Inlet Gap Effect on Tonal Noise Generated from a Voluteless Centrifugal Fan

: In this study, three voluteless centrifugal fans are compared for their aeroacoustic performances. The tonal noise is predicted by coupling the IDDES with Formulation 1A of Farassat. The sources of the tonal noise at the blade passing frequency ( BPF ) are identiﬁed. It is found that the sources are related to the fan inlet gap, which introduces higher velocity intensities and turbulent ﬂuctuations interacting with the blade leading edge. By redesigning the gap, the tonal noise at the BPF is reduced effectively.


Introduction
Today, most people spend the majority of their time indoors.The indoor environmental quality (IEQ) has become more and more important.When considering IEQ, we usually think about temperature, CO 2 level, and humidity.However, it has been noticed that sound quality is an important factor for good comfort in the indoor environment [1,2].
Nowadays, a ventilation system is usually driven by a voluteless centrifugal fan, which has a gap between the rotating fan front shroud and the stationary inlet duct.The pressure difference between the inner and outer sides of the fan drives air to pass through the gap.As clarified by Hariharan and Govardhan [3], increasing the gap width worsens the blade aerodynamic performance.
There are some previous studies on voluteless centrifugal fans.It was found in both simulations and experiments for a voluteless fan [4] that the tonal noise at BPF is generated from a helical unsteady inlet vortex that interacts with the rotating blades near the fan backplate.Another cause is inflow distortion, which leads to flow separation at the blade root near the backplate [5].To solve the inflow distortion, flow obstructions were suggested to be placed upstream of the fan inlet [6].The shape and location of obstructions were found as the key parameters for noise reduction.Schaefer and Boehle [7] found, using the Lattice-Boltzmann solver, that the accuracy of the noise prediction, especially at BPF, is improved when the mesh is refined at the gap and the fan outlet.However, they provided no discussions on the physical mechanisms that are associated with the improved accuracy.In a recent study, the present authors found that for a voluteless centrifugal fan, the tonal frequencies associated with the BPF are related to wall pressure fluctuations at the blade leading edge (BLE) [8].This is produced when turbulent structures, which are generated by the flow through the gap, interact with the BLE near the shroud.To resolve these turbulent structures, the mesh had to be refined at the gap region.
The numerical simulations in the current study are carried out using a hybrid method coupling the improved delayed detached eddy simulation (IDDES) [9] for the flow simulation with the Ffowcs Williams and Hawkings (FW-H) equation [10] for the noise prediction.The IDDES is used in the flow simulation, and the FW-H is used for the noise prediction.
This study aims to investigate how the tonal noise at the BPF is affected when the gap is modified.Two different gap designs are compared with a reference fan (Case 1), which is the same fan in the previous study [8].

Configuration
The baseline fan (Case 1) and the two different designs (Case 2 and Case 3) are illustrated in Figure 1.Case 1 is the same fan as the one examined in [8].The fan geometry is the same for all three cases, meaning that the blades, shroud, and backplate are the same.There are seven blades in the fan.A clearance (i.e., the gap) is located between the stationary (brown) and rotating parts (gray).Case 2 has larger gap width (w case2 ), and Case 3 has smaller gap width (w case3 ) than Case 1.The gap width is varied by changing the wall thickness of the inlet duct.The rounded edge of the inlet duct is shorted for Case 2 and extended for Case 3. The inlet duct is moved in the axial direction for the different cases so that the axial overlap between the fan shroud and the inlet duct is the same for all cases.As shown in Figure 1d, the fan and inlet duct are placed in a downstream duct, and the inlet duct is connected to an upstream duct.This simple geometry layout is designed for the numerical simulations.This simplification reduces the geometry complexity but retains the principal flow and acoustic characteristics.The fan and case parameters are listed in Table 1.Here, d 1 is the fan intake diameter at the BLE, and d 2 is the fan diameter at the blade trailing edge.For the downstream duct, h 2 denotes the duct length and d 4 the outlet diameter.For the upstream duct, h 1 is the duct length and d 3 the inlet diameter.In addition, h 3 is the distance between the inlet duct and the microphone M1. simulation with the Ffowcs Williams and Hawkings (FW-H) equation [10] for the noise prediction.The IDDES is used in the flow simulation, and the FW-H is used for the noise prediction.
This study aims to investigate how the tonal noise at the  is affected when the gap is modified.Two different gap designs are compared with a reference fan (Case 1), which is the same fan in the previous study [8].

Configuration
The baseline fan (Case 1) and the two different designs (Case 2 and Case 3) are illustrated in Figure 1.Case 1 is the same fan as the one examined in [8].The fan geometry is the same for all three cases, meaning that the blades, shroud, and backplate are the same.There are seven blades in the fan.A clearance (i.e., the gap) is located between the stationary (brown) and rotating parts (gray).Case 2 has larger gap width (wcase2), and Case 3 has smaller gap width (wcase3) than Case 1.The gap width is varied by changing the wall thickness of the inlet duct.The rounded edge of the inlet duct is shorted for Case 2 and extended for Case 3. The inlet duct is moved in the axial direction for the different cases so that the axial overlap between the fan shroud and the inlet duct is the same for all cases.As shown in Figure 1d, the fan and inlet duct are placed in a downstream duct, and the inlet duct is connected to an upstream duct.This simple geometry layout is designed for the numerical simulations.This simplification reduces the geometry complexity but retains the principal flow and acoustic characteristics.The fan and case parameters are listed in Table 1.Here, d1 is the fan intake diameter at the BLE, and d2 is the fan diameter at the blade trailing edge.For the downstream duct, h2 denotes the duct length and d4 the outlet diameter.For the upstream duct, h1 is the duct length and d3 the inlet diameter.In addition, h3 is the distance between the inlet duct and the microphone M1.The fan rotation speed is 2800 rpm.Given that the fan has seven blades, the BPF is 326.7 Hz.The operation condition is the same as in [7], where the mass flow rate was set to 0.467 kg/s, and this gave a pressure rise of 270 Pa for Case 1.

CFD Method
The air is considered as an ideal gas.The flow is compressible.A finite volume method is utilized to discretize the continuity, momentum, and energy equations.The method employs a segregated flow solver accomplished with the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm.The under-relaxation factors for the velocity and pressure in the segregated flow solver are set to 0.7 and 0.4, respectively.The under-relaxation factor for the turbulence equations is 0.7.All simulations are performed using the commercial software STAR-CCM+ [11].The turbulence is simulated using the IDDES [12] combined with the k-ω SST turbulence model.This setup has been tested in several studies on rotating machinery [13,14].The normal wall sizes of the first layer cells near all walls fulfill ∆y + < 1.

Numerical Settings
The entire computational domain is divided into stationary and rotating parts.The meshes of the stationary and rotating meshes are not conformable at the interfaces between them.
The mass-flow boundary condition is set at the inlet with a uniform velocity distribution.The modeled turbulence intensity is set to I = 4% according to I = 0.16(R e ) −1/8 , where the Reynolds number is calculated based on the inlet diameter, d 3 [11].The modeled turbulence length scale is set to = 0.5 m based on = 0.7d 3 , where d 3 is the upstream duct diameter.The pressure-outlet boundary condition is set at the outlet with the static pressure of 101,325 Pa, which is the reference pressure (p re f ) in the ambient air.The no-slip boundary condition is specified on all walls.
To ensure the fan performance, the incoming flow at the fan inlet should not be distorted.Furthermore, there should be no flow separation near the fan inlet, especially on the wall in front of the fan gap.These conditions are satisfied in the present study by aligning a straight duct upstream with the fan axis and inserting a small section of the duct into the fan inlet to form the clearance between the duct and the fan.
The time step is set to ∆t = 2.0 × 10 −5 s.This gives a maximum convective Courant number of around 10, which is observed at the blade trailing edges.This value fulfills the numerical stability required for the implicit time-marching method.The convective Courant number at the gap region is below 1.The maximum number of inner iterations per time step is set to 12.
The sampling period of the noise is 0.3 s for all cases, corresponding to 14 fan revolutions.The sound pressure level (SPL) is calculated using the von Hann window for 3000 samples per signal section, which leads to a frequency resolution of around 3 Hz.The signal sections do not overlap each other.

FW-H Equation
A hybrid approach is adopted to predict the noise generated from the flow.In this approach, the IDDES is coupled with Formulation 1A of Farassat [15].The ambient air density is set to ρ 0 = 1.225 kg/m 3 and the speed of sound c 0 = 340 m/s.
According to Neise [16], the fan noise generation at low Mach numbers is dominated by dipole noise sources that are derived based on the FW-H equation.Hence, the noise prediction in this study considers only an impermeable integral surface for Formulation 1A.The selected integral surface are the fan blades, shroud, and backplate (see Figure 1), while the upstream and downstream ducts as well as the fan inlet duct are neglected.Indeed, there is a limited acoustic reflection from these walls [17].

Mesh
We adopt the same mesh generation strategy that was developed and evaluated in [8], where the fan is the same as Case 1.A polyhedral mesh generation method was used to produce prism layers near the walls and polyhedral cells in the rest of the computational domain.The use of polyhedral cells for turbomachines was demonstrated in [17,18].The growth rate is set to 1.05, as suggested in [19].The most important finding in the mesh study performed in [8] was that a local mesh refinement has to be made at the regions extending from the gap to the BLE and along the shroud to the blade trailing edge (colored in dark grey), as illustrated in Figure 2. The same mesh distribution and refinement are used for the three cases.The mesh parameters are listed in Table 2.
approach, the IDDES is coupled with Formulation 1A of Farassat [15].The ambien density is set to  = 1.225 kg/m 3 and the speed of sound  = 340 m/s.
According to Neise [16], the fan noise generation at low Mach numbers is domin by dipole noise sources that are derived based on the FW-H equation.Hence, the prediction in this study considers only an impermeable integral surface for Formul 1A.The selected integral surface are the fan blades, shroud, and backplate (see Figu while the upstream and downstream ducts as well as the fan inlet duct are negle Indeed, there is a limited acoustic reflection from these walls [17].

Mesh
We adopt the same mesh generation strategy that was developed and evaluat [8], where the fan is the same as Case 1.A polyhedral mesh generation method was to produce prism layers near the walls and polyhedral cells in the rest o computational domain.The use of polyhedral cells for turbomachines was demonst in [17,18].The growth rate is set to 1.05, as suggested in [19].The most important fin in the mesh study performed in [8] was that a local mesh refinement has to be made regions extending from the gap to the BLE and along the shroud to the blade trailing (colored in dark grey), as illustrated in Figure 2. The same mesh distribution refinement are used for the three cases.The mesh parameters are listed in Table 2.

Identifying Sources for Tonal Noise for Case 1
The contours of vorticity magnitudes ‖ ⃗‖ at the blade leading edge for Cas shown in Figure 3a.There are regions with high vorticity magnitude upstream of the Here, the BLE is marked out with a dashed line.They are generated when the gap fl mixed with the main flow.This phenomenon was also observed in previous studies [ The black dashed line is a monitoring line positioned at the BLE of one blade, a extends from the backplate to the shroud.The monitoring line follows the blade as th rotates, and the wall pressure is monitored for 12 fan revolutions.The root mean sq (RMS) of the wall pressure fluctuations is shown in Figure 3b.At the position neare shroud, the RMS has its highest value.As the distance to the shroud increases

Identifying Sources for Tonal Noise for Case 1
The contours of vorticity magnitudes → ω at the blade leading edge for Case 1 is shown in Figure 3a.There are regions with high vorticity magnitude upstream of the BLE.Here, the BLE is marked out with a dashed line.They are generated when the gap flow is mixed with the main flow.This phenomenon was also observed in previous studies [8,20].The black dashed line is a monitoring line positioned at the BLE of one blade, and it extends from the backplate to the shroud.The monitoring line follows the blade as the fan rotates, and the wall pressure is monitored for 12 fan revolutions.The root mean square (RMS) of the wall pressure fluctuations is shown in Figure 3b.At the position nearest the shroud, the RMS has its highest value.As the distance to the shroud increases, the pressure RMS value decays.At the backplate, the pressure RMS value is 5 Pa, which is approximately 4% of the value at the shroud (129 Pa).
pressure RMS value decays.At the backplate, the pressure RMS value is 5 Pa, which is approximately 4% of the value at the shroud (129 Pa).Moreover, a periodic low-frequency fluctuation in relation to the fan revolution is observed, which was also found in [21].By comparing the three monitoring points, highfrequency fluctuations decay rapidly with increased distance to the shroud.The periodic low frequency is predominant at the middle position and at the backplate.
Based on the band-filtered power spectral density(PSD) of the wall pressure fluctuations, the location and magnitudes of dominant tonal noise sources are evaluated.The results at the tonal frequency  (326.7 Hz) are illustrated in Figure 5.The PSD is calculated using the von Hann window for 3000 samples per signal section, which leads to a frequency resolution of around 3 Hz.The signal sections do not overlap each other.The location of the highest wall pressure fluctuations is at the same position (the BLE close to the shroud) as the region with high vorticity magnitudes, high-pressure RMS value, and largest pressure fluctuation.Hence, the interaction between inlet-gap turbulence and the BLE is responsible for the tonal noise generation.This was also found for Case 1 in a   Moreover, a periodic low-frequency fluctuation in relation to the fan revolution is observed, which was also found in [21].By comparing the three monitoring points, highfrequency fluctuations decay rapidly with increased distance to the shroud.The periodic low frequency is predominant at the middle position and at the backplate.
Based on the band-filtered power spectral density(PSD) of the wall pressure fluctuations, the location and magnitudes of dominant tonal noise sources are evaluated.The results at the tonal frequency  (326.7 Hz) are illustrated in Figure 5.The PSD is calculated using the von Hann window for 3000 samples per signal section, which leads to a frequency resolution of around 3 Hz.The signal sections do not overlap each other.The location of the highest wall pressure fluctuations is at the same position (the BLE close to the shroud) as the region with high vorticity magnitudes, high-pressure RMS value, and largest pressure fluctuation.Hence, the interaction between inlet-gap turbulence and the BLE is responsible for the tonal noise generation.This was also found for Case 1 in a Moreover, a periodic low-frequency fluctuation in relation to the fan revolution is observed, which was also found in [21].By comparing the three monitoring points, highfrequency fluctuations decay rapidly with increased distance to the shroud.The periodic low frequency is predominant at the middle position and at the backplate.
Based on the band-filtered power spectral density(PSD) of the wall pressure fluctuations, the location and magnitudes of dominant tonal noise sources are evaluated.The results at the tonal frequency BPF (326.7 Hz) are illustrated in Figure 5.The PSD is calculated using the von Hann window for 3000 samples per signal section, which leads to a frequency resolution of around 3 Hz.The signal sections do not overlap each other.The location of the highest wall pressure fluctuations is at the same position (the BLE close to the shroud) as the region with high vorticity magnitudes, high-pressure RMS value, and largest pressure fluctuation.Hence, the interaction between inlet-gap turbulence and the BLE is responsible for the tonal noise generation.This was also found for Case 1 in a previous study [8].This approach is therefore selected when the three cases are compared in the following analysis.
previous study [8].This approach is therefore selected when the three cases are compared in the following analysis.

Fan Performance Comparison
The static pressure excluding the reference pressure ( = 101,325 Pa) is displayed along the axial symmetric line for the three cases in Figure 6.All cases show similar pressure amplitudes in the upstream duct of the fan, while differences are seen downstream.Case 3 has the highest pressure, and Case 2 has the lowest.The difference in the pressure rise downstream of the fan is only due to the gap design.Hariharan and Govardhan clarified [3] that when the gap width is increased, the blade aerodynamics performance is worsened.In addition, when the gap width is decreased, the performance is improved, which is seen in Figure 6.According to Lee [20], a smaller gap improves the flow when it turns from axial to radial, and it also improves the flow separation conditions at the blade trailing edge.

Wall Pressure Fluctuations Comparison
The RMS of pressure fluctuations at the monitoring line at the BLE (described in Figure 2a) are illustrated for all cases in Figure 7a.At the position nearest the shroud, the highest pressure RMS value is observed for all cases.The pressures' RMS values have the same physical behavior for all three cases, and when the distance to the shroud increases, the pressure fluctuations decay.From the shroud to the backplate, Case 2 has the lowest pressure RMS value, and Cases 1 and 3 have the highest.At the backplate, the cases have almost the same pressure RMS value, approximately 4% of the maximum value.

Fan Performance Comparison
The static pressure excluding the reference pressure p re f = 101, 325 Pa is displayed along the axial symmetric line for the three cases in Figure 6.All cases show similar pressure amplitudes in the upstream duct of the fan, while differences are seen downstream.Case 3 has the highest pressure, and Case 2 has the lowest.The difference in the pressure rise downstream of the fan is only due to the gap design.Hariharan and Govardhan clarified [3] that when the gap width is increased, the blade aerodynamics performance is worsened.In addition, when the gap width is decreased, the performance is improved, which is seen in Figure 6.According to Lee [20], a smaller gap improves the flow when it turns from axial to radial, and it also improves the flow separation conditions at the blade trailing edge.
previous study [8].This approach is therefore selected when the three cases are compare in the following analysis.

Fan Performance Comparison
The static pressure excluding the reference pressure ( = 101,325 Pa) displayed along the axial symmetric line for the three cases in Figure 6.All cases show similar pressure amplitudes in the upstream duct of the fan, while differences are see downstream.Case 3 has the highest pressure, and Case 2 has the lowest.The differenc in the pressure rise downstream of the fan is only due to the gap design.Hariharan an Govardhan clarified [3] that when the gap width is increased, the blade aerodynamic performance is worsened.In addition, when the gap width is decreased, the performanc is improved, which is seen in Figure 6.According to Lee [20], a smaller gap improves th flow when it turns from axial to radial, and it also improves the flow separation condition at the blade trailing edge.

Wall Pressure Fluctuations Comparison
The RMS of pressure fluctuations at the monitoring line at the BLE (described i Figure 2a) are illustrated for all cases in Figure 7a.At the position nearest the shroud, th highest pressure RMS value is observed for all cases.The pressures' RMS values have th same physical behavior for all three cases, and when the distance to the shroud increase the pressure fluctuations decay.From the shroud to the backplate, Case 2 has the lowes pressure RMS value, and Cases 1 and 3 have the highest.At the backplate, the cases hav almost the same pressure RMS value, approximately 4% of the maximum value.

Wall Pressure Fluctuations Comparison
The RMS of pressure fluctuations at the monitoring line at the BLE (described in Figure 2a) are illustrated for all cases in Figure 7a.At the position nearest the shroud, the highest pressure RMS value is observed for all cases.The pressures' RMS values have the same physical behavior for all three cases, and when the distance to the shroud increases, the pressure fluctuations decay.From the shroud to the backplate, Case 2 has the lowest pressure RMS value, and Cases 1 and 3 have the highest.At the backplate, the cases have almost the same pressure RMS value, approximately 4% of the maximum value.The time-average of the wall pressures at the BLE are shown at different positions for all cases in Figure 7b.The maximum values for all cases occur at the shroud, whereas at the backplate, the values are negative.The amplitudes of the maximum and minimum pressure (error bar) are the largest at the shroud, and it becomes smaller when the distance from the shroud increases for all cases.At the shroud, Case 1 has the highest amplitude.Case 2 has the lowest amplitudes at all positions.At point 2, the maximum positive fluctuations are larger than the magnitude of the negative ones, and it is the same for all cases.
Figure 8 shows the time history of the wall pressure for all cases at point 2 during 12 fan periods.The dashed lines indicate the average value for each case, and as illustrated in Figure 7b, it is lowest for Case 2. For Cases 1 and 3, pressure fluctuations with large amplitudes and high frequencies are observed.Fluctuations are also obvious for Case 2, but the amplitudes are smaller.Moreover, a periodic low-frequency fluctuation in relation to the fan rotation is seen for Case 2, which was also found in [21].For Cases 1 and 3, there is no clear periodic low-frequency fluctuation.For all the cases, the maximum absolute values of the positive fluctuations are larger than those of the negative fluctuations.The reason is due to the upstream turbulent vortex impingement.Figure 8 shows the time history of the wall pressure for all cases at point 2 during 12 fan periods.The dashed lines indicate the average value for each case, and as illustrated in Figure 7b, it is lowest for Case 2. For Cases 1 and 3, pressure fluctuations with large amplitudes and high frequencies are observed.Fluctuations are also obvious for Case 2, but the amplitudes are smaller.Moreover, a periodic low-frequency fluctuation in relation to the fan rotation is seen for Case 2, which was also found in [21].For Cases 1 and 3, there is no clear periodic low-frequency fluctuation.For all the cases, the maximum absolute values of the positive fluctuations are larger than those of the negative fluctuations.The reason is due to the upstream turbulent vortex impingement.
The contours of the velocity magnitudes, |v R |, defined in Equation ( 1), in the y-z plane (see Figure 1) and the streamlines (colored in gray) of the relative velocity vectors in Plane 1 (location see Figure 1a) are shown for all cases in Figure 9.Note that the axial velocity component along the fan rotation axis is excluded from the vectors.For Case 1 and 3, there are regions with large velocity magnitudes near the shroud.These regions appear periodically in relation to the blade positions.These regions cannot be observed for Case 2. This suggests that the flow near the shroud is highly fluctuating due to the gap turbulence and that Case 2 has less turbulence above the blades (near the fan inlet).The contours of the velocity magnitudes, | |, defined in Equation ( 1), in th plane (see Figure 1) and the streamlines (colored in gray) of the relative velocity ve in Plane 1 (location see Figure 1a) are shown for all cases in Figure 9.Note that the velocity component along the fan rotation axis is excluded from the vectors.For C and 3, there are regions with large velocity magnitudes near the shroud.These re appear periodically in relation to the blade positions.These regions cannot be obse for Case 2. This suggests that the flow near the shroud is highly fluctuating due to th turbulence and that Case 2 has less turbulence above the blades (near the fan inlet).In the previous studies of Case 1 where the numerical prediction was compared  The contours of the velocity magnitudes, | |, defined in Equation ( 1), in the y-z plane (see Figure 1) and the streamlines (colored in gray) of the relative velocity vectors in Plane 1 (location see Figure 1a) are shown for all cases in Figure 9.Note that the axial velocity component along the fan rotation axis is excluded from the vectors.For Case 1 and 3, there are regions with large velocity magnitudes near the shroud.These regions appear periodically in relation to the blade positions.These regions cannot be observed for Case 2. This suggests that the flow near the shroud is highly fluctuating due to the gap turbulence and that Case 2 has less turbulence above the blades (near the fan inlet).In the previous studies of Case 1 where the numerical prediction was compared with experimental data [17,21], it was shown that SPL at  was best-predicted upstream of the fan.The SPL predicted upstream of the fan (microphone M1) for the tonal frequency  is compared between the cases in Figure 10.Case 1 and 3 have almost the same  In the previous studies of Case 1 where the numerical prediction was compared with experimental data [17,21], it was shown that SPL at BPF was best-predicted upstream of the fan.The SPL predicted upstream of the fan (microphone M1) for the tonal frequency BPF is compared between the cases in Figure 10.Case 1 and 3 have almost the same BPF amplitude.The lowest BPF amplitude has Case 2, where the level decreased by 5.7 dB compared with Case 1.These results agree with the results from Figures 7-9, where Case 2 had the lowest wall pressure on the BLE.

|𝒗 | = 𝑣 𝑣
The cases are compared for their aerodynamic and acoustic performance in Table 3. Increasing the gap size reduces the tonal noise at the BPF, and the static pressure rise also decreases.
amplitude.The lowest  amplitude has Case 2, where the level decreased by 5.7 dB compared with Case 1.These results agree with the results from Figures 7-9, where Case 2 had the lowest wall pressure on the BLE.The cases are compared for their aerodynamic and acoustic performance in Table 3. Increasing the gap size reduces the tonal noise at the BPF, and the static pressure rise also decreases.The results of the wall pressure fluctuations at the  are illustrated for the three cases in Figure 11.Here, only magnitudes above 1.5 Pa 2 /Hz are visualized with colorful contours.The location of the highest wall pressure fluctuations is at the same position (the BLE close to the shroud) for all cases.The differences are the magnitude and the size of the area with high magnitude.Case 1 and Case 3 have the largest sound pressure (see Figure 10), and they have also the largest area and magnitude for the tonal frequency.The high-energy locations are consistent with the wall pressure fluctuations indicated in Figure 7.The high energy is caused by the interaction between inlet-gap vortices associated with the gap turbulence and the BLE [8].According to Lee [20], the gap gives rise to a local jet, and the velocity magnitude of the air flowing through the gap increases with decreased gap size.Higher velocity leads to more inlet-gap vortices that interact with the BLE, which is also seen in Figure 11, where Cases 1 and 3 have higher PSD value for the  compared with Case 2.  The results of the wall pressure fluctuations at the BPF are illustrated for the three cases in Figure 11.Here, only magnitudes above 1.5 Pa 2 /Hz are visualized with colorful contours.The location of the highest wall pressure fluctuations is at the same position (the BLE close to the shroud) for all cases.The differences are the magnitude and the size of the area with high magnitude.Case 1 and Case 3 have the largest sound pressure (see Figure 10), and they have also the largest area and magnitude for the tonal frequency.The high-energy locations are consistent with the wall pressure fluctuations indicated in Figure 7.The high energy is caused by the interaction between inlet-gap vortices associated with the gap turbulence and the BLE [8].According to Lee [20], the gap gives rise to a local jet, and the velocity magnitude of the air flowing through the gap increases with decreased gap size.Higher velocity leads to more inlet-gap vortices that interact with the BLE, which is also seen in Figure 11, where Cases 1 and 3 have higher PSD value for the BPF compared with Case 2.

Conclusions
The tonal noise at the BPF is compared for three different gap sizes for a voluteless centrifugal fan.The present study is motivated by a previous study [8], where the gap flow was found to play an important role in the tonal noise generation.The flow is

Figure 1 .
Figure 1.The fan configurations.Gray indicates the rotating fan and brown the stationary inlet duct.(a) Case 1 (baseline), (b) Case 2 (with a larger gap width = w), and (c) Case 3 (with a smaller gap

Figure 1 .
Figure 1.The fan configurations.Gray indicates the rotating fan and brown the stationary inlet duct.(a) Case 1 (baseline), (b) Case 2 (with a larger gap width = w), and (c) Case 3 (with a smaller gap width = w).(d) The simple geometry layout for the numerical simulations.M1 is the microphone position.The rotation axis of the fans is the x-axis.

Figure 2 .
Figure 2. Important mesh regions: (a) Mesh refinement regions (dark grey) at the inlet ga blades top region.Mesh cells near (b) the blade trailing edge and (c) the inlet gap.

Figure 2 .
Figure 2. Important mesh regions: (a) Mesh refinement regions (dark grey) at the inlet gap and blades top region.Mesh cells near (b) the blade trailing edge and (c) the inlet gap.

Figure 3 .
Figure 3. Turbulence at the BLE: (a) Vorticity magnitude ‖ ⃗‖ at one blade.The black dashed line marks the monitoring line at the BLE; (b) the RMS of the pressure fluctuations with respect to the normalized length along the monitoring line.The time history of the wall pressure at the monitoring line is shown in Figure 4.At the position on the shroud, the pressure fluctuates with large amplitudes and high frequencies.As the distance from the shroud increases, the amplitudes of pressure fluctuations decrease.Small fluctuations are observed at the middle position.At the backplate, fluctuations are almost negligible.

Figure 4 .
Figure 4.The time history of wall pressure fluctuations at three locations along the BLE.

Figure 3 .
Figure 3. Turbulence at the BLE: (a) Vorticity magnitude → ω at one blade.The black dashed line marks the monitoring line at the BLE; (b) the RMS of the pressure fluctuations with respect to the normalized length along the monitoring line.The time history of the wall pressure at the monitoring line is shown in Figure 4.At the position on the shroud, the pressure fluctuates with large amplitudes and high frequencies.As the distance from the shroud increases, the amplitudes of pressure fluctuations decrease.Small fluctuations are observed at the middle position.At the backplate, fluctuations are almost negligible.

Figure 3 .
Figure 3. Turbulence at the BLE: (a) Vorticity magnitude ‖ ⃗‖ at one blade.The black dashed line marks the monitoring line at the BLE; (b) the RMS of the pressure fluctuations with respect to the normalized length along the monitoring line.The time history of the wall pressure at the monitoring line is shown in Figure 4.At the position on the shroud, the pressure fluctuates with large amplitudes and high frequencies.As the distance from the shroud increases, the amplitudes of pressure fluctuations decrease.Small fluctuations are observed at the middle position.At the backplate, fluctuations are almost negligible.

Figure 4 .
Figure 4.The time history of wall pressure fluctuations at three locations along the BLE.

Figure 4 .
Figure 4.The time history of wall pressure fluctuations at three locations along the BLE.

Figure 5 .
Figure 5. PSD of the wall the wall pressure fluctuations at BPF (326.7 Hz).

Figure 6 .
Figure 6.The pressure along the axial axis of the fan across the computational domain.Here, x = −2 corresponds to the location near the outlet and x = 4 near the inlet.The fan location is marked out with the red zone.

Figure 5 .
Figure 5. PSD of the wall the wall pressure fluctuations at BPF (326.7 Hz).

Figure 5 .
Figure 5. PSD of the wall the wall pressure fluctuations at BPF (326.7 Hz).

Figure 6 .
Figure 6.The pressure along the axial axis of the fan across the computational domain.Here, x = − corresponds to the location near the outlet and x = 4 near the inlet.The fan location is marked ou with the red zone.

Figure 6 .
Figure 6.The pressure along the axial axis of the fan across the computational domain.Here, x = −2 corresponds to the location near the outlet and x = 4 near the inlet.The fan location is marked out with the red zone.

Figure 7 .
Figure 7. Pressure at BLE.(a) The RMS of the pressure fluctuations for 12 fan revolutions, on one blade at the monitoring line; (b) the time-average pressure and error bars showing the minimum and maximum pressure.

Figure 7 .
Figure 7. Pressure at BLE.(a) The RMS of the pressure fluctuations for 12 fan revolutions, on one blade at the monitoring line; (b) the time-average pressure and error bars showing the minimum and maximum pressure.The time-average of the wall pressures at the BLE are shown at different positions for all cases in Figure7b.The maximum values for all cases occur at the shroud, whereas at the backplate, the values are negative.The amplitudes of the maximum and minimum pressure (error bar) are the largest at the shroud, and it becomes smaller when the distance from the shroud increases for all cases.At the shroud, Case 1 has the highest amplitude.Case 2 has the lowest amplitudes at all positions.At point 2, the maximum positive fluctuations are larger than the magnitude of the negative ones, and it is the same for all cases.Figure8shows the time history of the wall pressure for all cases at point 2 during 12 fan periods.The dashed lines indicate the average value for each case, and as illustrated in Figure7b, it is lowest for Case 2. For Cases 1 and 3, pressure fluctuations with large amplitudes and high frequencies are observed.Fluctuations are also obvious for Case 2, but the amplitudes are smaller.Moreover, a periodic low-frequency fluctuation in relation to the fan rotation is seen for Case 2, which was also found in[21].For Cases 1 and 3, there is no clear periodic low-frequency fluctuation.For all the cases, the maximum absolute values of the positive fluctuations are larger than those of the negative fluctuations.The reason is due to the upstream turbulent vortex impingement.The contours of the velocity magnitudes, |v R |, defined in Equation (1), in the y-z plane (see Figure1) and the streamlines (colored in gray) of the relative velocity vectors in Plane 1 (location see Figure1a) are shown for all cases in Figure9.Note that the axial velocity component along the fan rotation axis is excluded from the vectors.For Case 1 and 3, there are regions with large velocity magnitudes near the shroud.These regions appear periodically in relation to the blade positions.These regions cannot be observed for Case 2. This suggests that the flow near the shroud is highly fluctuating due to the gap turbulence and that Case 2 has less turbulence above the blades (near the fan inlet).

Figure 8 .
Figure 8.The time history of wall pressures along the BLE at point 2: (a) Case 1, (b) Case 2, a Case 3. The dashed lines indicate the average value for each case.

Figure 8 .
Figure 8.The time history of wall pressures along the BLE at point 2: (a) Case 1, (b) Case 2, and (c) Case 3. The dashed lines indicate the average value for each case.

Figure 8 .
Figure 8.The time history of wall pressures along the BLE at point 2: (a) Case 1, (b) Case 2, and (c) Case 3. The dashed lines indicate the average value for each case.

Figure 10 .
Figure 10.SPL of the sound upstream of the fan.The tonal frequency  = 326.7 Hz.

Figure 10 .
Figure 10.SPL of the sound upstream of the fan.The tonal frequency BPF = 326.7 Hz.

Table 1 .
The fan parameters.

Table 2 .
The mesh parameters.

Table 2 .
The mesh parameters.

Table 3 .
Aerodynamic and acoustic performance.

Table 3 .
Aerodynamic and acoustic performance.