## 1 Introduction  
The interaction between impeller and diffuser in centrifugal compressors is characterized by complex flow phenomena, which are still the focus of several research studies. Such an interaction, characterized by a set of mechanisms which causes distortions in the flow pattern, cannot be pinpointed if the respective components are studied independently. The impeller, because of secondary flows, gives rise to an outlet flow with a strongly unsteady character; the flow coming from the impeller would be non-uniform even after an ideal circumferential averaging, due to the above secondary flows which generate strong differences of relative velocity and flow angle along the hub to shroud direction. When the effects of viscous transport phenomena are considerable, the flow at impeller exit is typically characterized with a jet and wake pattern ([PERSON] and [PERSON], 1960) [1]. As long as the mixing process takes place rapidly, it cannot be completed in the vaneless space, so that unsteady effects are more or less significant, depending on the radial gap. Despite the mixing process taking place rapidly, in the vaneless space the unsteady effects of the impeller are not damped before the diffuser blade; these effects are significant and depend on the radial gap between impeller and diffuser vane. On the other hand, the diffuser vanes cause a speed decrease to ascribe to a potential flow effect ([PERSON], 1971) [2], unsteady pressure disturbances, and Mach number at the rotor-stator interface ([PERSON] and [PERSON], 2009) [3].  
Numerical investigations provide good chances to split the different phenomena, especially through the comparison between a steady calculation with a mixing plane at impeller-diffuser interface and a transient rotor-stator calculation ([PERSON] et al., 1993 [4], [PERSON], 1994 [5], [PERSON] and [PERSON], 1999 [6], [PERSON] et al., 2000 [7], [PERSON] et al., 2012 [8], [PERSON] et al., 2017 [9]). Many numerical simulations have been developed and applied to study the flow unsteadiness in vaned diffusers ([PERSON], 2002) [10], in vaneless diffusers ([PERSON] and [PERSON], 2013) [11], and to compare the two types ([PERSON], 2005) [12]. The effect of the radial gap between the impeller and vaned diffuser has been investigated also ([PERSON] et al., 2017) [13]. In the last years, particular attention has been focused on the near surge conditions to better understand the stall mechanism ([PERSON] et al., 2014 [14], [PERSON] and [PERSON], 2017 [15]) and to devise some criteria to predict the surge limit in centrifugal compressors ([PERSON] et al., 2017 [16], [PERSON] and [PERSON], 2017 [17]). The interaction between rotor and stator in a centrifugal compressor has also been extensively investigated through experimental analysis ([PERSON] and [PERSON], 1981 [18], [PERSON], 1982 [19]).  
A previous study on this centrifugal compressor was presented in 2002 by [PERSON] [20], who conducted an experimental investigation to understand the flow interaction between the impeller and the vaned diffuser. In this paper, a CFD analysis of the compressor object of the [PERSON]'s studies is shown; the detailed geometry and experimental data are available for the test case called Radiver. The [PERSON]'s studies show that a better performance is obtained when the radial gap is small; in this condition, the rotor wake ingested by the diffuser channel is capable of energizing the diffuser boundary layer, preventing flow separation. For larger radial gaps the mixing processes cause a more radial wake flow in the vaneless space, reducing the above positive effect. In the following, the paper begins with an overview of the reference test case, then the CFD modelling is explained with mesh generation, pre-processing, and steady simulations. The second half of the paper is dedicated to the description of the numerical results, the comparison with experimental data and the discussion of the flow patterns, with special reference to the unsteady effects.