Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T13:51:46.475Z Has data issue: false hasContentIssue false
  • English
  • Français

Complex dynamics in a stratified lid-driven square cavity flow

Published online by Cambridge University Press:  20 September 2018

Ke Wu ,
Bruno D. Welfert  and
Juan M. Lopez Open the ORCID record for Juan M. Lopez [Opens in a new window]
Ke Wu
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
Bruno D. Welfert
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
Juan M. Lopez*
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The dynamic response to shear of a fluid-filled square cavity with stable temperature stratification is investigated numerically. The shear is imposed by the constant translation of the top lid, and is quantified by the associated Reynolds number. The stratification, quantified by a Richardson number, is imposed by maintaining the temperature of the top lid at a higher constant temperature than that of the bottom, and the side walls are insulating. The Navier–Stokes equations under the Boussinesq approximation are solved, using a pseudospectral approximation, over a wide range of Reynolds and Richardson numbers. Particular attention is paid to the dynamical mechanisms associated with the onset of instability of steady state solutions, and to the complex and rich dynamics occurring beyond.

JFM classification

Geophysical and Geological Flows: Internal waves Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Geophysical and Geological Flows: Stratified flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 855 , 25 November 2018 , pp. 43 - 66
Check for updates
Copyright
© 2018 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

Ansong, J. K. & Sutherland, B. R. 2010 Internal gravity waves generated by convective plumes. J. Fluid Mech. 648, 405–434.Google Scholar
Auteri, F., Parolini, N. & Quartapelle, L. 2002 Numerical investigation of the stability of singular driven cavity flow. J. Comput. Phys. 183, 1–25.Google Scholar
Blanchette, F., Peacock, T. & Cousin, R. 2008 Stability of a stratified fluid with a vertically moving sidewall. J. Fluid Mech. 609, 305–317.Google Scholar
Boyland, P. L. 1986 Bifurcations of circle maps: Arnol’d tongues, bistability and rotation intervals. Commun. Math. Phys. 106, 353–381.Google Scholar
Brezillon, A., Girault, G. & Cadou, J. M. 2010 A numerical algorithm coupling a bifurcating indicator and a direct method for the computation of Hopf bifurcation points in fluid mechanics. Comput. Fluids 39, 1226–1240.Google Scholar
Cheng, T. S. & Liu, W.-H. 2010 Effect of temperature gradient orientation on the characteristics of mixed convection flow in a lid-driven square cavity. Comput. Fluids 39, 965–978.Google Scholar
Cohen, N., Eidelman, A., Elperin, T., Kleerin, N. & Rogachevskii, I. 2014 Sheared stably stratified turbulence and large-scale waves in a lid driven cavity. Phys. Fluids 26, 105106.Google Scholar
Dohan, K. & Sutherland, B. R. 2003 Internal waves generated from a turbulent mixed region. Phys. Fluids 15, 488–498.Google Scholar
Fernando, H. J. S. 1991 Turbulent mixing in stratified fluids. Annu. Rev. Fluid Mech. 23, 455–493.Google Scholar
Fortin, A., Jardak, M., Gervais, J. J. & Pierre, R. 1997 Localization of Hopf bifurcations in fluid flow problems. Intl J. Numer. Meth. Fluids 24, 1185–1210.Google Scholar
Hanratty, T. J., Rosen, E. M. & Kabel, R. L. 1958 Effect of heat transfer on flow field at low Reynolds numbers in vertical tubes. Ind. Engng Chem. 50, 815–818.Google Scholar
Hugues, S. & Randriamampianina, A. 1998 An improved projection scheme applied to pseudospectral methods for the incompressible Navier–Stokes equations. Intl J. Numer. Meth. Fluids 28, 501–521.Google Scholar
Ivey, G. N., Winters, K. B. & Koseff, J. R. 2008 Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech. 40, 169–184.Google Scholar
Iwatsu, R., Hyun, J. H. & Kuwahara, K. 1993 Mixed convection in a driven cavity with a stable vertical temperature gradient. Intl J. Heat Mass Transfer 36, 1601–1608.Google Scholar
Ji, T. H., Kim, S. Y. & Hyun, J. M. 2007 Transient mixed convection in an enclosure driven by a sliding lid. Heat Mass Transfer 43, 629–638.Google Scholar
Knobloch, E. & Proctor, M. R. E. 1988 The double Hopf bifurcation with 2 : 1 resonance. Proc. R. Soc. Lond. A 415, 61–90.Google Scholar
Koseff, J. R. & Street, R. L. 1985 Circulation structure in a stratified cavity flow. J. Hydraul. Engng 111, 334–354.Google Scholar
Kuznetsov, Y. A. 2004 Elements of Applied Bifurcation Theory, 3rd edn. Springer.Google Scholar
Lopez, J. M. & Marques, F. 2009 Centrifugal effects in rotating convection: nonlinear dynamics. J. Fluid Mech. 628, 269–297.Google Scholar
Lopez, J. M., Welfert, B. D., Wu, K. & Yalim, J. 2017 Transition to complex dynamics in the cubic lid-driven cavity. Phys. Rev. Fluids 2, 07441.Google Scholar
Maiti, M. K., Gupta, A. S. & Bhattacharyya, S. 2008 Stable/unstable stratification in thermosolutal convection in a square cavity. Trans. ASME J. Heat Transfer 130, 122001.Google Scholar
Majda, A. J. & Shefter, M. G. 1998 Elementary stratified flows with instability at large Richardson number. J. Fluid Mech. 376, 319–350.Google Scholar
Majda, A. J. & Shefter, M. G. 2000 Nonlinear instability of elementary stratified flows at large Richardson number. Chaos 10, 3–27.Google Scholar
Michaelian, M. E., Maxworthy, T. & Redekopp, L. G. 2002 The coupling between turbulent, penetrative convection and internal waves. Eur. J. Mech. (B/Fluids) 21, 1–28.Google Scholar
Munroe, J. R. & Sutherland, B. R. 2014 Internal wave energy radiated from a turbulent mixed layer. Phys. Fluids 26, 096604.Google Scholar
Nicolás, A. & Bermúdez, B. 2005 2D thermal/isothermal incompressible viscous flows. Intl J. Numer. Meth. Fluids 48, 349–366.Google Scholar
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135–167.Google Scholar
Pham, H. T., Sarkar, S. & Brucker, K. A. 2009 Dynamics of a stratified shear layer above a region of uniform stratification. J. Fluid Mech. 630, 191–223.Google Scholar
Poliashenko, M. & Aidun, C. K. 1995 A direct method for computation of simple bifurcations. J. Comput. Phys. 121, 246–260.Google Scholar
Sherman, F. S., Imberger, J. & Corcos, G. M. 1978 Turbulence and mixing in stably stratified waters. Annu. Rev. Fluid Mech. 10, 267–288.Google Scholar
Stevens, C. & Imberger, J. 1996 The initial response of a stratified lake to a surface shear stress. J. Fluid Mech. 312, 39–66.Google Scholar
Turner, J. S. 1979 Buoyancy Effects in Fluids. Cambridge University Press.Google Scholar
Vanel, J.-M., Peyret, R. & Bontoux, P. 1986 A pseudo-spectral solution of vorticity-stream function equations using the influence matrix technique. In Numerical Methods for Fluid Dynamics II (ed. Morton, K. W. & Baines, M. J.), pp. 463–475. Clarendon Press.Google Scholar

Wu Supplementary Movie 1

Animations over one period of the deviations of the streamfunction from its mean for the limit cycle L5, L3 and L2 at Re and Ri as indicated.

Download Wu Supplementary Movie 1(Video)
Video 2.7 MB

Wu Supplementary Movie 2

Animations over one period of the deviations of the horizontal temperature from its means for the limit cycles L1 and L2 at Re and Ri as indicated.

Download Wu Supplementary Movie 2(Video)
Video 5.4 MB

Wu Supplementary Movie 3

Animations over 0.01 viscous time of the horizontal temperature of non-periodic states at Re=5000 and Ri as indicated.

Download Wu Supplementary Movie 3(Video)
Video 25.2 MB