The Determination of Formation Number for Starting Buoyant Jets

Starting buoyant jets are widely observed in nature as well as in engineering applications. The interactions between the leading vortex ring and the trailing stem play a significant role on the development of the staring processes, and the Formation Number is established to be the criterion that demarcates the presence of the trailing stem and thus, the occurrence of pinch-off. In this study, the buoyant formation number for a starting buoyant jet which includes the momentum inducement due to presence of buoyancy is examined numerically. The investigation is based on the results of a series of numerical simulations with the Large-Eddy Simulation (LES) approach to reproduce the starting buoyant jet in a wide range of conditions from pure jets to lazy plumes. Based on the results, the buoyant formation number can be obtained following the occurrence of a step-jump in the vortex ring circulation in the following manner. First, the vorticity is integrated through the half central plane of the computational domain, which a trough can be observed to follow the head vortex ring in the vertical distribution. The trailing stem and the head vortex ring are differentiated based on this trough location. Subsequently, if and just before a pitch off occurs, a step-jump in the circulation of the head vortex ring is typically observed. The jump value is then traced back to the total circulation, and the non-dimensional time that it occurs. This non-dimensional time is found to the same as the formation number for the runs conducted. Using this method, a comparison of the numerical results with the experimental data for a starting pure jet is performed, and the widely accepted formation number (approximate to 4.0) is obtained which verifies that the method is satisfactory. The effect of buoyancy on the formation number is then investigated for two turbulent discharge conditions of Re = 2000 and 2500 and with a wide range of buoyancy fluxes.