  Wall-models are essential for enabling large-eddy simulations (LESs) of
realistic problems at high Reynolds numbers. The present study is focused on
approaches that directly model the wall shear stress, specifically on filling
the gap between models based on wall-normal ordinary differential equations
(ODEs) that assume equilibrium and models based on full partial differential
equations (PDEs) that do not. We develop ideas for how to incorporate
non-equilibrium effects (most importantly, strong pressure-gradient effects) in
the wall-model while still solving only wall-normal ODEs. We test these ideas
using two reference databases: an adverse pressure-gradient turbulent
boundary-layer and a shock/boundary-layer interaction problem, both of which
lead to separation and re-attachment of the turbulent boundary layer.
