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Abstract

In turbulent Rayleigh–Bénard (RB) convection with regular, mono-scale, surface
roughness, the scaling exponent β in the relationship between the Nusselt number
Nu and the Rayleigh number Ra, Nu ∼ Raβ
can be ≈1/2 locally, provided that Ra is
large enough to ensure that the thermal boundary layer thickness λθ
is comparable to
the roughness height. However, at even larger Ra, λθ becomes thin enough to follow
the irregular surface and β saturates back to the value for smooth walls (Zhu et al.,
Phys. Rev. Lett., vol. 119, 2017, 154501). In this paper, we prevent this saturation
by employing multiscale roughness. We perform direct numerical simulations of
two-dimensional RB convection using an immersed boundary method to capture the
rough plates. We find that, for rough boundaries that contain three distinct length
scales, a scaling exponent of β = 0.49 ± 0.02 can be sustained for at least three
decades of Ra. The physical reason is that the threshold Ra at which the scaling
exponent β saturates back to the smooth wall value is pushed to larger Ra, when the
smaller roughness elements fully protrude through the thermal boundary layer. The
multiscale roughness employed here may better resemble the irregular surfaces that
are encountered in geophysical flows and in some industrial applications.