Abstract
Based on the n-family of laminar vortex formulation, a new generalized model applicable to the turbulent kind is presented. The self-similarity of the phenomenon allows, through the application of Vatistas and Aboelkassem (2006) simple variable transformation, to simulate its decay phase.
For the steady-state case, given the Vatistas et al. (1991) exponent n, the value of the turbulent intensity parameter  , intrinsic in the azimuthal velocity formula, is found by fitting the analytical tangential velocity to various experimental profiles with different effective Reynolds numbers using the Least Square Error (LSE) method. Alike to the laminar n-family, n = 2 gives the smallest error and thus the best approximation. Also taking  to be constant or varying with the radius produces insignificant differences in the velocity profile. Thus, in order to close the system, the tangential velocity with n = 2 and a constant  that minimizes the error is inducted into the analysis. When  is plotted against the effective Reynolds number, a coherent relationship amongst the two emerges. An empirical equation, which connects the two properties, is then constructed. This gives the ability to researchers to approximate the velocity (tangential, axial and radial components) using only three parameters: the effective Reynolds number, the core radius, and the maximum tangential velocity.
Application of the abovementioned variable transformation to the steady turbulent vortex yields its corresponding decaying version. The validity of the model is tested for several laminar and turbulent cases. The tangential velocity decay of fixed wing aircraft wake, and rotating helicopter blade tip turbulent vortices, approximated using the new model provides more realistic results than the traditional circulation approach. The profiles of the last property, that is routinely used in aviation to define the hazard threshold in order to provide a safe aircraft separation distance in large airports, is found to be lacking in representing the real cases of diminishing vortices. The previous lies on the fact that the assumed flattening of the circulation curve at large radii, applicable to laminar cases it is not true when the vortex is turbulent. Consequently, the prescribed value of the radius (e.g. 7 times the core radius) to represent the circulation at “infinity” proposed by Squire (1965) and Iversen (1976), implemented also in numerous other models like Burnham and Hallock (1982) and Proctor (2000), must be reconsidered. Future work should focus on the definition of the hazard threshold based on the tangential velocity instead of its circulation signature.