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. Examples of such situations are the traveling 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, explicit time advancing schemes with global time stepping are very costly. In order to overcome 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 can 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 possibly equipped with turbulence models. The method relies on a prediction step where large time steps are performed with an evaluation of the fluxes on macro-cells for the smaller elements for stability purpose, and on a correction step in which small time steps are employed only for the smaller elements. The efficiency of the proposed method is evaluated on several benchmarks flows: the problem of a moving contact discontinuity (inviscid flow), the computation with hybrid turbulence model of flows around bluff bodies like a tandem cylinders at Reynolds number $1.66 × 10 5$ , a circular cylinder at Reynolds number $8.4 × 10 6$ , and a flow around a space probe model at Reynolds number $1 × 10 6$ .