Turbulent natural convection scaling in a vertical channel

• DNS mean temperature data for Ra⩽2.0×107Ra⩽2.0×107 continue to support a −1/3 power-law.
• Implicit Nu–Ra equation is acceptable for 5.4×105⩽Ra⩽2.0×1075.4×105⩽Ra⩽2.0×107.
• Inner temperature fluctuations scale with inner temperature and length scale.
• Inner mean velocity and normal velocity variances scale with outer velocity scale, uo.
• Incipient proportional relationship for uτ/uo for 5.4×105⩽Ra⩽2.0×1075.4×105⩽Ra⩽2.0×107.

Using direct numerical simulation (DNS) data, this study appraises existing scaling laws in literature for turbulent natural convection of air in a differentially heated vertical channel. The present data is validated using past DNS studies, and covers a range of Rayleigh number, Ra between 5.4 × 105 and 2.0 × 107. We then appraise and compare the various scaling laws proposed by Versteegh and Nieuwstadt, 1999, Hölling and Herwig, 2005, Shiri and George, 2008 and George and Capp, 1979 with the profiles of the mean temperature defect, mean streamwise velocity, normal velocity fluctuations, temperature fluctuations and Reynolds shear stress. Based on the arguments of an inner (near-wall) and outer (channel centre) region, the data is found to support a minus one-third power law for the mean temperature in an overlap region. Using the inner and outer temperature profiles, an implicit heat transfer equation is obtained and we show that a correction term is non-negligible for the present Ra range when compared with explicit equations found in literature. In addition, we determined that the mean streamwise velocity and normal velocity fluctuations collapse in the inner region when using the outer velocity scale. We also find that the temperature fluctuations scale in inner coordinates, in contrast to the outer scaling behaviour reported in the past. Lastly, we show evidence of an incipient proportional relationship between friction velocity, uτ, and the outer velocity scale, uo, with increasing Ra.