Effect of impeller inlet diameter on saddle-shaped positive slope and non-uniform flow patterns at low flow rates of a mixed-flow pump

ABSTRACT Most mixed-flow pumps obtain a saddle-like Q-P curve with a backflow, owing to the increased incidence angle at low flow rates. The backflow was developed near the shroud and followed downstream again at its end to form a recirculating flow. The rotating stall, which could be a part of the recirculating flow, followed the impeller’s rotational direction, and its properties affected the local stability. The reattaching flow became strong when the upstream flow from the blade leading edge deviated from the same circumferential degree as the dominant flow of the rotating stall heading downstream. The fluctuation in the total pressure rise decreased when the average incidence angle was smaller than that of the design flow rate. As a passive control to suppress the saddle and the above flow patterns, the impeller inlet diameter was reduced from the shroud, and the inlet blade angle was further adjusted to maintain the incidence angle. From the reduced inlet diameter, the backflow was mostly suppressed, and the saddle was improved with a wider operating range. Here, the performance near the design flow rate was almost maintained. The stability was evaluated using the fast Fourier transform, and the numerical method was validated through experimental tests.


Introduction
An ideal characteristic of Q-P ( -) curve of a mixedflow pump would be a negative slope with increasing pressure as the flow rate decreases. However, when the flow rate decreases beyond an acceptable range of incidence angles at which a pump can maintain stable operation, flow separation occurs on the blade suction surface (SS). In the lower flow rate range, irregular and non-uniform flow patterns can intensify, which is known to cause performance degradation, pressure fluctuation, vibration, and noise (Gulich, 2008;Stepanoff, 1957). As a result, the Q-P curve forms a horse's saddle-like positive slope at low flow rates (Stepanoff, 1957); this slope is described as 'saddle' in this study. The saddle is unavoidable in most fluid machinery and is mainly observed in the mixed and axial flow types. If the saddle is completely suppressed, one pump unit can allow stable operation over a wider flow rate range.
Some previous studies have been reported to focus on the saddle (Kan et al., 2018;Li et al., 2021;Miyabe et al., 2009;Ye et al., 2020). They commonly described the performance degradation and internal instability based on qualitative data, in which it was observed that the internal flow pattern was congested with the decrease in CONTACT Young-Seok Choi yschoi@kitech.re.kr flow rate. In particular, Li et al. (2021) visualized rotating stalls inside the inlet passage at low flow rates. Although the results were obtained from steady-state simulation, the rotating stall was observed over an axially longer and radially thicker region as the flow rate decreased.
Here, the rotating stall was distributed in the inlet passage, whereas it was often distributed between the blade passage (pitch) in a centrifugal type. Zhang et al. (2017) and Wang et al. (2018) confirmed that the pressure fluctuation, vibration, and noise could intensify due to the separation and vortex developed in the blade passage even if the Q-P curve did not include the saddle; these were the cases of a low-specific-speed mixed-flow pump and a centrifugal pump, respectively. As a method to control the saddle, some extradesigned components such as double-inlet-nozzle (Cao & Li, 2020;Perez Flores et al., 2008), inlet groove (Mu et al., 2020), and inlet guide vane (IGV)  can be applied; however, doing so can complicate one pump unit. It would therefore be more effective and economical to restrain the saddle using only an impeller. Accordingly, the number of blades (Ji et al., 2022), blade setting angle (Lei et al., 2017), tip clearance (Li et al., 2013), and rotational speed can be adjusted in accordance with empirical bases. In general cases, the increase in number of blades and rotational speed is correlated with a higher pressure rise, and it can be expected that the saddle is improved. The decrease in tip clearance implies that the saddle is reformed with a higher pressure rise. The best efficiency point can also be selected from the adjustment of blade setting angle. Nevertheless, these ways must be accompanied by performance change near the design flow rate point, which means the dissolution of guaranteed performance.
The saddle and stability at low flow rates must be considered. These points need to be analyzed in relation to broader and more diverse perspectives using quantitative indicators. This study can be understood as a numerical attempt on passive control to suppress the saddle and non-uniform flow patterns at low flow rates. Based on the rotating stall and vortex developed at low flow rates, which were mainly distributed near the shroud at the impeller inlet (Kan et al., 2018;Li et al., 2021;Miyabe et al., 2009;Ye et al., 2020;Zhang et al., 2017), the impeller inlet diameter was adjusted to be reduced from the shroud tip. Here, the incidence angle was maintained to minimize the performance change near the design flow rate. This premise came from the previous studies (Ardizzon & Pavesi, 1995;Kim et al., 2019;Suh et al., 2019). Although the focus of this study was to suppress the saddle, the overriding goal was to not lose its designed performance. A key fact that can be considered as a principle to suppress the saddle was expected to be the inlet velocity component. From the reduced inlet diameter, the circumferential velocity (u 1 ) must be decreased near the shroud, and the rotational component of the backflow would be weakened. At the same time, the meridional component of absolute velocity (c m1 ) became faster and might be able to suppress the backflow developing upstream. If the relative velocity (w 1 ) could be slower, the backflow would have been more difficult to be developed. The inlet diameter might be one of the critical variables that could affect the recirculating flow near the inlet passage (Yedidiah, 2008). Meanwhile, the reduced inlet diameter can lead to additional advantages, such as lower torque consumption and reduced model size. The results were quantitatively and qualitatively analyzed from the internal flow patterns such as backflow, rotating stall, recirculating flow, and reattaching flow, and the classification for each flow was consistently described in all sections below.

Design specification and method
The impeller inlet diameter was reduced from a base model, which had been optimized with design variables such as blade angle and meridional coordinates. The base model is a part of the database accumulated in commercial software (PumpON, ANFLUX Co., Ltd.), and it performed the best efficiency at the design flow rate with an optimal degree of incidence angle. In general, the incidence angle of fluid machinery, such as a mixedflow pump, has a non-zero degree at the best efficiency point  and is a sensitive design variable that causes performance change even in the order of 1 degree (Suh et al., 2019). Among the design parameters of a pump, the incidence angle is the most important one and must be carefully considered. In this study, the effect of the impeller inlet diameter was examined while maintaining the incidence angle in order to minimize the performance change near the design flow rate. Meanwhile, the design specifications of the base model are presented in Table 1. The specific speed in the concept of type number (N s ), flow coefficient ( ), and head coefficient ( ) are defined as: where w, Q, g, H, c m2 , and u 2 denote the angular velocity, volumetric flow rate, acceleration due to gravity, total head, meridional component of absolute velocity at the impeller outlet, and circumferential velocity at the impeller outlet, respectively. Figure 1 shows the inlet velocity triangle (schematic) without an inlet swirl. Thus, the meridional component of absolute velocity (c m1 ) is equal to the absolute velocity (c 1 ) and is parallel to the axis. For the same flow rate but different inlet diameter, if the orange absolute velocity is the base model's vector, the green one becomes the reduced model's vector. Accordingly, the incidence angle (i 1 ; β 1b − β 1 ), defined as the difference between the inlet blade angle (β 1b ) and inlet flow angle (β 1 ), no longer has the base model's degree. As the inlet diameter decreased, the inlet flow angle gradually increased due to the reduced inlet area, where the angular definition in this study is the tangential direction. The blade angle was adjusted to compensate for this deviation, i.e. the reduced  model's inlet blade angle had a larger degree than that of the base model. It was adjusted only near the leading edge (LE), and that of the trailing edge (TE) was unaltered. The details for blade angle profile could not be disclosed, while it was in quasi-gradual distribution. Finally, the models, in which the inlet blade angle was further adjusted from that of the base model, were prepared as shown in Figure 2. On the meridional plane, the inlet diameter was adjusted from the base model's shroud tip vertex. The linear slope of the LE line was unaltered for each model. The LE length for each model was reduced every 2% based on the base model's LE length while the other three vertices were fixed: the fixation for the hub vertex on the LE line allows the same shaft diameter to be used; the fixation for the two vertices on the TE line, which is directly related to the pressure rise of a pump, also enables to obtain the expected performance. The hub and shroud lines were fitted using the Bezier definition. As a reference, the length corresponding to 2% of the LE line was approximately 0.006 D 2s , and its axial length was approximately 0.004 D 2s , where D 2s denotes the outlet diameter of shroud.

Computational domain and grid system
As shown in Figure 3, the stationary inlet, rotating impeller, and stationary outlet domains had their own interfaces. In the steady-state analysis, a single passage including one blade was applied, whereas in the transient analysis, a full passage including all five blades was adopted. A steady-state analysis with a full passage was additionally performed for the initial state of the transient analysis. The outlet passage was extended by 1.5 times the impeller outlet diameter to not affect the numerical results (Choi et al., 2001). The inlet passage was extended for a longer length to account for a recirculating flow which would occur under the saddle condition. Meanwhile, the hexahedral grids formed a flow passage, and the average y + on the blade surface was approximately 7. The grid test (Figure 4) was performed while maintaining the y + and was for the design flow rate of the base model (single passage). As the total number of nodes increased, the head coefficient ( ) gradually converged to 0.808, and the grid system was applied with a topology corresponding to approximately 553,000 nodes. This test method has been commonly applied in our field .

Governing equation and turbulence model
The details of numerical setup are listed on Table 2. In governing equation, mass and momentum equations were solved under iso-thermal conditions. The time variation terms were considered in the transient analysis and not in the steady-state analysis. Commonly, k-ω-based shear stress transport (SST) standard model is recommended for relatively weak separation levels and its onset (ANSYS User manual, 2016). If a questionable prediction was confirmed in the low-flow-rate range where strong separation would be expected, the turbulence model might be substituted for another model, SST reattachment modification (Kim et al., 2021a), to increase the accuracy in separating shear layers emanating from the   wall. However, the application of the SST standard model in this study was suitable.

Boundary condition
The total pressure (stable) and mass flow rate were applied to the inlet and outlet boundaries, and the turbulence intensity at the inlet boundary was selected as the 'medium' level (1-5%, actually 3.24% at the design flow rate of the base model). The counter-rotating condition was applied to the shroud wall of the rotating domain. The working fluid was water at 25°C. In the steady-state analysis, the frozen-rotor method was applied with the periodic condition. This method makes the data transmit to the next domain without averaging, when there is only one rotor for a simulation. In the transient analysis, the transient rotor-stator method was applied, and the internal flow patterns inside a full passage were observed in depth. The total time for one revolution was approximately 0.034 s, and the transient data were obtained every 3 degrees.

Performance characteristics
First, the numerical setup was validated experimentally. The experimental facility had been constructed at the Korea Institute of Industrial Technology (KITECH), Korea (Kim et al., 2021b). As a lab scale, measuring instruments such as pressure gauges, temperature sensors, magnetic flow meters, and torque meter formed a closed-loop. The maximum standard deviation of the measuring instruments was ±0.2% for each. The international standards for pump tests, ISO 5198 (1987) and ANSI/HI 1.6 (2000), were applied to obtain annual certification from a certificate authority. The flow rate was controlled using two gate-type valves. In Figure 5, the experimental and numerical results for the base model did not contain any notable differences in the tendency (slope) of each performance curve, where the numerical results in Figure 5 were based on the steadystate. The upper-lower deviation between the experimental and numerical results was mainly based on the present/absence of a diffuser. In the feasibility study, the diffuser, which had a geometry for the experimental tests, could degrade the overall performance. At the design flow rate, the total pressure rise and efficiency decreased by approximately 6.9% and 7.5%, respectively, where the numerical setup was almost the same as in this study.
Here, the efficiency was still predicted to be somewhat higher than in the experimental tests, and this was due to mechanical loss. At the design flow rate, the torque in the experimental tests was predicted to be approximately 7.2% higher than that of the numerical simulation. The leakage, roughness, and friction (bearing and seal) might be understood as incidental factors, which were unavoidable in the experimental tests. Moreover, the inlet and outlet passages in the numerical simulation were extended in a hydraulic-friendly geometry (straight pipe) while the experimental facility had a curved pipe. Nevertheless, it seemed reasonable to analyze the slope only with an impeller.
The base model showed a positive slope from 0.8 d ( ≈ 0.135) in the Q-P curve of Figure 5, and the deepest saddle point was located at 0.7 d ( ≈ 0.118); this range was enlarged as [A]. The efficiency followed the Table 3. Increase and decrease in the total head and total efficiency (%, not an absolute difference, from the base model to the M8 model). same tendency; this range was enlarged as [B]. On the other hand, the M2 and M4 models immediately started to recover the positive slope, and the M8 model showed the best improvement. The operating range was further extended by suppressing the positive slope. A more encouraging result was the slope near the design flow rate (0.8-1.2 d ). All the models that were considered in this study (M2-12) almost maintained their own performance, and it was relevant to apply the base (reference) model's impeller outlet diameter and optimized incidence angle. From the base model, the increase and decrease in the total head and total efficiency (η) were listed in Table 3, which could be obtained with the M8 model. Here, the reduced models could obtain somewhat lower torque for the impeller and hub (rotor) so that the total efficiency could be compensated, and the torque was reduced by approximately −0.1-0.2% per 2% deduction. Figure 6 shows the Q-P curve (0.6-1.0 d ) obtained from the transient analysis of the base and M8 models: the steady-state results are the same as those presented in   Figure 5; the white square symbols for transient points represent the total average within each of the last 6 revolutions, and the bar symbols are the maximum and minimum values of the fluctuation; all the points of the transient analysis were from the last 6 revolutions, which generally consisted of repetitive fluctuation patterns, except for revolutions that showed irregular fluctuation patterns from the first revolution; in severe saddle conditions, there were cases in which a regular pattern could not be observed within the last 1-2 revolutions (Berten et al., 2009;Xue et al., 2019). From each curve, the averaged data generally showed a similar slope, slightly below the steady-state data. In terms of pressure fluctuation, the base model showed that the range increased at 0.8-0.9 d compared to the design flow rate (1.0 d ), but subsided at the deepest saddle point (0.7 d ) and further decreased at 0.6 d . In the M8 model, the fluctuation was the most pronounced at the point where the saddle improved (0.7 d ), and sharply decreased again at 0.6 d . From the viewpoint of a general expectation, the separation accompanying instability would gradually increase owing to the increase in the incidence angle at low flow rates (Li et al., 2016). Nevertheless, pressure fluctuations were not always inversely proportional to the flow rate (Cheng et al., 2012;Zhou et al., 2018;Zhou et al., 2019); in this study, the abovementioned tendency was linked to the actual distribution of the incidence angle (see Section 4.3).

Recirculating flow and rotating stall (inlet passage)
This section describes the internal flow patterns inside the inlet passage. Figure 7(a) shows the projected 3D streamline and the contour for the meridional component of absolute velocity (c m = v st , streamwise velocity) on the meridional plane of the base model: the x-axis represents the distance normalized to the outlet diameter of shroud (D 2s ); the results were based on the steady-state analysis; if the contour had a negative value, it should be identified as a backflow. From the deepest saddle point (0.7 d ), a fairly strong backflow developed near the shroud span of the inlet passage and blade LE. This backflow followed downstream at its end and entered the blade, i.e. a 'recirculating flow' was formed. Eventually, these flow patterns affected the outlet flow passage through the mid-and hub spans of the blade TE. The intensity was further exacerbated at the lower flow rate (0.6 d ). The positive slope in Figure 5 was directly related to this backflow. Meanwhile, the backflow could act as a blockage in the inlet passage, increasing the flow velocity into the impeller; it seemed to be faster than the design flow rate. No singularity was observed from the design flow rate to the flow rate just before the saddle (0.8 d ), and only a low contour level was confirmed near the shroud vertex of the blade LE at 0.8 d .
Figures 7(b and c) relate to the lower flow rate (0.6 d ) and the deepest saddle point (0.7 d ) for models (M2-12) with a reduced inlet diameter. At 0.7 d , the backflow of the M2 and M4 models appeared to be weaker than the base model, but were still active. On the other hand, the backflow of the M6-12 models was mostly suppressed. The tendency of the suppressed backflow was exactly the same as the improved range of the saddle in Figure 5. At 0.6 d , the M2-12 models did not completely suppress the backflow. However, its axial length gradually decreased as the inlet diameter decreased. In Figure 5, the performance at 0.6 d gradually improved.
Further, as shown in Figure 8, the backflow identified in Figure 7 was quantified from the boundary at which the streamwise velocity became zero, where l z , l s , and l LE denote the axial length (maximum) of backflow, spanwise length (maximum) of backflow, and blade LE length on the meridional plane, respectively. Each length was based on the Cartesian coordinate system and point-to-point measurement. This can be an indicator that relies on a qualitative description. The spanwise length was measured on the LE line of the meridional plane because it was always the longest on the LE line for all the models, including the base model. In particular, the M8 model, having the best improvement in the saddle slope, contained the shortest backflow length in the axial and spanwise directions at 0.7 d . Meanwhile, from the axial and spanwise lengths at 0.6 d , the variable closely related to the performance could be inferred as the axial length with a gradual distribution rather than the spanwise length. For 0.8-1.0 d of the base model, l z /D 2s was approximately 0.13, 0.06, and 0.00, and l s /l LE was approximately 0.05, 0.01, and 0.00, respectively.
Mechanisms of the recirculating flow, including backflow, were subdivided from Figures 9(b-e) in the viewpoint of Figure 9(a): the planes A, B, and LE in Figure 9(a) were indicated for description in the subsequent sections; the axial gap (0.03D 2s ) between planes B and LE was for the base model; the point (particle) seeding of the 3D streamline with water velocity was performed on the free edge of the inlet surface; the arrows in Figures 9(b and e) were according to the detailed vectors; the results were based on the full passage domain in the steady-state analysis. First, Figure 9(b) was for 0.6 d of the base model. The non-uniform flow pattern at this low flow rate began to form from R1, owing to the increase in the incidence angle. R1 climbed up along the shroud wall as a backflow, which was the same as described in the 2D perspective of Figures 7 and 8. From the 3D perspective of Figure 9(b), the rotational component was additionally confirmed in the backflow, i.e. R1-2 was the backflow with a rotational direction as same as that of the impeller. This region grew wider as the flow rate decreased, as shown in Figures 7 and 8. The backflow faced at its end with the inlet flow (F1). Here, F1 followed downstream at lower spans than the shroud span due to the thick and long distribution of R1-2 near the shroud wall and also had a swirl component in the same direction as the impeller's rotational direction: this flow was indicated as F2, which seemed to obtain a slower velocity than R1-2; the case where F2 formed a bundle might be referred to as 'rotating stall'. Finally, F2 entered the impeller blade, and it was divided into a flow (F3) that was transferred to the impeller outlet and a flow (F4) that joined R1 (backflow). The above loop could be defined as the recirculating flow that started at R1 and returned to R1, and would indefinitely repeat if its energy was not dissipated. R3 and R4 could be classified as a 'reattaching flow', which occurred on the blade SS at low flow rates (see Section 4.4), affecting R1 in the process. Meanwhile, the flow pattern might be described from F1 in terms of particle tracking in the numerical simulations. Figures 9(c and d) were for 0.6 d of the M8 model and 0.7 d of the base model, respectively. Based on the results of Figures 7 and 8, each region of recirculating flow was decreased compared to Figure 9(b), but the mechanism was the same. Figure 9(e) was for 0.7 d of the M8 model. As expected, no recirculating flow was observed in the inlet passage. However, the flow entering the impeller (F1) directly affected the blades, leading to separation corresponding to R1 with a weak backflow. This means that the incidence angle was still not appropriate. Here, R1 did not develop into the recirculating flow and was directly headed to the next blade's LE. It seemed that the recirculating flow was about to be developed. Meanwhile, F2 along the circumferential direction near the blade LE was opposite to the impeller's rotational direction because this was the relative velocity component.
The above paragraphs confirm that the recirculating flow in the inlet passage could consist of a backflow and rotating stall at low flow rates. Figures 10 and 12 show the streamwise velocity (v st = c m ) contour and limiting streamline on the cross-sectional planes A and B, which are orthogonal to the axis in the inlet passage (see Figure 9a): the white dotted line denotes an outline of the projected blade's LE and hub; the blade numbering was assigned to only one; the results were based on the last revolution of the transient analysis. First, Figures 10(a,b) correspond to 0.6 d of the base model ( Figure 9b). The streamlines, which appeared to form concentric circles on plane A, were backflow (negative contours) as a definite blockage from the shroud wall to approximately 0.75 span and were directed downstream (positive contours) in spans below approximately 0.75. Below approximately 0.45 span, the streamlines along the circumferential direction were subsided, and the flow was directed toward the impeller blade with a relatively high velocity. The streamlines, which were straightened in the radial direction, implied the mainstream area. This flow pattern was steady over time. No specific formations other than streamlines along the circumferential direction were observed at any time, even just before the impeller (plane B). This implies that F2 in Figure 9(b) was evenly spread out toward the impeller. The intensity of the backflow increased with separation just near the blade LE. Figures 10(c,d), corresponding to 0.6 d of the M8 model (Figure 9c), obtained a visual difference. On plane A, five bundles of rotating stalls were identified as forward flows. It was located inside the circled streamlines, forming the backflow. Stall's bundles appeared to form over the entire span within the flow passage. Accordingly, the main flow passage formed a pentagonal shape under the influence of the rotating stalls. The velocity contour inside the main flow passage was fairly uniform Figure 11. Velocity vector of rotating stall (enlarged view of dotted box in Figure 10c). over time. This implies that each bundle maintained a uniform intensity. The circumferential movement (rotational direction) of each bundle was identical to the impeller's rotational direction. One bundle of rotating stalls moved approximately 122 degrees during one revolution of the impeller (122/360 ≈ 0.34), with a uniform speed of approximately 30 degrees per 0.25 revolutions of the impeller. Figure 11 shows the vector distribution plotted as an enlarged scale for the dashed box in Figure 10(c). The vector component in the rotating stall was also identified in a clockwise direction (impeller's rotational direction). However, just before the impeller (plane B), the rotating stall's bundle almost declined, repetitively deforming over time so that the rotational speed was not easy to be assumed. From the center of some remaining stalls, it could be confirmed that a substantially constant speed was maintained. Figures 12(a,b), corresponding to 0.7 d of the base model (Figure 9d), presented four bundles of rotating stalls. The mainstream was passed through a rotating rectangular flow passage owing to the rotating stalls. However, because the intensity of each stall's bundle was not uniform, the mainstream area showed a distorted quadrangular shape and uneven velocity contour over time. On the other hand, the circumferential movement of the bundle followed a rotational direction as same as the impeller. One bundle of rotating stalls moved approximately 142 degrees during one revolution of the impeller (142/360 ≈ 0.39). It showed a uniform moving speed of approximately 36 degrees per 0.25 revolutions of the impeller. The internal vectors making up the stall's bundle were also clockwise. Just before the impeller (plane B), the stall's bundle seemed to exhibit repeated deformations and almost declined, if not as much as at 0.6 d of the M8 model (Figure 10d). The circumferential speed from a vague formation was more non-uniform than in Figure 10(d). Figures 12(c,d), corresponding to 0.7 d of the M8 model (Figure 9e), were under the condition without any recirculating flow. Hence, plane A showed an almost constant contour without any blockage. On plane B, the contour repeatedly contained positive and negative levels near the shroud wall, which was attributed to R1 in Figure 9(e). Here, the pattern was irregular and non-uniform over time.
From the results presented in Figures 10 and 12, the rotating stall followed downstream under the condition with a strong backflow, but it was not necessarily formed under that premise, i.e. 'non-uniform' flow did not always lead to 'instability'. If the backflow region (blockage) was distributed thicker in the spanwise direction, the internal flow that had to pass through the narrower area faster inhibited the formation of stall's bundle. Accordingly, it was difficult to confirm that the rotating stall had an effect on the 'performance'. The stall's bundle was identified to have more counts (five) and stable patterns at a lower flow rate (0.6 d ) of the saddle-improved model (M8) and fewer counts (four) and unstable patterns at a higher flow rate (0.7 d , deep saddle point) of the base model. Although it might be difficult to describe the stability only with the range of the pressure fluctuation, in Figure 6, the pressure fluctuation was lower at 0.6 d of the M8 model than 0.7 d of the base model. The rotating stall was, therefore, more closely related to the stability of the internal flow than to the performance. Here, the stability must be evaluated using additional quantitative indicators (see Section 4.5). Most obviously, it was important to prevent the recirculating flow, including the backflow, for performance.

Inlet velocity triangle (impeller inlet)
The internal flow patterns were quantified as data on the velocity vector at the impeller inlet. The recirculating flow with backflow and rotating stalls developing in the flow passage would cause the impeller to no longer face the ideal flow. Under the premise that the impeller's rotational speed and diameter are the same, if the inlet flow uniformly enters the impeller as at the design flow rate, it contains an absolute flow angle (α 1 ) of 90 degrees, as shown in the red velocity triangle in Figure 13; in this case, the incidence angle (i 1 = β 1b − β 1 ) becomes positive. On the other hand, if the inlet flow has a swirl component, the absolute flow angle is not 90 degrees.  If the swirling direction is the same as the impeller's rotational vector (u 1 ), the yellow velocity triangle can be obtained; in this case, the incidence angle becomes negative; depending on the flow rate, the incidence angle may be restored to a positive value. The blue velocity triangle denotes the case of backflow with a swirl component, and all the values except for the incidence angle become negative.
Under the above definition in Figure 13, Figures 15-19 was presented from the guide of Figure 14: the LE plane in Figure 14 was the same as indicated in Figure 9(a), which tangentially connected the LEs; a total of 21 points were inserted in a normalized pitch (S * ) from the pressure surface (PS, 0) to the suction surface (SS, 1), which were on the LE plane; the normalized span (r * ) from the hub span (0) to the shroud span (1) was divided into ten sectors on the LE plane; the results were based on the full passage in the steady-state analysis, and the data of each pitch was averaged. Figure 15 shows the meridional component of absolute velocity at each point in the pitch. At the design flow rate (1.0 d ) of the base model (Figure 15a), the flow was the fastest toward downstream near the SS and formed a backflow near the PS. The distributions in the hub and shroud spans were nearly identical, which could be considered as an ideal flow. This tendency was maintained at 0.9 d , but the effect of separation that occurred as the incidence angle gradually increased on the 0.9-1.0 spans of 0.8 d was confirmed near the SS (S * ≈ 0.65-0.95 for 0.9 span, 0.55-0.95 for 1.0 span). This effect was also observed near the PS (S * ≈ 0.0-0.2) of 1.0 span. This flow rate (0.8 d ) was just before the onset of a positive slope (saddle) on the performance curve. At the deepest saddle point (0.7 d ), the effect of separation disappeared near the SS, while backflow developed near the PS of 0.7-1.0 spans, and its intensity became stronger and irregular as it approached the shroud. Based on the normalized pitch, the circumferential lengths of the backflow at spans of 0.8, 0.9, and 1.0 were approximately 0.18, 0.35, and 0.60, respectively. Moreover, the region above the mid-span was affected by the non-uniform flow. At the lower flow rate (0.6 d ), the velocity level over the spans slightly increased near the PS and SS, but a downward shift was confirmed in 0.4-1.0 spans with a larger range near the shroud. Here, the circumferential lengths of backflow at spans of 0.8, 0.9, and 1.0 were approximately 0.10, 0.65, and 1.00, respectively. Meanwhile, the pattern at the design flow rate (1.0 d ) of the M8 model (Figure 15b) was maintained without any significant difference at 0.8-0.9 d , i.e. the flow rate range for the stable operation had been expanded by 0.1 d than the base model. The pattern similar to 0.8 d of the base model was confirmed at the point where the saddle was improved (0.7 d ). Obviously, the slope near the SS (S * ≈ 0.5-0.95) of 0.9 span became sharper, and the entire pitch of 1.0 span was filled with backflow. This pattern might have been due to R1 in Figure 9(e). Here, F2 in Figure 9(e) was a flow located slightly upstream from the blade LE and was actually confirmed as a flow toward downstream, but there was only backflow in 1.0 span on the blade LE plane; upstream and downstream were encountered near the impeller inlet. In terms of the same flow rate (0.7 d ), the backflow that the base model contained over 0.7-1.0 spans near the PS was almost suppressed (except for 1.0 span) in the M8 model. Thus, the correlation between backflow and performance was still valid. The following pattern of 0.6 d was quite similar to that of 0.6 d of the base model, i.e. it was difficult for the rotating stall's bundle to significantly affect the flow pattern near the impeller inlet. At 0.6 d of each model, the base model did not include the rotating stall, whereas the M8 model did (Figures 10a and  c). At 0.6 d of the M8 model, the rotating stall almost declined immediately before the impeller (Figure 10d), and the pattern just at the impeller inlet could be inferred to be similar to Figure 10(b). Here, even if the rotating stall is eliminated, its streamwise velocity component must continue to be a part of the recirculating flow in the inlet passage. The flow pattern at the impeller inlet was not unaffected by the backflow and recirculating flow. Meanwhile, the pattern contained at 0.7 d of the base model was omitted within the flow rate range of the M8 model. Figure 16 shows each point averaged over 21 data in each span of Figure 15. Because the meridional component of absolute velocity was expressed as an approximation, it was more appropriate to analyze the circumferential quantification of backflow as shown in Figure 15 rather than in Figure 16. Nevertheless, additional quantitative analysis was possible as follows: the base model was affected by the recirculating flow with backflow in spans 0.4-1.0, 0.5-1.0, 0.9-1.0 at 0.6, 0.7, and 0.8 d ; the M8 model was subjected to recirculating flow with backflow in spans 0.5-1.0 and 0.9-1.0 (obviously 0.7-1.0) at 0.6 and 0.7 d ; within the spans where each model was not affected by the recirculating flow with backflow, the meridional component of absolute velocity did not decrease gradually as the flow rate decreased, and in particular, a faster velocity than that of each design flow rate was obtained at 0.6-0.7 d of the base model and 0.6 d of the M8 model.
The incidence angle at each point within the pitch was plotted in Figure 17. The backflow region could be identified through cross-validation with Figure 15, e.g. at all the flow rates in the base and M8 models, the incidence angle near the PS increased drastically because of the backflow. The incidence angle distribution generally followed a tendency to increase as the meridional component of absolute velocity decreased; however, there could be some irregular flow patterns that were difficult to infer only based on the incidence angle and the meridional component of absolute velocity, as shown in S * ≈ 0.5-0.95 of 0.9 span at 0.7 d of the M8 model, and in such case, the absolute flow angle, as described in the next paragraph, must also be considered. In this paragraph, the data of Figure 17 was averaged for each span and represented in Figure 18. Except for the region identified as affected by the recirculating flow with the backflow from Figure 16, the incidence angle had lower positive degrees at 0.6-0.7 d for the base model and 0.6 d for the M8 model, compared to each design flow rate. In particular, the base model had lower positive degrees at 0.6 d than 0.7 d , which seemed to be reasonable from the tendency shown in Figure 16. This tendency could be linked to the range of pressure fluctuation contained at 0.6-0.7 d of the base model and 0.6 d of the M8 model in Figure 6, i.e. if the averaged incidence angle at the impeller inlet had smaller degrees than the design flow rate, the fluctuation of total pressure rise could be suppressed. Meanwhile, in Figure 18, i 1 > β 1b denotes backflow, but it would be correct to quantify the exact backflow from Figure 15. Figure 19 shows the distribution of the absolute flow angle within the pitch. From the definition of Figure 13, the absolute flow angle of ±90 degrees refers to free swirl (circumferential component of absolute velocity), and the negative degree must be the backflow. The absolute flow     (Figure 17), hence, there was no need to focus on the absolute flow angle at the moment when the streamwise direction changed. Nevertheless, from an obvious perspective, an absolute flow angle other than ±90 degrees implied a swirl, and the subsidiary focus was given to the points where the recirculating flow was active. The data within S * ≈ 0.2-0.4 of 0.8 span at 0.7 d of the base model (Figure 19a) implied that the flow entering the blade contained swirls opposite to the impeller's rotational direction. It was confirmed over a wider pitch (S * ≈ 0.2-0.65) at 0.6 d . Here, this swirl was not a flow corresponding to the rotating stall, as the rotating stall disappeared near the blade LE, which hardly affected the flow pattern, as shown in Figures 15 and 17. In Figure 19(a) again, the backflow was confirmed with a swirl toward the impeller's rotational direction within S * ≈ 0.2-0.65 of 0.9 span at 0.6 d . However, the meridional component of absolute velocity in this pitch was close to zero. At 0.6 d of the M8 model (Figure 19b), the pattern was similar to 0.6 d of the base, as expected.

Reattaching flow (blade passage)
The reattaching flow is a type of flow pattern that occurs separation on the blade SS, when the incidence angle increases at low flow rates and has been reported as a cause of internal instability and performance degradation (Krause et al., 2005;Posa et al., 2016;Zhao et al., 2021). In this study, the reattaching flow was mentioned as R3 and R4 in Figure 9(b) and was shown for each flow rate in Figure 20: the iso-surfaces were overlapped to distinguish the low-velocity region (v/u 2 = 0.000-0.087) inside the blade passage; the blade's shroud camber was normalized to 0 and 1 for LE and TE, and indicated every 0.1; the results were based on the full passage in the steady-state analysis. As expected, the internal flow pattern at 0.9-1.0 d of the base model and 0.8-1.0 d of the M8 model contained almost no low-velocity region, and it appeared just on the hub wall. At 0.8 d of the base model, S1 was confirmed near the shroud span on the SS with a length of approximately 0.2 from the LE; it could be related to Figure 15(a), Figure 17(a), and Figure 19(a). At 0.7 d , S1 seemed to be suppressed and decreased, while P1 was observed near the shroud span on the PS with a length of approximately 0.4 from the LE. Subsequently, the reattaching flow, S2, proliferated near the shroud span on the SS at approximately 0.4-0.8. P1 and S2 were already in contact at the pitches of some blade passages. The pattern deteriorated at 0.6 d . The reattaching flow occupied the shroud span of approximately 0.2-0.8 from the LE, and the region of P1 + S2 looked like a pillar. At 0.7 d of the M8 model, S1 was longer with a length of approximately 0.3, and P1 began to develop. The pattern such as 0.7 d of the base model was omitted in the M8 model, and the pattern of 0.6 d was similar to the base model. Here, in the region of P1+S2 (pillar) for both models, the internal flow vectors had a congested direction with a low magnitude. Figure 21 was plotted to visualize the uniformity of internal flow in the blade passage: a streamwise velocity contour was presented on the blade-to-blade plane of 0.9 span; each blade was numbered as #1-5 only to indicate their tangential position over time and had no relation with the other numberings in Figures 10 and 12; the diagonal white dotted lines were drawn to connect the blade's LE over time; based on the streamwise direction, the vertical black dotted lines were drawn in the  inlet passage, just before the blade's LE, and just after the blade's TE, respectively, and the dotted lines corresponding to the inlet passage and just before the blade's LE had the same axial coordinates as planes A and B in Figure 9(a); the results were from the last revolution of transient analysis. At 0.6 d of the base (Figure 21a) and M8 models (Figure 21b), and 0.7 d of the base model (Figure 21c), a strong backflow (dark blue) was observed near the blade's LE and passage. This region was related to P1+S2 (pillar) in Figure 20, which was involved with the reattaching flow. At 0.6 d of the base model (Figure 21a), there was no change over time. Accordingly, regions (1)-(5) were maintained over time, i.e. the backflow was stably directed upstream with the reattaching flow. At 0.6 d of the M8 model (Figure 21b), the formation of a strong backflow near each blade's LE and passage was slightly different at a specific time and slightly decreased or increased over time. Here, regions A-E were due to the rotating stall with five bundles in Figure 10(c), which maintained almost the same circumferential speed with approximately 0.34 revolutions per a revolution of the impeller. In fact, since the rotating stall predominantly exhibited a downstream direction, it would have caused a sawtooth-shaped contour corresponding to the number of bundles, such as regions A-E. At 0.7 d of the base model (Figure 21c), the strong backflow near each blade's LE and passage showed significant non-uniformity at a specific time, even over time.
Regions A-D could be confirmed due to the effect of the rotating stall with four bundles in Figure 12(a), which was found to be approximately 0.39 revolutions per a revolution of the impeller. However, region C was generally the strongest, and regions A and D were relatively weak. At 0.7 d of the M8 model (Figure 21d), the effect of the rotating stall was not found, but the flow patterns near the blade's LE and passage were quite congested.
From the time variation in Figure 21, the rotating stall in the inlet passage could be correlated with the backflow in the reattaching and recirculating flows. First, at 0.6 d of the M8 model (Figure 21b), when the upstream flow from the blade's LE (e.g. R1-2 in Figure 9b) deviated as far as possible from the same circumferential degree (θ) as the downstream flow of the rotating stall, i.e. when the stem of backflow could be the longest, it was the fastest and widest. Under this condition, the reattaching flow in the blade passages became stronger together; this was marked as 'concord'. From the reattaching flow on the blade SS to the end of the backflow in the recirculating flow, a long stem of backflow was formed in the axial direction. However, the upstream flow from the blade's LE was weakened when the circumferential degree coincided with the dominant flow of the rotating stall, and then reattaching flow in the blade passage was also weakened; this was marked as 'discord'. This tendency was pronounced at 0.7 d of the base model (Figure 21c), where the axial length of the recirculating flow was relatively short. Although Figure 15 was from the steady-state analysis, it was confirmed that the rotating stall did not have a significant influence on the flow pattern near the impeller inlet. However, from an obvious perspective, the meridional component of absolute velocity for 0.9 span at 0.6 d of each model showed a difference in slope near the PS; the other slopes were quite similar. Hence, the correlation in this paragraph should be localized around the shroud (0.9 span).

Fast Fourier transform
As a final point, the stability was evaluated using the fast Fourier transform (FFT), which was performed with pressure fluctuations at certain monitoring points, as depicted in Figure 22: each monitoring point was inserted near the shroud wall; points 1-4 were on the same axial coordinate as plane A in Figure 9(a) and the black dotted line inside the inlet passage in Figure 21; points 5-8 were on the same axial coordinate as plane B in Figure 9(a) and the black dotted line just before the blade's LE in Figure 21; points 9-12 were on the same axial coordinate as the black dotted line just after the blade's TE in Figure 21; the results were from the last 6 revolutions of the transient analysis in Figure 6; the magnitude of pressure fluctuation was non-dimensionalized as in equation (3); the base and M8 models had the same blade passing frequency (BPF; f n = ZN/60 ≈ 148.3 Hz) because the number of blades (Z) and rotational speed (N) were constant. At 0.6 d of the base model (Figure 23a), harmonic peaks were observed every 0.2f n at points 9-12. The peaks at 0.2 and 0.8f n were also observed at points 1-4. These peaks showed a harmonic pattern every 0.2f n , indicating its dependency on the number of blades. Moreover, its magnitude was too weak to be classified as a factor affecting the internal flow pattern. The peaks with low frequencies (f < 30 Hz) at points 5-8 were also weak in magnitude. Although the base model actually contained the thickest and longest backflow and caused disorder in the internal flow pattern at 0.6 d , it was the most stable. At 0.6 d of the M8 model (Figure 23b), points 1-4 showed meaningful peaks at approximately 0.33f n . A multiple of approximately 0.33 was based on the circumferential speed of the rotating stall's bundle, as shown in Figure 10(c). The magnitude for each point was similar, with the highest at point 4, and irregular peaks other than 0.33f n were difficult to be considered as a meaningful factor, i.e. the rotating stall inside the inlet passage was fairly stable. Meanwhile, because the rotating stall in Figure 10(d) had almost disappeared, the peaks at points 5-8 usually contained a lower magnitude than points 1-4. Here, if the rotating stall disappeared with a constant speed along the circumferential direction, the frequencies at which all four points obtained peaks, such as approximately 0.43 and 0.67f n , could be related to the rotating stall. For both 0.43 and 0.67f n , the magnitude was the highest at point 7. Here, 0.43 and 0.67f n were approximately 1.3 and 2 times 0.33f n , respectively.
At 0.7 d of the base model (Figure 23c), the peaks at approximately 0.32f n on points 1-4 should be related to the circumferential speed of the rotating stall. The magnitudes were similar to those of the M8 model's rotating stall. In particular, the peaks were higher at frequencies lower than 0.32f n . This could be understood as non-uniformity of the rotating stall (Figure 12a). Accordingly, irregular peaks appeared with greater magnitudes at points 5-8. All four points had peaks at 0.5f n , which was approximately 1.6 times the value of 0.32f n , but their  magnitudes were stronger at points 6 and 7, and weaker at 5 and 8, with the largest deviation between each point. This implies that the intensity of the rotating stall, which had to face extinction, was quite unstable. As a result, the stability of the internal flow at 0.7 d of the base model was evaluated to be more unstable than 0.6 d of the M8 model despite the higher flow rate, which was attributed to the characteristic of the rotating stall. At 0.7 d of the M8 model (Figure 23d), points 1-4 no longer showed significant peaks. On the contrary, points 5-8 contained irregular peaks in the widest band up to approximately 3f n . The inter-blade interference in Figure 9(e) (or 12d) could contain more unstable characteristics than the effect of the rotating stall. Meanwhile, the FFT results above were based on the last 6 revolutions, and the qualitative figures were for the last revolution. Thus, Figure 23 could be understood as showing instability within 6 revolutions. Nevertheless, the stability evaluation was verified without any arbitrariness.

Conclusion
The main focus of this study was to suppress the saddleshaped positive slope and non-uniform flow patterns, which could be obtained from the effect of the impeller inlet diameter. Based on the predicted performance, each flow pattern in the inlet passage, impeller inlet, and blade passage was observed along the streamwise direction and then correlated with each other. The length of the backflow was quantified, and its tendency was directly correlated with performance changes. The circumferential speed of the rotating stall was estimated, which was consistent with the FFT results. In addition, the velocity profile at the impeller blade inlet was appropriately applied to evaluate the internal flow pattern and stability. The level of low-velocity was presented for the discrimination of the reattaching flow, and it was linked with each result. Subsequently, for general applications, dimensions related to speed (velocity), length, and frequency were fully normalized, and dimensionless variables were employed. Above all, the most valuable dimensions for this study would be the optimal degree of the incidence angle and blade angle profile. However, the details could not be disclosed because it is a part of the database accumulated in commercial software (PumpON, ANFLUX Co., Ltd.); the blade angle profile was in quasi-gradual distribution. On the other hand, the meridional plane was explicitly presented for whole models. It is expected that the readers can easily apply this design method to any of their reference models because they just need to redesign the blade angle to have the same degree of incidence angle as their reference one. The results can be summarized as follows: (1) Using the design concept for the inlet diameter, the saddle was sufficiently controlled, and the effect was the most optimal when the inlet diameter was reduced by 8% (2.4% based on the outlet diameter of shroud) along the base model's LE line on the meridional plane. Even with a 2%-reduced inlet diameter, the saddle was immediately recovered and significantly improved from 6% or more. However, excessive reduction could have the opposite effect. The improvement of the saddle was due to the suppression of backflow inside the flow passage, and the axial length of the backflow was the most sensitive to the saddle. Although the pump was redesigned with a reduced inlet diameter, the performance was almost maintained, except for the saddle range. The attempt to maintain the optimal incidence angle was therefore a reasonable solution.
(2) At the deepest saddle point, the recirculating flow could develop from the backflow in the inlet passage and cause the rotating stall, which was more irregular and unstable inside the passage. At the lower flow rate point, no rotating stalls were formed because the backflow area (blockage) was distributed thicker in the spanwise direction, and the inside of the pump became rather stable. On the contrary, when the inlet diameter was reduced, the rotating stall was identified at the lower flow rate so that the pump's operating range could be expanded. The rotating stalls at the flow rate other than the saddle point exhibited uniform intensity and stability. The circumferential speed of the rotating stall was slower in the pump where the inlet diameter was reduced with a lower circumferential velocity (u) at the shroud. Both types of rotating stalls followed the impeller's rotational direction and disappeared before reaching the blade's LE, such that the blade's inlet flow pattern was not significantly affected. The rotating stall was more closely related to the stability than the performance. (3) The pressure fluctuation for the total pressure rise was further decreased at the deepest saddle point and lower flow rate point. As the flow rate decreased, the meridional component of absolute velocity did not gradually decrease because of the blockage formed by the backflow area. Moreover, it was faster at the deepest saddle point and lower flow rate point, than the design flow rate. The fluctuation in the total pressure rise could be reduced when the average incidence angle was smaller than that of the design flow rate. (4) The reattaching flow on the blade SS at low flow rates, such as the deepest saddle point, should be distinct from the recirculating flow with backflow and rotating stalls. The reattaching flow became stronger when the upstream flow from the blade's LE deviated from the same circumferential degree (θ ) as the dominant flow of the rotating stall. Further, the formation was irregular at the deepest saddle point and regular at the non-saddle point. Although the rotating stall was dissipated near the impeller inlet, the reattaching flow had a local correlation with the rotating stall near the shroud. (5) Although no rotating stalls developed, the interblade interference due to the incidence angle beyond a certain limit could contain more unstable characteristics than the effect of the rotating stall.