Abstract
Abstract This paper presents the effects of natural convection on laminar heat transfer in the Graetz problem. The fluid with a fully developed velocity profile is cooled or heated isothermally in the thermal entrance region of a circular pipe. Because of the density difference near the pipe wall, vortices superimposed on the fully developed velocity profile are generated, therefore the Graetz solution is no longer valid since the heat transfer rate and the pressure drop vary. This paper employs the Patankar's power law scheme, the boundary‐vorticity method and the Du Fort‐Frankel method to solve the present problem. At Ra=105, the present solution agrees well with the existing solution. The Nusselt number and the variations of flow and temperature fields are found at higher Rayleigh numbers, Ra=2.5×105∼106. It is observed from the present result that an oscillatory flow pattern of one‐pair and two‐pairs of counter rotating vortices appears at Ra≥5×105 along the flow direction. Eventually, a pair of count...
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