Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-27T21:17:00.154Z Has data issue: false hasContentIssue false
  • English
  • Français

Scaling in concentration-driven convection boundary layers with transpiration

Published online by Cambridge University Press:  17 September 2020

G. V. Ramareddy Open the ORCID record for G. V. Ramareddy [Opens in a new window] ,
P. J. Joshy ,
Gayathri Nair  and
Baburaj A. Puthenveettil Open the ORCID record for Baburaj A. Puthenveettil [Opens in a new window]
G. V. Ramareddy*
Affiliation:
Department of Mechanical Engineering, University of Melbourne, Parkville, VIC 3010, Australia
P. J. Joshy
Affiliation:
Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai600036, Tamil Nadu, India
Gayathri Nair
Affiliation:
Chennai Mathematical Institute, SIPCOT IT Park, Kelambakkam, Siruseri603103, Tamil Nadu, India
Baburaj A. Puthenveettil
Affiliation:
Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai600036, Tamil Nadu, India
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study concentration-driven natural convection boundary layers on horizontal surfaces, subjected to a weak, surface normal, uniform blowing velocity $V_i$ for three orders of range of the dimensionless blowing parameter $10^{-8}\le J=Re_x^3/Gr_x\le 10^{-5}$, where $Re_x$ and $Gr_x$ are the local Reynolds and Grashof numbers at the horizontal location $x$, based respectively on $V_i$ and ${\rm \Delta} C$, the concentration difference across the boundary layer. We formulate the integral boundary layer equations, with the assumption of no concentration drop within the species boundary layer, which is valid for weak blowing into the thin species boundary layers that occur at the high Schmidt number ($Sc \simeq 600$) of concentration-driven convection. The equations are then numerically solved to show that the species boundary layer thickness $\delta _d = 1.6\,x(Re_x/Gr_x)^{1/4}$, the velocity boundary layer thickness $\delta _v=\delta _d Sc^{1/5}$, the horizontal velocity $u = V_i(Gr_x/Re_x)^{1/4}f(\eta )$, where $\eta =y/\delta _v$, and the drag coefficient based on $V_i$, $C_D = 2.32/\sqrt {J}$. We find that the vertical profile of the horizontally averaged dimensionless concentration across the boundary layer becomes, surprisingly, independent of the blowing and the species diffusion effects to follow a $Gr_y^{2/3}$ scaling, where $Gr_y$ is the Grashof number based on the vertical location $y$ within the boundary layer. We then show that the above profile matches the experimentally observed mean concentration profile within the boundary layers that form on the top surface of a membrane, when a weak flow is forced gravitationally from below the horizontal membrane that has brine above it and water below it. A similar match between the theoretical scaling of the species boundary layer thickness and its experimentally observed variation is also shown to occur.

JFM classification

Convection: Convection in porous media
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 903 , 25 November 2020 , A3
Check for updates
Copyright
© The Author(s), 2020. 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.)

Footnotes

‡

Now independent researcher.

References

REFERENCES

Atkinson, P. M. & Lloyd, D. R. 2000 Anisotropic flat sheet membrane formation via tips: atmospheric convection and polymer molecular weight effects. J. Membr. Sci. 175 (2), 225–238.CrossRefGoogle Scholar
Chen, T. S., Buchanan, W. P. & Armaly, B. F. 1993 Natural convection on vertical and horizontal plates with vectored surface mass transfer. Intl J. Heat Mass Transfer 36 (2), 479–487.CrossRefGoogle Scholar
Chen, T. S., Tien, H.-C. & Armaly, B. F. 1986 Natural convection on horizontal, inclined, and vertical plates with variable surface temperature or heat flux. Intl J. Heat Mass Transfer 29 (10), 1465–1478.CrossRefGoogle Scholar
Clarke, J. F. & Riley, N. 1975 Natural convection induced in a gas by the presence of a hot porous horizontal surface. Q. J. Mech. Appl. Maths 28, 373–395.CrossRefGoogle Scholar
Clarke, J. F. & Riley, N. 1976 Free convection and the burning of a horizontal fuel surface. J. Fluid Mech. 74 (3), 415–431.CrossRefGoogle Scholar
Clausing, A. M. & Berton, J. J. 1989 An experimental investigation of natural convection from an isothermal horizontal plate. Trans. ASME: J. Heat Transfer 111, 904–908.CrossRefGoogle Scholar
Coussot, P. 2000 Scaling approach of the convective drying of a porous medium. Eur. Phys. J. B 15 (3), 557–566.CrossRefGoogle Scholar
Gebhart, B., Jaluria, Y., Mahajan, R. L. & Sammakia, B. 1988 Buoyancy-Induced Flows and Transport. Hemisphere Publishing.Google Scholar
Gill, W. F., Zeh, D. W. & Casal, E. D. 1965 Free convection on a horizontal plate. Z. Angew. Math. Phys. 16, 387–389.CrossRefGoogle Scholar
Goldstein, R. J., Sparrow, E. M. & Jones, D. C. 1973 Natural convection mass transfer adjacent to horizontal plates. Intl J. Heat Mass Transfer 16 (5), 1025–1035.CrossRefGoogle Scholar
Gunasegarane, G. S. & Puthenveettil, B. A. 2014 Dynamics of line plumes on horizontal surfaces in turbulent convection. J. Fluid Mech. 749, 37–78.CrossRefGoogle Scholar
Huppert, H. E. & Neufeld, J. A. 2014 The fluid mechanics of carbon dioxide sequestration. Annu. Rev. Fluid Mech. 46, 255–272.CrossRefGoogle Scholar
Kitamura, K. & Kimura, F. 1995 Heat transfer and fluid flow of natural convection adjacent to upward-facing horizontal plates. Intl J. Heat Mass Transfer 38 (17), 3149–3159.CrossRefGoogle Scholar
Kneafsey, T. J. & Pruess, K. 2010 Laboratory flow experiments for visualizing carbon dioxide-induced, density-driven brine convection. Transp. Porous Med. 82 (1), 123–139.CrossRefGoogle Scholar
Kozanoglu, B. & Lopez, J. 2007 Thermal boundary layer and the characteristic length on natural convection over a horizontal plate. Heat Mass Transfer 43 (4), 333–339.CrossRefGoogle Scholar
Lide, D. R. 2003 CRC Handbook of Chemistry and Physics, 84th edn, chap. 5. CRC Press.Google Scholar
Lin, H. T. & Yu, W. S. 1988 Free convection on a horizontal plate with blowing and suction. Trans. ASME: J. Heat Transfer 110, 793–796.CrossRefGoogle Scholar
Lloyd, J. R. & Moran, W. R. 1974 Natural convection adjacent to horizontal surface of various planforms. Trans. ASME: J. Heat Transfer 96 (4), 443–447.CrossRefGoogle Scholar
Merkin, J. H. 1975 The effects of blowing and suction on free convection boundary layers. Intl J. Heat Mass Transfer 18 (2), 237–244.CrossRefGoogle Scholar
Merkin, J. H. 1985 Free convection above a uniformly heated horizontal circular disk. Intl J. Heat Mass Transfer 28 (6), 1157–1163.CrossRefGoogle Scholar
Pera, L. & Gebhart, B. 1973 Natural convection boundary layer flow over horizontal and slightly inclined surfaces. Intl J. Heat Mass Transfer 16 (6), 1131–1146.CrossRefGoogle Scholar
Puthenveettil, B. A. 2004 Investigations on high Rayleigh number turbulent free convection. PhD thesis, Indian Institute of Science, Bangalore.Google Scholar
Puthenveettil, B. A. & Arakeri, J. H. 2004 Integral analysis of laminar indirect free convection boundary layers with weak blowing for Schmidt no. $\sim$1. Intl Commun. Heat Mass Transfer 31 (8), 1199–1208.CrossRefGoogle Scholar
Puthenveettil, B. A. & Arakeri, J. H. 2005 Plume structure in high Rayleigh-number convection. J. Fluid Mech. 542, 217–249.CrossRefGoogle Scholar
Puthenveettil, B. A. & Arakeri, J. H. 2008 Convection due to an unstable density gradient across a permeable membrane. J. Fluid Mech. 609, 139–170.CrossRefGoogle Scholar
Puthenveettil, B. A., Gunasegarane, G. S., Agrawal, Y. K., Schmeling, D., Bosbach, J. & Arakeri, J. H. 2011 Length of near-wall plumes in turbulent convection. J. Fluid Mech. 685, 335–364.CrossRefGoogle Scholar
Radziemska, E. & Lewandowski, W. M. 2001 Heat transfer by natural convection from an isothermal downward-facing round plate in unlimited space. Appl. Energy 68 (4), 347–366.CrossRefGoogle Scholar
Ramareddy, G. V. 2009 Investigations on convection across a horizontal permeable membrane. Master's thesis, Indian Institute of Technology Madras, Chennai.Google Scholar
Ramareddy, G. V. & Puthenveettil, B. A. 2011 The $Pe\sim 1$ regime of convection across a horizontal permeable membrane. J. Fluid Mech. 679, 476–504.CrossRefGoogle Scholar
Rotem, Z. & Claassen, L. 1969 Natural convection above unconfined horizontal surfaces. J. Fluid Mech. 39 (1), 173–192.CrossRefGoogle Scholar
Samanta, S. & Guha, A. 2012 A similarity theory for natural convection from a horizontal plate for prescribed heat flux or wall temperature. Intl J. Heat Mass Transfer 55 (13–14), 3857–3868.CrossRefGoogle Scholar
Schlichting, H. & Gersten, K. 2017 Boundary-Layer Theory. Springer.CrossRefGoogle Scholar
Seuntiens, H. J., Kieft, R. N., Rindt, C. C. M. & van Steenhoven, A. A. 2001 2D temperature measurements in the wake of a heated cylinder using LIF. Exp. Fluids 31, 588–595.Google Scholar
Shemilt, L. W. & Sedahmed, G. H. 1976 Natural convection mass transfer at horizontal screens. J. Appl. Electrochem. 6, 471–476.CrossRefGoogle Scholar
Slezak, A., Grzegorczyn, S., Jasik-Slezak, J. & Michalska-Malecka, K. 2010 Natural convection as an asymmetrical factor of the transport through porous membrane. Transp. Porous Med. 84, 685–698.CrossRefGoogle Scholar
Stewartson, K. 1958 On the free convection from a horizontal plate. Z. Angew. Math. Phys. 9 (3), 276–282.CrossRefGoogle Scholar
Tanaka, Y. 1991 Concentration polarization in ion exchange membrane electrodialysis. J. Membr. Sci. 57 (2–3), 217–235.CrossRefGoogle Scholar
Theerthan, S. A. & Arakeri, J. H. 1998 A model for near-wall dynamics in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 373, 221–254.CrossRefGoogle Scholar
Umemura, A. & Law, C. K. 1990 Natural-convection boundary-layer flow over a heated plate with arbitrary inclination. J. Fluid Mech. 219, 571–584.CrossRefGoogle Scholar
Youm, K. H., Fane, A. G. & Wiley, D. E. 1996 Effects of natural convection instability on membrane performance in dead-end and cross-flow ultrafiltration. J. Membr. Sci. 116 (2), 229–241.CrossRefGoogle Scholar

Ramareddy et al. supplementary movie 1

Spatial and temporal evolution of sheet plumes over a horizontal permeable membrane of area 150 mm x 150 mm, which separates brine above it from water below it so as to have an unstable density gradient across it, when water is forced through the membrane from below with a uniform velocity of 0.002 cm/s. The lighter green colour streaks are the line plumes which evolve as sheets into the bulk fluid above them. The density difference across the membrane decreases with time as the experiment progresses. The Rayleigh number, based on the density difference and the height of the liquid layer above the membrane, of the first frame of the movie = 8.1*10^11. The movie plays with the same frame rate (25 Hz) as it was acquired.
Download Ramareddy et al. supplementary movie 1(Video)
Video 89.1 MB

Ramareddy et al. supplementary movie 2

Front view of the spatial and temporal evolution of near-membrane sheet plumes when water with a constant, uniform velocity of 0.002 cm/s is forced through the membrane from below, with the membrane separating brine above it from water below it. The membrane is 150 mm wide, of which a region of 20 mm width is shown in the movie. The red regions at the bottom of the movie, of approximately up to 0.3 mm height, are the boundary layers, while the swaying near-vertical red streaks are the sheet plumes, with the yellow background being the bulk fluid. The Rayleigh number, based on the density difference and the height of the liquid layer above the membrane, of the first frame of the movie = 8.4*10^11. The movie plays with the same frame rate (14 Hz) as it was acquired.

Download Ramareddy et al. supplementary movie 2(Video)
Video 15.3 MB