Scaling of maximum probability density function of velocity increments in turbulent Rayleigh-Bénard convection

Abstract

In this paper, we apply a scaling analysis of the maximum of the probability density function (pdf) of velocity increments, i.e., ${p_{\max }}\left( \tau \right) = {\max _{\Delta {u_\tau }}}p\left( {\Delta {u_\tau }} \right) \sim {\tau ^{ - \alpha }}$, for a velocity field of turbulent Rayleigh-Bénard convection obtained at the Taylor-microscale Reynolds number Re_λ ≈ 60. The scaling exponent α is comparable with that of the first-order velocity structure function, ζ(1), in which the large-scale effect might be constrained, showing the background fluctuations of the velocity field. It is found that the integral time T(x/D) scales as T(x/D) ~ (x/D)^−β, with a scaling exponent β = 0.25 ± 0.01, suggesting the large-scale inhomo-geneity of the flow. Moreover, the pdf scaling exponent α(x, z) is strongly inhomogeneous in the x (horizontal) direction. The vertical-direction-averaged pdf scaling exponent $\tilde \alpha \left( x \right)$ obeys a logarithm law with respect to x, the distance from the cell sidewall, with a scaling exponent ξ ≈ 0.22 within the velocity boundary layer and ξ ≈ 0.28 near the cell sidewall. In the cell’s central region, α(x, z) fluctuates around 0.37, which agrees well with ζ(1) obtained in high-Reynolds-number turbulent flows, implying the same intermittent correction. Moreover, the length of the inertial range represented in decade ${{\tilde T}_I}\left( x \right)$ is found to be linearly increasing with the wall distance x with an exponent 0.65 ± 0.05.