Series solutions for turbulent plumes evolving in a natural environment

Closed form solutions to the boundary layer equations for turbulent point and line thermal plumes evolving in natural convection have been obtained in the form of power series for the case when the turbulent viscosity is assumed to be proportional to the vertical height above the source. The initial values needed to generate the coefficients of the power series for different turbulent Prandtl number have been obtained numerically. To compliment these values the constants of proportionality between the turbulent viscosity and the height were determined using different methodologies including analysis of experimental data, computational fluid dynamics and numerical considerations. Evaluation of the results is primarily carried out by comparison to experimentally determined profiles of the temperature and velocity that are found in the literature. The best agreement was obtained when the Reynolds analogy—giving the turbulent Prandtl number as unity—was adopted. While the range of validity of the power series is limited, its radius of convergence can be extended using a suitable transformation.