6 research outputs found
High-wavenumber steady solutions of two-dimensional Rayleigh--B\'enard convection between stress-free boundaries
Recent investigations show that steady solutions share many features with
turbulent Rayleigh--B\'enard convection (RBC) and form the state space skeleton
of turbulent dynamics. Previous computations of steady roll solutions in
two-dimensional (2D) RBC between no-slip boundaries reveal that for fixed
Rayleigh number and Prandtl number , the heat-flux-maximizing solution
is always in the high-wavenumber regime. In this study, we explore the
high-wavenumber steady convection roll solutions that bifurcate supercritically
from the motionless conductive state for 2D RBC between stress-free boundaries.
Our computations confirm the existence of a local heat-flux-maximizing solution
in the high-wavenumber regime. To elucidate the asymptotic properties of this
solution, we perform computations over eight orders of magnitude in the
Rayleigh number, , and two orders of magnitude in
the Prandtl number, . The numerical results
indicate that as , the local heat-flux-maximizing aspect ratio
, the Nusselt number , and the Reynolds number ,
with all prefactors depending on . Moreover, the interior flow of the local
-maximizing solution can be well described by an analytical heat-exchanger
solution, and the connection to the high-wavenumber asymptotic solution given
by Blennerhassett & Bassom is discussed. With a fixed aspect ratio
at , however, our computations show that as
increases, the steady rolls converge to the semi-analytical asymptotic
solutions constructed by Chini & Cox, with scalings and
. Finally, a phase diagram is delineated to gain a
panorama of steady solutions in the high-Rayleigh-number-wavenumber plane