# Time-Frequency Characterization of Rotating Instabilities in  a Centrifugal Pump with a Vaned Diffuser

Authors: G. Pavesi; G. Cavazzini; G. Ardizzon
Journal: International Journal of Rotating Machinery
 (2008)

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## Abstract

This paper presents acoustic and flowdynamic investigations of large-scale instabilities in a radial pump with a vaned diffuser. Pressure fluctuations were measured with transducers placed flush at the inlet duct, at the impeller discharge, and in the vane diffuser walls. Two impeller rotation speeds were analyzed in the study, at design, and at off-design flow rates.  A spectral analysis was carried out on the pressure signals in frequency and in time-frequency domains to identified precursors, inception, and evolution of the pressure instabilities. The results highlighted the existence of a rotating pressure structure at the impeller discharge, having a fluid-dynamical origin and propagating both in the radial direction and inside the impeller. The experimental data were then compared with the results obtained with help of ANSYS CFX computer code; focusing on the changing flow field at part load. Turbulence was reproduced by DES model.


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## Body

## 1. Introduction

Flow in the diffuser
pump is dominated by unsteady interactions due to the relative motion and close
proximity between the rotating impeller and stationary diffuser, thus,
resulting in pressure and velocity fluctuations both upstream and downstream.  In the literature, different analytical and
experimental approaches were used to investigate the mutual influence of the
impeller and its surroundings.Experiments were
conducted to measure pressure fluctuations in diffuser radial pumps by
Arndt et al. [1, 2],
Furukawa et al. [3] and
Guo and Maruta [4], and the details of
wake transport across the rotor were tracked in continuously running facilities
thanks to LDV  and PIV [5–8].The noise spectral content for studying the
unsteady phenomena developing in the impeller passages and at the flow
discharge was also considered by researchers.
Cumpsty [9] and Neise
[10] focused the attention on the tonal pulse
generation, whereas
Mongeau [11],
Mongeau et al. [12],
Choi [13],
Choi et al. [14],
and Bent [15]  showed that the flow structure was associated with
the dominant unsteadiness of the spectral content at the impeller discharge.Notwithstanding Ferrara et al. [16] and
Horodko [17, 18] demonstrated the applicability of the
signal processing technique waterfalls, based on the time-frequency analysis,
to the turbomachinery field,  few
researchers utilized the most advanced time-frequency analysis technique in the
field of turbomachinery.The objective of this research was to identify
and characterize the unsteady phenomena produced in a centrifugal pump,
utilizing the Fourier approach and the time-frequency analysis. A correlation
between the fluid dynamics and the emitted noise was investigated in order to
define the characteristic frequencies of the unsteady stator/rotor interaction.The paper
reports time-resolved fluctuations of pressure upstream and downstream of a
radial vaned pump at design flow rate and at low-flow rates. Particular
emphasis was dedicated to understand the source of the pitch-wise variations at
the impeller outlet and in the diffuser channels.Furthermore, this paper
discusses the numerical data obtained with the ANSYS CFD 11 code, and shows
comparison between the theoretical and experimental results on the impeller
model equipped with vaned diffuser.


## 2. Test Facilities

The experimental analysis was performed in the Open
Turbomachinery  Facility (OTF), in the Department of Mechanical Engineering, University of Padua. The OTF is a water test rig
designed for testing the influence of several parameters on the overall
performances of both pumps and turbines. The analyzed pump had an original
configuration, constituted by a 3D impeller, coupled with a vaned diffuser and
a vaned return channel, reproducing the first stage of a two-stage pump turbine
(ns=37.6m0.75s−1.5)
[19]. The pump
configuration was modified to operate with an axial symmetric vaneless return
channel, to remove the noise sources due to two stator rows interaction.
Table 1 reports the diameter, the width, and the blade angle of the tested pump, as
well as the flow coefficient at design point.Table 1Impeller dataD2/mmB2/mmnbβ2c/°φDes40040726.5.125Adjustable
guide-diffuser vanes dataD3/mmB3/mmnbα3c/°410402221.8Pressure transducers, placed flush with the
wall, at different positions carried out measurements of the unsteady pressure.
Figure1 shows a schematic view of the tested configuration with the reference
marks of the measurement locations as follows:Figure 1
Scheme of the
impeller and the vaned diffuser.(i)
three
traverse planes, 68 mm upstream the impeller, in the inlet duct (positions 10,
11, and 12—white marks in Figure1);(ii)
a
traverse plane, 2 mm downstream the impeller (position 20—green mark in
Figure1);(iii)
twelve
pressure points in the mid height (Figure2) of a diffuser blade (yellow one in
Figure 1).Figure 2
Scheme of the diffuser
blade with the 12 micro transducers, 7 of them numbered.Piezoelectric dynamic pressure transducers
(sensitivity of 76.89 V/Pa) were placed circumferentially equidistant in the
inlet duct (positions 10, 11, and 12); and a piezoelectric transducer
(sensitivity of 7.62 V/Pa) was located 2 mm downstream the impeller (position
20).  Miniaturized piezoresistive
transducers (sensitivity of about 0.29 V/Pa) were placed in the diffuser vane.
The sensors’ combined nonlinearity, hysteresis, and repeatability were better
than ±0.1%.The vibrations of the rig were also measured by
accelerometers fixed at the inflow and outflow pump casing, and in the dynamo
that powered the pump (charge sensitivity of0.295pC/ms−2).The pump inflow was connected to a water
reservoir, filled continuously to a weir brim by an external pump, so that the
water level change during operations could be considered negligible. The
vibrations of the tank were monitored by an accelerometer.The data were simultaneously acquired from all
the channels at a sampling rate of 2048 Hz and recorded by a workstation
controlled transient data recorder, with a dynamic range of 24 bit. The outputs
from the transducers channel were conditioned with a low-pass filter at 1024 Hz.Design and off-design conditions, and two
impeller rotation speeds (n=500rpm; 600 rpm) were investigated.


## 3. Signal Processing

The pressure signals were analyzed both in the
frequency domain and in the time-frequency domains.Auto-spectral and cross-spectral matrices were
computed by partitioning each time signal into 256, nonoverlapping segments of215 samples, filtered with a Hanning window for avoiding aliasing
and leakage errors. The frequency resolution was 0.125 Hz. The auto-spectra and
the cross-spectra were determined from the following equation:(1)Gxy(f)=1NWH∑k=1N[Xk*(f)Yk(f)],where N is the number of
segments (i.e., N=256), WH is weighting constant corresponding to
the Hanning window, Yk(f) is the fast Fourier transform of the kth data segment, and of the signal y and Xk(f)* is the
complex conjugate of the fast Fourier transform of the kth data
segment of the signal x. The number of the time segments (N=256) was
established, as twice the lowest values whose increase did not involve a
significant variation in  the resulting function Gij(f).The coherence functionγ2(f)(2)γ2(f)=|Gxy(f)|2Gxx(f)⋅Gyy(f) between two signals was also
considered. The square cross-spectrum Gxy was divided by the product
of the auto-spectra (Gxx,Gyy) of the signals to measure
the degree of linearity between the two signals versus the frequency in the
normalized range 0 and 1.The signal processing in the frequency domain
allows finding and analyzing the spectral components contained in the measured
pressure signals, but it does not allow an assignment of these spectral
components to time. In order to provide information about their time evolution,
a time-frequency analysis was carried out by means of the wavelet transforms.
For the time-frequency analysis the continuous wavelet transformWn(s)
of the discrete sampled pressure signal xn was computed via the
FFT-based fast convolution: (3)W(s,n)=∑k=0N−1Xk(2πsδtΨ0*(sωk)eiωknδt),where Xk is the discrete
Fourier transform (DFT) of xn,k is the frequency index, N is the
data series length, s is the wavelet scale, δt is the sampling interval, n is the localized
time index, Ψ0*(sωk) is the complex conjugate of the Fourier
transform of the translated and scaled version of the “mother wavelet” Ψ(t), and ωk is the angular frequency (if
k≤N/2,ωk=2πk/Nδt,
else ωk=−2πk/Nδt). The equation contains a normalization
factor 2πs/δt to obtain unit energy at each scale. The choice
of the best mother wavelet, which is the basic form of the wavelet function
from which dilated and translated versions are derived and used in the wavelet
transform, depends on several factors [20]. In this paper, the complex
Morlet wavelet (with 2πf0=6) was chosen, since it provided a
good balance between time and frequency localization and it returned
information about both amplitude and phase.In order to determine the relation between two
pressure signalsx(t) and y(t) in the time-frequency domains, the wavelet
cross-correlation function, defined by Li [21], was introduced: (4)WCx,y(s,τ)=limT→∞1T∫−T/2T/2Wx*(s,b)Wy(s,b+τ)db,where τ is the
time delay between the two signals, Wy is the continuous wavelet
transform of y(t),Wx* is the complex conjugate of the
continuous wavelet transform of x(t),b is the location, and s is the scale. In
particular, for two sampled pressure signals xn and yn with wavelet transforms Wx(s,n) and Wy(s,n), the
cross-wavelet spectrum was determined as (5)Wxy(s,n)=Wx(s,n)Wy*(s,n),where Wy*(s,n) is the complex conjugate of Wy(s,n) [22].Finally, the wavelet coherence function,
measuring the coherence between two sampled signalsxn and yn both in frequency bands and in time intervals, was considered: (6)R2(s,n)=|〈Wxy(s,n)〉|2〈|Wx(s,n)|2〉⋅〈|Wy(s,n)|2〉,where 〈⋅〉 indicates a smoothing, done using a weighted
running convolution in both the time and scale directions, that depends on the
mother wavelet [22, 23]. For the
complex Morlet wavelet function, the time smoothing was done using the absolute
value of the wavelet function at each scale, normalized to have a total weight
of unity, with a Gaussian e−t2/(2s2).
The scale smoothing was done using a boxcar filter of width δj0=0.60, that is the empirically
determined scale decorrelation length. The smoothed real (ℜ) and imaginary (ℑ) parts of the numerator of (5) were used for
determining the wavelet-coherence phase difference: (7)ϕ(s,n)=tan−1{ℑ[〈Wxy(s,n)〉]ℜ[〈Wxy(s,n)〉]}.


## 4. Numerical Procedure

The commercial software
package ANSYS CFX 11.0 was used for performing the numerical simulations on the
entire machine. On both blades and wall surfaces, the boundary layer was
assumed fully turbulent.The detached eddy simulation (DES)
model  was chosen as turbulence
model.  The shear stress transportk-ω model covered the boundary layer while the
Smagorinsky-lilly model was applied in detached regions.Since one of the interesting analysis aspects is
the possible prediction of noise and vibrations due to stator-rotor
interaction, the LES peculiarity of providing information on turbulent flow
structures and spectral distribution is useful.An unsteady model was used for all the computations.
For the interface between stator/rotor blocks, the standard transient sliding
interface approach was chosen.The scheme adopted for the time discretization
was a second-order implicit time stepping. The time step definition was based
on the impeller rotation and it was of about one degree. So, the RMS courant number
was CFL = 2.04. A maximum number of five iterations were fixed for each time step,
resulting in a mass residue of10−6,
momentum residues of 10−4, and turbulence kinetic energy and energy
dissipation residues of 10−4.The numerical data were acquired between the 2nd and 3rd impeller revolution subsequent to the cyclic behavior of the
residual plots obtained after about 3 impeller revolutions.An H-type grid was used for the impeller,
whereas an O-type grid was adopted for the diffuser. The leakage from the
labyrinth seal was also considered and several H-blocks were built to describe
the cavities. The grid,  globally of5.2⋅106 points, (stage inlet 700,000 grid
points, impeller 1700,000 grid points, vaned diffuser 1250,000 grid points,
return channel 1100,000 grid points, and leakage 400,000) was used to compute
the flow in the impeller passages, with y+ values below 80 in the
whole computational region.As regards the boundary conditions, mass flow
rates were prescribed with stochastic fluctuations of the velocities with 5%
free-stream turbulence intensity at the pump inlet. At the pump outlet the
average static pressure was fixed.


## 5. Results

Stator rotor interaction gave rise to a complex
pressure pulsation system.Figures3, 4, and 5 report the power spectra of
the unsteady pressure, measured in the impeller inlet and outlet and in the some
of microtransducers placed in the diffuser blade versus the Strouhal number (St=πD2/nbf/U2).Power spectrum of
the pressure measured at the pump inlet and outlet at 500 rpm.
(a)(b)Power spectrum of
the pressure measured in the diffuser vane at 500 rpm.
(a)(b)(c)(d)Power spectrum
of the pressure measured in the diffuser vane at 600 rpm.
(a)(b)(c)(d)(e)(f)The spectra were dominated by
the blade passage frequency BPF(StBPF=1,2,3,…) and two subtonal peaks:
the main one at StF=0.664 for both impeller speeds, and the second
one at a frequency of about 5 Hz (StS=0.086 at 500 rpm and StS=0.071
at 600 rpm). All other peaks in the spectra were harmonics of these
fluctuations or nonlinear components, generated by the interaction between the
subtonal pressure pulsations’ frequency and the BPF: (8)a⋅StF,S±b⋅StBPF,(a,binteger),or between the same two
subtonal pressure pulsations frequency StF=0.664 and StS=0.086
or StS=0.071(9)c⋅StF±d⋅StS,(c,dinteger).Table2 reports some of the identified nonlinear
interaction components.Table 2
Some identified nonlinear
interaction components.
500 rpm600 rpmStNonlinear
componentsStNonlinear
componentsStNonlinear
componentsStNonlinear
components0.064StF−7StS1.093StF+5StS0.164StF−7StS1.021StF+5StS0.336StBPF−StF1.264StF+7StS0.336StBPF−StF1.164StF+7StF0.493StF−2StS1.1572StF−2StS0.521StF−2StS1.1852StF−2StS0.750StF+StS1.664StF+StBPF0.736StF+StS1.664StF+StBPF0.835StF+2StS2.2274StF−5StS0.807StF+2StS2.3004StF−5StS0.9712StF−5StS2.328StBPF+2StF0.9712StF−5StS2.328StBPF+2StF0.8995StBPF−6StF2.6642StBPF+StF1.0145StBPF−6StF2.6642StBPF+StFThe pressure pulsation at a frequency of 5 Hz
was independent of the impeller velocity and it was also identified with the
impeller not running (Figure6). The intensity of the phenomenon was observed
mainly in the inlet duct with a wavelet magnitude characterized by a variable
value for the whole time interval. For these reasons, it appeared to  be  associated
with  system  fluctuations,  and in  particular, with  the
random  fluctuation of the water
level in the accumulation tank of the pump.Figure 6
Wavelet
magnitude|Wn| in the inlet duct (traverse 10) with impeller no running.The pressure pulsation atStF=0.664 (38.75 Hz at 500 rpm and
46.5 Hz at 600 rpm) appeared to have an internal  fluid-dynamical origin, due to the absence of
these frequencies in the spectra of the not running pump (Figure 6) and in the dynamo
vibrations (Figures 7 and 8).Figure 7
Wavelet
magnitude|Wn| in the dynamo for Q/QDes=100%(n=500rpm).Figure 8
Wavelet
magnitude|Wn| in the dynamo for Q/QDes=100%(n=600rpm).The fluid-dynamical unsteadiness is well
highlighted by the time-frequency analysis at all the flow rates. ForQ/QDes=100%,
the wavelet transforms were characterized by great and quite constant values
for the frequency at StF=0.664 for all the transducers positioned in
the diffuser blade (Figure 9). At part load, the magnitude progressively became
not constant in the time domain (Figure 10). The differences of the power level
were greater at low flow rates and diminished increasing the flow rate up to
the design conditions.Wavelet
magnitude|Wn| for Q/QDes=100%(n=500rpm).
(a)(b)(c)Wavelet
magnitude|Wn| for Q/QDes=48%(n=500rpm).
(a)(b)(c)The passage of an energy core, having a varying
energy level was supposed to generate the pulsating behavior of the structure,
as seen by the wavelet analysis. The energy core was associated with one of the
zones of jet and wake, localized at the impeller discharge. As these zones pass
over the transducers, they could be perceived as pressure fluctuations,
pulsating in time and rotating around the impeller discharge.The numerical analysis showed  that at low flow rate the wake zone was
characterized by sequences of  vortices.
The vortices, originated at about mid cord, evolved growing along the blade
length up to a maximum to be reabsorbed subsequently (Figure11). This
sequence, which is comparable with a stall, generated a remarkable pressure
pulsation with a period of about 3/14 of the impeller revolution, that is, a
Strouhal number equal to about 0.66.Vector plot at 10% blade heightQ/QDes=50%
(n=600rpm).
(a)(b)(c)(d)(e)(f)It seemed that the modes, constituting the
unsteady pattern, interfered with one another alternately in a constructive and
destructive way. The resulting structure was not frozen in time, but it changed
with a pulsating behavior, appearing and disappearing, depending on the
interference of its modes (Figure10).At design flow rate, the numerical analysis did
not highlight a well-defined wake zone, but the investigation of the impeller
blade load showed pressure fluctuations due to both the stator rotor
interaction and an independent source. In Figure12, the pressure on the
impeller blade pressure side is plotted as a function of the time and of the
radius. The passage in front of the diffuser blades (DB) is marked with broken
lines. The pressure evolution could be explained, making the hypothesis of a
superimposition between the diffuser blade passage frequency (DBF = 220 Hz at 600rpm) and a pressure pulsation with a frequency that is quite a half of the DBF.Figure 12
Pressure
evolution in the impeller mid height forQ/QDes=100(%n=600rpm) (DB:
relative diffuser blade passage).In the relative system of reference, the Strouhal
number could be expressed by(10)StRelative=mnb−StAbsolute.Cavazzini [24],
demonstrated that the number of modes m, matching St=0.664, was 15. Therefore,
the relative frequency corresponding to these pulsations is 103.8 Hz at 600 rpm,
that is, about 47% of the DBF. It appears to be in good accord with the
identified pattern in the numerical analysis.The mode of each
blade interfered weakly with the others constituting a well-defined global
pattern (Figure9).The frequency of the unsteady
pattern (StF=0.664), as well as several of its nonlinear interaction
components, had high cross-spectrum values and coherence levels greater than
0.6 for all the flow rates (Figure 13), confirming a continuous propagation of
the pressure pulsations of the unsteady pattern inside the impeller.Figure 13
Coherence
level between inlet (traverse 10) and impeller discharge (traverse 20)(n=600rpm).The presence of the BPF was not always well
defined, and its magnitude was often less than that of the other frequencies (Figures4, 5,
9, and
10). A possible explanation for the low BPF amplitude could be, on
one hand, the shadow effect of the diffuser blade on the transducer located on
the suction side and close to the trailing edge, on the other, the existence of
the unsteady pattern that determined a “natural” manipulation of the spectral
content, lowering the BPF intensity [25]. Moreover, when the
BPF was identifiable, it appeared to be pulsating as well as the frequencies of
the unsteady pattern. This apparent pulsating behavior could be explained by
the proximity of harmonics of the unsteady pattern and of a nonlinear interaction
component (see Table 2) collectively with the limited resolution in the
frequency domain of the wavelet transform due to the Heisenberg uncertainty.In regards to the phase displacement, Figure14
highlights a time-independent phase coincidence of the first harmonic of the
pattern (St=0.664) and an antiphase behavior
in the range between St=0.91 and St=1.13. In this figure the arrows
represent the relative phase relationship. The in-phase relation was presented
as arrows pointing right, the antiphase relation as arrows pointing left, and the
90° phase displacement as arrows pointing straight on.Figure 14
Cross-wavelet transform between the pressure transducers in
traverse 20 and in position 4 forQ/QDes=120%(n=600rpm).


## 6. Conclusions

Experimental and numerical analyses were carried
out on the flow field instability in an impeller, coupled with a vaned diffuser
to study the characteristics and the development of the unsteady phenomena.
Acoustic measurements were acquired by means of pressure transducers, placed in
the inlet duct and at the impeller discharge, and  in
a  diffuser vane, for two
different impeller rotation velocities, in order to identify the pulsating
phenomena, to define their characteristics, and to evaluate their evolution and
the influence of the impeller rotation velocity on their characteristics.The pressure signals were processed by the
spectral analysis in the frequency and in the time-frequency domains to obtain
information about the evolution at the time of the phenomena.The analysis of the pressure signals highlighted
the presence of a first pulsating phenomenon with a low frequency (St=0.071),
identified without the pump running. For these characteristics, it was
associated with the system fluctuations and in particular, with the
fluctuations of the water level in the accumulation tank of the pump. This
hypothesis was strengthened by its greater intensity in the inlet duct.On the other side, at the impeller discharge, a
rotating structure of pressure pulsations was identified with the fundamental
frequency atSt≅0.664 that was not associated with the blade passage
frequency (BPF). It was probably due to the pulsating wake zone at the impeller
discharge resulting in a continuous propagation of the pressure pulsations of
the unsteady pattern inside the impeller.Interaction with both the blade passage
frequency and with the system fluctuations generated nonlinear interaction
components in the spectra, appearing and disappearing in time with regular
intervals. The resulting structure was not frozen in time; it changed with a
pulsating behavior, depending on the constructive or destructive interference
of its constituting modes.The numerical analyses seem to confirm the
existence of this unsteady pattern and to detect the fluid-dynamical origin in
the blade load as in the jet-wake  instability.



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*Source: 202179-2008-11-30.xml*