Skip to main content
    • Aa
    • Aa

Penetrative internally heated convection in two and three dimensions

  • David Goluskin (a1) and Erwin P. van der Poel (a2)

Convection of an internally heated fluid, confined between top and bottom plates of equal temperature, is studied by direct numerical simulation in two and three dimensions. The unstably stratified upper region drives convection that penetrates into the stably stratified lower region. The fraction of produced heat escaping across the bottom plate, which is one half without convection, initially decreases as convection strengthens. Entering the turbulent regime, this decrease reverses in two dimensions but continues monotonically in three dimensions. The mean fluid temperature, which grows proportionally to the heating rate ( $H$ ) without convection, grows proportionally to $H^{4/5}$ when convection is strong in both two and three dimensions. The ratio of the heating rate to the fluid temperature is likened to the Nusselt number of Rayleigh–Bénard convection. Simulations are reported for Prandtl numbers between 0.1 and 10 and for Rayleigh numbers (defined in terms of the heating rate) up to $5\times 10^{10}$ .

Corresponding author
Email address for correspondence:
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

G. Ahlers , S. Grossmann  & D. Lohse 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81 (2), 503537.

F. J. Asfia  & V. K. Dhir 1996 An experimental study of natural convection in a volumetrically heated spherical pool bounded on top with a rigid wall. Nucl. Engng Des. 163 (3), 333348.

A. A. Emara  & F. A. Kulacki 1980 A numerical investigation of thermal convection in a heat-generating fluid layer. Trans. ASME J. Heat Transfer 102, 531537.

D. Goluskin  & E. A. Spiegel 2012 Convection driven by internal heating. Phys. Lett. A 377 (1–2), 8392.

G. Grötzbach  & M. Wörner 1999 Direct numerical and large eddy simulations in nuclear applications. Intl J. Heat Fluid Flow 20 (3), 222240.

P. Irwin 2009 Giant Planets of Our Solar System: Atmospheres, Composition, and Structure, 2nd edn. Springer.

H. Johnston  & C. R. Doering 2009 Comparison of turbulent thermal convection between conditions of constant temperature and constant flux. Phys. Rev. Lett. 102 (6), 064501.

R. H. Kraichnan 1962 Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5 (11), 13741389.

S. D. Lee , J. K. Lee  & K. Y. Suh 2007 Boundary condition dependent natural convection in a rectangular pool with internal heat sources. Trans. ASME J. Heat Transfer 129 (5), 679682.

L. Lu , C. R. Doering  & F. H. Busse 2004 Bounds on convection driven by internal heating. J. Math. Phys. 45 (7), 29672986.

W. V. R. Malkus 1954 Discrete transitions in turbulent convection. Proc. R. Soc. Lond. A 225 (1161), 185195.

R. R. Nourgaliev , T. N. Dinh  & B. R. Sehgal 1997 Effect of fluid Prandtl number on heat transfer characteristics in internally heated liquid pools with Rayleigh numbers up to $10^{12}$. Nucl. Engng Des. 169, 165184.

S. J. Peale , P. Cassen  & R. T. Reynolds 1979 Melting of Io by tidal dissipation. Science 203 (4383), 892894.

R. S. Peckover  & I. H. Hutchinson 1974 Convective rolls driven by internal heat sources. Phys. Fluids 17 (7), 13691371.

E. P. van der Poel , R. Ostilla-Mónico , J. Donners  & R. Verzicco 2015 A pencil distributed finite difference code for strongly turbulent wall-bounded flows. Comput. Fluids 116, 1016.

Lord Rayleigh 1916 On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Phil. Mag. 32 (192), 529546.

J. Schmalzl , M. Breuer  & U. Hansen 2004 On the validity of two-dimensional numerical approaches to time-dependent thermal convection. Europhys. Lett. 67 (3), 390396.

J. M. Straus 1976 Penetrative convection in a layer of fluid heated from within. Astrophys. J. 209, 179189.

M. Tveitereid 1978 Thermal convection in a horizontal fluid layer with internal heat sources. Intl J. Heat Mass Transfer 21, 335339.

R. Verzicco 1996 A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates. J. Comput. Phys. 123, 402414.

X. M. Wang 2004 Infinite Prandtl number limit of Rayleigh–Bénard convection. Commun. Pure Appl. Maths 57, 12651285.

X. M. Wang 2008 Stationary statistical properties of Rayleigh–Bénard convection at large Prandtl number. Commun. Pure Appl. Maths 61, 789815.

M. Wörner , M. Schmidt  & G. Grötzbach 1997 Direct numerical simulation of turbulence in an internally heated convective fluid layer and implications for statistical modeling. J. Hydraul. Res. 35 (6), 773797.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Type Description Title

Goluskin et al. supplementary movie
Temperature field in a 3D simulation with Pr=1 and R=5×108. The coolest fluid is blue. Warmer fluid is orange, and the hottest fluid is transparent to aid visualization.

 Video (4.1 MB)
4.1 MB


Full text views

Total number of HTML views: 2
Total number of PDF views: 69 *
Loading metrics...

Abstract views

Total abstract views: 162 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 27th June 2017. This data will be updated every 24 hours.