Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-13T08:29:43.885Z Has data issue: false hasContentIssue false
  • English
  • Français

Homogenization-based design of microstructured membranes: wake flows past permeable shells

Published online by Cambridge University Press:  29 September 2021

Pier Giuseppe Ledda Open the ORCID record for Pier Giuseppe Ledda [Opens in a new window] ,
E. Boujo Open the ORCID record for E. Boujo [Opens in a new window] ,
S. Camarri Open the ORCID record for S. Camarri [Opens in a new window] ,
F. Gallaire Open the ORCID record for F. Gallaire [Opens in a new window]  and
G.A. Zampogna Open the ORCID record for G.A. Zampogna [Opens in a new window]
Pier Giuseppe Ledda*
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
E. Boujo
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
S. Camarri
Affiliation:
Dipartimento di Ingegneria Civile e Industriale, Università degli Studi di Pisa, 56122 Pisa, Italy
F. Gallaire
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
G.A. Zampogna
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A formal framework to characterize and control/optimize the flow past permeable membranes by means of a homogenization approach is proposed and applied to the wake flow past a permeable cylindrical shell. From a macroscopic viewpoint, a Navier-like effective stress jump condition is employed to model the presence of the membrane, in which the normal and tangential velocities at the membrane are respectively proportional to the so-called filtrability and slip numbers multiplied by the stresses. Regarding the particular geometry considered here, a characterization of the steady flow for several combinations of constant filtrability and slip numbers shows that the flow morphology is dominantly influenced by the filtrability and exhibits a recirculation region that moves downstream of the body and eventually disappears as this number increases. A linear stability analysis further shows the suppression of vortex shedding as long as large values of the filtrability number are employed. In the control/optimization phase, specific objectives for the macroscopic flow are formulated by adjoint methods. A homogenization-based inverse procedure is proposed to obtain the optimal constrained microscopic geometry from macroscopic objectives, which accounts for fast variations of the filtrability and slip profiles along the membrane. As a test case for the proposed design methodology, a cylindrical membrane is designed to maximize the resulting drag coefficient.

JFM classification

Flow Control: Flow Control Biological Fluid Dynamics: Membranes Low-Reynolds-number flows: Porous media
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 927 , 25 November 2021 , A31
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

Abderrahaman-Elena, N. & García-Mayoral, R. 2017 Analysis of anisotropically permeable surfaces for turbulent drag reduction. Phys. Rev. Fluids 2 (11), 114609.CrossRefGoogle Scholar
Asadzadeh, S.S., Nielsen, L.T., Andersen, A., Dölger, J., Kiørboe, T., Larsen, P.S. & Walther, J.H. 2019 Hydrodynamic functionality of the lorica in choanoflagellates. J. R. Soc. Interface 16 (150), 20180478.CrossRefGoogle ScholarPubMed
Baddoo, P.J, Hajian, R. & Jaworski, J.W 2021 Unsteady aerodynamics of porous aerofoils. J. Fluid Mech. 913, A16.CrossRefGoogle Scholar
Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75 (5), 750–756.CrossRefGoogle Scholar
Barkley, D. & Henderson, R.D. 1996 Three-dimensional floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215–241.CrossRefGoogle Scholar
Barta, E. & Weihs, D. 2006 Creeping flow around a finite row of slender bodies in close proximity. J. Fluid Mech. 551, 1–17.CrossRefGoogle Scholar
Bongarzone, A., Bertsch, A., Renaud, P. & Gallaire, F. 2021 Impinging planar jets: hysteretic behaviour and origin of the self-sustained oscillations. J. Fluid Mech. 913, A51.CrossRefGoogle Scholar
Boujo, E., Ehrenstein, U. & Gallaire, F. 2013 Open-loop control of noise amplification in a separated boundary layer flow. Phys. Fluids 25 (12), 124106.CrossRefGoogle Scholar
Boujo, E. & Gallaire, F. 2014 Controlled reattachment in separated flows: a variational approach to recirculation length reduction. J. Fluid Mech. 742, 618–635.CrossRefGoogle Scholar
Boujo, E. & Gallaire, F. 2015 Sensitivity and open-loop control of stochastic response in a noise amplifier flow: the backward-facing step. J. Fluid Mech. 762, 361–392.CrossRefGoogle Scholar
Brinkman, H.C. 1949 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Flow Turbul. Combust. 1 (1), 27.CrossRefGoogle Scholar
Camarri, S. & Iollo, A. 2010 Feedback control of the vortex-shedding instability based on sensitivity analysis. Phys. Fluids 22 (9), 094102.CrossRefGoogle Scholar
Castro, I.P. 1971 Wake characteristics of two-dimensional perforated plates normal to an air-stream. J. Fluid Mech. 46 (3), 599–609.CrossRefGoogle Scholar
Chomaz, J-M 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37 (1), 357–392.CrossRefGoogle Scholar
Crabill, J., Witherden, F.D. & Jameson, A. 2018 A parallel direct cut algorithm for high-order overset methods with application to a spinning golf ball. J. Comput. Phys. 374, 692–723.CrossRefGoogle Scholar
Cummins, C., Seale, M., Macente, A., Certini, D., Mastropaolo, E., Viola, I.M. & Nakayama, N. 2018 A separated vortex ring underlies the flight of the dandelion. Nature 562 (7727), 414–418.CrossRefGoogle ScholarPubMed
Cummins, C., Viola, I.M., Mastropaolo, E. & Nakayama, N. 2017 The effect of permeability on the flow past permeable disks at low Reynolds numbers. Phys. Fluids 29, 097103.CrossRefGoogle Scholar
Dalwadi, M.P., Bruna, M. & Griffiths, I.M. 2016 A multiscale method to calculate filter blockage. J. Fluid Mech. 809, 264–289.CrossRefGoogle Scholar
Darcy, H. 1856 Les fontaines publiques de la ville de Dijon: exposition et application des principes a suivre et des formules a employer dans les questions de distribution d'eau. Victor Dalmont.Google Scholar
Elimelech, M. & Phillip, W. 2011 The future of seawater desalination: energy, technology, and the environment. Science 333, 712–717.CrossRefGoogle ScholarPubMed
Ellington, C.P. 1980 Wing mechanics and take-off preparation of thrips (thysanoptera). J. Expl Biol. 85 (1), 129–136.CrossRefGoogle Scholar
Fornberg, B. 1980 A numerical study of steady viscous flow past a circular cylinder. J. Fluid Mech. 98 (4), 819–855.CrossRefGoogle Scholar
Fritzmann, C., Löwenberg, J., Wintgens, T. & Melin, T. 2007 State-of-the-art of reverse osmosis desalination. Desalin. 216, 1–76.CrossRefGoogle Scholar
Garcia-Mayoral, R. & Jiménez, J. 2011 Drag reduction by riblets. Phil. Trans. R. Soc. A 369 (1940), 1412–1427.CrossRefGoogle ScholarPubMed
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167–197.CrossRefGoogle Scholar
Hajian, R. & Jaworski, J.W 2017 The steady aerodynamics of aerofoils with porosity gradients. Proc. R. Soc. A 473 (2205), 20170266.CrossRefGoogle ScholarPubMed
Hornung, U. 1997 Homogenization and Porous Media. Springer.CrossRefGoogle Scholar
Icardi, M., Boccardo, G., Marchisio, D.L., Tosco, T. & Sethi, R. 2014 Pore-scale simulation of fluid flow and solute dispersion in three-dimensional porous media. Phys. Rev. E 90, 013032.CrossRefGoogle ScholarPubMed
Iungo, G.V., Viola, F., Camarri, S., Porté-Agel, F. & Gallaire, F. 2013 Linear stability analysis of wind turbine wakes performed on wind tunnel measurements. J. Fluid Mech. 737, 499–526.CrossRefGoogle Scholar
Jackson, C.P. 1987 A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 23–45.CrossRefGoogle Scholar
Jaworski, J.W. & Peake, N. 2020 Aeroacoustics of silent owl flight. Annu. Rev. Fluid Mech. 52 (1), 395–420.CrossRefGoogle Scholar
Jensen, K.H., Berg-Sørensen, K., Bruus, H., Holbrook, N.M., Liesche, J., Schulz, A., Zwieniecki, M.A. & Bohr, T. 2016 Sap flow and sugar transport in plants. Rev. Mod. Phys. 88, 035007.CrossRefGoogle Scholar
Jones, S.K., Yun, Y.J.J., Hedrick, T.L., Griffith, B.E. & Miller, L.A. 2016 Bristles reduce the force required to ‘fling’ wings apart in the smallest insects. J. Expl Biol. 219 (23), 3759–3772.CrossRefGoogle ScholarPubMed
Labbé, R & Duprat, C 2019 Capturing aerosol droplets with fibers. Soft Matt. 15 (35), 6946–6951.CrossRefGoogle ScholarPubMed
Lācis, U. & Bagheri, S. 2017 A framework for computing effective boundary conditions at the interface between free fluid and a porous medium. J. Fluid Mech. 812, 866–889.CrossRefGoogle Scholar
Lācis, U., Zampogna, G.A. & Bagheri, S. 2017 A computational continuum model of poroelastic beds. Proc. R. Soc. A 473 (2199), 20160932.CrossRefGoogle ScholarPubMed
Lācis, U., Sudhakar, Y., Pasche, S. & Bagheri, S. 2020 Transfer of mass and momentum at rough and porous surfaces. J. Fluid Mech. 884, A21.CrossRefGoogle Scholar
Ledda, P.G., Siconolfi, L., Viola, F., Camarri, S. & Gallaire, F. 2019 Flow dynamics of a dandelion pappus: A linear stability approach. Phys. Rev. Fluids 4, 071901.CrossRefGoogle Scholar
Ledda, P.G., Siconolfi, L., Viola, F., Gallaire, F. & Camarri, S. 2018 Suppression of von kármán vortex streets past porous rectangular cylinders. Phys. Rev. Fluids 3, 103901.CrossRefGoogle Scholar
Lemke, M., Reiss, J. & Sesterhenn, J. 2014 Adjoint based optimisation of reactive compressible flows. Combust. Flame 161 (10), 2552–2564.CrossRefGoogle Scholar
Lin, C.-J. 2007 Projected gradient methods for nonnegative matrix factorization. Neural Comput. 19 (10), 2756–2779.CrossRefGoogle ScholarPubMed
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46 (1), 493–517.CrossRefGoogle Scholar
Marquet, O., Sipp, D. & Jacquin, L. 2008 Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221–252.CrossRefGoogle Scholar
Matin, A., Khan, Z., Zaidia, S.M.J. & Boyce, M.C. 2011 Biofouling in reverse osmosis membranes for seawater desalination: phenomena and prevention. Desalin. 281, 1–16.CrossRefGoogle Scholar
Meliga, P., Boujo, E., Pujals, G. & Gallaire, F. 2014 Sensitivity of aerodynamic forces in laminar and turbulent flow past a square cylinder. Phys. Fluids 26 (10), 104101.CrossRefGoogle Scholar
Meliga, P., Chomaz, J.-M. & Sipp, D. 2009 Unsteadiness in the wake of disks and spheres: instability, receptivity and control using direct and adjoint global stability analyses. J. Fluids Struct. 25 (4), 601–616.CrossRefGoogle Scholar
Meliga, P., Sipp, D. & Chomaz, J.-M. 2010 Open-loop control of compressible afterbody flows using adjoint methods. Phys. Fluids 22 (5), 054109.CrossRefGoogle Scholar
Monkewitz, P.A. 1988 The absolute and convective nature of instability in two dimensional wakes at low Reynolds numbers. Phys. Fluids 31 (5), 999–1006.CrossRefGoogle Scholar
Nemili, A., Özkaya, E., Gauger, N., Thiele, F. & Carnarius, A. 2011 Optimal control of unsteady flows using discrete adjoints. In 41st AIAA Fluid Dynamics Conference And Exhibit, AIAA paper 2011-3720.Google Scholar
Nicolle, A. & Eames, I. 2011 Numerical study of flow through and around a circular array of cylinders. J. Fluid Mech. 679, 1–31.CrossRefGoogle Scholar
Norberg, C. 2003 Fluctuating lift on a circular cylinder: review and new measurements. J. Fluids Struct. 17 (1), 57–96.CrossRefGoogle Scholar
Olivier, J. 2004 Fog harvesting: an alternative source of water supply on the west coast of South Africa. GeoJ. 61 (2), 203–214.CrossRefGoogle Scholar
Park, H.B., Kamcev, J., Robeson, L.M., Elimelech, M. & Freeman, B.D. 2017 Maximizing the right stuff: the trade-off between membrane permeability and selectivity. Science 356, eaab0530.CrossRefGoogle ScholarPubMed
Park, K-C, Chhatre, S.S., Srinivasan, S., Cohen, R.E. & McKinley, G.H. 2013 Optimal design of permeable fiber network structures for fog harvesting. Langmuir 29 (43), 13269–13277.CrossRefGoogle ScholarPubMed
Pasche, S., Avellan, F. & Gallaire, F. 2017 Part load vortex rope as a global unstable mode. Trans. ASME: J. Fluids Engng 139 (5), 051102.Google Scholar
Pezzulla, M., Strong, E.F., Gallaire, F. & Reis, P.M. 2020 Deformation of porous flexible strip in low and moderate Reynolds number flows. Phys. Rev. Fluids 5, 084103.CrossRefGoogle Scholar
Provansal, M., Mathis, C. & Boyer, L. 1987 Bénard-von kármán instability: transient and forced regimes. J. Fluid Mech. 182, 1–22.CrossRefGoogle Scholar
Quarteroni, A. 2017 Domain Decomposition Methods, pp. 555–612. Springer International Publishing.Google Scholar
Rahardianto, A., McCool, B.C. & Cohen, Y. 2010 Accelerated desupersaturation of reverse osmosis concentrate by chemically-enhanced seeded precipitation. Desalin. 264, 256–267.CrossRefGoogle Scholar
Schulze, J. & Sesterhenn, J. 2013 Optimal distribution of porous media to reduce trailing edge noise. Comput. Fluids 78, 41–53, lES of turbulence aeroacoustics and combustion.CrossRefGoogle Scholar
Gómez-de Segura, G. & García-Mayoral, R. 2019 Turbulent drag reduction by anisotropic permeable substrates – analysis and direct numerical simulations. J. Fluid Mech. 875, 124–172.CrossRefGoogle Scholar
Shannon, M.A., Bohn, P.W., Elimelech, M., Georgiadis, J.G., Mari nas, B.J. & Mayes, A.M. 2008 Science and technology for water purification in the coming decades. Nature 452, 301–310.CrossRefGoogle ScholarPubMed
Shi, W., Anderson, M.J., Tulkoff, J.B., Kennedy, B.S. & Boreyko, J.B. 2018 Fog harvesting with harps. ACS Appl. Mater. Interfaces 10 (14), 11979–11986.CrossRefGoogle ScholarPubMed
Steiros, K. & Hultmark, M. 2018 Drag on flat plates of arbitrary porosity. J. Fluid Mech. 853, R3.CrossRefGoogle Scholar
Steiros, K., Kokmanian, K., Bempedelis, N. & Hultmark, M. 2020 The effect of porosity on the drag of cylinders. J. Fluid Mech. 901, R2.CrossRefGoogle Scholar
Strong, E.F., Pezzulla, M., Gallaire, F., Reis, P. & Siconolfi, L. 2019 Hydrodynamic loading of perforated disks in creeping flows. Phys. Rev. Fluids 4, 084101.CrossRefGoogle Scholar
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43 (1), 319–352.CrossRefGoogle Scholar
Viola, F., Iungo, G.V., Camarri, S., Porté-Agel, F. & Gallaire, F. 2014 Prediction of the hub vortex instability in a wind turbine wake: stability analysis with eddy-viscosity models calibrated on wind tunnel data. J. Fluid Mech. 750, R1.CrossRefGoogle Scholar
Wagner, H., Weger, M., Klaas, M. & Schröder, W. 2017 Features of owl wings that promote silent flight. Interface Focus 7 (1), 20160078.CrossRefGoogle ScholarPubMed
Willert, C., Schulze, M., Waltenspül, S., Schanz, D. & Kompenhans, J. 2019 Prandtl's flow visualization film c1 revisited. In 13th Int. Symp. on Particle Image Velocimetry. arXiv: https://elib.dlr.de/128984/1/ISPIV2019_Willert_Paper161.pdf.Google Scholar
Williamson, C.H.K. 1996 Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28 (1), 477–539.CrossRefGoogle Scholar
Zampogna, G.A. & Gallaire, F. 2020 Effective stress jump across membranes. J. Fluid Mech. 892, A9.CrossRefGoogle Scholar
Zampogna, G.A., Pluvinage, F., Kourta, A. & Bottaro, A. 2016 Instability of canopy flows. Water Resour. Res. 52 (7), 5421–5432.CrossRefGoogle Scholar
Zampogna, G.A. & Bottaro, A. 2016 Fluid flow over and through a regular bundle of rigid fibres. J. Fluid Mech. 792, 5–35.CrossRefGoogle Scholar
Zampogna, G.A., Magnaudet, J. & Bottaro, A. 2019 Generalized slip condition over rough surfaces. J. Fluid Mech. 858, 407–436.CrossRefGoogle Scholar
Zong, L. & Nepf, H. 2012 Vortex development behind a finite porous obstruction in a channel. J. Fluid Mech. 691, 368–391.CrossRefGoogle Scholar