Round buoyant laminar and turbulent plumes

Two approximate solutions, which become exact for Prandtl numbers 1 and 2, are given for laminar round plumes for arbitrary Prandtl number σ. The first of these gives a very accurate solution for laminar found plumes in air, for which σ = 0.73, and the second provides a good solution for laminar round plumes in water, for which σ = 6.7. Two approximate solutions, which become exact for turbulent Prandtl numbers 1.1 and 2, respectively, are given for turbulent round plumes. The one which becomes exact at σ = 1.1 is used to give a highly accurate solution for σ = 1, which, for an eddy-viscosity coefficient λ equal to 0.0156, provides a remarkably good agreement with the experimental data of Beuther, Capp, and George, Jr.