This paper investigates flow through variable permeability, two-dimensional circular cylinders using a pseudospectral numerical model. Two types of permeability (K) distributions are considered: constant with a lower permeability blockage, and constant with a higher permeability duct. Boundary conditions set by external flow with high Reynolds number lead to streamwise flow asymmetry and more short length scale variability within the cylinder when compared to conditions set by potential flow. High permeability belts are observed to guide flow around regions of lower permeability, while low permeability belts are observed to impede flow from reaching areas surrounded by the low permeability region. Inward surface flux is used to quantify changes in flow through variable permeability cylinders relative to the constant permeability cylinder. For blocking cases, the relationship between and the largest possible change in relative surface flux is nearly linear. In ducting simulations, where to , this relationship is no longer linear. Simple polynomial fits are derived for both situations, allowing for the calculation of the change in permeability required to achieve a given increase or reduction in inward flux. Finally, the numerical results are contrasted with theoretical perturbation results for the case of azimuthal variations in permeability, which lead to a fundamentally different pressure distribution.
References
1.
M.
Huettel
, W.
Ziebis
, and S.
Forster
, “Flow-induced uptake of particulate matter in permeable sediments
,” Limnol. Oceanogr.
41
, 309
–322
(1996
).2.
J.
Olsthoorn
, M.
Stastna
, and N.
Soontiens
, “Fluid circulation and seepage in lake sediment due to propagating and trapped internal waves
,” Water Resour. Res.
48
, W11520
, (2012
).3.
R. T.
Amos
, D. W.
Blowes
, L.
Smith
, and D.
Sego
, “Measurement of wind-induced pressure gradients in a waste rock pile
,” Vadose Zone J.
8
, 953
–962
(2009
).4.
E. M.
Hinton
and A. J.
Hogg
, “Modeling the influence of a variable permeability inclusion on free-surface flow in an inclined aquifer
,” Water Resour. Res.
57
, e2020WR029195
, (2021
).5.
F.
Tatti
, M. P.
Papini
, G.
Sappa
, M.
Raboni
, F.
Arjmand
, and P.
Viotti
, “Contaminant back-diffusion from low-permeability layers as affected by groundwater velocity: A laboratory investigation by box model and image analysis
,” Sci. Total Environ.
622–623
, 164
–171
(2018
).6.
O.
Anterrieu
, M.
Chouteau
, and M.
Aubertin
, “Geophysical characterization of the large-scale internal structure of a waste rock pile from a hard rock mine
,” Bull. Eng. Geol. Environ.
69
, 533
–548
(2010
).7.
W.
Robertson
, N.
Yeung
, P.
vanDriel
, and P.
Lombardo
, “High-permeability layers for remediation of ground water; go wide, not deep
,” Groundwater
43
, 574
–581
(2005
).8.
R.
Lefebvre
, D.
Hockley
, J.
Smolensky
, and P.
Gélinas
, “Multiphase transfer processes in waste rock piles producing acid mine drainage 1: Conceptual model and system characterization
,” J. Contam. Hydrol.
52
, 137
–164
(2001
).9.
P. D.
Noymer
, L. R.
Glicksman
, and A.
Devendran
, “Drag on a permeable cylinder in steady flow at moderate Reynolds numbers
,” Chem. Eng. Sci.
53
, 2859
–2869
(1998
).10.
M.
Bovand
, S.
Rashidi
, M.
Dehesht
, and J. A.
Esfahani
, “Effect of fluid-porous interface conditions on steady flow around and through a porous circular cylinder
,” Int. J. Numer. Methods Heat Fluid Flow
25
, 1658
(2015
).11.
S.
Yu
, T.
Tang
, J.
Li
, and P.
Yu
, “Effect of Prandtl number on mixed convective heat transfer from a porous cylinder in the steady flow regime
,” Entropy
22
, 184
(2020
).12.
Y.
Zhuang
, H.
Yu
, and Q.
Zhu
, “Experimental and numerical investigations on the flow around and through the fractal soft rocks with water vapor absorption
,” Int. J. Heat Mass Transfer
96
, 413
–429
(2016
).13.
J.
Sun
and Q.
Zhu
, “Investigation on flow around and through a hygroscopic porous cylinder with consideration of compressibility of moist air
,” AIP Adv.
11
, 095316
(2021
).14.
E. M.
Hinton
and A. W.
Woods
, “The effect of vertically varying permeability on tracer dispersion
,” J. Fluid Mech.
860
, 384
–407
(2019
).15.
S. M.
Hassanizadeh
and W. G.
Gray
, “Boundary and interface conditions in porous media
,” Water Resources Res.
25
, 1705
–1715
(1989
).16.
M.
Chandesris
and D.
Jamet
, “Derivation of jump conditions for the turbulence k-ε model at a fluid/porous interface
,” Int. J. Heat Fluid Flow
30
, 306
–318
(2009
).17.
U.
Lācis
and S.
Bagheri
, “A framework for computing effective boundary conditions at the interface between free fluid and a porous medium
,” J. Fluid Mech.
812
, 866
–889
(2017
).18.
K.
Nandakumar
and J. H.
Masliyah
, “Laminar flow past a permeable sphere
,” Can. J. Chem. Eng.
60
, 202
–211
(1982
).19.
A. K.
Jain
and S.
Basu
, “Flow past a porous permeable sphere: Hydrodynamics and heat-transfer studies
,” Ind. Eng. Chem. Res.
51
, 2170
–2178
(2012
).20.
R. D.
Anne
and G.
Pantelis
, “Coupled natural convection and atmospheric wind forced advection in above ground reacting heaps
,” in International Conference on Computational Fluid Dynamics in Mineral and Metal Processing and Power Generation
(Melbourne, Australia
, 1997
) pp. 453
–458
.21.
G. S.
Beavers
and D. D.
Joseph
, “Boundary conditions at a naturally permeable wall
,” J. Fluid Mech.
30
, 197
–207
(1967
).22.
P. G.
Saffman
, “On the boundary condition at the surface of a porous medium
,” Stud. Appl. Mathematics
50
, 93
–101
(1971
).23.
Z.-G.
Feng
and E. E.
Michaelides
, “Motion of a permeable sphere at finite but small Reynolds numbers
,” Phys. Fluids
10
, 1375
–1383
(1998
).24.
M.
Carr
, “Penetrative convection in a superposed porous-medium-fluid layer via internal heating
,” J. Fluid Mech.
509
, 305
–329
(2004
).25.
26.
J.
Ochoa-Tapia
and S.
Whitaker
, “Momentum transfer at the boundary between a porous medium and a homogeneous fluid–I. Theoretical development
,” Int. J. Heat Mass Transfer
38
, 2635
–2646
(1995
).27.
J.
Ochoa-Tapia
and S.
Whitaker
, “Momentum jump condition at the boundary between a porous medium and a homogeneous fluid: Inertial effects
,” J. Porous Media
1
, 31
–217
(1998
).28.
A.
Bhattacharyya
and G.
Raja Sekhar
, “Viscous flow past a porous sphere with an impermeable core: Effect of stress jump condition
,” Chem. Eng. Sci.
59
, 4481
–4492
(2004
).29.
P.
Yu
, Y.
Zeng
, T. S.
Lee
, X. B.
Chen
, and H. T.
Low
, “Steady flow around and through a permeable circular cylinder
,” Comput. Fluids
42
, 1
–12
(2011
).30.
P.
Yu
, Y.
Zeng
, T. S.
Lee
, X. B.
Chen
, and H. T.
Low
, “Numerical simulation on steady flow around and through a porous sphere
,” Int. J. Heat Fluid Flow
36
, 142
–152
(2012
).31.
S.
Rashidi
, A.
Nouri-Borujerdi
, M.
Valipour
, R.
Ellahi
, and I.
Pop
, “Stress-jump and continuity interface conditions for a cylinder embedded in a porous medium
,” Transp. Porous Media
107
, 171
–186
(2015
).32.
J.
Penney
, “A computational study of pressure driven flow in waste rock piles
,” M.S. thesis (University of Waterloo
, 2012
).33.
S.
Bhattacharyya
, S.
Dhinakaran
, and A.
Khalili
, “Fluid motion around and through a porous cylinder
,” Chem. Eng. Sci.
61
, 4451
–4461
(2006
).34.
Q.
Zhu
, Y.
Chen
, and H.
Yu
, “Numerical simulation of the flow around and through a hygroscopic porous circular cylinder
,” Comput. Fluids
92
, 188
–198
(2014
).35.
A.
Roshko
, “Experiments on the flow past a circular cylinder at very high Reynolds number
,” J. Fluid Mech.
10
, 345
(1961
).36.
C. H. K.
Williamson
, “Vortex dynamics in the cylinder wake
,” Annu. Rev. Fluid Mech.
28
, 477
–539
(1996
).37.
P.
Catalano
, M.
Wang
, G.
Iaccarino
, and P.
Moin
, “Numerical simulation of the flow around a circular cylinder at high Reynolds numbers
,” Int. J. Heat Fluid Flow
24
, 463
–469
(2003
).38.
I.
Rodríguez
, O.
Lehmkuhl
, J.
Chiva
, R.
Borrell
, and A.
Oliva
, “On the flow past a circular cylinder from critical to super-critical Reynolds numbers: Wake topology and vortex shedding
,” Int. J. Heat Fluid Flow
55
, 91
–103
(2015
).39.
P. K.
Kundu
and I. M.
Cohen
, Fluid Mechanics
, 4th ed. (Academic Press
, Amsterdam
, 2008
).40.
G.
Neale
, N.
Epstein
, and W.
Nader
, “Creeping flow relative to permeable spheres
,” Chem. Eng. Sci.
28
, 1865
–1874
(1973
).41.
J. H.
Masliyah
and M.
Polikar
, “Terminal velocity of porous spheres
,” Can. J. Chem. Eng.
58
, 299
–302
(1980
).42.
J. P.
Boyd
and F.
Yu
, “Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan-Shepp ridge polynomials, Chebyshev-Fourier Series, cylindrical Robert functions, Bessel-Fourier expansions, square-to-disk conformal mapping and radial basis functions
,” J. Comput. Phys.
230
, 1408
– 1438
(2011
).43.
L. N.
Trefethen
, Spectral Methods in Matlab
(Society for Industrial and Applied Mathematics
, 2000
).44.
MathWorks, see https://www.mathworks.com/help/matlab/ref/quiver.html for “
quiver — Matlab Documentation (2021)
” (last accessed September 10, 2021).45.
M. H.
Protter
and H. F.
Weinberger
, Maximum Principles in Differential Equations, Partial Differential Equations
(Prentice-Hall
, 1967
).46.
H. H.
Gerke
and M. T.
van Genuchten
, “Macroscopic representation of structural geometry for simulating water and solute movement in dual-porosity media
,” Adv. Water Resour.
19
, 343
–357
(1996
).© 2021 Author(s). Published under an exclusive license by AIP Publishing.
2021
Author(s)
You do not currently have access to this content.
