Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-21T01:17:44.307Z Has data issue: false hasContentIssue false
  • English
  • Français

Spatio-temporal structure of the ‘fully developed’ transitional flow in a symmetric wavy channel. Linear and weakly nonlinear stability analysis

Published online by Cambridge University Press:  29 December 2014

S. Blancher ,
Y. Le Guer  and
K. El Omari
S. Blancher*
Affiliation:
Université de Pau et des Pays de l’Adour, Laboratoire SIAME, Fédération CNRS IPRA, Avenue de l’Université, 64000 Pau, France
Y. Le Guer
Affiliation:
Université de Pau et des Pays de l’Adour, Laboratoire SIAME, Fédération CNRS IPRA, Avenue de l’Université, 64000 Pau, France
K. El Omari
Affiliation:
Université de Pau et des Pays de l’Adour, Laboratoire SIAME, Fédération CNRS IPRA, Avenue de l’Université, 64000 Pau, France
*
†Email address for correspondence: serge.blancher@univ-pau.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

This work addresses the transition from 2D steady to 2D unsteady laminar flow for a fully developed regime in a symmetric wavy channel geometry. We investigate the existence and characteristics of the spatio-temporal structure of the fully developed unsteady laminar flow for those particular geometries for which the steady flow presents a periodic variation of the main stream velocity component. We perform a 2D global linear stability analysis of the fully developed steady laminar flow, and we show that, for all the geometries studied, the transition is triggered by a Hopf bifurcation associated with the breaking of the symmetries and the invariance of the steady flow. Critical Reynolds numbers, most unstable modes and their characteristics are presented for large ranges of the geometric parameters, namely wavenumber ${\it\alpha}$ from 0.3 to 5 and amplitude from 0 (straight channel) to 0.5. We show that it is possible to define geometries for which the wavenumber is proportional to the most unstable mode wavenumber for the critical Reynolds number. From this modal study we address a weakly nonlinear stability analysis with a view to obtaining the Landau coefficient $g$, and then the sub- or supercritical nature of the first bifurcation characterising the transition. We show that a critical geometric amplitude beyond which the first bifurcation is supercritical is associated with each geometric wavenumber.

JFM classification

Instability: Instability Instability: Parametric instability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 764 , 10 February 2015 , pp. 250 - 276
Check for updates
Copyright
© 2014 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

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
Amon, C. H., Guzmán, A. M. & Morel, B. 1996 Lagrangian chaos, Eulerian chaos, and mixing enhancement in converging–diverging channel flows. Phys. Fluids 8, 1192–1206.Google Scholar
Asai, M. & Floryan, J. M. 2006 Experiments on the linear instability of flow in a wavy channel. Eur. J. Mech. (B/Fluids) 25 (6), 971–986.CrossRefGoogle Scholar
Blancher, S.1991 Transfer convectif stationnaire et stabilité hydrodynamique en géométrie périodique. PhD thesis, Université de Pau et des Pays de l’Adour, UPPA, Pau.Google Scholar
Blancher, S., Creff, R., Batina, J. & André, P. 1994 Hydrodynamic stability in periodic geometry. Finite Elem. Anal. Des. 16, 261–270.Google Scholar
Blancher, S., Creff, R. & Le Quéré, P. 1998 Effect of Tollmien Schlichting wave on convective heat transfer in a wavy channel. Part 1: linear analysis. Intl J. Heat Fluid Flow 19, 39–48.Google Scholar
Blancher, S., Creff, R. & Le Quéré, P. 2004 Analysis of convective hydrodynamic instabilities in a symmetric wavy channel. Phys. Fluids 16, 3726–3737.Google Scholar
Cabal, A., Szumbarski, J. & Floryan, J. M. 2002 Stability of flow in a wavy channel. J. Fluid Mech. 457 (1), 191–212.CrossRefGoogle 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, 349–370.Google Scholar
Chomaz, J. M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.Google Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Floryan, J. M. 2003 Vortex instability in a diverging–converging channel. J. Fluid Mech. 482, 17–50.CrossRefGoogle Scholar
Floryan, J. M. 2005 Two-dimensional instability of flow in a rough channel. Phys. Fluids 17, 044101.CrossRefGoogle Scholar
Ghaddar, N. K., Korczak, K. Z., Mikic, B. B. & Patera, A. T. 1986 Numerical investigation of incoimpressible flow in grooved channel. Part 1: stability and self sustained oscillations. J. Fluid Mech. 163, 99–127.Google Scholar
Greiner, M., Chen, R. F. & Wirtz, R. A. 1990 Heat transfer augmentation through wall shape induced destabilisation. Trans. ASME J. Heat Transfer 112, 337–341.Google Scholar
Gschwind, P. & Kottke, V. 2000 Regular flow structures in channels with symmetric and asymmetric walls. In Proceedings of the Sixth Triennial International Symposium on Flow Control, Measurement and Flow Visualization, Flucome 2000, Sherbrooke (ed. Laneville, A.), vol. 3, pp. 387–392. IOS Press.Google Scholar
Guzman, A. M. & Amon, C. H. 1994 Transition to chaos in converging–diverging channel flows: Ruelle–Takens–Newhouse scenario. Phys. Fluids 6, 1994–2002.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 (1), 25–57.Google Scholar
Herman, C. V. & Mayinger, F. 1990 Experimental investigation of the heat transfer in laminar forced convection flow in a grooved channel. In Proceedings of the 9th International Heat Transfer Conference, Jerusalem, vol. 3, pp. 387–392. ASME.Google Scholar
Kim, S. K.2001 An experimental study of flow in a wavy channel by PIV. In Proceedings of the Sixth Asian Symposium on Visualization, Pusan, Korea. J. Vis. 5, 105–111.Google Scholar
Manneville, P. 1990 Dissipative Structures and Weak Turbulence. Academic.Google Scholar
Monkewitz, P. A., Huerre, P. & Chomaz, J. M. 1993 Global linear stability analysis of weakly non-parallel shear flows. J. Fluid Mech. 251, 1–20.CrossRefGoogle Scholar
Newell, J. M., Passot, T. & Lega, J. 1993 Order parameter equations for patterns. Annu. Rev. Fluid Mech. 25, 399–453.CrossRefGoogle 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., Bian, Y. N., Matsumoto, Y. & Kunitsugu, K. 2003 Fluid flow and mass transfer characteristics in a sinusoidal wavy-walled tube at moderate Reynolds numbers for steady flow. J. Heat Mass Transfer 39, 239–248.Google Scholar
Nishimura, T., Kajimoto, Y., Tarumoto, A. & Kawamura, Y. 1986 Flow structure and mass transfer for a wavy wall in transitional flow regime. J. Chem. Engng Japan 19, 449–455.CrossRefGoogle Scholar
Nishimura, T., Murakami, S., Arakawa, S. & Kawamura, Y. 1990 Flow observations and mass transfer characteristics in symmetrical wavy-walled channels at moderate Reynolds number for steady flow. Intl J. Heat Mass Transfer 33, 835–845.Google Scholar
Nishimura, T., Ohori, Y. & Kawamura, Y. 1983 Flow characteristics in a channel with symmetric wavy wall for steady flow. J. Chem. Engng Japan 17, 466–471.Google Scholar
Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50, 683–703.CrossRefGoogle Scholar
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall bounded shear flows. J. Fluid Mech. 128, 347–385.Google Scholar
Plaut, E.2008 Modélisation d’instabilités – méthodes non-linéaires. Lecture Master 2, INPL, Nancy University.Google Scholar
Rivera-Alvarez, A. & Ordonez, J. C. 2014 Global stability of flow in symmetric wavy channels. J. Fluid Mech. 733, 625–649.Google Scholar
Rush, T. A., Newell, T. A. & Jacobi, A. M. 1999 An experimental study of flow and heat transfer in sinusoidal wavy passages. Intl J. Heat Mass Transfer 42, 1541–1553.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flow. Springer.Google Scholar
Selvarajan, S., Tulapurkara, E. G. & Ram, V. V. 1999 Stability characteristics of wavy walled channel flows. Phys. Fluids 11, 579–589.Google Scholar
Sobey, I. J. 1980 On flow through furrowed channels. Part 1: calculated flow patterns. J. Fluid Mech. 96, 1–26.Google Scholar
Stephanoff, K. D. 1986 Self-excited shear-layer oscillations in a multi-cavity channel with a steady mean velocity. Trans. ASME J. Fluids Engng 108, 338–342.Google Scholar
Stephanoff, K. D., Sobey, I. J. & Bellhouse, B. J. 1980 On flow through furrowed channels. Part 2: observed flow patterns. J. Fluid Mech. 96, 27–32.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, 024105.Google Scholar
Theofilis, V. 2003 Advances in global linear instability analysis of non-parallel and three-dimensional flows. Prog. Aerosp. Sci. 39 (4), 249–315.Google Scholar
Wang, G. & Vanka, S. P. 1995 Convective heat transfer in periodic wavy passages. Intl J. Heat Mass Transfer 38, 3219–3230.CrossRefGoogle 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, 3114–3133.Google Scholar