We use the background method to find an upper bound on the nondimensional temperature dissipation, $〈{‖\mathbf{\nabla }T‖}_2^2〉$, for Rayleigh-Bénard convection with the temperature of one boundary modulated in time. The resulting bound depends on characteristics of the temperature modulation profile, $f\left(t\right)$, including the nondimensional parameter $\omega$, which is defined as the supremum of $|f^{\prime }\left(t\right)|$. It is found that the resulting bound on $〈{‖\mathbf{\nabla }T‖}_2^2〉$ grows like $\sqrt{\omega }$ for $\omega \gg \mathrm{Ra}$, with $\mathrm{Ra}$ fixed, and like $\sqrt{\mathrm{Ra}}$ for $\mathrm{Ra}\gg \omega$, with $\omega$ fixed. Asymptotically, the bound for large $\mathrm{Ra}$ has the same leading order behavior as the nonmodulated case, with modulation effects appearing only at $O\left({\mathrm{Ra}}^{-1/2}\right)$.

Close

PrevNext

View figure in article

• Dissipative dynamics
• Flow instability
• Rayleigh-Bénard convection