Abstract
An equation for the evolution of mean kinetic energy,
$E$
, in a two-dimensional (2-D) or 3-D Rayleigh–Bénard system with domain height
$L$
is derived. Assuming classical Nusselt-number scaling,
$Nu \sim Ra^{1/3}$
, and that mean enstrophy, in the absence of a downscale energy cascade, scales as
$\sim E/L^2$
, we find that the Reynolds number scales as
$Re \sim Pr^{-1}Ra^{2/3}$
in the 2-D system, where
$Ra$
is the Rayleigh number and
$Pr$
the Prandtl number. Using the evolution equation and the Reynolds-number scaling, it is shown that
$\tilde {\tau } \gtrsim Pr^{-1/2}Ra^{1/2}$
, where
$\tilde {\tau }$
is the non-dimensional convergence time scale. For the 3-D system, we make the estimate
$\tilde {\tau } \gtrsim Ra^{1/6}$
for
$Pr = 1$
. It is estimated that the total computational cost of reaching the high
$Ra$
limit in a simulation is comparable between two and three dimensions. The predictions are compared with data from direct numerical simulations.
Publisher
Cambridge University Press (CUP)
Subject
Mechanical Engineering,Mechanics of Materials,Condensed Matter Physics,Applied Mathematics