Energy consistent Gaussian integral model for jet with off-source heating

Jets with volumetric off-source heating have been studied extensively in the past few decades, especially since they mimic cumulus clouds, where diabatic heating of updrafts occurs due to condensation of droplets. In such systems, a pure jet, entering the flow domain via a round inlet in a neutrally buoyant environment, is heated within a specified "heat injection zone" (HIZ). The heating rate inside the HIZ is proportional to the concentration of tracer species, injected at the inlet. In this work, we construct one-dimensional Gaussian integral models for this system, in which we evolve the mean flow rates of volume, specific momentum, mean temperature, and mean tracer concentration, along the axial direction. We additionally evolve the mean flow rates of the squares of mean velocity, mean temperature, and mean tracer concentration, leading to an "energy consistent" closure for the entrainment rate coefficient and the integral flow variables. Unlike conventional integral models for jets and plumes, our formulation allows the widths for mean velocity, mean temperature, and mean tracer concentration profiles to evolve independently along the jet axis. We also perform large eddy simulations (LESs) of the system at different heating rates. A priori estimates of the source terms in the Gaussian integral equations, as well as a posteriori predictions from the one-dimensional integral model, compare reasonably well with LES statistics. Our results demonstrate the importance of evolving the dispersion ratios for velocity, temperature, and tracer concentration independently in integral models of such systems. Extending such an energy consistent model to an ensemble of cloudy atmospheric plumes is not straighforward. Data from LES of clouds is used to examine the validity of energy consistency, as well as the way forward in terms of implementing such a scheme in General Circulation Models.