Max Planck Institute for Dynamics and Self-Organization
Horizontal Convection
In horizontal convection (HC), heating and cooling are applied to different parts of the same horizontal surface of a fluid layer. This setup is relevant in many geophysical systems, in particular, in the large-scale ocean circulation, as heat is supplied to and removed from the ocean predominantly through its upper surface, where the ocean contacts the atmosphere. It is also important in process engineering, e.g., in glass melting.
Figure 1: Sketch of classical and symmetrical Horizontal convection. Shown is a velocity component along the cell: a warm fluid moves away from the heated part of the bottom and a cold fluid moves away from the cooled part of the bottom for Pr=0.1 and Ra=3x1011.
Figure 1: Sketch of classical and symmetrical Horizontal convection. Shown is a velocity component along the cell: a warm fluid moves away from the heated part of the bottom and a cold fluid moves away from the cooled part of the bottom for Pr=0.1 and Ra=3x1011.
Before our study, Shishkina, Grossmann & Lohse, Geophys. Res. Lett. 43 (2016), it was commonly agreed that the convective heat transport, measured by the Nusselt number Nu, follows the Rossby scaling with the Rayleigh number Ra, i.e., Nu ~ Ra1/5, see Rossby, Deep Sea Res. 12 (1965). The Rossby scaling is based on the assumptions that the HC flows are laminar and determined by their boundary layers. However, the universality of this scaling is very questionable, as these flows are known to become more turbulent with increasing Ra. In Shishkina, et al., Geophys. Res. Lett. 43 (2016) we proposed a theoretical model for heat and momentum transport scalings with Ra, which is based on the ideas of Grossmann & Lohse, J. Fluid Mech. 407 (2000), applied to HC flows. The theory suggests multiple scaling regimes, including the Rossby scaling, see the figure below.
Figure 2: A schematic sketch of the phase diagram in a (Ra, Pr)-plane of the main theoretically possible limiting regimes of the scaling Nu ~ Raβ in horizontal convection, as proposed in Shishkina, et al., Geophys. Res. Lett. 43 (2016) The scaling exponent β for each regime is given in a magenta box. The boundaries between neighbour regimes, Pr ~ Raγ, are determined by matching Nu in these regimes; the exponent γ is written close to each corresponding boundary. Dash lines denote the boundaries between the laminar and turbulent viscous BLs. Only slopes of the regime boundaries are relevant in these diagrams, not their exact locations. Adopted from Shishkina, et al., Geophys. Res. Lett. 43 (2016).
Figure 2: A schematic sketch of the phase diagram in a (Ra, Pr)-plane of the main theoretically possible limiting regimes of the scaling Nu ~ Raβ in horizontal convection, as proposed in Shishkina, et al., Geophys. Res. Lett. 43 (2016) The scaling exponent β for each regime is given in a magenta box. The boundaries between neighbour regimes, Pr ~ Raγ, are determined by matching Nu in these regimes; the exponent γ is written close to each corresponding boundary. Dash lines denote the boundaries between the laminar and turbulent viscous BLs. Only slopes of the regime boundaries are relevant in these diagrams, not their exact locations. Adopted from Shishkina, et al., Geophys. Res. Lett. 43 (2016).
In order to check, whether the derived scalings hold in the case of laminar and transitional HC flows, DNS were conducted for Ra from 3⋅108 to 3⋅1012 and Pr from 0.05 to 50, i.e., more than 3 decades in Ra and Pr, in a container of the aspect ratio Γ=10, see Shishkina, et al. Phys. Rev. Lett. 116 (2016). In perfect agreement with the theoretical predictions it was found that Re ~ Ra 2/5 Pr –4/5, Nu ~ Ra 1/5 Pr 1/10 in the laminar regime Iℓ, and Re ~ Ra 1/2 Pr –1, Nu ~ Ra 1/4 Pr 0 in the regime Iℓ∗, see the figure below. At larger Ra a transition to Nu ~ Ra 1/4 and Re ~ Ra 1/3 was found, which hints toward a transition to a turbulent regime Reiter,et al. J. Fluid Mech. accepted (2020).
Figure 3: (a) Nu vs. Ra for Pr=0.1 (circles), Pr=1 (squares) and Pr=10(diamonds) for classical (closed symbols) and symmetrical HC (open symbols). (b) Re based on kinetic energy in complete volume (circles) and only volume above heated plates (squares). (c),(d) shows the Pr dependence for Ra=109 (green pentagons) and Ra=1010 (violet pentagons). The DNS results support the scalings of the regime Iℓ and transition to the regime Iℓ∗. Adopted from Shishkina, et al. Phys. Rev. Lett. 116 (2016).
Figure 3: (a) Nu vs. Ra for Pr=0.1 (circles), Pr=1 (squares) and Pr=10(diamonds) for classical (closed symbols) and symmetrical HC (open symbols). (b) Re based on kinetic energy in complete volume (circles) and only volume above heated plates (squares). (c),(d) shows the Pr dependence for Ra=109 (green pentagons) and Ra=1010 (violet pentagons). The DNS results support the scalings of the regime Iℓ and transition to the regime Iℓ∗. Adopted from Shishkina, et al. Phys. Rev. Lett. 116 (2016).
In general, the HC dynamics are rich in flow structures and instability transitions. For our setup, the analysis reveals three different unsteady flow regimes: detached plume regime, oscillatory regime in SHC and chaotic regime (see figure 3 a, b). The onset of the former two instabilities have been obtained theoretically up to a constant and were confirmed by our DNS data (see figure 4 a,b). The oscillations takes place at RaPr -1 ≈ 5x109 and the onset of detaching plumes at RaPr 5/4 ≈ 9x1010 . They coincide with the onsets of scaling transitions (figure 2).
Figure 4: Snapshots of the temperature field for (a) detaching plumes (Pr = 10, Ra =1010)and (b) oscillations (Pr = 0.1, Ra = 3 × 108).
Figure 4: Snapshots of the temperature field for (a) detaching plumes (Pr = 10, Ra =1010)and (b) oscillations (Pr = 0.1, Ra = 3 × 108).
Figure 5: Ra-Pr phase space of the flow dynamics: steady (diamonds), oscillations (opens quares), plumes (triangles) and chaotic (open circles). The solid lines in (a,b) show thetheoretical predicted onsets of oscillation and plume regime, the red dashed line thesemi-empirical predicted plume regime onset.
Figure 5: Ra-Pr phase space of the flow dynamics: steady (diamonds), oscillations (opens quares), plumes (triangles) and chaotic (open circles). The solid lines in (a,b) show thetheoretical predicted onsets of oscillation and plume regime, the red dashed line thesemi-empirical predicted plume regime onset.
