Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-23T06:33:49.469Z Has data issue: false hasContentIssue false
  • English
  • Français

Decomposition of available potential energy for networks of connected volumes

Published online by Cambridge University Press:  09 June 2021

John Craske Open the ORCID record for John Craske [Opens in a new window]
John Craske*
Affiliation:
Department of Civil and Environmental Engineering, Imperial College London, LondonSW7 2AZ, UK
*
†Email address for correspondence: john.craske07@imperial.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

A decomposition of available potential energy is derived for Boussinesq fluid flow in networks of connected control volumes. The two constituent parts of the decomposition are positive definite and therefore meaningful representations of available energy. The first (inner) part accounts for available potential energy that is intrinsic to each control volume, while the second (outer) part accounts for the context provided by the larger parent volume to which each smaller control volume belongs. While the intended application casts the control volumes as connected rooms in a building, the formulation can be applied to any domain that is partitioned by either physical boundaries or abstract zones and can be invoked recursively to clarify the hierarchical dependence of available potential energy on scale and context. By deriving budgets for the decomposition, two ways in which available potential energy can be redistributed between its inner and outer parts are identified. The first accounts for an apparent generation of available potential energy due to diapycnal mixing within a control volume that is constrained by removable boundaries. The second involves the reversible conversion between inner and outer parts that occurs when mass or heat is transported between control volumes and accounts for the concomitant change in context. Analytical expressions are derived for the hierarchy of contributions to available potential energy in an example involving three connected spaces, before budgets for the decomposition from a direct numerical simulation are analysed. Finally, the dependence of mixing efficiency on remote regions that was identified by Davies Wykes et al. (J. Fluid Mech., vol. 781, 2015, pp. 261–275) is revisited to demonstrate the precise way in which the proposed decomposition quantifies context.

JFM classification

Mixing: Turbulent mixing Turbulent Flows: Stratified turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 920 , 10 August 2021 , A17
Check for updates
Copyright
© The Author(s), 2021. 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

Andrews, D.G. 1981 A note on potential energy density in a stratified compressible fluid. J. Fluid Mech. 107, 227–236.CrossRefGoogle Scholar
Andrews, D.G. 2006 On the available energy density for axisymmetric motions of a compressible stratified fluid. J. Fluid Mech. 569, 481–492.CrossRefGoogle Scholar
Axley, J. 2007 Multizone airflow modeling in buildings: history and theory. HVAC&R Res. 13 (6), 907–928.CrossRefGoogle Scholar
Bhagat, R.K., Davies Wykes, M.S., Dalziel, S.B. & Linden, P.F. 2020 Effects of ventilation on the indoor spread of COVID-19. J. Fluid Mech. 903, F1.CrossRefGoogle Scholar
Caulfield, C.P. 2020 Open questions in turbulent stratified mixing: do we even know what we do not know? Phys. Rev. Fluids 5, 110518.CrossRefGoogle Scholar
Craske, J. & Hughes, G. 2019 On the robustness of emptying filling boxes to sudden changes in the wind. J. Fluid Mech. 868, R3.CrossRefGoogle Scholar
Craske, J. & van Reeuwijk, M. 2015 Energy dispersion in turbulent jets. Part 1. Direct simulation of steady and unsteady jets. J. Fluid Mech. 763, 500–537.CrossRefGoogle Scholar
Davies Wykes, M.S., Hogg, C., Partridge, J. & Hughes, G.O. 2019 Energetics of mixing for the filling box and the emptying-filling box. Environ. Fluid Mech. 19, 819–831.CrossRefGoogle Scholar
Davies Wykes, M.S., Hughes, G.O. & Dalziel, S.B. 2015 On the meaning of mixing efficiency for buoyancy-driven mixing in stratified turbulent flows. J. Fluid Mech. 781, 261–275.CrossRefGoogle Scholar
Dewar, W.K. & McWilliams, J.C. 2019 On energy and turbulent mixing in the thermocline. J. Adv. Model. Earth Sy. 11 (3), 578–596.CrossRefGoogle Scholar
Fernando, H.J.S. 1991 Turbulent mixing in stratified fluids. Annu. Rev. Fluid Mech. 23 (1), 455–493.CrossRefGoogle Scholar
Gaggioli, R. 1998 Available energy and exergy. Intl J. Thermodyn. 1, 1–8.Google Scholar
Haywood, R.W. 1974 A critical review of the theorems of thermodynamic availability, with concise formulations. J. Mech. Engng Sci. 16 (4), 258–267.CrossRefGoogle Scholar
Holliday, D. & Mcintyre, M.E. 1981 On potential energy density in an incompressible, stratified fluid. J. Fluid Mech. 107, 221–225.CrossRefGoogle Scholar
Ivey, G., Winters, K. & Koseff, J. 2008 Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech. 40 (1), 169–184.CrossRefGoogle Scholar
Kang, D. & Fringer, O. 2010 On the calculation of available potential energy in internal wave fields. J. Phys. Oceanogr. 40 (11), 2539–2545.CrossRefGoogle Scholar
Keenan, J.H. 1951 Availability and irreversibility in thermodynamics. Brit. J. Appl. Phys. 2 (7), 183–192.CrossRefGoogle Scholar
Kucharski, F. 1997 On the concept of exergy and available potential energy. Q. J. R. Meteorol. Soc. 123 (543), 2141–2156.CrossRefGoogle Scholar
Kuesters, A.S. & Woods, A.W. 2012 On the competition between lateral convection and upward displacement in a multi-zone naturally ventilated space. J. Fluid Mech. 707, 393–404.CrossRefGoogle Scholar
Lamb, K. 2008 On the calculation of the available potential energy of an isolated perturbation in a density-stratified fluid. J. Fluid Mech. 597, 415–427.CrossRefGoogle Scholar
Linden, P.F. 1979 Mixing in stratified fluids. Geophys. Astrophys. Fluid Dyn. 13 (1), 3–23.CrossRefGoogle Scholar
Linden, P.F. 1999 The fluid mechanics of natural ventilation. Annu. Rev. Fluid Mech. 31 (1), 201–238.CrossRefGoogle Scholar
Littlejohn, R.G. 1982 Singular Poisson tensors. AIP Conf. Proc. 88 (1), 47–66.Google Scholar
Lorenz, E. 1955 Available potential energy and the maintenance of the general circulation. Tellus 7 (2), 157–167.CrossRefGoogle Scholar
MacLane, S. & Birkhoff, G. 1999 Algebra. Chelsea Publishing Company.Google Scholar
Margules, M. 1903 Über die Energie der Stürme. Jahrb. Zent.-Anst. für Meteorol. und Erdmagnet. 48, 1–26.Google Scholar
Peltier, W.R. & Caulfield, C.P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35 (1), 135–167.CrossRefGoogle Scholar
Roullet, G. & Klein, P. 2009 Available potential energy diagnosis in a direct numerical simulation of rotating stratified turbulence. J. Fluid Mech. 624, 45–55.CrossRefGoogle Scholar
Samsel, F., Wolfram, P., Bares, A., Turton, T.L. & Bujack, R. 2019 Colormapping resources and strategies for organized intuitive environmental visualization. Environ. Earth Sci. 78, 269.CrossRefGoogle Scholar
Scotti, A., Beardsley, R. & Butman, B. 2006 On the interpretation of energy and energy fluxes of nonlinear internal waves: an example from Massachusetts Bay. J. Fluid Mech. 561, 103–112.CrossRefGoogle Scholar
Scotti, A. & Passaggia, P.-Y. 2019 Diagnosing diabatic effects on the available energy of stratified flows in inertial and non-inertial frames. J. Fluid Mech. 861, 608–642.CrossRefGoogle Scholar
Scotti, A. & White, B. 2014 Diagnosing mixing in stratified turbulent flows with a locally defined available potential energy. J. Fluid Mech. 740, 114–135.CrossRefGoogle Scholar
Shepherd, T.G. 1993 A unified theory of available potential energy. Atmosphere-Ocean 31 (1), 1–26.CrossRefGoogle Scholar
Smith, P.J. 1973 The net generation of large-scale available potential energy by subgrid-scale processes. J. Atmos. Sci. 30 (8), 1714–1717.2.0.CO;2>CrossRefGoogle Scholar
Smith, R.K., Montgomery, M.T. & Zhu, H. 2005 Buoyancy in tropical cyclones and other rapidly rotating atmospheric vortices. Dyn. Atmos. Oceans 40 (3), 189–208.CrossRefGoogle Scholar
Stewart, K., Saenz, J.A., Hogg, A.M., Hughes, G.O. & Griffiths, R.W. 2014 Effect of topographic barriers on the rates of available potential energy conversion of the oceans. Ocean Model. 76, 31–42.CrossRefGoogle Scholar
Tailleux, R. 2009 On the energetics of stratified turbulent mixing, irreversible thermodynamics, Boussinesq models and the ocean heat engine controversy. J. Fluid Mech. 638, 339–382.CrossRefGoogle Scholar
Tailleux, R. 2013 a Available potential energy and exergy in stratified fluids. Annu. Rev. Fluid Mech. 45 (1), 35–58.CrossRefGoogle Scholar
Tailleux, R. 2013 b Available potential energy density for a multicomponent Boussinesq fluid with arbitrary nonlinear equation of state. J. Fluid Mech. 735, 499–518.CrossRefGoogle Scholar
Tailleux, R. 2018 Local available energetics of multicomponent compressible stratified fluids. J. Fluid Mech. 842, R1.CrossRefGoogle Scholar
Thorpe, S.A. 1977 Turbulence and mixing in a Scottish Loch. Phil. Trans. R. Soc. Lond. A 286 (1334), 125–181.Google Scholar
Tseng, Y. & Ferziger, J.H. 2001 Mixing and available potential energy in stratified flows. Phys. Fluids 13 (5), 1281–1293.CrossRefGoogle Scholar
Van Mieghem, J. 1956 The energy available in the atmosphere for conversion into kinetic energy. Beitr. Zur Physik der Atmosphäre 29 (2), 129–142.Google Scholar
Weinstein, A. 1983 The local structure of Poisson manifolds. J. Differ. Geom. 18 (3), 523–557.Google Scholar
Winters, K.B. & Barkan, R. 2013 Available potential energy density for Boussinesq fluid flow. J. Fluid Mech. 714, 476–488.CrossRefGoogle Scholar
Winters, K.B., Lombard, P.N., Riley, J.J. & D'Asaro, E.A. 1995 Available potential energy and mixing in density-stratified fluids. J. Fluid Mech. 289, 115–128.CrossRefGoogle Scholar