Numerical simulation of pressure pulsation and transient flow field in an axial flow fan

Energy 129 (2017) 185e200

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Numerical simulation of pressure pulsation and transient flow field in an axial flow fan Xuemin Ye*, Xueliang Ding, Jiankun Zhang, Chunxi Li School of Energy Power and Mechanical Engineering, North China Electric Power University, Baoding 071003, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 3 November 2016 Received in revised form 29 March 2017 Accepted 14 April 2017 Available online 19 April 2017

An abnormal regulation of the stagger angle deteriorates the internal flow field and pressure pulsation, leading to an augmentation of aero-acoustic noise and vibrations in variable-pitch axial fans. To evaluate the effects of an abnormal stagger angle, the pressure pulsation and transient flow field under normal and abnormal regulations of the stagger angle were simulated using unsteady 3D modelling. The characterization capabilities of the approximate entropy and sample entropy for identifying an abnormal deviation were examined by extracting the features from the static pressure signals. The results indicate that, after an abnormal deviation of stagger angle, the periodic and quasi-periodic pulsation distributions of the static pressure are distinctly hindered, and the impacts of an abnormal deviation angle on the pressure distributions in the time and frequency domains are intensified with increasing deviation degree, resulting in increased pressure fluctuation intensity. The transient flow field clearly changes with time and degree of deviation, and abnormal high- and low-pressure regions are developed. Both the approximate entropy and sample entropy can be used to identify an abnormal blade deviation, but the sample entropy is more suitable for characterizing the effects of deviation degree on the static pressure at the impeller and guide vane outlets. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Variable-pitch axial fan Abnormal stagger angle Pressure pulsation Transient flow field Approximate entropy Sample entropy

1. Introduction A variable-pitch axial flow fan has inherently diverse merits, including a compact structure, a wide operating scope, and high efficiency under varying-load conditions; hence, it is widely employed in many engineering applications, such as the energy, mining, power, and transportation fields. In particular, a variablepitch axial fan is a preferred choice for primary fans, induced draft fans, and forced draft fans in modern large power-generation units [1]. However, a stagger angle anomaly of one or several rotating blades induced through a mechanical fault frequently appears in varying-load processes, leading to a degradation of the operating performance, an aggravation of vibrations and noises, and even the emergence of a stall or stoppage [2]. These faults include a rotor unbalance caused by an inaccurate assembly of the balance components, a poor sliding of the gear sleeve associated with the deeper installation depth of spare bolts in the feedback apparatuses, an increased regulating resistance of a petiole bearing

* Corresponding author. P O Box 29, Yonghuabei Street 619, Baoding 071003, China. E-mail address: [email protected] (X. Ye). http://dx.doi.org/10.1016/j.energy.2017.04.076 0360-5442/© 2017 Elsevier Ltd. All rights reserved.

owing to the penetration of flyash into the bearing clearance, and a fixation anomaly between the gear sleeve and adjustment lever resulting from a bearing failure or lever break [2e4]. Previous researches have indicated that one of the important reasons for the aforementioned performance variations is a deterioration of the pressure pulsation in the unsteady flow fields induced by an abnormal deviation of the stagger angle [2e5]. Thus, it is important to investigate the characteristics of the pressure pulsation and the internally transient dynamics of such a fan with a stagger angle anomaly of the rotating blades. Significant advances in capturing the internal flow field and pressure pulsation have been achieved both experimentally and numerically. Regarding the internal flow field, using hot wire anemometry (HWA), Chen et al. [6] conducted an experiment study on the flow field of a variable-pitch axial fan under different stagger angles and evaluated the effects of the stagger angle variation on the flow patterns at the inlet and outlet of the impeller; their results showed that the performance and surge margin are improved by regulating the stagger angle, whereas the maximum efficiency is reduced to a certain extent. Similarly, using the HWA technique, ndez Oro et al. [7] measured the unsteady flow structure of a Ferna low-speed axial fan at different operating points and presented the

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underlying mechanism in both wake-transport phenomena and stator-rotor interactions. Zhang et al. [8] numerically simulated the transient flow field of an axial flow fan under rotating stall condition; their results showed that a stalled cell exists in the impeller, with its propagation direction being the same as the rotation direction of the impeller, and the equivalent stress distribution is greatly influenced by the centrifugal force. Nouri et al. [9] examined the effects of the rotor structure and relative axial spacing on the performance of counter-rotating axial fans both numerically and experimentally; their results showed that increasing the rotation rate ratio can improve the overall performance and steady operation domain, and that increasing the axial spacing causes only a small decrease in efficiency. Using a steady modelling approach, Ye and Li [10e11] simulated the influence of an abnormal stagger angle of single and multiple rotating blades on the aerodynamic performance and aero-acoustics of an axial flow fan; their results showed that both the aerodynamics and acoustics tend to deteriorate, and that the effect of the number of abnormal blades on the efficiency is more pronounced than that on the total pressure rise. A variation in the transient flow field is inevitably accompanied with an alteration of the pressure pulsation. Hirata et al. [12] carried out an experiment study on an unsteady pressure on a rotating flat-blade surface of a simple cross-flow impeller in an open space without any casings. Funaki et al. [13] revealed the minute fluctuating pressure features on the 3D blade surfaces of a basic propeller fan, and illustrated the spatial distribution on the blade based on both the time-mean pressure and pressure-fluctuation intensity. Hurault et al. [14] tested the steady and unsteady wall pressure fluctuations in an automotive cooling fan, and predicted the surface-pressure power spectra using 3D Reynolds averaged Navier-Stokes (RANS) equation with Reynolds stress model and semi-empirical aero-acoustic models; their results showed that the wall pressure spectra levels simulated using the semi-empirical models are undervalued within a low frequency range of 100 to 2000 Hz, but agree quite well within a high frequency range of 2e10 kHz. Recent investigations show that the large eddy simulation (LES) method is an efficient scheme for accurately predicting highly unsteady features of transient flow fields, and the Ffowcs Williams-Hawkings (FW-H) model based on the Lighthill sound analogy was successfully applied to an estimation of aerodynamic noise [2,15e17]. Carolus et al. [16] predicted the fluctuating forces on the blades and the broadband noise of low-pressure axial fans using LES and a simple semi-empirical noise model, and the turbulence statistics were verified with HWA; their findings showed that the predicted effects of the ingested turbulence on the fluctuating blade forces and fan noise agree with the experiment results. Fern andez Oro et al. [17] conducted 3D simulations of dynamic and periodic stator-rotor interactions in a low-speed axial fan using the RANS and LES techniques, and characterized the unsteady flow structures involved in an axial flow blower, as well as the mechanism related to the blade-passing frequency in a single rotor-stator interaction. Li et al. [2] assessed the effects of a deviation in the stagger angle of an abnormal blade on the aerodynamics and aero-acoustics of an axial fan using LES and FW-H models, and extracted the energy features of the sound pressure using wavelet packet decomposition (WPD) and empirical mode decomposition (EMD); their results showed that the degree of deviation of an abnormal stagger angle has a crucial influence on the range and amplitude of the sound pressure level, and that the features extracted from WPD and EMD provide important references for recognizing an abnormal angle deviation. Although many admirable achievements on the pressure pulsation and transient flow field of axial fans have been gained over the past two decades, few studies have focused on the impact of an abnormal stagger angle of rotating blades on the aerodynamic and

aero-acoustic performance of a variable-pitch axial flow fan [2,10,11]. In addition, it should be pointed out that in Li et al.’s investigation [2], noise monitoring probes were placed in the bell mouth, impeller, guide vane, and diffuser, and only the averaged total pressure distributions at the outlet stream surface and in the span-wise cross section at 10% of the blade height were presented. However, published studies have revealed that the pressure pulsations at the tip clearance of the rotating blades, as well as before and after the impeller, are extremely important sources of noise; hence, to effectively highlight the influence of an abnormal stagger angle, the monitoring probes should be arranged at key positions of a severe pressure pulsation. Additionally, transient flow fields were not provided in the results of Li et al. [2], and thus the effect of an abnormal blade on the variation of the transient flow field and the characterization of the pressure pulsation are not well understood. Therefore, for such an axial fan under an abnormal regulation of the stagger angle, some crucial characteristics including the pressure pulsation and transient flow field at the key positions and cross sections, as well as a characterization of the pressure pulsation feature, need to be further investigated. This paper is organized as follows: an axial fan model is described in Section 2; the numerical methodology is elaborated in Section 3. The distributions of pressure pulsation and transient flow field, and the characterization of pressure pulsation with the approximate entropy and sample entropy are discussed in Section 4. Finally, the conclusions of this work are summarized in Section 5. 2. Formulation of fan model In the present modelling, a OB-84 type variable-pitch axial fan was used to explore the effects of an abnormal stagger angle of a single rotating blade on the pressure pulsation and transient flow fields. The axial fan model consists of four components: a bell mouth, an impeller, an outlet guide vane, and a diffuser, as shown in Fig. 1. The crucial characteristics are given as follows: (1) the fan has 14 rotating blades and 15 guide vanes, (2) the impeller diameter and tip clearance are 1500 and 4.5 mm, respectively, (3) the fan is driven at a constant rotating speed of 1200 rpm, (4) under the design conditions, the normal stagger angle of the rotating blades is 32
, and the volumetric flow rate and total pressure rise are 37.12 m3/s and 2254 Pa, respectively, and (5) the blade-passing frequency is 280 Hz. The fan performance curves under three blade angles of 29
, 32
, and 35
are given in Ref. [18], and were used to assess the accuracy and reliability of the present modelling. The aero-acoustics of an axial flow fan is mainly dominated by the pressure pulsation and internal unsteady flow. An abnormal

Fig. 1. Diagram of fan structure and arrangement of monitoring probes.

X. Ye et al. / Energy 129 (2017) 185e200

regulation of the stagger angle of the rotating blades frequently occurs, leading to a deterioration of the internal flow field and an augmenting of the pressure pulsation. Not only is the aeroacoustic noise severely enlarged, the safe operation of the fan is also seriously affected. Practical operations of axial flow fans revealed that a positive deviation of the stagger angle frequently occurs [2,10e11]. Hence, the purpose of this study is to investigate the features of the pressure pulsation and transient flow fields of an axial flow fan under a positive deviation of the stagger angle. To clarify the description of the stagger angle deviation, Db is designated as the deviation angle of an abnormal blade from the normal angle (rotating along the airfoil axis in a counter-clockwise direction), where Db ¼ 0
denotes a normal blade angle of 32
, as shown in Fig. 2. The effects of Db ¼ 10
, 20
, 30
, 40
, and 50
on the pressure pulsation and transient flow field were examined in this study. 3. Computational methodology 3.1. Simulation method The computational fluid dynamics software Fluent 6.3.26 was employed to capture the characteristics of the pressure pulsation and transient flow field of the axial fan. The realizable k-ε turbulence model with a rotation correction was initially selected for simulating a steady flow field through a coupling with the 3D steady RANS equations. Many studies have verified the feasibility of applying the realizable k-ε turbulence model to solve a rotating flow, jet flow, mixing flow, and boundary layer separation under an adverse pressure gradient, and good agreement between simulated and experiment results have been found [1,2,10,11,19e21]. No slip and penetration conditions were exerted on the wall, and a standard wall function was selected for the near-wall region. The SIMPLEC algorithm was applied to couple the pressure and velocity fields. A multiple rotating reference frame (MRF) was used to couple the rotating impeller and stationary casing because the MRF considers not only the influence of a stator on the flow field, but also the interaction between the rotating impeller and stationary zones [2,10,11,20,21]. A discretisation of the convective terms, diffusive terms, and turbulent viscosity coefficient was addressed using a second-order upwind scheme. The effects of the wall roughness and gravity on the flow field were not considered in this study. A data exchange at the interfaces of different domains was accomplished using the Interface scheme. The convergence of the present modelling is guaranteed when the residuals of the velocity in all directions and k, ε are less than 104, and when the values of

Fig. 2. Diagram of the deviation degree of abnormal stagger angle.

187

the total pressure at the inlet and outlet remain simultaneously unchanged. Next, the flow field obtained from the steady modelling was introduced as an initial flow field for the unsteady modelling. For the unsteady calculation, a moving mesh model was employed for the data exchange among the various interfaces. This scheme was chosen because it only requires a calculation of the flux on both sides of the interface and an equality of the flux, and the information transmission between dynamic-static interfaces is therefore favourably achieved. The modelling domain covers the entire flowing passage from the bell mouth to the diffuser, as indicated in Fig. 1. The inlet at the bell-mouth and the outlet of the diffuser serve as the entrance and exit of the entire modelling domain, respectively. The boundary conditions of the inlet and outlet are the velocity and free outflow, respectively. In actual operation of an axial fan, the airflow harvests mechanical energy in the impeller region and changes the flow direction in the guide vane region, leading to a more violent pressure pulsation of the airflow in the impeller and guide vane when compared with those occurring in both the bell mouth and diffuser regions [2,10e13,17,22e25]. Consequently, the present study mainly focused on the pressure pulsation characteristics in the time- and frequency-domains in the impeller and guide vane regions, which differs from the results given by Li et al. [2]. The monitoring points in the same radial cross-section have identical pressure characteristics owing to a symmetrical structure; accordingly, a representative monitoring probe was arranged in each radial cross-section. The monitoring probes were placed at the inlet of the impeller (P1), the blade tip clearance (P2) of a rotating blade, the outlet of the impeller (P3), and the outlet of the guide vane (P4), as shown in Fig. 1. Fig. 3 shows the phase positions of the monitoring probe and the abnormal blade at the beginning of one rotation period (t ¼ 0). It can be seen that the phase difference between monitoring probe P2 and the abnormal blade is one-fourth of the rotation period; therefore, the sampled signals of the pressure pulsation of P2 are posterior to the impact of the abnormal blade. In this study, one rotation period of the fan is 0.05 s, and the time step was set to 5  104 s in the unsteady modelling to accurately capture the transient flow fields [22]; hence, there was a total number of 100 time steps during one rotation period. The present study simulated

Fig. 3. Phase positions of the monitoring probe and abnormal blade at t ¼ 0.

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Fig. 4. The distribution of the static pressure of P2 in two rotation periods.

the transient flow fields during a total time of ten rotation periods. The simulation results show that the distributions of the pressure pulsation in the time- and frequency-domains during the different rotation periods are the same (as shown in Fig. 4), that is, the present simulation reached a statistically steady state, and consequently, the data for a single period of 0e0.05 s were collected for the following analysis. 3.2. Grid strategy According to a multi-block topology meshing method and the complex structure of the axial fan, the computational domain was first divided into the bell-mouth, impeller, guide vane, and diffuser. The grids were created using a T-Grid type, and the Tet/Hybrid elements were refined to meet the minute flow demands, particularly in the impeller region. Considering the complicated flow field in the tip clearance, the grids near the tip clearance are dense, whereas the grids in the other regions are sparser. All grids were created based on reference to the impeller grids, and the other parts were subsequently meshed. To ensure the quality of the grid, a size function was introduced to densify the grid in the tip clearance. This was completed by meshing the blade surface as a priority, and then meshing the impeller volume through reference to the meshes of the blade surface. The specific parameters of the size function are as follows: the type is meshed, the source and attachment entities are the

Fig. 5. Independent verification of grid number.

surfaces of all blades and the impeller volume, the growth rate is 1.1, and the maximum size is 15. Other regions of the fan were then meshed by referring to the mesh of the impeller. The maximum skewness ratios were 0.789, 0.891, 0.757, and 0.785 for the bellmouth, impeller, guide vane, and diffuser, respectively. It should be noted that all skewness ratios were less than 0.97, and hence a high-quality grid was assured [1,2]. To eliminate the effect of the grid number on the modelling, and to satisfy the requirements of the computational accuracy, a verification of the grid independence was conducted using six groups of grid numbers, as shown in Fig. 5. It can be seen in the figure that the variations in the total pressure rise and efficiency are quite small when the grid number is over 2.35 million. Although the modelling accuracy was further promoted by increasing the grid numbers, the promotion amplitude was significantly limited, and the computational time was notably increased. Taking the computation time and the illustration of the flow details into account, a grid number of 2.35 million was selected for the present study. This total grid number was divided into 0.35 million for the bell mouth, 1.36 million for the impeller, 0.21 million for the guide vane, and 0.43 million for the diffuser. The grid diagram is shown in Fig. 6, including the grid of the fan in Fig. 6a, and the grids of the impeller and guide vane in Fig. 6b. 4. Results and discussion 4.1. Aerodynamic performance Limited by the complex field conditions during actual operation, the present simulation was not compared with the experiment results owing to the fact that the noise sources include not only the fan but also other rotating devices, and because it is extremely

Fig. 6. The grid diagram of the axial fan.

X. Ye et al. / Energy 129 (2017) 185e200

difficult to conduct field tests under a stagger angle anomaly of the rotating blades during actual operation; in addition, the deviation of an abnormal stagger angle occurring during actual operation is not immediately available, and hence the measurement of noise becomes infeasible under field conditions. However, the simulated performance under normal conditions was compared with a sample curve provide by the manufacturer [18], as shown in Fig. 7. It can be seen that, under a grid number of 2.35 million, the average relative offset of the total pressure rise was 1.5% for a volumetric flow rate of 33e44 m3/s, indicating that the present results are in good agreement with the conducted experiments. Therefore, the present modelling can be considered viable and accurate. Fig. 8 shows a variation in the total pressure rise under different values of Db. It can be clearly seen that the fan performance varies appreciably under a small Db of 10
, but deteriorates distinctly when Db reaches beyond 20
. For the designed flow rate, when Db ¼ 10
, the total pressure rise reduces slightly by 0.7%, and once Db  20
is reached, the total pressure rise declines considerably, e.g., compared with Db ¼ 0
, the total pressure rise drops by 25%, 46%, 48% and 53% for Db ¼ 20
, 30
, 40
and 50
, respectively. These variations indicate that the deviation angle has a clearly adverse effect on the fan performance. In addition, a minor reduction of the total pressure rise under Db ¼ 30
e50
is possibly caused by the constrained change in the flow field exerted by the large deviation angle [2]. 4.2. Pressure pulsation characteristics 4.2.1. Time-domain characteristics To investigate the static pressure variation over time under different deviations in the stagger angles, the time-domain distribution of the static pressure within a rotation period at the designed flow rate is shown in Fig. 9. A close inspection of this figure shows that under normal conditions of the stagger angle regulation, the static pressure in each monitoring point presents a regularly periodic feature, or a similarly periodic feature with 14 significant peaks. This is due to the fact that the rotating blades strike the ambient air periodically, resulting in periodic excitations on the rotating blades at the monitoring point. Additionally, it should be pointed out that, as shown in Fig. 9, along the airflow direction, the time-averaged static pressure for all monitoring points, P1eP4, shows an increasing tendency, whereas the pulsation amplitude shows a decreasing tendency. This is because, along the flow direction from P1 to P4, the airflow obtains

Fig. 7. Comparison of experimental and numerical performance.

189

Fig. 8. The effect of different Db on the total pressure rise.

mechanical energy continuously, leading to an increment of the static pressure; meanwhile, when the airflow moves away from the impeller, which provides energy, the pulsation amplitude is subsequently reduced. Under abnormal conditions, the static pressure of P1 in the impeller inlet shows abnormal fluctuations during the period of 0.01e0.02 s, whereas the number of peak points does not change, as shown in Fig. 9a. The main features of this are highlighted as follows: the static pressure of the peak points increases significantly, whereas that of the valley points declines remarkably; in addition, the pulsation amplitude between the peak and valley points is more prominent with an increase in Db. Under the case of Db ¼ 50
, in particular, the pulsation amplitude is approximately 11-times that under normal conditions during the period of 0.01e0.02 s; however, the variation in pulsation amplitude is not distinct in the other period. For monitoring point P2 at the tip clearance, as shown in Fig. 9b, an abnormal increment of the static pressure appears roughly at 0.01 s and is notably enlarged with an increase in Db; when Db is 10
, an abnormal peak emerges at 0.01 s, whereas the influence of the abnormal deviation in the stagger angle is expanded to t ¼ 0.015 s when Db ¼ 50
. The effects of Db on the static pressure at P3 and P4 are basically similar, as shown in Fig. 9c and d. Under normal conditions, the static pressure is positive, whereas under abnormal conditions, the static pressure tends to be negative, and the time-domain distributions display severe fluctuations. The distinguishing feature here is that the prominent fluctuation is expanded to the time period of 0.01e0.03 s, and presents a V-type profile. The reason for this is possibly the attack angle of the airflow being increased by the change in flow angle when the airflow passes the abnormal blade, and the blade then violently striking the airflow passing the impeller, leading to an enlargement of the pulsation amplitude of the static pressure with an increase in Db, and the crucial influence on the pressure pulsation during the subsequent time period. It can be concluded from the above analysis that the time-domain characteristics of the static pressure at each monitoring point are distinctly different after the occurrence of an abnormal deviation in the stagger angle, whereas the ranges of the abnormal fluctuation are analogous, and the influencing degree is aggravated with an increase in Db. 4.2.2. Distribution of pressure pulsation intensity To further examine the transient flow characteristics, the pressure pulsation intensity at each monitoring point under various

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Fig. 9. Characteristics of the static pressure in the time domain.

values of Db is presented in Fig. 10. The static pressure coefficient Cp is expressed as

Cp ¼

p ; 0:5ru22

where p and r denote the static pressure and fluid density, respectively. The averaged pressure coefficient in a rotation period and the pressure pulsation intensity are formulated as follows [26]:

Cp ¼

1X Cp ðtÞ N

Cp*

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 X Cp ðtÞ  C p ; ¼ N

where N is the length of the data. Fig. 10 shows that the averaged pressure pulsation intensity (APPI) of P2 is much greater than that of the other monitoring points. This possibly results from the impact of the complex flows, including the tip leakage vortex and secondary flow that emerge at the tip clearance, thereby exhibiting a severe pressure pulsation. Once the stagger angle is regulated abnormally, the APPI variation of P2 is not significant when the value of Db changes, whereas for the other monitoring points, P1, P3, and P4, an obvious increase in the APPI is observed. This is because the static pressure level at the tip clearance is much higher than that at the other monitoring

X. Ye et al. / Energy 129 (2017) 185e200

Fig. 9. (Continued).

191

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X. Ye et al. / Energy 129 (2017) 185e200

These results show that an abnormal blade has a weak influence on the APPI at the tip clearance, but has a significant impact on the APPI at the downstream impeller outlet.

Fig. 10. Pressure pulsation intensity under different deviation degrees.

points (as shown in Fig. 9), and hence an abnormal blade has a limited effect on the static pressure at P2; however, in the inlet and outlet domains of the impeller, as well as at the outlet of the guide vane, an abnormal blade has a pronounced impact on the local static pressure with an increase in Db, leading to an augmentation of the APPI and an enhanced interference effect on the downstream flow. Furthermore, the APPI increment at P3 is the largest among all of the points, and the increments at P1 and P4 are relatively small.

4.2.3. Time-frequency characteristics A time-frequency analysis can be used to describe the variations in signal energy in the time and frequency domains, including linear and quadratic time-frequency distributions. A quadratic time-frequency distribution can highlight the characteristics of a signal more intuitively and rationally [27]. As a representative distribution, although a Wigner-Vill distribution has a good timefrequency resolution, it cannot eliminate the cross-interference terms. By addressing the elimination of the cross-interference terms in a Wigner-Vill distribution, a Choi-Williams distribution exhibits an excellent resolution and evidently does not increase the computational burden. Hence, a Choi-Williams distribution was selected for the present study. Fig. 11 illustrates the Choi-Williams time-frequency distribution of pressure signals at different monitoring points under normal conditions. It can be seen from Fig. 11(a)e(c) that the pressure pulsation energy (PPE) of P1eP3 converges within a frequency band of 200e400 Hz, and the maximum PPE appears near 280 Hz, which is consistent with the blade-passing frequency and is the main frequency component of a pressure pulsation. Similar results were shown by Liu and Huang [28], which is largely due to the fact that, when rotating blades strike the airflow periodically, a periodical pulsation of static pressure is generated, causing a clear peak at the blade-passing frequency of 280 Hz. A close inspection of Fig. 11 shows that PPE peaks of P2 and P3 emerge at the double

Fig. 11. Time-frequency distribution at designed flow rate.

X. Ye et al. / Energy 129 (2017) 185e200

193

Fig. 12. Time-frequency distribution of P2 under different deviation degrees.

blade-passing frequency, but their value is far less than that at 280 Hz. The PPE of P4 is mainly gathered within the frequency band of 0e400 Hz, and the maximum PPE still emerges within the vicinity of 280 Hz; however, a high PPE is also observed when the frequency is below 280 Hz. This indicates that the rotation of the impeller is not the only factor affecting the pressure pulsation. Within the frequency band of a higher PPE, the energy clearly fluctuates over time; in addition, the variations in the amplitude of the PPE of P2 and P3 are notable, whereas such a variation for P1 is not apparent. It should be mentioned that the PPE evolution of P4 over time exhibits distinctly diverse characteristics, namely, the PPE is higher in the most time domain with the exception of 0.03e0.04 s, which has a lower PPE. The PPE of P2 is generally higher than that of the other

monitoring points, as shown in Fig. 10. Hence, the time-frequency distribution of P2 under an abnormal regulation of the stagger angle was examined, as shown in Fig. 12. After the abnormal deviation of the stagger angle, the effect of the deviation degree on the time-frequency distribution is distinct. Compared with normal conditions, when Db is 10
, an abnormal PPE distribution is generated at t ¼ 0.01e0.02 s, which is coincident with the range of abnormal fluctuation indicated in Fig. 9b. In addition, a low PPE emerges in the other frequency domains exclusive of the 1- to 2fold increase in the blade-passing frequency. When Db ¼ 20
, the following prominent features are highlighted: the PPE decreases distinctly at t ¼ 0.013e0.025 s, whereas the reduced amplitude is relatively small at t ¼ 0e0.013 s, and the energy distribution shows no obvious change at t > 0.025s. When Db ¼ 30
, the maximum PPE

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Fig. 13. Static pressure distribution at the outlet stream surface.

X. Ye et al. / Energy 129 (2017) 185e200

195

Fig. 13. (continued).

is trigged at t ¼ 0.012 s; meanwhile, the PPE is observed over the entire frequency band, whereas the PPE is reduced when t < 0.012s. When Db increases to 40
and 50
, the PPE distributions show little change at t ¼ 0.01e0.02 s, whereas in the other time domains they clearly diminish. It can be drawn from the above discussion that the maximal PPE is increased as Db increases; therefore, the larger the deviation degree of the stagger angle that occurs, the more remarkable the effect of an abnormal deviation of the blade on the time-frequency distribution. 4.3. Characteristics of transient flow field 4.3.1. Distribution of static pressure In this section, an internal transient flow field is investigated to explain the features mentioned above. The results shown in Fig. 9 indicate that, after an abnormal deviation of the stagger angle, abnormal fluctuations of the static pressure pulsation are evolved, and the start and end times of the abnormal fluctuations are approximately 0.01 and 0.02 s, respectively. Hence, the following investigation is focused on the transient flow field at t ¼ 0.01, 0.015, and 0.02 s. Fig. 13 shows the static pressure distribution at the outlet stream surface of the impeller, and marks the positions of monitoring point P3 and the abnormal blade, as well as the rotating direction. Under the normal conditions shown in Fig. 13a, the static pressure presents a periodical and symmetrical distribution in the circumferential direction. The static pressure increases from the hub to the casing in the radial direction, and the high-pressure and low-

pressure regions are alternately distributed in the circumferential direction; in addition, the static pressure in the middle and upper parts of the stream surface is much greater than that at the bottom. Fig. 13a shows that the static pressure distributions at three moments are basically similar, and the differences in the contour lines are not easily identifiable. Under abnormal conditions, the periodical distribution of the static pressure is clearly hindered with an increase in Db, and an abnormal variation is developed along the reverse rotation direction. When Db ¼ 10
, the static pressure in the flow passage adjacent to the abnormal blade in the reverse rotating direction is reduced, and the increasing tendency of the static pressure from the hub to the casing is clearly eliminated; additionally, a small low-pressure region appears in the middle and upper parts of the stream surface, and both the maximum and minimum static pressures are decreased compared with those under normal conditions. When Db ¼ 20
, both the maximum and minimum static pressures are further reduced, with a significant reduction occurring in the minimum static pressure, and an evident low-pressure region emerges, indicating an expansion of the influence of the abnormal blade. Once Db reaches 30
e50
, one of the prominent features is a reduction in the minimum static pressure with the increase in Db, whereas the maximum static pressure remains constant. Another feature is a visible increase in the pressure gradient in the abnormal static pressure region. The static pressure variation at P3, shown in Fig. 13bef, is in accord with the pressure pulsation in Fig. 9c, which can be explained as follows: At t ¼ 0.01 s, the monitoring point P3 is in front of the abnormal blade, and P3 starts to enter the abnormal region of the static pressure. At

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Fig. 14. Total pressure distribution in the spanwise cross section at 90% of the blade height.

t ¼ 0.015 s, P3 is in the vicinity of the centre of the abnormal static pressure region, and the static pressure at this point is relatively lower. Finally, when t ¼ 0.02 s, P3 starts to move away from the abnormal pressure region. 4.3.2. Distribution of total pressure To clarify the heterogeneity of the flow field induced by the abnormal blade, Fig. 14 depicts the transient distribution of the total pressure in a span-wise cross section at 90% of the blade height, and both the flow and rotation direction are labelled. As mentioned above, the flow field adjacent to the abnormal blade in the reverse rotation direction is considerably affected; therefore, the flow field including the abnormal blade and its adjacent blades

(blades A and B in Fig. 10) was selected to examine the transient effect of the abnormal blade. Under normal conditions, the total pressure is generally increased owing to the work effect on the airflow exerted by the impeller, and the maximum total pressure appears in the minor regions of the leading and trailing edges, whereas the minimum negative value occurs at the suction side. With an increase in time, the transient distribution of the total pressure is not altered as a whole. Under the abnormal conditions of Db ¼ 10
, the high-pressure region at the trailing edge disappears, whereas the low-pressure region at the suction side expands. In addition, the total pressure level is weakened, and the total pressure distribution is highlighted with a tiny change at the impeller outlet. When Db ¼ 20
, as compared with normal

X. Ye et al. / Energy 129 (2017) 185e200

197

Fig. 15. Contour of time-averaged static pressure at the mid-section of the stream surface.

conditions, the low-pressure region at the suction side moves towards the leading edge, the maximum total pressure only occurs around the leading edge of the abnormal blade, and the maximum and minimum total pressures increase and decrease, respectively. At t ¼ 0.01 s, a suction area develops at the pressure side of blade B, whereas at t ¼ 0.015 and 0.02 s, the suction area formed completely vanishes. When Db increases to 30
, the low-pressure region moves towards the leading edge as well, and not only the high-pressure region emerges at the pressure side of blade A, the region also shrinks with an increase in time. Compared with the case of Db ¼ 30
, the high-pressure region around the abnormal blade is enlarged when Db ¼ 40
, and basically remains unchanged over time. When Db ¼ 50
, the maximum and minimum total pressures increase and decrease, respectively; in addition, the high-pressure region at the pressure side of the abnormal blade is further expanded, and the low-pressure region at the impeller inlet moves from blade A towards blade B. 4.3.3. Distribution of time-averaged static pressure The time-averaged static pressure can reflect the changes in static pressure over time and the unsteady characteristics of an

internal flow field. Thus, Fig. 15 shows the distribution of timeaveraged static pressure at the mid-section of the stream surface. As shown in Fig. 15a, under normal conditions, the time-averaged static pressure in the flow passage decreases from the pressure side to the suction side, and the minimum static pressure appears near the blade tip. After abnormally deviating from Db ¼ 10
, as shown in Fig. 15b, the high-pressure region at the pressure side of the abnormal blade expands, and the low-pressure region emerges at the suction side near the hub, the reason for which is the interaction between the vortex separating from boundary layer owing to the abnormal blade and the strengthening of the main flow, leading to a visible variation in the time-averaged static pressure distribution at the pressure and suction sides of the abnormal blade. However, this variation is not clearly observed in other flow passages exclusive of the abnormal blade. When Db increases further, as shown in Fig. 15cef, the low-pressure region at the suction side of the abnormal blade is amplified and the lowpressure value is gradually diminished; when Db reaches 50
, in particular, the pressure within the entire flow passage of the suction side of the abnormal blade is negative. Meanwhile, the flow field in the flow passage adjacent to the abnormal blade is also

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Fig. 16. Approximate entropy distribution under different deviation degrees.

influenced to a certain extent, and the low-pressure region on the suction side of the adjacent blade moves from the middle part to the bottom part of the blade. Consequently, the impact of the abnormal blade on the time-averaged static pressure is intensified with an increase in Db. 4.4. Feature extraction 4.4.1. Approximate entropy The concept of approximate entropy was first proposed by Pincus, and can be used to characterize the complexity of the signals and quantify the unpredictability of fluctuations in a time series [29]. The more complex the signal is, the greater the approximate entropy that occurs. This can provide a reliably estimated value based on a small amount of data, and has strong antinoise and anti-jamming capabilities. The approximate entropy is expressed as follows:

Ea ðm; r; NÞ ¼ Fm ðrÞ  Fmþ1 ðrÞ; where r is the tolerance and m is the pattern dimension. In this study, m ¼ 2 and r ¼ 0.2S are selected, where S is the standard deviation of the original data [30]. Fig. 16 shows the approximate entropy distribution at each monitoring point under different deviations of the stagger angles and flow rates. It can be seen from Fig. 16 that the approximate entropy of each monitoring point has different features. Under normal conditions, the approximate entropy of P1 at the impeller inlet is at a low level and changes slightly with the flow rate. Under

abnormal conditions, however, the averaged approximate entropy increases significantly, and with an increase in Db, it tends to initially increase and then decrease. As for P2 at the tip clearance, the approximate entropy under abnormal conditions is clearly greater than that under normal conditions, and the approximate entropy is increased as a whole as Db increases. For P3 at the impeller outlet, the approximate entropy at Db ¼ 10
is similar to that under normal conditions; when Db increases further, the approximate entropy is clearly reduced. For P4 at the outlet of the guide vane, the averaged approximate entropy after an abnormal deviation is less than that under normal conditions, and roughly diminishes with an increase in Db. The features mentioned above indicate that the approximate entropy of P1 and P2 under normal conditions is smaller than that under abnormal conditions, whereas that of P3 and P4 is adverse, which is due to the fact that the approximate entropy level is closely dependent on the order of complexity. The results shown in Fig. 9 reveal that the pressure signals of P1 and P2 present periodic fluctuations under normal conditions, and that the variations in the values of the peaks and valleys clearly do not change over time; however, the pressure signals after a blade abnormally begins deviating show abnormal fluctuations, resulting in an amplification of the complexity order of the pressure signals and an aggravation of the approximate entropy. As for P3 and P4, under normal conditions, the values of both the peak and valley vary significantly over time, resulting in a notable complexity order of the pressure signals. After a blade begins to abnormally deviate, the pressure signals show a regular V-type profile, and hence the approximate entropy tends to decrease. It can be inferred that the approximate

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Fig. 17. Sample entropy distribution under different deviation degrees.

entropy can be used to identify an abnormal deviation of the stagger angle of an axial fan. 4.4.2. Sample entropy As a modification of the approximate entropy, the sample entropy is also used to evaluate the complexity of the time-series signals [31]. Because the sample entropy excludes self-matches of the statistical vectors, the uncertainty given by the sample entropy is reduced and the characterization capability is promoted. The features used to assess the complexity of the static pressure signals of an axial fan using the sample entropy are presented as follows: the smaller the sample entropy is, the higher the self-similarity of the time-series pressure signals, whereas the greater the sample entropy is, the more prominent the complexity becomes [32]. The sample entropy is expressed as follows:

Es ðm; r; NÞ ¼ Ln½Am ðrÞ=Bm ðrÞ; where A and B denote the number of template vector pairs of data length m and mþ1 having d[Xm(i),Xm(j)] < r, respectively. Here, m ¼ 2 and r ¼ 0.2 were selected. Fig. 17 shows the sample entropy distribution at each monitoring point under different deviations in the stagger angles and flow rates. A close comparison of Figs. 16 and 17 shows that, for the monitoring points of P1 and P2, the variation in sample entropy is similar to that of the approximate entropy, namely, under abnormal conditions, the sample entropy is apparently more than that under normal conditions, and the sample entropy varies distinctly with an increase in the flow rate, which differs from normal conditions.

Roughly, the sample entropy of P3 and P4 tends to reduce with an increase in Db, and the approximate entropy values of P3 and P4 under Db ¼ 10
(shown in Fig. 16) are slightly different from those under normal conditions, whereas the sample entropy is obviously reduced when compared with normal conditions. It can be seen from Figs. 16 and 17 that, for the monitoring points of P1 and P2, both the approximate entropy and the sample entropy are able of characterizing an abnormal deviation well; however, for the monitoring points of P3 and P4, the sample entropy is a good tool for identifying any abnormal deviations.

5. Conclusions The pressure pulsation and transient flow field of a variablepitch axial fan are clearly affected by an abnormal blade and a deviation of the stagger angle. Under normal conditions, the static pressure at each monitoring point appears to be periodic or to have quasi-periodic fluctuations. After an abnormal deviation of the stagger angle, the periodic or quasi-periodic distribution of the pressure fluctuation is clearly hindered, and abnormal fluctuations in the static pressure emerge, as well as the effect of which is enlarged with an increase in the deviation angle. The pressure pulsation at the impeller outlet presents a V-type profile, and the time domain influenced is distinctly expanded. The pressure pulsation intensity of monitoring points P1, P3, and P4 is clearly promoted with an increase in Db, whereas that of monitoring point P2 changes slightly, and the average pressure pulsation intensity at P2 is the largest among all monitoring points. Under normal

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conditions, the maximum pressure pulsation energy of P2 is emerged at the blade-passing frequency, whereas the timefrequency distribution of P2 is altered from the abnormally regulating blade, and the maximum pressure pulsation energy is increased when the deviation in the stagger angle increases. Under abnormal conditions, the periodic distribution of the transient flow field including the static pressure at the impeller outlet, the total pressure at a 90% cross-section of the blade height, and the time-averaged static pressure at mid-section stream surface are all hindered to a certain extent. The features of the transient flow field clearly change over time based on the degree of deviation, and abnormal high- and low-pressure regions are developed. The scope of the abnormal pressure region is basically unchanged over time, whereas the position of the abnormal highor low-pressure region is clearly altered. The results of a feature extraction indicate that the levels of both the approximate entropy and the sample entropy at each monitoring point are changed in the presence of an abnormal stagger angle, particularly under the conditions of a severe angle deviation. Both the approximate entropy and the sample entropy can be used to identify an abnormal blade deviation at all monitoring points; however, the sample entropy is more suitable for a characterization of the effect of the angle deviation on the pressure signals at the impeller and guide vane outlets. Acknowledgements This work has been supported by Fundamental Research Funds for the Central Universities of Ministry of Education of China (Grant No. 13MS98). References [1] Ye XM, Li PM, Li CX, Ding XL. Numerical investigation of blade tip grooving effect on performance and dynamic of an axial flow fan. Energy 2015;82: 556e69. [2] Li CX, Li Q, Ding XL, Ye XM. Performance, aeroacoustics and feature extraction of an axial flow fan with abnormal blade angle. Energy 2016;103:322e39. [3] Song B. Cause analysis of blade angle drifting in a two-stage axial-flow primary fan. Electr Eng 2009;11:67e9 (in Chinese). [4] Hao SY. Preventive measures and influencing factors on reliability of axial flow fan in power stations. Cogener Power Technol 2010;3:52e3 (in Chinese). [5] Xu C, Amano RS. Unsteady pressure field investigation of an axial fan-blade unsteady pressure field measurement. Int J Rotat Mach 2002;8:385e95. [6] Chen HS, Liang XZ, Tan QC, Kang S. Experimental study at the exit flow field of an axial flow fan with variable setting angle. J China Coal Soc 2000;25:412e5 (in Chinese). ndez Oro JM, Argüelles Diaz KM, Santolaria Morros C, Blanco [7] Ferna Marigorta E. Unsteady flow and wake transport in a low-speed axial fan with inlet guide vanes. J Fluid Eng T ASME 2007;129:1015e29. [8] Zhang L, Wang R, Wang SL. Simulation of broadband noise sources of an axial fan under rotating stall conditions. Adv Mech Eng 2014:507079. [9] Nouri H, Ravelet F, Bakir F, Rey R. Design and experimental validation of a ducted counter-rotating axial-flow fans system. J Fluid Eng T ASME 2012;134:

104504e9. [10] Ye XM, Li J, Li CX, Wang SL. Aerodynamics and operating performance of a variable pitch axial fan with asynchronous regulation of installation angles of multiple blades. J Proc CSEE 2010;30:77e83 (in Chinese). [11] Li CX, Yin P, Ye XM. Numerical simulation on performance and acoustic characteristics with deeply abnormal installation angle of single blade in a variable pitch axial fan. J Proc CSEE 2012;32:122e8 (in Chinese). [12] Hirata K, Iida Y, Takushima A, Funaki J. Instantaneous pressure measurement on a rotating blade of cross-flow impeller. J Environ Eng 2008;3(2):261e71. [13] Funaki J, Fuchi T, Onishi Y, Hirata K. Measurements of minute unsteady pressure on three-dimensional fan. J Environ Eng 2010;5(2):298e310. [14] Hurault J, Kouidri S, Bakir F. Experimental investigations on the wall pressure measurement on the blade of axial flow fans. Exp Therm Fluid Sci 2012: 29e37. [15] Ghasemian M, Nejat A. Aero-acoustics prediction of a vertical axis wind turbine using Large Eddy Simulation and acoustic analogy. Energy 2015;88: 711e7. [16] Carolus T, Schneider M, Reese H. Axial flow fan broad-band noise and prediction. J Sound Vib 2007;300:50e70. ndez Oro JM, Argüelles Diaz KM, Santolaria Morros C, Galdo Vega M. [17] Ferna Numerical simulation of the unsteady stator-rotor interaction in a low-speed axial fan including experimental validation. Int J Numer Method H 2011;21: 168e97. [18] Солоу Mahuo ребенком TC. Ventilator pneumatic schematic and performance curve. Beijing, China: China Coal Industry Press; 1986. p. 336e48 (in Chinese). [19] Li CX, Wang SL, Jia YK. The performance of a centrifugal fan with enlarged impeller. Energy Convers Manag 2011;52:2902e10. [20] Zhang L, Wang R, Yuan W, Wang SL. Simulation of air jets for controlling stall in a centrifugal fan. Proc IMechE Part C J Mech Eng Sci 2015;29:2045e55. [21] Li Y, Liu J, Ouyang H, Du ZH. Internal flow mechanism and experimental research of low pressure axial fan with forward-skewed blades. J Hydrodyn 2008;20:299e305. [22] Ye Li, Zhou SQ, Wang J, Li H, Fu GJ. Analysis of aerodynamic noise of a sweptcurved axial flow fan with different rotate speed. J Eng Thermophys 2014;35(1):51e5 (in Chinese). [23] Kameier F, Neise F. Experimental study of tip clearance losses and noise in axial turbomachines and their reduction. J Turbomach 1997;119(3):460e71. [24] Fang HH, Xie J, Li LJ. Numerical study on energy saving and noise reduction for axial flow fans. Min Process Eq 2010;3:25e8 (in Chinese). [25] Chen QG, Fang F, Zhang ZD, Jia XX. Analysis on pressure fluctuation of the interior flow field in impellers of a counter-rotating axial fan. J Shandong Univ Sci Tech Nat Sci 2011;30:81e5 (in Chinese). [26] Wang WJ, Yuan SQ, Pei J, Zhang J, Yuan J. Numerical analysis of the clocking effect on the pressure fluctuation in the centrifugal pump with vaned diffuser. J Mech Eng 2015;51:185e92 (in Chinese). [27] Hu JS. The rotating machinery faults diagnosis oriented empirical mode decomposition time-frequency analysis method and experiment study (PhD dissertation). Hangzhou, Zhejiang: College of Mechanical and Energy Engineering, Zhejiang University; April, 2003. p. 13e34 (in Chinese). [28] Liu T, Huang QB. Characteristic information extraction of pressure pulsation signal in axis-flow pump based on EEMD and HT. J Mech Elec Eng 2012;29: 278e81 (in Chinese). [29] Pincus SM. Approximate entropy as a measure of system complexity. Proc Natl Acad Sci 1991;88:2297e301. [30] Kaffashi F, Foglyano R, Wilson CG, Loparo K. The effect of time delay on approximate and sample entropy calculations. Phys D Nonlinear Phenom 2008;237:3069e74. [31] Richman JS, Moorman JR. Physiological time-series analysis using approximate entropy and sample entropy. Am J Physiol Heart Circ Physiol 2000;278: H2039e49. [32] Zurek S, Guzik P, Pawlak S, Kosmider M, Piskorski J. On the relation between correlation dimension, approximate entropy and sample entropy parameters, and a fast algorithm for their calculation. Phys A Stat Mech Appl 2012;391: 6601e10.