Suppression of Stall-Induced Instability and Positive Slope at Low Flow Rates of an Axial Fan With Two-Dimensional Anti-Stall Fin

Axial fans are in high demand for air conditioning systems such as vehicles, aircraft, home appliances, electronic devices, and underground spaces, with their structural simplicity. Especially for applications in subways or tunnels, where demands cannot be even, the axial fans need to secure stable characteristics in a wide operating range through an antistalling process. Here, “stall” is referred to as a phenomenon that can be frequently contained in fluid machinery and is due to an increase in incidence angle at the low flow rates. Figure 1 shows the inlet velocity triangle under the ideal condition, where $u$⁠, $c$⁠, $w$⁠, $i$⁠, and $β$ denote circumferential velocity, absolute velocity, relative velocity, incidence angle, and flow angle, and subscripts 1, $m$⁠, and $b$ denote blade inlet, meridional component, and blade, respectively. If the incidence angle increases beyond a specific limit, the fluid machinery can no longer guarantee stable operation, i.e., in the stall range, the performance curve (flowrate-pressure; $Q$-$P$⁠) exhibits a positive slope with a decrease in pressure [1,2] and is generally characterized by noise [3,4], vibration [5,6], and system instability [7–9]. In our field, the stall stands as a profound problem.

Some previous studies [10–12] experimentally and numerically focused on the internal flow field to visualize the inlet flow pattern in the stalling flow rates, and “rotating stall” was pointed in common. The rotating stall could be defined as a vortex's cluster which was following the blade's rotational direction in the inlet passage (front part of the blade) and was facing the downstream: it could be confirmed when “backflow” was premised from the blade tip; here, the backflow should be located at the spans higher than the rotating stall's location and could obtain a circumferential velocity component. This flow pattern could cause performance degradation in the stalling flow rates and be closely related to noise, vibration, and system instability. In particular, the backflow became more assertive at lower flow rates [13–17]. Here, in the case of centrifugal fluid machinery, the rotating stall may be observed only between the blade passages rather than the inlet passage; this study relates to an axial fan.

The researchers have been trying to control the stall for decades; the main focus was to disrupt the formation of backflow or rotating stalls near the shroud. Lim et al. [18] configured the sliding rollers as a part of the casing treatment and provided them outside the casing to be arranged as an annular row so that grooves (or slots) were formed at the moment of stalling condition. The groove, which had been proposed since the 1950s, allowed the high-pressure air at the rear of the blade to be transferred back to the front of the blade; however, it could increase the clearance between the casing and blade tip, which could affect the performance near the design flowrate [19]. Thus, they had to drive the sliding rollers again to obliterate the grooves in case of no stall. Gentry et al. [20] made the grooves permanent with minimized depth and length. Jung et al. [21] perforated the casing at the front and rear of the blade and constructed a seesaw-type link to control the perforation. The perforation could be opened to release the high-pressure air at the front of the blade. McKelvey [22] provided an annular chamber and padded it outside the casing so that the high-pressure air at the rear of the blade could be retransferred to the front of the blade; this recirculation method was reviewed with Bianchi et al. [23] and was similar to the proposal of Dygert and Bushnell [24]. Khaleghi [25] and Salunkhe et al. [26] introduced an air injection method to supply high pressure to the low-pressure region. In terms of blades, Jung et al. [27] suggested a recessed cavity inside the blade tip; this can be understood as the reverse concept against the grooves above. The inlet guide vane could be considered as an additional system so that the absolute flow angle at the blade inlet could be adjusted to secure the stable operation [28]. A bump-shaped obstacle was added on the blade surface [29] or on the hub surface [30] to affect the flow patterns locally, and a sinusoidal-like blade leading edge (LE) was applied to enhance the lift coefficient [31]. These deep efforts eventually paid off in the antistalling performance; however, each of them might face major or minor disadvantages in the case by case:

• operating devices and systems; cost and time

• complicated design

• installation space and maintenance

• performance degradation (or change) from design specification

The highly inspired focus of this study was a “simple” and “original” method to compensate for the above disadvantages based on the antistalling process of an axial fan. Thus, two-dimensional plates were attached inside the casing and toward the shaft. Since the plates should be fixed and semipermanent as a stator, it could be a kind of passive control for the stall, and it has been named as “anti-stall fin (ASF)” by ourselves. First, the sensitivity analysis for design variables was conducted through the $2k$ full factorial design method, and the optimization was performed using the response surface method (RSM). The optimized ASF was directly applied to an axial fan and was compared to the non-ASF state. The analysis was focused on the internal flow field and the fast Fourier transform (FFT) results: it was mainly based on the numerical simulations; the performance before/after application of the ASF was validated through experimental tests. If performance degradation in the stalling flow rates is improved without any instability, an efficient operation can be secured through expansion of the operating range (stall margin). Additionally, noise reduction can be expected through suppression of the unfavorable flow patterns near the blade LE. Meanwhile, the antistalling method with ASF of this study was recognized for its originality from a patent application [32].

where $ω$⁠, $Q$⁠, $P$⁠, $ρ$⁠, and subscript 2 denote the angular velocity, volume flowrate, total pressure, air density, and blade outlet, respectively and $u2$ was assigned for the blade tip. Meanwhile, the axial fan showed a peak point at 0.8$Φd$ on the $Q$-$P$ (or $Φ$-$Ψ$⁠) curve and contained a positive slope in the lower flow rates than 0.8$Φd$ [33].

The ASF was designed to be attached inside the casing as a stator so that no additional space was required for installation (see Fig. 3). Since the ASF's axial directionality (angle; $βASF$⁠) inevitably causes the absolute flow angle at the blade inlet and leads to the decrease (or change) in performance even near the design flowrate, the ASF was designed so that its surface was not visible in front of an axial fan, i.e., the ASF exhibited two-dimensional geometry just toward the shaft. The basic principle was to prevent the development of backflow and redirected the circumferential velocity component to the axial direction. The features that could be considered from the above concept are:

• no operating devices and systems

• simple configuration

• immediacy (on-site welding or fastening); semi-permanent

• guaranteed performance based on design specification

• regardless of material (iron, rubber, plastic, etc.)

where any material, including a flexible one, would be available under the premise that the ASF and casing are completely attached. Nevertheless, before applying the ASF, a previous consideration on the meridional plane is strongly recommended to prevent contact between the ASF's TE and rotor's LE; the same is true in the case of high-pressure fans (or fluid machinery) that may have arguments with thrust.

Table 2 lists the design parameters of ASF. From an empirical perspective, the axial gap (⁠$δ$⁠) was selected as the most critical variable; it was based on the fact that the backflow from the blade LE mainly causes instability at low flow rates. In addition, since the backflow occupies a wider region inside the flow passage as the flowrate decreases, the radial and axial length (⁠$lr$ and $la$⁠) were considered as the variables: the above three parameters were selected as the main variables for ASF (see Fig. 3) and were normalized to the fan diameter (⁠$D2$⁠). The number of fins (13) was not a variable; the number of fins was considered to be more than the number of blades (10) to sufficiently control the unstable flow associated with the backflow from each blade LE and to avoid overlap with the number of blades and guide vanes (11). The ellipse ratio was selected as 1 (semicircle) for both LE and TE, and the angular distribution (⁠$θASF$ or $βASF$⁠) should not be considered with two-dimensional geometry. The thickness was designed to be constant in terms of general application; it was the same dimension as that of the guide vane in this study. Besides the main variables (⁠$lr$⁠, $la$⁠, $δ$⁠), the parameters that can affect the function of the ASF may be the number of fins and the tangential angle (⁠$θASF$⁠); these may be addressed in further studies.

Most of all, the experimental process and facility of this study fully complied with the international standard, ANSI/AMCA 210-07 [34]. Figure 4(a) shows a photograph of each axial fan connected to the test facility: the case of “none” represents a typical assembly as shown in Fig. 2; in the case of “ASF attached,” only the ivory-colored duct was replaced with the ASF. The rotor (including blades and hub cap) was manufactured using five-axis machining technology. The 0.5 hp (0.373 kW/h) class motor was selected with three-phase electric conversion and four poles and was placed inside the hub duct.

Regarding further details on the experimental test, the outlet chamber setup was adopted as shown in Fig. 4(b). A straight duct that has twice the axial length for the fan diameter was connected between the fan outlet and chamber inlet. The flow settling means had secured the proper porosity [35]. Relative humidity, barometric pressure, and dry-bulb temperature were measured to calculate the density; here, the density and rotational speed were respectively converted as the same values with the numerical setup. The flowrate was adjusted with nozzles and was calculated from the differential pressure (⁠$ΔPs$⁠). The flowrate range that could not be measured with a combination of nozzles was handled with a servoblower. The pressure and rotational speed were measured with pressure manometers and a laser tachometer (or a stroboscope). Actually, all measuring instruments in this test facility have been subject to annual certification so that the uncertainty for pressure manometers, stroboscope, and dry-bulb temperature detectors was 0.001–0.005 kPa for the range of 0–1.33 kPa, 0.1–1 rpm for the range of 40–35,000 rpm, and 0.07 °C for the range of 0–60 °C, respectively. The data acquisition system was incorporated with an averaging function for all outputs (measured data) for 6 s. The average measurements during the test for dry-bulb temperature at the front of the nozzles inside the chamber, dry-bulb temperature at the rear of the nozzles inside the chamber, dry-bulb temperature in the laboratory, barometric pressure in the laboratory, and relative humidity in the laboratory were 24.24 °C, 24.08 °C, 24.92 °C, 101.1 kPa, and 46.01%, respectively.

where $t$⁠, $U$ (could be substituted as $V$ or $W$⁠), $x$ (could be substituted as $y$ or $z$⁠), and $Fi$ denote the time, velocity, coordinate, and body force, respectively, and the terms in square brackets denote the viscous stress tensor (⁠$τij$⁠). Here, there was no change in density over time; the maximum Mach number at the blade tip was estimated as 0.09 at 25 °C (subsonic flow; Mach number $<$ 0.3). Meanwhile, a high-resolution discretization method was adopted based on the second-order approximation, and the root-mean-square (RMS) residuals were kept within $1.0×10−4$ and $1.0×10−5$⁠.

The $k$-$ω$-based shear stress transport standard (SST Std) model is known to be quite suitable for rotating machinery: it had been developed to provide accurate predictions in adverse pressure gradients, especially for onset and amount of the flow separation; however, it included transport effects to the eddy-viscosity formulation [36], and flow separation from smooth surfaces could be exaggerated under the influence of adverse pressure gradients. To enhance turbulence levels in the separating shear layers emanating from walls, a modified SST Std model was suggested and was referred to as “shear stress transport reattachment modification (SST RM)” [37–39]. The SST RM model considered an additional source term for $k$-equation to secure the ratio of turbulence production to dissipation, which might be greatly exceeded in large flow separation; it means that the SST RM model is more suitable when focusing on the range of low flow rates as in this study. Meanwhile, the SST Std and RM models had little effect on weak separation such near the design flowrate (0.8–1.2$Φd$⁠) [40]. As further presented in Sec. 4.2, the SST RM model predicted relatively accurate performance curves at 0.5–1.2$Φd$⁠, including the stalling flow rates. Obviously, there was a deviation in the stalling flow rates; it was not enough to be of concern, while the numerical accuracy may be enhanced with an adjusted ratio (turbulence production to dissipation).

where $U$⁠, $V$⁠, $W$⁠, $U¯$⁠, $V¯$⁠, $W¯$⁠, and $k$ denote the instantaneous and averaged velocity of $x$⁠, $y$⁠, $z$-direction in the orthogonal coordinate system, and turbulence kinetic energy, respectively. According to a general recommendation, it was appropriate that the level of turbulence intensity was selected as “medium,” which corresponds to the range of 1–5%; this range is recommended for flow in not-so-complex devices such as large pipes, fans, wind tunnels, or ventilation flows [34].

Figure 5(a) depicts the whole flow passage: the inlet passage included the ASFs and was sufficiently extended to account for rotating and recirculating flow under the stalling condition; the rotating passage included the blades, and the counter-rotating condition was given to the shroud wall; the outlet passage included the guide vanes. Here, the bell mouse and hub cap could not be considered because their effects were insignificant compared to the straightly extended passage [41]. The atmospheric pressure and mass flowrate were given to the inlet and outlet boundary, respectively, and the working fluid was air at 25 °C. The wall function was selected as automatic, and the boundary walls were treated as smooth and nonslip conditions. The stage (mixing-plane) method with the physical time-step (0.001 s) was applied to each interface of the steady-state simulation, while transient rotor–stator method was given to the transient one. The total time for one revolution was approximately 0.0408 s, and transient data were obtained for each time-step (every 3 deg) with a maximum loop of 5; i.e., 5 iterations per time-step.

where $L$ and $A2$ denote shaft power and fan outlet area (⁠$πrs2−πrh2$⁠). The evaluated values above (0.000297 and 0.000006) were considerably lower than the self-proposed criteria [43]; the numerical results were hardly affected with the N1 set, and the grid system was applied with the same topology corresponding to N1 set.

A commercial software, ansyscfx 19.1, was basically used for simulations. minitab 17 (Minitab Inc., Pennsylvania, PA) and origin 2019b (OriginLab Corp., Northampton, MA) were utilized to extract the optimized point with the RSM and perform the FFT from the fluctuation signal. The workstation has the specifications as follows: Intel^® Xeon^® CPU E5-2680 v2; clocked at 2.80 GHz with dual processor; random access memory with 80 GB; 64-bit operating system; parallel computations. The computational time for a set of steady-state and a revolution of transient simulations was approximately 26 and 30 h, respectively.

where subscripts $x1$ and $x2$ denote the multiple of normalized flowrate based on the design flowrate; according to the descriptions below, $x1$ and $x2$ would be designated as 0.7 and 0.8, respectively.

As the first step for designing ASF, the low-high levels, which were derived from a feasibility study, were assigned to the reference set (center; see Table 2). The total run time (number of sets) based on the $2k$ full factorial design method was 8. Including the set of none and reference, each set was subjected to the steady-state simulations in the flowrate range, 0.5–1.0$Φd$⁠. Table 3 lists the dimensions of the main variables for each set, total pressure rise at each flowrate point, and the values of the objective function: 0 and n in the “-” column mean the set of center and none; the objective function should be recognized as negatively maximizing the gradient in 0.7–0.8$Φd$⁠. From the sensitivity analysis (Fig. 6(a)) for the total pressure rise, it was difficult to confirm the effect of the main variables in 0.8–1.0$Φd$⁠; this suggests that ASF had little effect near the design flowrate where backflow or stall is not generally expected; at 0.8, 0.9, and 1.0$Φd$⁠, each mean of pressure coefficient decreased only by $−$1.25, $−$0.20, and $−$0.08% compared to the set of none. However, at 0.7$Φd$⁠, the higher rise was obtained with the longer radial length and the narrower axial gap; the ASF played some role in 0.7–0.8$Φd$⁠; the axial gap was the most sensitive. In 0.5–0.6$Φd$⁠, the radial and axial length showed more sensitive effects.

where “∼” imply normalization to $D2$⁠, and a second-order regression came from the RSM for $2k$ sets including the center point; the center point implies that the response surface can be curved (nonlinear) rather than linear, and in general RSM, it is preferable to examine second-order regression rather than first-order one, which can reflect more detailed interactions. Here, the optimized dimensions for $lr̃$⁠, $lã$⁠, and $δ̃$ were 0.035, 0.15, and 0.01, respectively, and the gradient (⁠$a$⁠; objective function) was estimated as $−$0.442 with $R2=$ 99.34.

The optimized ASF was attached to the axial fan and was compared with the case of none. Figure 7(a) shows $Q$-$P$ (or $Φ$–$Ψ$⁠) curves for each case of none and ASF attached; the transient data represent the average (solid-square symbol) and maximum-minimum value (bar symbol) of time-traced total pressure fluctuations in the rotor's last revolution (120 data per set) exhibiting an adequately repeated pattern, and the description would be included in Sec. 4.4. In the case of ASF attached, the positive slope contained in the case of none was completely reversed to become negative. The ASF-attached axial fan stably recovered pressure degradation in the stalling flow rates and allowed to form a negative slope to 0.5$Φd$⁠. The gradient in 0.7–0.8$Φd$⁠, which was selected as an objective function, was $−$0.428, and a deviation from the estimated value was only 0.014. Moreover, the ASF had little effect near the design flowrate. Here, transmitted data on each interface could be more detailed with the transient simulation because the time-step was more dense (see Sec. 3.2.3); this can be a main cause for the small deviation between the steady-state and transient results. Figure 7(b) depicts the shaft power coefficient (⁠$λ$⁠) in the same flowrate range. For each shaft power (⁠$L$⁠) of experimental and numerical data, it was substituted as $3$VIcos$θ$ and $Tω$⁠, respectively. Accordingly, the experimental data had to take motor efficiency into account. Here, motor efficiency might not be constant depending on the load (flowrate or rotational speed), and this fluctuation is generally not easy to be guaranteed as the motor was small size. Nevertheless, the overall tendency seemed to be almost the same as Fig. 7(a). It could additionally be considered that the motor efficiency was approximately 71% (datasheet by HIGEN Motors Inc.) in this experimental condition. Table 4 lists the increase and decrease in the total pressure rise, shaft power, and total efficiency that could be obtained from the ASF. The increase in pressure could compensate for the increase in shaft power in the stalling flow rates, sufficiently.

Obviously, the ASF caused a slight degradation in 0.8–1.0$Φd$⁠, and it could be more stressed at 0.8$Φd$ from the CFD results, i.e., the ASF might be detrimental to performance near the peak point where the positive slope began to be formed in the case of none; since the peak point implies the minimum limit for the stable operation (or incidence angle) of an axial fan, it seemed reasonable that the performance was affected by ASF near this point. Nevertheless, it was not at a level of concern compared to the huge earnings on stall margin. The degradation in 0.9–1.0$Φd$ seemed to be the blockage effect concerning the thickness of ASF, and it may be advantageous to design the thickness as thin as possible to minimize the blockage effect.

Meanwhile, especially for the case of none, which obtained the pressure fluctuations with the positive slope, the expanded uncertainty for measured pressure during the experimental tests was the greatest at 0.3$Φd$ for the whole flowrate range, but it was only ±0.55%. In the stalling flow rates (0.4, 0.5, 0.6, and 0.7$Φd$⁠), it was ±0.43, 0.24, 0.24 and 0.20%, respectively. Here, the evaluation of extended uncertainty was performed in a previous study [33].

First, Fig. 8 depicts circumferentially averaged turbulence kinetic energy (⁠$k$⁠) contours and circumferentially projected three-dimensional streamlines on the meridional plane: the $x$-axis represents the axial coordinate normalized to the fan diameter (⁠$D2$⁠); the $y$-axis represents the normalized span (⁠$r*$⁠); the white-dotted perpendicular line near the blade LE indicates a mark for descriptions below. In the case of none, a backflow region was identified near the shroud in 0.5–0.7$Φd$⁠, where the positive slope was formed, and it gradually increased in the spanwise direction toward the hub and the streamwise direction to the inlet passage (upstream) as the flowrate decreased. On the other hand, in the case of ASF attached, most of the backflow was suppressed. Here, the maximum length of backflow for each direction was quantified as presented in Fig. 9(a). At 0.5, 0.6, and 0.7$Φd$⁠, the axial length (⁠$ζa$⁠) decreased by $−$72.8, $−$69.4, and $−$58.7% with ASF, and the radial length (⁠$ζr$⁠) decreased by $−$62.9, $−$75.7, and $−$73.8% with ASF, respectively. Figure 9(b) shows the maximum length of backflow only for the radial direction under the concept of normalized span. Meanwhile, in Figs. 9(a) and 9(b), it was confirmed that the ASF caused some backflow at 0.8$Φd$⁠; this could be the reason why the case of ASF attached showed a lower pressure rise at 0.8$Φd$ than that of none.

The stalling and antistalling mechanisms were visualized with Fig. 10; a single particle was seeded near the shroud to track over the whole flow passage. Under the stalling condition, the particle entering the blade (2) was distorted while receiving the same directionality as the blade's rotational direction. In short course, it entered the blade at a span much lower than the shroud; this was because of the occupancy of backflow, which was shown in Fig. 8(a) and would be described later. The particle could not face downstream and headed to the inlet passage as a backflow (3). Subsequently, it showed circling movement along the shroud wall (4); at this time, the circling direction was the same as the blade's rotational direction. The circling particle tried to enter the blade again; however, it became a backflow (5) as in pattern (3). The particle repeated circling movement along the shroud wall (6–7). The ASF blocked this unfavorable flow pattern. The particle entering the blade could be straight without distortion, and the backflow lost its circumferential velocity component, close to zero. The ASF caught the circling movement of backflow and forced it back into the blade.

The flow pattern near the blade LE was further observed from the transient analysis in Figs. 11 and 12: the sectional plane is located at the coordinates corresponding to the white-dotted perpendicular line in Fig. 8; inside each box for a time-step, the left and right one show the circumferential and axial velocity contour; the streamlines are the same for both left and right one; the left one includes the inlet passage with transparency, the blades (black), and a vortex identification method with isosurface (⁠$Q$-criterion) [45]; the right one contains only an arrow pointing to the center of a certain vortex core identified from the streamlines; the circumferential velocity contour indicates a higher value as it becomes stronger against the blade's rotational direction, and the axial velocity contour indicates a higher value as the component toward the downstream becomes stronger. First, in the case of none at 0.7$Φd$⁠, the backflow and rotating stall did not fully occupy the shroud wall in the flow passage. There were two cores of rotating stall in accordance with the blade's rotational direction. As expected from Fig. 7, backflow and rotating stall near the blade LE appeared to be in the quickening period at 0.7$Φd$⁠. At 0.6$Φd$⁠, backflow and rotating stall entered the growing period. It was confirmed that two cores were strongly developed to be the farthest from each other, and another two cores repeated formation and extinction with weak intensity, i.e., stall cores were rotating with an annular symmetry (every 90 deg); however, the intensity and rotating speed seemed uneven. At 0.5$Φd$⁠, it could be understood as the setting period. Although the backflow was distributed thicker from the shroud wall, the imaginary line connecting the center of stall core was nearly pentagonal. Moreover, the rotating speed seemed to be uniform. As a qualitative analysis, the rotating speed for each type of rotating stall was approximately 1/3 or less to the blade's rotational speed, and it gradually increased as the flowrate decreased, i.e., at 0.7, 0.6, and 0.5$Φd$⁠, 85/360 $≈$ 0.23, 103/360 $≈$ 0.28, and 112.5/360 $≈$ 0.31, respectively. Meanwhile, all the stall cores mentioned above had their swirling directionality even while rotating in the flow passage, which was counterclockwise based on the captured figure. On the other hand, in the case of ASF attached, only small eddies due to the filtered flow were confirmed on each pressure side of the ASF. These eddies became progressively active at a lower flowrate; however, the ASF still controlled them. The directionality inside each eddy was quite complicated, and it did not rotate along the circumferential direction. The ASF remarkably stabilized the velocity contour near the blade LE, which was distorted with the backflow and rotating stall in the case of none.

The fluctuations with respect to the pressure rise (differential pressure between the inlet and outlet) in Fig. 7(a) were further indicated in Fig. 13; the $y$-axis represents the difference of maximum-minimum level (⁠$ΔΨ*$⁠, bar symbol in Fig. 7(a)). In the case of none, the pressure fluctuation increased sharply in the flowrate range lower than 0.8$Φd$ (0.5–0.7$Φd$⁠). However, it was slightly decreased at 0.5$Φd$⁠: at the flowrate point where backflow would be actively developed, the magnitude of pressure fluctuation could be subsided [46]; the tendency in Fig. 11 could be referred in this case. On the other hand, the pressure fluctuations were thoroughly suppressed with the ASF. The ASF eventually secured stability in the whole flow passage of the axial fan.

The stability near the blade LE was evaluated from the FFT results performed with total pressure fluctuation at the four monitoring points (P1–4) in Fig. 5(a): the points were located every 90 deg slightly inward from the shroud wall (0.001$D2$ far), and the axial coordinates were the same as the white-dotted perpendicular line in Fig. 8; the data extracted here is time-traced total pressure during approximately one revolution of rotating stall's core, here, since stalls are generally unstable factors, the characteristics of each revolution may be different, but data from one revolution was adequate to detect stalls; although the static wall pressure can evaluate the stability of the internal flow, it is considered that the characteristics of the rotating stall cannot be clearly captured because it does not include dynamic pressure. Figure 14 contains the FFT results at each monitoring point. Since the blade passing frequency (BPF) of this axial fan is 245 Hz, it could be confirmed that the ASF completely suppressed instability in the stalling flow rates (0.5–0.7$Φd$⁠). Even the small eddies in Fig. 12 could not contain any meaningful peaks, and this could be additional proof that the eddies were not rotating. On the other hand, the case of none included quite complicated peaks at lower frequencies than the first BPF. In particular, based on the qualitative analysis of the rotating speed and the number of cores for a rotating stall, its passing frequencies could be inferred as 0.02, 0.11, and 0.15$fn$ at 0.7, 0.6, and 0.5$Φd$⁠, respectively, where $fn$ denote the BPF. This inference was based on the following two factors: first, the rotating speed was uniform for each core at 0.7, 0.6, and 0.5$Φd$⁠; second, the number of cores was 1, 4, and 5 at 0.7, 0.6, and 0.5$Φd$⁠, respectively; e.g., at 0.5$Φd$⁠, the stall core's rotating speed was estimated as 112.5/360 $≈$ 0.31 times to the blade's rotational speed, then the stall core's passing frequency could be [5$×$(1470 $×$ 0.31)]/60 $≈$ 37.98 Hz, thus 0.15$fn$ (37.98/245) was indicated in Fig. 14(a). Here, the number of cores at 0.7$Φd$ (Fig. 11(a)) appeared to be two; however, it was not annularly symmetric and was adjacent to each other. Thus, it should be understood as a combined component in terms of the frequency domain. Accordingly, as shown in Fig. 14(a), the unfavorable peaks appeared at 0.02, 0.07–0.12, and 0.15$fn$ for 0.7, 0.6, and 0.5$Φd$⁠, respectively. It can prove that these peaks were directly originated with the rotating stall. The reason that the peaks at 0.6$Φd$ were confirmed at rather sporadic frequencies for each monitoring point is because each core's rotating speed was not uniformly maintained. At 0.5$Φd$⁠, the pattern for each monitoring point was almost identical, meaning that the rotating stall had a fairly uniform characteristic. Meanwhile, the magnitude of fluctuations was the highest at 0.7$Φd$⁠, and those of 0.5$Φd$ and 0.6$Φd$ were similar.

The ASF was designed and optimized using the $2k$ full factorial design and response surface method. In terms of the objective function, the axial gap was the most sensitive variable and should be less than 4% of the fan diameter to avoid a positive slope. The radial and axial length could be designed at least 1 and 5% to the fan diameter. The optimized dimensions for radial length, axial length, and axial gap were 3.5, 15, and 1%, respectively, based on the fan diameter. An ASF-attached axial fan stably recovered performance degradation in the stalling flow rates and allowed to form a negative slope to 0.5$Φd$⁠. From the transient simulations, the total pressure in 0.5–0.7$Φd$ increased by +48.2, +30.0, and +10.4%, respectively. Moreover, the ASF had little effect on the performance near the design flowrate. In the case of an axial fan without the ASF, the backflow gradually increased in the spanwise direction toward the hub and the streamwise direction to the inlet passage (upstream), as the flowrate decreased. The rotating stall was developed through the process of quickening-growing-setting period and was rotating faster, as the flowrate decreased. The ASF blocked the circling backflow near the blade LE and forced it back into the blade. The backflow lost its circumferential velocity component, close to zero. The flow entering the blade could be straight without distortion. The rotating stall could not be formed with the ASF. The ASF resolved the instability inside the flow passage, which was contained in the stalling flow rates. In 0.5–0.7$Φd$⁠, the total pressure fluctuation was reduced by $−$92.6, $−$95.6, and $−$67.2%, respectively. It did not allow instability even near the blade LE.

As an additional consideration, the optimized dimensions for each variable were limited to each low-high level. From the results of this study, if the variables were considered at a wider low-high level, the radial and axial length could be longer than 3.5 and 15%, and the axial gap could be narrower than 1% to the fan diameter, respectively. However, the sufficiently secured radial and axial length had little effect on the objective function. From a practical point of view, the authors tried to avoid presenting results for the range where the effect of radial and axial length on the objective function was insignificant. Since the axial gap could not be zero, 1% to the fan diameter was selected as the low level, which could be an intuitive value with a nonzero. The authors expect the selection for each low-high level to be understood as an implicit constraint.

The antistalling method of this study was recognized for its originality from a patent application (1071231, 10-2022-0099333) and the patent contained most of the results.

• Korea Institute of Energy Technology Evaluation and Planning (KETEP), Ministry of Trade, Industry and Energy (MOTIE) (Award No. 2021202080026D; Funder ID: 10.13039/501100003052).

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The data that support the findings of this study are available from the first and corresponding author, Y.-I.K. and Y.-S.C., upon reasonable request.

$a$ =

objective function (gradient on $Q$-$P$ curve)

$A2$ =

fan outlet area (⁠$πrs2−πrh2$⁠), $m2$

ANSI =

american national standards institute

ASF =

anti-stall fin

BPF =

blade passing frequency, 245 hertz

$c$ =

absolute velocity, $m/s$

$C$ =

blade chord length, m

CFD =

computational fluid dynamics

$D2$ =

fan diameter, m

Exp. =

experimental test

$Fi$ =

body force, $kg·m/s2$

$fn$ =

blade passing frequency, $Hz$

FFT =

fast Fourier transform

GCI =

grid convergence index

$i$ =

incidence angle, deg

I =

electric current, $A$

$k$ =

turbulence kinetic energy, $m2/s2$

$la$ =

axial length of ASF, $m$

$lr$ =

radial length of ASF, $m$

$lhexa$ =

maximum length of a hexahedral grid cell, $m$

$ltetra$ =

maximum length of a tetrahedral grid cell, $m$

$L$ =

shaft power, $W$

LE =

leading edge

$n$ =

number of grid cells

$N$ =

rotational speed, $rpm$

$Ns$ =

specific speed (type number)

NACA =

national advisory committee for aeronautics

$P$ =

total pressure, $Pa$

$Q$ =

volume flow rate, $m3/s$ or invariant of velocity gradient tensor, $1/s2$

$r$ =

fan radius, $m$

$r*$ =

hub-to-shroud normalized span

$R2$ =

coefficient of determination

RMS =

root-means-square

RSM =

response surface method

$S$ =

inlet blade-to-blade pitch, $m$

SST RM =

shear stress transport reattachment modification

SST Std =

shear stress transport standard

$T$ =

torque, $N·m$

$Tu$ =

turbulence intensity

$t$ =

time, $s$

TE =

trailing edge

$u$ (⁠$vθ$⁠) =

circumferential velocity, $m/s$

$U$ (could be substituted

as $V$ or $W$⁠) =

velocity, $m/s$

V =

voltage, $V$

$va$ =

axial velocity, $m/s$

$vθ$ (⁠$u$⁠) =

circumferential velocity, $m/s$

$w$ =

relative velocity, $m/s$

$x$ (could be substituted as $y$ or $z$⁠) =

coordinate, $m$

$z$ =

axial coordinate, $m$

$αgrid$ =

grid ratio

$β$ =

flow angle, $deg$

$βASF$ =

axial directionality (angle) of ASF, $deg$

$ΔPs$ =

differential pressure for nozzles, $Pa$

$ΔΨ*$ =

difference of maximum-minimum level for pressure fluctuations

$δ$ =

axial gap of ASF, $m$

$δI←LE$ =

axial distance between upstream interface and blade LE, $m$

$δTE→I$ =

axial distance between blade TE and downstream interface, $m$

$δt$ =

tip clearance, $m$

$ζa$ =

axial length of backflow, $m$

$ζr$ =

radial length of backflow, $m$

$θASF$ =

tangential directionality (angle) of ASF, $deg$

$λ$ =

shaft power coefficient

$ρ$ =

air density, $kg/m3$

$τij$ =

viscous stress tensor, $N/m2$

$Φ$ =

flow coefficient

$Ψ$ =

pressure coefficient

$ω$ =

angular velocity, $rad/s$ or turbulence eddy frequency, $1/s=Hz$

subscript $b$ =

blade

subscript $h$ =

hub

subscript $m$ =

meridional component

subscript $s$ =

shroud or static

subscript $x1$ =

multiple of normalized flow rate based on the design flow rate, 0.7

subscript $x2$ =

multiple of normalized flow rate based on the design flow rate, 0.8

subscript 1 =

blade inlet

subscript 2 =

blade outlet

where $ltetra$ and $lhexa$ denote the maximum length that one tetrahedral and hexahedral cell can have in each direction (circumferential, streamwise, or spanwise). The grid system was reviewed with a total of nine sets from G1-I1 to G7-I6: the review was performed at the design flowrate for the case of none; the abbreviation G and I denote grid and interface, respectively; the apostrophe implies it had only adjusted in spanwise direction; the axial distance for upstream interface-blade LE (⁠$δI←LE$⁠) and blade TE-downstream interface (⁠$δTE→I$⁠) is listed in Table 5; the total pressure rise was normalized based on G7-I6. From the results, G7-I6 was predicted as the highest rise among the reviewed grid systems, which showed a deviation of only +0.47% compared to the experimental test. At the interface, the grid ratio in each direction (circumferential, streamwise, and spanwise) appeared to have a minimal effect when it was within approximately 0.8 (1:5 based on the number of cells, $n$⁠): this was consistent with a general recommendation [37]; the grid ratios outside the recommendation led to low predictions. For example, the increase from G2 s to G3 s and from G5-I4 to G6-I5 could be obtained because $δI←LE$ in shroud and hub were adjusted to improve the grid ratio in each streamwise and circumferential direction. On the other hand, the increase from G3 s to G4-I3 seemed to be due to the adjustment of $δTE→I$ in shroud; however, it could not be analyzed as a specific reason because Fig. 15 contained interactions between the grid size and axial coordinates of the interface. Among the sets that maintained the proper grid ratio, the interface condition that allowed the ASF to be attached in the inlet passage was I5, thus the interface corresponding to I5 was combined with the grid system, satisfying the grid ratio within 0.8. Meanwhile, the results in Fig. 15 are just a process and reference, not absolute.