Turbopumps operating at reduced flow rates experience significant separation and backflow phenomena. Although Reynolds-Averaged Navier–Stokes (RANS) approaches proved to be usually able to capture the main flow features at design working conditions, previous numerical studies in the literature verified that eddy-resolving techniques are required in order to simulate the strong secondary flows generated at reduced loads. Here, highly resolved large-eddy simulations (LES) of a radial pump with a vaned diffuser are reported. The results are compared to particle image velocimetry (PIV) experiments in the literature. The main focus of the present work is to investigate the separation and backflow phenomena occurring at reduced flow rates. Our results indicate that the effect of these phenomena extends up to the impeller inflow: they involve the outer radii of the impeller vanes, influencing significantly the turbulent statistics of the flow. Also in the diffuser vanes, a strong spanwise evolution of the flow has been observed at the reduced load, with reverse flow, located mainly on the shroud side and on the suction side (SS) of the stationary channels, especially near the leading edge of the diffuser blades.

## Introduction

Stall phenomena and reverse flow at reduced flow rates affect substantially the flow physics in turbopumps, which becomes significantly more complex than that observed at nominal conditions, for which the flow is smoothly guided between rotating and stationary parts. These phenomena have a negative impact on the performance and the structural integrity of such machines, increasing the mechanical stress acting on their elements, especially because of the periodic nature of the flow instabilities that are generated, tied to the interaction between rotating and stationary elements. Being able to predict accurately the behavior of a specific design solution at off-design is critical, since usually turbopumps need to operate at conditions far from the nominal ones.

The highly unsteady nature of the flow produced in radial pumps working at off-design was carefully analyzed by Sinha et al. [1], who adopted PIV and pressure fluctuations measurements to investigate the onset of dynamic stall at reduced flow rates in a vaned centrifugal pump. In the diffuser vanes close to the tongue of the volute, the flow alternated between outward and backflow. Decreasing the flow rate, the backflow increased and the stall extended from one to two diffuser vanes. The strong rotating stall was attributed to the large radial gap between impeller and diffuser blades (20% of the impeller radius). PIV and pressure measurements at off-design conditions were also carried out by Miyabe et al. [2] on a mixed-flow pump, equipped with a vaned diffuser: the rotating stall occurring at low flow rates was associated with reverse flow from the outlet of the vaned diffuser to the outlet of the impeller. The backflow was correlated to large vortices generated at the inlet of the diffuser vanes. Thus, the operation at reduced flow rates is characterized by an increased interaction between moving and stationary parts, as pointed out also by Akhras et al. [3], who performed laser Doppler velocimetry measurements on a radial pump with a vaned diffuser at various operating conditions. They verified that at low flow rates, there was a stronger dependence of the flow field inside the impeller vanes on the relative location of the stator blades. Also, the PIV measurements in the impeller of a radial pump with vaneless diffuser by Wuibaut et al. [4] showed instabilities from the diffuser, affecting the outer region of the impeller and causing increased level of unsteadiness, compared to the design working conditions.

Although the experiments have been a valuable tool to study the flow through turbopumps at off-design conditions, they require sophisticated and expensive instrumentations, due to the complexity of the system. Measurements, especially near the wall, are not trivial, due to the presence of elements in relative motion. Computational tools can address some of these shortcomings and complement experimental studies. Most of the computational works in this class of flows utilizes the RANS equations, coupled with a turbulence model, which represents all turbulent scales (see, for example, Refs. [5] and [6]). In particular, in Ref. [6], the study was focused on the noise produced due to backflow, which at flow rates lower than 70% of the nominal one reached the impeller inlet. The reverse flow and the intensity of the noise generated by the machine increased as the flow rate decreased. Flow visualizations were utilized to validate the numerical results, which were in broad qualitative agreement with the experiments. The production of stall cells and an increased dependence of the flow field on the relative position between impeller and diffuser at reduced flow rates were also evident in the RANS simulations by Stel et al. [7] and Feng et al. [8]. Several additional RANS studies dealing with centrifugal pumps working at reduced flow rates are available in the literature (see, for instance, Refs. [9–14]). However, due to the highly unsteady nature of the flow in those conditions, RANS methodologies do not perform as well as at nominal conditions, as pointed out by Barrio et al. [15,16], who attributed a decreased accuracy of their RANS results at very low flow rates, compared to the nominal ones, to strong separation phenomena, whose treatment is typically more problematic. Also, Gao et al. [17] verified in their coupled experimental/numerical study of a centrifugal pump a decreased agreement between computations and experiments away from design conditions for their steady RANS computations, while unsteady RANS performed better. Eddy-resolving techniques, such as the LES approach, are more suitable, as discussed by Kato et al. [18,19], who utilized LES to investigate turbopumps flows at off-design conditions. The superiority of LES in the simulation of off-design working conditions was proven by Byskov et al. [20], who performed RANS and LES of the flow in a centrifugal pump impeller at both nominal and quarter-load flow conditions. The comparison with the PIV measurements by Pedersen et al. [21] demonstrated the improved accuracy of the LES results, especially in capturing the nonrotating stall phenomena and the asymmetry of the flow between the rotor channels at the reduced flow rate. LES, together with an overset grids framework, was also utilized by Yamanishi et al. [22] to analyze the backflow vortices at the inlet of an axial inducer, providing new insight into the properties of such structures and a good agreement with experimental measurements. Feng et al. [23] utilized the detached-eddy simulation methodology to investigate the reverse flow phenomena in the impeller of a centrifugal pump with a vaned diffuser, with a lower computational cost, compared to LES. For similar reasons, the scale-adaptive simulation approach was adopted by Lucius and Brenner [24,25], who compared their prediction of rotating stall in a centrifugal pump at part load to experimental measurements. Overall, highly resolved LES are not frequently adopted in studies of turbopumps flows, due to their relatively high computational cost. In the case of mixed-flow and centrifugal pumps with vaned diffusers, the complexity of the flow field and separation/backflow phenomena are enhanced by the highly three-dimensional geometry of the impeller vanes and the interaction between rotating and stationary blades, rendering high-fidelity computations even more challenging.

In this work, we report high-resolution LES of the flow in a radial pump with a vaned diffuser. To overcome the challenges introduced by the geometric complexity and the simultaneous presence of moving and stationary parts within the computational domain, we utilize an immersed boundary (IB) method. In such case, the geometry is immersed in a fixed cylindrical grid. This eliminates the need to use sliding mesh techniques, overlapping grids or grid deformation/regeneration, utilized in most of computations reported above, enhancing accuracy and stability. In particular, we use the same configuration as in the PIV experiments by Boccazzi et al. [26]. Detailed comparisons to the experiments for design and off-design conditions are reported in Refs. [27] and [28], respectively. In this study, we will focus on the analysis of the reverse flow occurring in both the impeller and diffuser vanes at a low flow rate, corresponding to 40% of the nominal one, the role of the impeller/diffuser interaction in the generation of stall cells, and the effect of separation phenomena on turbulence.

In the following, the numerical method will be discussed in Sec. 2, together with details of the machine geometry and its working conditions. In Sec. 3, separation and reverse flow will be analyzed, together with their influence on the turbulent kinetic energy distribution. Finally, Sec. 4 will provide the conclusions of the present work.

## Problem Formulation and Methods

*is the flow velocity,*

**u**_{i}*p*is a modified pressure, including the isotropic part of the subgrid scale (SGS) stress tensor,

*x*is the spatial coordinate,

_{i}*is a forcing term, used to impose boundary conditions,*

**f**_{i}*t*is the time variable, and Re =

*UL*/

*ν*is the Reynolds number, based on a reference velocity,

*U*, a reference length,

*L*, and the kinematic viscosity of the fluid,

*ν*. The symbol ∼ indicates spatially filtered quantities. The filtering operation gives rise to an additional term, the SGS stress tensor, $\tau ij=uiuj\u0303\u2212u\u0303iu\u0303j$, representing the scales smaller than the adopted computational grid, which are parameterized with an eddy viscosity model

where *δ _{ij}* is the Kronecker delta and $S\u0303ij$ is the resolved strain rate tensor. In the computations reported in this paper, the eddy viscosity is estimated using the wall-adapting local eddy-viscosity model by Nicoud and Ducros [29], which is based on the square of the velocity gradient tensor. Details on the implementation and validation of this SGS model for the present case are provided in our earlier work [28]. The geometry of the turbopump has been immersed in a fixed cylindrical grid, composed of 1602 × 718 × 402 nodes along the azimuthal, radial, and axial directions, respectively. With such grid, the average normal distance of the first Eulerian node away from the impeller and diffuser blades was equal to about five wall units. Note also that, due to the nonconforming nature of the approach, resolution along each direction was roughly uniform, except for the azimuthal stretching associated to the cylindrical topology of the grid. In this regard, cells were designed to be isotropic at the impeller/diffuser interface. An IB method is utilized to impose boundary conditions on both moving and stationary parts, avoiding the costly task of grid deformation/regeneration, which typically has an adverse impact on the accuracy and efficiency of the solver. Indeed, the requirement for the grid to conform to the body is relaxed and the no-slip boundary condition is defined at the interface nodes by a local linear interpolation between the surface of the “immersed” body and the fluid region of the computational domain. Note that the interface nodes are those outside of the body, but having at least one adjacent node inside the body along any of the grid directions. All other outside nodes of the Eulerian grid are tagged as fluid. Details on the method, as well as applications, including rotating machinery cases, can be found in Refs. [30–37], while the implementation in the present configuration is outlined in Refs. [27] and [28].

The governing equations are solved by a projection method [38]. Third-order Runge–Kutta has been adopted for the discretization in time, except for the convective and viscous terms of azimuthal derivatives, which have been treated implicitly using a Crank–Nicolson scheme, to relax the stability constraints on the time-step at small radial coordinates. However, the resolution in time was very high, requiring an average of 2255 steps per impeller revolution, equivalent to a rotation of 0.16 deg per time-step. The discretization in space has been performed by second-order central differences on a staggered grid. Computations were conducted using 400 cores on a CRAY XT5 supercomputer, with a cost of approximately 16 hrs/rotation.

The geometry of the pump considered in this study is shown in Fig. 1, where different capital letters highlight different components. The impeller is composed of six backswept blades (B). Its inflow is axial and its outflow is along the radial/azimuthal directions. The geometry of the impeller is closed, which means that the shroud (C) is part of the impeller. Therefore, there is no gap between the impeller blades and the shroud, which decreases the computational challenge in that region. The diffuser blades (F) are seven. Their geometry is two-dimensional and their design is based on the circular arc criterion [39]. The diffuser walls (E) have no expansion: their span is constant along the radius. The cross section of the volute (G) is rectangular; therefore, it is not designed to have a smooth expansion of the flow at the outlet of the diffuser vanes. Note that the gap between the impeller shroud/hub and pump casing was not simulated, which would be computationally very expensive. At the same time, the leakage flows were not discussed by Boccazzi et al. [26], but we expect that sealing limits such flows. Therefore, we modeled a simplified zero-clearance configuration, as shown in detail in Fig. 2, where a meridional section of the pump is plotted, together with a snapshot of the vorticity magnitude field. There the main flow direction is represented by arrows, through the impeller (** i**) and the diffuser (

**) vanes, up to the volute (**

*d***). Observe that for clarity, the impeller and diffuser blades are not shown, but their locations are distinguishable in the areas of 0 vorticity within the rotor and stator channels. Note also that the zones indicated by**

*v***were tagged as solid regions in the present computations, being obviously not part of the fluid domain.**

*s*The inflow into the computational domain has been enforced using a Dirichlet condition on a circular area corresponding to the cross section of the suction pipe (A in Fig. 1). The outflow was defined by means of a convective condition, whose convective velocity was estimated based on the flow rate and the outflow surface, which is the intersection of the volute channel with the cylindrical radial boundary of the computational domain.

In Table 1, a summary of the geometrical parameters and operating conditions is given. To better characterize the machine, the values at the best efficiency point (BEP) of the specific speed $\omega sopt$, flow coefficient $\varphi opt$, and head coefficient *ψ*^{opt} are given. The definitions of specific speed, flow coefficient, and head coefficient are as follows:

Rotational speed Ω | 55.4 rad/s |

Impeller diameter at the inlet D_{1} | 154 mm |

Impeller diameter at the outlet D_{2} | 224 mm |

Impeller span at the outlet b_{2} | 41 mm |

Impeller outflow angle β_{2} | 29 deg |

Number of impeller blades n_{r} | 6 |

Diffuser diameter at the inlet D_{3} | 233 mm |

Diffuser diameter at the outlet D_{4} | 361 mm |

Diffuser span (constant) b_{3} | 44 mm |

Diffuser inflow angle β_{3} | 18 deg |

Number of diffuser blades n_{d} | 7 |

Volute span (constant) b_{4} | 138 mm |

Design flow rate Q_{d} | 25 l/s |

Design Reynolds number | |

$Red=(QdD2)/(2A1\nu )$ | 1.5 × 10^{5} |

Off-design flow rate Q_{off} | 10 l/s |

Off-design Reynolds number | |

$Reoff=(QoffD2)/(2A1\nu )$ | 6.0 × 10^{4} |

BEP specific speed $\omega sopt$ | 1.08 |

BEP flow coefficient $\varphi opt$ | 0.291 |

BEP head coefficient ψ^{opt} | 0.443 |

Rotational speed Ω | 55.4 rad/s |

Impeller diameter at the inlet D_{1} | 154 mm |

Impeller diameter at the outlet D_{2} | 224 mm |

Impeller span at the outlet b_{2} | 41 mm |

Impeller outflow angle β_{2} | 29 deg |

Number of impeller blades n_{r} | 6 |

Diffuser diameter at the inlet D_{3} | 233 mm |

Diffuser diameter at the outlet D_{4} | 361 mm |

Diffuser span (constant) b_{3} | 44 mm |

Diffuser inflow angle β_{3} | 18 deg |

Number of diffuser blades n_{d} | 7 |

Volute span (constant) b_{4} | 138 mm |

Design flow rate Q_{d} | 25 l/s |

Design Reynolds number | |

$Red=(QdD2)/(2A1\nu )$ | 1.5 × 10^{5} |

Off-design flow rate Q_{off} | 10 l/s |

Off-design Reynolds number | |

$Reoff=(QoffD2)/(2A1\nu )$ | 6.0 × 10^{4} |

BEP specific speed $\omega sopt$ | 1.08 |

BEP flow coefficient $\varphi opt$ | 0.291 |

BEP head coefficient ψ^{opt} | 0.443 |

The definition of the Reynolds number is based on the flow velocity at the impeller inlet and the impeller outer radius.

where Ω is the impeller rotational speed, *Q* is the volumetric flow rate, *g* is the acceleration of gravity, *H* is the head of the pump, and *A*_{1} is the cross-sectional area at the impeller inflow, while *u*_{1} and *u*_{2} are the tangential speeds of the impeller blades, respectively, at the outer radius of the inlet and at the outlet (see also Table 1). Note that the flow rate at the computed off-design working condition is equivalent to 40% of the nominal one. The rotational speed for the two simulated cases is the same, equal to 55.4 (rad/s). Some additional details can be found in Refs. [26] and [40].

In Fig. 3, the locations of the pressure side (PS) and SS of both impeller and diffuser vanes are shown, and the diffuser blades are numbered, in order to simplify the following discussion of the results: for both blades rows, the SS corresponds to the concave one and the PS to the convex one. As shown in Fig. 1, the impeller blades are characterized by a complex three-dimensional geometry; therefore in Fig. 3, their section on the $r\u2212\u03d1$ plane of representation is plotted, for visibility purposes. Also, the experimental window of the PIV measurements by Boccazzi et al. [26] is reported.

## Results

*f*is any physical quantity, dependent on the three coordinates in space, ϑ,

*r*and

*z*, and time

*t*. The variable $t0(\phi )$ represents the instant in time when the sampling process of the phase-averaged statistics is started for the particular rotor/stator configuration described by the angle $\phi $, while

*n*is the number of rotor blades.

_{r}*N*is the number of instantaneous fields utilized to evaluate the statistics, given by the number of revolutions multiplied by the number of impeller blades. For each flow condition considered here, after developing the flow, the sampling process was carried out during five impeller revolutions. Thus in the present case,

*N*was equal to 30. The passage averages are evaluated from the phase-averaged fields

where *m* stands for the number of phase-averaged fields, that is, the number of considered impeller/stator configurations. For all passage averages presented below, the value of *m* is equal to 12.

where $u\u03d1\u2032,\u2009ur\u2032$, and $uz\u2032$ are the fluctuations of the instantaneous values of tangential, radial, and axial velocity components, relative to the phase-averaged ones for the particular impeller/diffuser configuration defined by the angle $\phi $. For dependence on angle $\phi $, space coordinates, and time, see Eq. (4). The passage-averaged turbulent kinetic energy is then computed using Eq. (5).

The phase-averaged velocity statistics are in good agreement with the PIV experiments by Boccazzi et al. [26]. Detailed comparisons for the design and off-design conditions have been presented in Refs. [27] and [28]. In terms of global performance of the turbopump, we computed at the particular off-design condition a head coefficient *ψ* = 0.515 and an efficiency *η* = 49.9%, while Boccazzi et al. [26] found *ψ* = 0.502 and *η* = 48.5%, with the efficiency defined by $\eta =(g\rho QH)/(T\Omega )$, where *ρ* is the fluid density and *T* is the torque. For completeness in Fig. 4, we present passage-averaged statistics for the magnitude of the velocity in the $r\u2212\u03d1$ plane, $ur\u03d1=(ur2+u\u03d12)0.5$, at 77% of the diffuser span from the hub side for both design (Fig. 4(a)) and off-design (Fig. 4(b)) working conditions. Here, a spanwise location of the diffuser closer to the shroud side is considered, where the highest adverse pressure gradients are experienced by the flow, especially at off-design conditions. In Figs. 4(c) and 4(d), comparisons to the PIV experiments by Boccazzi et al. [26] are included for the streamwise velocity along the line *f* shown in Fig. 3. At design condition (Figs. 4(a) and 4(c)), the velocity field is rather uniform along the cross-stream direction, in contrast with the off-design case (Figs. 4(b) and 4(d)). In the latter condition, the flow on the convex side of the diffuser blades is close to separation, due to a stronger streamwise pressure gradient. The LES results match fairly well the experiments (see Figs. 4(c) and 4(d)), with only a slight overestimate of the velocity near the PS at off-design conditions.

### Reverse Flow Inside the Impeller.

The pressure measurements by Boccazzi et al. [26] at off-design conditions highlight the presence of backflow inside the impeller at the shroud, which has a profound effect on the flow upstream, extending up to the impeller inlet. The present computations capture these phenomena. In order to provide a measure of the distribution of the reverse flow in the rotating channels, the ensemble average of the velocity −*u _{z}* is represented in Fig. 5(b). Observe that the minus sign was adopted to have positive values of

*w*for a flow going from the inlet toward the outlet. The ensemble averages were evaluated as average of the passage averages along the azimuthal direction, giving the mean radial distribution at four cross sections inside the impeller (see Fig. 5(a) for locations)

The horizontal axis in Fig. 5(b) is normalized based on the local radius of the impeller, *R _{i}*, which is increasing along the streamwise direction, while

*w*is scaled using the velocity

*u*

_{2}. It is clear that at design conditions, the flow keeps attached. Velocity declines at the outer radii, due to the strong adverse pressure gradient experienced near the shroud, where the curvature of the streamlines is larger. This effect becomes more obvious moving downstream, but the flow does not separate. The behavior at off-design conditions is strongly modified from that at the nominal flow rate and consistent with the findings in Ref. [26]. At the upstream locations

**and**

*a***, the flow is separated at the shroud and significant backflow phenomena are produced. It is likely that the massive separation at the impeller shroud seen for this case of a small radial pump is associated to the simultaneous effects of high head coefficient and relatively low Reynolds number, making the boundary layer especially prone to separation, under the action of a strong unfavorable pressure gradient. Moving downstream, the reverse flow shifts to inner radii and reattachment occurs at the shroud, as shown by the positive values of**

*b**w*at the outer radii on the sections

**and**

*c***. As a consequence of the discussed reverse flow near the shroud, regions of positive**

*d**w*populate mainly small radii, although this trend becomes weaker moving downstream.

The distribution of the backflow cells also affects the turbulent kinetic energy, as shown in Fig. 5(c). At design conditions, the values of turbulent kinetic energy are relatively small. They are significantly higher at the outer radii, due to development of a thick boundary layer on the shroud. In agreement with the literature on turbulent boundary layers under the effect of an adverse pressure gradient [41,42], the peak of turbulent kinetic energy moves away from the wall and becomes less distinguishable, evolving toward a broader region of high turbulence. Again, at off-design conditions, the distributions are significantly altered: the values of turbulent kinetic energy are much higher than at the design flow rate; their peaks are located about at the radii where the flow changes its streamwise direction and their values decrease moving downstream, which is consistent with a reduction of the negative *w* velocities.

The flow physics summarized by the ensemble-averaged plots in Fig. 5 is well represented in Fig. 6, where the phase averages of the turbulent kinetic energy for a particular impeller configuration are shown at both design and off-design conditions on the section ** d** in Fig. 5(a). At the nominal flow rate (Fig. 6(a)), the development of a thick turbulent boundary layer at the shroud is obvious. Higher values are generated on the SS of the impeller vanes, due to the destabilizing effect of the rotation of the channels [43]. Lower levels of turbulence are visible on the PS of the impeller blades at the outer radii. At the reduced flow rate (Fig. 6(b)), the flow physics is drastically different. As observed in Fig. 5(b), separation and backflow affect a significant radial extent of the impeller: this is shown in Fig. 6(b) by the early detachment of the shear layer from the SS of the rotor blades, moving toward the center of the channels, where the values of turbulent kinetic energy are even higher than those in the backflow region at the outer radii of the impeller. The phase-averaged fields in Fig. 6 highlight the profound modifications of the flow physics at off-design conditions. Indeed at design conditions, the flow keeps attached on the SS of the impeller blades, while at the reduced flow rate, stall cells are generated, especially on the shroud side. This is shown in Fig. 7, where local streamlines through markers on the surface of the blades are plotted for a detail of the impeller. For visibility, the upstream casing of the pump is not represented in the figure. Obviously, at the nominal flow rate, the flow keeps following the geometry of the impeller blades, and it is in general well guided through the rotating channels. In contrast, at off-design, large stall cells are produced over the SS of the impeller blades, especially on the shroud (upstream) side, where larger adverse pressure gradients are experienced by the flow. The locations of those cells correspond to the positions where the shear layer seen in Fig. 6(b) separates from the surface of the blades. Further details on turbulent kinetic energy fields at additional locations through the impeller are discussed by Posa et al. [28].

### Backflow Between Impeller and Diffuser.

In Fig. 8, the azimuthal distribution of the angle *β*, defined as the angle of the flow relative to the azimuthal direction, has been represented at 77% of the span (closer to the shroud) and at 23% of the span (closer to the hub) at the radial location corresponding to the trailing edge of the impeller blades. The horizontal dashed line represents the inlet angle of the diffuser vanes. The vertical lines show the positions of the trailing edges of the impeller blades (dashed lines) and leading edges of the diffuser blades (dotted–dashed lines). The distributions in Fig. 8 refer to the phase average relative to a particular impeller/diffuser configuration. However, it is worth noting that such configuration was found representative of several cases, due to the asymmetry between impeller and diffuser geometries, having different numbers of vanes.

At the spanwise location closer to the shroud in Fig. 8(a) for both design and off-design conditions, the fluid dynamic angle *β* is smaller than the geometric angle of the diffuser blades at their leading edge, which means that the incidence to the diffuser blades is mainly positive, especially in the azimuthal areas associated with the SS of the impeller channels. However, the incidence is much larger at the reduced flow rate. In particular, at off-design conditions, the fluid dynamic angle *β* becomes even negative at the locations of the leading edge of the diffuser blades, due to the incorrect incidence from the impeller. Negative values are produced also on the SS of the impeller blades, which is on the left of the vertical dashed lines representing their trailing edges, although there this behavior is much less prominent. Note that at the design condition, the flow angle *β* never becomes negative.

The passage of the flow from the impeller to the diffuser is much smoother at 23% of the span, as shown in Fig. 8(b). At design conditions, the incidence of the flow on the diffuser blades can be considered practically correct and is mainly affected by the blockage produced by their leading edge. At off-design conditions, the incidence is again positive almost everywhere, especially for the flow coming from the SS of the rotating channels, since the reduced flow rate implies a decreased radial velocity component. However, the values of incidence are significantly lower than in Fig. 8(a) and the reverse flow phenomena are almost missing, limited to small regions close to the leading edge of the stationary blades, when they face the SS of the impeller channels.

The considerations above are reinforced by Fig. 9, referring to the phase-averaged radial velocity component at the same radius as Fig. 8 and confirming the presence of regions of backflow at the impeller/diffuser interface. At 77% of the span in Fig. 9(a), the reverse flow, not occurring at the nominal flow rate, is obvious at off-design, especially at the leading edge of the diffuser blades and in some areas also on the SS of the impeller blades, depending on the relative position between moving and stationary vanes. At 23% of the span in Fig. 9(b), the backflow phenomena at off-design are limited to much narrower regions, associated again with the leading edge of the diffuser blades. However, also at this spanwise location, the azimuthal gradients are substantially smaller at the nominal flow rate.

The impeller/diffuser interaction has a significant impact on the evolution of the turbulent kinetic energy at their interface, shown in Fig. 10 at the same locations as in Fig. 8. Note that two different vertical scales have been adopted for the design (right) and off-design conditions (left), due to the striking turbulence increase at the reduced flow rate. In Fig. 10(a), referring to the spanwise position closer to the shroud, the design condition is characterized by the presence of peaks of turbulent kinetic energy on the SS of the trailing edge of the impeller blades. No significant increase is visible near the leading edge of the diffuser blades. Therefore, turbulence comes mainly from the SS of the impeller vanes, due to the destabilizing effect of the rotation of the channel, as discussed by Kristoffersen and Andersson [43], and the interaction between impeller and diffuser is not source of strong turbulent dissipative phenomena. In Fig. 10(a), at off-design conditions, the highest values are associated with the leading edge of the diffuser blades, on their SS, because of the high positive incidence on those blades (see Fig. 8(a)). Such incidence causes separation and backflow phenomena, as shown at the same locations in Fig. 9(a). Such reverse flow is also highly unsteady, being dependent on the relative position between trailing edge of impeller blades and leading edge of diffuser blades. The smaller the azimuthal clearance between them, the higher the turbulence peak. Lower maxima are visible on the SS of the trailing edge of the impeller blades. Therefore, from the comparison between Figs. 9(a) and 10(a), it is quite obvious that the highest values of turbulent kinetic energy occur where backflow phenomena are produced, especially at the azimuthal locations where the trailing edges of the impeller blades are approaching the leading edges of the diffuser blades. Actually, a further confirmation of the stronger impeller/diffuser interaction at reduced flow rate is provided by the increased dependence of the turbulence distribution on the relative position between rotor and stator vanes. Overall, the above conclusions are consistent with those relative to the diffuser midspan reported in Ref. [28].

In Fig. 10(b), the side closer to the hub is considered. The distribution at design condition is not substantially modified, compared to Fig. 10(a), both qualitatively and quantitatively, although lower values can be noticed at the impeller outlet at the midangles ϑ between the PS and the SS. This behavior is due to the lower adverse pressure gradient experienced by the flow closer to the hub: the boundary layer on the SS of the impeller blades undergoes a milder growth, compared to the locations closer to the shroud. At off-design condition, the distribution of turbulent kinetic energy shows more striking variations along the span. In agreement with the strong reduction of reverse flow, closer to the hub turbulence is substantially decreased, as well as its dependence on the relative position between impeller and diffuser blades. The evident peaks associated with the leading edges of the diffuser blades in Fig. 10(a) are practically missing in Fig. 10(b), where the azimuthal distribution is much more uniform. As proven in Fig. 8(b), the incidence on the diffuser blades is decreased toward the hub and the reverse flow is almost missing, which implies a smoother interaction between rotating and stationary parts.

### Stall Phenomena Inside the Diffuser.

The presence of stall and backflow at low flow rates in the stator vanes can be well represented via the distribution of the skin-friction coefficient on the suction and pressure sides of the diffuser blades. Its trends and in particular its sign can be utilized to discuss the areas of massive separation at reduced flow rate and the differences in terms of stall between design and off-design conditions.

where *τ _{w}* is the wall stress. The distribution of the skin friction in Fig. 11 refers to a global average, both in time and space. The average in space has been estimated among the seven vanes of the diffuser and along the span. The chord fraction

*x*on the horizontal axis was defined as described in Fig. 11(a). For instance, for a probe on the PS $NPS$ its projection $PPS$ along the chord is found. Then, the chord fraction

_{c}*x*is computed as ratio between the length of the segment $LPPS$ and the chord length

_{c}*c*. The same approach applies to any probe on the SS, such as $NSS$ in the same figure. In Figs. 11(b) and 11(c), the cases at the nominal and reduced flow rates are shown, respectively. Since the normalization of the wall stress in Fig. 11 is based on the tangential speed at the trailing edge of the blades, which is the same at design and off-design conditions, it is not surprising to verify lower values at the reduced flow rate, which implies reduced velocities through the stator vanes. In both cases on the PS (solid lines), a monotonic decrease of the wall stress is observed, with the production of a plateau at the lower flow rate in Fig. 11(c).

The most significant differences between the two operating conditions occur on the SS, corresponding to the dashed lines in Figs. 11(b) and 11(c). In both cases, an evident minimum is produced near the leading edge. However, while at the nominal flow rate the skin-friction keeps positive, at the off-design condition, the separation of the flow is shown by its negative values. At both flow rates, this minimum of *c _{f}* is due to the positive incidence on the diffuser blades (see Fig. 8); however, the deviation between the geometric and the fluid dynamic angles is much larger at the lower flow rate, which justifies the separation occurring in that case and missing at design conditions.

Moving downstream through the diffuser vanes, the value of *c _{f}* on the SS rises both in Figs. 11(b) and 11(c), becoming significantly larger than the one on the PS: the flow rate is displaced toward the SS, especially at the off-design condition. Indeed, the blockage generated by the large thickness of the boundary layer on the PS is such that in Fig. 11(c), the skin friction on the SS increases downstream of 60% of the chord length, although the cross section of the diffuser vanes is increasing along the streamwise direction.

where *p _{∞}* is the pressure at the inflow of the computational domain. Figure 12 shows the ensemble-averaged distribution of

*c*. At design condition (Fig. 12(a)), the pressure growth involves the whole vane, both on the PS and SS, although the pressure gradient decreases moving downstream. Note that, in agreement with the presence of a minimum for the skin-friction coefficient near the leading edge (see Fig. 11(b)), on the SS the pressure gradient is initially at its maximum and relaxes significantly between 10% and 30% of the chord length, in the region where the flow on the SS experiences an acceleration. Figure 12(b) shows that at off-design condition, the pressure gradient on the SS is higher than at design condition, which justifies the separation of the flow immediately downstream of the leading edge of the diffuser blades. Also in this case, the pressure gradient fades out substantially moving along the streamwise direction. On the PS, the pressure recovery is much weaker and stops practically at about 30% of the chord length, which is an additional clue of the substantial boundary layer growth in this area.

_{p}Figure 13 shows the ensemble-averaged contours of *c _{p}*, where the passage-averaged statistics were further averaged over the diffuser span. The impeller region was blanked out, since these fields do not refer to any specific impeller configuration. At the design condition, the pressure rise is azimuthally rather uniform. In Fig. 13(a), the differences among diffuser vanes are marginal and unaffected by the asymmetry of the volute geometry. It is evident that the pressure recovery stops quickly on the PS of the diffuser vanes and is practically limited to the upstream region. This is even more obvious at the off-design condition in Fig. 13(b). Furthermore, pressure recovery is faster in the channels away from the volute tongue, with more distinguishable deviations among stator vanes, compared to the design case.

Although Fig. 11 provides a global overview relative to the evolution of the flow in the diffuser channels, the complexity of the flow physics in the present machine, especially at the reduced load, requires a detailed study of the flow features at different azimuthal and spanwise locations, where different behaviors have been verified. Therefore, Fig. 14 is utilized to discuss the dependence of the flow through the diffuser vanes on the spanwise location, with a focus on the backflow phenomena. Figure 14 refers to the passage averages on the SS of the diffuser blade 4, which is that located on the side opposite to the tongue of the volute. Note that, although the authors analyzed the distributions of the skin-friction coefficient at several locations, both on the PS and SS of the diffuser blades, for clarity in Fig. 14 only three spanwise positions on the SS of the diffuser blade 4 have been considered, where the most interesting phenomena were observed.

At design conditions (Fig. 14(a)), the distributions of the skin-friction coefficient display a dependence on the spanwise location, but the flow keeps attached over the whole span of the channel. This is not the case at the off-design condition. Actually in Fig. 14(b), the separation at the leading edge of the blade occurs at all spanwise positions, but it is stronger on the shroud side, affecting significantly also the diffuser midspan, and the evolution of *c _{f}* downstream highlights the strong displacement of the flow rate toward the hub side, confirmed by the PIV experiments by Boccazzi et al. [26]. It was verified that, moving even closer to the shroud side of the diffuser channel, the reverse flow undergoes a further increase, affecting the SS behind the midchord locations. Figure 14(b) shows again that over the entire diffuser span, an acceleration is generated on the SS near the trailing edge, due to the increased thickness of the boundary layer on the PS of the channel, but this effect is more obvious on the shroud side, at 77% of the span.

Figure 14 referred to the diffuser blade located away from the tongue of the volute. However, a strong dependence of the flow physics was verified over the azimuth of the diffuser at off-design conditions: the asymmetry of the volute affects substantially the evolution of the flow on the suction and pressure sides of the diffuser blades. To further elaborate on this, Fig. 15 shows the evolutions of the passage-averaged skin-friction coefficient at the midspan location on the blades 4, 6, and 0. It is clear that at design conditions, the evolution of the skin-friction coefficient on the SS is practically independent of the position relative to the volute. On the PS, the dependence on the azimuthal location is increased, but still quite marginal. The flow is slightly decelerated in the vanes closer to the tongue of the volute. Note that the evolution on the blade 4 is representative of that on the other blades not shown in Fig. 15(a). In general at the nominal flow rate, the influence of the asymmetry of the volute geometry on the flow through the stator channels is weak. No stall phenomena were observed moving along the azimuth of the diffuser.

At off-design condition (Fig. 15(b)), the dependence of the distributions on the azimuth is highly increased, both on PS and SS. On the SS, the backflow near the leading edge of the blade 4 is not observed for the blades 6 and 0. The values of *c _{f}* are also higher on the PS of the blades closer to the tongue of the volute. Actually, this behavior was verified associated with a stronger spanwise unbalance of the flow rate in the diffuser vanes near the volute outlet. Although not reported here, it was found that the backflow on the shroud side is increased there, compared with that on the blade 4, affecting even the upstream region of the PS. As a consequence, for continuity, moving toward the midspan locations higher flow rates are delivered, due to the stronger blockage generated near the shroud side of the diffuser vanes. Note finally that, although for clarity only three blades were considered in Fig. 15(b), at off-design condition the distributions on the PS and SS of the blade 4 cannot be anymore assumed representative of those on the other blades away from the volute tongue.

The discussed dependence of the flow physics on the azimuthal coordinate is described more qualitatively in Fig. 16, with the advantage of a more global and intuitive overview of the flow, as well as additional details. There the passage-averaged fields of velocity magnitude in the $r\u2212\u03d1$ plane, $ur\u03d1$, at 50% of the diffuser span are plotted. The choice to consider just the tangential and radial velocity components in Fig. 16 is motivated by the fact that the skin-friction coefficient in the above discussion was evaluated using the same criterion, assuming the streamwise direction tangential to the surface of the blade in the $r\u2212\u03d1$ planes, being the geometry of the diffuser blades two-dimensional. In Fig. 16, the velocity fields are represented for both design (*a*) and off-design (*b*) flow rates. The impeller region was blanked out, since the passage averages have no meaning there, referring to several configurations of the rotor blades. The field at design condition in Fig. 16(a) highlights that the flow is azimuthally quite uniform. The flow decelerates smoothly through the diffuser vanes: the kinetic energy of the flow is properly converted into pressure energy. The only small difference that one can notice among diffuser vanes is represented by the lower values of velocity on the PS of the diffuser blades closer to the volute outlet. At the reduced flow rate, the variations of the flow field in the different vanes are more obvious, as shown in Fig. 16(b). This is due to both the asymmetry of the volute geometry and the more significant role of low-frequency phenomena, although both in the experiments by Boccazzi et al. [26] and the present computations it was not possible to identify a specific low frequency, typical of rotating stall, and the spectra were still largely dominated by the frequency of the blade passage. This conclusion was verified also by means of the instantaneous fields made available by the eddy-resolving approach, where again rotating stall was not distinguishable. In Fig. 16(b), it is also interesting to see an increased velocity near the trailing edge of the diffuser blades on their SS. This behavior can be justified based on the widening blockage on the PS, where the thickness of the boundary layer grows moving downstream, producing a substantial displacement of flow rate toward the SS of the diffuser channels.

## Summary

In this paper, a study of the reverse flow phenomena at a reduced flow rate, equal to 40% of the design one, is presented for a small radial pump. The machine is composed of an impeller with six backswept blades and a constant-span diffuser with seven two-dimensional circular arc blades; the volute has a rectangular cross section. The flow has been solved using the LES methodology, coupled with an immersed-boundary method to handle the moving geometry in a fixed cylindrical grid.

At the reduced flow rate, separation is produced at the impeller shroud, where significant reverse flow occurs. The blockage in that region implies a displacement toward the hub of the flow rate delivered by the pump. It was also shown that the distribution of the turbulent kinetic energy is highly affected, due to the shear layer separating from the shroud and reaching the central region of the rotating channels. The modification of the flow physics in comparison with the design condition is substantial. At the nominal flow rate, the flow on the shroud keeps attached, although the thickness of its boundary layer increases substantially, as well as the values of turbulent kinetic energy, which are however significantly lower than those at the off-design condition.

The flow at the interface between impeller and diffuser shows at the reduced flow rate an increased unbalance along the span, compared to the case at the nominal flow rate. Referring to the off-design condition, the backflow phenomena are almost missing on the hub side of the diffuser vanes, while on the shroud side, reverse flow is produced at the inlet of all diffuser channels, with values of turbulent kinetic energy three times higher, due to the incorrect interaction between moving and stationary blades.

Through the diffuser channels, separation at off-design involves especially the SS immediately downstream of the leading edge of the blades, because of the large values of positive incidence by the flow delivered from the impeller. The boundary layer on the PS of the diffuser blades experiences a substantial deceleration, as proven by the distributions of skin-friction and pressure coefficients. However, a strong dependence of these phenomena has been verified on both the spanwise and azimuthal positions. The reverse flow occurs mainly on the shroud side: it was observed that close to the diffuser wall negative values of skin-friction coefficient on the SS of the stator blades extend almost up to their trailing edge. The distribution of the skin friction is also strongly variable among diffuser vanes. For instance, a stronger backflow was seen on the shroud side in the vanes closer to the tongue of the volute, with higher velocities and skin-friction moving toward the hub, due to blockage effects. In contrast, at design conditions, the skin-friction coefficient keeps always positive both on the PS and SS of the diffuser blades, showing a smaller variability along the span and especially along the azimuth.

## Acknowledgment

The authors are grateful to A. Boccazzi and R. Miorini for providing their experimental results. This work has been partially supported by the Office of Naval Research (Award No. N000141110588).

## Nomenclature

*A*_{1}=cross section at the impeller inlet

*b*_{2}=impeller outlet span

*b*_{3}=diffuser span

*b*_{4}=volute span

*c*=blade chord

*c*=_{f}skin-friction coefficient

*c*=_{p}pressure coefficient

- ·
=_{d}quantity at design condition

*D*_{1}=impeller inlet diameter

*D*_{2}=impeller outlet diameter

*D*_{3}=diffuser inlet diameter

*D*_{4}=diffuser outlet diameter

*f*=generic physical quantity

=**f**_{i}forcing term along the

*i*direction*g*=acceleration of gravity

*H*=pump head

*k*=turbulent kinetic energy

*L*=length reference scale

*m*=number of phase-locked configurations

*N*=sample size of the phase-averaged statistics

*n*=_{d}number of diffuser blades

*n*=_{r}number of rotor blades

- ·
_{off}=quantity at off-design condition

- ·
^{opt}=quantity at the best efficiency point

*p*=pressure

*p*=_{∞}pressure at the domain inflow

*Q*=volumetric flow rate

*r*=radial coordinate

*R*=_{i}local outer radius of the impeller

- Re =
Reynolds number

=**S**_{ij}deformation tensor

*t*=time variable

*T*=torque

*t*_{0}=start time of the sampling process

*U*=velocity reference scale

=**u**_{i}velocity component along the

*i*direction*u*=_{t}velocity tangential to the blade surface

*u*_{1}=outer tangential velocity at the impeller inlet

*u*_{2}=tangential velocity at the impeller outlet

- $ur\u03d1$ =
magnitude of the velocity in the $r\u2212\u03d1$ plane

*w*=ensemble average of −

*u*_{z}*x*=_{c}chord fraction

*x*=_{i}coordinate along the

*i*direction*z*=axial coordinate

*β*=angle of the fluid velocity relative to the azimuthal direction

*β*_{2}=impeller outlet angle

*β*_{3}=diffuser inlet angle

*δ*=_{ij}Kronecker delta

*η*=efficiency

- ϑ =
azimuthal coordinate

*ν*=molecular viscosity

*ν*=_{t}eddy viscosity

*ρ*=fluid density

*τ*=_{w}wall stress

=**τ**_{ij}subgrid scale stress tensor

- $\varphi $ =
flow coefficient

- $\phi $ =
angular position of the impeller

*ψ*=head coefficient

*ω*=_{s}specific speed

- Ω =
impeller rotational speed

- ·$\xb7\u0303$ =
filter operator

- $\xb7\u2032$ =
fluctuation in time

- $\u2329\xb7\u232a$ =
phase average operator

- $\u2329\u2329\xb7\u232a\u232a$ =
passage average operator