Warm discharges in cold fresh water. Part 1. Line plumes in a uniform ambient

Abstract

Turbulent buoyant plumes in cold fresh water are analysed, assuming a quadratic dependence of density on temperature. The model is based on the assumption that entrainment velocity is proportional to vertical velocity in the plume. Numerical and asymptotic solutions are obtained for both rising and descending plumes from virtual sources with all possible combinations of buoyancy, volume and momentum fluxes. Physical sources can be identified as points on trajectories of plumes from virtual sources.The zero-buoyancy condition, at which the plume and the ambient have equal densities but their temperatures are on opposite sides of the temperature of maximum density, is of particular importance. If an upwardly buoyant plume rising through a body of water reaches the surface before passing through its zero-buoyancy level, it will form a surface gravity current; otherwise, the plume water will return to the source as a fountain. The height at which zero buoyancy is attained generally decreases as the source momentum flux increases: greater plume velocity produces greater entrainment and hence more rapid temperature change. Descending plumes, if ejected downwards against upward buoyancy, may be classified as strongly or weakly forced according to whether they reach the zero-buoyancy condition before being brought to rest. If they do, they continue to descend with favourable buoyancy; otherwise, they may form an inverted fountain. Once a descending plume has attained downward buoyancy, it can continue to descend indefinitely, ultimately behaving like a plume in a fluid with a linear equation of state. In contrast, a rising plume will eventually come to rest, however large its initial upward buoyancy and momentum fluxes are.