Experimental and numerical investigation of noise generation due to acoustic resonance in a cavitating valve

Journal of Sound and Vibration 463 (2019) 114956

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Experimental and numerical investigation of noise generation due to acoustic resonance in a cavitating valve Stefan Semrau a, *, Romuald Skoda b, Walther Wustmann a, Klaus Habr a a b

Bosch Rexroth AG, Partensteiner Straße 24, 97816, Lohr am Main, Germany €t Bochum, Universita €tsstraße 150, 44801, Bochum, Germany Chair of Hydraulic Fluid Machinery, Ruhr Universita

a r t i c l e i n f o

a b s t r a c t

Article history: Received 1 March 2019 Received in revised form 9 September 2019 Accepted 11 September 2019 Available online 18 September 2019 Handling Editor: R.E. Musafir

In order to study cavitation induced noise generation in a hydraulic system, cavitation is generated at a planar orifice. The sudden condensation (implosion) of vapor results in shock waves which excite the connected hydraulic pipes with pressure fluctuations. The synchronization between the continuous condensation and evaporation process and shock wave reflection can result in a stationary wave with superelevated pressure amplitudes. The fluid-borne sound is transferred through the mechanical structure into the air, where it can be perceived as a distracting whistling noise. A high-speed camera is used to visualize the void volume. Dynamic pressure signals are recorded in the vicinity of the orifice to analyze the pressure pulsation. The operating point is varied, the orifice geometry is changed and the reflection properties of the pipe on the up- and downstream sides are changed too. In addition, CFD (Computational Fluid Dynamics) simulations of the test bench are used to investigate the root cause of the whistling noise. With regards to representative operating points, the numerical results confirm the measured shedding vapor frequency, particularly the pressure pulsation frequency in the pipe and its amplitude. The conjunction of the experimental and numerical investigations provides the following findings: By reducing the discharge pressure, the void volume increases, which leads to a reduction of the resonance frequency of the pipe downstream of the orifice. For large void volumes, the pressure wave reflection at the void volume can be identified in the amplitude spectrum. The whistling noise depends on the history of the flow field. So, the increase and the reduction of the discharge pressure level leads to different whistling ranges. The whistling range decreases with the increasing length of the pipe resonator. Besides that, the order of the dominant resonance frequency increases. The experimental and numerical results indicate that the whistling noise only occurs, when the jet downstream of the orifice and hence the vapor is perpendicular to the sound propagation in the pipe resonator. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Hydraulic valve Whistling noise Cavitation Self-sustained resonance frequency High-speed flow visualization CFD

1. Introduction In addition to performance-related and functional aspects, acoustic properties play an ever-increasing role in the development of hydraulic systems. This is attributed to the fact that statutory requirements have become more stringent, the acceptance on the customers' part has increased, and damage to one's health should be avoided. There is a great variety of * Corresponding author. E-mail address: [email protected] (S. Semrau). https://doi.org/10.1016/j.jsv.2019.114956 0022-460X/© 2019 Elsevier Ltd. All rights reserved.

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mechanisms which can be responsible for the development of noise in hydraulic systems. Ning and Zhang [1] analyzed sound sources exemplary based on hydraulic hoists. As Schmid [2] showed in early studies, valves may be the main generator of noise in a hydraulic system. According to Schmid, unsteady flow processes are responsible for the noise development. Reethof [3] provides a detailed summary of mechanisms in which turbulence is the origin of the noise emitted from hydraulic lines. Eich [4] detects that flow cavitation is another essential reason for the development of noise at valves. Flow cavitation, also referred to as hydrodynamic cavitation, occurs downstream of constricted cross-sections where high flow velocities are developing. Due to high dynamic pressure, the static pressure locally decreases to a value below a critical pressure close to the saturated vapor pressure which will cause the fluid to vaporize. The downstream flow velocity decreases, the static pressure increases, and the vapor condenses abruptly (implodes). The subsequent compression shocks result in broadband excitation of structure-borne noise which is transferred to the air by mechanical vibrations and will be perceived as airborne noise. The intensity of the noise is rated considerably higher than that of the turbulence noise. From today's perspective, Eich's recommendation to prevent cavitation by gradually decreasing the pressure difference is inefficient for many applications. With regards to present problems, cavitation cannot always be prevented due to the constantly increasing performance demands. However, without fully understanding of the physical mechanisms involved in the sound generation, a purposive development of noise-reduced hydraulic solutions is not possible. This paper cites scientific works of Ueno [5], Leonhard [6], Müller [7] and Okita [8], which are representative for the numerous research groups exploring the development of noise caused by cavitation in the context of valves. Based on experimental and numerical studies of a pressure reduction valve, Ueno [5] shows that cavitation is the main reason for noise development. As recommended by Ueno, the formation of stationary vortices should be supported to avoid instabilities and therefore reduce the noise emitted. Using experimental and numerical methods, Leonhard [6] analyzes the noise emitted by a steering valve. According to Leonhard, flow excitation is caused by flow separation at the point of the cross-sectional expansion resulting from friction effects. This produces downstream flow instabilities in the form of vortices which support the instationary cavitation process. Müller [7] and Okita [8] use additional flow visualization means in their experimental studies. Their high-speed recordings confirm that the gas and vapor fractions produced by flow cavitation are dominating the instationary flow field and the pressure distribution and therefore the noise emitted. In addition, instationary flow conditions at valves in combination with cavitation may excite and change the resonance frequency of the connected pipe. Resonance results in superelevated pressure amplitudes and this enormously intensifies the pressure load. This interdependency may not be neglected in the design, for example, for water pipes in the energy sector. Selected papers on this topic are those of Hassis [9], Testud [10,11] and Ruchonnet [12]. Hassis examines the influence of cavitation on the resonance behavior of a pipeline in which the pressure drop is realized with an orifice. Hassis uses the cavitation number s to estimate the cavitation intensity for the respective operating mode. With decreasing cavitation number, the vapor volume increases and therefore the noise emitted also increases. The sound velocity in the vaporized fluid is considerably lower than that in the homogeneous pure liquid. The resonance frequency of the pipe therefore drops with increasing cavitation. Hassis developed a hydroacoustic model which allows an analysis of the decrease in frequency. In this model, he relates the sound velocity and the extension of the void to the resonance frequency of the pipe section. Testud and Ruchonnet use similar test setups with which they can confirm Hassis’ basic findings and derive analytical models to describe the resonance frequency of their systems. Testud additionally documents different cavitation forms which result in a change of the acoustic properties of the cavitation zone. When hydrodynamic cavitation has formed, he documents a whistling noise which he attributes to acoustic resonance. The origin of the whistling noise is not the focus of his studies. Compared to available studies, this paper aims to improve the understanding of the cavitation-induced valve whistling noise caused by resonance between hydrodynamic cavitation and the eigenfrequency of the connected hydraulic system, using experimental and numerical studies. For the experimental studies a test bench is used to generate the whistling noise. It also provides a precise adjustment of the frequency and loudness of the whistling noise for certain configurations and operating points. The considered parameters in this present paper are listed below:
The pressure difference across a generic orifice is set by changing the discharge pressure level.
The history of the flow field is changed by switching between reducing and increasing the discharge pressure level.
The resonance frequencies of the pipe upstream and downstream of the orifice are set in a predefined range by positioning cross-sectional expansions in the pipes.
The jet shape downstream of the orifice and the flow resistance are adjusted by varying the orifice geometry. The experimental results are completed by compressible two-phase flow CFD (Computational Fluid Dynamics) simulations of selected operating points at which the whistling noise was measured. The objective of the simulations is to reproduce the results of the test and to deepen the findings gained from the test bench. The outline of the present paper is as follows: Section 2 describes the setup of the test bench. Section 3 provides a detailed description of the whistling noise and subsequently presents influencing parameters. Section 4 describes the numerical setup of the test bench. Section 5 describes the analysis with CFD simulations. The findings and conclusions are summarized in Section 6.

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2. Experimental setup 2.1. Test rig Fig. 1 shows the hydraulic circuit diagram (a) and the design (b) of the test rig. The power unit consists of a reservoir HC1, a pressure-controlled motor-pump unit HC2, a filter HC3 with a pore size of 3 mm, a high-pressure accumulator HC4, and a pressure relief valve HC5. Mineral oil HLP ISO VG 46 is used for all experiments. Pressure is relieved by the orifices PO1 or PO2. This results in a high-pressure (HP) section pHP and a low-pressure (LP) section pLP. The heat released in this process is sufficient to heat the oil up to a temperature as high that the target temperature can be obtained without additional heating. The precise target temperature is set by the connected air and water cooling unit HC6. The static pressure upstream and downstream of the orifice is set using the manual throttle valves HC7, HC8. The timeaveraged characteristic values are measured using the pressure sensors PSHP, PSLP, the temperature sensors TSHP, TSLP and the gear-type volume flow sensor QS. These parameters are measured with a recording frequency of f ¼ 100 Hz. An oxygen sensor OST installed in the reservoir is used for measuring the partial oxygen pressure. This allows to monitor the saturation of the liquid with air. It is ensured for all the measurements, that the oil in the reservoir is saturated with air. The sensors PSHP1, PSLP1 and PSLP2 are strain gauge pressure transducers of type XP5 by Althen. These sensors are dedicated to measure the instationary (transient) static pressures. The dynamic properties of these transducers are promoted by the comparatively small circular measurement area with a diameter of Ød ¼ 3.6 mm. In addition, the sensor membrane is at the top of the sensor shaft and the measurement area is flush with the flow. The natural frequency of the sensor is f ¼ 700 kHz. The sensors are used to register the frequency of the pressure changes in the pipe. The fluid-borne sound is transferred through the mechanical structure into the air, where it can be perceived as whistling noise. The sensors therefore must be able to cover at least the frequency range that is perceived by humans. According to Kinsler et al. [13], the hearing of young people responds to a frequency range of 16 Hz  f  16 kHz. A sampling rate of f ¼ 100 kHz is selected for the sensors so that frequencies in the ultrasonic range can additionally be measured. The downstream section is visualized based on the transmitted light method. This method is a noncontact optical measuring method which allows visualizing flow details and effects in a two-dimensional view. The fluid detail is located between the light source MVL, see Fig. 1 and the CCD camera. The camera is on the same optical axis as the light source. A mercury vapor lamp is used as a permanent light source. The light beams are parallelized with a lens system. This way density changes in the fluid can be recorded with the high-speed camera. Cavitation clouds consist of multiple vapor and gas microbubbles which reflect, refract and absorb the individual light beams. Snapshots are taken with a high-sensitivity CCD camera of type Speed Star from LaVision, which has a long-distance microscope. The optical measuring section is shown in Fig. 1b. The snapshots shown in this paper were taken with a maximum recording rate of fHS ¼ 60 kHz. 2.2. Tested orifices The transmitted light method is used to visualize the flow field. For this reason, a planar orifice geometry is used. Fig. 2 shows the low-pressure section of the flow geometry downstream of the orifice.

Fig. 1. (a) Hydraulic circuit diagram; (b) design of the test rig.

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Fig. 2. (a) Sectional view of the planar orifice geometry, the adjacent hydraulic system and the first reflection location; (b) detailed view with orifice PO1, visualization zone and pressure sensor PSLP1.

The whole core region near the orifice is planar and optically accessible (oa) up to the distance where the cross-sectional geometry is changed (Fig. 2a, A-A). The planar geometry is grouted with sapphire disks to seal the system. Contrary to acrylic glass, sapphire is resistant to pressure and erosion. The used visualization zone (vis) is shown in the detail Z-Z. The downstream fluid volume between the orifice and the cross-sectional expansion (cse) is intended to operate as a pipe resonator (pr). The cross-sectional expansion is purposefully introduced into the system. The high-pressure region is vertically offset but otherwise identical in design. The wire erosion method is used to produce different gap heights and throttle designs which can be inserted into a fitting mold made of heat-treated steel. Two orifices were used with different gap heights, thus narrowest flow cross-sections smin (Fig. 2b, Z:Z). For a comparison with a circular cross-sectional shape the hydraulic diameter dh ¼ 2sminb/(smin þ b) is used. Table 1 shows the flow cross-section Ae ¼ sminb, the cross-section rate l0/dh ratio and the main jet orientations (1)e(3) for the respective orifice. The l0/dh ratio is used for the volume flow characterization of orifices. A value below 1.5 is applied for both orifices. Due to a double angle in the orifice cross-section, the separation edge of the jet changes as a function of the narrowest flow crosssection. As far as orifice PO1 is concerned, the jet is always aligned with the vertical wall R12 (jet direction (1)). As far as orifice PO2 is concerned, the jet propagation varies between positions (1)e(3), depending on the operating point. A decisive factor for the validity of the instationary static pressure data is the distance of the sensor PSLP1 from the vaporization zone. The locally high-pressure peaks [14] developing during the sudden condensation of vapor decrease in inverse proportion with the distance from the point of their origin. It is therefore appropriate to position the sensor PSLP1 as close to the vapor zone as possible. The allowed mean maximum pressure of the sensors, however, is p  350 bar. This pressure is regionally exceeded many times over during the implosion of vapor. If positioned inside the collapse center, the sensor would be destroyed over time. The position of the sensor PSLP1, as shown in Fig. 2b therefore is a compromise. The distance of the sensor from the wall R12 is lLP1 ¼ d3. For certain operating points, when the discharge pressure level is low and, at the same time, the pressure difference across the orifice is large, the sensor PSLP1, is directly located in the cavitation zone. The strongly compressible void zone causes the sensor to record a broadband frequency noise with low amplitude. Under these circumstances the sensor PSLP2, whose distance to the orifice is lLP2 ¼ 2.8d3, ensures a better signal as it is situated outside of the void region. 2.3. Acoustic boundary conditions of the pipe resonator If undisturbed, a wave uniformly propagates in all spatial directions in spherical form. Subject to the test configuration, the shock wave propagates through the pipe having a large length/cross-section ratio downstream of the orifice: 7.9  lpr/ d3  17.9. It is therefore assumed to be acceptable that the wave propagates in one dimension only. The shock waves developing during the vapor condensation propagate as downstream longitudinal waves. This results in local pressure changes dp and compression and decompression processes of the fluid volume V. The volume change of the compressible 0 0 fluid dV ¼ V=Ef l dp is dependent on a fluid-specific replacement bulk modulus Ef l [15]. If wall effects are neglected, the 0 0:5 propagation speed of the pressure wave c ¼ ðEf l =rfl Þ is only dependent on the density r and the bulk modulus of the fluid 0 Ef l . The maximum flow velocity reached in the tests is considerably lower than the sound speed of the homogeneous pure liquid. Under these boundary conditions, a constant factor Z is used, which describes the ratio of the pressure change Dp to the Table 1 Characteristics of the two orifices. Orifice

PO1

PO2

Flow cross-section, Ae (mm2) Length cross-section rate, lo/dh Jet orientation, see Fig. 2b

0.69 0.68 (1)

0.92 0.42 (1)e(3)

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volume flow change DQ [16]. This factor is referred to as wave impedance Z. The volume flow Q is the product of the timeaveraged velocity v and the flow cross-section Ae. By inserting the volume flow and the pressure defined by the pressure surge [17], the wave impedance can be expressed as:

Z¼

Dp rc ¼ DQ Ae

(1)

According to Equation (1), the wave impedance depends on the density, the sound speed c and the flow cross-section. A change in the wave impedance results in wave reflection. Based on the ratio of the impedances upstream Zi and downstream Ziþ1 of the reflection point, the reflection coefficient r is defined according to Eq. (2):

r¼

1  Zi =Ziþ1 1 þ Zi =Ziþ1

(2)

The reflection coefficient is a measure for the type and the degree of the reflection. If r ¼ 0, the wave impedance does not change at the point of observation so that waves will not be reflected. If r ¼ |1|, which is the other extreme case, total reflection will occur. In the test setup (Figs. 1 and 2), cross-sectional expansions can be specifically introduced upstream (VCHP) and downstream (VCLP) of the orifice. Fig. 2a shows the wall R12, the cross-sections R23 and R34 which represent potential discontinuities for the shock wave propagation. Table 2 summarizes the acoustic properties for these locations. If the pressure difference is high enough, the complete flow cross-section downstream of the orifice may be filled with vapor. In this case, the position R12 is referred to as R12*. As mentioned above, the low-pressure and the high-pressure section are arranged offset. As compared with the wall area R12, the cross-section of the orifice is extremely small. Consequently, the downstream shock waves are almost only influenced by the wall R12. At this location, the reflection factor is approximately r z 1, which means it is a volume flow node and an almost total reflection does occur. The amplitude doubles directly at the wall and the pressure is reflected in phase. The shape of the flow cross-section changes at the transition R23. Since the flow cross-section remains constant, the reflection factor is r ¼ 0. No reflection is expected here. With the cross-sectional expansion R34 that was used, the pipe diameter becomes larger by a factor of Ød4/Ød3 ¼ 9, thus the expansion is a sound-absorbing reflection location. The reflection coefficient is r ¼ 0.98. This location is referred to as a pressure node. 98% of the incident pressure amplitude is reflected at the cross-sectional expansion. The expansion ratio was specially chosen for this study to ensure that the pressure amplitude is almost completely reflected at R34. The case of R12* exclusively results in reflection from the void. The density of the vapor phase is considerably smaller than that of the homogeneous pure liquid. The decrease in density is accompanied by a reduction in sound speed. The reflection coefficient is nearly r ¼ 1. Thus, the reflection behavior is equivalent as for location R34. When determining the running length of the wave, it must be considered that the wave is not directly reflected at the cross-sectional expansion R34. Subject to the smaller flow cross-section at the reflection point, the shock wave propagates beyond this point. The expansion chamber has a depth of lcse ¼ 2.5d3. Fletcher [18] proposes various analytical and empirical approaches to determine a length correction lec. The used approach lec ¼ (p/8)d3 is based on [19]. The proportion of the length correction on the total length ltot ¼ lpr þ lec decreases with the length of the pipe resonator lpr. 2.4. Case dependent behavior of the pipe resonator The downstream section, according to Fig. 2a is the part between the wall R12 and the cross-sectional expansion R34. It is a one-dimensional wave guide and represents a resonator if it is excited at its eigenfrequency. The resonance frequencies of an oscillator are frequencies at which a sound or vibration field develops in loss-free cases, even if the source is arbitrarily weak [15]. However, this is only the case assuming that the sound waves are weakened neither when they propagate along the medium nor when they are reflected from acoustic boundaries. At the test rig viscous friction on pipe walls enforced by frequency dependent fluid friction and non-ideal reflection locations, for example, impair the development of resonance. In case of resonance, incoming and reflected waves superimpose and intensify each other in phase. There will be stationary waves which are characterized by stationary minima and maxima pressure and velocity values. For the one-dimensional wave guide, two potential resonator types and thus frequencies may occur, dependent upon the reflection properties downstream of the orifice.

Table 2 Summary of the acoustic properties for the potential discontinuities for the shock wave propagation between the orifice and the enlargement of the crosssection. Position

R12 wall

R23 cross-sectional transition

R34 cross-sectional extension

R12* vapor

Event Reflection coefficient, r

Ae1 / 0 r12 z 1

Ae2 ¼ Ae3 r23 ¼ 0

Ae3 ≪ Ae4 r34 ¼ 0.98

r2, c2 [ r1*, c1* r12* z 1

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Fig. 3. (a), (b) Simplified (identical) geometry without and with vapor region; (c), (d) pressure amplitude b p for one period tP.l/4 resp. tP.l/2 over the length of the resonator ltot resp. ltot* and the relative pressure amplitude b p R12/ b p resp. b p R23/ b p plotted over the time t; (e), (f) expected characteristics of the stationary wave for the first-order kl/4 ¼ 1 resp. kl/2 ¼ 1 and second-order kl/4 ¼ 3 resp. kl/2 ¼ 3 resonance frequency.

According to Table 2, in case of a small vapor volume, the resonator is limited by a sound-reflecting R12 and a soundabsorbing R34 reflection boundary (Fig. 3a). The sound wave must traverse the pipe resonator four times to connect to a new emitted shock wave in phase (Fig. 3b). Thus, it is called l/4-resonator because the resonator length is a quarter of the wavelength l. The resonance frequency of the l/ 4-resonator fl/4 is determined according to the following equation:

fl=4 ¼

kl=4 c 4ltot

(3)

Fig. 3c shows the deviation of the pressure amplitude for the first-order (kl/4 ¼ 1) and second-order (kl/4 ¼ 3) resonance frequency. The higher-order harmonic frequencies are integer uneven multiples kl/4 > 1 of the fundamental frequency kl/4 ¼ 1. The number of pressure nodes increases with the order of the resonance frequency. In case that the wall R12 is covered with vapor R12*, the change of the reflection property of the boundary causes the change of the resonator type to a l/2-resonator (Fig. 3d). In addition, the void shortens the total length ltot* of the resonator. The inphase connection of a wave is achieved after it has traversed the pipe resonator two times (Fig. 3e):

fl=2 ¼

kl=2 c 2ltot

(4)

Consequently, the resonance frequency fl/2 is twice the number as for the l/4-resonator. This is one of the main differences between these two resonator types. The second main difference is, that the higher-order harmonic frequencies of the l/2resonator are integer multiples kl/2 > 1 of the fundamental frequency kl/2 ¼ 1 (Fig. 3f). 3. Experimental results and discussion 3.1. Cavitation regimes and whistling phenomenon The test bench, introduced in Section 2, is used to generate a cavitation-induced whistling noise. In this section the focus is on the origin of the noise and particularly on the pressure pulsation frequency in the pipe. The configuration presented below is used as reference for the comparison of the whistling noise for different settings in Section 3.2. At a high-pressure level of pHP ¼ 200 bar, an oil temperature of TLP ¼ 349 K and the orifice PO1 (Table 1), the pressure difference is continuously increased

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Dp[ during the measurement, by decreasing the discharge pressure level pLP until ambient pressure p0 is reached. The length of the pipe resonator, which is the distance between the wall R12 downstream of the orifice and the cross-sectional expansion R34, is set to lpr ¼ 7.9d3. This is the shortest length which is realizable with the test bench. (lpr ¼ lpr.min). The Reynolds number Re ¼ vmdh/y is calculated using the time-averaged flow velocity vm, the hydraulic diameter dh and the kinematic viscosity y. The maximum Reynolds number in the orifice is Re ¼ 1400 for the examined operating range. Fig. 4 shows the measured volume flow Qexp (a), the amplitude spectra of the pressure sensors PSLP1 and PSLP2 (b) and transmitted light snapshots of characteristic operating points (c). The measured volume flow, see Fig. 4a is time-averaged. For the diagram these data are taken every Dp ¼ 10 bar. To analyze the transient pressure oscillations downstream of the orifice, the pressure signals PSLP1 and PSLP2 are transferred from time to frequency range. Every DpLP ¼ 0.5 bar a spectral analysis of the pressure signal is done by using the fast Fourier transformation (FFT) and flat top windowing [20]. The pressure amplitude root-mean-square values associated with the frequencies and the discharge pressure pLP are colored with logarithmic scaling (Fig. 4b). The operating points are subdivided into characteristic ranges for different discharge pressure levels to allow systematic analysis. While decreasing the discharge pressure level pLPY, starting at pLP ¼ 200 bar, the cavitation process begins in range RA. A whistling noise can be heard within a discharge pressure range of 17 bar  pLP  38 bar. Based on different dominant frequencies and loudness, this range is divided into the ranges RB - RD. For a constant operating point, the frequency component with the highest-pressure amplitude is called the dominant frequency. The whistling noise stops after further pLP decrease, leading to range RE which extends to discharge pressure level down to ambient pressure p0. 3.1.1. Cavitation inception and the beginning of the choked flow regime (range RA) Range RA: according to the Bernoulli equation, the analytical volume flow Qth across an orifice rises with increasing pressure difference as follows: Q2 ~ Dp. Fig. 4a shows that the measured volume flow Qexp corresponds to the analytical flow rate Qth up to a pressure difference of approximately 120 bar. At operating point A3, a volume flow plateau is reached Q 2exp =

Fig. 4. Measurement for the reference for decreasing discharge pressure level pLPY: (a) measured time-averaged volume flow Q 2exp and the analytical volume flow of an ideal orifice Q 2th relative to the measured volume flow maximum Q 2exp:max ; (b) amplitude spectra b p LP1(fLP1) and b p LP2(fLP2) of the pressure sensors PSLP1 and PSLP2; (c) snapshots of the cavitation zone for characteristic operating points.

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Q 2exp:max ¼ 1 which does not rise any further with increasing pressure difference, due to chocking as a consequence of void inside the gap. The snapshots for the two operating points A1 and A2 (Fig. 4c) show the flow field, before reaching the plateau. A fluid jet can be seen, which is aligned with the vertical wall (Fig. 2b, R12, (1)) and diverted towards the outlet. With the increase of the volume flow, the jet velocity also rises until the local static pressure drops to a value below the saturated vapor pressure psat. Locally, the static pressure decreases by approximately two orders of magnitude below the ambient pressure. Then, the fluid vaporizes. At the transition to the volume flow plateau, snapshot A3 in Fig. 4c clearly shows a cavitation zone that is concentrated around the vicinity of the gap. The vapor region is almost stationary. In addition, the jet fans out when cavitation occurs. When the low-pressure is decreased from pLP z 80 bar (A3) to pLP z 55 bar (A4), the pressure drop increases but the flow rate remains constant. As compared to A3, the snapshot A4 in Fig. 4c shows a larger almost stationary vapor region. By the further increase of the pressure drop, the additional energy leads to more vapor. Starting at operating point A4, the amplitude spectrum of the pressure sensor PSLP1 (Fig. 4b) contains the two highfrequency components fLP1 z 4.8 kHz and fLP1 z 14.4 kHz. It is known that cavitation excites the fluid system with broadband shock waves which mainly have high-frequency contents [21]. Below, it will be shown that the measured frequencies correspond to the resonance frequency of the downstream pipe section between the orifice R12 and the cross-sectional expansion R34 (Fig. 2a) for which the resonator type of the pipe was determined. Since, at pLP z 55 bar (A4), the void concentrates around the gap, it can be assumed that the shock waves are essentially reflected from the wall R12 and not from the void R12*. As described in Section 2.4, the behavior of the downstream pipe is that of a l/4-resonator due to the different reflection coefficients of the wall r12 z 1 and the cross-sectional expansion r34 ¼ 0.98. So, the analytical fundamental frequency (kl/4 ¼ 1) is calculated with Eq. (3) by inserting the total length ltot (Sec. 2.3) and the sound speed of the homogeneous pure liquid. The analytically determined frequency fl/4 z 4.95 kHz is 3% larger than the measured fundamental frequency. The second dominant measured frequency fLP1 z 14.4 kHz corresponds to the second-order analytical resonance frequency of a l/ 4-resonator (kl/4 ¼ 3), which is three times larger than the fundamental frequency. The second-order resonance frequency is characteristic for systems with stable vibrations in which non-linear effects occur, such as cavitation [11]. The observation that the measured and the analytical frequency perfectly correspond to each other confirms firstly the assumption that the cavitation process causes the excitation of the pipe resonator with high-frequency contents and secondly that the resonator response corresponds to a l/4-resonator. When the discharge pressure level of approx. pLP ¼ 55 bar (A4) is further reduced to pLP z 38 bar, which is the boundary between range RA and RB, the resonance frequency continuously decreases slightly (Fig. 4b). This is due to the reduction of the sound speed of the homogeneous pure liquid when the pressure level is reduced. The reduction of the measured resonance frequency corresponds to the analytical relationship according to Eq. (3). Within range RA, the measured pressure amplitudes, which are a measure for the loudness of the noise emission of the test bench, are below b p LP < 0.5 bar. The level of pressure amplitudes above which a whistling noise can be perceived by humans strongly depends on the design of the hydraulic system. In range RA no whistling noise differing from the usual noise of the test bench can be perceived. 3.1.2. Occurrence of the whistling noise (range RB - RD) A whistling noise can be heard below a discharge pressure level of pLP z 38 bar. The noise emission ceases when the discharge pressure level is reduced to a value below pLP z 17 bar. Besides the whistling noise, the essential difference to the ranges RA and RE can be seen in the flow field. The void fraction is not constant. Instead, the vapor condensates periodically and then builds up again abruptly. For the whistling range (RB - RD) each snapshot B1, B2, C1, C2, D1 and D2 in Fig. 4c shows the frame with the maximum void fraction for each operating point. To confirm the assumption that the shock wave reflection and the periodical vapor behavior are in phase, Fig. 5 shows the normalized pressure signal pLP1=pLP over time t in combination with corresponding snapshots of the flow field for operating point D1: The six snapshots are taken with a time interval of Dt ¼ 5$105 s. When snapshot D1.1 was taken, the static pressure at the sensor PSLP1 reaches its maximum for which reason the vapor volume is at its minimum. The shock wave which was developed during the previous abrupt condensation of the vapor propagates towards the reservoir. The pressure downstream of the orifice decreases while the vapor amount increases (D1.2). According to Section 2.3, the pressure waves are reflected at the cross-sectional expansion with phase shift. According to Fig. 3b, the pressure level downstream of the orifice decreases after half of the period has elapsed. As can be seen from snapshots D1.3 and D1.4, this supports the vaporization process. After an integer period duration, the pressure level downstream of the orifice is increased again. The jet-induced vapor collapses abruptly (D1.5, D1.6) and generates a new shock wave. The synchronization of the shock wave reflection with the pulsating cavitation process results in a superelevation of the pressure amplitudes. In the case of resonance, the downstream pressure changes support the formation of vapor. As described in Section 3.1.1, the operating points are subdivided into characteristic ranges for different discharge pressure levels to allow systematic analysis. Based on different dominant frequencies and loudness, the whistling range is divided into the sections RB - RD. The amplitude spectra in Fig. 4b give an overview of the resonance frequencies and pressure amplitudes depending on the discharge pressure level. In order to point out the differences within the whistling range, the measured data are reduced and plotted in Fig. 6b. While the amplitude spectra in Fig. 4b contain all frequency components for an operating point, Fig. 6b only shows the frequency fLP1 corresponding to the highest-pressure amplitude b p LP1. The considered range is

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Fig. 5. (a) Amplitude spectra b p LP1(fLP1) of the pressure sensor PSLP1 for decreasing discharge pressure level pLPY; (b) pressure pLP1(t) of the sensor PSLP1 relative to the time-averaged discharge pressure level pLP within the operating point D1; (c) corresponding snapshots of the cavitation zone.

Fig. 6. (a) Amplitude spectra b p LP1(fLP1) of the pressure sensor PSLP1 for decreasing discharge pressure level pLPY; (b) dominant resonance frequency fLP1 scaled to the analytical fundamental l/4-frequency fl/4(kl/4 ¼ 1) and the pressure amplitude b p LP1 with respect to the discharge pressure level pLP.

marked with an arrow in the amplitude spectrum in Fig. 6a. Furthermore, the discharge pressure level range in Fig. 6b is smaller because only operating points ( b p LP1  0.5 bar) are considered, where a whistling noise can be perceived. In order to focus on the order of the resonance frequency kl/4, the dominant frequency in Fig. 6b is related to the operating-pointdependent analytical fundamental frequency fl/4(kl/4 ¼ 1). The range of operating points for which the highest-pressure amplitudes correspond to the second-order resonance frequency (kl/4 ¼ 3), are defined as range RB. For the ranges RC and RD the fundamental frequency (kl/4 ¼ 1) is dominant. The range is subdivided because of an abrupt drop of the resonance frequency (pLP z 20.5 bar). Hereinafter the different behavior of the hydraulic system is discussed for each range, in which the whistling noise occurs: Range RB: as discussed in Section 3.1.1, in range RA a stationary wave has developed between the wall R12 and the crosssectional expansion R34 at a discharge pressure level of pLP z 55 bar. As has already been shown for range RA, the void fraction increases with decreasing discharge pressure level. As a result, the pressure amplitude of the stationary wave also increases.

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b LP1. At the transition from range According to theory, see Fig. 3b the pressure amplitude at wall R12 is pulsating between ±2 p RA to RB (pLP z 38 bar) the pressure changes at wall R12 are large enough to excite a pulsation of the vapor amount. As a result, the pressure amplitudes for the second-order resonance frequencies (kl/4 ¼ 3) rise to b p LP1 z 6 bar and a whistling noise can be heard. According to theory, see Fig. 3c, a pressure node ( b p LP1 ¼ 0 bar) is situated between the wall R12 and the cross-sectional extension R34. According to Fig. 4c, the snapshots (B1) for pLP z 35 bar and (B2) for pLP z 24 bar demonstrate that the maximum vapor amount increases while the discharge pressure is reduced. A deeper insight into the vapor behavior is provided by the snapshots of the flow field. At operating point B2 the image shows that the cavitation zone is so large, that vapor filled vortices are detaching in the vicinity of the gap. Range RC: due to the vapor-filled vortices that are detaching continuously according to the snapshot B2, a large part of the vertical wall R12 is covered with void. It is assumed that the influence of the reflection from the newly formed vapor zone R12* at the transition from range RB to RC has become so dominant, that it disturbs the development of the pressure node between the wall R12 and the cross-sectional extension R34. As a result, the pressure amplitudes of the fundamental frequency (kl/4 ¼ 1) increase significantly (Fig. 6) and the maximum expansion of the vapor zone (Fig. 4c, C1) increases as well. The hearable whistling noise rises and its frequency drops by factor of ~3. The amplitude spectrum in Fig. 4b contains two further superimposed resonance frequencies, beside the first and secondorder resonance frequency of the l/4-resonator for the ranges RC and RD. To point this out, Fig. 7 shows amplitude spectra of the operating point C1 and C2. The mechanism for the additional frequency components shall be described here: in the grown cavitation zone, the shock waves are now also reflected from the vapor to a perceptible extent. In comparison to the wall R12, the shock waves are reflected from the vapor volume R12* with a phase shift (Table 2). In the event of resonance, the discharge pressure line therefore responds as a l/2-resonator (Fig. 3d). According to Section 2.4, the fundamental frequency (kl/2 ¼ 1) of the l/2resonator is twice the frequency of a l/4-resonator. The second-order resonance frequency (kl/2 ¼ 2) of a l/2 resonator is twice its fundamental frequency. The distribution of the pressure amplitudes in Fig. 7 shows that the reflection from the vapor R12* is clearly subordinate to the reflection from the wall R12. When the pressure is reduced from pLP z 23 bar (C1) to pLP z 20 bar (C2), the maximum vapor volume increases further (Fig. 4c) until the vapor covers almost the whole vertical wall R12. Range RD: the transition from range RC to RD is defined by an abrupt decrease of the fundamental frequency of the measured l/4-resonator, which can be seen in the amplitude spectrum in Fig. 4b. In addition, the pressure amplitude of the l/ 2-resonator increases significantly. Figure 8 shows amplitude spectra of the operating points D1 and D2. For the operating points D1 and D2, the pressure amplitude of the fundamental l/4-resonator frequency is three times larger, compared to the fundamental l/2-resonator frequency. The amplitudes of the higher-order resonance frequencies are suppressed almost completely. A reason for the shift of the amplitude spectra is assumed in the position of the void fraction. As written above, at the transition from range RC to RD the vertical wall R12 is covered with vapor. From this operating point on, the decrease of the discharge pressure level is accompanied by a growth of the maximum vapor volume towards the crosssectional expansion R34 (Fig. 4c, D1 and D2). It can be concluded, that the location of the vapor zone, in relation to the direction of the propagation of sound, has an essential influence on the amplitude spectrum. 3.1.3. Developed cavitation and degassing air regime (range RE) Range RE: for a discharge pressure level lower than pLP < 17 bar, the pressure pulsation amplitude measured with sensor PSLP1 suddenly decreases (Figs. 4b and 6). This indicates that the shock wave reflection is no longer in phase with the pulsating cavitation process. This assumption is confirmed by the visualization of the flow field. The void volume remains constant. As a result, no whistling noise can be perceived any longer. Although the pressure difference for operating point E1 is higher than for operating point D2, the void fraction is lower (Fig. 4c). For operating point E1, the measured resonance frequency fLP1(kl/ 4 ¼ 1) is again significantly closer to the analytical fundamental fl/4(kl/4 ¼ 1) frequency of the l/4-resonator with pure homogeneous liquid. The comparison between measured and analytically determined fundamental frequency fl/4(kl/4 ¼ 1) for the characteristic operating points according to Fig. 4 is listed in Table 3. In conjunction with the snapshots from Fig. 4c, Table 3 shows the influence of the vapor fraction on the mixture sound velocity of the pipe resonator. The increase of the vapor fraction is followed by an increase of the frequency deviation. That is

Fig. 7. Amplitude spectra of the operating points C1 and C2 (Fig. 4b).

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Fig. 8. Amplitude spectra of the operating points D1 and D2 (Fig. 4b).

because the reduction of the sound speed caused by the vapor fraction is not analytically considered according to Eq. (3). The deviation reduction from an operating point within the whistling range (D2) to an operating point outside of the whistling range (E1) again shows, that the synchronization of the shock wave reflection with the pulsating cavitation process within the whistling range supports the formation of vapor. Regarding the amplitude spectrum of the sensor PSLP2 (Fig. 4b), the further drop of the discharge pressure up to the ambient pressure p0 leads to a significant reduction of the resonance frequency. This is caused by the increasing influence of the dissolving air. This assumption is discussed below using the snapshots (Fig. 4, E1 - E4). As has been described in Section 2.1, the fluid is saturated with dissolved air. The saturation pressure of air is significantly higher than the saturated vapor pressure. For that reason, the air dissolved in the oil already diffuses from the fluid before vaporization starts. As a consequence micro air bubbles are formed. In the ranges RA - RD, the discharge pressure level is high enough, so that the free air dissolves again immediately. Below a discharge pressure level of pLP  15 bar (E1), the influence of the air release clearly increases [22]. Below a pressure level of pLP  10 bar (E2), the release of air bubbles can be observed in addition to a growing vapor area. Air bubbles can be seen as dark spots in the snapshots. When the pressure level is further reduced to pLP z 5 bar (E3), the observed flow field is gradually filled with further vapor and air bubbles. The sound velocity of air is about four times lower than that of the homogeneous pure mineral oil liquid. The numerous phase boundaries of the vapor and air bubbles additionally result in a reduction in the mean sound velocity in the pipe resonator. As a result, the resonance frequency (Fig. 4b) in the pipe resonator drops considerably. At a discharge pressure level of pLP z 5 bar (E4), the content of vapor and air is high enough to distract the light beams from the mercury vapor lamp. As a consequence, no further visualization can be generated below that pressure level.

3.2. Influence quantities for the whistling phenomenon 3.2.1. Pressure wave propagation upstream of the orifice The instationary (transient) static pressure upstream of the orifice is recorded using the sensor PSHP1 (Section 2.1 and Fig. 1). It is the equivalent one to the sensor PSLP1 (Fig. 2b, Z:Z), which is installed downstream of the orifice. Within the range of the examined operating points in Section 3.1, there are no resonance frequencies in the high-pressure section with amplitudes b p HP1  0.5 bar. That is in contrast to the low-pressure section. The cavitation process which excites the propagation of shock waves takes place downstream of the orifice. The measurements show that the sound propagation in the supply pipe to the orifice is almost separated from its discharge. The main causes are the very small gap of the orifice and the geometrical offset at the orifice. 3.2.2. History of the flow field For the reduction of the discharge pressure level pLPY from pLP ¼ 200 bar to pLP z p0 a whistling noise occurs between 17 bar  pLP  38 bar, see Fig. 6. Subsequently, the discharge pressure level is increased pLP[ to the initial pressure of pLP ¼ 200 bar (Dp ¼ 0 bar). The oil temperature is set to TLP ¼ 349 K. During the pressure increase, the whistling noise can be heard between 27 bar  pLP < 38 bar, see Fig. 9a. The comparison between decreasing pLPY (blue dotted curve) and increasing pLP[ (red dotted curve) discharge pressure level is displayed in Fig. 9b: The dominant resonance frequency and the pressure amplitude correspond to the values for the decrease of the discharge pressure level. The range in which the noise is generated, corresponds to the specified measurement range RB which relates to the second-order resonance frequency (kl/4 ¼ 3). Consequently, the fundamental frequency (kl/4 ¼ 1) cannot be heard during the increase of the discharge pressure level. That supports the assumption of Section 3.1.2 that the system excitation in range Table 3 Measured fundamental frequency fLP1(kl/4 ¼ 1), analytical l/4-resonance frequency fl/4(kl/4 ¼ 1) and its deviation for characteristic operating points (Fig. 4b). Operating point

A4

B1

B2

C1

C2

D1

D2

E1

Measured fundamental frequency, fLP1(kl/4 ¼ 1) (kHz) Analytical l/4-resonance frequency, fl/4(kl/4 ¼ 1) (kHz) Frequency devitation, fLP1(kl/4 ¼ 1)/fl/4(kl/4 ¼ 1) (%)

4.81 4.95 3

4.65 4.92 6

4.21 4.90 14

4.20 4.90 14

4.05 4.89 19

3.42 4.89 31

3.32 4.89 33

4.00 4.88 18

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Fig. 9. (a) Amplitude spectra b p LP1(fLP1) of the pressure sensor PSLP1 for increasing discharge pressure level pLP[; (b) dominant resonance frequency fLP1 scaled to the analytical fundamental l/4-frequency fl/4(kl/4 ¼ 1) and the pressure amplitude b p LP1 plotted over the discharge pressure level pLP for decreasing pLPY and increasing pLP[ discharge pressure level.

RB, with the dominant second-order resonance frequency (kl/4 ¼ 3), is required for the development of the dominant fundamental frequency (kl/4 ¼ 1) in the pipe resonator. The whistling noise depends on the history of the flow field. So, the increase and the reduction of the discharge pressure level leads to different whistling ranges. 3.2.3. Length of the pipe resonator Based on the reference settings (lpr ¼ lpr.min), measurements are taken with different distances between the wall R12 downstream of the orifice and the cross-sectional expansion R34. Therein, the length of the pipe resonator is gradually increased within a range of 1.0  lpr/lpr.min  2.27. The discharge pressure level is reduced from pLP ¼ 200 bar to pLP z p0. The oil temperature is set to TLP ¼ 349 K. According to the measurements, the maximum volume flow is independent of the length of the pipe resonator lpr. For operating point A4 (Fig. 4b), the vapor zone is in the vicinity of the gap and quasi-stationary. There is no perceivable whistling noise. Table 4 shows the respective dominant measured resonance frequency for each pipe resonator length at operating point A4 and a comparison with the analytical fundamental frequency of a l/4-resonator according to Eq. (3): The measured resonance frequency decreases according to the theoretical relationship fLP1 ~ 1/ltot. The deviation between the analytically determined frequency and the measured value does not exceed 5%. For every resonator length, the pressure signal PSLP1 is evaluated analogous to the measurement data for lpr/lpr.min ¼ 1 in Fig. 6b. Fig. 10 shows the summarized measured whistling ranges for each resonator length with respect to the order of the measured dominant l/4-resonance frequency: The whistling range is marked with the upper (UB) and lower bound (LB). Fig. 10 illustrates, that the whistling range becomes smaller as the resonator length gets longer. This can be explained by the losses which increase along the resonance path and therefore attenuate the propagation of the shock waves. The order of the dominant resonance frequency rises with the resonator length and thus the number of vibration nodes in the pipe resonator increases as well. For a resonator length of lpr/lpr.min ¼ 2.27 the highest-pressure amplitude corresponds to the fourth-order resonance frequency (kl/4 ¼ 7). The measurements again confirm that the cavitation process excites the downstream section with broadband pressure amplitudes and that the void volume in the whistling range pulsates with the dominant resonance frequency of the pipe resonator. 3.2.4. Direction of the jet To examine the influence of the jet position on the whistling noise, the orifice PO2 according to Table 1 is used, instead of the orifice PO1 (reference). The cross-section of the orifice PO2 is larger and the jet behavior changes, depending on the operating point. Table 4 Comparison of the measured resonance frequency fLP1(kl/4 ¼ 1) with the analytical resonance frequency fl/4(kl/4 ¼ 1) for the resonator lengths 1.0  lpr/lpr.min  2.27 at the operating point A4 (Fig. 4b). Relative length of the pipe resonator, lpr/lpr.min

1.00

1.32

1.63

1.95

2.27

Measured fundamental frequency, fLP1(kl/4 ¼ 1) (kHz) Analytical l/4-resonance frequency, fl/4(kl/4 ¼ 1) (kHz) Frequency deviation, fLP1(kl/4 ¼ 1)/fl/4(kl/4 ¼ 1) (%)

4.81 4.95 3

3.61 3.81 5

2.96 3.09 5

2.47 2.60 5

2.17 2.24 3

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Fig. 10. Upper (UB) and lower (LB) bound of the whistling noise for the resonator lengths 1.0  lpr/lpr.min  2.27 and the order of the measured dominant resonance frequency fLP1 for a decreasing discharge pressure level pLPY.

The discharge pressure level is reduced from pLP ¼ 200 bar to pLP z p0. The oil temperature is set to TLP ¼ 349 K. In Fig. 11, the measured data are illustrated like the data of the reference according to Fig. 4: The maximum value of the volume flow is reached at a discharge pressure level of pLP z 60 bar (Fig. 11a). The maximum volume flow is about 35% higher than that of the reference. When the discharge pressure level is decreased pLPY, no whistling noise can be heard. According to the amplitude spectrum of the pressure signal PSLP1 (Fig. 11b) only relatively small pressure amplitudes ( b p LP1  0.5 bar) are measured. The void region for each operating point is almost stationary. Snapshots of the void fraction for the selected operating points APO2 - HOP2 are shown in Fig. 11c.

Fig. 11. Measurement with the orifice PO2 for decreasing discharge pressure level pLPY: (a) measured time-averaged volume flow Q 2exp and the analytical volume flow of an ideal orifice Q 2th relative to the measured volume flow maximum Q 2exp:max ; (b) amplitude spectra b p LP1(fLP1) of the pressure sensor PSLP1; (c) snapshots of the cavitation zone for characteristic operating points.

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Fig. 12. Measurement with the orifice PO2 for increasing discharge pressure level pLP[: (a) amplitude spectra b p LP1(fLP1) of the pressure sensor PSLP1; (b) snapshots of the cavitation zone for characteristic operating points.

Subsequently, the discharge pressure level is increased again to pLP ¼ 200 bar. Fig. 12 shows the amplitude spectrum (a) and the snapshots of the void fraction for the selected operating points IPO2 - LPO2 (b): While increasing the discharge pressure level pLP[, a whistling noise can be heard between 14 bar  pLP  20 bar. And again the occurrence of the whistling noise depends on the history of the flow field, see Section 3.2.2. It is assumed that the variation of the jet direction causes the different whistling behavior for the increase and decrease of the discharge pressure for the setup with the orifice PO2. For further analysis, the snapshots of the operating points APO2 - LPO2 (Figs. 11c and 12b) are evaluated below: The snapshot APO2 shows that the jet points into the downstream section at an angle of about 45 to the horizontal wall for this operating point. While lowering the discharge pressure level, the jet fans out (Fig. 11c, BPO2). In contrast to the reference, the jet changes its position subject to the operating point. The first amendment can be seen for operating point CPO2: the jet is aligned with the horizontal wall. This changes at operating point DPO2: the jet is directed upwards to the vertical wall R12. The measured dominant frequencies are attributed to the resonance frequency of the pipe resonator, analogous to the reference. The amplitude spectrum (Fig. 11b) contains the fundamental frequency of the l/4-pipe resonator (kl/4 ¼ 1) for the two operating points CPO2 and DPO2. In comparison to the reference, the second-order l/4-resonance frequency (kl/4 ¼ 3) has a smaller extent as compared qualitatively with the reference. For the operating points EPO2 and FPO2, the vicinity of the orifice is filled with vapor. By decreasing the discharge pressure level below operating point FPO2, the jet is aligned with the vertical wall R12 (Fig. 11c, GPO2). It is assumed that the propagating pressure waves in the pipe, with abruptly increasing pressure amplitudes below operating point FPO2 (Fig. 11b), redirect the jet. The jet direction is similar to the reference now. As can be seen in Fig. 11b, the fraction of the pressure amplitude of the second-order l/4-resonance frequency (kl/4 ¼ 3) significantly rises, so that the distribution of the orders of the resonance frequency is comparable with the range RA of the reference (Fig. 4b). It is presumed that the change of the jet direction causes the shift in the amplitude spectrum. In analogy with operating point E4 (Fig. 4c) from the reference, the snapshot HPO2 is almost completely black because of the large void fraction. For the subsequent increase of the discharge pressure, the jet remains aligned with the vertical wall (Fig. 12b I). As expected, the void fraction decreases as the discharge pressure level rises initially (Fig. 12b J). After the discharge pressure level at operating point FPO2 (Fig. 11c) is exceeded again, a whistling noise can be heard. The void fraction is not stationary anymore and its maximum grows (Fig. 12b, KPO2). The frequency of the periodically forming vapor is equal to the fundamental resonance frequency measured with the pressure sensor PSLP1. The operating points EPO2 (pLPY, no whistling sound) and KPO2 (pLP[, whistling sound) are about the same, even though the jet direction is different. In contrast to the decrease of the discharge pressure, the jet remains aligned with the vertical wall R12 when the pressure level is increased. Similar to the reference case, the whistling noise does only occur, when the jet direction is perpendicular to the direction of propagation of the pressure waves. So, the contact surface between oil and vapor in the direction of pressure propagation is very large at a specific moment. This encourages the condensation of the vapor, which in turn leads to a stronger vaporization. In this case, relatively small pressure amplitudes are required to excite the pulsation of the vapor volume. The amplitude spectrum (Fig. 12a) of the pressure signal PSLP1 for operating point KPO2 corresponds with that from range RD of the reference (Fig. 4b). It contains the first-order and second-order resonance frequency of the reflection from the wall R12 (l/4-resonator) and from the vapor R12* (l/2-resonator). The fundamental l/4-resonance frequency (kl/4 ¼ 1) of fLP1 z 2.2 kHz, for operating point KPO2 is smaller than that of the reference at a comparable discharge pressure level. As the cross-section of the orifice PO2 is larger than that of the reference PO1, more vapor is generated for PO2 at the same operating point. As it has already been shown in Section 3.1, the vapor amount is responsible for the reduction of the resonance frequency of the pipe resonator.

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In contrast to the reference, see Section 3.2.2, the pressure amplitudes of the fundamental l/4-resonance frequency (kl/ are dominant without a further dominant higher-order resonance frequency. The low-pressure amplitudes for the operating point LPO2 in spectrum Fig. 12a indicate that the shock wave reflection is no longer in phase with the pulsating cavitation process. The snapshot in Fig. 12b shows that for operating point LPO2, the jet has been detached again from the vertical wall. The comparison of the measurements with different orifices PO1 and PO2 indicates, that a jet direction, which is perpendicular to the direction of propagation of the pressure waves is especially susceptible to the whistling noise.

4 ¼ 1)

4. Numerical setup 4.1. Numerical method and physical model A compressible and transient CFD flow algorithm is used to capture cavitation-induced shock waves [23]. It is extended by particular convective flux formulation for cavitation [24]. To solve the Navier-Stokes equations, a compressible density-based flow algorithm with an explicit time step procedure (Runge-Kutta fourth-order method) and a fixed Courant-Friedrichs-Lewy (CFL < 1) number is used. A homogenous mixture of liquid and vapor is assumed, as well as the kinematic and thermodynamic equilibrium for the liquid and vapor. The relative motion between the phases is neglected and the transition takes place immediately. Turbulence is neglected, because cavitation is an inertia-dominated phenomenon [25]. Phase change is modelled based on an approach using barotropic equation of state (EOS), following an isentropic path [24]. As a result, the density and therefore the speed of sound and viscosity are only a function of the pressure [26]. So, the temperature is fixed in the simulation. The data for the density, the dynamic viscosity and the sound velocity are specified for the flow solver in tabular form based on measurements. In the past, it has already been shown, that the above-mentioned modelling approach is able to simulate threedimensional instationary compressible cavitating flows with shock wave propagation that is induced by collapsing bubbles [25,27e31]. The development of the CFD code including the cavitation model was not part of the presented work. 4.2. Simulated operating points In order to confirm and deepen the experimental findings for the reference case from Section 3, a simulation is executed for one operating point of each range RA - RE, see Fig. 4. Fig. 13a shows the amplitude spectrum of the pressure signal PSLP1 for decreasing discharge pressure pLPY, with the discharge pressure pLP of the simulated operating points AOP - EOP. Fig. 13b presents corresponding snapshots taken with the high-speed camera, each with maximum void volume: 4.3. Simulation model Fig. 14 shows the geometry of the 3D flow simulation model of the test bench (a) and a microscope image of the orifice PO1 with a 6.31-fold magnification (b): All numerical results are identified with the index S. For each operating point, the static pressure is defined for the inlet and outlet boundary condition. The pressure difference is determined by the time-averaged static pressure values of the high (PSHP) and discharge (PSLP) pressure monitoring points. The time-averaged volume flow QS is recorded area-averaged at the inlet boundary. The static pressure and volume flow are averaged over a simulation time of tS ¼ 1.5$102 s.

Fig. 13. Measurement with the orifice PO1 for decreasing discharge pressure level pLPY: (a) amplitude spectra b p LP1(fLP1) of the pressure sensor PSLP1; (b) snapshots of the cavitation zone for the simulated operating points AOP - EOP.

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Fig. 14. (a) Geometry of the simulation model; (b) a microscopic image of the planar orifice PO1.

The position and size of the monitoring surface PSLP1 is equal to the membrane surface of sensor PSLP1. The pressure values pS.LP1 are area-averaged. The sampling frequency f ¼ 100 kHz and the settings for the frequency analysis are equal to the measurement, see Section 3.1.1. The geometry of the orifice PO1, see Fig. 14b is consistent with the simulation model in detail. The hexahedral mesh of the fluid domain has no non-matching interfaces. Approx. 93% of the cells are located in the vicinity of the orifice and on the discharge pressure side, where cavitation is expected. In the other parts of the simulation model, the propagation and reflection of the pressure waves are of interest. Because of the minor CPU resources saving potential, no 3D - 1D coupling [32e34] was used and the whole flow volume was detailed using CFD. To save CPU resources, a symmetry plane according to Fig. 14a is used to analyze the flow in one half. For the operating points BOP and COP, simulations were done for the entire domain as well as the half-section model. The evaluation of the volume flow, the pressure signal PSLP1 and the structure of the vapor volume fraction show no substantial effect of the model reduction on the simulation results. By using the half-section model, the CPU resources decreased by half compared to the model with the entire flow domain. That's why all presented numerical results are obtained with the half-section model. To ensure that the numerical result doesn't depend on the spatial discretization of the model, a grid study is exemplarily conducted for operating point BOP. Starting from the coarsest grid I (0.26 Mio. cells, Dt z 5.4$109 s), the refinement is achieved by doubling the resolution in each direction to obtain grid II (0.84 Mio. cells, Dt z 3$109 s) and grid III (5.54 Mio. cells, Dt z 1.4$109 s). The same CFL number is used for each simulation. Due to the explicit time step procedure, by changing the shortest edge length Dx, the time step size Dt ¼ (CFLDx)/c varies too. The numerical results with the coarse grid I differ significantly from the measurements. In the simulation, the jet aligned with the horizontal wall and the vapor volume is almost stationary. It is assumed, that the low number of cell layers in the narrowest gap causes the deviation of the jet direction. According to the experimental results described in Section 3.2.4 the different jet direction is seen as the reason for the retaining almost stationary vapor volume. Conversely, the medium II and fine grid III, match the measured time-averaged volume flow, the frequency and amplitude of the pressure signal PSLP1, the shedding frequency of the vapor fraction and its shape. Because of the higher demands on CPU resources for the fine grid III, the presented numerical results are obtained with the medium grid II. Based on this study, it can be assumed that the spatial discretization of the fluid domain for the medium grid II has no influence on the simulation results. 5. Numerical results and discussion 5.1. Identification of the whistling phenomenon (range RA - RE) The simulations were conducted for the operating points AOP - EOP. The time-averaged volume flow deviation between simulation QS and experiment Qexp is less than 2%. The respective volume flow for every simulated operating point match the measured volume flow in the chocked flow region QS z Qexp.max, see Fig. 4a. For the operating points AOP and EOP the maximum pressure amplitudes b p S in the pipe resonator are less than b p S.LP1 < 0.5 bar and the vapor volume fraction is nearly stationary. According to the measurements, the simulation results do not indicate a whistling noise for these operating points. Fig. 15 shows the frequency analysis of the simulated and measured pressure signal PSLP1 for the points BOP - DOP (Fig. 13a), where the whistling noise occurs at the test bench: The percentage deviation of the dominant resonance frequency fS.LP1/fLP1 and the corresponding pressure amplitude b p S.LP1/ b p LP1 ( b p S.LP1 > 0.5 bar) are provided in Table 5: For the operating points BOP - DOP, which are within the whistling range, the dominant pressure and shedding frequency of the vapor match the measurement. The maximum deviation between measurement and simulation for the characteristic values is less than 4%. This is assumed to be a very good correlation. Apart from this, the frequency analysis in Fig. 15 shows differences for subordinate frequency components. This deviation is not further pursued in this paper, since the model is able to represent the dominant pressure frequency and amplitude in the downstream pipe. The dominant pressure frequency is decisive for the prediction of the frequency and volume of the whistling noise.

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Fig. 15. FFT of the measured pressure signal b p LP1 in comparison with the numerical results b p S.LP1 for the operating points (a) BOP, (b) COP and (c) DOP.

5.2. Analysis of the whistling phenomenon 5.2.1. Second-order resonance frequency (range RB) The evaluation methodology is extended for operating point BOP below, in order to verify the conclusions for the pressure and vapor coupling, which were obtained from the measurement data in Section 3.1.2. The operating point COP therefore is analyzed in a comparable manner in the subsequent Section 5.2.2. As extension to the previous evaluation methods, the pressure signal pLP1 is scaled to the time-averaged discharge pressure pLP and plotted with respect to the time. This is done for the simulation and the measurement data in Fig. 16a. The simulated vapor volume fraction Vvap, scaled to the time-averaged vapor volume fraction V vap is also plotted in the graph. In the second graph Fig. 16b, for six characteristic points of time t1 - t6 of one period of the dominant frequency the simulated static pressure pMP is plotted from 0 < l/lpr < 1.25 along a center line in the pipe resonator for six characteristic points of time t1 - t6 of one period of the dominant frequency. The figure also contains the length of the visualization zone lvis, the dimension of the pressure sensor lLP1, the position of the cross-sectional shape change loa (Fig. 2a) and the estimated length correction lec (Section 2.3). The dominant pressure frequency for operating point BOP is fS.LP1 z 13.5 kHz. The snapshots were taken with a recording rate of fHS ¼ 60 kHz. The presentation of detailed information about the flow field in Fig. 17 are comparable to that in Fig. 5. The measurements provide a cross-sectional view of the flow field and thus of the void volume. To assure comparability, the simulation data are evaluated in a similar way. While the void volume in the snapshots appears nearly black, the simulated vapor volume fraction of h  10% is grayed. The camera snapshots show density differences from which the flow field can only be derived. The simulated velocity magnitude vmag of the fluid whereas is colorized from blue (0 m/s) to red (50 m/s) on the symmetry plane. The snapshots and the simulation images at the times t1 - t6 are arranged one below the other. The recording frequency of the snapshots for operating point BOP is fHS ¼ 60 kHz. Fig. 16a shows, that the course of the measured pressure signal pLP1 is confirmed by the simulation. Additionally, the snapshots in Fig. 17 along with the comparable images of the simulation show a good correspondence of the dynamic vapor behavior and distribution. The simulation provides, in advantage over the measurements, the correlation between the change of the vapor volume fraction and the pressure amplitudes in the pipe resonator. The simulation therefore confirms the

Table 5 Percentage deviations of the dominant resonance frequency fS.LP1/fLP1 and those of the corresponding amplitude b p S.LP1/ b p LP1 ( b p S.LP1 > 0.5 bar) for the operating points BOP - DOP (Fig. 13a). Operating point

BOP

COP

DOP

Measured dominant frequency, fLP1 (kHz) Simulated dominant frequency, fS.LP1 (kHz) Frequency deviation, fS.LP1/fLP1 (%) Measured dominant amplitude, b p LP1 (bar) Simulated dominant amplitude, b p S.LP1 (bar) Amplitude deviation, b p S.LP1/ b p LP1 (%)

13.18 13.47 þ2 6.5 6.6 þ2

4.10 4.00 2 11.6 11.9 þ3

3.34 3.32 1 12.4 12.0 3

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Fig. 16. Operating point BOP: (a) normalized simulated pS.LP1/pS.LP and measured pLP1/pLP static pressure signal PSLP1 in conjunction with normalized simulated vapor volume fraction Vvap/V vap as a function of time; (b) simulated static pressure pMP(t) monitored on a center line in the downstream pipe and plotted over the resonator length l/lpr at the time t1 - t6 according to the left side diagram.

conclusion from Section 3.1.1, that the vapor shedding frequency is synchronized with the dominant pressure frequency in the pipe resonator. The static pressure curves pMP(t1) - pMP(t6) visualize the stationary pressure wave along the pipe resonator. The characteristic stationary pressure nodes and stationary alternating pressure amplitudes are displayed in Fig. 16b. The pressure amplitude at the wall downstream of the orifice R12 (l/lpr ¼ 0.0) alternates between positive and negative superelevated values (sound-reflecting boundary condition). In contrast, the static pressure at the cross-sectional expansion R34 (l/lpr ¼ 1.0) remains almost constant (sound-absorbing boundary condition). Therefore, the simulation confirms the acoustic boundary assumptions from Section 2.3. Additionally, an influence of the change of the flow cross-section from a rectangular to a circular form at R23 cannot be found. The static pressure curves confirm that the pressure node is a bit further downstream of the cross-sectional expansion R34. This known effect has been introduced in Section 2.4. The simulation results match with the analytically estimated length correction lec for most of the pressure curves pMP(t). The simulation data also display the intersection of all static pressure curves at l/lpr z 0.25 at the time-averaged discharge pressure level. This pressure node is characteristic for the higher-order resonance frequency (kl/4 ¼ 3) as introduced in Fig. 3c. 5.2.2. First-order resonance frequency (range RC) The dominant pressure frequency for operating point COP is fS.LP1 ¼ 4.0 kHz. The snapshots were taken with a recording rate of fHS ¼ 20 kHz. The evaluation methodology for operating point COP in Fig. 18 and Fig. 19 is adopted from BOP:

Fig. 17. Snapshots for the operating point BOP, with a frequency of fHS ¼ 60 kHz, in comparison with the numerical results at the points of time t1 - t6 (Fig. 16).

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Fig. 18. Operating point COP: (a) normalized simulated pS.LP1/pS.LP and measured pLP1/pLP static pressure signal PSLP1 in conjunction with normalized simulated vapor volume fraction Vvap/V vap as a function of time; (b) simulated static pressure pMP(t) monitored on a center line in the downstream pipe and plotted over the resonator length l/lpr at the time t1 - t6.

The dominant frequency of the pressure signal PSLP1 is three times higher for operating point BOP compared to COP. According to this, the duration between the evaluation points of time t1 - t6 is three times larger for COP. So, the recording frequency for the snapshots in Fig. 19 was set to fHS ¼ 20 kHz: The comparison of the pressure signal plotted with respect to the time in Fig. 18a and the snapshots shown in Fig. 19 confirm the consistency between the simulation and the measurement in a very good manner. According to the pressure curves in Fig. 18b, there is no pressure node between the location R12 (l/lpr ¼ 0.0) and R34 (l/ lpr ¼ 1.0). The characteristic of the stationary pressure wave correlates with the fundamental frequency (kl/4 ¼ 1) in Fig. 3c. The calculated length correction lec is marked in Fig. 18b. The correlation with the simulated values is even better, than it has been for BOP. The pressure amplitudes are almost zero at the analytically calculated position in the pipe. The normalized pressure change (Fig. 18b) is higher, than that for BOP (Fig. 16b), which is consistent with the frequency analysis (Fig. 15). The comparison of the maximum vapor volume at the points of time t1 - t6 between the snapshots for operating point BOP (Fig. 17) and COP (Fig. 19) illustrates a significantly higher vapor volume for COP. This correlation between pressure amplitude and vapor volume confirms the assumption of Section 3.1.1, that in case of resonance the increase of both is mutually dependent. The comparison of the vapor volume fraction for BOP (Fig. 17) and COP (Fig. 19) demonstrate that the vapor volume increases with higher pressure amplitudes in the pipe resonator.

Fig. 19. Snapshots for the operating point COP, with a frequency of fHS ¼ 20 kHz, in comparison with the numerical results at the points of time t1 - t6 (Fig. 18).

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The evaluation of the simulation results for operating point BOP (Fig. 16a) and COP (Fig. 18a) also confirm the assumption of Section 3.1.1, that the vapor shedding frequency is synchronized with the dominant pressure frequency in the pipe resonator. The pressure curves for BOP (Fig. 16b) and COP (Fig. 18b) verify the acoustic boundary conditions definition from Section 2.3. 6. Conclusions To analyze the mechanism for the occurrence of an acoustic whistling signal at valves, experimental studies of planar and optically accessible orifices were conducted. The pressure wave propagating downstream of the orifice interferes with an adjustable cross-section expansion. The flow field and the void volume downstream of the orifice were recorded with a highspeed camera. Pressure sensors were used to record highly dynamic signals in order to measure the pressure change processes in the connected pipes. The analysis of the whistling noise is supported by CFD simulations of the test bench. The measurements confirm the assumption that cavitation is an essential cause for the development of noise at valves. A distinct whistling noise can be heard for configurable operating points. The volume of the whistling noise is proportional to the static pressure amplitudes in the downstream pipe of the orifice. The whistling noise occurs when these pressure changes are in phase with periodically generated cavitation zones. In the so-called resonance case, the maximum expansion of the vapor volume and the pressure amplitudes of the stationary wave increase abruptly in the pipe resonator. The factors with an impact on the noise are summarized below:
By reducing the discharge pressure, the void volume increases, which leads to a reduction of the resonance frequency of the pipe downstream of the orifice. The void expansion in the direction of sound propagation particularly reduces the resonance frequency.
With a higher order of the dominant pressure resonance frequency, the maximum void volume decreases and thus both, the amplitudes of the stationary wave and the volume of the whistling noise are decreasing.
For large void volumes, the pressure wave reflection at the void volume can be identified in the amplitude spectrum.
The whistling noise depends on the history of the flow field. So, the increase and the reduction of the discharge pressure level lead to different whistling ranges.
The whistling range decreases with increasing length of the pipe resonator. Besides that, the order of the dominant resonance frequency increases.
The experimental and numerical results indicate that the whistling noise only occurs, when the jet downstream of the orifice and hence the vapor is perpendicularly oriented to the sound propagation in the pipe resonator. The findings of the presented work can be used to avoid the development of the acoustic whistling noise at valves by excluding critical operating points and changing the design of the hydraulic system. The results presented here are certainly limited and call for further research. In particular, the influence of the supply pressure and air content of the oil should be examined in future works. Acknowledgments The authors gratefully acknowledge Bosch Rexroth AG Germany for the funding of this research. References [1] N. Chenxiao, Z. Xushe, Study on vibration and noise for the hydraulic system of hydraulic hoist, in: Proceedings of 2012 International Conference on Mechanical Engineering and Material Science, 2012, pp. 126e128. Shanghai, China, http://doi.org/10.2991/mems.2012.95. €glichkeiten der Ger€ [2] G. Schmid, Mo auschminderung an Hydroventilen im Werkzeugmaschinenbau (possibilities of noise reduction for hydro-valves in the machine tool industry), Olhydraulik Pneum. 21 (1977), 23-25 and 89-97. [3] G. Reethof, Turbulence-generated noise in pipe flow, Annu. Rev. Fluid Mech. 10 (1978) 333e367. https://doi.org/10.1146/annurev.fl.10.010178.002001. € [4] O. Eich, Entwicklung ger€ auscharmer Ventile in der Olhydraulik (development of low-noise valves in the field of oil hydraulics), Ph.D. thesis, RWTH Aachen University, 1979. [5] H. Ueno, A. Okajima, H. Tanaka, T. Hasegawa, Noise measurement and numerical simulation of oil flow in pressure control valves, JSME Int. J. B-Fluid T. 37 (2) (1994) 336e341. https://doi.org/10.1299/jsmeb.37.336. € mungsger€ [6] A. Leonhard, F. Rüdiger, S. Helduser, Stro ausche in Lenkventilen (flow-induced noise in steering valves - analysis and reduction), Olhydraulik Pneum. 4 (2005) 346e350. [7] L. Müller, S. Helduser, F. Rüdiger, Visualization of cavitation in oil hydraulic spool valves, in: 8th International Conference on Fluid Power Transmission and Control, 2013. Hangzhou, China. [8] K. Okita, Y. Miyamoto, T. Kataoka, S. Takagi, H. Kato, Mechanism of noise generation by cavitation in hydraulic relief valve, JPCS 656 (1) (2015) 12104e12107. https://doi.org/10.1088/1742-6596/656/1/012104. [9] H. Hassis, Noise caused by cavitating butterfly and monovar valves, J. Sound Vib. 225 (3) (1999) 515e526. https://doi.org/10.1006/jsvi.1999.2254. gan, Cavitating orifice: flow regime transitions and low frequency sound production, in: Proceedings of [10] P. Testud, A. Hirschberg, P. Moussou, Y. Aure ASME 2005 Pressure Vessel and Piping Division Conference, Denver, USA, 2005, pp. 437e445, 71232, https://doi.org/10.1115/PVP2005-71232. gan, Noise generated by cavitating single-hole and multi-hole orifices in a water pipe, J. Fluids Struct. 23 (2) [11] P. Testud, P. Moussou, A. Hirschberg, Y. Aure (2007) 163e189. https://doi.org/10.1016/j.jfluidstructs.2006.08.010. , C. Nicolet, F. Avellan, Cavitation influence on hydroacoustic resonance in pipe, J. Fluids Struct. 28 (2012) 180e193. https://doi. [12] N. Ruchonnet, S. Alligne org/10.1016/j.jfluidstructs.2011.10.001. [13] L.E. Kinsler, A.R. Frey, A.B. Coppens, J.V. Sanders, Fundamentals of Acoustics, fourth ed., Wiley, New York, 2000. [14] J.P. Franc, J.M. Michel, Fundamentals of Cavitation, Springer, New York, 2005.

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