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Dynamics in a stably stratified tilted square cavity

Published online by Cambridge University Press:  29 November 2019

Hezekiah Grayer Open the ORCID record for Hezekiah Grayer [Opens in a new window] ,
Jason Yalim Open the ORCID record for Jason Yalim [Opens in a new window] ,
Bruno D. Welfert Open the ORCID record for Bruno D. Welfert [Opens in a new window]  and
Juan M. Lopez Open the ORCID record for Juan M. Lopez [Opens in a new window]
Hezekiah Grayer
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, TempeAZ85287, USA
Jason Yalim
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, TempeAZ85287, USA
Bruno D. Welfert
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, TempeAZ85287, USA
Juan M. Lopez*
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, TempeAZ85287, USA
*
†Email address for correspondence: juan.m.lopez@asu.edu
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Abstract

The dynamics of a fluid flow in a differentially heated square container is investigated numerically. Two opposite conducting walls are maintained at constant temperatures, one hot and the other cold, and the other two walls are insulated. When the conducting walls are horizontal with the lower one cold, the static linearly stratified state is stable. When the container is tilted, the static equilibrium ceases to exist and the fluid flows due to the baroclinic torque arising from the bending of isotherms near the tilted insulated walls. This flow is found to be steady for tilt angles less than $45^{\circ }$, regardless of the relative balance between buoyancy and viscous effects (quantified by a buoyancy number $R_{N}$). For tilt angles above $45^{\circ }$, the flow becomes unsteady above a critical $R_{N}$ with localized boundary layer undulations at the conducting walls, at the heights of the horizontally opposite corners. From these corners emanate horizontal shear layers, which become thinner and more intense with increasing $R_{N}$. As the tilt angle approaches $90^{\circ }$, the nature of the instability changes, corresponding to that of the well-studied laterally heated cavity flow.

JFM classification

Convection: Buoyancy-driven instability Geophysical and Geological Flows: Baroclinic flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 883 , 25 January 2020 , A62
Copyright
© 2019 Cambridge University Press

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Grayer et al. supplementary movie 1

Isotherms and vorticity of the steady states. Unstable steady states were computed using selective frequency damping.

Download Grayer et al. supplementary movie 1(Video)
Video 9.2 MB

Grayer et al. supplementary movie 2

Pointwise and setwise invariant limit cycles L1.

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Video 472.4 KB

Grayer et al. supplementary movie 3

States near the double-Hopf bifurcation dH12.

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Video 3.7 MB

Grayer et al. supplementary movie 4

States near the double-Hopf bifurcation dH23.

Download Grayer et al. supplementary movie 4(Video)
Video 11.8 MB