Numerical Analysis of Bionic Inlet Nozzle Effects on Squirrel-Cage Fan Flow Characteristics

Abstract

In order to improve the inlet distortion of the squirrel-cage fan, this study proposes a parametric design method for the bionic structure of the inlet nozzle generatrix, which is spliced by multiple sinusoidal curves, based on the bionic structure of the humpback whale flipper leading-edge nodule. The geometric shape of the bionic generatrix is controlled by three parameters: the number of segments n, the amplitude ratio T_m, and the amplitude of the last curve A_n. These parameters are optimized through orthogonal tests and numerical simulations, with the aim of improving the fan’s aerodynamic efficiency. Based on the selected solution, a comparative analysis is conducted to examine the impact of cylindrical, conical, and bionic inlet nozzles on inlet distortion and flow evolution within the centrifugal fan. Numerical calculations demonstrate that the fan’s maximum total efficiency, with a bionic inlet nozzle designed in a rational manner, is 5.46% higher than that of the original fan and is 2.01% higher than that of the fan with a conical inlet nozzle. The proposed bionic structure can create a buffer zone at the fan’s inlet, thereby reducing the region of high vorticity caused by the separated flow. Consequently, this improvement leads to enhanced uniformity at the impeller’s inlet. Furthermore, the design method proposed in this study for the inlet nozzle’s bionic structure effectively regulates the airflow angle near the impeller shroud, thereby enhancing the fan’s inlet distortion and improving its overall aerodynamic performance.

1. Introduction

As one of the important components in HVAC (heating, ventilation, and air conditioning) systems, the squirrel-cage fan is widely used in refrigeration, construction, environmental protection, and other fields [1,2]. Improving its pneumatic efficiency is crucial for saving energy and reducing emissions in these industries [3]. However, due to the mechanical structure of squirrel-cage fans, the flow direction undergoes an axial-to-radial transition when the airflow enters the moving blade from the inlet channel. This transition results in a separated flow at the fan’s inlet, causing a strong inhomogeneity in the airflow when it enters the impeller, which is commonly referred to as inlet distortion [4,5].

Inlet distortion is identified as the main cause of complex flow in a centrifugal impeller [6,7]. Zhang [8] conducted a study comparing the flow characteristics of a single-stage reversing centrifugal fan under distorted inlet conditions (DIC) and uniform inlet conditions (UIC). The results revealed that inlet distortion exacerbated the effects of the front-stage impeller as a leading impeller on the rear-stage impeller; this detrimental effect of the front impeller as a leading impeller on the internal flow of the rear-stage impeller resulted in significantly higher entropy loss within the two-stage impeller compared to uniform inlet conditions. Additionally, the degree of turbulence within the diffuser was found to be intensified under distorted inlet conditions. Zhang [9] conducted an analysis on the impact of inlet distortion on the aerodynamic stability of aero-engine fan blades. The results indicated that the stall margin of the blade diminished as the corrected rotational speed decreased, given the same level and mode of inlet distortion. Nevertheless, the total pressure loss at stall also decreased with the reduction in speed. Schnell R [10] reviewed the effects of inlet distortion on centrifugal pressurizer performance and stall margins in aircraft propulsion systems. The study demonstrated that inlet distortion significantly impacted the aerodynamic performance of the fan, particularly in the integration of aircraft propulsion systems, where inlet distortion directly influenced the compressor’s performance and stability. With emerging trends in aircraft design, such as hybrid wing-body design and stealth capabilities, the inlet-engine flow-matching problem has become increasingly complex, necessitating new research directions and solutions. To address the issue of inlet distortion and enhance performance, researchers have proposed different control structures or techniques for the fan inlet flow. In their investigations of centrifugal fans, Madhavan and Wright [11], Kassens and Rautenberg [12], and Coppinger and Swain [13] utilized inlet guide vanes to minimize inlet distortion. Similarly, P Chen [14] examined the rotational stall and inlet vortex of a centrifugal fan with inlet vane control. The experimental results indicated that the positive precession generated by the inlet vanes contributed to delaying the onset of the stall. Additionally, the frequency of the inlet vortex was always found to be linearly associated with the flow velocity, even when inlet guide vanes were present. Researchers have also discovered that structural and geometric parameters, such as the shape, size, and eccentricity of the fan inlet, impact inlet distortion. Vadari VR [15] designed an annular inlet guide (AIG) for a centrifugal fan to reduce the size and intensity of the separation region, improve the flow quality into the impeller, and reduce noise. Yamamoto S [16] experimentally investigated the effects of fan efficiency and noise caused by inserted, extruded, and combined inlet nozzles. The results indicated that optimal lengths existed for the inserted and extruded types, while the combined type was more effective in improving efficiency. In a study conducted by Wen [17], the effects of inlet nozzle geometry and mounting on the aerodynamic and acoustic performance of a small squirrel-cage fan were investigated. The results demonstrated that optimizing elliptical inlet nozzles could enhance the fluid flow quality into the impeller channel, resulting in reduced flow loss and the formation of a tip flow pattern. This pattern was found to be beneficial in increasing the volumetric flow rate and suppressing leakage flow loss. Wang [18] numerically analyzed the impact of vortices and eccentric inlets in the blade channel of a multi-blade centrifugal fan for air conditioning, and the results showed that, near the rotor shroud, separation phenomena occurred in almost all of the blade channels, with the airflow predominantly being in a stagnant state below the vortex channel. However, the strength of the vortex was weakened when the fan inlet was eccentrically mounted.

When improving the internal flow characteristics of fans or impeller ducts, researchers often draw inspiration from bionics. Xiong [19] designed a bionic blade for multi-wing centrifugal fans that imitates the curved posture of the fish body in the family Mollusca. They analyzed the internal flow and discovered that the intensity of swirls in the impeller ducts of the fish-like blade was significantly smaller than that of the prototype fan, resulting in a more homogeneous distribution of the flow field. The adoption of fish-like blades effectively reduced the pressure pulsation at the snail tongue of the fan and weakened the non-stationary interaction between the blades and the snail tongue. Dong X [20] applied the contour of the leading edge of an owl’s wings to the snail tongue of a multi-blade centrifugal fan. They found that the bio-mimetic design of the snail tongue could improve the aerodynamic performance of the fan under different flow conditions. Wang K [21] conducted a numerical and experimental analysis of the aerodynamic performance of a squirrel-cage fan with a bionic volute. The findings indicated that the bio-mimetic design of the vortex tongue offered benefits in terms of enhancing flow quality and reducing sudden pressure changes near the vortex tongue, as well as alleviating axial inhomogeneity in the flow. Bionic design is predominantly utilized in centrifugal fans to directly manage or enhance complex flow, but limited research has been carried out on bionic design of the inlet nozzle to indirectly enhance the inward flow characteristics by improving inlet distortion.

In this study, a bionic inlet nozzle structure is proposed for the traditional squirrel-cage fan inlet distortion and resulting non-uniform internal flow issues. This structure combines the hydrodynamic improvement principle of the humpback whale flipper’s leading-edge nodule. Unlike the traditional method of extracting and applying bionic structures, this study extracts the bionic structure in a figurative form and applies it to the generatrix of the inlet nozzle using multiple half-periodic sinusoidal curves spliced into a wave line. Control parameters are proposed to achieve parametric design of the bionic structure, allowing for accurate control of the design shape. This design method can be applied to various structures and sizes to address different degrees of inlet distortion in squirrel-cage fans. Furthermore, this study compares and analyzes the effects of traditional cylindrical and conical inlet nozzle structures on the inlet distortion and internal flow characteristics of the fan. The mechanism of the bionic inlet nozzle’s impact is analyzed qualitatively and quantitatively using parameters such as flow velocity, vortex volume, airflow angle, and entropy generation rate as indices. The results of this study can serve as a reference for improving non-uniform air inlets in squirrel-cage fans and centrifugal impellers.

2. Design Methodology and Modeling

2.1. Bionic Inlet Nozzle’s Structural Design and Parameterization

The leading-edge nodules on the flippers of humpback whales are raised structures that alter the flow of water (as shown in Figure 1), enabling the whales to maneuver and control their movements more effectively in the water [22]. Although the exact mechanism and effects of these nodules are still being researched, they have been found to play a crucial role in humpback whales’ swimming. This structure has been examined by researchers in the fields of engineering and aerospace, leading to advancements such as new turbine blades on wind turbines tested by Whalepower and specialized surfboards with corrugated fronts produced by Fluid Earth [23].

In this study, the parametric design of this structure was achieved by connecting multiple sinusoidal curves, as depicted in Figure 2. This resulting wavy line was then controlled through parameters to create a more structured design.

Figure 2 illustrates the 3D structure of the bionic inlet nozzle and the parametric design of its bionic generatrix. The bionic inlet nozzle’s structure was achieved by designing the generatrix of the inlet nozzle as a wavy line, mimicking the nodal bumps on the leading edge of the humpback whale’s flippers (as shown in Figure 2). This design was implemented to utilize the benefits of the bionic structure throughout the entire circumference of the inlet nozzle. More specifically, the design of the wave line allows for precise control of the generatrix parameters through the use of multiple segments of sinusoidal curves, as depicted in Figure 2. These curves are spliced with “n” (where n = 5, as shown in Figure 2) segments of sinusoidal curves with varying amplitudes of the half-periods. The amplitude decreases along the mainstream direction, and the amplitude of the last segment of the curves (A_n) is limited to minimize the formation of separating vortices at the fan inlet. Based on the above idea, precise control of the bionic structure is achieved by three parameters: the number of segments of the half-periodic sinusoidal curves (n), the amplitude change control parameter (T_m), and the amplitude of the last segment of the sinusoidal curves (A_n). The inlet nozzle’s generatrix consists of a patchwork of n (where n is odd) segments of sinusoidal curves. These n segments are composed of sinusoidal curves from the 1st to the nth segments, based on the domain of definition of [(i − 1), i]. The amplitude decreases sequentially from the 1st to the nth segments, where i is an integer value ranging from 1 to n. The amplitude change control parameter, T_m, is defined as T_m = A_m/A_m+1, where m is an integer ranging from 1 to n − 1. The amplitude of the last sinusoidal curve is represented as A_n. During the design process, it is necessary to set values for n and A_n, and then to calculate the entire section of the bionic inlet nozzle’s generatrix based on the chosen value of T_m. For example, if n = 9 and A₉ = 0.02, and T_m = 1.8 is chosen, the amplitudes of A₁ to A₈ will be 1.29, 1.07, 0.90, 0.75, 0.62, 0.52, 0.43, and 0.36, respectively.

In order to investigate the optimal structure for a reasonable bionic inlet nozzle generatrix, three main parameters were selected and optimized through an orthogonal test. The specific scheme is presented in

Table 1. Cases of the orthogonal experimental design.

No.	n (−)	Tm (−)	An (m)
1	5	1.2	0.02
2	5	1.3	0.03
3	5	1.4	0.025
4	7	1.2	0.03
5	7	1.3	0.025
6	7	1.4	0.02
7	9	1.2	0.025
8	9	1.3	0.02
9	9	1.4	0.03
10	0	0	0
0	-	-	-

. In this table, scheme No. 10 refers to the conical inlet nozzle structure, while scheme No. 0 represents the original machine’s cylindrical inlet nozzle structure.

2.2. Model and Validation

This study was conducted on a 48-blade squirrel-cage fan with a speed of 960 rpm. The fan primarily consisted of an impeller, an inlet nozzle, and a volute, as represented schematically in Figure 3. The primary parameters of the squirrel-cage fan are presented in

Table 2. Main parameters of squirrel-cage fan.

Parameter	Value
Number of blades, Z	48
Impeller inlet diameter, D1	380 mm
Impeller outlet diameter, D2	500 mm
Impeller width, b	275 mm
Volute width, B	325 mm
$\mathrm{Entrance} \mathrm{blade} \mathrm{angle}, {\alpha }_1$	60°
$\mathrm{Exit} \mathrm{blade} \mathrm{angle}, {\alpha }_2$	150°

.

The schematic diagram of the mesh model is shown in Figure 4, where the impeller and the volute are structurally meshed. The mesh-independence verification is shown in Figure 5a. The total number of meshes was chosen to be approximately 11,148,237, in order to take into account the computational accuracy and efficiency, and a boundary layer mesh was applied to the solid surface to calculate the viscous flux within the boundary layer. Experimental tests were conducted on No. 0, and the total fan efficiency of the experimental data was compared with that of the simulated data, as shown in Figure 5b. The performance curves obtained from the simulated data generally align with the trends observed in the experimental measurement data.

In this study, Fluent was utilized to solve the incompressible Reynolds-averaged Navier–Stokes (RANS) equations for the 3D turbulence simulation. The relationship between the Reynolds stress term and the mean volume of the flow field is based on the Boussinesq vortex viscosity assumption. The three-dimensional Reynolds-averaged Navier–Stokes equations were resolved using the shear stress transport model (SST $k$-$\mathsf{\omega }$), due to its high computational accuracy in regions of large velocity gradients near the vortex [24]. Air was considered as the medium, with constant values for temperature, density, and viscosity. The inlet and outlet boundary conditions were defined as the mass flow inlet and pressure outlet, respectively. Among them, the mass flow inlet was determined according to the flow rate and inlet area, and the outlet pressure was set to 0. A coupled velocity and pressure model was employed, utilizing the versatile SIMPLE algorithm. The momentum equation, dissipation rate equation, and turbulent kinetic energy equation were discretized using a second-order upwind scheme. In this study, the simulation utilized the Moving Reference Frame (MRF) to address the flow problem of a rotating centrifugal impeller, allowing for the transformation of an unsteady problem in a stationary reference system to a steady problem in a moving reference system, thereby simplifying the computation process. To ensure the accuracy and convergence of the numerical calculations, the root mean square (RMS) value of the residuals of the control equations was set to be less than 10⁻⁵ [25].

The Grid Convergence Index (GCI) test is an important method for evaluating grid independence and convergence [26].

Table 3. Grid independence summary.

Grid Properties	Grid 1	Grid 2	Grid 3
Number of elements × 106	8.85	11.15	15.53
Volute element size (mm)	8	7	6
Impeller element size (mm)	4	3	2
Number of inflation layers	5	5	5
First layer thickness (mm)	0.05	0.05	0.05
Inflation growth rate	1.2	1.2	1.2
Maximum skewness	0.872	0.881	0.861
Impeller averaged y+	0.534	0.482	0.215
η (−)	0.6287	0.6277	0.6262

provides grid setup parameters for three groups of grids (coarse grid 1, medium grid 2, and fine grid 3) to determine the GCI and the simulated aerodynamic efficiency values. Grid tests were performed at the best efficiency points. The results show that increasing the number of grids to more than 8.85 million does not significantly affect efficiency. As described by Roache [27], the GCIs for the coarse and fine grids were calculated with a safety factor of F_s = 1.25. Specifically, GCI_Grid1 = 5.74 × 10⁻³, and GCI_Grid3 = 3.75 × 10⁻³. The accuracy of the solution does not rely on the number of grids used, as indicated by the respective GCIs. In addition, the average y⁺ value on the blade surface was less than 1, which satisfies the turbulence modeling requirements. Therefore, the simulation results can be considered reliable when analyzed alongside Figure 5.

2.3. Control Equations and Turbulence Model

The flow control equations are three-dimensional incompressible Reynolds-averaged Navier–Stokes (RANS) equations coupled with the SST $k$-$\mathsf{\omega }$ turbulence model. The RANS equations are written as follows [28]:

$${\displaystyle \frac{\partial u_i}{\partial x_i}}=0$$

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho u_i\right)+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho u_i{u}_j\right)=-{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}\left[\mu \right({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}})-{\displaystyle \frac{2}{3}}{\displaystyle \frac{\partial u_l}{\partial x_l}}{\delta }_{ij})]+{\displaystyle \frac{\partial }{\partial x_j}}(-\rho \overline{u_i^{\prime }}\overline{u_j^{\prime }})$$

where the Reynolds stress $-\rho \overline{u_i^{\prime }}\overline{u_j^{\prime }}$ is modeled by the Boussinesq hypothesis [27] and related with turbulent viscosity $u_t$.

$$-\rho \overline{u_i^{\prime }}\overline{u_j^{\prime }}=u_t({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}})-{\displaystyle \frac{2}{3}}(\rho k+u_t{\displaystyle \frac{\partial u_i}{\partial x_i}}){\delta }_{ij}$$

k is the turbulent kinetic energy:

$$k={\displaystyle \frac{\overline{u_i^{\prime }}\overline{u_j^{\prime }}}{2}}$$

The SST $k$-$\mathsf{\omega }$ model combines the advantages of the $k$-$\mathsf{\epsilon }$ and $k$-$\mathsf{\omega }$ models to accurately predict turbulence in varying flow regions [29]. The governing equations of the SST $k$−$\mathsf{\omega }$ turbulence model mainly consist of the turbulent kinetic energy (k) equation and the turbulent dissipation rate ($\omega$) equation:

$${\displaystyle \frac{\partial \rho k}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho u_{j}k\right)={\displaystyle \frac{\partial }{\partial x_j}}\left[\right(\mu +{\sigma }_k{\mu }_t\left){\displaystyle \frac{\partial k}{\partial x_j}}\right]+{\tau }_{ij}{S}_{ij}-{\beta }^{*}\rho \omega k$$

$${\displaystyle \frac{\partial \rho \omega }{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho u_j\omega \right)={\displaystyle \frac{\partial }{\partial x_j}}\left[\right(\mu +{\sigma }_{\omega }{\mu }_t\left){\displaystyle \frac{\partial \omega }{\partial x_j}}\right]+{\displaystyle \frac{C_{\omega }\rho }{{\mu }_t}}{\tau }_{ij}{S}_{ij}-\beta \rho {\omega }^2+2(1-F_1){\displaystyle \frac{\rho {\sigma }_{\omega 2}}{\omega }}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}}$$

The turbulent viscosity coefficient (${\mu }_t)$ is given by

$${\mathsf{\mu }}_{\mathrm{t}}=\mathsf{\rho }{\displaystyle \frac{a_{1}k}{max(a_1\omega ,\Omega F_2)}}$$

Constant: $a_1=0.31$; ${\beta }^{\mathrm{*}}$= 0.09; ${\sigma }_{k1}=1.176$; ${\sigma }_{\omega 1}=2.0$; ${\beta }_1=0.075$; $C_{\omega 1}=0.533$; ${\sigma }_{k2}=1.0$; ${\sigma }_{\omega 2}=0.856$; ${\beta }_2=0.0828$; $C_{\omega 2}=0.440$, where $\mathrm{y}$ is the height of the first grid layer, $\Omega$ is the strain rate, and $F_1$ and $F_2$ are mixed functions.

$$F_1={tanh\left\{\right\{min\left[max\right({\displaystyle \frac{\sqrt{k}}{{\beta }^{*}\omega y}},{\displaystyle \frac{500\nu }{y^2\omega }}),{\displaystyle \frac{4\rho {\sigma }_{\omega 2}k}{C{D}_{k\omega }{y}^2}}]\}}^4\}$$

$$F_2={tanh\left\{\right[max({\displaystyle \frac{2\sqrt{k}}{0.09\omega y}},{\displaystyle \frac{500\nu }{y^2\omega }})]}^2\}$$

$$C{D}_{k\omega }=max(2\rho {\sigma }_{\omega 2}{\displaystyle \frac{1}{\omega }}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}},{10}^{-20})$$

3. Results and Discussion

3.1. Analysis of Fan Aerodynamic Performance

Figure 6 displays the calculation results for the design working conditions of the 10 schemes, as well as the aerodynamic efficiency curves of the cylindrical (No. 0), conical (No. 10), and optimized bionic generatrix inlet nozzles (No. 9) under all working conditions. Figure 6 indicates that the fan equipped with the No. 9 scheme bionic inlet nozzle, as determined through the orthogonal design, exhibits the highest aerodynamic efficiency. Numerical calculations reveal that the No. 9 scheme improves maximum efficiency by 5.46% compared to the original machine, and by 2.01% compared to the conical inlet nozzle. Furthermore, Figure 6 shows that the maximum efficiency conditions for both the conical and No. 9 bionic inlet nozzles have shifted to larger flow rates compared to the cylindrical inlet nozzle. Within the flow rate range of 3000~8500 m³/h, the aerodynamic efficiency of No. 9 is higher than that of No. 10. However, its efficiency drops below that of No. 10 for flow rates exceeding 8500 m³/h, which indicates that the impact of the aforementioned bionic structure on inlet distortion may be influenced by the flow rate at the inlet.

3.2. Analysis of the Internal Flow

To comprehend the impact of various inlet nozzle types on the internal flow within the fan, it is imperative to compare and analyze the flow field and vortex diagrams for different span sections of No. 0, No. 9, and No. 10. Figure 7 displays the absolute velocity contours and streamlines for the various cross-sections, while Figure 8 illustrates the vortex contours plotted based on the Q-criterion for the three groups of models. It is evident that the separation vortex at inlet of No. 9 exhibits the smallest volume among the three groups, followed by No. 10, and the most prominent inlet separation vortex is caused by No. 0. Analyzing the absolute velocity cloud plots of the different span sections, it is apparent that the outflow absolute velocities of the blades for both No. 9 and No. 10 are significantly higher compared to No. 0. In particular, the effects of span = 0.8 and span = 0.6 are more pronounced, and the absolute flow velocities of the impeller outlets of No. 9 and No. 10 are more uniformly distributed for the span = 0.4 and span = 0.2 segments. Additionally, the flow fields of models No. 9 and No. 10 are similar in most regions, but there are noticeable differences in some specific locations. The low-speed flow region at the volute tongue of No. 9 is smaller, and the flow lines in the flow path between the impeller and the volute of No. 9 are smoother for the span = 0.4 and span = 0.2 segments. When combined with Figure 8, it is evident that the flow channel between the impeller and the volute of No. 9 exhibits fewer vortices at the same scale. This indicates that the bionic inlet nozzle structure provides better control of the inlet distortion of the fan compared to the cylindrical and conical shapes, resulting in improved internal flow characteristics of the fan.

The blade runner profiles of the three inlet nozzle structures with different span sections are illustrated in Figure 9. It can be observed that, in the region close to the fan inlet (span = 0.8), both the No. 9 and No. 10 schemes exhibit better flow-through compared to the original structure. In contrast, the original structure experiences evident vortex blockage at both the blade suction and pressure surfaces throughout the circumferential range. In the middle section of the impeller (span = 0.6/0.4), No. 0 shows a significant improvement in ventilation flow. However, near the fan outlet position (250~280° for span = 0.6 and span = 0.4), No. 9 and No. 10 demonstrate higher impeller inlet and outlet flow velocities. This indicates that both configurations can achieve higher flow rates under the same operating conditions. The disparity in the impeller flow field at the bottom section of the impeller (span = 0.2) is less pronounced for all three configurations.

In addition, the No. 9 blade runner exhibits better flow characteristics, especially at 240~300° for span = 0.8 and span = 0.6. However, the improvement compared to the No. 10 scheme is significantly smaller than the improvement compared to No. 0. Furthermore, the vortices near the blade pressure surface are noticeably weakened in the No. 9 scheme. This suggests that the inlet nozzle’s bionic structure effectively weakens the separating vortex at the fan inlet, thereby increasing the blade work area near the shroud and improving the attack angle of the airflow in that impeller inlet. After comparing the specific flow details of span = 0.8, it is evident that the inlet low-velocity region of the blade channel has been significantly reduced from 60~300° (No. 0) to 30~240° (No. 9, No. 10). Additionally, the area of the low-velocity region in the middle of the blade channel, which is situated between the pressure and suction surfaces, has also been significantly reduced. These observations suggest that the bionic structure greatly enhances the inflow characteristics of the blade channel near the shroud across a substantial circumferential range. In terms of schemes No. 9 and No. 10, the flow characteristics of the blade channel display minimal differences over most of the span range (span = 0.2~0.6). The main disparity is observed in the front span (span = 0.8), therefore necessitating a more detailed analysis of the flow at the impeller inlet.

3.3. Analysis of the Effect on Inlet Distortion

Figure 10 displays the vortices at the fan inlet for the three structures to better illustrate the impact of various inlet nozzle structures on inlet distortion. Firstly, it is evident that all three structures produce more vortices in this region due to the interference effect at the volute tongue. Secondly, the cylindrical inlet nozzle structure (No. 0) produces the most distinct separating vortices near the shroud of the impeller, which leads to decreased airflow into the blade channel, reducing the blade area available for work. In contrast, No. 9 and No. 10 exhibit significantly fewer vortices at the fan inlet, with notable reductions in columnar and striated vortices. Moreover, No. 9 demonstrates narrower vortex bands in the impeller inlet area near the shroud compared to No. 10, which suggests that the bionic structure enables better control of the inlet airflow. Importantly, due to the bionic inlet nozzle’s structural characteristics, No. 9 features several low-speed vortices in the “concave” region. These vortices store a portion of the inlet air’s kinetic energy, thus providing improved cushioning for the fan’s inlet. This unique aspect is absent in the conical inlet nozzle (No. 10). Furthermore, the findings also demonstrate that a well-designed bionic inlet nozzle structure can more precisely manage the flow at the fan’s inlet.

To further analyze the specific effect of the bionic structure on the inlet distortion, the variation in the mass circ average values of the airflow angle and mounting angle at the impeller inlet for the entire span range of the cylindrical (No. 0), conical (No. 10), and bionic (No. 9) modeling schemes is plotted (Figure 11a), as well as the change curve of the mass circ average values of the radial velocity (Figure 11b).

In Figure 11, the results for the variation in the blade setting angle and velocity flow angle, along with their differences with respect to the relative radius ($r^{\prime })$, are represented by $\beta , {\beta }_f$. The $r_1$ and $r_2$ are the impeller inlet and outlet radius, respectively, and $r^{\prime }$ is the relative radius of the impeller after becoming dimensionless, which is defined as follows:

$${\mathrm{r}}^{\prime }={\displaystyle \frac{\mathrm{r}-{\mathrm{r}}_1}{{\mathrm{r}}_2-{\mathrm{r}}_1}}$$

The velocity flow angle ${\beta }_f$ refers to the angle between the flow direction of the incoming airflow in relation to the blade surface and the normal to the blade surface.

It can be observed that No. 0 exhibits the largest difference between the airflow angle and the mounting angle, particularly near the forward span (span = 0.62~1.00), which results in poor impeller inflow characteristics. On the other hand, both No. 9 and No. 10 have a smaller difference between the airflow angle and the mounting angle at this location. Specifically, in the range of span = 0.56~0.82 and span = 0~0.27, the airflow angle for No. 9 is closer to the mounting angle compared to No. 10. This suggests that No. 9 demonstrates better flow-through as it enters the blade channel. The difference in airflow angle between No. 9 and No. 10 is largest at span = 0.75, which is approximately 20°, and the uniformity of the inlet airflow angle of No. 9 outperforms the other two configurations across the entire span range. Furthermore, as displayed in Figure 11b, the radial velocities of the impeller inlet inflow for different span ranges of the No. 9 configuration exhibit a higher level of uniformity, particularly in the vicinity of the shroud (span = 0.8~1.0). The mass circ average values highlight a significant disparity among the radial velocities of the three structures, with No. 9 > No. 10 > No. 0. This suggests that the bionic structure yields the most favorable impact on the impeller inlet, particularly near the shroud, in terms of the airflow’s improvement in inlet characteristics. In other words, it provides enhanced inlet alignment and higher radial inlet velocities.

The entropy production value assesses the degree of fluid chaos and energy loss inside the fan [30]. Analyzing the entropy production value near the shroud of the impeller inlet allows for a more precise quantification of variations in inlet distortion across different configurations.

Figure 12a shows the entropy generation values at the impeller inlet ($r^{\prime }=0$.01). According to Figure 12a, it can be seen that there is an obvious entropy generation peak at the angle range of 270~350°, which can be further proven by the internal flow field diagram, showing that there are many kinds of complex flows at the location of the volute tongue, and the peak value of No. 9 is obviously lower than that of No. 0 and No. 10 there, which corresponds to the flow field diagram of Figure 7, span = 0.8 (Figure 12b). In the whole range of the annular direction, the fluctuations in the entropy production value of No. 9 are smaller (Figure 12a), indicating that the bionic structure has a better effect on reducing the entropy yield of the impeller near the inlet of the front disk and improving the flow characteristics there compared to both the cylindrical and conical structures.

A comparative analysis of Figure 12c–e reveals that No. 0 has fewer high-entropy-generation regions (EGR > 30,000 W·m⁻³ k⁻¹) in the impeller channel at the span = 0.8 cross-section, due to the blades’ poorer power capacity near the shroud. On the other hand, the entropy generation distributions of No. 9 and No. 10 are more similar. Further comparison shows that the high-entropy-generation region at the volute tongue for No. 9 is closer to the wall, resulting in a smaller annular spanning range due to interference of the high-entropy-generation region at the impeller inlet (which is depicted in the red boxed area in Figure 12c–e), and fewer blades are affected by the complex flow caused by the interference action of the volute tongue. Specifically, corresponding to Figure 12a, the higher-entropy-generation region (EGR > 15,000 W·m⁻³ k⁻¹) in the annular region at the impeller inlet is mainly concentrated at the volute tongue position (310~360°) for No. 9, and the range is larger for No. 10 (270~360° and 0~30°). Meanwhile, for No. 0, multiple irregular regions are observed throughout the annular direction, indicating more severe inlet distortion. Additionally, it is evident that the entropy yield value in the range of 7000~10,000 for No. 9 has a smaller area in the flow channel between the impeller and volute, suggesting lower flow loss in that area, which is consistent with the phenomenon shown in Figure 8. The comprehensive analysis indicates that different inlet nozzle structures result in different interference effects at the shroud volute tongue, leading to varied flow states. This variation may be attributed to differences in the impeller inlet velocity flow angle in this region.

The ${\beta }_f^{\prime }$ is determined by the difference between the velocity flow angle and the blade setting angle:

$${\beta }_f^{\prime }\left(r^{\prime }\right)={\beta }_f \left(r^{\prime }\right)-\beta \left(r^{\prime }\right)$$

Figure 13 depicts the curves for the mounting angle, velocity flow angle, and their difference in the blade flow channel for No. 0, No. 9, and No. 10. It is evident that the radial distributions of the flow angles and blade mounting angles differ greatly among the three schemes. However, the trends in velocity flow angles are essentially the same. The velocity airflow angles and the differences between them for No. 9 and No. 10 are noticeably smaller compared to those for No. 0, which is particularly evident at the impeller inlet ($r^{\prime }=0$.01), where the velocity flow angles of No. 9 and No. 10 are significantly reduced to around 20°, highlighting the positive impact of the conical and bionic inlet nozzle structures on enhancing the impeller inlet flow. These structures enable the airflow to enter the flow channel at an angle that closely matches the blades, thereby improving flow separation within the blades. Additionally, the airflow angle and difference curves for No. 9 and No. 10 exhibit a high level of similarity, indicating a consistent overall airflow pattern within the blade channel. Considering that the values given in Figure 13 are averaged over the entire span, the difference in airflow between No. 9 and No. 10 in the vicinity of the shroud (as analyzed in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12) is not sufficiently large to affect the averages over the entire axial range.

4. Conclusions

Considering the need to improve the inlet distortion of the squirrel-cage fan, a bionic inlet nozzle structure and corresponding parametric design method are proposed here. This method is based on multiple-segment sinusoidal curves spliced together to imitate the humpback whale flipper limb’s leading-edge nodal structure. The number of segments (n), the amplitude ratio (T_m), and the amplitude of the last curve (A_n) were used as control parameters for the multivariate design of the bionic inlet nozzle structure. The purpose of this design is to control and improve the separation vortex and airflow angle at the fan’s inlet, increase the uniformity of the impeller inflow, and improve the work capacity of the shroud impeller. To analyze the optimization effect of the bionic inlet nozzle, we assessed the above three key parameters through orthogonal tests and numerical simulations, and then we compared and analyzed the effects of the optimized bionic inlet nozzle structure on the inlet distortion and internal flow of the fan with those of the cylindrical and conical inlet nozzles. The main conclusions are as follows:

• The bionic structure of the humpback whale flipper’s leading-edge nodule can be achieved by splicing multiple sinusoidal curves, and the three control parameters (n, T_m, A_n) allow for the multivariate design of the bionic structure, enabling it to adapt to fans of varying sizes and operating conditions.
• The inlet nozzle bionic structure, if designed properly, can significantly improve the inlet distortion and enhance the fan’s aerodynamic performance. Simulation results demonstrate that the fan’s total efficiency when equipped with a bionic inlet nozzle, using control parameters of n = 9, T_m = 1.4, and A_n = 0.3, is 5.46% higher compared to that of a cylindrical inlet nozzle and 2.01% higher compared to that of a conical inlet nozzle.
• The bionic design of the inlet nozzle allows the airflow to receive sufficient buffer in the “concave area” on the inner surface of the inlet nozzle before entering the fan. This helps to reduce the separation vortex caused by the fan inlet due to steering, and the optimized design of the bionic structure improves the radial flow velocity in the front span of the impeller. It also enhances the axial uniformity of the inlet airflow angle of the impeller and reduces the entropy generation rate of inlet air at the shroud impeller.
• The bionic inlet nozzle structure improves the flow characteristics of the shroud impeller on the basis of improving the inlet distortion of the fan, and it also improves the uniformity of the flow velocity at different spans of the impeller outlet, as well as reducing the flow loss of the flow channel between the impeller and volute.

In conclusion, the design of the bionic inlet nozzle structure can improve the inlet distortion and internal flow state of the squirrel-cage fan. The parameterized design method of the inlet nozzle bionic structure proposed in this study can serve as a valuable reference for practical engineering applications.