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Taylor–Couette–Poiseuille flow with a weakly permeable inner cylinder: absolute instabilities and selection of global modes

  • Nils Tilton (a1) and Denis Martinand (a2)

Variations in the local stability of the flow in a Taylor–Couette cell can be imposed by adding an axial Poiseuille flow and a radial flow associated with one or both of the cylinders being permeable. At a given rotation rate of the inner cylinder, this results in adjacent regions of the flow that can be simultaneously stable, convectively unstable, and absolutely unstable, making this system fit for studying global modes of instability. To this end, building on the existing stability analysis in absolute modes developing over axially invariant base flows, we consider the case of axially varying base flows in systems for which the outer cylinder is impermeable, and the inner cylinder is a weakly permeable membrane through which the radial flow is governed by Darcy’s law. The frameworks of linear and nonlinear global modes are used to describe the instabilities and assess the results of direct numerical simulations using a dedicated pseudospectral method. Three different axially evolving set-ups are considered. In the first, fluid injection occurs along the full inner cylinder. In the second, fluid extraction occurs along the full inner cylinder. Besides its fundamental interest, this set-up is relevant to filtration devices. In the third, fluid flux through the inner cylinder evolves from extraction to injection as cross-flow reversal occurs. In agreement with the global mode analyses, the numerical simulations develop centrifugal instabilities above the predicted critical rotation rates and downstream of the predicted axial locations. The global mode analyses do not fully explain, however, that the instabilities observed in the numerical simulations take the form of axial stacks of wavepackets characterized by jumps of the temporal frequency.

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Altmeyer, S., Hoffmann, Ch. & Lücke, M. 2011 Islands of instability for growth of spiral vortices in the Taylor–Couette system with and without axial through flow. Phys. Rev. E 84, 046308.
Appelquist, E., Alfredsson, P., Schlatter, P. H. & Lingwood, R. J. 2016 On the global nonlinear instability of the rotating-disk flow over a finite domain. J. Fluid Mech. 803, 332355.
Appelquist, E., Schlatter, P. H., Alfredsson, P. & Lingwood, R. J. 2015 Global linear instability of the rotating-disk flow investigated through simulations. J. Fluid Mech. 765, 612631.
Babcock, K. L., Ahlers, G. & Cannell, D. S. 1991 Noise-sustained structure in Taylor–Couette flow with through-flow. Phys. Rev. Lett. 67, 33883391.
Babcock, K. L., Ahlers, G. & Cannell, D. S. 1994 Noise amplification in open Taylor–Couette flow. Phys. Rev. E 50 (5), 36703692.
Beaudoin, G. & Jaffrin, M. Y. 1989 Plasma filtration in Couette flow membrane devices. Artif. Organs 13 (1), 4351.
Büchel, P., Lücke, M., Roth, D. & Schmitz, R. 1996 Pattern selection in the absolutely unstable regime as a nonlinear eigenvalue problem: Taylor vortices in axial flow. Phys. Rev. E 53 (5), 47644776.
Carrière, Ph. & Monkewitz, P. A. 2001 Transverse-roll global modes in a Rayleigh–Bénard–Poiseuille convection. Eur. J. Mech. (B/Fluids) 20, 751770.
Chomaz, J.-M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357392.
Chomaz, J.-M. & Couairon, A. 1999 Against the wind. Phys. Fluids 11 (10), 29772983.
Chung, K. C. & Astill, K. N. 1977 Hydrodynamic instability of viscous flow between rotating coaxial cylinders with fully developed axial flow. J. Fluid Mech. 81, 641655.
Cotrell, D. L. & Pearlstein, A. J. 2004 The connection between centrifugal instability and Tollmien–Schlichting-like instability for spiral Poiseuille flow. J. Fluid Mech. 509, 331351.
Couairon, A. & Chomaz, J.-M. 1997 Absolute and convective instabilities, front velocities and global modes in nonlinear systems. Physica D 108, 236276.
Dee, G. & Langer, J. 1983 Propagating pattern selection. Phys. Rev. Lett. 50, 383386.
DiPrima, R. C. 1960 The stability of a viscous fluid between rotating cylinders with an axial flow. J. Fluid Mech. 9, 621631.
Donnelly, R. J. & Fultz, D. 1960 Experiments on the stability of spiral flow between rotating cylinders. Proc. Natl Acad. Sci. USA 46, 11501154.
Grandjean, E. & Monkewitz, P. A. 2009 Experimental investigation into localized instabilities of mixed Rayleigh–Bénard–Poiseuille convection. J. Fluid Mech. 640, 401419.
Hallström, B. & Lopez-Leiva, M. 1978 Decription of a rotating ultrafiltration module. Desalination 24 (1–3), 273279.
Healey, J. J. 2010 Model for unstable global modes in rotating-disk boundary layer. J. Fluid Mech. 663, 148159.
Huerre, P. 2000 Open shear flow instabilities. In Perspectives in Fluid Dynamics (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G.), pp. 159229. Cambridge University Press.
Huerre, P. & Monkewitz, P. A. 1985 Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159, 151168.
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473537.
Imayama, S., Alfredsson, P. H. & Lingwood, R. J. 2016 Experimental study of rotating-disk boudary-layer flow with surface roughness. J. Fluid Mech. 786, 528.
Johnson, E. C. & Lueptow, R. M. 1997 Hydrodynamic stability of flow between rotating porous cylinders with radial and axial flow. Phys. Fluids 9 (12), 36873696.
Koshmieder, E. L. 1993 Bénard Cells and Taylor Vortices. Cambridge University Press.
Lee, S. & Lueptow, R. M. 2004 Rotating membrane filtration and rotating reverse osmosis. J. Chem. Engng Japan 37 (4), 471482.
Lueptow, R. M., Docter, A. & Min, K. 1992 Stability of axial flow in an annulus with a rotating inner cylinder. Phys. Fluids A 4 (11), 24462455.
Martinand, D., Carrière, Ph. & Monkewitz, P. A. 2006 Three-dimensional global instability modes associated with a localized hot spot in Rayleigh–Bénard–Poiseuille convection. J. Fluid Mech. 551, 275301.
Martinand, D., Serre, E. & Lueptow, R. M. 2009 Absolute and convective instability of cylindrical Couette flow with axial and radial flows. Phys. Fluids 21, 104102.
Martinand, D., Serre, E. & Lueptow, R. M. 2017 Weakly nonlinear analysis of cylindrical Couette flow with axial and radial flows. J. Fluid Mech. 824, 438476.
Meseguer, A. & Marques, F. 2002 On the competition between centrifugal and shear instability in spiral Poiseuille flow. J. Fluid Mech. 455, 129148.
Monkewitz, P. A., Huerre, P. & Chomaz, J.-M. 1993 Global linear stability analysis of weakly non-parallel shear flows. J. Fluid Mech. 288, 120.
Nicolas, X. 2002 Bibliographical review on the Poiseuille–Rayleigh–Bénard flows: the mixed convection flows in horizontal rectangular ducts heated from below. Intl J. Therm. Sci. 41 (10), 9611016.
Pier, B. 2003 Finite-amplitude crossflow vortices, secondary instability and transition in the rotating-disk boundary layer. J. Fluid Mech. 487, 315343.
Pier, B. 2013 Transition near the edge of a rotating disk. J. Fluid Mech. 737, R1.
Pier, B. & Huerre, P. 1996 Fully nonlinear global modes in spatially developing media. Physica D 97, 206222.
Pier, B., Huerre, P. & Chomaz, J.-M. 2001 Bifurcation to fully nonlinear synchronized structures in slowly varying media. Physica D 148, 4996.
Pier, B., Huerre, P., Chomaz, J.-M. & Couairon, A. 1998 Steep nonlinear global modes in spatially developing media. Phys. Fluids 10, 24332435.
Provansal, M., Mathis, C. & Boyer, L. 1987 Bénard–von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 122.
Rayleigh, Lord 1916 On convection currents in a horizontal layer of fluid when the higher temperature is on the under side. Phil. Mag. 32, 529546.
Recktenwald, A., Lücke, M. & Müller, H. W. 1993 Taylor vortex formation in axial through-flow: linear and weakly nonlinear analysis. Phys. Rev. E 48 (6), 44444454.
van Saarloos, W. & Hohenberg, P. 1992 Fronts, pulses, sources and sinks in generalized complex Ginzburg–Landau equations. Physica D 56, 303367.
Schwille, J. A., Mitra, D. & Lueptow, R. M. 2002 Design parameters for rotating filtration. J. Membr. Sci. 204 (1–2), 5365.
Soward, A. M. & Jones, C. A. 1983 The linear stability of the flow in the narrow gap between two concentric rotating spheres. Q. J. Mech. Appl. Math. 36, 1942.
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223, 289343.
Theofilis, V. 2003 Advances in global linear instability analysis of non-parallel and three-dimensional flows. Prog. Aerosp. Sci. 39, 289315.
Tilton, N., Martinand, D., Serre, E. & Lueptow, R. M. 2010 Pressure-driven radial flow in a Taylor–Couette cell. J. Fluid Mech. 660, 527537.
Tilton, N., Serre, E., Martinand, D. & Lueptow, R. M. 2014 A 3D pseudospectral algorithm for fluid flows with permeable walls. Application to filtration. Comput. Fluids 93, 129145.
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Journal of Fluid Mechanics
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