Effect of natural convection on stability of flow in a vertical pipe

Abstract

If water is heated or cooled while flowing through a vertical pipe with a laminar motion, the velocity profile will differ from the parabolic shape for isothermal flow due to density variations in the fluid. If a constant heat flux is used at the wall and if the changes in temperature affect only the density appearing in the gravity term of the equations of motion, a condition is attained far downstream in the heat-transfer section such that there is no further change in the velocity profile. The shape of this fully developed velocity profile depends on the ratio of the heat flux to the flow rate. The stability of flow in an electrically heated pipe 762 diameters long was studied by detecting temperature fluctuations in the effluent. By use of a carefully designed entry and a long isothermal section prior to the heat exchange section, inlet disturbances were eliminated and transition to an unsteady flow resulted from a natural instability of the distorted profiles. It was found that the stability depends primarily on the shape of the velocity profile and only secondarily on the value of the Reynolds number, if at all. For upflow heating the flow first becomes unstable when the velocity profiles develop points of inflexion. Transition to an unsteady flow involves the gradual growth of small disturbances and therefore it is quite possible to have unstable flows without observing transition because the pipe is not long enough for the disturbances to attain a measurable amplitude. For downflow heating the flow instability is associated with separation at the wall. Transition to an unsteady flow is sudden and therefore transition occurs shortly after an unstable flow occurs. It is suggested that a change from a steady symmetrical to a steady unsymmetrical flow occurs in downflow when the profile develops points of inflexion.