Skip to main content

Linear stability of confined flow around a 180-degree sharp bend

  • Azan M. Sapardi (a1) (a2), Wisam K. Hussam (a1) (a3), Alban Pothérat (a4) and Gregory J. Sheard (a1)

This study seeks to characterise the breakdown of the steady two-dimensional solution in the flow around a 180-degree sharp bend to infinitesimal three-dimensional disturbances using a linear stability analysis. The stability analysis predicts that three-dimensional transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number $\mathit{Re}<1150$ and bend opening ratio (ratio of bend width to inlet height) $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 5$ . This range of $\mathit{Re}$ and $\unicode[STIX]{x1D6FD}$ captures both steady-state two-dimensional flow solutions and the inception of unsteady two-dimensional flow. For $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$ , the two-dimensional base flow transitions from steady to unsteady at higher Reynolds number as $\unicode[STIX]{x1D6FD}$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely a spanwise oscillating mode for $\unicode[STIX]{x1D6FD}=0.2$ and a spanwise synchronous mode for $\unicode[STIX]{x1D6FD}\geqslant 0.3$ . The critical Reynolds number and the spanwise wavelength of perturbations increase as $\unicode[STIX]{x1D6FD}$ increases. For $1<\unicode[STIX]{x1D6FD}\leqslant 2$ both the critical Reynolds number for onset of unsteadiness and the spanwise wavelength decrease as $\unicode[STIX]{x1D6FD}$ increases. Finally, for $2<\unicode[STIX]{x1D6FD}\leqslant 5$ , the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different three-dimensional modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.

Corresponding author
Email address for correspondence:
Hide All
Abu-Nada, E. 2008 Application of nanofluids for heat transfer enhancement of separated flows encountered in a backward facing step. Intl J. Heat Fluid Flow 29 (1), 242249.
Alam, M. & Sandham, N. D. 2000 Direct numerical simulation of ‘short’ laminar separation bubbles with turbulent reattachment. J. Fluid Mech. 403, 223250.
Albarède, P. & Provansal, M. 1995 Quasi-periodic cylinder wakes and the Ginzburg–Landau equation. J. Fluid Mech. 291, 191222.
Armaly, B. F., Durst, F., Pereira, J. C. F. & Schonung, B. 1983 Experimental and theoretical investigation of backward-facing step flow. J. Fluid Mech. 127, 473496.
Astarita, T. & Cardone, G. 2000 Thermofluidynamic analysis of the flow in a sharp 180° turn channel. Exp. Therm. Fluid Sci. 20 (3–4), 188200.
Barkley, D., Blackburn, H. M. & Sherwin, S. J 2008 Direct optimal growth analysis for timesteppers. Intl J. Numer. Meth. Fluids 57 (9), 14351458.
Barkley, D., Gomes, M. G. M. & Henderson, R. D. 2002 Three-dimensional instability in flow over a backward-facing step. J. Fluid Mech. 473, 167190.
Barkley, D. & Henderson, R. D. 1996 Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215242.
Barleon, L., Casal, V. & Lenhart, L. 1991 MHD flow in liquid–metal-cooled blankets. Fusion Engng Des. 14 (3), 401412.
Barleon, L., Mack, K. J. & Stieglitz, R. 1996 The MEKKA-facility: A Flexible Tool to Investigate MHD-flow Phenomena. Forschungszentrum Karlsruhe.
Barton, I. E. 1997 The entrance effect of laminar flow over a backward-facing step geometry. Intl J. Numer. Meth. Fluids 25 (6), 633644.
Blackburn, H. M., Barkley, D. & Sherwin, S. J. 2008 Convective instability and transient growth in flow over a backward-facing step. J. Fluid Mech. 603, 271304.
Boccaccini, L. V., Giancarli, L., Janeschitz, G., Hermsmeyer, S., Poitevin, Y., Cardella, A. & Diegele, E. 2004 Materials and design of the European DEMO blankets. J. Nucl. Mater. 329, 148155.
Brede, M., Eckelmann, H. & Rockwell, D. 1996 On secondary vortices in the cylinder wake. Phys. Fluids 8 (8), 21172124.
Bühler, L. 2007 Liquid metal magnetohydrodynamics for fusion blankets. In Magnetohydrodynamics: Historical Evolution and Trends (ed. Molokov, S., Moreau, R. & Moffatt, H. K.), Fluid Mechanics and its Applications, vol. 80, pp. 171194. Springer.
Carmo, B. S., Sherwin, S. J., Bearman, P. W. & Willden, R. H. J. 2008 Wake transition in the ow around two circular cylinders in staggered arrangements. J. Fluid Mech. 597, 129.
Chung, Y. M., Tucker, P. G. & Roychowdhury, D. G. 2003 Unsteady laminar flow and convective heat transfer in a sharp 180° bend. Intl J. Heat Fluid Flow 24 (1), 6776.
Cruchaga, M. A. 1998 A study of the backward-facing step problem using a generalized streamline formulation. Commun. Numer. Meth. Engng 14 (8), 697708.
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability. Cambridge University Press.
Duŝek, J., Le Gal, P. & Fraunié, P. 1994 A numerical and theoretical study of the first Hopf bifurcation in a cylinder wake. J. Fluid Mech. 264, 5980.
Erturk, E. 2008 Numerical solutions of 2-D steady incompressible flow over a backward-facing step. Part I: high Reynolds number solutions. Comput. Fluids 37 (6), 633655.
Ghia, K. N., Osswald, G. A. & Ghia, U. 1989 Analysis of incompressible massively separated viscous flows using unsteady Navier–Stokes equations. Intl J. Numer. Meth. Fluids 9 (8), 10251050.
Griffith, M. D., Leweke, T., Thompson, M. C. & Hourigan, K. 2008 Steady inlet flow in stenotic geometries: convective and absolute instabilities. J. Fluid Mech. 616, 111133.
Griffith, M. D., Thompson, M. C., Leweke, T., Hourigan, K. & Anderson, W. P. 2007 Wake behaviour and instability of flow through a partially blocked channel. J. Fluid Mech. 582 (1), 319340.
Hammond, D. A. & Redekopp, L. G. 1998 Local and global instability properties of separation bubbles. Eur. J. Mech. (B/Fluids) 17 (2), 145164.
Henderson, R. D. 1997 Nonlinear dynamics and pattern formation in turbulent wake transition. J. Fluid Mech. 352, 65112.
Henderson, R. D. & Barkley, D. 1996 Secondary instability in the wake of a circular cylinder. Phys. Fluids 8, 1683.
Hirota, M., Fujita, H., Syuhada, A., Araki, S., Yoshida, T. & Tanaka, T. 1999 Heat/mass transfer characteristics in two-pass smooth channels with a sharp 180-deg turn. Intl J. Heat Mass Transfer 42 (20), 37573770.
Hussam, W. K., Thompson, M. C. & Sheard, G. J. 2012a Enhancing heat transfer in a high Hartmann number magnetohydrodynamic channel flow via torsional oscillation of a cylindrical obstacle. Phys. Fluids 24 (11), 113601.
Hussam, W. K., Thompson, M. C. & Sheard, G. J. 2012b Optimal transient disturbances behind a circular cylinder in a quasi-two-dimensional magnetohydrodynamic duct flow. Phys. Fluids 24 (2), 024105.
Johnson, T. A. & Patel, V. C. 1999 Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech. 378, 1970.
Kaiktsis, L., Karniadakis, G. E. & Orszag, S. A. 1991 Onset of three-dimensionality, equilibria, and early transition in flow over a backward-facing step. J. Fluid Mech. 231, 501528.
Karniadakis, G. E., Israeli, M. & Orszag, S. A. 1991 High-order splitting methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 97 (2), 414443.
Kirillov, I. R., Reed, C. B., Barleon, L. & Miyazaki, K. 1995 Present understanding of MHD and heat transfer phenomena for liquid metal blankets. Fusion Engng Des. 27, 553569.
Krall, K. M. & Sparrow, E. M. 1966 Turbulent heat transfer in the separated, reattached, and redevelopment regions of a circular tube. Trans. ASME J. Heat Transfer 88 (1), 131136.
Landau, L. D. & Lifshitz, E. M. 1976 Mechanics, p. 93. Pergamon.
Lanzerstorfer, D. & Kuhlmann, H. C. 2012 Global stability of the two-dimensional flow over a backward-facing step. J. Fluid Mech. 693, 127.
Larson, H. K. 1959 Heat transfer in separated flows. J. Aero. Sci. 26 (11), 731738.
Le Gal, P., Nadim, A. & Thompson, M. 2001 Hysteresis in the forced Stuart–Landau equation: application to vortex shedding from an oscillating cylinder. J. Fluids Struct. 15 (3), 445457.
Lehoucq, R. B., Sorenson, D. C. & Yang, C. 1998 ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM.
Leweke, T. & Williamson, C. H. K. 1998 Cooperative elliptic instability of a vortex pair. J. Fluid Mech. 360, 85119.
Liou, T.-M., Chen, C.-C., Tzeng, Y.-Y. & Tsai, T.-W. 2000 Non-intrusive measurements of near-wall fluid flow and surface heat transfer in a serpentine passage. Intl J. Heat Mass Transfer 43 (17), 32333244.
Liou, T.-M., Tzeng, Y.-Y. & Chen, C.-C. 1999 Fluid flow in a 180 deg sharp turning duct with different divider thicknesses. Trans. ASME J. Turbomach. 121 (3), 569576.
Marquillie, M. & Ehrenstein, U. W. E. 2003 On the onset of nonlinear oscillations in a separating boundary-layer flow. J. Fluid Mech. 490, 169188.
Metzger, D. E. & Sahm, M. K. 1986 Heat transfer around sharp 180-deg turns in smooth rectangular channels. Trans. ASME J. Heat Transfer 108 (3), 500506.
Moffatt, H. K. 1985 Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 1. Fundamentals. J. Fluid Mech. 159, 359378.
Mullin, T., Seddon, J. R. T., Mantle, M. D. & Sederman, A. J. 2009 Bifurcation phenomena in the flow through a sudden expansion in a circular pipe. Phys. Fluids 21 (1), 014110.
Natarajan, R. & Acrivos, A. 1993 The instability of the steady flow past spheres and disks. J. Fluid Mech. 254, 323344.
Neild, A., Ng, T. W., Sheard, G. J., Powers, M. & Oberti, S. 2010 Swirl mixing at microfluidic junctions due to low frequency side channel fluidic perturbations. Sensors Actuators B 150 (2), 811818.
Provansal, M., Mathis, C. & Boyer, L. 1987 Bénard–von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 122.
Ryan, K., Butler, C. J. & Sheard, G. J. 2012 Stability characteristics of a counter-rotating unequal strength Batchelor vortex pair. J. Fluid Mech. 696, 374401.
Schumm, M., Berger, E. & Monkewitz, P. 1994 Self-excited oscillations in the wake of two-dimensional bluff bodies and their control. J. Fluid Mech. 271, 1753.
Sheard, G. J. 2011 Wake stability features behind a square cylinder: focus on small incidence angles. J. Fluids Struct. 27 (5), 734742.
Sheard, G. J., Fitzgerald, M. J. & Ryan, K. 2009 Cylinders with square cross-section: wake instabilities with incidence angle variation. J. Fluid Mech. 630, 4369.
Sheard, G. J., Thompson, M. C. & Hourigan, K. 2003 A coupled Landau model describing the Strouhal–Reynolds number profile of a three-dimensional circular cylinder wake. Phys. Fluids 15 (9), L68L71.
Sheard, G. J., Thompson, M. C. & Hourigan, K. 2004a Asymmetric structure and non-linear transition behaviour of the wakes of toroidal bodies. Eur. J. Mech. (B/Fluids) 23 (1), 167179.
Sheard, G. J., Thompson, M. C. & Hourigan, K. 2004b From spheres to circular cylinders: non-axisymmetric transitions in the flow past rings. J. Fluid Mech. 506, 4578.
Taneda, S. 1979 Visualization of separating Stokes flows. J. Phys. Soc. Japan 46, 19351942.
Thompson, M. C., Hourigan, K. & Sheridan, J. 1996 Three-dimensional instabilities in the wake of a circular cylinder. Exp. Therm. Fluid Sci. 12 (2), 190196.
Thompson, M. C. & Le Gal, P. 2004 The Stuart–Landau model applied to wake transition revisited. Eur. J. Mech. (B/Fluids) 23 (1), 219228.
Thompson, M. C., Leweke, T. & Williamson, C. H. K. 2001 The physical mechanism of transition in bluff body wakes. J. Fluids Struct. 15 (3), 607616.
Tomboulides, A. G. & Orszag, S. A. 2000 Numerical investigation of transitional and weak turbulent flow past a sphere. J. Fluid Mech. 416, 4573.
Vo, T., Montabone, L. & Sheard, G. J. 2014 Linear stability analysis of a shear layer induced by differential coaxial rotation within a cylindrical enclosure. J. Fluid Mech. 738, 299334.
Vo, T., Montabone, L. & Sheard, G. J. 2015 Effect of enclosure height on the structure and stability of shear layers induced by differential rotation. J. Fluid Mech. 765, 4581.
Wang, T.-S. & Chyu, M. K. 1994 Heat convection in a 180-deg turning duct with different turn configurations. J. Thermophys. Heat Transfer 8 (3), 595601.
Wee, D., Yi, T., Annaswamy, A. & Ghoniem, A. F. 2004 Self-sustained oscillations and vortex shedding in backward-facing step flows: simulation and linear instability analysis. Phys. Fluids 16 (9), 33613373.
Williamson, C. H. K. 1988 Defining a universal and continuous Strouhal–Reynolds number relationship for the laminar vortex shedding of a circular cylinder. Phys. Fluids 31 (10), 27422744.
Zhang, L. & Pothérat, A. 2013 Influence of the geometry on the two- and three-dimensional dynamics of the flow in a 180° sharp bend. Phys. Fluids 25, 053605.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Altmetric attention score

Full text views

Total number of HTML views: 4
Total number of PDF views: 146 *
Loading metrics...

Abstract views

Total abstract views: 279 *
Loading metrics...

* Views captured on Cambridge Core between 9th June 2017 - 22nd March 2018. This data will be updated every 24 hours.