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Unstable mixed convection in a heated inclined porous channel

Published online by Cambridge University Press:  06 August 2015

L. A. Sphaier ,
A. Barletta  and
M. Celli
L. A. Sphaier*
Affiliation:
Department of Mechanical Engineering, Universidade Federal Fluminense – PGMEC/UFF, Rua Passo da Pátria 156, bloco E, sala 216, Niterói, RJ, 24210-240, Brazil
A. Barletta
Affiliation:
Department of Industrial Engineering, Alma Mater Studiorum Università di Bologna, Viale Risorgimento 2, 40136, Bologna, Italy
M. Celli
Affiliation:
Department of Industrial Engineering, Alma Mater Studiorum Università di Bologna, Viale Risorgimento 2, 40136, Bologna, Italy
*
†Email address for correspondence: [email protected]
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Abstract

This paper presents a stability analysis of a mixed convection problem in an inclined parallel-plate channel with uniform heating (or cooling) from the top and bottom. The channel is filled with a saturated homogeneous porous medium and the momentum equation is given by Darcy’s model. A forced through-flow is prescribed across the channel. Linear stability analysis is thus employed to determine the onset of thermoconvective instability. The channel inclination is shown to play an important role in the stability of the problem, where two different regimes can be present: a buoyancy-assisted regime and a buoyancy-opposed regime. The analysis of the problem leads to a differential eigenvalue problem composed of a system of four complex-valued equations that are used to determine the critical values of the Rayleigh number leading to an instability under different problem configurations. This eigenproblem is solved by employing the generalised integral transform technique (GITT), in which simpler real eigenfunction bases are used to expand the complex eigenproblem. The results indicate that the longitudinal rolls are always more unstable than oblique and transverse rolls. For a buoyancy-opposed regime, even with a very small channel inclination angle, the basic through-flow is always unstable. This result has an important implication for experimental research, as it shows that a perfect alignment must be employed for horizontal mixed-convection experiments to avoid instabilities that arise in the buoyancy-opposed regime.

JFM classification

Convection: Buoyancy-driven instability Mathematical Foundations: Computational methods Convection: Convection in porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 778 , 10 September 2015 , pp. 428 - 450
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Copyright
© 2015 Cambridge University Press 

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References

Barletta, A. 2011 Thermal instabilities in a fluid saturated porous medium. In Heat Transfer in Multi-Phase Materials (ed. Öchsner, A. & Murch, G. E.), pp. 381–414. Springer.Google Scholar
Barletta, A. 2012 Thermal instability in a horizontal porous channel with horizontal through flow and symmetric wall heat fluxes. Trans. Porous Med. 92, 419–437.CrossRefGoogle Scholar
Barletta, A. 2013 Instability of mixed convection in a vertical porous channel with uniform wall heat flux. Phys. Fluids 25, 084108.Google Scholar
Barletta, A. & Rees, D. A. S. 2012 Linear instability of the Darcy–Hadley flow in an inclined porous layer. Phys. Fluids 24, 074104.CrossRefGoogle Scholar
Barletta, A. & Storesletten, L. 2011 Thermoconvective instabilities in an inclined porous channel heated from below. Intl J. Heat Mass Transfer 54, 2724–2733.CrossRefGoogle Scholar
Bories, S. A. & Combarnous, M. A. 1973 Natural convection in a sloping porous layer. J. Fluid Mech. 57, 63–79.Google Scholar
Caltagirone, J. P. & Bories, S. 1985 Solutions and stability criteria of natural convective flow in an inclined porous layer. J. Fluid Mech. 155, 267–287.Google Scholar
Cotta, R. M. 1990 Hybrid numerical/analytical approach to nonlinear diffusion problems. Numer. Heat Transfer B 17, 217–226.Google Scholar
Cotta, R. M. 1993 Integral Transforms in Computational Heat and Fluid Flow. CRC Press.Google Scholar
Cotta, R. M. 1994 Benchmark results in computational heat and fluid flow: the integral transform method. Intl J. Heat Mass Transfer 37, 381–394.Google Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability, 2nd edn. Cambridge University Press.Google Scholar
Horton, C. W. & Rogers, F. T. 1945 Convection currents in a porous medium. J. Appl. Phys. 16, 367–370.Google Scholar
Karimi-Fard, M., Charrier-Mojtabi, M. C. & Mojtabi, A. 1999 Onset of stationary and oscillatory convection in a tilted porous cavity saturated with a binary fluid: linear stability analysis. Phys. Fluids 11, 1346–1358.Google Scholar
Knupp, D. C., Naveira-Cotta, C. P. & Cotta, R. M. 2012 Theoretical analysis of conjugated heat transfer with a single domain formulation and integral transforms. Intl Commun. Heat Mass Transfer 39, 355–362.Google Scholar
Knupp, D. C., Naveira-Cotta, C. P. & Cotta, R. M. 2014 Theoretical–experimental analysis of conjugated heat transfer in nanocomposite heat spreaders with multiple microchannels. Intl J. Heat Mass Transfer 74, 306–318.Google Scholar
Lapwood, E. R. 1948 Convection of a fluid in a porous medium. Proc. Camb. Phil. Soc. 44, 508–521.Google Scholar
Mikhailov, M. D. & Cotta, R. M. 1994 Integral transform solution of eigenvalue problems. Commun. Numer. Meth. Engng 10, 827–835.Google Scholar
Naveira-Cotta, C. P., Cotta, R. M., Orlande, H. R. B. & Fudym, O. 2009 Eigenfunction expansions for transient diffusion in heterogeneous media. Intl J. Heat Mass Transfer 52, 5029–5039.Google Scholar
Nield, D. A., Barletta, A. & Celli, M. 2011 The effect of viscous dissipation on the onset of convection in an inclined porous layer. J. Fluid Mech. 679, 544–558.Google Scholar
Nield, D. A. & Bejan, A. 2013 Convection in Porous Media, 4th edn. Springer.Google Scholar
Rees, D. A. S. 2000 The stability of Darcy–Bénard convection. In Handbook of Porous Media (ed. Vafai, K. & Hadim, H. A.), pp. 521–558. CRC Press, chap. 12.Google Scholar
Rees, D. A. S. & Barletta, A. 2011 Linear instability of the isoflux Darcy–Bénard problem in an inclined porous layer. Trans. Porous Med. 87, 665–678.CrossRefGoogle Scholar
Rees, D. A. S. & Bassom, A. P. 2000 Onset of Darcy–Bénard convection in an inclined layer heated from below. Acta Mechanica 144, 103–118.CrossRefGoogle Scholar
Rees, D. A. S. & Postelnicu, A. 2001 The onset of convection in an inclined anisotropic porous layer. Intl J. Heat Mass Transfer 44, 4127–4138.Google Scholar
Rees, D. A. S., Storesletten, L. & Postelnicu, A. 2006 The onset of convection in an inclined anisotropic porous layer with oblique principal axes. Trans. Porous Med. 62, 139–156.Google Scholar
Sphaier, L. A. & Barletta, A. 2014 Unstable mixed convection in a heated horizontal porous channel. Intl J. Therm. Sci. 78, 77–89.Google Scholar
Sphaier, L. A. & Cotta, R. M. 2000 Integral transform analysis of multidimensional eigenvalue problems within irregular domains. Numer. Heat Transfer B 38, 157–175.Google Scholar
Sphaier, L. A. & Cotta, R. M. 2002 Analytical and hybrid solutions of diffusion problems within arbitrarily shaped regions via integral transforms. Comput. Mech. 29, 265–276.Google Scholar
Storesletten, L. & Tveitereid, M. 1999 Onset of convection in an inclined porous layer with anisotropic permeability. Appl. Mech. Engng 4, 575–587.Google Scholar
Tyvand, P. A. 2002 Onset of Rayleigh–Bénard convection in porous bodies. In Transport Phenomena in Porous Media II (ed. Ingham, D. B. & Pop, I.), pp. 82–112. Pergamon, chap. 4.Google Scholar
Weber, J. E. 1975 Thermal convection in a tilted porous layer. Intl J. Heat Mass Transfer 18, 474–475.Google Scholar