Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-27T03:47:52.089Z Has data issue: false hasContentIssue false
  • English
  • Français

Global stability of flow in symmetric wavy channels

Published online by Cambridge University Press:  26 September 2013

Alejandro Rivera-Alvarez  and
Juan C. Ordonez
Alejandro Rivera-Alvarez*
Affiliation:
Department of Mechanical Engineering and Center for Advanced Power Systems, Florida State University, Tallahassee, FL 32310, USA
Juan C. Ordonez
Affiliation:
Department of Mechanical Engineering and Center for Advanced Power Systems, Florida State University, Tallahassee, FL 32310, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A two-dimensional global stability analysis is numerically conducted for the basic fully developed steady flow inside symmetric wavy channels. The relative amplitude of channel modulation, defined as ratio of wall modulation amplitude to mean hydraulic diameter, is fixed via the analysis at a large value of 0.15. The relative channel wavelength, defined as the ratio of wall modulation wavelength to mean hydraulic diameter, is varied between 1 and 5. An important feature of the present approach is the detailed consideration of the streamwise conditions imposed on the flow, which allows a tailored restriction of the possible disturbances by modifying the number of channel sections that set the periodicity. Stability of the base flow is determined on the basis of the spectral structure analysis conducted, possible after calculation of the complete eigenvalue spectrum and facilitated by the spectral method employed. Two different destabilization mechanisms have been identified for the geometry studied. For small relative channel wavelengths, with values below approximately 2.7, the flow is destabilized via a Hopf bifurcation produced by a Tollmien–Schlichting wave at Reynolds numbers in the approximated range from 265 to 324. For large relative channel wavelengths, with values above approximately 2.5, a pitchfork bifurcation (or a Hopf bifurcation with very small oscillation frequency), not previously reported in the literature and produced by a symmetric stationary disturbance, is found at Reynolds numbers in the approximated range from 146 to 230.

JFM classification

Instability: Instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 733 , 25 October 2013 , pp. 625 - 649
Copyright
©2013 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

Abu-Ramadan, E. & Khayat, R. E. 2007 Stability of vortex flow in a modulated channel. Phys. Rev. E 75 (2), 026305.Google Scholar
Adachi, T. & Hasegawa, S. 2006 Transition of the flow in a symmetric channel with periodically expanded grooves. Chem. Engng Sci. 61 (8), 2721–2729.Google Scholar
Bahaidarah, H. M. S. 2007 A numerical study of fluid flow and heat transfer characteristics in channels with staggered wavy walls. Numer. Heat Transfer A 51 (9), 877–898.CrossRefGoogle Scholar
Bahaidarah, H. M. S., Anand, N. K. & Chen, H. C. 2005 Numerical study of heat and momentum transfer in channels with wavy walls. Numer. Heat Transfer A 47 (5), 417–439.Google Scholar
Battaglia, F. & Papadopoulos, G. 2006 Bifurcation characteristics of flows in rectangular sudden expansion channels. Trans. ASME: J. Fluids Engng 128 (4), 671–679.Google Scholar
Blancher, S., Creff, R. & Quéré, P. Le 1998 Effect of Tollmien Schlichting wave on convective heat transfer in a wavy channel. Part 1. Linear analysis. Intl J. Heat Fluid Flow 19 (1), 39–48.Google Scholar
Blancher, S., Creff, R. & Quéré, P. Le 2004 Analysis of convective hydrodynamic instabilities in a symmetric wavy channel. Phys. Fluids 16 (10), 3726–3737.CrossRefGoogle Scholar
Challis, N. V. & Burley, D. M. 1982 A numerical method for conformal mapping. IMA J. Num. Anal. 2 (2), 169–181.Google Scholar
Cho, K. J., Kim, M.-U. & Shin, H. D. 1998 Linear stability of two-dimensional steady flow in wavy-walled channels. Fluid Dyn. Res. 23 (6), 349–370.Google Scholar
Cliffe, K. A., Spence, A. & Tavener, S. J. 2000 The numerical analysis of bifurcation problems with application to fluid mechanics. Acta Numerica 9, 39–131.Google Scholar
Cotrell, D. L., McFadden, G. B. & Alder, B. J. 2008 Instability in pipe flow. Proc. Natl Acad. Sci. USA 105 (2), 428–430.Google Scholar
Fani, A., Camarri, S. & Salvetti, M. V. 2012 Stability analysis and control of the flow in a symmetric channel with a sudden expansion. Phys. Fluids 24.Google Scholar
Fearn, R. M., Mullin, T. & Cliffe, K. A. 1990 Nonlinear flow phenomena in a symmetric sudden expansion. J. Fluid Mech. 211, 595–608.Google Scholar
Floryan, J. M. 2005 Two-dimensional instability of flow in a rough channel. Phys. Fluids 17 (4), 044101.Google Scholar
Floryan, J. M. 2007 Three-dimensional instabilities of laminar flow in a rough channel and the concept of hydraulically smooth wall. Eur. J. Mech. B 26 (3), 305–329.Google Scholar
Floryan, J. M. & Asai, M. 2011 On the transition between distributed and isolated surface roughness and its effect on the stability of channel flow. Phys. Fluids 23 (10), 104101.Google Scholar
Floryan, J. M. & Floryan, C. 2010 Traveling wave instability in a diverging–converging channel. Fluid Dyn. Res. 42 (2), 025509.CrossRefGoogle Scholar
Guzman, A. M. & Amon, C. H. 1996 Dynamical flow characterization of transitional and chaotic regimes in converging–diverging channels. J. Fluid Mech. 321, 25–57.Google Scholar
Hasan, M. N., Saha, S. & Saha, S. C. 2012 Effects of corrugation frequency and aspect ratio on natural convection within an enclosure having sinusoidal corrugation over a heated top surface. Intl Commun. Heat Mass 39 (3), 368–377.CrossRefGoogle Scholar
Hossain, Z. M. & Islam, A. K. M. S. 2007 Numerical investigation of fluid flow and heat transfer characteristics in sine, triangular, and arc-shaped channels. Therm. Sci. 11 (1), 17–26.Google Scholar
Kirby, R. M. & Yosibash, Z. 2003 Chebyshev-collocation method for bi-harmonic problems with homogeneous Dirichlet boundary conditions. Retrieved from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.83.35378&rep=rep1&type=pdf.Google Scholar
Kopriva, D. A. 2009 Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers. Springer.Google Scholar
Mahmud, S., Islam, A. K. M. S. & Mamun, M. A. H. 2002 Separation characteristics of fluid flow inside two parallel plates with wavy surface. Intl J. Engng Sci. 40 (13), 1495–1509.CrossRefGoogle Scholar
Mejia-Alvarez, R. & Christensen, K. T. 2010 Low-order representations of irregular surface roughness and their impact on a turbulent boundary layer. Phys. Fluids 22 (1), 015106.Google Scholar
Mizushima, J., Okamoto, H. & Yamaguchi, H. 1996 Stability of flow in a channel with a suddenly expanded part. Phys. Fluids 8 (11), 2933–2942.Google Scholar
Mizushima, J. & Shiotani, Y. 2001 Transitions and instabilities of flow in a symmetric channel with a suddenly expanded and contracted part. J. Fluid Mech. 434, 355–369.Google Scholar
Mohammadi, A. & Floryan, J. M. 2012 Mechanism of drag generation by surface corrugation. Phys. Fluids 24 (1), 013602.Google Scholar
Mullin, T., Shipton, S. & Tavener, S. J. 2003 Flow in a symmetric channel with an expanded section. Fluid Dyn. Res. 33 (5), 433–452.Google Scholar
Niavarani, A. & Priezjev, N. V. 2009 The effective slip length and vortex formation in laminar flow over a rough surface. Phys. Fluids 21 (5), 052105.Google Scholar
Ničeno, B. & Nobile, E. 2001 Numerical analysis of fluid flow and heat transfer in periodic wavy channels. Intl J. Heat Fluid Flow 22 (2), 156–167.Google Scholar
Nishimura, T. & Kojima, N. 1995 Mass transfer enhancement in a symmetric sinusoidal wavy-walled channel for pulsatile flow. Intl J. Heat Mass Transfer 38 (9), 1719–1731.Google Scholar
Nishimura, T., Ohori, Y. & Kawamura, Y. 1984 Flow characteristics in a channel with symmetric wavy wall for steady flow. J. Chem. Engng Japan 17 (5), 466–471.Google Scholar
Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50, 689–703.Google Scholar
Papamichael, N. & Stylianopoulos, N. 2010 Numerical Conformal Mapping: Domain Decomposition and the Mapping of Quadrilaterals. World Scientific.Google Scholar
Quartapelle, L. 1993 Numerical Solution of the Incompressible Navier–Stokes Equations. Birkhäuser.Google Scholar
Sobey, I. J. 1980 On flow through furrowed channels. Part 1. Calculated flow patterns. J. Fluid Mech. 96, 1–26.Google Scholar
Sobieski, W. 2009 Numerical and experimental analyses of Hopf bifurcations in a locally expanded channel. Tech. Sci. 12 (1), 272–285.Google Scholar
Stalio, E. & Piller, M. 2007 Direct numerical simulation of heat transfer in converging–diverging wavy channels. Trans. ASME: J. Heat Transfer 129 (7), 769–777.Google Scholar
Szumbarski, J. & Floryan, J. M. 2006 Transient disturbance growth in a corrugated channel. J. Fluid Mech. 568, 243–272.Google Scholar
Takaoka, M., Sano, T., Yamamoto, H. & Mizushima, J. 2009 Convective instability of flow in a symmetric channel with spatially periodic structures. Phys. Fluids 21 (2), 024105.Google Scholar
Trefethen, L. N. 2000 Spectral Methods in MATLAB. SIAM.Google Scholar
Wang, C.-C. & Chen, C.-K. 2002 Forced convection in a wavy-wall channel. Intl J. Heat Mass Transfer 45 (12), 2587–2595.Google Scholar
Wang, G. & Vanka, S. P. 1995 Convective heat transfer in periodic wavy passages. Intl J. Heat Mass Transfer 38 (17), 3219–3230.Google Scholar
Wang, H. L. & Wang, Y. 2007 Flow in microchannels with rough walls: flow pattern and pressure drop. J. Micromech. Microengng 17 (3), 586–596.Google Scholar
Zhou, H., Martinuzzi, R. J., Khayat, R. E., Straatman, A. G. & Abu-Ramadan, E. 2003 Influence of wall shape on vortex formation in modulated channel flow. Phys. Fluids 15 (10), 3114–3133.Google Scholar