Large eddy simulation on the vortex evolution in a squirrel-cage fan based on a slice computational model

ABSTRACT To investigate the hard-to-measure complex flow in a squirrel-cage fan, a slice model was established in the present study to perform a partial large eddy simulation calculations on an r-θ sectional flow field instead of the large-scale and time-consuming full three-dimensional flow calculations. The method used to build the slice model was first introduced, and then the model was verified by the results obtained by other models and flow measurements. It is shown that to calculated the correct flow field and detailed vortex evolution, the Courant number of the slice model should be below 1, and the model thickness should be at least 2.4% of the impeller outer diameter. Through this simplified model, vortex evolutions rarely observed in other studies were successfully captured, such as vortex shedding caused by leading-edge separation, periodic backflow in the blade cascade, and vortex cluster traveling across the impeller. The flow separations were also illustrated on the inner surface of the volute tongue, accompanied by tiny and high-frequency eddies. The evolution process and frequency characteristics of these flow patterns were analyzed in detail, providing references for the design and noise reduction of this type of fan.

Impeller inner diameter D 2 Impeller outer diameter f 0 Fundamental rotating frequency f 1, f 2 Peak frequencies below 100Hz f b Blade passing frequency f s Passage seperation frequency b Impeller thickness b Thickness of the slice model b Dimensionless thickness of the slice model n Rotation speed Q Q-criterion vorticity Q Volume flow rate Q Volume flow rate of the slice model t Time t 0 Impeller rotation period t b Blade period u Airflow velocity u 2 Blade tip speed

Introduction
As a widely applied and typical rotating machine, the internal flow field of a squirrel-cage fan contains various complex flow structures like separation, secondary flow, and backflows (Eck, 1973). The relatively special geometrical features of the squirrel-cage fans, such as the forward-curved blades and the crowed and short blade passages, make their internal flow more complicated than the other centrifugal fans (Kind & Tobin, 1990). Over the past century, researchers and engineers have made countless efforts (Gholamian et al., 2013;Kim et al., 2015;Xiao et al., 2019;Yang et al., 2020) to make these fans operate more powerfully and quietly, and a consensus was formed that an in-depth understanding and exploration of fan transient internal flow is the essential prerequisite for a further leap in aerodynamic and noise performance. Raj and Swim (1981) were the first to measure the passage flow field in a squirrel-cage rotor. Their results showed that the mismatch between the intake airflow and the blade angle results in a large, permanent separation region on the blade SS. The corresponding 'jet-wake' at the exit of the blade passages was captured by Akbari et al. (2012) using a particle image velocimetry (PIV) method. Existing measured results also reported a blocking area at the rotor shroud, about front 1/4 of the blade is not effectively working (Raj & Swim, 1981), and the experimental results from Yamazaki and Satoh (1986) and He and Sato (2001) show that the valid width of the rotor only accounts for about 2/3 of its total width. This axial non-uniformity of the rotor outflow results in varying separation states on different blade span sections (Kadota et al., 1994) and causes secondary flow in the fan volute (Kitadume et al., 2007). In addition, the internal flow state of the fan under different operating conditions is quite different. The measured flow field illustrated that the flow in the blade passages passing the volute tongue is relatively low (Adachi et al., 2004) and even turns reverse when the fan flow rate is considerably reduced (Kadota et al., 1994). This reverse airflow passes through the blade passages upstream of the volute tongue and then re-passes the rotor blades downstream, creating a cross-impeller flow (Kind & Tobin, 1990;Gyeong et al., 2005).
With the development of computational fluid dynamics (CFD), flow measurement is no longer the only method to explore the complex fan internal flow. The three-dimensional (3D) viscous flow of the different turbomachines, such as fans (Meyer et al., 2021;Zhou et al., 2021), pumps (Shadab et al., 2022), and compressors (Li et al., 2019) could be solved easily by solving Reynold averaged Naiver-Stokes (RANS) equations. Through CFD, flow states that were difficult to measure or were overlooked in the past can be examined in detail. Jiang et al. (2017) successfully determined the relationship between the volute tongue backflow and nozzle reverse blow. Wang et al. (2018) calculated the flow field in a squirrel-cage fan and detailed analyzed the axial and circumferential flow non-uniformity. Since the effects of structural changes on the fan performance can be quickly and accurately predicted, CFD is widely used in designing different fan components, such as inlet nozzle size (Gholamian et al., 2013), rotor width (Jung & Baek, 2010), and the volute profile (Samarbakhsh & Alinejad, 2011;Wen et al., 2013). Transient simulations were also reported to gain a deep insight into the turbulence mechanism and evolving transient vortexes. Younsi et al. (2007) studied the pressure pulsation on the volute through unsteady calculations of two-dimensional (2D) and 3D models. Velarde-Suárez et al. (2009) performed URANS simulations to determine the pressure fluctuations inside the fan volute. Fernandez Oro et al. (2013) performed unsteady calculations and comprehensively assessed the impeller-tongue interactions under different working conditions. The flow field obtained by unsteady simulation contains more transient details than that obtained by steady simulation, but the corresponding solution time and calculation cost also increase exponentially.
To obtain a more accurate and detailed flow field, large eddy simulations (LES) were gradually adopted for refined analysis of the internal flow of various rotating machinery (Kye et al., 2018;Lim et al., 2020) in recent years. The main idea of LES is to separate the large-scale turbulence fluctuations from the small-scale turbulence fluctuations, so as to capture as much turbulence information as possible compared with the RANS method (Liu et al., 2018;Su et al., 2012). However, the LES calculation requires a more refined grid model, making the calculation very expensive and time-consuming (Yao et al., 2016;Zhou et al., 2020). Pogorelov et al. (2015) conducted an LES analysis on the development of the vortical flow structures in the tip-gap region of an axial-flow fan. The grid quantity of the single-passage model even reached 1 billion. Iwase et al. (2020) investigated the effect of grid resolution on the flow field and noise prediction of a centrifugal fan and suggested a model with 500 million girds. Such large-scale calculations are unaffordable and relatively inefficient for the engineers and developers of average companies. Therefore, there is a need for a simplified simulation method that allows the capture of the instantaneous flow field information of interest to fan designers with a relatively acceptable calculation scale.
In the present study, the 3D flow field was simplified to a slice model through periodic boundaries, and  the transient flow on the r-θ cross-section was then calculated by LES. The effects of grid resolution, timestep size, and model thickness on the results were analyzed in detail. The flow field and aerodynamic performance were also compared with the simulated results by 3D RANS model and valid by test results. Through this simulation method, the separation, backflow, and instantaneous flow characteristics that have not been measured or even considered were successfully obtained; the corresponding formation mechanism and separation characteristics were then analyzed in detail.

Test fan and the range hood
The object of this study was a small double-suction squirrel-cage fan used in a range hood. The internal structure of the entire hood, with the fan installed, is shown in Figure 1. There are baffles and a filter at the machine entrance to collect and filter the cooking lampblack, but these components were removed in both measurement and simulation to facilitate CFD modeling. The volume flow rate and rotation speed at the fan best efficiency point (BEP) are 13.42 m 3 ·min −1 and 1200 rpm, respectively, and the corresponding fan total pressure is nearly 300 Pa. The detailed geometry parameters are listed in Table 1.
In the previous study (Liu et al., 2021), a 3D grid model with almost 13 million elements was built for this fan (as shown in Figure 2), and the 3D flow was calculated by solving the RANS equations coupled with an SST k-ω turbulence model. Due to the simplification of the fan's intake chamber shape and the motor bracket, as well as the intrinsic error of both performance testing and CFD simulations, there was a certain difference between the CFD predicted performance and the measurement data. Despite these minor deviations, both the predicted pressure and fan efficiency show a good agreement with the test curves under most flow rate conditions, practical structural improvements were also successfully carried out. Therefore, this 3D grid model and the corresponding numerical setups are reliable.

Slice computational model
The radial velocity distribution along the axis under three working flow rates, representing how the mass flow passing the rotor distributes axially, is shown in Figure 3. Since a motor is installed at the rear inlet of the fan, the corresponding flow rate is significantly lower than that at the front, so the speed distribution at the front half of the impeller is the focus here. As previously mentioned by other researchers (Montazerion et al., 2000), the rapid turning of the airflow results in a low flow rate at the blade shroud. However, the average velocity at each section is relatively average between 0 and 60% span, which means that the volume flow rates in these sections are close. Based on this, the authors undertook to simplify the 3D model to a thin layer model across the fan r-θ section, which may improve the spatial resolution of the model when the grid number is equal, or speed up the calculation when the spatial resolution is the same.
The r-θ section perpendicular to the rotation axis, including all the key geometric shapes, such as blades and volute, was extracted in the middle of the test fan, and then stretched along the axial direction to a thickness of b' = 10 mm to form the so-called r-θ section slice model (slice model for short in the following studies). As shown in Figure 4, the slice model is also divided into three regions: inlet, impeller, and volute. The outlet of the volute area directly extends to form an outlet channel,  and the extended distance is consistent with the full 3D model. The width of the outlet channel is equal to the full 3D model outlet, the pipe diameter. The inlet area of the thin-layer model cannot be the same as that of the inlet chamber that reflects the real fan, and can only extend to the center of the slice model, taking the middle torus as the inlet boundary. Additionally, the front and back surfaces of the model are set as periodic boundaries, and the sliding grid model is used in the impeller domain for transient calculation.
Specifying the air velocity or flow rate in the inlet may lead to a quite different result in the impeller circumferential flow pattern from the actual intake condition. Therefore, the volume flow rate of the slice model was defined by an average velocity in the outlet, while the inlet of the zone adopts a pressure inlet with a total gauge pressure of 0 Pa. The volume flow rate of the slice model is calculated by the ratio between the thickness of the model and the actual width of the 3D rotor: (1) Dimensionless parameters, namely flow coefficient and pressure coefficient, were used as aerodynamic performance metrics of the test fan. Based on the fan flow coefficient at the best efficiency point, φ BEP , the CFD calculations were performed at the operating points of the fan flow rate from 0.4-1.9φ BEP .

Turbulence model
In this study, the LES method was used to calculate the transient flow field. The flow calculations were performed using the commercially available Fluent software (Ansys, Canonsburg, PA, USA). Using LES, the large and energetic scales of eddies were directly resolved but the small and dissipative scales of turbulence still require modeling. Therefore, the governing equations used for LES were obtained by filtering the time-dependent Navier-Stokes equations. The filtering process filters out the eddies whose scales are smaller than the specified filter width or grid spacing in the computations. Any flow variable ϕ could be filtered to large scaleφ and smaller part ϕ : The filtered form of the incompressible, unsteady continuity and transport equations for LES are: where u i is the flow velocity in the i direction (i,j = 1, 2, 3); p is pressure; t is time; ρ and ν are air density and kinematic viscosity, respectively. The τ S ij in equation (4), the subgrid-scale stress (SGS) term that represents the effects of small-scale vortices on the calculation of motion equations, is defined as: In order to close the control equations, mathematical modeling is required for this SGS tensor, which is the socall SGS model. In this study, a classical and widely used SGS model proposed by Smargorinsky (1963) was used, in which the SGS tensor is associated with the rate-ofstrain tensor for the resolved scale, S ij : where δ ij is the Kronecker delta function, andS ij is calculated by the gradient of the filtered velocity: Also, μ T is the SGS turbulent viscosity, which is modeled as: Lilly (1967) suggested the theoretical value of the isotropic turbulence with high Reynolds number as, C s = 0.18. However, C s varies when solving different fluid problems. In this study the default value for C s was set to 0.1.
Additionally, since a constant C s may cause a non-zero SGS tensor in the laminar region and near the walls, Moin and Kim (1982) introduced the Van Driest attenuation function when calculating the SGS turbulent viscosity, equation (8) is correspondingly modified as follows: where y + is the dimensionless height of the first grid layer; A + is a semi-empirical factor, taking A + = 25. The calculation medium is considered as ideal incompressible air. The control equations are discretized by the finite volume method and solved by velocity-pressure coupling. The discretization of the pressure term, convection term, and time term adopt the second-order scheme, the bounded central difference scheme, and the secondorder implicit scheme, respectively. Also, both the grid quantity and time-step configuration will affect the accuracy of the calculation, which will be discussed in detail in the following sections.

Grid independent
A sufficiently refined grid model is the key to the success of LES calculations. Piomelli and Balaras (2002) suggested that the node distance along different flow directions should satisfy: 50 ≤ x + ≤ 150, y + < 1, 15 ≤ z + ≤ 40. In the present study, special blocks were generated along the wall boundary to conveniently divide the boundary layer grids (As shown in Figure 5). Except the y + near the volute outlet is slightly higher than 1, the wall y + on most of the walls satisfies the Piomelli requirement. The average y + on the blade surface and the volute walls of the final mesh model are 0.12 and 0.77, respectively, which generally meets the requirements of y + .
However, the flow field in a fan is completely 3D and complex, the x + and z + are difficult to calculate. Therefore, in this study, the Grid Convergence Index (GCI) proposed by Roache (1994;1997) was used instead as the metric for the grid independence verification. This criterion required three sets of grid models, namely fine grid, medium grid, and coarse grid, based on the same topologies, which are denoted as model 1, 2, and 3 respectively (the detailed information are shown in Table 2). The grid feature scale is: where, V i is the volume of each single element, N is the total grid number. The grid scale of these grid models meets: h 1 < h 2 < h 3 , and the scale ratio between them The transcendental equation of the order p is iteratively solved as follows: where: ξ is the performance metric for assessing the convergence, which here are the total pressure and total pressure efficiency predicted by the LES method. Equations (11)-(13) were iteratively solved, initiated by q(p) = 0, and ended when the error of p reaches the specific threshold. The GCI of the three models were calculated as: GCI 32 = r p GCI 21 (15) where e 21 is the relative error of the convergence metrics between the refined and medium grids, which is defined as: Based on the premise that the height of the first grid layer meets the requirement of y+, three grid models with a grid quantity of 1.97, 4.5 and 10.98 million are generated. Using the same time-step t = 6.944×10−5 s for the LES calculation, the total pressure, total pressure efficiency of the fan, and the corresponding GCI and convergence index are obtained, as shown in Table 3. This illustrated that although the grid number of the coarse model is less than 20% that of the refined one, the difference in the performance of the wind turbines predicted by them is negligible. When the grid number reaches 10.98 million, the final errors of total pressure and efficiency are both below 0.5%, and the corresponding GCI is also less than 0.5%. Therefore, the calculation of the refined model can be considered as convergent. Figure 6 shows the comparison of the internal flow vorticity extracted from the three grid models. The vorticity is evaluated by Q-criterion (Hunt et al., 1988), which is common used to identify vortex structures in turbulence flow (Zhan et al., 2020;Zhao et al., 2017). The Q-criterion vorticity is calculated as: where: In the results of the coarse model, only one high vorticity region occupy the center of each blade channel and it is dragging from the blade leading edge (LE). However, how this high vorticity regions is generated cannot be understood due to the limitation set by the spatial resolution of the coarse model. The increase of the grid number revealed that the vortex in each blade channel is not one, but a series of vortexes shedding from the blade LE. The shape of the passage vortexes determined by the refined model is basically the same as that by the medium model, but the contour line seems clearer and smoother. Comprehensively weighing the calculation convergence, flow field resolution, and the calculation cost, the refined model with 10.98 million grids was ultimately selected for LES calculation. Figure 7 illustrates a cross-section of the final grid model. The size of the grid cells in the blade passage center is about 0.5 mm, and the circumferential resolution at the impeller outlet is about 0.26°.

Time discretization
In transient calculations, the time-step t will also affect the accuracy of both performance prediction and flow calculation. Thus, it needs to be carefully selected in combination with the grid complexity, spatial-temporal resolution requirements, and computer setup. Smaller time-steps are more conducive to obtaining the flow details but will lengthen the solution time, while too  coarse time-steps will produce larger truncation errors, which may lead to completely incorrect results. Darvish and Frank (2012; considered that a rotor rotation of 3.6°at each time-step is small enough to accurately predict the fan aerodynamic performance, and the flow field inside the fan can be accurately captured when the rotor rotates 1.5°at each time-step. The impeller parameters and the corresponding time-step of the unsteady calculations of squirrel-cage fans in recent studies are listed in Table 4. The number of time-steps ranges from 10 −4 -10 −5 s, and the rotation angle per time-step of most of the rotors blew 1°. In some studies, the time-step where the rotation pitch per step is less than 0.5°in order to capture the high frequency pressure pulsation to evaluate the fan noise level.
Base on the final gird model obtained in the previous section, LES calculations were performed at different time resolution: 2.777×10 −4 , 1.389×10 −4 , 6.944×10 −5 , and 3.472×10 −5 s, the results are shown as scheme A-D for convenient analysis. The fan total pressure and efficiency at high-flow condition calculated by these schemes are listed in Table 5, where C is the average Courant number of the rotor grid, and the 3D Courant number is defined as: It is shown that the Courant number exceeds 1 when the time resolution is coarse, and the corresponding fan total pressure and efficiency are over predicted. When the time-step reaches 6.944×10 −5 s, the performance prediction tends to converge. The Courant number values of schemes C and D are both below 1, and the calculation results are almost the same.
The velocity contours around the fan blades independently calculated by grids scheme A, B, and C are shown in Figure 8. Their velocity distribution is basically the same, except for a slight difference in the shape of the blade outflow. When the time-step is 2.777×10 −4 s, the jet wake character at the blade exit is successfully obtained but its shape is shapely and rigid, which is similar to the calculation result using the RANS equations. With the increase of the time resolution, the passage outflow starts to wave, which illustrates the interaction between the high-speed jet-flow and the low-speed wake.
The lump of Q-criterion vorticity magnitude in the blade passage calculated at different time-steps are shown in Figure 9. They reveal that when the time-step is as large as 2.777×10−4 s, the vorticity value of the blade passage flow field is relatively low, and there is only a single strip of vorticity bubble in the center of the passage. With the  shortening of the time-step, the instantaneous interaction between the separation zone and the main flow zone were solved accurately. The result revealed that the actual shape of this vorticity 'tail' is a series of vortices shed from the blade LE and gradually developed along the stream wise.
Considering the calculation convergence, accuracy and time-cost, the largest time-step satisfying the Courant number below 1, t = 6.944×10−5s is selected as the final configuration, which means that the rotor rotates 0.5°per time-step at the rotation speed of 1,200 rpm,

Model thickness
Different model thicknesses were calculated to verify their effects on performance prediction and flow field calculation. When keeping the total grid quantity unchange, the model thickness was reduced from 10 mm to 8, 6, 4, and 2 mm, respectively. Divided by the impeller outer diameter D 2 , the model thickness can be given in a dimensionless form:b = b /D 2 . The dimensionless thicknessesb of these five models are about 0.04, 0.032, 0.024, 0.016 and 0.008, respectively. The calculated fan pressure and efficiency are listed in Table 6. It is shown that both the calculated pressure and efficiency increase as the thickness decreases, which means an increasing error because the performance metrics predicted by the slice model are already over-predicted. However, these increments of predicted fan performance brought by the decreasing model thickness are negligible before the dimensionless thickness is below 0.024. Figure 10 shows the sectional flow field characteristics of schemes C1, C3, and C5. The sub-figures on the  left are the velocity contours, while the right shows the corresponding vorticity distribution. Consistent with the effect of model thickness on the predicted fan performance, the calculated sectional flow field changes tiny when the model thickness is reduced to 6 mm, and the flow difference appears when the model thickness is further reduced to 2 mm. The flow field of scheme C5 shows an abnormally thickened velocity boundary layer, and a series of vortices are rolled up on the volute wall. In addition, the passage vorticity calculated by the 2 mm model is also over predicted, which makes the vortex shape ambiguous compared with the results of the thicker schemes.
A too-thin slice model will cause over-prediction in the fan performance, while their too-close periodic boundary will lead to a deformed and wrong flow result. So the thickness of the slice model proposed in this study is recommended to be at least 2.4% of the impeller outer diameter. The following calculations were all carried out by a slice model with a thickness of 10 mm.

Fan pressure prediction
The flow field was initiated by zero-velocity and zero gauge pressure, and the calculations were performed for a total of 18 revolutions. The fan total pressure coefficient was calculated by subtracting the total pressure at the fan inlet from that at the outlet. The convergence curves of different working flow rate conditions are shown in Figure 11. The total pressure of the fan gradually increases in the initial stage, then enters a convergence state after 4 revolutions. The converged fan pressure is stable under the large-flow condition with a 130% BEP flow rate, but it shows a state of periodic fluctuation under BEP. This fluctuation is mainly caused by the unstable transient internal flow, and this state deteriorates and loses the fixed frequency as the flow rate continues to decrease. Due to these fluctuations, the calculation results need to be averaged along the time-step to obtain the exact fan performance. Averaging starts after revolution 4, and the dashed black line in each figure shows the change in averaged fan pressure over time. It is shown that the averaging results hardly change after the rotation period for averaging exceeds four revolutions, even in the case of the low-flow conditions. The time-averaged total pressure and total pressure efficiency during the 5th to 10th cycles are used for the final predicted fan performance. The performance results in the previous sections are all calculated by this averaging method.
The pressure coefficient predicted by the 3D model using the RANS method and slice model using LES are compared with the experimental data in Figure 12. The experiment platform and detailed test process can be found in the previous study (Liu et al., 2021); the uncertainty of the measured total pressure under most conditions was below 1%. The fan total pressure of the 3D model is obtained by subtracting the average pressure of the outlet and the average pressure of the inlet section of the intake chamber, which has been simplified in the slice model, so the predicted pressure of the slice model will obviously be higher than that of the 3D model and the experimental data. The changing trend of the total pressure with the flow rate of the two models is also different, especially at the low-flow condition. However, considering that many energy-consuming 3D flow structures, such as the axial vortex transmission in the blade passages and the secondary flow in the volute, are ignored  in the simplified model, it is not easy for the predicted performance of the two to achieve such agreements.

Comparison with RANS results
The comparison of the circumferential distribution of the radial velocity on the rotor inlet and outlet, obtained by the two simulation methods is shown in Figure 13. The Multi-Reference Frame (MRF) model is used in the 3D calculation, so the unevenness caused by the blade thickness is directly observed on the distribution curve. However, in the slice model simulation, the blades move during time-steps, so the velocity distribution at the same position changes with time, and the red line represents the time average velocity. Moreover, as both models are 3D, the velocity results are also averaged along the axis of the rotor.
Although the time-averaged velocity curve of the slice model is smooth, and completely different from the curve calculated by the 3D model using the RANS method, the distribution trend of the two is relatively similar. The u r is negative at around 60°and then quickly increases to a positive value, which means that backflow occurs in the blade passage when passing the volute tongue. It becomes negative for a second time at around 150°, mainly due to the special shape of the volute profile at such point. The region after 180°is the main flow area of the rotor, where the outflow is high speed and evenly distributed. The most significant difference between the two distribution curves lies near the volute tongue. The test fan contains an inclined volute tongue, but the 3D characteristics of the inclined volute tongue are simplified in the simplified model. Thus, the reflow region in the 3D results is wider than that of the slice model.

PIV measurement and validation
The flow field near the volute tongue was measured through a 2D-PIV method. The measurement system and the test photo are shown in Figure 14. The platform mainly consists of a double-cavity Nd-YAG laser, an synchronizer, and an sCMOS camera. Measuring windows were set at both the front and side of the machine by replacing part of the metal shells with transparent glasses. When measuring, a pair of lasers sheet is generated to illuminate the measure region through the side window, and the camera takes a pair of particle images through the front one simultaneously. The velocity in the measured region was obtained by calculating the displacement of the particle clusters in two images through a cross-correlation algorithm. The maximum velocity in the tongue region was expected to be about 30 m 3 /s, so the corresponding time interval between a pair of shots is 100 ns.
The test fan was adjusted to operate at BEP by outlet throttling, and seeding particles were released at the machine inlet after the fan operated stably. The seeding particles adapted in this measurement are DEHS with an average size of about 1μm, which follow the airflow well and own good reflectivity. The measurement started when the seeding particles and airflow were thoroughly mixed and lasted for 20 s. In each second, 15 instantaneous flow fields could be obtained, and the mean flow field was revealed by averaging these transient results.
As shown in Figure 14(c), the measure section locates at the 30% span of the front half impeller. The mean flow field calculated by the slice model is compared to the measured velocity field in Figure 15. It is shown that the simulated airflow is about 30% faster than the measured airflow. This deviation is mainly because the volute is wider than the impeller in the actual fan (as shown  in Table 1), but the thickness of the impeller and volute is equal in the present slice model. In the simulation, the impeller outflow will not diffuse and decelerate when entering the volute as in real, so the obtained velocity is higher than the measured results. Additionally, the calculation result by the slice model can be considered as an averaged r-θ sectional flow, which is also essentially different from the actual flow field. Coupled with the inevitable errors in the simulation and the experiment themself, such a significant difference between the measured and calculated velocity is still acceptable. After all, the calculated and the measured velocity field achieve good agreement on some key distribution characteristics, such as the gradual deceleration after the airflow leaving the impeller and the significant velocity decrease near the volute tongue.
In summary, the slice model is a reduced-order simplified model, and its calculation results of both the aerodynamic performance and fan flow field are inevitably different from that of the 3D numerical simulation. However, the slice model still predicts the performance difference of the fan under various working conditions and the circumferential distribution characteristics of the flow field around the rotor. Therefore, it is considered that the simulation results of the slice model can provide a reference for investigating the internal flow of squirrel cage fans.

LE separation vortex shedding
The velocity map and the vorticity distribution in the mid-span of the fan under 1.3φ BEP conditions has been shown in Figure 10(a). It is illustrated that the flow state in the blade passages differs along the circumferential direction. As the air taken in from below, the flow rate is relatively large when the blade passages rotate from 180°to the volute tongue, whereas the flow rate is low when passing through the upper side of the rotor. Correspondingly, there are also differences in the evolution pattern of passage vortexes under these two working conditions. A complete vortex shedding can be observed in the high-flow passages, while vortexes tear and merge in the blocked blade passages, showing a more complex movement pattern.
The movement patterns of passage vortexes when the blade channel passes through the high-flow region are clearly shown in Figure 16. The two representative blade passages from regions A and B selected to analyze are shown in Figure 16 (a). In Figure 11, the initial time T = 0.36 s, and the blade period t b is 8.33 × 10 −4 s. The vortex evolution pattern of the blade passage in region A, shown in Figure 16 (b), reveals that a vortex is shedding from the blade LE at the initial time. The head of this vortex moves downstream with the rotation, but its tail is still connected to the blade LE, and gradually deforms into a long strip shape. After rotation of about three t b , this vortex eventually detached from the blade LE, while a new vortex began to grow at the same place. The vortex evolution pattern of the blade passage in region B, displayed in Figure 16 (c), shows that since the blade passage flow here is higher than that in region A, the separation vortex shedding from the blade LE breaks into a series of continuous vortices. A tadpole-like vortex (vortex No.4) is about to fall off the blade LE at the initial time, and the vortexes 3, 2, and 1 that have fallen off before can be observed along the blade passage. Vortex No.4 gradually elongates and completely detaches from the blade LE after time t b , followed by the growth of a new vortex (No.5). About one more blade period later, the scale of vortex No.5 develops into a state similar to that of vortex No.4 at the initial time. Additionally, the results in Figure 16 also illustrates that the vortex evolution state in the following blade passage is also not consistent with that of the previous one. It even moves to the same position, which indicates that the shedding of the LE vortex is a relatively independent process from the rotation of either the rotor or blades.
Three monitoring points are set in the flow field, the locations of which are shown in Figure 17. M1 and M3 are static monitors, respectively located upstream and downstream of the blade cascade in the circumferential direction of 300°. M2 is a rotating monitor, the rotation speed of which is equal to that of the fan rotor, thus it is relatively stationary to the blades. Also, the rotating monitor M2 is located in the middle of the passage entrance, exactly in the shedding region of separation vortexes.
The dimensionless radial velocity pulsation at the above three monitoring points and the corresponding fast Fourier transform (FFT) spectrum, shown in Figure  18, reveal that the flow pattern at the upstream M1 point is relatively stable. The corresponding frequency spectrum has prominent peaks at a low-frequency 70 Hz, as well as the blade passing frequency (BPF) and its multiple-harmonics. In comparison, the velocity pulsation at the downstream M3 point is more severe, and the amplitude is significantly higher than that of M1 in the frequency band above 100 Hz, especially above 1kHz. The BPF is still the dominant frequency, but the 2nd harmonic is already submerged in the broadband noises. The pulsation curve of the radial velocity at the blade passage monitor point M2 shows significant rotating periodicity. The radial velocity climbs to a peak when passing the high-flow region, but falls to below zero when passing the volute tongue. Compared with the FFT spectrum of point M3, the pulsation amplitude of the low-frequency bands below 1kHz is significantly increased, and the peak frequency is replaced by the rotation frequency of the rotor and its multiples. In addition, a tiny peak is observed around 700 Hz. The shedding period of the LE separation vortexes observed in Figure 16 is approximate twice the blade period, which coincides with the peak frequency here.

Tongue backflow and cross-impeller vortex cluster
The curve of the change in total pressure with time under the BEP, showing an apparent periodic law is displayed in Figure 19 (a). The flow results at the valley-pressure and peak-pressure moments were selected for further comparison, and their pressure coefficient distributions on the mid-span section of the fan are illustrated in Figure  19 (b)-(c). In general, the pressure in the volute is higher than that in the inlet zone, and there is a stagnation point at the tip of the tongue, due to the highest pressure. At the peak moment, the pressure in the outlet pipeline and diffuser increases significantly, creating a reverse pressure gradient with the lower pressure region upstream of the impeller. This reverse gradient drives the airflow through the blade cascade, producing a backflow in the blade passages. The small windows show the corresponding radial velocity contours, and reveal that the reverse airflow dominates the blade passages near the volute tongue at the peak-pressure moment.
The radial velocity at different positions on the impeller inlet during the 5th-10th revolutions were calculated from the transient results. A negative radial velocity at the impeller inlet indicates that the corresponding passage has been penetrated by the backflow. Taking the time-step as the abscissa and the circumferential angle as the ordinate, the evolution pattern of the rotor backflow with time and space were obtained. The ordinate is split at 180°, in order to ensure the backflow regions' shape continuity, The results under φ BEP operating conditions, shown in Figure 20, indicate that the impeller backflow has occurred as early as around 240°. At the beginning of the backflow, the scope is tiny, and only occurs in 1-2 blade passages. However, the time and space range of the backflow region gradually expands when it approaches the volute tongue. In addition, the flow direction in the passages is restored to radial shortly after the volute tongue, but it soon begins to flow back again until around 180°.
The two backflow regions before and after the volute tongue are connected end to end, and the slope of their front edge is exactly the same. From the perspective of time, backflow regions with similar shape appear periodically along the time axis. There are at least two backflow regions that exist on the blade cascade at the same time. The evolution map of the backflow region of the impeller under 0.7φ BEP conditions is similar to that of BEP and is also composed of periodic back regions. Each backflow region is also split into two sub-regions by the volute tongue. The shape of each region is an inverted and inclined triangle, which represents the development process of the backflow. Compared with BEP results, each backflow region under the 0.7φ BEP condition is broader in both time and space dimensions, indicating that the tongue backflow under this working condition is significantly exacerbated. At the same time, the repetition frequency of the impeller backflow under the 0.7φ BEP condition is higher than that of the BEP, and the repeatability is also worse.
In order to further examine the flow state of the backflow after it returns to the upstream of the blade cascade, the 3D iso-surface lumps of Q-criterion vorticity in the inlet zone is extracted and analyzed under 0.7φ BEP condition. Figure 21 (a) shows the selected moments of 5 consecutive time-steps, and Figure 21 (b)-(f) shows the corresponding results. In order to facilitate the judgment of the relative position between the blade and the backflow vortexes, a blade was marked with a red dot. At t 1 , the blade passages on the left have already been penetrated before they approach the volute tongue, and a vortex cluster can be found at the blade inlet. Columnar vortices fell off from the LE of the leading blades, while some tiny vortices occupied the entrances of the following ones. With the impeller rotates, the columnar vortex begins to lag behind the blade and gradually moves apart from the cascade, while the tiny vortexes follow the rotation more closely. The changes in the relative positions of the two shedding vortexes seem to be self-rotating, the direction of which is the same as the impeller rotation. The entire backflow vortex cluster was significantly lagged behind the marked blade at t 4 and then quickly dissipated after passing through the volute tongue. The vortex cluster significantly dissipated at t 5 , while a new vortex cluster is forming on the left. Additionally, the vortex cluster at t 5 is similar to that at t 1 , which illustrates the periodic characteristic of the backflow evolution shown in Figure 20.
In summary, the backflow does not constantly occur in the blade passages near the volute tongue, nor does it occur completely fixed in specific blade passages, but periodically appears, moves, and disappears in different blade passages. The periodic backflow  correspondingly forms self-rotating vortex cluster traveling across the cascade at a slower speed than the impeller rotation.  Figure 22 shows the flow state near the volute tongue under the volume flow rate of 1.3φ BEP . It is illustrated that the momentum layers on both sides of the tongue are thickening, especially along the inner side closer to the impeller. The airflow is accelerated to a high-speed 'jet-like' state when passing through the small tongueimpeller clearance, and it is inevitable to separate from the volute wall that has started to expand and apart from the impeller. In the separation region, the volute wall close to the tongue is covered with a large number of small low-energy eddied.

Flow separation and pulsation characteristics at volute tongue
The power spectral density (PSD) values at monitor points T1-T3 on the volute tongue and T4 in the separation region were calculated using a reference pressure of 2×10 −5 Pa, and the results are shown in Figure 23. All the broadband amplitudes at points T1-T3 on the volute tongue peak at a low-frequency f 1 = 70 Hz, but their specific spectral distribution, especially the discrete characteristics at the BPF, is still different. T1 locates farthest from the blades among the three points, and there is no peak at the BPF. It shows a prominent BPF peak in the spectrum of point T2, and there is even a second harmonic in that of point T3. The radial position of point T4 is approximately equal to point T3, and their peak amplitude at the BPF and its second harmonic are also almost the same. However, due to the highfrequency eddies inside the volute tongue separation zone shown in Figure 22(b), the pulsation intensity of T4 in the range between 1-2 kHz is significantly enhanced, resulting in the peak at BPF being not conspicuous.
The flow state of tongue separation at several moments illustrated in Figure 20(a) under 0.7φ BEP conditions is shown in Figure 24. The reverse pressure gradient at the volute tongue is large under this working condition than that of the large-flow conditions, results in a increase of the clearance airflow. The velocity passing the clearance is correspondingly increased, the separation angle and scope on the inner surface of the tongue are also significantly enlarged. At t 4 , the tongue separation is fully developed, and both the scope of separation and wake regions reached the peak. However, the separation state at t 5 soon improved and returned to a similar state of t 1 , showing a similar periodic separation pattern at the volute tongue.
The pressure pulsation at the monitoring points T5-T7 downstream of the volute tongue under different working conditions, as well as the corresponding PSD distribution are shown in Figure 25. In the large-flow conditions of 1.3φ BEP , the flow state at each point is basically stable, and the pressure pulsation amplitude is relatively low. In this working condition, the PSD shows a significant peak at the BPF, as well as a lower peak at about f 1 = 70 Hz. When the flow rate is reduced to the φ BEP , the pulsation at each point starts to pulsate periodically with the entire fan flow field. At this time, the PSD of each point is significantly enhanced at all frequencies, and an obvious peak appears at about f 2 = 47 Hz instead. The PSD Figure 25. Pressure pulsation and corresponding PSD at monitoring points downstream the volute tongue under different working conditions. Pressure pulsation at (a) 1.3φ BEP ; (c) φ BEP and (e) 0.7φ BEP ; PSD at (b) 1.3φ BEP ; (d) φ BEP and (f) 0.7φ BEP amplitude below 1kHz of each monitor point continues to increase when the fan flow rate further decreased to 0.7φ BEP . However, as the backflow no longer has a fix time period, the PSD of the monitor point no longer significantly peaks at neither f 1 nor f 2 , and instead is replaced by a series of small peaks below 100 Hz.

Conclusion
A slice model was established for LES calculation in this study by extracting the geometric features of sections and using periodic boundary conditions. After evaluating the effect of time and spatial resolution on the simulation results, the numerical configuration of the slice model was suggested with a Courant number below 1, which means a temporal resolution below 0.5°/timestep, and a model thickness at least 2.4% of the impeller outer diameter. Then, the transient fan flow fields under various working conditions were calculated and refined analyzed. The results showed that: (1) The mismatch between the blade leading edge and the flow direction of the intake air leads to severe flow separation at the blade suction surface. A series of vortexes are periodically generated on the blade leading edge and fall off one by one. The domain frequency of the shedding vortex is about 60% of the blade passing frequency, and the motion of the vortexes correspondingly results in low-frequency pulsations below 1 kHz in the blade passages.
(2) The instantaneous reverse pressure gradient leads to backflow in part of the blade passages, forming a vortex cluster consisting of tiny eddies upstream of the blade cascade. The vortex cluster self-rotates slowly in the same direction as the rotor and transfers across the impeller at a circumferential speed slightly lower than the speed of the rotor. (3) The pressure pulsation at the volute tongue is dominated by the blade passing frequency when the fan operates in the large-flow conditions, while the vortexes generated by the flow separation on the inner side of the tongue contribute additional pulsation between 1-2 kHz. However, the pulsation frequency on the volute tongue turns to be dominated by the periodic fluctuation of the entire flow field as the fan flow rate decreases, characterized by low frequency and wide bands.
A practical approach is provided in the present study to finely analyze the internal flow of a squirrel-cage fan. The evolution mechanism obtained also provides new references for the design and noise reduction of this type of fan. However, as a simplified model, the results calculated by the slice model differ from the actual flow field, so whether the obtained flow patterns and frequency characters can be captured in the actual machine internal flow requires further experiment verification. Additionally, there is a possibility that the parameter configuration when constructing a slice model varies for fans with different sizes or intake types. Therefore this method needs to be verified in more squirrel-cage fans' simulations.