Explicit formulations of widely used wall models for large-eddy simulation

Algebraic wall models for large-eddy simulations may suffer from robustness issues when applied in complex flow configurations. On the other hand, models based on ordinary differential equations (ODE-based models) are robust, but are slower to converge, computationally more expensive, and are harder to implement. The latter is exacerbated by the plethora of hardware back-ends a modern code is expected to support. Here, approximate explicit formulations for, arguably, the three most widely used wall models are provided: the equilibrium ODE model, Spalding's model, and Reichardt's model. Using these explicit versions of the models makes them trivial to implement and the computation of the stress unconditionally stable. The resulting expressions are compact, yet introduce at most a 1% relative error in the models' velocity profile for all physically realizable values of the von Karman constant and intercept of the logarithmic law. Furthermore, tuned approximations with at most 0.05% relative error are provided for two sets of logarithmic law parameters: modern values based on high-Reynolds-number data and classical ones used in the original formulation of the wall models. In addition to using explicit models directly, they can be utilized to provide an initial guess for the iterative methods used to solve the respective original counterparts. We demonstrate that this leads to increased robustness and, for the ODE model, to faster convergence. While our work covers three wall models, the provided approximation approach is expected to be applicable to any model based on a one-dimensional coupling of the linear and logarithmic laws of the wall.