Reconsidering turbulent natural convection in a differentially heated vertical channel

The turbulent natural convection boundary layer next to heated vertical surfaces was originally considered by George and Capp (IJHMT 1979). By recognizing the existence of a constant heat flux layer, but the absence of a constant stress layer (due to gravity), they postulated the existence of a buoyant sublayer in which the buoyancy flux from the wall, g$\beta $F$_{o}$, was the crucial parameter. Using it they postulated the existence of inner and outer scales for temperature and velocity. Asymptotic matching led to cube root and inverse cube root profiles for the mean velocity and temperature profiles respectively, and Nu $\sim $ C(Pr) Ra$^{1/3}$ for the wall heat transfer. The theory has generally been found to be in excellent agreement with numerous studies over the past three decades for the temperature and heat transfer relations, but the velocity scaling has always been problematical. This work reconsiders the velocity scaling, and shows the GC result to be inconsistent with the momentum integral. We argue instead that the proper inner velocity should be u$_{i}$= (g$\beta \Delta $T$_{w}$ h)$^{1/2}$, where h is the channel width. This is in turn proportional to the friction velocity, hence sets the friction law.