Scaling of velocity and temperature fluctuations in turbulent thermal convection

Abstract

The scaling arguments presented by [J. Fluid Mech. 204 (1989) 1] to justify the Nu∼Ra 2/7 power law observed in high Rayleigh number thermal convection also have implications regarding the distribution of velocity and temperature fluctuations within the turbulent layer. Asymptotic matching of the inner scaling law and the outer scaling law implies that the root-mean-square (r.m.s.) vertical velocity fluctuation varies as σ w∼ln z, and the r.m.s. temperature fluctuation varies as σ θ ∼ln z or σ θ ∼ z −1/2 depending on the details of the modeling assumptions [Exp. Fluids 4 (1986) 121]. These results differ markedly from the Priestley similarity theory [Turbulent Transport in the Lower Atmosphere, 1959], in which σ w∼ z 1/3 and σ θ ∼ z −1/3. Consequently, the r.m.s. velocity and temperature profiles offer the possibility to distinguish between the two theories. We present measurements in wide-aspect-ratio convection over the range Ra=10 7–10 9 that support the logarithmic variation for both the temperature and velocity.