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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]
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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
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

‡

Now independent researcher.

References

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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