The three-dimensional distributions of tangential velocity and total- temperature in vortex tubes

The axial and radial gradients of the tangential velocity distribution are calculated from prescribed secondary flow functions on the basis of a zero-order approximation to the momentum equations developed by Lewellen. It is shown that secondary flow functions may be devised which meet pertinent physical requirements and which at the same time lead to realistic tangential velocity gradients. The total-temperature distribution in both the axial and radial directions is calculated from such secondary flow functions and corresponding tangential velocity results on the basis of an approximate turbulent energy equation. The method employed for the solution of this equation stresses the equivalence of the vortex tube to counter-current systems with transverse diffusion such as distillation columns and heat exchangers. An availability function is derived that permits the evaluation of vortex tube performance on the basis of velocity data. Turbulent diffusivities resulting from the quantitative use of the tangential velocity approximation are shown to agree with those derived from the total-temperature calculations.