  A large eddy simulation wall model is developed based on a formal
interpretation of quasi-equilibrium that governs the momentum balance
integrated in the wall-normal direction. The model substitutes the
law-of-the-wall velocity profile for a smooth surface into the wall-normal
integrated momentum balance, leading to a Lagrangian relaxation towards
equilibrium (LaRTE) transport equation for the friction velocity vector
$\mathbf{u}_\tau(x,z,t)$. This partial differential equation includes a
relaxation timescale governing the rate at which the wall stress can respond to
imposed fluctuations due to the inertia of the fluid layer from the wall to the
wall-model height. A-priori tests based on channel flow direct numerical
simulation (DNS) data show that the identified relaxation timescale ensures
self-consistency with assumed quasi-equilibrium conditions. The new approach
enables us to formally distinguish quasi-equilibrium from additional,
non-equilibrium contributions to the wall stress. A particular model for
non-equilibrium contributions is derived, motivated by laminar Stokes layer
dynamics in the viscous sublayer when applying fast varying pressure gradients.
The new wall modeling approach is first tested in standard equilibrium channel
flow in order to document various properties of the approach. The model is then
applied in LES of channel flow with a suddenly applied spanwise pressure
gradient (SSPG). The resulting mean wall stress evolution is compared to DNS
with good agreement. At the onset of the SSPG, the laminar Stokes layer
develops rapidly while the LaRTE portion of the stress has a delayed response
due to its inherent relaxation dynamics. Results also highlight open challenges
such as modeling the response of near-wall turbulence occurring above the
viscous sublayer and at timescales faster than quasi-equilibrium conditions.
