A frequent configuration in computational fluid mechanics combines an explicit time advancing scheme for accuracy purposes and a computational grid with a very small portion of much smaller elements than in the remaining mesh. Two examples of such situations are the travel of a discontinuity followed by a moving mesh, and the large eddy simulation of high Reynolds number flows around bluff bodies where together very thin boundary layers and vortices of much more important size need to be captured. For such configurations, multi-stage explicit time advancing schemes with global time stepping are very accurate, but also very CPU consuming. In order to reduce this problem, the multirate time stepping approach represents an interesting improvement. The objective of such schemes, which allow to use different time steps in the computational domain, is to avoid penalizing the computational cost of the time advancement of unsteady solutions which would become large due to the use of small global time steps imposed by the smallest elements such as those constituting the boundary layers. In the present work, a new multirate scheme based on control volume agglomeration is proposed for the solution of the compressible Navier-Stokes equations equipped with turbulence models of flows around bluff bodies like a flow around a space probe model at Reynolds number 106, a circular cylinder at Reynolds number 8.4 × 106, and two tandem cylinders at Reynolds number 1.66 × 105 and 1.4 × 105.