A new ODE-based turbulence wall model accounting for pressure gradient and Reynolds number effects

In wall-modeled large-eddy simulations (WMLES), the near-wall model plays a significant role in predicting the skin friction, although the majority of the boundary layer is resolved by the outer large-eddy simulation (LES) solver. In this work, we aim at developing a new ordinary differential equation (ODE)-based wall model, which is as simple as the classical equilibrium model yet capable of capturing non-equilibrium effects and low Reynolds number effects. The proposed model reformulates the classical equilibrium model by introducing a new non-dimensional mixing-length function. The new mixing-length function is parameterized in terms of the boundary layer shape factor instead of the commonly used pressure-gradient parameters. As a result, the newly introduced mixing-length function exhibits great universality within the viscous sublayer, the buffer layer, and the log region (i.e., $0 < y < 0.1\delta$, where the wall model is typically deployed in a WMLES setup). The performance of the new model is validated by predicting a wide range of canonical flows with the friction Reynolds number between 200 and 5200, and the Clauser pressure-gradient parameter between -0.3 and 4. Compared to the classical equilibrium wall model, remarkable error reduction in terms of the skin friction prediction is obtained by the new model. Moreover, since the new model is ODE-based, it is straightforward to be deployed for predicting flows with complex geometries and therefore promising for a wide range of applications.