Implied models approach for turbulence model form physics-based uncertainty quantification

Model form uncertainty arises from physical assumptions made in constructing models either to model physical processes that are not well understood or to reduce the physical complexity. Understanding these uncertainties is important for both quantifying prediction uncertainty and unraveling the source and nature of model errors. Physics-based uncertainty quantification utilizes inherent physical model assumptions to estimate and ascertain the sources of model form uncertainty or error. Compared to data-based approaches, physics-based approaches can be extrapolated beyond available data and go beyond strictly uncertainty estimation. In this work, an implied models approach is developed where the transport equation for the model error is derived by taking the difference between the exact transport equation for a quantity of interest and the transport equation implied by a particular model form. The implied models approach is then specifically applied to the modeling of the Reynolds stresses by the Boussinesq eddy viscosity model. Budgets of the model error transport are analyzed to better understand the sources of error in two-equation Reynolds-averaged Navier-Stokes models focusing on the relative contributions from the Boussinesq hypothesis and the specific form of the eddy viscosity in turbulent channel flow at various friction Reynolds numbers. The results indicate that the errors are largely due to the misalignment of the mean strain rate tensor and the Reynolds stress tensor as well as the high degree of anisotropy near the wall, with errors in the shear component being dominant. An exploration of the k-É and k-ω models reveals that both models benefit from error cancellation. In particular, the improved results of the k-ω model over the k-É model are shown to be the direct result of this error cancellation. An exploration of the effect of friction Reynolds number on the error budgets reveals that the errors saturate with increasing Reynolds number owing to the relative decrease of anisotropy.