Effects of Acoustic Resonance and Volute Geometry on Phase Resonance in a Centrifugal Fan

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International Journal of Fluid Machinery and Systems DOI: http://dx.doi.org/10.5293/IJFMS.2013.6.2.075 Vol. 6, No. 2, April-June 2013 ISSN (Online): 1882-9554

Original Paper

Effects of Acoustic Resonance and Volute Geometry on Phase Resonance in a Centrifugal Fan

Yoshinobu Tsujimoto

¹

, Hiroshi Tanaka

²

, Peter Doerfler

³

, Koichi Yonezawa

¹

Takayuki Suzuki

⁴

and Keisuke Makikawa

¹

1

Graduate School of Engineering Science, Osaka University 1-3, Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan tujimoto@me.es.osaka-u.ac.jp, yonezawa@me.es.osaka-u.ac.jp

2

Toshiba Corporation (Retired)

3

R&D Department, Andritz Hydro Ltd.

Postfach 2602, 8021 Zurich, Switzerland, Peter.Doerfler@andritz.com

4

Department of Mechanical Engineering, Kobe City College of Technology 8-3, Gakuenhigashimachi, Kobeshi-kitaku, 651-2194, Japan

taka8170@kobe-kosen.ac.jp

Abstract

The effects of acoustic resonance and volute geometry on phase resonance are studied theoretically and experimentally using a centrifugal fan. One dimensional theoretical model is developed taking account of the reflection from the discharge pipe end. It was found that the phase resonance occurs, even with the effects of acoustic resonance, when the rotational speed of rotor-stator interaction pattern agrees with the sound velocity. This was confirmed by experiments with and without a silencer at the discharge pipe exit. The pressure wave measurements showed that there are certain effects of the cross-sectional area change of the volute which is neglected in the one dimensional model. To clarify the effects of area change, experiments were carried out by using a ring volute with a constant area. It was demonstrated that the phase resonance occurs for both interaction modes travelling towards/away from the volute. The amplitude of travelling wave grows towards the volute exit for the modes rotating towards the volute exit, in the same direction as the impeller. However, a standing wave is developed in the volute for the modes rotating away from the volute exit in the opposite direction as the impeller, as a result of the interaction of a growing wave while travelling towards the tongue and a reflected wave away from the tongue.

Keywords:

Phase resonance, acoustic resonance, rotor-stator interaction, centrifugal fan

1. Introduction

Phase resonance was pointed out by Den Hartog [1] in 1929 as a phenomenon that the pressure fluctuation in the penstock of hydraulic turbine installation can become very large when the pressure waves from each guide vane caused by the interaction with the runner vane reach the penstock with the same phase. The condition for phase resonance was theoretically studied for general cases by Chen [2] and examined in detail by Doerfler [3]. Through on site measurements of pressure patterns in the volute [4], it was found that the phase resonance occurs when the mode of rotor-stator interaction rotates at a sound speed. This was confirmed by model tests and CFD using a centrifugal fan and a one-dimensional theoretical model [5]. However, the reflection from the exit pipe end has been neglected in the study. The present study is intended to clarify the effects of the reflected wave from the exit pipe end which causes acoustic resonance.

2. One-dimensional theory of phase resonance

2.1 Rotor-stator interaction

We consider that a rotor with R blades is rotating with an angular velocity Ω in a stator with S blades. As a result of rotor-stator interaction, a circumferencial pressure wave of the form

Received May 28 2013; accepted for publication March 28 2013: Review conducted by Prof. Youn-Jea Kim. (Paper number O12012K)

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Fig. 1 One dimensional theoretical model

( ) ( )

, , , cos

m n m n

p θ t =a mθ−nR tΩ (1)

occurs [5],[6] when m=nR+kS where m is the circumferencial mode number, n is the order of harmonics, and k is an arbitrary integer number. The rotational speed of the mode is nRΩ/m which can be much larger than the rotor speed Ω when nR is large and m is small.

2.2 One-dimensional model

The acoustic wave generation and propagation in the volute and exit pipe is modeled by those in a half infinite duct with the width b, as shown in Fig.1. The effect of steady flow is totally neglected. The volute tongue is located at x=0, the volute exit at x=L, and the exit pipe end at x=L_p. The disturbance due to the rotor-stator interaction is represented by the introduction of the fluid into the duct with the velocity v x t( , )=v₀cos (ω t−x U/ )in the region of volute 0 < x < L, with ω=2πmU/L. In the model, we consider that m>0 and U can be positive or negative. The continuity equation, the momentum equation and the isentropic equation results in the following non- homogeneous wave equation for the velocity disturbance u(x,t) in the duct:

2 2

2 2 2

1 1 ( , )

u u v x t

b x

x a t

∂ − ∂ = ∂

∂

∂ ∂ ⁽²⁾

where, a is the velocity of sound.

If we apply the boundary condition that the pressure fluctuation becomes zero at the pipe exit x=L_p, the solutions are represented as follows using the Mach number M=U/a of the translational velocity of the disturbance.

In 0 < x < L

( / ) ( / ) ( / )

0 2

/ 1

/ 2 1

j t x U j t x a j t x a

u v L b e Ae Be

m M

ω ω ω

π ^ ⁻ ⁻ ⁺ ^

= − − + +  (3)

( / ) ( / ) ( / )

2 0

/ 1

2 1

j t x U j t x a j t x a

p L b

Me Ae Be

av m M

ω ω ω

ρ ⁼ π − ^^⁻ ⁻ ⁺ ⁻ ⁻ ⁺ ^^ (4)

In L < x < L_p

2 /

( / ) ( / )

0 2

/ /

2 1

j Lp a

j t x a j t x a

L b C

u v e e e

m M

ω ω ω

π

− − +

 

= −  +  (5)

2 /

( / ) ( / )

2 0

/

2 1

j Lp a

j t x a j t x a

p L b C

e e e

av m M

ω ω ω

ρ π

− − +

 

= −  −  (6)

The solution for the case without the reflecting wave from the pipe exit at x=L_p can be obtained by putting formally

2 / 2 ( 2 / )

p p 0

j L a jMm L L

e⁻ ^ω =e^π ⁻ = throughout the paper. The boundary conditions that u=0 at x=0, that the pressure and the velocity is continuous at x=L are used to determine the value of the constants A, B and C:

2 (1 2 / )

2

2 ( 2 / )

1 ( 1) / 2 ( 1) / 2

1

p p

jMm L L

jMm

jMm L L

M e M e

A

e

π π π

− −

−

+ − + +

= + (7)

2 ( 2 / ) 2 2 (1 2 / )

2 ( 2 / )

( 1) / 2 ( 1) / 2

1

p p

p

jMm L L jMm jMm L L

jMm L L

e M e M e

B

e

π π π

π

− − −

−

− − − +

= + (8)

2 2

2 ( 2 / )

1 ( 1) / 2 (1 ) / 2

1 ^p

jMm jMm

jMm L L

M e M e

C

e

π π

π

−

+ − − +

= + (9)

The pressure and velocity disturbances can formally become infinite under the following conditions:

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i) Phase resonance: M→±1. In the limits to these conditions, the value of [ ] in Eqs. (3)- (6) tends to zero and finite pressure and velocity disturbances are obtained as will be shown later.

ii) Acoustic resonance in the entire region0< <x L_p. This occurs when

2 ( 2 / )

1 + e

^π^jMm⁻ ^L^p ^L

= 0

⁽¹⁰⁾

Since ω=2πmU/L=2πf and the wavelength is λ =a f/ =aL mU/( )=L Mm/( ), the condition of 1+e²^π^jMm^{( 2}⁻ ^L^p^{/ )}^L = leads to 0 2πjMm( 2− L_p/ )L =2πj l( +1/ 2) and

( / 2 1/ 4) /( ) ( '/ 2 1/ 4)

Lp = −l + L Mm = l − λ (11)

and

( ' 1/ 2) /(2 _p/ )

M = −l mL L (12)

where, l and l’ are integers. Equation (11) shows that the acoustic resonance occurs when L equals to the multiples of half wavelength _p minus a quarter wavelength. This corresponds to the resonance of an open-closed pipe with the length ofL , associated with the _p boundary conditions (u=0at x=0) and (p=0at x=L_p) applied in the present analysis. Equation (12) gives the Mach number at which the acoustic resonance occurs. When L_p/Lis large, the pitch 1/(2mL_p/ )L of the resonant Mach numbers becomes small.

2.3 Acoustic modes under phase resonance

Under phase resonance, clearer expressions of the pressure and velocity are obtained as follows:

For the case of M→1 In 0< <x L

2 ( 2 / )

( / )

0 2 ( 2 / )

2 1

/ ( ) ( ) sin(2 )

2 1 2

p p jm L L

j t x U j t

jm L L

L j x je x

u v e m e

b L e m L

π

ω ω

π π

π

−

 

= − − − + 

 + 

  (13)

2 ( 2 / )

( / )

2 ( 2 / )

0

( ) (2 cos(2 ) 1 sin(2 ))

2 1 2

p p jm L L

j t x U j t

jm L L

p L j x e x x

e m m e

av b L e L m L

π

ω ω

π π π

ρ π

−

 

= − − + + 

 + 

  (14)

Equations (13) and (14) show that the disturbance in the volute is composed of a growing and travelling wave shown by

( / )

( / )x L e^j^ω^{t x U}⁻ and standing waves represented by other terms. Without reflection (with e^±²^π^jm^{( 2}⁻ ^L^p^{/ )}^L = in Eqs.(13) and (14)), the 0 amplitude of the standing wave is much smaller than the traveling wave since1/ 2π <<m 1. Under acoustic resonance with

2 ( 2 / )

1+e^π^jm⁻^L^p ^L → , the amplitude of corresponding standing wave becomes larger than other components . 0

In

L< <x L_p

(

y=L_p−x, 0< <y L_p−L

, )

2 ( 2 / )

( / ) ( / )

0 2 ( 2 / )

/ 1

2 1

cos(2 / )

2 cos(2 / )

p p

jm L L

j t x U j t x U

jm L L

j t p

u v L j e e e

b e

L j my L

b mL L e

ω π ω

π

π ω

π

− − +

−  

= +  + 

= (15)

2 ( 2 / )

( / ) ( / )

2 ( 2 / )

0

1 2 1

( 1) sin(2 / )

2 cos(2 / )

p p

jm L L

j t x U j t x U

jm L L

j t p

p L j

e e e

av b e

L my L

b mL L e

ω π ω

π

ω

ρ

π π

− − +

−  

= +  − 

= −

(16)

Equations (15) and (16) show that the disturbance in the exit pipe is a standing wave and the amplitude tends to infinity under acoustic resonance, for the case with reflective wave. Without reflections (with e⁻²^{j L}^ω ^p^/^a =e²^π^jMm^{( 2}⁻^L^p^{/ )}^L = ), the upper expressions of Eqs.(15) 0 and (16) show that only travelling wave exists.

For the case of M → −1 In 0< <x L

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2 ( 2 / )

( / ) ( / )

0 2 ( 2 / )

2 1

/ ( ) ( 1) ( ) sin(2 )

2 1 2

p

p jm L L

j t x U j t x U j t

jm L L

L j x je x

u v e e m e

b L e m L

π

ω ω ω

π π

π

− −

− +

− −

 − 

=  − + + + 

 + 

  (17)

2 ( 2 / )

( / ) ( / )

2 ( 2 / )

0

2 1

( ) (1 ) ( cos(2 ) sin(2 ))

2 1 2

p

p jm L L

j t x U j t x U j t

jm L L

p L j x e x x

e e m m e

av b L e L m L

π

ω ω ω

π π π

ρ π

− −

− +

− −

 − 

=  − + + + 

 + 

  (18)

Equation (17) and (18) show that the disturbance in the volute is composed of a wave growing while propagating in negative x- direction, (1−x L e/ ) ^j^ω⁽^{t x U}⁻ ^/ ⁾(note that U<0 for M=-1) , a wave propagating in positive x-direction e^j^ω⁽^{t x U}⁺ ^/ ⁾reflected at x=0, and standing waves represented by other terms. Under acoustic resonance with 1+e⁻²^π^jm^{( 2}⁻^L^p^{/ )}^L →0, the standing wave amplitude tends to infinity.

In

L< <x L_p

(

y=L_p−x, 0< <y L_p−L

, )

2 (2 / )

( / ) ( / )

0 2 (2 / )

/ 1

2 1

cos(2 / )

2 cos(2 / )

p p

jm L L

j t x U j t x U

jm L L

j t p

u v L j e e e

b e

L j my L

b mL L e

ω π ω

π

π ω

π

+ −

 

= +  + 

= (19)

2 (2 / )

( / ) ( / )

2 (2 / )

0

1 2 1

1 sin(2 / )

2 cos(2 / )

p p

jm L L

j t x U j t x U

jm L L

j t p

p L j

e e e

av b e

L my L

b mL L e

ω π ω

π

ω

ρ

π π

+ −

 

= +  − 

=

(20)

Equations (13)-(20) show that finite amplitude standing waves appear in the volute and exit pipe under phase resonance. However, the amplitude grows to infinity when approaching the acoustic resonant condition. We should note that a free oscillation term in

0< <x L_pwith an arbitrary amplitude has been omitted here assuming that it is damped by viscosity.

3. Experimental facility

Figure 2 shows the impeller, diffuser, volute, and silencer used for the experiments. Two types of diffusers with the blade number 16 and 20, and two types of volutes, spiral and ring volute were used. Specifications of the impeller and diffuser are shown in Table 1.

The experimental facility is shown in Fig.3. The flow from the exit pipe with 17.5mm x 40mm cross section is discharged to a settling tank with the inner diameter 145mm and the length 750mm directly, or after passing through a silencer shown in Fig.2. The wall impedance of the silencer was changed linearly by covering the perforated wall by plastic tapes to minimize the reflection from the pipe exit. The pressure fluctuation in the volute was measured with flush mounted pressure transducers at the locations 1-40 shown in Fig.2, with reference to the pressure signal from the tap 1 for spiral volute and tap 6 for ring volute.

Figure 4 shows the static pressure performance ψ =(p_exit−p_inlet) /(0.5ρU_T²) with the 16 vane diffuser and the spiral volute, plotted against the flow coefficient φ=Q U D/( _T _T²)where D_T, U_T are the impeller tip diameter and speed. All measurements were carried out at the flow coefficient φ =0.015.

(a) Top view of impeller, vaned diffuser, and volute (b) Cross section and top view of silencer Fig. 2 Impeller, vaned diffuser, volute, and silencer

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Table 1 Specifications of impeller and diffuser

Fig. 3 Experimental facility

Fig. 4 Performance curve

4. Effects of acoustic resonance

Figure 5 compares the spectrum of pressure fluctuation at position 5, with the 16 blade diffuser and the spiral volute, with and without the silencer. The pressure fluctuation ∆p is normalized as ψ_e = ∆p/(0.5ρU_T²). The data was taken with slowly increasing the rotational speed while keeping the opening of the throttle valve to be constant, corresponding to φ =0.015. With the combination of the rotor blade number R=6 and the stator blade number S=16, we have the following modes for m=nR+kS: 2= × − ×3 6 1 16 and − = × − ×2 5 6 2 16 . With m=2, two lobes are rotating in the same direction as the impeller at the speed of

/ 3 6 / 2 9

nRΩ m= × Ω = Ω, with the frequency nRΩ = × Ω = Ω3 6 18 measured in the stationary frame. With m=-2, two lobes are rotating in the opposite direction as the impeller at the speed of nRΩ/m= − × Ω5 6 / 2= − Ω15 , with the frequency

5 6 30

nRΩ = × Ω = Ω. The components of these modes are shown in the figure.

We focus on the mode with m=2 appearing more clearly. In order to evaluate the amplitude of the pressure wave transmitted to the exit pipe, the peak values corresponding to m=2 mode measured at the pressure taps 27-40 in the exit pipe are plotted against the rotational frequency in Fig.6. We observe several peaks. The number of peaks is larger for the case without the silencer. The value of

Number of vanes, R 6 Number of vanes, S 16,20 Inlet vane angle 45 [deg] Inner vane angle 4 [deg]

Outlet vane angle 35.5 [deg] Outlet vane angle 25 [deg]

Inner diameter, D1 37 [mm] Inner diameter, D3 91 [mm]

Outer diameter, D2 90 [mm] Outer diameter, D4 121 [mm]

Inner width, b1 12.8 [mm] Inner width, b3 7.2 [mm]

Outer width, b² 6 [mm] Outer width, b⁴ 12.5 [mm]

Impeller Diffuser

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( / ) / 2L b C ^

π

m(1−M2)^, representing the non-dimensional pressure amplitude ψ_m = p/(ρav₀)from the 1-D model (see Eq.(6)) is plotted by assuming that M=1 at 92Hz where the experimental pressure fluctuation becomes the maximum for the case with the silencer. For this plot, the value of L b/ =12.0 has been used. The rotational speed of the mode 9Ω×r_vat r=r_v becomes 340 m/sat

0.0654

rv = mwhich is slightly larger than the diffuser outer radius of 0.0605m. The blue line represents the case without the reflection from the exit pipe end obtained by putting formally e²^π^jMm⁽⁻²^Lp^/^L⁾

= 0

in Eq. (16). It becomes the maximum at M=1 and represents the phase resonance. The red line represents the result with reflection, which tends to infinity at the Mach numbers satisfying

2 ( 2 / )

1+e^π^jMm⁻^L^p ^L =0. This is caused by the acoustic resonance in the volute + exit pipe with the tongue as a closed end and the exit pipe end as an open end. The minimums of the red curve with reflection become one half of the blue curve representing the case without reflection.

(a) Without silencer (b) With silencer

Fig. 5 Pressure fluctuation at position 5, spiral volute

(a) Without silencer (b) With silencer

Fig. 6 Amplitude of pressure fluctuation in exit pipe, spiral volute

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(a) Spiral volute without silencer at 92Hz (b) Theory with reflection at 98Hz

(c) Spiral volute with silencer at 92Hz (d) Theory without reflection at 98Hz Fig. 7 Amplitude and phase of pressure fluctuation at phase resonance

(a) Spiral volute without silencer at 78Hz (b) Theory with reflection at 75Hz

(c) Spiral volute with silencer at 78Hz (d) Theory without reflection at 75Hz Fig. 8 Amplitude and phase of pressure fluctuation below phase resonance

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Qualitative comparison between

ψ

_e^and

ψ

_msuggests that multiple peaks observed for the case without the silencer are caused by the acoustic resonance. However, the peak values are finite showing that the resonance is highly damped. This shows that the phase resonance is important even with the acoustic resonance.

Figure 7 shows the amplitude and the phase of the pressure fluctuation in the volute around 92Hz, near phase resonance. The conditions of the plots are indicated in Fig.6. The numbers on the amplitude plot shows the location of the pressure tap in Fig.2(a). The broken line shows the phase

− ω x a /

of an acoustic wave travelling from the tongue to the pipe exit. The amplitude and phase of the theoretical results were determined by applying the same FFT tool as used for the experiments on the pressure fluctuation of Eq.(4). A standing wave is observed for the case without the silencer ((a) and (b)). A pressure loop at x=0 does not exist for

ψ

_esuggesting that the boundary condition u=0 applied at the tongue is not adequate for the spiral volute. Four peaks exist in the volute region corresponding with m=2. With the silencer, as shown in (c), the amplitude is more flat with continuous phase delay suggesting a wave propagating from the tongue to the exit pipe with the velocity of sound. Some peaks suggest that some part of the wave is reflected from the silencer. The result from the model without reflection shown in (d) shows the growth of the amplitude in x-direction. The difference from the experiment is considered to be caused by the assumption of constant cross-sectional area of the volute.

Figure 8 shows the mode around 78Hz, below phase resonance. The dash-dot line in the figure shows the phase

− ω x U /

of the interaction pattern. Unlike the results of the theoretical model, the disturbance rotates at the speed U of the interaction mode in the volute. In the previous study[5], the pressure fluctuation in the volute was discussed based on CFD. Although CFD succeeds in predicting the phase resonance, it fails to predict the acoustic resonance and the pressure propagation in the volute perhaps caused by higher damping of acoustic waves in the volute.

Figure 9 shows the mode with m=2 obtained with the ring diffuser. The pressure amplitude becomes the maximum around 88Hz with the ring diffuser. The rotational speed of the mode becomes 340m/s at the radius r_v=0.0687m which is slightly larger than

0.0605m

for the case of spiral volute. The pressure loop is at the tongue both with and without the silencer. Unlike the case with spiral volute, the amplitude is somewhat increased towards the volute exit. These are closer to the theoretical model shown in Fig.7 and shows the area change in the spiral volute has an effect to increase the amplitude of pressure fluctuation near the tongue. At lower speed than the phase resonance, the disturbance in the volute propagates with the interaction mode velocity

U

rather than the sound velocity

a

.

(a) Without silencer at 86Hz (b) With silencer at 88Hz

(c) Without silencer at 78Hz (d) With silencer at 78Hz Fig. 9 Amplitude and phase of pressure fluctuation with ring volute

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5. Effects of the rotational direction of the mode

In order to examine the effects of the rotational direction of the mode, experiments were carried out with S=20 diffuser. For this case we have

− = × − × 2 3 6 1 20

and

4 = × − × 4 6 1 20

for

m = nR + kS

. Figure 10 shows the example of the spectrum for the case with the spiral and ring volutes, both with the silencer. We focus on the mode with m=-2. The amplitude of pressure fluctuation in the volute is shown in Fig.11. At 80 Hz where the amplitude of m=-2 mode becomes the maximum, the rotational speed of the mode becomes 340m/s at r_v=0.0751m. With the ring volute, we do have other peaks even with silencer, caused by acoustic resonance.

(a) Spiral volute at point 5 (b) Ring volute at point 11 Fig. 10 Pressure fluctuation with S=20 diffuser

(a) Spiral volute (b) Ring volute

Fig. 11 Amplitude of pressure fluctuation in volute with silencer

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(a) Spiral volute with silencer at 80Hz (b) Theory without reflection at 80Hz

(c) Ring volute with silencer at 80Hz (d) Ring volute without silencer at 80Hz

(e) Ring volute with silencer at 68Hz (f) Ring volute without silencer at 68Hz Fig. 12 Amplitude and phase with spiral or ring volute, S=20, m=-2

Figure 12 shows the amplitude and the phase of the pressure fluctuation of the mode of m=-2. We observe a clear standing wave with decreasing amplitude with the increase of x. This is caused by the term (1 x) ^j ⁽^{t x U}^/ ⁾ ^j ⁽^{t x U}^/ ⁾

e e

L

ω − ω +

− + in Eq. (18), showing a wave

growing while propagating towards the tongue at x=0 (U<0, for the present case) and a wave reflected at the tongue. The pressure fluctuation is much larger in the volute, as compared with that in the exit pipe. With the spiral casing, the pressure loop does not appear at x=0 and the maximum amplitude in the volute is about 13 times of that in the exit pipe. With the ring volute, the pressure loop appears at the tongue and the maximum pressure in the volute is about 4 times of that in the exit pipe. So, better agreement with the theoretical model is obtained with the ring volute. The results with ring volute without the silencer is shown in (d). The amplitude in the exit pipe is larger than the case with the silencer and we find 5+1/2 loops in the volute+exit pipe. At lower speed 68Hz, the mode

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rotates with the sound velocity and we find 4+1/2 loops both with (e) and without (f) the silencer. This shows that the peak at 68Hz is caused by the acoustic resonance even with the silencer.

Figure 13 shows the mode from the theoretical model. At 80Hz with reflection, 5.5 loops are found as in the experiments. The experimental mode shapes with and without silencer are something between the theoretical results with and without wave reflection. At 68Hz, the number of loops is getting closer to the experimental number of 4.5.

(a) Without reflection at 80Hz (b) With reflection at 80Hz

(c) Without reflection at 68Hz (a) With reflection at 68Hz Fig. 13 Amplitude and phase of pressure fluctuation, S=20, m=-2

6. Conclusion

(1) Considering the effect of reflected wave from the end of the exit pipe, 1-D theoretical model of phase resonance was developed for the case with finite exit pipe length.

(2) Even with the effects of acoustic resonance, the main character of the phase resonance, that it occurs when the rotational speed of the interaction pattern agrees with the velocity of sound, is not changed.

(3) The phase resonance occurs for both rotor-stator interaction modes travelling towards/away from the volute exit.

(4) With an interaction mode rotating towards the volute exit (in the same direction as the rotor), a standing wave occurs for the case with the reflection from the exit pipe end. Without the reflection, a wave growing while travelling from the tongue to the volute exit appears. The magnitude of the pressure fluctuation is of the same order in the volute and in the exit pipe.

(5) With an interaction mode rotating away from the volute exit to the tongue (opposite to the rotor rotation), a standing wave with larger amplitude near the tongue appears, caused by the interaction of a pressure wave growing while propagating to the tongue, with a wave reflected at the tongue. The magnitude of the pressure fluctuation is larger in the volute as compared with that in the exit pipe.

(6) With a spiral volute, the amplitude of the pressure fluctuation near the tongue becomes larger than that in a ring volute and the 1-D theoretical model. The pressure loop which occurs near the tongue of the ring volute and the 1-D theoretical model does not occur with the spiral volute. Although there are these differences, the fundamental characteristics obtained from the 1-D are verified by experiments.

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Acknowledgements

The authors would like to acknowledge the suggestions by Prof.Zoltan Spakovszky of MIT GTL, who pointed out the possibility of acoustic resonance. Acknowledgements are also to Mr.Shingo Toyahara for his supports in experiments and data processing.

Nomenclature

a b b₁ b₂ b₃ b₄ D₁ D₂, D_T D₃ D₄ f j m λ L L₀ M n

Velocity of sound [m/s]

Mean width of volute [m]

Inner width of impeller [mm]

Outer width of impeller [mm]

Inner width of diffuser [mm]

Outer width of diffuser [mm]

Inner diameter of impeller [mm]

Outer diameter of impeller [mm]

Inner diameter of diffuser [mm]

Outer diameter of diffuser [mm]

Frequency [Hz]

Complex unit

Circumferential mode number Wave length [m]

Distance between volute tongue and exit [m]

Distance between volute tongue and pipe exit [m]

Mach number Order of harmonics

p p_inlet p_exit R S t u U U_T v Ω ω x ϕ ψ ψe

ψm

Pressure disturbance [Pa]

Inlet pressure of fan [Pa]

Outlet pressure of downstream pipe [Pa]

Impeller blade number Stator blade number Time [s]

Velocity disturbance [m/s]

Propagation velocity [m/s]

Tip speed of impeller [m/s]

Velocity of flow introduction into volute [m/s]

Rotating angular frequency of impeller [rad/s]

Angular frequency [rad/s]

Distance from volute tang [m]

Flow coefficient Pressure coefficient

Non-dimensional pressure amplitude

Non-dimensional pressure amplitude from 1-D model

References

[1] Den Hartog, J.P., Mechanical Vibrations in Penstocks of Hydraulic Turbine Installations, Transactions of ASME, 1929, pp.

101-110.

[2] Chen, Y.N., 1961, “Oscillation of Water Pressure in the Spiral Casing of Storage Pumps,” Techn. Review Sulzer, Research Issue (Turbomachinery), pp. 21-24.

[3] Doerfler, P., 1984, “On the Role of Phase Resonance in Vibration Caused by Blade Passage In Radial Hydraulic Turbomachines,” Proceedings of 12th IAHR Symposium, Stirling.

[4] Ohura, H., Fujii, M., Sugimoto, O., Tanaka, H., and Yamagata, I., 1990, “Vibration of Powerhouse Structure of a Pumped Storage Power Plant,” Proceedings of IAHR Symposium, Belgrade, Yugoslavia.

[5] Nishiyama, Y., Suzuki, T., Yonezawa, K.,Tanaka,H., Doerfler, P., and Tsujimoto, Y., “Phase Resonance in a Centrifugal Compressor,” International Journal of Fluid Machinery and Systems, Vol.4, No.3, July-September, 2011, pp. 325-333.

[6] Tyler, J. M., and Sofrin, T. G., “Axial Flow Compressor Noise Studies,” SAE Trans., Vol. 70, 1962, pp. 309-332.