Transition and turbulence in horizontal convection: linear stability analysis

Abstract

The linear instability mechanisms of horizontal convection in a rectangular cavity forced by a horizontal buoyancy gradient along its surface are investigated using local and global stability analyses for a Prandtl number equal to unity. The results show that the stability of the base flow, a steady circulation characterized by a narrow descending plume and a broad upwelling region, depends on the Rayleigh number, $Ra$. For free-slip boundary conditions at a critical value of $Ra\approx 2\times 10^{7}$, the steady base flow becomes unstable to three-dimensional perturbations, characterized by counter-rotating vortices originating within the plume region. A Wentzel–Kramers–Brillouin (WKB) method applied along closed streamlines demonstrates that this instability is of a Rayleigh–Taylor type and can be used to accurately reconstruct the global instability mode. In the case of no-slip boundary conditions, the base flow also becomes unstable to a self-sustained two-dimensional instability whose critical Rayleigh number is $Ra\approx 1.7\times 10^{8}$. Beyond this critical $Ra$, two-dimensional equilibrium stationary states of the Navier–Stokes equations are computed using the selective frequency damping method. The two-dimensional onset of instability is shown to be characterized by a family of modes also originating within the plume. A local spatio-temporal stability analysis shows that the flow becomes absolutely unstable at the origin of the plume. Taken together, these results illustrate the mechanisms behind the onset of turbulence that has been observed in horizontal convection.

Classical and symmetrical horizontal convection: detaching plumes and oscillations

“…At higher Ra, plumes will detach faster, and for sufficiently large Ra one finds multiple plumes detaching from the thermal BL. This phenomenon was reported in Passaggia et al (2017) for Ra = 9 × 10 14 , where plumes were visible immediately after entering the convectively unstable region.…”

“…Gayen et al (2014) showed, for Pr = 5 and varying Ra, that the flow goes through a sequence of stability transitions, starting with the growth of plumes in the BL, followed by convective rolls at higher Ra, and finally show fully 3-D turbulence within a region above the hot BL at Ra ≈ 5 × 10 11 . The linear stability analysis of Passaggia, Scotti & White (2017) for Pr = 1 supports these findings and suggests that there exists a competition between 3-D rolls around a streamwise axis and two-dimensional (2-D) Rayleigh-Taylor (RT) instability. The former seem to be dominant in wider cells, as found by Sukhanovsky et al (2012), whereas the latter seems to be most relevant for no-slip BC in narrow cells.…”

The Nusselt numbers of horizontal convection

“…(3.10) (Shishkina, Grossmann & Lohse 2016;Passaggia, Scotti & White 2017). The advantage of the profile (3.10) is that the horizontal buoyancy flux through the discontinuity in b s (x) -that is J(0) -is distinguished and provides a 'natural' definition of the horizontal-convective Nusselt number.…”

Turbulent horizontal convection at high Schmidt numbers

“…At early times, the dye travels from the fresh side in a laminar and stratified layer toward the salty side where it sinks as a turbulent plume [see Figs. 1(a)-1(f)] [2]. The bottom of the tank is progressively filled with dyed/salty water which has mixed in the plume and drives a deep circulation [see Fig.…”