Analytical relations have been derived for calculation of developing thick, axisymmetric, turbulent boundary layers in a pressure gradient from two simultaneous differential equations: momentum and shape parameter. An entrainment method is used to obtain the shape parameter equation. Both equations incorporate the velocity similarity laws that provide a two-parameter velocity profile general enough to include any range of Reynolds numbers. Newly defined "quadratic" shape parameters which arise from the geometry of the thick axisymmetric boundary layer are analytically related to the two-dimensional shape parameter by means of these velocity similarity laws. The variation of momentum loss, boundary-layer thickness, local skin friction, and local velocity profile may be calculated for the axisymmetric turbulent boundary layers on underwater bodies, including the thick boundary layers on the tails. The various formulations are shown to correlate well with available experimental data.
A robust post-processing method to determine skin friction in turbulent boundary layers from the velocity profile