Hostname: page-component-848d4c4894-p2v8j Total loading time: 0 Render date: 2024-06-07T20:22:58.898Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear disturbance growth induced by viscous dissipation in Darcy–Bénard convection with throughflow

Published online by Cambridge University Press:  31 October 2023

P.V. Brandão Open the ORCID record for P.V. Brandão [Opens in a new window] ,
L.S. de B. Alves Open the ORCID record for L.S. de B. Alves [Opens in a new window] ,
D. Rodríguez Open the ORCID record for D. Rodríguez [Opens in a new window]  and
A. Barletta Open the ORCID record for A. Barletta [Opens in a new window]
P.V. Brandão
Affiliation:
Department of Industrial Engineering, Alma Mater Studiorum Università di Bologna, Bologna, Italy
L.S. de B. Alves*
Affiliation:
Department of Mechanical Engineering, Universidade Federal Fluminense, Niterói, RJ, Brazil
D. Rodríguez
Affiliation:
Universidad Politécnica de Madrid, ETSIAE-UPM, Plaza del Cardenal Cisneros 3, 28040 Madrid, Spain
A. Barletta
Affiliation:
Department of Industrial Engineering, Alma Mater Studiorum Università di Bologna, Bologna, Italy
*
†Email address for correspondence: leonardo.alves@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Modal and non-modal linear stability analyses are employed to investigate the effect of internal and external heating on disturbance temporal growth for the Darcy–Bénard convection with throughflow. A matrix-forming approach is employed for both purposes, where the generalised eigenvalue problem is built using the generalised integral transform technique. Although the disturbance equations are not self-adjoint, the non-modal analysis indicates that there is no transient growth. Hence, any disturbance growth in time must be induced by modal mechanisms. An absolute instability analysis reveals that viscous dissipation has a destabilising effect and introduces new modes that are eventually destabilised by increasing the Péclet number. Beyond critical values of the Péclet number, where codimension-two absolutely unstable points exist, these modes become more unstable than the classical mode found in the absence of viscous dissipation, which is stabilised by an increasing Péclet number. This internal heating mechanism generated by viscous dissipation is so strong at high enough Péclet numbers that instability becomes possible through heating from above.

JFM classification

Convection: Convection in porous media Instability: Absolute/convective instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 974 , 10 November 2023 , A15
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press

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

Alves, L.S.d.B., Barletta, A., Hirata, S. & Ouarzazi, M.N. 2014 Effects of viscous dissipation on the convective instability of viscoelastic mixed convection flows in porous media. Intl J. Heat Mass Transfer 70, 586–598.CrossRefGoogle Scholar
Alves, L.S.d.B., Hirata, S.C., Schuabb, M. & Barletta, A. 2019 Identifying linear absolute instabilities from differential eigenvalue problems using sensitivity analysis. J. Fluid Mech. 870, 941–969.CrossRefGoogle Scholar
Barletta, A. 2019 Routes to Absolute Instability in Porous Media. Springer.CrossRefGoogle Scholar
Barletta, A. & Celli, M. 2011 Local thermal non-equilibrium flow with viscous dissipation in a plane horizontal porous layer. Intl J. Therm. Sci. 50 (1), 53–60.CrossRefGoogle Scholar
Barletta, A., Celli, M. & Alves, L.S.d.B. 2020 Wavepacket instability in a rectangular porous channel uniformly heated from below. Intl J. Heat Mass Transfer 147, 118993.CrossRefGoogle Scholar
Barletta, A., Celli, M. & Kuznetsov, A.V. 2011 a Transverse heterogeneity effects in the dissipation-induced instability of a horizontal porous layer. Trans. ASME J. Heat Transfer 133 (12), 122601.CrossRefGoogle Scholar
Barletta, A., Celli, M. & Nield, D.A. 2010 Unstably stratified Darcy flow with impressed horizontal temperature gradient, viscous dissipation and asymmetric thermal boundary conditions. Intl J. Heat Mass Transfer 53 (9), 1621–1627.CrossRefGoogle Scholar
Barletta, A., Celli, M. & Rees, D.A.S. 2009 a The onset of convection in a porous layer induced by viscous dissipation: a linear stability analysis. Intl J. Heat Mass Transfer 52 (1), 337–344.CrossRefGoogle Scholar
Barletta, A., Celli, M. & Rees, D.A.S. 2009 b Darcy–Forchheimer flow with viscous dissipation in a horizontal porous layer: onset of convective instabilities. Trans. ASME J. Heat Transfer 131 (7), 072602.CrossRefGoogle Scholar
Barletta, A., di Schio, E.R. & Celli, M. 2011 b Instability and viscous dissipation in the horizontal Brinkman flow through a porous medium. Transp. Porous Media 87 (1), 105–119.CrossRefGoogle Scholar
Barletta, A. & Mulone, G. 2021 The energy method analysis of the Darcy–Bénard problem with viscous dissipation. Contin. Mech. Thermodyn. 33, 25–33.CrossRefGoogle Scholar
Barletta, A. & Storesletten, L. 2010 Viscous dissipation and thermoconvective instabilities in a horizontal porous channel heated from below. Intl J. Therm. Sci. 49 (4), 621–630.CrossRefGoogle Scholar
Bénard, H. 1901 Les tourbillons cellulaires dans une nappe liquide.-méthodes optiques d'observation et d'enregistrement. J. Phys. Théor. Appl. 10 (1), 254–266.Google Scholar
Biau, D. & Bottaro, A. 2004 The effect of stable thermal stratification on shear flow stability. Phys. Fluids 16 (12), 4742–4745.CrossRefGoogle Scholar
Brandão, P.V., Alves, L.S.d.B. & Barletta, A. 2014 Onset of absolute instability induced by viscous dissipation in the Poiseuille–Darcy–Bénard convection of a newtonian fluid. In Journal of Physics: Conference Series, vol. 547, p. 012039. IOP Publishing.CrossRefGoogle Scholar
Brevdo, L. 1991 Three-dimensional absolute and convective instabilities, and spatially amplifying waves in parallel shear flows. Z. Angew. Math. Phys. 42, 911–942.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Drazin, P.G. & Reid, W.H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Gebhart, B. 1962 Effects of viscous dissipation in natural convection. J. Fluid Mech. 14 (2), 225–232.CrossRefGoogle Scholar
Hanifi, A., Schmid, P.J. & Henningson, D.S. 1996 Transient growth in compressible boundary layer flow. Phys. Fluids 8 (3), 826–837.CrossRefGoogle Scholar
Herbert, T. 1997 Parabolized stability equations. Annu. Rev. Fluid Mech. 29, 245–283.CrossRefGoogle Scholar
Hirata, S.C. & Ouarzazi, M.N. 2010 Three-dimensional absolute and convective instabilities in mixed convection of a viscoelastic fluid through a porous medium. Phys. Lett. A 374 (26), 2661–2666.CrossRefGoogle Scholar
Horton, C.W. & Rogers, F.T. Jr. 1945 Convection currents in a porous medium. J. Appl. Phys. 16 (6), 367–370.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P.A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.CrossRefGoogle Scholar
Jerome, J.J.S., Chomaz, J.-M. & Huerre, P. 2012 Transient growth in Rayleigh–Bénard–Poiseuille/ Couette convection. Phys. Fluids 24 (4), 044103.CrossRefGoogle Scholar
Lapwood, E.R. 1948 Convection of a fluid in a porous medium. Math. Proc. Camb. Philos. Soc. 44 (4), 508–521.Google Scholar
Lesshafft, L. & Marquet, O. 2010 Optimal velocity and density profiles for the onset of absolute instability in jets. J. Fluid Mech. 662, 398–408.CrossRefGoogle Scholar
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46, 493–517.CrossRefGoogle Scholar
Magyari, E., Rees, D.A.S. & Keller, B. 2005 Effect of viscous dissipation on the flow in fluid saturated porous media. In Handbook of Porous Media II (ed. K. Vafai), pp. 373–406. Marcel-Dekker.CrossRefGoogle Scholar
Murthy, P.V.S.N. 1998 Thermal dispersion and viscous dissipation effects on non-Darcy mixed convection in a fluid saturated porous medium. Heat Mass Transfer 33 (4), 295–300.CrossRefGoogle Scholar
Nakayama, A. & Pop, I. 1989 Free convection over a nonisothermal body in a porous medium with viscous dissipation. Intl Commun. Heat Mass Transfer 16 (2), 173–180.CrossRefGoogle Scholar
Nield, D.A. 2000 Resolution of a paradox involving viscous dissipation and nonlinear drag in a porous medium. Transp. Porous Media 41 (3), 349–357.CrossRefGoogle Scholar
Nield, D.A. & Barletta, A. 2010 Extended Oberbeck–Boussinesq approximation study of convective instabilities in a porous layer with horizontal flow and bottom heating. Intl J. Heat Mass Transfer 53 (4), 577–585.CrossRefGoogle 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.CrossRefGoogle Scholar
Nield, D.A. & Bejan, A. 2006 Convection in Porous Media, 3rd edn. Springer.Google Scholar
Pearson, J.R.A. 1958 On convection cells induced by surface tension. J. Fluid Mech. 4, 489–500.CrossRefGoogle Scholar
Prats, M. 1966 The effect of horizontal fluid flow on thermally induced convection currents in porous mediums. J. Geophys. Res. 71 (20), 4835–4838.CrossRefGoogle Scholar
Rapaka, S., Chen, S., Pawar, R.J., Stauffer, P.H. & Zhang, D. 2008 Non-modal growth of perturbations in density-driven convection in porous media. J. Fluid Mech. 609, 285–303.CrossRefGoogle Scholar
Rapaka, S., Pawar, R.J., Stauffer, P.H., Zhang, D. & Chen, S. 2009 Onset of convection over a transient base-state in anisotropic and layered porous media. J. Fluid Mech. 641, 227–244.CrossRefGoogle Scholar
Rayleigh, Lord 1916 Lix. on convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Lond. Edinb. Dublin Philos. Mag. J. Sci. 32 (192), 529–546.CrossRefGoogle Scholar
Requilé, Y., Hirata, S.d.C. & Ouarzazi, M.N. 2020 Viscous dissipation effects on the linear stability of Rayleigh–Bénard–Poiseuille/couette convection. Intl J. Heat Mass Transfer 146, 118834.CrossRefGoogle Scholar
Roy, K. & Murthy, P.V.S.N. 2015 Soret effect on the double diffusive convection instability due to viscous dissipation in a horizontal porous channel. Intl J. Heat Mass Transfer 91, 700–710.CrossRefGoogle Scholar
Roy, K. & Murthy, P.V.S.N. 2017 Effect of viscous dissipation on the convective instability induced by inclined temperature gradients in a non-Darcy porous medium with horizontal throughflow. Phys. Fluids 29 (4), 044104.CrossRefGoogle Scholar
Sameen, A. & Govindarajan, R. 2007 The effect of wall heating on instability of channel flow. J. Fluid Mech. 577, 417–442.CrossRefGoogle Scholar
Schmid, P.J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.CrossRefGoogle Scholar
Schuabb, M., Alves, L.S.d.B. & Hirata, S.d.C. 2020 Two-and three-dimensional absolute instabilities in a porous medium with inclined temperature gradient and vertical throughflow. Transp. Porous Media 132 (1), 135–155.CrossRefGoogle Scholar
Storesletten, L. & Barletta, A. 2009 Linear instability of mixed convection of cold water in a porous layer induced by viscous dissipation. Intl J. Therm. Sci. 48 (4), 655–664.CrossRefGoogle Scholar
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43, 319–352.CrossRefGoogle Scholar
Turcotte, D.L., Hsui, A.T., Torrance, K.E. & Schubert, G. 1974 Influence of viscous dissipation on Bénard convection. J. Fluid Mech. 64 (2), 369–374.CrossRefGoogle Scholar