Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T06:25:29.309Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal convection cooling flows in general 2D geometries

Published online by Cambridge University Press:  08 February 2017

S. Alben Open the ORCID record for S. Alben [Opens in a new window]
S. Alben*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We generalize a recent method for computing optimal 2D convection cooling flows in a horizontal layer to a wide range of geometries, including those relevant for technological applications. We write the problem in a conformal pair of coordinates which are the pure conduction temperature and its harmonic conjugate. We find optimal flows for cooling a cylinder in an annular domain, a hot plate embedded in a cold surface, and a channel with a hot interior and cold exterior. With a constraint of fixed kinetic energy, the optimal flows are all essentially the same in the conformal coordinates. In the physical coordinates, they consist of vortices ranging in size from the length of the hot surface to a small cutoff length at the interface of the hot and cold surfaces. With the constraint of fixed enstrophy (or fixed rate of viscous dissipation), a geometry-dependent metric factor appears in the equations. The conformal coordinates are useful here because they map the problems to a rectangular domain, facilitating numerical solutions. With a small enstrophy budget, the optimal flows are dominated by vortices that have the same size as the flow domain.

JFM classification

Convection: Convection Flow Control: Mixing enhancement Mathematical Foundations: Variational methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 814 , 10 March 2017 , pp. 484 - 509
Check for updates
Copyright
© 2017 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

Ablowitz, M. J. & Fokas, A. S. 2003 Complex Variables: Introduction and Applications. Cambridge University Press.CrossRefGoogle Scholar
Acheson, D. J. 1990 Elementary Fluid Dynamics. Oxford University Press.CrossRefGoogle Scholar
Açıkalın, T., Garimella, S. V., Raman, A. & Petroski, J. 2007 Characterization and optimization of the thermal performance of miniature piezoelectric fans. Intl J. Heat Fluid Flow 28 (4), 806–820.CrossRefGoogle Scholar
Ahlers, M. F. 2011 Aircraft thermal management. In Encyclopedia of Aerospace Engineering. Wiley.Google Scholar
Alben, S. 2015 Flag flutter in inviscid channel flow. Phys. Fluids 27 (3), 033603.CrossRefGoogle Scholar
Bazant, M. Z. 2004 Conformal mapping of some non-harmonic functions in transport theory. Proc. R. Soc. Lond. A 460, 1433–1452.CrossRefGoogle Scholar
Bazant, M. Z. & Crowdy, D. 2005 Conformal mapping methods for interfacial dynamics. In Handbook of Materials Modeling, pp. 1417–1451. Springer.CrossRefGoogle Scholar
Bejan, A. 2013 Convection Heat Transfer. Wiley.CrossRefGoogle Scholar
Bird, R. B., Stewart, W. E. & Lightfoot, E. N. 2007 Transport Phenomena. Wiley.Google Scholar
Biswas, G., Torii, K., Fujii, D. & Nishino, K. 1996 Numerical and experimental determination of flow structure and heat transfer effects of longitudinal vortices in a channel flow. Intl J. Heat Mass Transfer 39 (16), 3441–3451.CrossRefGoogle Scholar
Boussinesq, J. 1902 Sur le pouvoir refroidissant d’un courant liquide ou gazeux. J. Phys. Theor. Appl. 1 (1), 71–75.CrossRefGoogle Scholar
Brown, J. W., Churchill, R. V. & Lapidus, M. 1996 Complex Variables and Applications, vol. 7. McGraw-Hill.Google Scholar
Camassa, R., Lin, Z., McLaughlin, R. M., Mertens, K., Tzou, C., Walsh, J. & White, B. 2016 Optimal mixing of buoyant jets and plumes in stratified fluids: theory and experiments. J. Fluid Mech. 790, 71–103.CrossRefGoogle Scholar
Campbell, M. I., Amon, C. H. & Cagan, J. 1997 Optimal three-dimensional placement of heat generating electronic components. J. Electronic Packag. 119 (2), 106–113.CrossRefGoogle Scholar
Caulfield, C. P. & Kerswell, R. R. 2001 Maximal mixing rate in turbulent stably stratified Couette flow. Phys. Fluids 13 (4), 894–900.CrossRefGoogle Scholar
Chen, Q., Liang, X.-G. & Guo, Z.-Y. 2013 Entransy theory for the optimization of heat transfer – a review and update. Intl J. Heat Mass Transfer 63, 65–81.CrossRefGoogle Scholar
Chien, W.-L., Rising, H. & Ottino, J. M. 1986 Laminar mixing and chaotic mixing in several cavity flows. J. Fluid Mech. 170, 355–377.CrossRefGoogle Scholar
Choi, J., Margetis, D., Squires, T. M. & Bazant, M. Z. 2005 Steady advection–diffusion around finite absorbers in two-dimensional potential flows. J. Fluid Mech. 536, 155–184.CrossRefGoogle Scholar
Crowdy, D. 2007 Schwarz–Christoffel mappings to unbounded multiply connected polygonal regions. In Mathematical Proceedings of the Cambridge Philosophical Society, vol. 142, pp. 319–339. Cambridge University Press.Google Scholar
Crowdy, D. 2012 Conformal slit maps in applied mathematics. ANZIAM J. 53 (03), 171–189.Google Scholar
Da Silva, A. K., Lorente, S. & Bejan, A. 2004 Optimal distribution of discrete heat sources on a wall with natural convection. Intl J. Heat Mass Transfer 47 (2), 203–214.CrossRefGoogle Scholar
Dipprey, D. F. & Sabersky, R. H. 1963 Heat and momentum transfer in smooth and rough tubes at various Prandtl numbers. Intl J. Heat Mass Transfer 6 (5), 329–353.CrossRefGoogle Scholar
Doering, C. R., Otto, F. & Reznikoff, M. G. 2006 Bounds on vertical heat transport for infinite-Prandtl-number Rayleigh–Bénard convection. J. Fluid Mech. 560, 229–241.CrossRefGoogle Scholar
Eagle, A. & Ferguson, R. M. 1930 On the coefficient of heat transfer from the internal surface of tube walls. Proc. R. Soc. Lond. A 127, 540–566.Google Scholar
Estrada, R. & Kanwal, R. P. 2012 Singular Integral Equations. Springer.Google Scholar
Fiebig, M., Kallweit, P., Mitra, N. & Tiggelbeck, S. 1991 Heat transfer enhancement and drag by longitudinal vortex generators in channel flow. Exp. Therm. Fluid Sci. 4 (1), 103–114.CrossRefGoogle Scholar
Foures, D. P. G., Caulfield, C. P. & Schmid, P. J. 2014 Optimal mixing in two-dimensional plane Poiseuille flow at finite Péclet number. J. Fluid Mech. 748, 241–277.CrossRefGoogle Scholar
Gerty, D. R.2008 Fluidic driven cooling of electronic hardware. Part I: channel integrated vibrating reed. Part II: active heat sink. PhD thesis, Georgia Institute of Technology, GA.Google Scholar
Golberg, M. A. 1990 Numerical Solution of Integral Equations. Plenum Press.CrossRefGoogle Scholar
Gopinath, D., Joshi, Y. & Azarm, S. 2005 An integrated methodology for multiobjective optimal component placement and heat sink sizing. IEEE Trans. Compon. Packag. Technol. 28 (4), 869–876.CrossRefGoogle Scholar
Hassanzadeh, P., Chini, G. P. & Doering, C. R. 2014 Wall to wall optimal transport. J. Fluid Mech. 751, 627–662.CrossRefGoogle Scholar
Hidalgo, P., Herrault, F., Glezer, A., Allen, M., Kaslusky, S. & Rock, B. S. 2010 Heat transfer enhancement in high-power heat sinks using active reed technology. In 2010 16th International Workshop on Thermal Investigations of ICs and Systems (THERMINIC), pp. 1–6. IEEE.Google Scholar
Jha, S., Hidalgo, P. & Glezer, A. 2015 Small-scale vortical motions induced by aeroelastically fluttering reed for enhanced heat transfer in a rectangular channel. Bull. Am. Phys. Soc. 60, 95.Google Scholar
Karniadakis, G. E. 1988 Numerical simulation of forced convection heat transfer from a cylinder in crossflow. Intl J. Heat Mass Transfer 31 (1), 107–118.CrossRefGoogle Scholar
Karniadakis, G. E., Mikic, B. B. & Patera, A. T. 1988 Minimum-dissipation transport enhancement by flow destabilization: Reynolds analogy revisited. J. Fluid Mech. 192, 365–391.CrossRefGoogle Scholar
Kotouč, M., Bouchet, G. & Dušek, J. 2008 Loss of axisymmetry in the mixed convection, assisting flow past a heated sphere. Intl J. Heat Mass Transfer 51 (11), 2686–2700.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Lienhard, J. H. 2013 A Heat Transfer Textbook. Courier Corporation.Google Scholar
McGlen, R. J., Jachuck, R. & Lin, S. 2004 Integrated thermal management techniques for high power electronic devices. Appl. Therm. Engng 24 (8), 1143–1156.CrossRefGoogle Scholar
Mohammadi, B., Pironneau, O., Mohammadi, B. & Pironneau, O. 2001 Applied Shape Optimization for Fluids, vol. 28. Oxford University Press.Google Scholar
Nakayama, W. 1986 Thermal management of electronic equipment: a review of technology and research topics. Appl. Mech. Rev. 39 (12), 1847–1868.CrossRefGoogle Scholar
Ockendon, J. R. 2003 Applied Partial Differential Equations. Oxford University Press.CrossRefGoogle Scholar
Otero, J., Dontcheva, L. A., Johnston, H., Worthing, R. A., Kurganov, A., Petrova, G. & Doering, C. R. 2004 High-Rayleigh-number convection in a fluid-saturated porous layer. J. Fluid Mech. 500, 263–281.CrossRefGoogle Scholar
Ozisik, M. N. 2000 Inverse Heat Transfer: Fundamentals and Applications. CRC Press.Google Scholar
Raschke, K. 1960 Heat transfer between the plant and the environment. Annu. Rev. Plant Physiol. 11 (1), 111–126.CrossRefGoogle Scholar
Rohsenow, W. M., Hartnett, J. P. & Cho, Y. I. 1998 Handbook of Heat Transfer. McGraw-Hill.Google Scholar
Rohsenow, W. M., Hartnett, J. P. & Ganic, E. N. 1985 Handbook of Heat Transfer Applications. McGraw-Hill.Google Scholar
Sharma, A. & Eswaran, V. 2004 Heat and fluid flow across a square cylinder in the two-dimensional laminar flow regime. Numer. Heat Transfer A 45 (3), 247–269.CrossRefGoogle 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.CrossRefGoogle Scholar
Sondak, D., Smith, L. M. & Waleffe, F. 2015 Optimal heat transport solutions for Rayleigh–Bénard convection. J. Fluid Mech. 784, 565–595.CrossRefGoogle Scholar
Souza, A. N.2016 An optimal control approach to bounding transport properties of thermal convection. PhD thesis, University of Michigan, MI.Google Scholar
Souza, A. N. & Doering, C. R. 2015a Maximal transport in the Lorenz equations. Phys. Lett. A 379 (6), 518–523.CrossRefGoogle Scholar
Souza, A. N. & Doering, C. R. 2015b Transport bounds for a truncated model of Rayleigh–Bénard convection. Physica D 308, 26–33.Google Scholar
Tang, W., Caulfield, C. P. & Kerswell, R. R. 2009 A prediction for the optimal stratification for turbulent mixing. J. Fluid Mech. 634, 487–497.CrossRefGoogle Scholar
Thomases, B., Shelley, M. & Thiffeault, J.-L. 2011 A Stokesian viscoelastic flow: transition to oscillations and mixing. Physica D 240 (20), 1602–1614.Google Scholar
Waleffe, F., Boonkasame, A. & Smith, L. M. 2015 Heat transport by coherent Rayleigh–Bénard convection. Phys. Fluids 27 (5), 051702.CrossRefGoogle Scholar
Wang, X. & Alben, S. 2015 The dynamics of vortex streets in channels. Phys. Fluids 27 (7), 073603.CrossRefGoogle Scholar
Zerby, M. & Kuszewski, M.2002 Final report on next generation thermal management (NGTM) for power electronics. NSWCCD Tech. Rep. TR-82-2002012.Google Scholar
Zimparov, V. D., Da Silva, A. K. & Bejan, A. 2006 Thermodynamic optimization of tree-shaped flow geometries. Intl J. Heat Mass Transfer 49 (9), 1619–1630.CrossRefGoogle Scholar