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Turbulence and mixing in a near-bottom convectively driven flow are examined by numerical simulations of a model problem: a statically unstable disturbance at a slope with inclination $$\unicode[STIX]{x1D6FD}$$ in a stable background with buoyancy frequency $$N$$ . The influence of slope angle and initial disturbance amplitude are quantified in a parametric study. The flow evolution involves energy exchange between four energy reservoirs, namely the mean and turbulent components of kinetic energy (KE) and available potential energy (APE). In contrast to the zero-slope case where the mean flow is negligible, the presence of a slope leads to a current that oscillates with $$\unicode[STIX]{x1D714}=N\sin \unicode[STIX]{x1D6FD}$$ and qualitatively changes the subsequent evolution of the initial density disturbance. The frequency, $$N\sin \unicode[STIX]{x1D6FD}$$ , and the initial speed of the current are predicted using linear theory. The energy transfer in the sloping cases is dominated by an oscillatory exchange between mean APE and mean KE with a transfer to turbulence at specific phases. In all simulated cases, the positive buoyancy flux during episodes of convective instability at the zero-velocity phase is the dominant contributor to turbulent kinetic energy (TKE) although the shear production becomes increasingly important with increasing $$\unicode[STIX]{x1D6FD}$$ . Energy that initially resides wholly in mean available potential energy is lost through conversion to turbulence and the subsequent dissipation of TKE and turbulent available potential energy. A key result is that, in contrast to the explosive loss of energy during the initial convective instability in the non-sloping case, the sloping cases exhibit a more gradual energy loss that is sustained over a long time interval. The slope-parallel oscillation introduces a new flow time scale $$T=2\unicode[STIX]{x03C0}/(N\sin \unicode[STIX]{x1D6FD})$$ and, consequently, the fraction of initial APE that is converted to turbulence during convective instability progressively decreases with increasing $$\unicode[STIX]{x1D6FD}$$ . For moderate slopes with $$\unicode[STIX]{x1D6FD}<10^{\circ }$$ , most of the net energy loss takes place during an initial, short ( $$Nt\approx 20$$ ) interval with periodic convective overturns. For steeper slopes, most of the energy loss takes place during a later, long ( $Nt>100$ ) interval when both shear and convective instability occur, and the energy loss rate is approximately constant. The mixing efficiency during the initial period dominated by convectively driven turbulence is found to be substantially higher (exceeds 0.5) than the widely used value of 0.2. The mixing efficiency at long time in the present problem of a convective overturn at a boundary varies between 0.24 and 0.3.
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The Nusselt numbers of horizontal convection
In the problem of horizontal convection a non-uniform buoyancy, $$b_{s}(x,y)$$ , is imposed on the top surface of a container and all other surfaces are insulating. Horizontal convection produces a net horizontal flux of buoyancy, $$\boldsymbol{J}$$ , defined by vertically and temporally averaging the interior horizontal flux of buoyancy. We show that $$\overline{\boldsymbol{J}\boldsymbol{\cdot }\unicode[STIX]{x1D735}b_{s}}=-\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$$ ; the overbar denotes a space–time average over the top surface, angle brackets denote a volume–time average and $$\unicode[STIX]{x1D705}$$ is the molecular diffusivity of buoyancy $$b$$ . This connection between $$\boldsymbol{J}$$ and $$\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$$ justifies the definition of the horizontal-convective Nusselt number, $Nu$ , as the ratio of $$\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$$ to the corresponding quantity produced by molecular diffusion alone. We discuss the advantages of this definition of $Nu$ over other definitions of horizontal-convective Nusselt number. We investigate transient effects and show that $$\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$$ equilibrates more rapidly than other global averages, such as the averaged kinetic energy and bottom buoyancy. We show that $$\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$$ is the volume-averaged rate of Boussinesq entropy production within the enclosure. In statistical steady state, the interior entropy production is balanced by a flux through the top surface. This leads to an equivalent ‘surface Nusselt number’, defined as the surface average of vertical buoyancy flux through the top surface times the imposed surface buoyancy $$b_{s}(x,y)$$ . In experimental situations it is easier to evaluate the surface entropy flux, rather than the volume integral of $$|\unicode[STIX]{x1D735}b|^{2}$$ demanded by $$\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$$ .
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- Award ID(s):
- 1657041
- PAR ID:
- 10169194
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 894
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2020.269
