Inner, meso, and outer scales in a differentially heated vertical channel

Abstract

Reynolds shear stress and Reynolds normal stress in a differentially heated vertical channel were shown in a previous paper to scale by a mixed scale of the friction velocity uτ and the maximum mean vertical velocity Umax. This scale was empirically determined; this work presents a physical understanding of the new scaling. A new dimensional analysis is developed that involves a simple modification of the traditional inner–outer scaling. The functional dependencies in the new dimensional analysis are determined using direct numerical simulation (DNS) data. The DNS data confirm the new physical reasoning and reveal, as follows: (i) The velocity fluctuation variance at the centerline scales with an outer scale, which is the mixed scale uτUmax. (ii) The location of the maximum mean vertical velocity scales as an Obukhov-style length scale, which, in turn, scales as the geometric mean of the inner length scale and the outer length scale. (iii) The temperature fluctuation variance at the centerline scales with an outer temperature scale. (iv) The temperature fluctuation variance peaks in the inner layer, and its peak value scales with an inner temperature scale. The new outer scales obtained from the dimensional analysis are shown to be consistent with the properties of the mean momentum balance equation.