Solving high Reynolds number flows on Cartesian cut-cell meshes using an ODE wall function with momentum balance

• Wall functions can resolve high Reynolds number RANS flows on cut-cell meshes • Including pressure momentum balance terms improve the accuracy of wall functions • Adding gradient quantities to forcing data can introduce noise on cut-cell meshes • ODE wall functions can be adapted using 1D MOESS Computational fluid dynamics is essential for designing aircraft, turbines, and other engineering systems. However, generating suitable computational meshes for complex geometries remains the primary bottleneck in analysis workflows, often requiring days of manual effort. Traditional boundary-conforming meshes excel at capturing near-wall physics in viscous flows but demand specialized expertise and extensive preprocessing time. Cartesian cut-cell methods provide automatic mesh generation for complex geometries in minutes, yet they struggle with high Reynolds number viscous flows where boundary layers exhibit rapid velocity changes that require prohibitively fine resolution for isotropic elements. The fundamental challenge is accurately modeling boundary layer physics on automatically generated meshes without sacrificing the computational efficiency that makes such methods attractive. In this work, we show that an ordinary differential equation (ODE) wall function incorporating pressure-momentum balance enables accurate high Reynolds number viscous flow predictions on coarse Cartesian cut-cell meshes. Our approach solves a one-dimensional boundary value problem at each wall boundary face that accounts for the transition from the viscous dominated near-wall region to the inviscid wake region, allowing forcing points to operate effectively at y + > 600 . Unlike traditional wall functions, the ODE is not limited to the logarithmic layer and maintains accuracy in strong pressure gradient environments typical of aerodynamic applications. The ODE can achieve correct skin friction predictions on meshes more than four times coarser than analytical wall functions require. The ODE wall functions are solved with a robust Newton–Krylov implementation that utilizes adaptive mesh refinement. It converges reliably across diverse flow conditions while solving hundreds of degrees of freedom in fewer than ten linear iterations. These results demonstrate that automatic high-fidelity viscous flow analysis is achievable without manual mesh generation expertise.