This paper deals with the turbulence closure problem associated with the Reynolds-averaged Navier–Stokes equations for compressible flows. Due to the known limitations faced by the turbulent closure model based on the Boussinesq hypothesis, additional tensors are needed in order to capture the complete Reynolds stress tensor. Models such as this are in the group of the so-called nonlinear eddy viscosity models. The methodology here presented consists mainly in translating the coefficients obtained via orthogonal projection of the Reynolds stresses that are calculated from a large eddy simulation onto a given basis of tensors. In addition, these coefficients can be seen as turbulent parameters which have analogous interpretation in a class of non-Newtonian fluids. The proposed closure, expressed in terms of the mean strain-rate and its Gordon–Schowalter convected derivative, has a concise and clear physical interpretation. In addition to the turbulent viscosity, parameters associated with the Gordon–Schowalter convected derivative are incorporated to help the interpretation of the nonlinear turbulent model. The methodology is applied to a compressible flow through a narrow channel with a series of holes of cylindrical shape. This configuration is representative of what is generally found in a Hole-Pattern turbomachinery seal. Different number of holes are tested in order to evaluate the influence of neighborhood holes in the analysis. The comparison between the Boussinesq hypothesis and the extended model shows how the latter one outperforms the former. Even though the nonlinear model has shown some limitations, there are still tensorial subspaces of the Reynolds stress tensor obtained from the LES approach that the present nonlinear model could not reach. In addition, the physical coefficients did not present strong similarities for the different configurations tested, what can be a challenge for constructing a complete closure model.