Fluid flows that are driven by rotor-stator interactions form the basis of many technical applications and are frequently analyzed and tuned for the purpose of optimizing process conditions. Although detailed insights may be obtained by experiments, practical measurements are not only costly and labor-intensive, but also tied to a particular flow configuration, operational scale and spatial or temporal resolution. In this contribution, we therefore focus on a computational method for the calculation of flow fields in rotor-stator configurations. The relative movement of the rotor and the stator is accommodated by decomposing the flow domain into two disjoint subdomains and associating the subdomains with a moving and a resting frame of reference, respectively. Our solution approach is based on a finite volume mesh-tying method and targets transitional and turbulent flows. Adopting the large eddy simulation framework, the resulting residual stresses are closed with the aid of a linear eddy viscosity model. At the subdomains' tied interface, the tractions and diffusive scalar fluxes are treated as Lagrange multipliers and determined in such a way that both the velocity field and the pressure remain continuous across the interface. A particular technical challenge we address is the accommodation of staggered grids. For both the interface-normal velocity and the pressure, the multiplier-carrying interface is offset from the interface on which the kinematic continuity constraint is defined. In order to reduce the computational cost incurred by the interface treatment, the multipliers and continuity constraints are collocated at discrete points along the interface. While the interfacial continuity constraints determine only the diffusive face-wise flow rates, we show that the advective flow rates are conserved exactly across the interface. A current exception is the advective radial momentum flow rate for which an imbalance may result. The mesh-tying method is embedded into a segregated projection-based flow solver and the multipliers are eliminated by augmentation, thus permitting the application of iterative linear system solvers. Besides providing a detailed analysis of the solver's accuracy, convergence and conservation properties in laminar flows, we validate a large eddy simulation by comparison with experimental data for a baffled stirred tank operated in the transitional flow regime.