Asymptotic ultimate regime of homogeneous Rayleigh–Bénard convection on logarithmic lattices

Abstract

We investigate how the heat flux $Nu$ scales with the imposed temperature gradient $Ra$ in homogeneous Rayleigh–Bénard convection using one-, two- and three-dimensional simulations on logarithmic lattices. Logarithmic lattices are a new spectral decimation framework which enables us to span an unprecedented range of parameters ( $Ra$ , $Re$ , $\Pr$ ) and test existing theories using little computational power. We first show that known diverging solutions can be suppressed with a large-scale friction. In the turbulent regime, for $\Pr \approx 1$ , the heat flux becomes independent of viscous processes (‘asymptotic ultimate regime’, $Nu\sim Ra ^{1/2}$ with no logarithmic correction). We recover scalings coherent with the theory developed by Grossmann and Lohse, for all situations where the large-scale frictions keep a constant magnitude with respect to viscous and diffusive dissipation. We also identify another turbulent friction-dominated regime at $\Pr \ll 1$ , where deviations from the Grossmann and Lohse prediction are observed. These two friction-dominated regimes may be relevant in some geophysical or astrophysical situations, where large-scale friction arises due to rotation, stratification or magnetic field.

Log-Lattices for Atmospheric Flows

“…The resulting heat transfer, Nu, as a function of the Rayleigh number Ra is shown in Figure 2. In that case, we observe that the convection starts as Ra c = 6 × 10 5 , which is larger than the value predicted by the linear theory at F = 0, and those observed in previous log-lattice simulations [11]. This means that friction stabilizes the convection in the non-rotating regime.…”

“…However, lower values of λ give rise to more interactions, and thus more fluctuations, which is more realistic for simulating a turbulent flow. As discussed in [11], the case λ = 2 should be avoided when simulating incompressible dynamics because it lacks backscatter. On the other hand, ref.…”

“…We now consider a rotating homogeneous fluid, with coefficient of thermal dilation α, viscosity ν and diffusivity κ, subject to a temperature gradient ∆T over a length H and vertical gravity g. Its dynamics are given by the HRB set of wequations [11,[16][17][18],…”

“…We will assume below that this friction is concentrated only on a large scale. As proven in [11], this friction is mandatory to allow the system to reach well-defined stationary states. Indeed, for periodic boundary conditions, exponential instabilities can grow because there is no wall to stop them.…”

“…We first consider the non-rotating case, E = ∞. In this case, the phenomenology of homogeneous convection distinguishes three regimes for the behavior of the heat flux as a function of the forcing [11,16]:…”