Exponentially growing solutions in homogeneous Rayleigh-Bénard convection

It is shown that homogeneous Rayleigh-Bénard flow, i.e., Rayleigh-Bénard turbulence with periodic boundary conditions in all directions and a volume forcing of the temperature field by a mean gradient, has a family of exact, exponentially growing, separable solutions of the full nonlinear system of equations. These solutions are clearly manifest in numerical simulations above a computable critical value of the Rayleigh number. In our numerical simulations they are subject to secondary numerical noise and resolution dependent instabilities that limit their growth to produce statistically steady turbulent transport.

Figure 1

Derivative with respect to time of the logarithm of $\mathrm{Nu}\left(t\right)$, in a homogeneous convective system at $\mathrm{Ra}=4.5\times {10}^4\simeq 30{\mathrm{Ra}}_c$. The direct numerical simulation (DNS) is implemented through a Lattice Boltzmann equation (LBE) algorithm on a cubic grid with resolution ${240}^3$, as in Refs. 5, 6. Horizontal lines, from top to bottom, correspond respectively to values $2\lambda (0,1)$, $2\lambda (1,1)$, $2\lambda (0,2)$, $2\lambda (0,2)$, and the zero level. The time, here and in the following figures, is dimensionless as for the set of Eqs. (3, 4), i.e., it has been normalized by the thermal diffusion time across the box, $H^2∕\kappa$.

Figure 2

Linear-log plot of the spatial rms value of the three velocity component and the thermal fluctuation in the DNS at resolution ${32}^3$, $w_{\mathit{rms}}$ (solid), $u_{\mathit{rms}}$ (dashed), and $v_{\mathit{rms}}$ (dotted).

Figure 3

A comparison between floating point (solid) and double precision (dashed) calculations for the root mean squared velocity component $w$, and ${⟨u^2+v^2⟩}^{1∕2}$ vs time, with spatial resolution ${32}^3$ and second order Adams-Bashforth as time marching algorithm.