Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-27T04:01:53.664Z Has data issue: false hasContentIssue false
  • English
  • Français

Natural convection in vertical enclosures with conjugate boundary conditions

Published online by Cambridge University Press:  03 August 2022

Tomas Solano Open the ORCID record for Tomas Solano [Opens in a new window] ,
Juan C. Ordonez Open the ORCID record for Juan C. Ordonez [Opens in a new window]  and
Kourosh Shoele Open the ORCID record for Kourosh Shoele [Opens in a new window]
Tomas Solano
Affiliation:
Department of Mechanical Engineering, Joint College of Engineering, Florida A&M University-Florida State University, Tallahassee, FL 32310, USA
Juan C. Ordonez
Affiliation:
Department of Mechanical Engineering, Joint College of Engineering, Florida A&M University-Florida State University, Tallahassee, FL 32310, USA
Kourosh Shoele*
Affiliation:
Department of Mechanical Engineering, Joint College of Engineering, Florida A&M University-Florida State University, Tallahassee, FL 32310, USA
*
†Email address for correspondence: kshoele@fsu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A numerical investigation and reduced-order modelling of natural convection in a cavity with differentially heated sidewalls is discussed. The effect of conjugate boundary conditions on the cavity's heat transfer and natural flow circulation with varying aspect ratios and Rayleigh numbers is examined. Validation of the canonical differentially heated cavity reveals that a modification to the definition of the Rayleigh number (Ra) is required to reconcile the $Nu\sim Ra^{1/4}$ scaling (where Nu is the Nusselt number) and validity of previously proposed correlations for the heat transfer in vertical enclosures. Dynamic mode decomposition is used to uncover the underlying time-dependent flow structures and the results are compared with stability bifurcation studies in the literature. A flow mode previously associated with smaller aspect ratio cavities is identified as the unstable mode for a larger aspect ratio of 4. The effect of conjugate boundary conditions is scaled based on the ratio of the internal and external boundary layers, wherein higher external Reynolds numbers aid in the heat transfer as the Nusselt number approaches the isothermal limit. A reduced-order theoretical model is proposed to predict the Nusselt number for the conjugate boundary conditions. The performance of conjugate boundary conditions is connected to the flow stability, with the parallel-flow configuration acting in a destabilizing manner while the counterflow configuration has a stabilizing effect and results in the highest heat transfer. We also found that relatively large heat transfer can be achieved with substantially lower external actuation with intermediate aspect ratios, translating to less power in practice.

JFM classification

Convection: Buoyancy-driven instability Convection: Convection in cavities Convection: Buoyant boundary layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 946 , 10 September 2022 , A17
Check for updates
Copyright
© The Author(s), 2022. 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

REFERENCES

Abourida, B., Hasnaoui, M. & Douamna, S. 1999 Transient natural convection in a square enclosure with horizontal walls submitted to periodic temperatures. Numer. Heat Tr. A: Appl. 36 (7), 737–750.Google Scholar
Anderson, R. & Bejan, A. 1980 Natural convection on both sides of a vertical wall separating fluids at different temperatures. Trans. ASME J. Heat Transfer 102, 630–631.10.1115/1.3244363CrossRefGoogle Scholar
Armenio, V. & Sarkar, S. 2002 An investigation of stably stratified turbulent channel flow using large-eddy simulation. J. Fluid Mech. 459, 1–42.10.1017/S0022112002007851CrossRefGoogle Scholar
Asgarian, A., Hossain, M.Z. & Floryan, J.M. 2016 Rayleigh–Bénard convection driven by a long wavelength heating. Theor. Comput. Fluid Dyn. 30 (4), 313–337.10.1007/s00162-015-0379-0CrossRefGoogle Scholar
Azzoune, M., Mammou, L., Boulheouchat, M.H., Zidi, T., Mokeddem, M.Y., Belaid, S., Salah, A.B., Meftah, B. & Boumedien, A. 2010 NUR research reactor safety analysis study for long time natural convection (NC) operation mode. Nucl. Engng Des. 240 (4), 823–831.10.1016/j.nucengdes.2009.11.040CrossRefGoogle Scholar
Batchelor, G.K. 1954 Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures. Q. Appl. Maths 12 (3), 209–233.10.1090/qam/64563CrossRefGoogle Scholar
Bejan, A. 1979 Note on Gill's solution for free convection in a vertical enclosure. J. Fluid Mech. 90 (3), 561–568.10.1017/S0022112079002391CrossRefGoogle Scholar
Bejan, A. 2013 Convection Heat Transfer. John Wiley & Sons.10.1002/9781118671627CrossRefGoogle Scholar
Buchberg, H., Catton, I. & Edwards, D.K. 1976 Natural convection in enclosed spaces: a review of application to solar energy collection. Trans. ASME J. Heat Transfer 98 (2), 182–188.10.1115/1.3450516CrossRefGoogle Scholar
Çengel, Y.A., Turner, R.H., Cimbala, J.M. & Kanoglu, M. 2008 Fundamentals of Thermal-Fluid Sciences, vol. 703. McGraw-Hill.Google Scholar
Chorin, P., Moreau, F. & Saury, D. 2018 Heat transfer modification induced by a localized thermal disturbance in a differentially-heated cavity. Intl J. Therm. Sci. 125, 101–110.10.1016/j.ijthermalsci.2017.11.018CrossRefGoogle Scholar
Christon, M.A., Gresho, P.M. & Sutton, S.B. 2002 MIT special issue on thermal convection. Issue edited by Mark A. Christon. Intl J. Numer. Meth. Fluids 40 (8), 953–980.10.1002/fld.395CrossRefGoogle Scholar
De Vahl Davis, G. 1983 Natural convection of air in a square cavity: a bench mark numerical solution. Intl J. Numer. Meth. Fluids 3 (3), 249–264.10.1002/fld.1650030305CrossRefGoogle Scholar
El Sherbiny, S.M., Raithby, G.D. & Hollands, K.G.T. 1982 Heat transfer by natural convection across vertical and inclined air layers. Trans. ASME J. Heat Transfer 104 (1), 96–102.10.1115/1.3245075CrossRefGoogle Scholar
Floryan, J.M., Shadman, S. & Hossain, M.Z. 2018 Heating-induced drag reduction in relative movement of parallel plates. Phys. Rev. Fluids 3 (9), 094101.10.1103/PhysRevFluids.3.094101CrossRefGoogle Scholar
Ganguli, A.A., Pandit, A.B. & Joshi, J.B. 2009 CFD simulation of heat transfer in a two-dimensional vertical enclosure. Chem. Engng Res. Des. 87 (5), 711–727.10.1016/j.cherd.2008.11.005CrossRefGoogle Scholar
Gill, A.E. 1966 The boundary-layer regime for convection in a rectangular cavity. J. Fluid Mech. 26 (3), 515–536.10.1017/S0022112066001368CrossRefGoogle Scholar
Hasan, N. & Sanghi, S. 2007 Proper orthogonal decomposition and low-dimensional modelling of thermally driven two-dimensional flow in a horizontal rotating cylinder. J. Fluid Mech. 573, 265–295.10.1017/S0022112006003806CrossRefGoogle Scholar
Howle, L.E. 1997 Active control of Rayleigh–Bénard convection. Phys. Fluids 9 (7), 1861–1863.10.1063/1.869335CrossRefGoogle Scholar
Kakac, S., Liu, H. & Pramuanjaroenkij, A. 2002 Heat Exchangers: Selection, Rating, and Thermal Design. CRC.10.1201/9781420053746CrossRefGoogle Scholar
Kays, W.M. & Alexander, L.L. 1998 Compact Heat Exchangers. Krieger.Google Scholar
Kim, B.S., Lee, D.S., Ha, M.Y. & Yoon, H.S. 2008 A numerical study of natural convection in a square enclosure with a circular cylinder at different vertical locations. Intl J. Heat Mass Transfer 51 (7–8), 1888–1906.10.1016/j.ijheatmasstransfer.2007.06.033CrossRefGoogle Scholar
Kwak, H.S. & Hyun, J.M. 1996 Natural convection in an enclosure having a vertical sidewall with time-varying temperature. J. Fluid Mech. 329, 65–88.10.1017/S0022112096008828CrossRefGoogle Scholar
Kwak, H.S., Kuwahara, K. & Hyun, J.M. 1998 Resonant enhancement of natural convection heat transfer in a square enclosure. Intl J. Heat Mass Transfer 41 (18), 2837–2846.10.1016/S0017-9310(98)00018-0CrossRefGoogle Scholar
Lage, J.L. & Bejan, A. 1993 The resonance of natural convection in an enclosure heated periodically from the side. Intl J. Heat Mass Transfer 36 (8), 2027–2038.10.1016/S0017-9310(05)80134-6CrossRefGoogle Scholar
Lappa, M. 2016 Control of convection patterning and intensity in shallow cavities by harmonic vibrations. Microgravity Sci. Technol. 28 (1), 29–39.10.1007/s12217-015-9467-4CrossRefGoogle Scholar
Le Quéré, P. 1990 A note on multiple and unsteady solutions in two-dimensional convection in a tall cavity. Trans. ASME J. Heat Transfer 112 (4), 965–974.10.1115/1.2910508CrossRefGoogle Scholar
Le Quéré, P. 1991 Accurate solutions to the square thermally driven cavity at high Rayleigh number. Comput. Fluids 20 (1), 29–41.10.1016/0045-7930(91)90025-DCrossRefGoogle Scholar
Lumley, J.L. & Poje, A. 1997 Low-dimensional models for flows with density fluctuations. Phys. Fluids 9 (7), 2023–2031.10.1063/1.869321CrossRefGoogle Scholar
Marella, S., Krishnan, S.L.H.H., Liu, H. & Udaykumar, H.S. 2005 Sharp interface Cartesian grid method I: an easily implemented technique for 3D moving boundary computations. J. Comput. Phys. 210 (1), 1–31.10.1016/j.jcp.2005.03.031CrossRefGoogle Scholar
Ng, C.S., Ooi, A., Lohse, D. & Chung, D. 2015 Vertical natural convection: application of the unifying theory of thermal convection. J. Fluid Mech. 764, 349–361.10.1017/jfm.2014.712CrossRefGoogle Scholar
Ojo, O. & Shoele, K. 2021 Load reduction and reconfiguration capabilities of branched trees. In Integrative and Comparative Biology, vol. 61, pp. E664–E664. Oxford University Press Inc.Google Scholar
Ortega, A. & Ramanathan, S. 2003 On the use of point source solutions for forced air cooling of electronic components–part I: thermal wake models for rectangular heat sources. J. Electron. Packaging 125 (2), 226–234.10.1115/1.1569506CrossRefGoogle Scholar
Ostrach, S. 1972 Natural convection in enclosures. In Advances in Heat Transfer, vol. 8, pp. 161–227. Elsevier.10.1016/S0065-2717(08)70039-XCrossRefGoogle Scholar
Paolucci, S. & Chenoweth, D.R. 1989 Transition to chaos in a differentially heated vertical cavity. J. Fluid Mech. 201 (1), 379–410.10.1017/S0022112089000984CrossRefGoogle Scholar
Penot, F., Skurtys, O. & Saury, D. 2010 Preliminary experiments on the control of natural convection in differentially-heated cavities. Intl J. Therm. Sci. 49 (10), 1911–1919.10.1016/j.ijthermalsci.2010.05.008CrossRefGoogle Scholar
Puragliesi, R. & Leriche, E. 2012 Proper orthogonal decomposition of a fully confined cubical differentially heated cavity flow at Rayleigh number $Ra=109$. Comput. Fluids 61, 14–20.10.1016/j.compfluid.2011.06.005CrossRefGoogle Scholar
Ramos, E.M., do Nascimento, F., Darze, G.M., Faccini, J.L.H. & Giraldi, G.A. 2019 Dynamic mode decomposition of numerical data in natural circulation. Braz. J. Radiat. Sci. 8 (3A), 18.Google Scholar
Rhie, C.M. & Chow, W.-L. 1983 Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA J. 21 (11), 1525–1532.10.2514/3.8284CrossRefGoogle Scholar
Rips, A., Shoele, K. & Mittal, R. 2020 Heat transfer enhancement in laminar flow heat exchangers due to flapping flags. Phys. Fluids 32 (6), 063603.10.1063/1.5142403CrossRefGoogle Scholar
Salvetti, M.V., Zang, Y., Street, R.L. & Banerjee, S. 1997 Large-eddy simulation of free-surface decaying turbulence with dynamic subgrid-scale models. Phys. Fluids 9 (8), 2405–2419.10.1063/1.869359CrossRefGoogle Scholar
Scheel, J.D. & Schumacher, J. 2014 Local boundary layer scales in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 758, 344–373.10.1017/jfm.2014.536CrossRefGoogle Scholar
Schmid, P.J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.10.1017/S0022112010001217CrossRefGoogle Scholar
Shah, R.K. & Sekulic, D.P. 2003 Fundamentals of Heat Exchanger Design. John Wiley & Sons.10.1002/9780470172605CrossRefGoogle Scholar
Shoele, K. & Mittal, R. 2014 Computational study of flow-induced vibration of a reed in a channel and effect on convective heat transfer. Phys. Fluids 26 (12), 127103.10.1063/1.4903793CrossRefGoogle Scholar
Shu, J.J. & Pop, I. 1999 Thermal interaction between free convection and forced convection along a vertical conducting wall. Heat Mass Transfer 35 (1), 33–38.10.1007/s002310050295CrossRefGoogle Scholar
Taira, K., Brunton, S.L., Dawson, S.T.M., Rowley, C.W., Colonius, T., McKeon, B.J., Schmidt, O.T., Gordeyev, S., Theofilis, V. & Ukeiley, L.S. 2017 Modal analysis of fluid flows: an overview. AIAA J. 55 (12), 4013–4041.10.2514/1.J056060CrossRefGoogle Scholar
Thiers, N., Gers, R. & Skurtys, O. 2020 Heat transfer enhancement by localised time varying thermal perturbations at hot and cold walls in a rectangular differentially heated cavity. Intl J. Therm. Sci. 151, 106245.10.1016/j.ijthermalsci.2019.106245CrossRefGoogle Scholar
Treviño, C., Mendez, F. & Higuera, F.J. 1996 Heat transfer across a vertical wall separating two fluids at different temperatures. Intl J. Heat Mass Transfer 39 (11), 2231–2241.10.1016/0017-9310(95)00315-0CrossRefGoogle Scholar
Turan, O., Poole, R.J. & Chakraborty, N. 2012 Influences of boundary conditions on laminar natural convection in rectangular enclosures with differentially heated side walls. Intl J. Heat Fluid Flow 33 (1), 131–146.10.1016/j.ijheatfluidflow.2011.10.009CrossRefGoogle Scholar
Vijayshankar, S., Nabi, S., Chakrabarty, A., Grover, P. & Benosman, M. 2020 Dynamic mode decomposition and robust estimation: case study of a 2D turbulent Boussinesq flow. In 2020 American Control Conference (ACC), pp. 2351–2356. IEEE.10.23919/ACC45564.2020.9147823CrossRefGoogle Scholar
Wan, D.C., Patnaik, B.S.V. & Wei, G.W. 2001 A new benchmark quality solution for the buoyancy-driven cavity by discrete singular convolution. Numer. Heat Tr. B: Fund. 40 (3), 199–228.Google Scholar
Xin, S. & Le Quéré, P. 2006 Natural-convection flows in air-filled, differentially heated cavities with adiabatic horizontal walls. Numer. Heat Tr. A: Appl. 50 (5), 437–466.10.1080/10407780600605039CrossRefGoogle Scholar
Xu, F. 2006 Transient natural convection in a differentially heated cavity with and without a fin on the sidewall. PhD thesis, James Cook University.Google Scholar
Yang, S., Pilet, T.J. & Ordonez, J.C. 2018 Volume element model for 3D dynamic building thermal modeling and simulation. Energy 148, 642–661.10.1016/j.energy.2018.01.156CrossRefGoogle Scholar
Yang, S. & Ordonez, J.C. 2019 Optimal cooling channel layout in a hot enclosure subject to natural convection. Trans. ASME J. Heat Transfer 141 (11), 112502.10.1115/1.4044510CrossRefGoogle Scholar
Ye, T., Mittal, R., Udaykumar, H.S. & Shyy, W. 1999 An accurate Cartesian grid method for viscous incompressible flows with complex immersed boundaries. J. Comput. Phys. 156 (2), 209–240.10.1006/jcph.1999.6356CrossRefGoogle Scholar
Yuan, L.L., Street, R.L. & Ferziger, J.H. 1999 Large-eddy simulations of a round jet in crossflow. J. Fluid Mech. 379, 71–104.10.1017/S0022112098003346CrossRefGoogle Scholar
Yucel, N. & Turkoglu, H. 1998 Numerical analysis of laminar natural convection in enclosures with fins attached to an active wall. Heat Mass Transfer 33 (4), 307–314.10.1007/s002310050194CrossRefGoogle Scholar
Zang, Y., Street, R.L. & Koseff, J.R. 1994 A non-staggered grid, fractional step method for time-dependent incompressible Navier–Stokes equations in curvilinear coordinates. J. Comput. Phys. 114 (1), 18–33.10.1006/jcph.1994.1146CrossRefGoogle Scholar
Zhou, Q., Stevens, R.J.A.M., Sugiyama, K., Grossmann, S., Lohse, D. & Xia, K.Q. 2010 Prandtl–Blasius temperature and velocity boundary-layer profiles in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 664, 297–312.10.1017/S0022112010003824CrossRefGoogle Scholar
Zhou, Q. & Xia, K.Q. 2010 Measured instantaneous viscous boundary layer in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 104 (10), 104301.10.1103/PhysRevLett.104.104301CrossRefGoogle ScholarPubMed