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

We 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 $Ra$ in the range $2.8\times10^{8}<Ra<5.6\times10^{9}$ and were conducted with the convection cell tilted with an angle $θ$ relative to gravity, at $θ=0.5^{o}$, $1.0^{o}$, $2.0^{o}$, and $3.4^{o}$, respectively. The fluid was water with Prandtl number $Pr=5.3$. It is found that at small tilt angles ($θ\le 1^{o}$), the measured viscous boundary layer thickness $δ_{v}$ scales with the Reynolds number $Re$ with an exponent close to that for a Prandtl-Blasius laminar boundary layer, i.e. $δ_{v} \sim Re^{-0.46\pm0.03}$. For larger tilt angles, the scaling exponent of $δ_{v}$ with $Re$ decreases with $θ$. The normalized mean horizontal velocity profiles measured at the same tilt angle but with different $Ra$ are found to have an invariant shape. But for different tilt angles, the shape of the normalized profiles is different. It is also found that the Reynolds number $Re$ based on the maximum mean horizontal velocity scales with $Ra$ as $Re \sim Ra^{0.43}$ and the Reynolds number $Re_σ$ based on the maximum rms velocity scales with $Ra$ as $Re_σ \sim Ra^{0.55}$, with both exponents do not seem to depend on the tilt angle $θ$. We also examined the dynamical scaling method proposed bys Zhou and Xia [Phys. Rev. Lett. 104, 104301 (2010)] and found that in both the laboratory and the dynamical frames the mean velocity profiles show deviations from the theoretical Prandtl-Blasius profile, with the deviations increase with $Ra$. But 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 $Ra$.