Viscous boundary layer properties in turbulent thermal convection in a cylindrical cell: the effect of cell tilting

Abstract

AbstractWe report an experimental study of the properties of the velocity boundary layer in turbulent Rayleigh–Bénard convection in a cylindrical cell. The measurements were made at Rayleigh numbers $\mathit{Ra}$ in the range $2. 4\times 1{0}^{8} \lt \mathit{Ra}\lt 5. 6\times 1{0}^{9} $ and were conducted with the convection cell tilted with an angle $\theta $ relative to gravity, at $\theta = 0. 5, 1. 0, 2. 0$ and $3. {4}^{\circ } $, respectively. The fluid was water with Prandtl number $\mathit{Pr}= 5. 3$. It is found that at small tilt angles ($\theta \leq {1}^{\circ } $), the measured viscous boundary layer thickness ${\delta }_{v} $ scales with the Reynolds number $\mathit{Re}$ with an exponent close to that for a Prandtl–Blasius (PB) laminar boundary layer, i.e. ${\delta }_{v} \sim {\mathit{Re}}^{- 0. 46\pm 0. 03} $. For larger tilt angles, the scaling exponent of ${\delta }_{v} $ with $\mathit{Re}$ decreases with $\theta $. The normalized mean horizontal velocity profiles measured at the same tilt angle but with different $\mathit{Ra}$ are found to have an invariant shape. However, for different tilt angles, the shape of the normalized profiles is different. It is also found that the Reynolds number $\mathit{Re}$ based on the maximum mean horizontal velocity scales with $\mathit{Ra}$ as $\mathit{Re}\sim {\mathit{Ra}}^{0. 43} $ and the Reynolds number ${\mathit{Re}}_{\sigma } $ based on the maximum root mean square velocity scales with $\mathit{Ra}$ as ${\mathit{Re}}_{\sigma } \sim {\mathit{Ra}}^{0. 55} $. Within the measurement resolution neither exponent depends on the tilt angle $\theta $. Several wall quantities are also measured directly and their dependencies on $\mathit{Re}$ are found to agree well with those predicted for a classical laminar boundary layer. These are the wall shear stress $\tau $ (${\sim }{\mathit{Re}}^{1. 46} $), the viscous sublayer ${\delta }_{w} $ (${\sim }{\mathit{Re}}^{0. 75} $), the friction velocity ${u}_{\tau } $ (${\sim }{\mathit{Re}}^{- 0. 86} $) and the skin friction coefficient ${c}_{f} $ (${\sim }{\mathit{Re}}^{- 0. 46} $). Again, all of these near-wall quantities do not exhibit a dependence on the tilt angle within the measurement resolution. We also examined the dynamical scaling method proposed by Zhou and Xia (Phys. Rev. Lett., vol. 104, 2010, p. 104301) and found that in both the laboratory and the dynamical frames the mean velocity profiles show deviations from the theoretical PB profile, with the deviations increasing with $\mathit{Ra}$. However, profiles obtained from dynamical scaling in general have better agreement with the theoretical profile. It is also found that the effectiveness of this method appears to be independent of $\mathit{Ra}$.