Li, S., Yu, W. and Thompson, L. J., Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles, Appl. Phys. Lett., 78 (2001), pp. 718–720.Google Scholar
[3]Xie, H., Wang, J., Xie, T. G., Liu, Y. and Ai, F., Thermal conductivity enhancement of suspensions containing nanosized alumina particles, J. Appl. Phys., 91 (2002), pp. 4568–4572.Google Scholar
[4]Ho, C. J., Chen, M. W. and L, Z.W., Numerical simulation of natural convection of nanofluid in a square enclosure: effects due to uncertainties of viscosity and thermal conductivity, Int. J. Heat Mass Trans., 51 (2008), pp. 4506–4516.CrossRefGoogle Scholar
[5]Abu-Nadaa, E. and Chamkha, A. J., Mixed convection flow in a lid-driven inclined square enclosure filled with a nanofluid, Euro. J. Mech., 29 (2010), pp. 472–482.Google Scholar
[6]Kumar, S., Prasad, S. K. and Banerjee, J., Analysis of flow and thermal field in nanofluid using a single phase thermal dispersion model, Appl. Math. Model., 34 (2010), pp. 573–592.CrossRefGoogle Scholar
[7]Das, S. K., Choi, S. U. S. and Patel, H. E., Heat transfer in nanofluids: a review, Heat Trans. Eng., 37 (2006), pp. 3–19.Google Scholar
[8]Wang, X. Q. and Mujumdar, A. S., Heat transfer characteristics of nanofluids: a review, Int. J. Thermal Sci., 46 (2007), pp. 1–19.CrossRefGoogle Scholar
[9]Khanafer, K., Vafai, K. and Lightstone, M., Buoyancy driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Trans., 46 (2003), pp. 3639–3653.Google Scholar
[10]Jo, R. Y. and Tzeng, S. C., Numerical research of nature convective heat transfer enhancement filled with nanofluids in rectangular enclosures, Int. Commun. Heat Mass Trans., 33 (2006), pp. 727–736.Google Scholar
[11]Imberger, J. and Hamblina, P. F., Dynamics of lakes, reservoirs and cooling ponds, Rev. Fluid Mech., 14 (1982), pp. 153–187.Google Scholar
[12]Ideriah, F. J. K., Prediction of turbulent cavity flow driven by buoyancy and shear, J. Mech. Eng Sci., 22 (1980), pp. 287–295.CrossRefGoogle Scholar
[13]Torrance, K., Davis, R., Elike, K., Gill, P., Gutman, D., Hsui, A., Lyons, S. and Zien, H., Cavity flows driven by buoyancy and shear, J. Fluid Mech., 51 (1972), pp. 221–231.Google Scholar
[14]Iwatsu, R., Hyun, J. M. and Kuwahara, K., Mixed convection in a driven cavity with a stable vertical temperature gradient, Int. J. Heat Mass Trans., 36 (1993), pp. 1601–1608.CrossRefGoogle Scholar
[15]Tiwari, R. K. and Das, M. K., Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids, Int. J. Heat Mass Trans., 50 (2007), pp. 2002–2018.CrossRefGoogle Scholar
[16]Muthtamilselvan, M., Kandaswamy, P. and Lee, J., Heat transfer enhancement of copper-water nanofluids in a lid-driven enclosure, Commun. Nonlinear Sci. Numer. Simul., 15 (2009), pp. 1501–1510.CrossRefGoogle Scholar
[17]Cheng, T. S. and Liu, W. H., Effect of temperature gradient orientation on the characteristics of mixed convection flow in a lid-driven square cavity, Comput. Fluids, 39 (2010), pp. 965–978.Google Scholar
[18]Iwatsu, R., Hyun, J. M. and Kuwahara, K., Mixed convection in a driven cavity with a stable vertical temperature gradient, Int. J. Therm. Sci., 36 (1993), pp. 1601–1608.Google Scholar
[19]Abu-Nada, E., Effects of variable viscosity and thermal conductivity of Al2O 3-water nanofluid on heat transfer enhancement in natural convection, Int. J. Heat Fluid Flow, 30 (2009), pp. 679–690.Google Scholar
[20]Spotz, W. F. and Carey, G. F., Texas Institute of Computational and Applied Mechanics Technical Report No. 94-03, 1994 (unpublished).Google Scholar
[21]Kalita, J. C., Dalal, D. C. and Dass, A. K., Fully compact higher-order computation of steady-state natural convection in a square cavity, DOI: 10.1103/PhysRevE.64.066703.Google Scholar
[22]Kalita, J. C., Dalal, D. C. and Dass, A. K., A class of higher order compact schemes for the unsteady two-dimensional convection-diffusion equation with variable convection coeffcients, Int. J. Numer. Methods Fluids, 38 (2002), pp. 1111–1131.Google Scholar
[23]Dennis, S. C. R. and Hudson, J. D., Compact h4 finite-difference approximations to operators of Navier-Stokes type, J. Comput. Phys., 85 (1989), pp. 390–416.Google Scholar
[24]Erturk, E. and Gökcöl, C., Fourth order compact formulation of Navier-Stokes equations and driven cavity flow at high Reynolds numbers, Int. J. Numer. Methods Fluids, 50 (2006), pp. 421–436.Google Scholar
[25]Davis, G. De Vahl and Jones, I. P., Natural convection in a square cavity: a comparison exercise, Int. J. Numer. Methods Fluids, 3 (1983), pp. 227–248.CrossRefGoogle Scholar
[26]Hortmann, M. and Periic, M., Finite volume multigrid prediction of laminar natural convection: benchmark solutions, Int. J. Numer. Methods Fluids, 11 (1990), pp. 189–207.CrossRefGoogle Scholar
[27]Tian, Z. F. and Ge, Y. B., A fourth-order compact finite difference scheme for the steady streamfunction-vorticity formulation of the Navier-Stokes/ Boussinesq equations, Int. J. Numer. Methods Fluids, 41 (2003), pp. 495–518.Google Scholar
[28]Erturk, E., Corke, T. C. and Gökcöl, C., Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers, Int. J. Numer. Methods Fluids, 48 (2005), pp. 747–774.Google Scholar
[29]Sekhar, T. V. S., Raju, B. H. S. and Sanyasiraju, Y. V. S. S., Higher-order compact scheme for the incompressible Navier-Stokes equations in spherical geometry, Commun. Comput. Phys., 11 (2012), pp. 99–113.CrossRefGoogle Scholar
[30]Chen, L., Shen, J. and Xu, C., Spectral direction splitting schemes for the incompressible Navier-Stokes equations, East Asian J. Appl. Math., 1 (2011), pp. 215–234.Google Scholar
[31]Chen, C., Ren, M., Srinivansan, A. and Wang, Q., 3-D numerical simulations of biofilm flows, East Asian J. Appl. Math., 1 (2011), pp. 197–214.CrossRefGoogle Scholar
[32]Peaceman, D. W. and Rachford, H. H., The numerical solution of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math., 3 (1995), pp. 28–41.Google Scholar
[33]Hirsh, R. S., Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique, J. Comput. Phys., 19 (1975), pp. 90–109.Google Scholar
[34]Shankar, P. N. and Deshpande, M. D., Fluid mechanics in the driven cavity, Annu. Rev. Fluid Mech., 32 (2000), pp. 93–136.Google Scholar
[35]Ghia, U., Ghia, K. N. and Shin, C. T., High-resolutions for incompressible Navier-Stokes equation and a multigrid method, J. Comput. Phys., 48 (1982), pp. 387–411.Google Scholar
[36]Zhang, J., Numerical simulation of 2D square driven cavity using fourth-order compact finite difference schemes, Comput. Math. Appl., 45 (2003), pp. 43–52.Google Scholar
[37]Botella, O. and Peyret, R., Benchmark spectral results on the lid-driven cavity flow, Comput. Fluids, 27 (1998), pp. 421–433.CrossRefGoogle Scholar
[38]Bruneau, C. H. and Saad, M., The 2D lid-driven cavity problem revisited, Comput. Fluids, 35 (2005), pp. 326–348.Google Scholar