Turbulent curtains of smoke rise initially as flat plumes and, above a certain height, they become round plumes. The same evolution of cross-sectional shape is exhibited by jets issuing from flat nozzles. Here we predict based on principle that all such flows should evolve their cross-sectional shapes from flat to round (and not the other way) at a critical distance downstream, which is predictable. The principle is that the prevailing flow architecture provides greater access to the flow of momentum from the moving core (plume, jet) to the still surroundings. For turbulent plumes and jets, the transition distance scales with the long dimensions (L) of the two-dimensional (flat) heat sources and nozzles that drive them. For laminar jets, the transition distance scales with L Re, where Re is the Reynolds number based on nozzle velocity and the smaller dimension of the nozzle cross section. These predictions are confirmed by full numerical experiments of the three-dimensional flow fields of turbulent and laminar jets covering the Re range 10-10(4).