Abstract
Flow structures and thermal aspects of buoyancy-driven flow occurring in the presence of protruded heater inside a square enclosure are investigated under different thermal boundary conditions of the enclosure for two common fluids (of Prandtl numbers Pr = 0.71 and 6.9) at various Rayleigh numbers (Ra = 103–106) numerically. The study is conducted considering a single heater as well as a set of two identical heaters located symmetrically on the bottom wall of the enclosure. Results reveals that the heat transfer, energy flow and fluid flow are markedly dependent on the heater aspect ratio, Ra, Pr and thermal boundary condition. Although with the single heater symmetric trends of fluid and heat flow are observed, the strong asymmetric flow features are found in case of the double heaters for Pr = 6.9 at higher Ra. Some interesting energy and fluid flow patterns are evolved depending upon the values of heater aspect ratio, interspatial heater distance and enclosure’s boundary condition. The visualization of mass and energy transports are properly appreciated using streamfunction and heatfunction. Total fluid flow inside the enclosure is analyzed by applying a novel approach based on streamfunctions of thermally induced vortices, which is also used to formulate a new streamfunction based symmetry indicator. It is seen that, with the increase in heater height although the total flow decreases, a consistent trend of increasing heat transfer is observed. The nature of steady-state solutions for the cavity is also analyzed by gradually increasing and decreasing Ra. Symmetry breaking bifurcation has been identified for Pr = 6.9. The bifurcation characteristics changes from supercritical to subcritical bifurcation depending upon thermal boundary conditions of the enclosure.
Similar content being viewed by others
Abbreviations
- A :
-
Heater aspect ratio
- d :
-
Distance between two heaters
- D :
-
Dimensionless interspatial distance
- g :
-
Acceleration due to gravity
- h :
-
Heater height
- H :
-
Dimensionless heater height
- \(I_{\theta }\) :
-
Symmetry indicator based on \(\theta\)
- \(I_{\psi }\) :
-
Symmetry indicator based on \(\psi\)
- L :
-
Length/height of the enclosure
- \({\text{Nu}}\) :
-
Nusselt number
- p :
-
Pressure
- P :
-
Dimensionless pressure
- Pr :
-
Prandtl number
- \(q_{i}\) :
-
Dimensionless total heat inflow
- \(q_{o}\) :
-
Dimensionless total heat outflow
- Ra :
-
Rayleigh number
- SSC:
-
Side-side cold condition
- STSC:
-
Side-top-side cold condition
- T :
-
Temperature
- u, v :
-
Velocity components
- U, V:
-
Dimensionless velocity components
- w :
-
Heater width
- X, Y :
-
Dimensionless Cartesian coordinates
- \(\alpha\) :
-
Thermal diffusivity
- \(\beta\) :
-
Volumetric expansion coefficient
- \(\theta\) :
-
Dimensionless temperature
- \(\nu\) :
-
Kinematic viscosity
- \(\varPi\) :
-
Dimensionless heat function
- \(\rho\) :
-
Fluid density
- \(\tau\) :
-
Dimensionless time
- \(\varPsi\) :
-
Dimensionless streamfunction
- 1H, 2H :
-
One heater, two heaters
- 1HE :
-
Single heater enclosure
- 2HE :
-
Double heater enclosure
- a :
-
Ambient
- avg :
-
Average
- c, h :
-
Cold wall, hot wall
- max :
-
Maximum
- min :
-
Minimum
- tot :
-
Total
References
Ostrach O (1988) Natural convection in enclosures. J Heat Transf 110:1175–1190
Incropera FP (1988) Convection heat transfer in electronic equipment cooling. J Heat Transf 110:1097–1111
GdeV Davies (1983) Natural convection of air in a square cavity: a bench-mark numerical solution. Int J Numer Methods Fluids 3:249–264
Narasimham GSVL (2010) Natural convection from discrete heat sources in enclosures: an overview. Vivechan Int J Res 1:63–78
Mukhopadhyay A (2010) Analysis of entropy generation due to natural convection in square enclosures with multiple discrete heat sources. Int Commun Heat Mass Transf 37:867–872
Banerjee S, Mukhopadhyay A, Sen S, Ganguly R (2008) Natural convection in a bi-heater configuration of passive electronic cooling. Int J Therm Sci 47(11):516–1527
Aydin O, Yang WJ (2000) Natural convection in enclosures with localized heating from below and symmetrical cooling from sides. Int J Numer Methods Fluids 10(5):519–529
Sharif MAR, Mohammad TR (2005) Natural convection in cavities with constant flux heating at the bottom wall and isothermal cooling from the sidewalls. Int J Therm Sci 44:865–878
Calcagni B, Marsili F, Paroncini M (2005) Natural convective heat transfer in square enclosures heated from below. Appl Therm Eng 25:22–31
Varol Y, Oztop HF, Yilmaz T (2007) Natural convection in triangular enclosures with protruding isothermal heater. Int J Heat Mass Transf 50:2451–2462
Paroncini M, Corvaro F (2009) Natural convection in a square enclosure with a hot source. Int J Therm Sci 48:1683–1695
Corvaro F, Paroncini M (2009) An experimental study of natural convection in a differentially heated cavity through a 2D-PIV system. Int J Heat Mass Transf 52:355–365
Corvaro F, Paroncini M (2009) The natural convective heat transfer in a partially divided enclosure: a study on the influence of the source position. J Thermodyn 2009:1–10
Nardini G, Paroncini M, Corvaro F (2014) Effect of heat transfer on natural convection in a square cavity with two source pairs. Heat Transf Eng 35(9):875–886
AlAmiri A, Khanafer K, Pop I (2009) Buoyancy-induced flow and heat transfer in a partially divided square enclosure. Int J Heat Mass Transf 52:3818–3828
Bakkas M, Hasnaoui M, Amahmid A (2010) Natural convective flows in a horizontal channel provided with heating isothermal blocks: effect of the inter blocks spacing. Energy Convers Manag 51:296–304
Baek C-I, Lee K-S, Kim W-S (1997) Study of combined heat transfer in a three dimensional enclosure with a protruding heat source. Numer Heat Transf A 32:733–747
Dagtekin I, Oztop HF (2001) Natural convection heat transfer by heated partitions within enclosure. Int Commun Heat Mass Transf 28(6):823–834
Kimura S, Bejan A (1983) The heatline visualization of convective heat transfer. J Heat Transf ASME 105(4):916–919
Costa VAF (2006) Bejan’s heatlines and masslines for convection visualization analysis. Appl Mech Rev 59:126–145
Basak T, Roy S (2008) Role of Bejan’s heatlines in heat flow visualization and optimal thermal mixing for differentially heated square enclosures. Int J Heat Mass Trans 51:3486–3503
Dalal A, Das MK (2008) Heatline method for the visualization of natural convection in a complicated cavity. Int J Heat Mass Transf 51:263–272
Biswas N, Mahapatra PS, Manna NK, Roy PC (2015) Influence of heater aspect ratio on natural convection in a rectangular enclosure. Heat Transf Eng 37(2):1–15
Alfieri F, Tiwari M, Zinovik KI, Brunschwiler T, Michel B, Poulikakos D (2012) On the significance of developing boundary layers in integrated water cooled 3D chip, stacks. Int J Heat Mass Transf 55:5222–5232
Kota K, Hidalgo P, Joshi Y, Glezer A (2009) Thermal management of a 3D chip stack using a liquid interface to a synthetic jet cooled spreader. In: 15th International workshop thermal investigations ICs and systems, Leuven, Belgium, pp 1–6
Kota K, Hidalgo P, Joshi Y, Glezer A (2010) A novel conduction-convection based cooling solution for 3d stacked electronics. In: 26th Annual IEEE semiconductor thermal measurement model management symposium, Santa Clara, CA, pp 1–8
Kota K, Hidalgo P, Joshi Y, Glezer A (2012) Hybrid liquid immersion and synthetic jet heat sink for cooling 3-D stacked electronics components. IEEE Transf Packag Manuf Technol 2(5):817–824
Bejan A (2004) Convection heat transfer, 3rd edn. Wiley, New York
Mahapatra PS, De S, Ghosh K, Manna NK, Mukhopadhyay A (2013) Heat transfer enhancement and entropy generation in a square enclosure in the presence of adiabatic and isothermal blocks. Numer Heat Transf A 64:576–597
Patankar SV (1980) Numerical heat transfer and fluid flow. Taylor and Francis, London
Bouafia M, Daube O (2007) Natural convection for large temperature gradients around a square solid body within a rectangular cavity. Int J Heat Mass Transf 50:3599–3615
Ridouane EH, Campo A (2006) Formation of a pitchfork bifurcation in thermal convection flow inside an isosceles triangular cavity. Phys Fluids 18:074102-1-8
Acknowledgments
The present authors would like to thank anonymous reviewers for their comments and valuable suggestions, which have improved the quality of this manuscript. The authors also wish to acknowledge helpful discussions with Prof. Dipankar Sanyal and Prof. Achintya Mukhopadhyay, Department of Mechanical Engineering, Jadavpur University.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Biswas, N., Mahapatra, P.S. & Manna, N.K. Buoyancy-driven fluid and energy flow in protruded heater enclosure. Meccanica 51, 2159–2184 (2016). https://doi.org/10.1007/s11012-016-0366-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11012-016-0366-6