Length of near-wall plumes in turbulent convection

We present planforms of line plumes formed on horizontal surfaces in turbulent convection, along with the length of line plumes measured from these planforms, in a six decade range of Rayleigh numbers ($1{0}^{5} \lt \mathit{Ra}\lt 1{0}^{11} $) and at three Prandtl numbers ($\mathit{Pr}= 0. 7, 5. 2, 602$). Using geometric constraints on the relations for the mean plume spacings, we obtain expressions for the total length of near-wall plumes on horizontal surfaces in turbulent convection. The plume length per unit area (${L}_{p} / A$), made dimensionless by the near-wall length scale in turbulent convection (${Z}_{w} $), remains constant for a given fluid. The Nusselt number is shown to be directly proportional to ${L}_{p} H/ A$ for a given fluid layer of height $H$. The increase in $\mathit{Pr}$ has a weak influence in decreasing ${L}_{p} / A$. These expressions match the measurements, thereby showing that the assumption of laminar natural convection boundary layers in turbulent convection is consistent with the observed total length of line plumes. We then show that similar relationships are obtained based on the assumption that the line plumes are the outcome of the instability of laminar natural convection boundary layers on the horizontal surfaces.