  Despite its limitations, Prandtl's mixing length model is widely applied in
modelling turbulent free shear flows. Prandtl's extended model addresses many
of the shortfalls of the original model, but is not so widely used, in part due
to additional mathematical complexities that arise in its derivation and
implementation. Furthermore, in both models Prandtl neglects the kinematic
viscosity on the basis that it is much smaller in magnitude than the turbulent
viscosity. Recent work has shown that including the kinematic viscosity in the
original model has both mathematical and physical advantages. In the present
work, a novel derivation of the extended model is provided, and it is
demonstrated that similar advantages are again obtained when the kinematic
viscosity is included. Additionally, through the use of scaling techniques,
similarity mean velocity profiles of the extended model are derived, resulting
in a single nonlinear ordinary differential equation that is solved numerically
with a Hermite spectral method. The computed profiles for the normalised
similarity mean velocity and shear stress are compared to experimental
observations and shown to be in excellent agreement.
