## Abstract

This paper describes an experimental investigation of heat transfer by free convection of a fluid in a heated vertical tube, sealed at its lower end. Heated fluid adjacent to the walls is discharged from the open end into a suitably cooled large reservoir, while a central core of cool fluid is continuously drawn into the tube by way of replacement. The system constitutes an unusual case of natural convection because the two streams of fluid, moving in opposite directions, are compelled to create their own internal boundary. Such an arrangement forms a static simulation of the Schmidt system (1951) for cooling high-temperature gas turbine blades, where sealed radial passages in the blades communicate with a reservoir in the rotor drum, and large centrifugal accelerations replace that due to gravity in the static system. The use of a scaled-up static tube in large measure compensates for the relatively small gravitational acceleration, when determining the working range of Rayleigh numbers, in this case from 10$^{7}$ to 10$^{13}$. These are based on tube length, the fluid property values being referred to tube-wall temperature. Separate assessments are made of the effect of fluid Prandtl number (covering values from 7600 to 0$\cdot $69) and tube length radius ratio (ranging from 7$\cdot $5 to 47$\cdot $5). In laminar flow the former is not found to be significant, but the quotient of the Rayleigh number (based on radius) and tube length-radius ratio determines the ranges of three laminar flow regimes. High values of the quotient correspond to 'boundary-layer flow' and greatest heat transfer. This is followed first by 'impeded non-similarity flow' and then by 'impeded similarity flow' as the quotient becomes smaller, where the two streams of fluid mingle. These findings are in close agreement with theoretical prediction (Lighthill 1953). Turbulence arises in two ways. For Prandtl numbers near unity, transition occurs during the laminar impeded-flow regimes, resulting in a mixing effect and reduced heat transfer. This is predicted by Lighthill, but his discussion of turbulent flow is restricted to a Prandtl number of unity. For larger Prandtl numbers, transition takes place during laminar boundary-layer flow, yielding a conventional turbulent boundary-layer regime with increased heat transfer. The mean transitional Grashof numbers (based on radius) are in the range 10$^{4\cdot 4}$ to 10$^{4\cdot 6}$; they compare favourably with a predicted range of from 10$^{4\cdot 0}$ to 10$^{4\cdot 3}$. The tendency for the cool entering fluid to become turbulent renders turbulent boundary-layer flow potentially unstable. Both modes of transition eventually lead to a stable 'fully mixed' regime where the two turbulent streams mix. This causes reduced circulation and heat transfer, the extent of the reduction varying directly with length-radius ratio and inversely with Prandtl number. The regime was predicted by Lighthill, but there are considerable discrepancies between estimated and experimental heat-transfer rates, and in the duration of the regime. In practice it appears to persist indefinitely, whereas Lighthill forecasts its replacement at high Rayleigh numbers by a stable boundary-layer flow. Empirical correlations show that fully mixed flow yields optimum heat transfer at a length-radius ratio, which is determined by the Rayleigh number. The suitability of the Schmidt system for blade cooling is briefly discussed in the light of the investigation.