Integral analysis based diagnostics of turbulence model errors in skin friction

Error diagnostics for turbulence models have traditionally focused on engineering quantities of interest, such as the skin-friction coefficient, $C_f$, most often by comparing the predicted $C_f$ against reference data. In wall-bounded turbulent boundary layers, however, $C_f$ results from several physical mechanisms -- viscous effects, turbulence, pressure gradients, and mean-flow development -- whose relative importance depends on the flow conditions. Modeling errors in these mechanisms vary across turbulence closures, and identifying them offers valuable physical insight for model evaluation and improvement. We propose a diagnostics framework that systematically isolates and quantifies such errors using the angular momentum integral (AMI) formulation. The method is applied to five transport-type Reynolds-averaged Navier-Stokes (RANS) models in two test cases: a canonical zero-pressure-gradient flat-plate boundary layer and flow over a three-dimensional hill. For the flat-plate case, comparison with direct numerical simulation (DNS) data shows that all models reproduce $C_f$ reasonably well, but often through strong error cancellation, particularly between the turbulent torque and mean-flux contributions; individual terms can deviate by more than 20% of $C_f$. For the hill case, where wall-resolved large-eddy simulation (WRLES) is used as the reference, errors are significantly larger. The dominant erroneous contribution differs by model and may exceed several times the local $C_f$, depending on streamwise position. In separated-flow regions, the error cancellation that was observed in the flat-plate case largely disappears for the hill case, and the leading source of error shifts between mechanisms. These results highlight the value of mechanism-resolved diagnostics and provide guidance for targeted turbulence-model improvements.