  The pressure gradient has a significant influence on the mean-flow properties
of turbulent boundary layers. Conventional analytical studies on PG turbulent
boundary layers are conducted separately for the viscous sub-layer and overlap
layer only, with the remaining large portion of the boundary layers less
described theoretically. Here a symmetry approach is proposed for modeling
whole mean velocity profiles of turbulent boundary layers with both PG and
historical turbulence effects. First, a modified defect law is constructed for
the total stress profile to capture the Reynolds stress overshoot owing to the
PG and historical turbulence effects. Second, the multi-layered power-law
formulation of the stress length function in the structural ensemble dynamics
theory of the canonical zero-PG turbulent boundary layer (J. Fluid Mechanics,
2017, Vol. 827, pp. 322-35) is extended to describe the PG turbulent boundary
layer. Comparing with that of the zero-PG turbulent boundary layers, the stress
length of the PG turbulent boundary layer possesses a variable (in magnitude
and extension) plateau in the defect layer, which are characterized by three
parameters: the buffer layer thickness, the (nominal) K'arm'an constant, and
the defect-law exponent. In the case of intense turbulence from upstream flow,
a newly-identified "bulk turbulent layer" replaces the conventional buffer
layer and overlap layer. With the above formulations, the entire mean velocity
profile is predicted analytically, and validated to accurately describe the
published direct numerical simulation data on a two-dimensional separation
bubble. The study provides a novel method to parameterize PG turbulent boundary
layers with high accuracy and sound physics, and paves a way for quantifying
complicated boundary-layer flows.
