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

TL;DR

This study experimentally investigates how tilting a cylindrical cell affects viscous boundary layer properties in turbulent thermal convection, revealing scaling behaviors, profile shapes, and deviations from classical laminar boundary layer theory.

Contribution

It provides new insights into boundary layer scaling, profile shapes, and the impact of cell tilt in turbulent Rayleigh-Bénard convection.

Findings

Boundary layer thickness scales with Reynolds number as Re^{-0.46} at small tilt angles.

Normalized velocity profiles are invariant with Ra but vary with tilt angle.

Reynolds numbers scale with Ra as Re ~ Ra^{0.43} and Re_sigma ~ Ra^{0.55}.

Abstract

We report an experimental study of the properties of the velocity boundary layer in turbulent Rayleigh-B\'{e}nard 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 $\theta$ relative to gravity, at $\theta=0.5^{o}$, $1.0^{o}$, $2.0^{o}$, and $3.4^{o}$, respectively. The fluid was water with Prandtl number $Pr=5.3$. It is found that at small tilt angles ($\theta \le 1^{o}$), the measured viscous boundary layer thickness $\delta_{v}$ scales with the Reynolds number $Re$ with an exponent close to that for a Prandtl-Blasius laminar boundary layer, i.e. $\delta_{v} \sim Re^{-0.46\pm0.03}$. For larger tilt angles, the scaling exponent of $\delta_{v}$ with $Re$ decreases with $\theta$. 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 $Re_{\sigma}$ based on the maximum rms velocity scales with $Ra$ as $Re_{\sigma} \sim Ra^{0.55}$, with both exponents do not seem to depend on the tilt angle $\theta$. 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 $Ra$.