  The geometrically different plan forms of near wall plume structure in
turbulent natural convection, visualised by driving the convection using
concentration differences across a membrane, are shown to have a common
multifractal spectrum of singularities for Rayleigh numbers in the range
$10^{10}- 10^{11}$ at Schmidt number of 602. The scaling is seen for a length
scale range of $2^5$ and is independent of the Rayleigh number, the flux, the
strength and nature of the large scale flow, and the aspect ratio. Similar
scaling is observed for the plume structures obtained in the presence of a weak
flow across the membrane. This common non trivial spatial scaling is proposed
to be due to the same underlying generating process of the near wall plume
structures.
