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Empirical modelling of noise from high aspect ratio rectangular jets
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International Journal of Heat and Fluid Flow 0 0 0 (2016) 1–9
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Empirical modelling of noise from high aspect ratio rectangular jets Kondwani Kanjere a,∗, Ludovic Desvard a, Frédéric Nicolas a, Ross H. Henrywood b, Anurag Agarwal b a b
Dyson Technology Ltd, Tetbury Hill, Malmesbury, Wiltshire, SN16 0RP, United Kingdom Cambridge University, Engineering Department, Trumpington Street, Cambridge, CB2 1PZ, United Kingdom
a r t i c l e
i n f o
Article history: Available online xxx Keywords: Jet noise Jet noise scaling laws High aspect ratio jet Rectangular jet
a b s t r a c t Noise measurements have been performed on rectangular jets of aspect ratios ranging from 49 to 987 with the aim of determining the appropriate velocity and length scaling to be used in an empirical noise prediction model. The results have shown that the velocity exponent is a function of the nozzle aspect ratio, decreasing with increasing nozzle aspect ratio. In an effort to establish a general prediction model, the velocity exponent of 7 was chosen as the best compromise to represent all the measured data. The analysis of the noise measurements from high aspect ratio nozzles of varying jet height and width has shown that, for the range of aspect ratios considered, the jet sound power level scales with the nozzle height to the power of 3 and the nozzle width to the power of 1. The derived jet noise scaling has been validated with independent experimental data and shows good agreement. © 2016 Elsevier Inc. All rights reserved.
1. Introduction High aspect ratio rectangular jets have seen a wide range of applications in aerospace (e.g. flow control to reduce noise from the flap side edge (Kanjere et al., 2012)), manufacturing (e.g. air knives), environmental control (e.g. air conditioning units) and recently on high speed hand dryers. On a high speed hand dryer, the air is accelerated through a high aspect ratio nozzle and is used to wipe water off the surface of the hands. The noise of the jet on a high speed hand dryer represents the main contribution to the overall product noise (Etaix et al., 2014). In order to reduce the jet noise a better understanding of the parameters that affect the noise is required. Unlike circular jets, the noise from high aspect ratio rectangular jets (here defined as having an aspect ratio greater than 100) has received less attention. Coles (1959) performed noise measurements on rectangular nozzles of aspect ratios of 14 and 100, with the same exit area. He hypothesised that the sound power generated by a high aspect ratio rectangular jet is half that of a circular jet of equal exit area. The results showed that this was true only for the rectangular jet of an aspect ratio of 100. Schrecker and Maus (1975) performed noise measurements on rectangular jets of aspect ratios ranging from 30 to 120. They found that the dependency of the jet noise on the jet velocity was a function of the
∗
Corresponding author. E-mail address: [email protected] (K. Kanjere).
nozzle aspect ratio. The jet noise scaled with the velocity to the power of 8 for the nozzle of aspect ratio of 30 and velocity to the power of 7 for the nozzle of aspect ratio 120. For the nozzle with aspect ratio between 30 and 120, a velocity exponent between 8 and 7 was found. Kouts and Yu (1975) found that the sound power scaled with the mean jet velocity to the power of 7 on a rectangular nozzle of aspect ratio of 10. Bjørnø and Larsen (1984) carried out a theoretical and experimental study of the noise of a jet issuing from rectangular slits of aspect ratios ranging from 33.3 to 100. They found that the sound power scaled with the velocity to the power of 8 following Lighthill (1952) and an equivalent length scale hL, where h and L are the rectangular jet height and width respectively. Munro and Ahuja (2003) performed flow and noise measurements on a rectangular nozzle with aspect ratios ranging from 100 to 30 0 0. They found that the jet noise data collapsed when scaled by the jet velocity to the power of 8 and an equivalent length scale h3/2 L1/2 . A review of the existing literature shows that there is no agreement on the velocity scaling and length scaling of the noise from high aspect ratio rectangular jets. In this work, noise measurements are performed on different rectangular nozzles of aspect ratios ranging from 49 to 987. The aim of the paper is to determine an appropriate velocity and length scale that can be applied to predict the noise from high aspect ratio jets. The paper is divided into four sections. Section 2 presents the details of the jet rigs and experimental set-up for the noise measurements. The results of the noise measurements, the derivation and validation of the empirical jet noise model are presented in Section 3. Finally,
http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.09.004 0142-727X/© 2016 Elsevier Inc. All rights reserved.
Please cite this article as: K. Kanjere et al., Empirical modelling of noise from high aspect ratio rectangular jets, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.09.004
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K. Kanjere et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–9 Table 1 Geometrical details of the rectangular jet rigs. Rig name Aluminium rig Aluminium rig Aluminium rig Aluminium rig Sintered metal Sintered metal Sintered metal Sintered metal
1 2 3 4 rig rig rig rig
1 2 3 4
h (mm)
L (mm)
AR = L/h
0.30 0.36 0.49 0.73 0.50 0.57 1.60 1.06
296 296 296 296 163 103 146 52
987 822 604 405 326 181 91 49
Fig. 2. Circular jet nozzle rigs.
In order to validate the experimental methods, noise measurements were performed on circular nozzles. Unlike high aspect ratio rectangular jets, there is a lot of literature pertaining to noise from circular jets (see a review paper by Karabasov, 2010) hence a good validation case. Fig. 2 shows the circular jet nozzles used in the study. The nozzles were machined from aluminium with a convergence angle of 30° and a sharp edge exit profile. Four circular nozzles were used in the test with diameters of 8.25, 12.70, 14.67 and 16.51 mm. 2.2. Flow apparatus set-up
Fig. 1. Rectangular jet rigs.
Section 4 presents the conclusions and recommendations for future work.
2. Experimental set-up 2.1. Jet rigs The jet rigs used in the current study consisted of rectangular jets of varying aspect ratios from 49 to 987 and circular jets of varying diameters from 8.25 to 16.51 mm. Fig. 1 shows the rectangular jet rigs used in the study. Both jet rigs consisted of a large plenum where the flow is diffused followed by a convergent section where the flow is accelerated towards the exit. For the aluminium jet rigs, the nozzle convergence angle, in the direction perpendicular to the span, is approximately 10° whereas for the sintered metal nozzle rigs it is 30°. The nozzle height of the aluminium rigs was adjusted using precision metal shims positioned between the two nozzle sections. Tie-down rods were used to hold the nozzle sections together under pressure. The nozzle height and width of the sintered metal rigs were fixed during prototyping. Table 1 summaries the nozzle dimensions of all the rectangular jet rigs.
The tests were performed in the hemi-anechoic chamber facility at Dyson HQ in Malmesbury. Air was supplied to the jet rigs using a 7.5 kW blower rig. A silencer was located downstream of the blower to attenuate any upstream noise coming from the blower and pipes. Both the blower rig and the silencer were located outside the semi-anechoic chamber. A venturi flow meter tube, located downstream of the silencer and upstream of the jet nozzle rig, was used to measure the volumetric air flow rate through the nozzle. Noise measurements were performed at jet velocities ranging from approximately 40 to 240 m/s. For the rectangular jets, this corresponds to a variation in the Reynolds number based on the nozzle height from Reh = 800–2.43 × 104 . For the circular jets, the Reynolds number based on the nozzle diameter, ReD , ranged from 2.09 × 104 to 2.5 × 105 . The jet velocity is calculated based on the static pressure and temperature measured behind the nozzle using the isentropic flow relationships. The jet Mach number, Mj , is given as,
M2j =
Pj Pamb
γ γ−1 −1
γ
2 , −1
(1)
where Pj and Pamb are the static pressure in the plenum behind the nozzle and the ambient pressure respectively, and γ is the specific heat ratio for ideal gas, equal to 1.4 for air. The jet Mach number is then used to compute the jet exit temperature, Tj , as follows,
Tj =
Tamb 1 + γ 2−1 M2j
,
(2)
Please cite this article as: K. Kanjere et al., Empirical modelling of noise from high aspect ratio rectangular jets, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.09.004
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3
Fig. 3. Measurement setup for determination of the sound power level following ISO3744. Figure extracted from reference Petersen (1994).
where Tamb is the ambient temperature. The speed of sound at the jet exit, cj , is then given as,
cj =
γ RTj ,
(3)
where R is the gas constant equal to 287 J/(kgK). Finally the jet exit velocity, Vj , is given as,
Vj = M j c j.
(4)
2.3. Microphone set-up The noise of the jets was determined from noise measurements following ISO3744 (1994). The hemi-anechoic chamber is designed for acoustic measurements at frequencies above approximately 250 Hz. Fig. 3 shows a schematic representation of the measurement set-up used. The jet nozzle was located in the centre (shown by the cuboid in Fig. 3) of a 2 m radius hemisphere of 10 microphones. The microphones are 1/4 inch condenser microphone, B&K type 4135. All microphones were calibrated at the start of each test session. Care was taken not to point the jet directly towards the microphones and all the microphones were covered with windshields. The sound power level was determined from the average sound pressure level of the 10 microphones over a hypothetical hemisphere of 2 m radius according to ISO3744. All measurements were corrected for background noise by applying a frequency dependent factor based on the measured background noise level at the start of each test session. A correction is also applied to the mean of sound pressure level over all the microphone positions on the measurement surface to account for the influence of reflected or absorbed sound. No weighting is applied to the measured sound pressure level spectra presented in this report. 2.4. Validation of experiment set-up In order to ensure that the measured noise was only from the jet, efforts were made to reduce the noise from the rig. Fig. 4 shows the 10 microphones’ average sound pressure level spectra before and after installation of the silencer at four jet velocities for the circular nozzle of diameter D = 16.51 mm (Fig. 4(a)) and aluminium rectangular nozzle of dimensions h = 0.49 mm and L = 296 mm (Fig. 4(b)). The silencer is effective at reducing the blower rig noise especially for frequencies less than 500 Hz. The tones observed in the sound pressure spectra, due to the harmonics of the blade passing frequency of the blower, were completely eliminated when the silencer was installed. Also shown in Fig. 4 is the background noise spectra of the hemi-anechoic chamber. The measured noise spectra are at approximately 20 dB above the background noise level for all frequencies greater than 300 Hz. This is true for jet velocities greater than 94 m/s and 112 m/s for
Fig. 4. Sound pressure spectra before (solid lines) and after (dashed lines) installation of the silencer at four jet velocities.
circular and rectangular jets respectively. Consequently care should be taken when interpreting the trends at lower velocities. Another source of rig noise is the flow noise in the duct upstream of the nozzle. The flow noise from the ducts is notoriously difficult to quantify and reduce, see Viswanathan (2003, 2010). The flow noise is not only a function of the air flow velocity inside the duct, but also the nozzle exit area. The acoustic impedance at the nozzle exit dictates how much internal flow noise will be reflected and transmitted out to the far field. Since the nozzle exit area is varied between the different nozzle geometry, it is difficult to quantify the flow noise impact for all nozzles. One strategy to reduce the influence of duct flow noise is to reduce the flow velocity by increasing the duct cross-sectional area. This option was not available to the authors as it would have required installation of new pipes through the existing walls of the hemianechoic chamber. The noise radiating directly from the pipe was reduced by wrapping all the pipes in thick dense acoustic foam. Efforts were also made to ensure the condition of the flow leaving nozzle was smooth. Fig. 5(a) shows results from hot-wire measurements of the velocity profile at non-dimensional streamwise location, x/h, away from the nozzle exit (where the jet cross-stream direction is denoted by y). Fig. 5(a) clearly shows the top hat exit velocity profile at x/h = 1 (where the jet axial direction is denoted by x). The velocity profile becomes wider as the jet spreads further downstream. Fig. 5(b) shows the decay of the non-dimensionalised centerline velocity confirming the constant velocity from the exit to x/h ≈ 3. This constant velocity region corresponds to the jet potential core. The rate of velocity decay measured for this jet agrees very well with the velocity decay rate of x/h−1/2 from
Please cite this article as: K. Kanjere et al., Empirical modelling of noise from high aspect ratio rectangular jets, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.09.004
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Fig. 5. Exit velocity profile and centerline velocity decay for a jet from the aluminium rig.
two-dimensional planar jets (Pope, 20 0 0). The above analysis shows that the rectangular jets are behaving as expected. 3. Results and discussion This section is divided into two main parts; Section 3.1 outlines the results from the validation test with the circular jet nozzles while the results for the rectangular jet nozzles are presented in Section 3.2. The derivation of the empirical model of rectangular jet noise is also presented in Section 3.2. Finally the derived rectangular jet noise scaling laws are compared with previous jet noise scaling laws in Section 3.3. 3.1. Circular jet The purpose of this validation was to check that the results obtained from the set-up agree with the jet noise scaling from literature (Lighthill, 1952). Fig. 6(a) shows the variation of the jet sound power level, Lw in dB (ref 10−12 W), with the jet velocity, Vj , (plotted in logarithmic scale) for different nozzle diameters. The slope of the measured data corresponds to the velocity exponent for scaling the noise. Fig. 6(a) also shows a solid line representing the jet sound power level variation with the velocity to the power 8 according to Lighthill (1952). For jet velocities greater than 100 m/s, the slopes of the measured data are in good agreement with the 8th power scaling. The sound power level at jet velocities less than 100 m/s is found to scale with the velocity to the power of 6 (see dashed
Fig. 6. Jet sound power level scaling with the jet velocity for all circular nozzles.
line in Fig. 6(a)). Nelson and Morfey (1981) have shown that the flow noise generated inside a duct scales with the flow velocity to the power of 6. The deviation from the 8th power law to the 6th power law observed for jet velocities less than 100 m/s may indicate the dominance of rig flow noise at low jet velocities. The variation of the velocity exponent with the nozzle diameter is shown in Fig. 6(b). The velocity exponent is calculated from the slope of the data for jet velocity greater than 100 m/s. The slope was calculated from different velocity thresholds from 50 to 200 m/s in steps of 20 m/s. The mean and standard deviation of the exponents are shown in Fig. 6(b) for each nozzle. The results show that the exponent does not vary significantly with the jet diameter and is approximately 8, in agreement with Lighthill’s analogy. Fig. 7(a) shows the results of scaling the sound power with the jet velocity to the power of 8. The scaled sound power level is collapsed for jet velocities greater than 100 m/s. Fig. 7(b) shows that scaling the sound power with the velocity to the power of 8 and the diameter to the power of 2 collapses all the curves (best collapse for Vj > 100 m/s). Fig. 8 shows the results of applying the velocity and length scaling obtained above to the 10 microphone average sound pressure level (SPL) spectra for 200 m/s jet from all four circular nozzles. Fig. 8(a) shows the measured average SPL spectra whereas Fig. 8(b) shows the scaled average SPL spectra. The scaled SPL spectra is plotted against the Strouhal number based on the jet diameter, StD = f D/V j , where D is the jet diameter and f is the frequency. The scaling collapses the different SPL spectra reasonably well over the entire frequency range. It is worth noting that the scaled SPL
Please cite this article as: K. Kanjere et al., Empirical modelling of noise from high aspect ratio rectangular jets, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.09.004
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Fig. 7. Jet sound power level, Lw , normalised with the velocity and diameter as a function of jet velocity, Vj .
spectra all peak between StD = 0.2 and 0.3 which shows good agreement with previous circular jet noise studies (Karabasov, 2010). This confirms that the jet noise measured using the set-up scales with jet velocity and diameter according to Lighthill’s theory and that the set-up could be used to investigate the noise scaling from the rectangular jets. 3.2. Rectangular jet The same experiment set-up used to measure the noise from the circular jets was used for the rectangular jets in this study. Fig. 9(a) shows a plot of the sound power level against the jet velocity, Vj , for varying nozzle heights. For clarity, only three results obtained from the aluminium jet rigs are shown. The slope of each curve corresponds to the velocity exponent. Varying degrees of agreement between the measured data with Lighthill (1952) 8th power law (solid line) can be observed. The agreement is good for high jet velocities (Vj > 150 m/s). For jet velocities less than 150 m/s the noise appears to scale well with the velocity to the power of 6 (see dashed line). This could be due to the influence of the flow noise inside the duct or the dominance of dipole sources generated at the surface of the nozzle (more discussion to follow). Note that the departure from the 8th power law occurs at a higher jet velocity (≈ 150 m/s) for the high aspect ratio rectangular jets compared to the circular jets (≈ 100 m/s see Fig. 6(a) in Section 3.1). Fig. 9(b) shows the velocity exponent plotted against the nozzle aspect ratio, AR = L/h. The velocity exponent is calculated from the slope of the curve for different velocity thresholds from 150 to
5
Fig. 8. Scaling of 10 microphones averaged sound pressure level spectra (no weighting) from the four different circular nozzles for V j = 200 m/s.
240 m/s. Fig. 9(b) shows the mean and standard deviation of the calculated exponents. The velocity threshold was set at 150 m/s to ensure that all the data points used were above the background noise level of the chamber, even for the nozzle with the smallest height. Fig. 9(b) shows that the velocity exponent decreases from approximately 7.7 to 6.5 as the nozzle aspect ratio is increased from AR = 49 to 987. The fact that the velocity exponent decreases with increasing aspect ratio suggests a change in the nature of the dominant noise sources or at least the noise generation mechanisms. The velocity exponent of 6.5 that occurs at high aspect ratio (AR ≈ 10 0 0) suggests that the dominant noise sources may be dipole in nature. In contrast, at low aspect ratios, the exponent of 7.7 suggests a dominance of noise sources that are quadrupole in nature. An increase in the nozzle aspect ratio corresponds to an increase in the nozzle perimeter, thus an increase in the nozzle wetted surface area where pressure fluctuations radiate noise of a dipole nature. It follows therefore that the overall exponent tends towards the power of 6 (fluctuating force dipole). As the nozzle aspect ratio reduces so does the nozzle perimeter and the wetted surface from which dipole sources may arise. Due to constraints in maintaining a very small nozzle height (< 0.1 mm) across the entire width of the nozzle, the authors were not able to extend the nozzle aspect ratio beyond AR = 987 to check whether the jet velocity exponent becomes asymptotic at higher aspect ratios. The velocity exponent is found to be approximately 7 from the ensemble average of the exponents. This exponent represents a good compromise to approximate the variety of nozzles tested. The value of the exponent suggests a combination of dipole and
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Fig. 10. Dependence of the jet noise on the nozzle height and width. The normalised jet sound power level for 150 < Vj < 240 m/s are averaged and shown with the standard deviation on the y-axis. Fig. 9. Jet sound power level scaling with the velocity for all the rectangular nozzles.
quadrupole noise sources which scale with the velocity to the power of 6 and 8 respectively. The existence of a noise source at the lip of the nozzle has been speculated previously by Ffowcs-Williams (1968). He suggested that the exit geometry can affect the noise by an additional component he termed “lip noise”. The noise from the lip radiates as a fluctuating force dipole source which typically follows the velocity to the power of 6. He further speculated that the combination of the turbulent mixing noise and the lip noise produced the V j7 scaling found in their investigation. It is very difficult to design an experiment to isolate the lip noise from the turbulent mixing noise. The only assumption we can draw is that the turbulent mixing noise dominates at very high jet velocity while the lip noise dominates at relatively low jet velocity. The cut-off point will depend on the nozzle geometry, where for circular nozzles it occurs at lower jet velocities (≈ 100 m/s) compared to high aspect ratio rectangular nozzles (≈ 150 m/s). The relationship between the jet noise and the nozzle height, h, is derived using the data from the aluminium nozzles, where the nozzle width remained fixed. Fig. 10(a) shows the variation of the ensemble averaged velocity scaled sound power level for Vj > 150 m/s on the y-axis with respect to the nozzle height on the x-axis. Note that the x-axis is in log-scale, hence the slope of the curve is the exponent of the nozzle height. A linear curve fit is performed on the measured data (shown by the red dashed line) which shows that the slope is approximately 3. The relationship between the jet noise and the nozzle width, L, is derived using all the data from all the nozzles. Fig. 10(b) shows
the variation of the ensemble averaged velocity and height scaled sound power level for Vj > 150 m/s with respect to the nozzle width. The x-axis is also in log-scale, which means that the slope of the curve is the nozzle width exponent. A linear curve fit is performed on the data (shown in red dashed line) which shows that the slope is approximately 1. The velocity and length scaling obtained above is applied to all of the data from all rectangular nozzles. Fig. 11(a) shows the data scaled by the velocity scaling only. The spread in the data represents the effect of changing the jet geometry for a given jet velocity. Fig. 11(b) shows that a good collapse of the data is obtained when the velocity and length scaling are applied. The sound power level collapses to within approximately 5 dB (for Vj > 150 m/s) when scaled with the velocity and length scaling derived in the current study. Based on the analysis above, the sound power level, Lw , from the high aspect ratio rectangular jets can be expressed as follows;
Lw ∝ ρ0 c0−4V j7 h3 L1 S0−1 ,
(5)
where ρ 0 [kg/m3 ] and c0 [m/s] are the ambient density and speed of sound respectively. Vj [m/s] is the jet exit velocity, h [m] and L [m] are the jet nozzle height and width respectively. S0 = 1 m2 is the reference surface area of the hemisphere used to compute the jet sound power level. The dimensions of the right hand side of Eq. (5) are [ML2 T −3 ], thus yielding units of Nm/s = Watt.
Please cite this article as: K. Kanjere et al., Empirical modelling of noise from high aspect ratio rectangular jets, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.09.004
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Fig. 12. Comparison of the jet noise scaling laws found from previous studies applied to the data from current study.
Fig. 11. Jet sound power level, Lw , normalised with the velocity and length scales as a function of jet velocity, Vj .
3.3. Comparison with previous jet noise scaling In this section the jet noise scaling found in the current study is compared to previous jet noise scaling for high aspect ratio rectangular jets. In the current study both the jet width and height were varied to obtain an empirical relationship between the noise and the nozzle geometry. In the study of Bjørnø and Larsen (1984), the jet width was fixed to 100 mm while the jet height was varied from 1 to 3 mm. The maximum aspect ratio from their study corresponds to the lower end of the aspect ratio range explored in the current study. In Fig. 9(b), it was shown that the velocity exponent approaches 8 for low aspect ratios, while at high aspect ratios close to 10 0 0 the exponent tends towards 6. This may explain the exponent of 8 found by Bjørnø and Larsen from their measurements on nozzles of aspect ratios AR ≤ 100. The decrease in the velocity exponent with increase in jet aspect ratio found in the current study is in good agreement with Schrecker and Maus (1975). They also found that the variation of jet noise with jet velocity departed from the 8th power laws at low jet velocities (Mj < 0.7) for the rectangular jets (similar to Fig. 9(a) in the current study). They found that the departure occurred at lower jet velocities, Mj < 0.5, for the circular nozzle of the same exit area as shown in the current study.
Fig. 12(a) shows the results of applying Bjørnø and Larsen’s scaling laws to the measured data in the current study. The normalised sound power level collapses to within approximately 13 dB for Vj > 150 m/s. Fig. 12(b) shows the normalised sound power level using the scaling law found by Munro and Ahuja (2003). The normalised sound power level collapses to within less than 10 dB (compared with the 5 dB spread obtained using the current scaling laws, see Fig. 11(b)). Munro and Ahuja derived their length scaling assuming similarity with round jet noise scaling based on Lighthill’s analogy, which contains a length dimension squared. In the current study no such assumption was made. The exponent of the jet height and width were derived through a linear regression and were allowed to take any value without constraints. Fig. 13 shows the effect of applying the current jet noise scaling laws to the overall sound pressure level (OASPL) extracted from figure 11 in appendix C of reference Ahuja (2003). Fig. 13(a) shows the OASPL normalised using Munro and Ahuja’s scaling laws whereas Fig. 13(b) shows the OASPL normalised using the current scaling laws. The OASPL collapses to within approximately 4 dB when scaled with both scaling laws. The comparison however shows that there is less variation in the normalised OASPL for changing nozzle height, h, using the current scaling laws compared with Munro and Ahuja’s scaling laws. The difference in the normalised OASPL from the minimum to maximum nozzle height for a given nozzle width is approximately 4.6 dB using Munro and Ahuja’s scaling laws compared to 2.4 dB using the current scaling laws.
Please cite this article as: K. Kanjere et al., Empirical modelling of noise from high aspect ratio rectangular jets, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.09.004
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Fig. 14. Jet noise rig set-up used by Henrywood et al. (2014).
Fig. 13. Comparison of the jet noise scaling laws applied to the data from reference Ahuja (2003) for three different nozzle widths of 6.5, 14.75 and 30 inches (165.1, 374.65 and 762 mm).
The authors have not yet found any physical justification for the scaling of the jet noise with jet dimensions found in the current study. Antonia et al. (1983) performed measurements of the spacetime correlations of longitudinal and normal velocity fluctuations and of temperature fluctuations in the self-preserving region of a turbulent plane jet. Their results supported the existence of quasiperiodic counter-rotating structures alternating on opposite side of the centreline. They estimated that the streamwise extent of the structures is two to three times as large as the lateral extent and spanwise extent. The lateral extent of the structures is proportional to the jet half width which is related to the nozzle height. The structures were found to be symmetric with respect to the centerline close to the nozzle exit x/h < 10. Coherent structures in the form of wave packets have been shown to be the mechanism of noise generation from at least circular jets (Jordan and Colonius, 2013). Flow measurements using hot-wire and Particle Image Velocimetry (PIV) are in progress that will shed more light on this matter. 3.4. Validation of jet noise scaling laws The velocity and length scaling obtained in the current study is applied to jet noise data from an independent experiment of Henrywood et al. (2014). The measurements were performed in a fully anechoic chamber. Fig. 14 shows the rig set-up in the anechoic chamber. The nozzle width was fixed to 250 mm and the height was varied from 0.3 to 1.2 mm yielding aspect ratios ranging from 208 to 833. Fig. 15 shows that the collapse of the data is reasonable with
Fig. 15. Jet sound power level, Lw , from Henrywood et al. (2014) normalised with the velocity and length scales as a function of jet velocity, Vj .
the maximum difference of approximately 2.5 dB for jet velocities greater than 150 m/s. 4. Conclusions An empirical model of the noise from rectangular jets of high aspect ratios ranging from 49 to 987 has been derived using noise measurements performed on nozzles with varying nozzle heights and widths. The Reynolds number ranged from Reh = 2.85 × 103 to
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2.43 × 104 based on the nozzle height. The experimental set-up has been validated by deriving the jet noise scaling law for circular jets. The sound power level from circular jets is found to scale with the jet velocity to the power of 8 and the jet diameter to the power of 2 in agreement with Lighthill’s theory. The above scaling is true only for velocities greater than 100 m/s. For the range of aspect ratios of the rectangular jet considered, the jet sound power level is found to scale with the jet velocity to the power of 7, the nozzle height to the power of 3 and the nozzle width to the power of 1. The scaling laws is found to be valid only for velocities greater than 150 m/s. The velocity exponent is shown to be a function of the nozzle aspect ratio. The velocity exponent decreases towards an exponent of 6 with increasing aspect ratio while reducing the aspect ratio (towards aspect ratio of 49) results in an exponent closer to 8. This behaviour suggests a change in the nature of the dominant noise sources or at least the noise generation mechanisms from a quadrupole type source for low aspect ratios to a combination of dipole and quadrupole type sources at high aspect ratios. The scaling is compared to previous jet noise scaling laws and it is shown to collapse the data better than or similar to the previous scaling laws. The noise scaling is applied to a set of independent experimental data and showed reasonable collapse of the data. No physical justification has been found for the scaling of jet noise with the nozzle height and width. Future work will involve verification of the jet noise scaling using more independent data. Measurements will be performed on rectangular jets of aspect ratios greater than 10 0 0 to find out whether the dependency of the jet velocity exponent on aspect ratio is asymptotic. Furthermore, flow measurements using hot-wire and PIV are in progress in order to understand the physical relationship between the unsteady flow and the noise generation from high aspect ratio jets. References
9
Antonia, R.A., Browne, L.W.B., Rajagopalan, S., Chambers, A.J., 1983. On the organized motion of a turbulent plane jet. J. Fluid Mech. 134, 49–66. doi:10.1017/ S0 0221120830 03213. Bjørnø, L., Larsen, P.N., 1984. Noise of air jets from rectangular slits. Acta Acustica united with Acustica 54 (5), 247–256. Coles, D.W., 1959. Jet-Engine Exhaust Noise from Slot Nozzles. Technical Report D-60. NASA. Etaix, N., Rodriguez, M.T., Desvard, L., Streeter, R.S., 2014. Sound quality improvement of a high speed hand dryer. Congres Francais d’Acoustique conference. Ffowcs-Williams, J.E., 1968. Some Open Questions on the Jet Noise Problem. Doc. DL-82-0730. Boeing Science Research Lab.. Henrywood, R., Agarwal, A., Babinsky, H., 2014. An experimental investigation into noise radiation from thin rectangular jets. AIAA Aviation. American Institute of Aeronautics and Astronautics doi:10.2514/6.2014-2889. ISO3744, 1994. Acoustics - Determination of sound power levels of noise sources using sound pressure - Engineering method for essentially free field over a reflecting plane. International Organisation for Standardisation. Jordan, P., Colonius, T., 2013. Wave packets and turbulent jet noise. Annu. Rev. Fluid Mech. 45 (1), 173–195. http://dx.doi.org/10.1146/annurev- fluid- 011212-140756. Kanjere, K., Angland, D., Zhang, X., 2012. A numerical study to optimise blowing flow control for flap side-edge noise reduction. In: 18th AIAA/CEAS Aeroacoustics Conferences (33rd AIAA Aeroacoustics Conference). American Institute of Aeronautics and Astronautics, Colorado Springs, CO. June. doi:10.2514/ 6.2012-2097. Karabasov, S.A., 2010. Understanding jet noise. Philos. Trans. R. Soc. London A 368 (1924), 3593–3608. doi:10.1098/rsta.2010.0086. Kouts, C.A., Yu, J.C., 1975. Far noise field of a two-dimensional subsonic jet. AIAA J. 13 (8), 1031–1035. doi:10.2514/3.60500. Lighthill., M.J., 1952. On sound generated aerodynamically.i. general theory. Proc. R. Soc. London. A211 (1107), 564–587. Munro, S.E., Ahuja, K.K., 2003. Aeroacoustics of a high aspect-ratio jet. In: 9th AIAA/CEAS Aeroacoustic Conference and Exhibit, May. American Institute of Aeronautics and Astronautics, doi:10.2514/6.2003-3323. Nelson, P.A., Morfey, C.L., 1981. Aerodynamic sound production in low speed flow ducts. J. Sound Vibr. 79 (2), 263–289. doi:10.1016/0022- 460X(81)90372- 2. Petersen, E.C., 1994. An Overview of Standards for Sound Power Determination. Bruel & Kjaer. Application Note Pope, S.B., 20 0 0. Turbulent Flows. Cambridge University Press, Cambridge Books Online. http://dx.doi.org/10.1017/CBO9780511840531. Schrecker, G.O., Maus, J.R., 1975. Noise Characteristics of Jet Flap Type Exhaust Flows. NGR 43-001-075. NASA Grant. Viswanathan, K., 2003. Jet aeroacoustic testing: issues and implications. AIAA J. 41 (9), 1674–1689. doi:10.2514/2.7313. Viswanathan, K., 2010. Best practices for accurate measurement of pure jet noise. Int. J. Aeroacoustics 9 (1–2), 145–206. doi:10.1260/1475-472X.9.1-2.145.
Ahuja, K.K., Sankar, L.N., Englar, R.J., Munro, ScottE., Li, Yi, Gaeta, R.J., 2003. Application of Circulation Control Technology to Airframe Noise Reduction: Appendix C - Aeroacoustics of a High Aspect-Ratio Jet. GTRI Report A5928/2003-1. Georgia Institute of Technology.
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International Journal of Heat and Fluid Flow 0 0 0 (2016) 1–9
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Empirical modelling of noise from high aspect ratio rectangular jets Kondwani Kanjere a,∗, Ludovic Desvard a, Frédéric Nicolas a, Ross H. Henrywood b, Anurag Agarwal b a b
Dyson Technology Ltd, Tetbury Hill, Malmesbury, Wiltshire, SN16 0RP, United Kingdom Cambridge University, Engineering Department, Trumpington Street, Cambridge, CB2 1PZ, United Kingdom
a r t i c l e
i n f o
Article history: Available online xxx Keywords: Jet noise Jet noise scaling laws High aspect ratio jet Rectangular jet
a b s t r a c t Noise measurements have been performed on rectangular jets of aspect ratios ranging from 49 to 987 with the aim of determining the appropriate velocity and length scaling to be used in an empirical noise prediction model. The results have shown that the velocity exponent is a function of the nozzle aspect ratio, decreasing with increasing nozzle aspect ratio. In an effort to establish a general prediction model, the velocity exponent of 7 was chosen as the best compromise to represent all the measured data. The analysis of the noise measurements from high aspect ratio nozzles of varying jet height and width has shown that, for the range of aspect ratios considered, the jet sound power level scales with the nozzle height to the power of 3 and the nozzle width to the power of 1. The derived jet noise scaling has been validated with independent experimental data and shows good agreement. © 2016 Elsevier Inc. All rights reserved.
1. Introduction High aspect ratio rectangular jets have seen a wide range of applications in aerospace (e.g. flow control to reduce noise from the flap side edge (Kanjere et al., 2012)), manufacturing (e.g. air knives), environmental control (e.g. air conditioning units) and recently on high speed hand dryers. On a high speed hand dryer, the air is accelerated through a high aspect ratio nozzle and is used to wipe water off the surface of the hands. The noise of the jet on a high speed hand dryer represents the main contribution to the overall product noise (Etaix et al., 2014). In order to reduce the jet noise a better understanding of the parameters that affect the noise is required. Unlike circular jets, the noise from high aspect ratio rectangular jets (here defined as having an aspect ratio greater than 100) has received less attention. Coles (1959) performed noise measurements on rectangular nozzles of aspect ratios of 14 and 100, with the same exit area. He hypothesised that the sound power generated by a high aspect ratio rectangular jet is half that of a circular jet of equal exit area. The results showed that this was true only for the rectangular jet of an aspect ratio of 100. Schrecker and Maus (1975) performed noise measurements on rectangular jets of aspect ratios ranging from 30 to 120. They found that the dependency of the jet noise on the jet velocity was a function of the
∗
Corresponding author. E-mail address: [email protected] (K. Kanjere).
nozzle aspect ratio. The jet noise scaled with the velocity to the power of 8 for the nozzle of aspect ratio of 30 and velocity to the power of 7 for the nozzle of aspect ratio 120. For the nozzle with aspect ratio between 30 and 120, a velocity exponent between 8 and 7 was found. Kouts and Yu (1975) found that the sound power scaled with the mean jet velocity to the power of 7 on a rectangular nozzle of aspect ratio of 10. Bjørnø and Larsen (1984) carried out a theoretical and experimental study of the noise of a jet issuing from rectangular slits of aspect ratios ranging from 33.3 to 100. They found that the sound power scaled with the velocity to the power of 8 following Lighthill (1952) and an equivalent length scale hL, where h and L are the rectangular jet height and width respectively. Munro and Ahuja (2003) performed flow and noise measurements on a rectangular nozzle with aspect ratios ranging from 100 to 30 0 0. They found that the jet noise data collapsed when scaled by the jet velocity to the power of 8 and an equivalent length scale h3/2 L1/2 . A review of the existing literature shows that there is no agreement on the velocity scaling and length scaling of the noise from high aspect ratio rectangular jets. In this work, noise measurements are performed on different rectangular nozzles of aspect ratios ranging from 49 to 987. The aim of the paper is to determine an appropriate velocity and length scale that can be applied to predict the noise from high aspect ratio jets. The paper is divided into four sections. Section 2 presents the details of the jet rigs and experimental set-up for the noise measurements. The results of the noise measurements, the derivation and validation of the empirical jet noise model are presented in Section 3. Finally,
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K. Kanjere et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–9 Table 1 Geometrical details of the rectangular jet rigs. Rig name Aluminium rig Aluminium rig Aluminium rig Aluminium rig Sintered metal Sintered metal Sintered metal Sintered metal
1 2 3 4 rig rig rig rig
1 2 3 4
h (mm)
L (mm)
AR = L/h
0.30 0.36 0.49 0.73 0.50 0.57 1.60 1.06
296 296 296 296 163 103 146 52
987 822 604 405 326 181 91 49
Fig. 2. Circular jet nozzle rigs.
In order to validate the experimental methods, noise measurements were performed on circular nozzles. Unlike high aspect ratio rectangular jets, there is a lot of literature pertaining to noise from circular jets (see a review paper by Karabasov, 2010) hence a good validation case. Fig. 2 shows the circular jet nozzles used in the study. The nozzles were machined from aluminium with a convergence angle of 30° and a sharp edge exit profile. Four circular nozzles were used in the test with diameters of 8.25, 12.70, 14.67 and 16.51 mm. 2.2. Flow apparatus set-up
Fig. 1. Rectangular jet rigs.
Section 4 presents the conclusions and recommendations for future work.
2. Experimental set-up 2.1. Jet rigs The jet rigs used in the current study consisted of rectangular jets of varying aspect ratios from 49 to 987 and circular jets of varying diameters from 8.25 to 16.51 mm. Fig. 1 shows the rectangular jet rigs used in the study. Both jet rigs consisted of a large plenum where the flow is diffused followed by a convergent section where the flow is accelerated towards the exit. For the aluminium jet rigs, the nozzle convergence angle, in the direction perpendicular to the span, is approximately 10° whereas for the sintered metal nozzle rigs it is 30°. The nozzle height of the aluminium rigs was adjusted using precision metal shims positioned between the two nozzle sections. Tie-down rods were used to hold the nozzle sections together under pressure. The nozzle height and width of the sintered metal rigs were fixed during prototyping. Table 1 summaries the nozzle dimensions of all the rectangular jet rigs.
The tests were performed in the hemi-anechoic chamber facility at Dyson HQ in Malmesbury. Air was supplied to the jet rigs using a 7.5 kW blower rig. A silencer was located downstream of the blower to attenuate any upstream noise coming from the blower and pipes. Both the blower rig and the silencer were located outside the semi-anechoic chamber. A venturi flow meter tube, located downstream of the silencer and upstream of the jet nozzle rig, was used to measure the volumetric air flow rate through the nozzle. Noise measurements were performed at jet velocities ranging from approximately 40 to 240 m/s. For the rectangular jets, this corresponds to a variation in the Reynolds number based on the nozzle height from Reh = 800–2.43 × 104 . For the circular jets, the Reynolds number based on the nozzle diameter, ReD , ranged from 2.09 × 104 to 2.5 × 105 . The jet velocity is calculated based on the static pressure and temperature measured behind the nozzle using the isentropic flow relationships. The jet Mach number, Mj , is given as,
M2j =
Pj Pamb
γ γ−1 −1
γ
2 , −1
(1)
where Pj and Pamb are the static pressure in the plenum behind the nozzle and the ambient pressure respectively, and γ is the specific heat ratio for ideal gas, equal to 1.4 for air. The jet Mach number is then used to compute the jet exit temperature, Tj , as follows,
Tj =
Tamb 1 + γ 2−1 M2j
,
(2)
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3
Fig. 3. Measurement setup for determination of the sound power level following ISO3744. Figure extracted from reference Petersen (1994).
where Tamb is the ambient temperature. The speed of sound at the jet exit, cj , is then given as,
cj =
γ RTj ,
(3)
where R is the gas constant equal to 287 J/(kgK). Finally the jet exit velocity, Vj , is given as,
Vj = M j c j.
(4)
2.3. Microphone set-up The noise of the jets was determined from noise measurements following ISO3744 (1994). The hemi-anechoic chamber is designed for acoustic measurements at frequencies above approximately 250 Hz. Fig. 3 shows a schematic representation of the measurement set-up used. The jet nozzle was located in the centre (shown by the cuboid in Fig. 3) of a 2 m radius hemisphere of 10 microphones. The microphones are 1/4 inch condenser microphone, B&K type 4135. All microphones were calibrated at the start of each test session. Care was taken not to point the jet directly towards the microphones and all the microphones were covered with windshields. The sound power level was determined from the average sound pressure level of the 10 microphones over a hypothetical hemisphere of 2 m radius according to ISO3744. All measurements were corrected for background noise by applying a frequency dependent factor based on the measured background noise level at the start of each test session. A correction is also applied to the mean of sound pressure level over all the microphone positions on the measurement surface to account for the influence of reflected or absorbed sound. No weighting is applied to the measured sound pressure level spectra presented in this report. 2.4. Validation of experiment set-up In order to ensure that the measured noise was only from the jet, efforts were made to reduce the noise from the rig. Fig. 4 shows the 10 microphones’ average sound pressure level spectra before and after installation of the silencer at four jet velocities for the circular nozzle of diameter D = 16.51 mm (Fig. 4(a)) and aluminium rectangular nozzle of dimensions h = 0.49 mm and L = 296 mm (Fig. 4(b)). The silencer is effective at reducing the blower rig noise especially for frequencies less than 500 Hz. The tones observed in the sound pressure spectra, due to the harmonics of the blade passing frequency of the blower, were completely eliminated when the silencer was installed. Also shown in Fig. 4 is the background noise spectra of the hemi-anechoic chamber. The measured noise spectra are at approximately 20 dB above the background noise level for all frequencies greater than 300 Hz. This is true for jet velocities greater than 94 m/s and 112 m/s for
Fig. 4. Sound pressure spectra before (solid lines) and after (dashed lines) installation of the silencer at four jet velocities.
circular and rectangular jets respectively. Consequently care should be taken when interpreting the trends at lower velocities. Another source of rig noise is the flow noise in the duct upstream of the nozzle. The flow noise from the ducts is notoriously difficult to quantify and reduce, see Viswanathan (2003, 2010). The flow noise is not only a function of the air flow velocity inside the duct, but also the nozzle exit area. The acoustic impedance at the nozzle exit dictates how much internal flow noise will be reflected and transmitted out to the far field. Since the nozzle exit area is varied between the different nozzle geometry, it is difficult to quantify the flow noise impact for all nozzles. One strategy to reduce the influence of duct flow noise is to reduce the flow velocity by increasing the duct cross-sectional area. This option was not available to the authors as it would have required installation of new pipes through the existing walls of the hemianechoic chamber. The noise radiating directly from the pipe was reduced by wrapping all the pipes in thick dense acoustic foam. Efforts were also made to ensure the condition of the flow leaving nozzle was smooth. Fig. 5(a) shows results from hot-wire measurements of the velocity profile at non-dimensional streamwise location, x/h, away from the nozzle exit (where the jet cross-stream direction is denoted by y). Fig. 5(a) clearly shows the top hat exit velocity profile at x/h = 1 (where the jet axial direction is denoted by x). The velocity profile becomes wider as the jet spreads further downstream. Fig. 5(b) shows the decay of the non-dimensionalised centerline velocity confirming the constant velocity from the exit to x/h ≈ 3. This constant velocity region corresponds to the jet potential core. The rate of velocity decay measured for this jet agrees very well with the velocity decay rate of x/h−1/2 from
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Fig. 5. Exit velocity profile and centerline velocity decay for a jet from the aluminium rig.
two-dimensional planar jets (Pope, 20 0 0). The above analysis shows that the rectangular jets are behaving as expected. 3. Results and discussion This section is divided into two main parts; Section 3.1 outlines the results from the validation test with the circular jet nozzles while the results for the rectangular jet nozzles are presented in Section 3.2. The derivation of the empirical model of rectangular jet noise is also presented in Section 3.2. Finally the derived rectangular jet noise scaling laws are compared with previous jet noise scaling laws in Section 3.3. 3.1. Circular jet The purpose of this validation was to check that the results obtained from the set-up agree with the jet noise scaling from literature (Lighthill, 1952). Fig. 6(a) shows the variation of the jet sound power level, Lw in dB (ref 10−12 W), with the jet velocity, Vj , (plotted in logarithmic scale) for different nozzle diameters. The slope of the measured data corresponds to the velocity exponent for scaling the noise. Fig. 6(a) also shows a solid line representing the jet sound power level variation with the velocity to the power 8 according to Lighthill (1952). For jet velocities greater than 100 m/s, the slopes of the measured data are in good agreement with the 8th power scaling. The sound power level at jet velocities less than 100 m/s is found to scale with the velocity to the power of 6 (see dashed
Fig. 6. Jet sound power level scaling with the jet velocity for all circular nozzles.
line in Fig. 6(a)). Nelson and Morfey (1981) have shown that the flow noise generated inside a duct scales with the flow velocity to the power of 6. The deviation from the 8th power law to the 6th power law observed for jet velocities less than 100 m/s may indicate the dominance of rig flow noise at low jet velocities. The variation of the velocity exponent with the nozzle diameter is shown in Fig. 6(b). The velocity exponent is calculated from the slope of the data for jet velocity greater than 100 m/s. The slope was calculated from different velocity thresholds from 50 to 200 m/s in steps of 20 m/s. The mean and standard deviation of the exponents are shown in Fig. 6(b) for each nozzle. The results show that the exponent does not vary significantly with the jet diameter and is approximately 8, in agreement with Lighthill’s analogy. Fig. 7(a) shows the results of scaling the sound power with the jet velocity to the power of 8. The scaled sound power level is collapsed for jet velocities greater than 100 m/s. Fig. 7(b) shows that scaling the sound power with the velocity to the power of 8 and the diameter to the power of 2 collapses all the curves (best collapse for Vj > 100 m/s). Fig. 8 shows the results of applying the velocity and length scaling obtained above to the 10 microphone average sound pressure level (SPL) spectra for 200 m/s jet from all four circular nozzles. Fig. 8(a) shows the measured average SPL spectra whereas Fig. 8(b) shows the scaled average SPL spectra. The scaled SPL spectra is plotted against the Strouhal number based on the jet diameter, StD = f D/V j , where D is the jet diameter and f is the frequency. The scaling collapses the different SPL spectra reasonably well over the entire frequency range. It is worth noting that the scaled SPL
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Fig. 7. Jet sound power level, Lw , normalised with the velocity and diameter as a function of jet velocity, Vj .
spectra all peak between StD = 0.2 and 0.3 which shows good agreement with previous circular jet noise studies (Karabasov, 2010). This confirms that the jet noise measured using the set-up scales with jet velocity and diameter according to Lighthill’s theory and that the set-up could be used to investigate the noise scaling from the rectangular jets. 3.2. Rectangular jet The same experiment set-up used to measure the noise from the circular jets was used for the rectangular jets in this study. Fig. 9(a) shows a plot of the sound power level against the jet velocity, Vj , for varying nozzle heights. For clarity, only three results obtained from the aluminium jet rigs are shown. The slope of each curve corresponds to the velocity exponent. Varying degrees of agreement between the measured data with Lighthill (1952) 8th power law (solid line) can be observed. The agreement is good for high jet velocities (Vj > 150 m/s). For jet velocities less than 150 m/s the noise appears to scale well with the velocity to the power of 6 (see dashed line). This could be due to the influence of the flow noise inside the duct or the dominance of dipole sources generated at the surface of the nozzle (more discussion to follow). Note that the departure from the 8th power law occurs at a higher jet velocity (≈ 150 m/s) for the high aspect ratio rectangular jets compared to the circular jets (≈ 100 m/s see Fig. 6(a) in Section 3.1). Fig. 9(b) shows the velocity exponent plotted against the nozzle aspect ratio, AR = L/h. The velocity exponent is calculated from the slope of the curve for different velocity thresholds from 150 to
5
Fig. 8. Scaling of 10 microphones averaged sound pressure level spectra (no weighting) from the four different circular nozzles for V j = 200 m/s.
240 m/s. Fig. 9(b) shows the mean and standard deviation of the calculated exponents. The velocity threshold was set at 150 m/s to ensure that all the data points used were above the background noise level of the chamber, even for the nozzle with the smallest height. Fig. 9(b) shows that the velocity exponent decreases from approximately 7.7 to 6.5 as the nozzle aspect ratio is increased from AR = 49 to 987. The fact that the velocity exponent decreases with increasing aspect ratio suggests a change in the nature of the dominant noise sources or at least the noise generation mechanisms. The velocity exponent of 6.5 that occurs at high aspect ratio (AR ≈ 10 0 0) suggests that the dominant noise sources may be dipole in nature. In contrast, at low aspect ratios, the exponent of 7.7 suggests a dominance of noise sources that are quadrupole in nature. An increase in the nozzle aspect ratio corresponds to an increase in the nozzle perimeter, thus an increase in the nozzle wetted surface area where pressure fluctuations radiate noise of a dipole nature. It follows therefore that the overall exponent tends towards the power of 6 (fluctuating force dipole). As the nozzle aspect ratio reduces so does the nozzle perimeter and the wetted surface from which dipole sources may arise. Due to constraints in maintaining a very small nozzle height (< 0.1 mm) across the entire width of the nozzle, the authors were not able to extend the nozzle aspect ratio beyond AR = 987 to check whether the jet velocity exponent becomes asymptotic at higher aspect ratios. The velocity exponent is found to be approximately 7 from the ensemble average of the exponents. This exponent represents a good compromise to approximate the variety of nozzles tested. The value of the exponent suggests a combination of dipole and
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Fig. 10. Dependence of the jet noise on the nozzle height and width. The normalised jet sound power level for 150 < Vj < 240 m/s are averaged and shown with the standard deviation on the y-axis. Fig. 9. Jet sound power level scaling with the velocity for all the rectangular nozzles.
quadrupole noise sources which scale with the velocity to the power of 6 and 8 respectively. The existence of a noise source at the lip of the nozzle has been speculated previously by Ffowcs-Williams (1968). He suggested that the exit geometry can affect the noise by an additional component he termed “lip noise”. The noise from the lip radiates as a fluctuating force dipole source which typically follows the velocity to the power of 6. He further speculated that the combination of the turbulent mixing noise and the lip noise produced the V j7 scaling found in their investigation. It is very difficult to design an experiment to isolate the lip noise from the turbulent mixing noise. The only assumption we can draw is that the turbulent mixing noise dominates at very high jet velocity while the lip noise dominates at relatively low jet velocity. The cut-off point will depend on the nozzle geometry, where for circular nozzles it occurs at lower jet velocities (≈ 100 m/s) compared to high aspect ratio rectangular nozzles (≈ 150 m/s). The relationship between the jet noise and the nozzle height, h, is derived using the data from the aluminium nozzles, where the nozzle width remained fixed. Fig. 10(a) shows the variation of the ensemble averaged velocity scaled sound power level for Vj > 150 m/s on the y-axis with respect to the nozzle height on the x-axis. Note that the x-axis is in log-scale, hence the slope of the curve is the exponent of the nozzle height. A linear curve fit is performed on the measured data (shown by the red dashed line) which shows that the slope is approximately 3. The relationship between the jet noise and the nozzle width, L, is derived using all the data from all the nozzles. Fig. 10(b) shows
the variation of the ensemble averaged velocity and height scaled sound power level for Vj > 150 m/s with respect to the nozzle width. The x-axis is also in log-scale, which means that the slope of the curve is the nozzle width exponent. A linear curve fit is performed on the data (shown in red dashed line) which shows that the slope is approximately 1. The velocity and length scaling obtained above is applied to all of the data from all rectangular nozzles. Fig. 11(a) shows the data scaled by the velocity scaling only. The spread in the data represents the effect of changing the jet geometry for a given jet velocity. Fig. 11(b) shows that a good collapse of the data is obtained when the velocity and length scaling are applied. The sound power level collapses to within approximately 5 dB (for Vj > 150 m/s) when scaled with the velocity and length scaling derived in the current study. Based on the analysis above, the sound power level, Lw , from the high aspect ratio rectangular jets can be expressed as follows;
Lw ∝ ρ0 c0−4V j7 h3 L1 S0−1 ,
(5)
where ρ 0 [kg/m3 ] and c0 [m/s] are the ambient density and speed of sound respectively. Vj [m/s] is the jet exit velocity, h [m] and L [m] are the jet nozzle height and width respectively. S0 = 1 m2 is the reference surface area of the hemisphere used to compute the jet sound power level. The dimensions of the right hand side of Eq. (5) are [ML2 T −3 ], thus yielding units of Nm/s = Watt.
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Fig. 12. Comparison of the jet noise scaling laws found from previous studies applied to the data from current study.
Fig. 11. Jet sound power level, Lw , normalised with the velocity and length scales as a function of jet velocity, Vj .
3.3. Comparison with previous jet noise scaling In this section the jet noise scaling found in the current study is compared to previous jet noise scaling for high aspect ratio rectangular jets. In the current study both the jet width and height were varied to obtain an empirical relationship between the noise and the nozzle geometry. In the study of Bjørnø and Larsen (1984), the jet width was fixed to 100 mm while the jet height was varied from 1 to 3 mm. The maximum aspect ratio from their study corresponds to the lower end of the aspect ratio range explored in the current study. In Fig. 9(b), it was shown that the velocity exponent approaches 8 for low aspect ratios, while at high aspect ratios close to 10 0 0 the exponent tends towards 6. This may explain the exponent of 8 found by Bjørnø and Larsen from their measurements on nozzles of aspect ratios AR ≤ 100. The decrease in the velocity exponent with increase in jet aspect ratio found in the current study is in good agreement with Schrecker and Maus (1975). They also found that the variation of jet noise with jet velocity departed from the 8th power laws at low jet velocities (Mj < 0.7) for the rectangular jets (similar to Fig. 9(a) in the current study). They found that the departure occurred at lower jet velocities, Mj < 0.5, for the circular nozzle of the same exit area as shown in the current study.
Fig. 12(a) shows the results of applying Bjørnø and Larsen’s scaling laws to the measured data in the current study. The normalised sound power level collapses to within approximately 13 dB for Vj > 150 m/s. Fig. 12(b) shows the normalised sound power level using the scaling law found by Munro and Ahuja (2003). The normalised sound power level collapses to within less than 10 dB (compared with the 5 dB spread obtained using the current scaling laws, see Fig. 11(b)). Munro and Ahuja derived their length scaling assuming similarity with round jet noise scaling based on Lighthill’s analogy, which contains a length dimension squared. In the current study no such assumption was made. The exponent of the jet height and width were derived through a linear regression and were allowed to take any value without constraints. Fig. 13 shows the effect of applying the current jet noise scaling laws to the overall sound pressure level (OASPL) extracted from figure 11 in appendix C of reference Ahuja (2003). Fig. 13(a) shows the OASPL normalised using Munro and Ahuja’s scaling laws whereas Fig. 13(b) shows the OASPL normalised using the current scaling laws. The OASPL collapses to within approximately 4 dB when scaled with both scaling laws. The comparison however shows that there is less variation in the normalised OASPL for changing nozzle height, h, using the current scaling laws compared with Munro and Ahuja’s scaling laws. The difference in the normalised OASPL from the minimum to maximum nozzle height for a given nozzle width is approximately 4.6 dB using Munro and Ahuja’s scaling laws compared to 2.4 dB using the current scaling laws.
Please cite this article as: K. Kanjere et al., Empirical modelling of noise from high aspect ratio rectangular jets, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.09.004
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Fig. 14. Jet noise rig set-up used by Henrywood et al. (2014).
Fig. 13. Comparison of the jet noise scaling laws applied to the data from reference Ahuja (2003) for three different nozzle widths of 6.5, 14.75 and 30 inches (165.1, 374.65 and 762 mm).
The authors have not yet found any physical justification for the scaling of the jet noise with jet dimensions found in the current study. Antonia et al. (1983) performed measurements of the spacetime correlations of longitudinal and normal velocity fluctuations and of temperature fluctuations in the self-preserving region of a turbulent plane jet. Their results supported the existence of quasiperiodic counter-rotating structures alternating on opposite side of the centreline. They estimated that the streamwise extent of the structures is two to three times as large as the lateral extent and spanwise extent. The lateral extent of the structures is proportional to the jet half width which is related to the nozzle height. The structures were found to be symmetric with respect to the centerline close to the nozzle exit x/h < 10. Coherent structures in the form of wave packets have been shown to be the mechanism of noise generation from at least circular jets (Jordan and Colonius, 2013). Flow measurements using hot-wire and Particle Image Velocimetry (PIV) are in progress that will shed more light on this matter. 3.4. Validation of jet noise scaling laws The velocity and length scaling obtained in the current study is applied to jet noise data from an independent experiment of Henrywood et al. (2014). The measurements were performed in a fully anechoic chamber. Fig. 14 shows the rig set-up in the anechoic chamber. The nozzle width was fixed to 250 mm and the height was varied from 0.3 to 1.2 mm yielding aspect ratios ranging from 208 to 833. Fig. 15 shows that the collapse of the data is reasonable with
Fig. 15. Jet sound power level, Lw , from Henrywood et al. (2014) normalised with the velocity and length scales as a function of jet velocity, Vj .
the maximum difference of approximately 2.5 dB for jet velocities greater than 150 m/s. 4. Conclusions An empirical model of the noise from rectangular jets of high aspect ratios ranging from 49 to 987 has been derived using noise measurements performed on nozzles with varying nozzle heights and widths. The Reynolds number ranged from Reh = 2.85 × 103 to
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2.43 × 104 based on the nozzle height. The experimental set-up has been validated by deriving the jet noise scaling law for circular jets. The sound power level from circular jets is found to scale with the jet velocity to the power of 8 and the jet diameter to the power of 2 in agreement with Lighthill’s theory. The above scaling is true only for velocities greater than 100 m/s. For the range of aspect ratios of the rectangular jet considered, the jet sound power level is found to scale with the jet velocity to the power of 7, the nozzle height to the power of 3 and the nozzle width to the power of 1. The scaling laws is found to be valid only for velocities greater than 150 m/s. The velocity exponent is shown to be a function of the nozzle aspect ratio. The velocity exponent decreases towards an exponent of 6 with increasing aspect ratio while reducing the aspect ratio (towards aspect ratio of 49) results in an exponent closer to 8. This behaviour suggests a change in the nature of the dominant noise sources or at least the noise generation mechanisms from a quadrupole type source for low aspect ratios to a combination of dipole and quadrupole type sources at high aspect ratios. The scaling is compared to previous jet noise scaling laws and it is shown to collapse the data better than or similar to the previous scaling laws. The noise scaling is applied to a set of independent experimental data and showed reasonable collapse of the data. No physical justification has been found for the scaling of jet noise with the nozzle height and width. Future work will involve verification of the jet noise scaling using more independent data. Measurements will be performed on rectangular jets of aspect ratios greater than 10 0 0 to find out whether the dependency of the jet velocity exponent on aspect ratio is asymptotic. Furthermore, flow measurements using hot-wire and PIV are in progress in order to understand the physical relationship between the unsteady flow and the noise generation from high aspect ratio jets. References
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Please cite this article as: K. Kanjere et al., Empirical modelling of noise from high aspect ratio rectangular jets, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.09.004