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Abstract

Measurements of far-field sound and surface pressure fluctuations in a tip clearance flow generated by a single, tapered airfoil blade placed in an open-jet flow were performed under both zero and adverse pressure gradient endwall conditions. The measurements were performed for seven different clearance heights (1.2 mm $\leq h \leq$ 20.4 mm), each smaller than the height of the incoming wall boundary layer. The Reynolds number, based on tip clearance height, ranged between 2600 and 45,400 and the blade geometric angle of attack was 12°. The results show that, regardless of the pressure gradient, tip clearance noise generally increases with clearance height, although the relationship is a complex function of both clearance height and frequency. It is shown that a change in clearance height not only changes the magnitude and shape of the spectra, but it also changes the spatial distribution of the dominant sound sources. The surface pressure fluctuations were measured both on the tip surface and on the rounded edge of the tip towards the suction-side, and show a behaviour largely consistent with that observed in the far-field. An increase in the Clauser pressure gradient parameter $(β)$ lowers the magnitude of both the near and the far-field pressure fluctuations for the same clearance height, with larger clearances showing a more significant reduction in the far-field sound. For smaller clearances, while the total tip clearance noise levels were found to be relatively less sensitive to $β$ , a large reduction in the surface pressure fluctuations and far-field sound generated in the mid-chord region of the clearance was observed.

Keywords

tip clearance noise adverse pressure gradient surface pressure fluctuations beamforming

Introduction

Tip clearance flow noise remains an important engineering problem due to its significant contribution to the overall acoustic signature of rotating machinery such as shrouded fans, compressors and turbines. Historically, the fluid dynamics of tip clearance flows has been studied extensively using both rotating rigs and cascade configurations^1–10 due to the losses produced by the flow. However, due to the difficulties associated with isolating the tip clearance noise in these measurement configurations, the fundamental nature of the tip clearance noise, its scaling and its dependence on the flow and geometrical parameters remains largely unknown. Recently, several studies^3,4,11–18 have employed what is referred to as an idealised tip clearance flow configuration to isolate and better understand the fundamental nature of the tip clearance noise. This idealised configuration consists of a single, stationary blade with sharp tip edges placed adjacent to a static flat wall. While this configuration lacks the relative blade/wall motion and blade-blade and rotor-stator interactions, it does produces the same complex tip clearance flow features (e.g., flow separation, three-dimensional boundary layers, separation and leakage vortices) as those found in representative rigs. The simplicity of this idealised configuration allows isolation of tip clearance flow noise sources and makes detailed flow and near-field pressure diagnostics feasible. As a result, past studies of idealised configuration have revealed several characteristics of tip clearance flow noise which were previously unknown. The present work builds upon this framework to study how the near and far-field pressure field is modified when a streamwise adverse pressure gradient (APG) is imposed on the approaching flow. In contrast to previous studies that primarily used airfoil blades with sharp tip edges, the current work employs a tapered airfoil with rounded tip geometry which is more representative of real-world blade designs.

Flow diagnostic studies of an idealised tip clearance flow configuration^11,12,18,19 have shown that the flow-field is dominated by the two vortex systems—the tip leakage vortex and the tip separation vortex. The leakage vortex is formed as the cross-flow across the gap from the pressure to the suction side rolls up after it interacts with the main passage flow. The tip separation vortex has been shown to reside closer to the suction side edge of the tip and it forms when the separated flow from the pressure side fails to reattach on the tip, resulting in vortex shedding. For typical airfoil sections, the separation vortex forms closer to the trailing-edge since that is where the airfoil is thin, preventing reattachment on the tip surface. Recently, it has been shown that these separation vortices are a significant source of low-frequency noise generation¹⁸ and can also lead to a broadband hump in pressure fluctuations on the blade surface.²⁰ Besides the separation vortices, the bulk flow across the tip which is typically strongest in the mid-chord region of the tip can also produce significant noise. Through conditional averaging of PIV snapshots triggered by strong events in the far-field,⁴ have shown that the dominant tip clearance sound-source was located approximately between 40% and 60% chord location. In a more recent study,¹⁸ found that this region is a significant source of mid-to-high frequency noise. In this same study, it was also revealed that for very small clearances, the interaction of the oncoming flow with the pressure side edge of the tip near the nose of the blade can generate significant turbulence intensity and intense high-frequency noise radiation. Of course, the three different sources described here are expected to be a strong function of clearance height, lift on the blade (the angle of attack), the geometry of the blade and even the incoming boundary layer characteristics. For instance, the presence of rounded edges on the tip could result in weaker flow separation vortices and the presence of an APG can modify the approach boundary layer statistics, resulting in a different aeroacoustic behaviour.

The effect of tip clearance height on the tip clearance noise can be complex and frequency-dependent. For instance,¹⁶ have shown that reducing the gap height below 10 mm results in suppression of low-frequency noise, whereas the high-frequency noise remains largely unaffected. Recently,¹⁷ measured the far-field sound using a microphone array for a large number of clearance heights, both smaller and larger than the incoming boundary layer height. They showed that immersing the tip inside the boundary layer generates a significant amount of low-frequency noise. For clearances larger than the boundary layer height, the dominant tip clearance sound-source resides closer to the trailing-edge, but for clearances smaller than the boundary layer height the leading-edge and the mid-chord regions can become significant noise radiators. In this study, they also showed that the tip clearance noise has a Reynolds number dependence and a Mach number scaling behaviour which is a strong function of the clearance height.

The effect of imposing a streamwise pressure gradient in the oncoming flow on the tip clearance noise remains largely unknown. Recent numerical investigations^21,22 have shown that imposing a streamwise APG can destabilise the leakage vortex and can lead to an earlier vortex breakdown as the low-momentum fluid near the vortex core cannot negotiate the pressure gradient. It is also reasonable to believe that the imposition of an APG can modify the vortex meandering behaviour of the leakage vortex in wind tunnels. Aeroacoustically, the fluid dynamic processes closer to the blade tip are more important as they are responsible for generating the near-field pressure fluctuations that are scattered to the far-field by the blade tip edges. It is possible that, similar to the leakage vortex, imposing a streamwise APG could also destabilise the tip separation vortices which would affect tip clearance noise generation, particularly at low frequencies. It can also be expected that the flow separation and reattachment behaviour inside the gap will also be affected by the pressure gradient due to a change in the oncoming turbulence intensity of the flow and the momentum through the gap. Past studies of the flow-field inside the gap^3,18,23,24 for similar configuration studied in the present work, but under ZPG conditions have shown the cross-flow velocity to be an important parameter that determines the behaviour of the tip clearance noise. The presence of a streamwise APG will likely reduce the strength of this cross flow and could result in lower near and far-field pressure fluctuations, when compared to the same flow under ZPG condition. It has recently also been shown that the tip clearance noise is significantly affected by the tip clearance to the incoming boundary layer height ratio,¹⁷ the thicker boundary layers under APG conditions will also change the tip clearance noise behaviour for the same clearance height. Finally, higher boundary layer turbulence intensities for an APG configuration can also enhance the unsteady surface pressure and noise production from the leading-edge of the tip.

This paper presents and discusses the results of near and far-field pressure fluctuation measurements in an idealised tip leakage flow under zero and adverse pressure gradient. A purpose-built test-rig capable of generating variable adverse pressure gradients with Clauser pressure gradient parameter $(β)$ values up to 1.11 was designed and characterised. Note that the $β$ values considered in this work correspond to attached flows with mild APG that are common on aircraft wings, turbine blades and tapered sections of naval vehicles. Utilising this rig, the far-field sound and surface pressure fluctuations on the blade tip were measured for seven clearance heights between 1.2 mm and 20.4 mm. Each clearance height considered is smaller than the incoming boundary layer height, regardless of the pressure gradient. The Reynolds number in the measurements (based on tip clearance height) ranged between 2600 and 45,400 and the measurements were performed at a free-stream velocity of 35 m/s and a blade geometric angle of attack $(α_{g})$ of 12°. The far-field sound was measured using a microphone phased array and the array response was beamformed to reveal the spatial distribution of tip clearance noise sources. The surface pressure fluctuations were measured at several locations on both the tip surface and the rounded edge of the blade tip towards the suction-side. The paper is organised as follows. The pressure gradient test-rig, the blade geometry and near and far-field instrumentation and data processing are discussed first, followed by the discussion of the near and far-field measurement results. The characteristics of the tip clearance far-field sound under zero pressure gradient (ZPG) conditions are discussed first, followed by a discussion on the effects of the APG on the noise radiation. This is followed by a discussion of the surface pressure fluctuations on the blade tip under ZPG and APG conditions. Finally, concluding remarks are provided.

Experimental setup and instrumentation

UNSW anechoic wind tunnel & pressure gradient tip clearance noise test-rig

The measurements were performed in the UNSW anechoic wind tunnel (UAT) which is an open-jet type facility where a 0.455 m $\times$ 0.455 m test section is surrounded by a 3.0 m $\times$ 3.2 m $\times$ 2.15 m anechoic chamber. The facility has low background noise levels and is anechoic above approximately 300 Hz.²⁵ The facility is capable of providing flow speeds up to approximately 60 m/s with low free-stream turbulence levels (approximately 0.25% at 30 m/s). In the present work, to generate tip leakage flows a modified version of the tip clearance test-rig discussed in Ref. 18 was utilised. The present test-rig installed downstream of the open-jet inlet is shown in Figure 1. The test-rig consisted of two 825 mm long endplates between which a single, tapered airfoil blade with rounded tip (discussed later in this section) was mounted. The tip leakage flow was formed between the blade tip and the top endplate, while the pressure gradient in the test-section was generated by pivoting the bottom endplate to create a ramp. The angle of this ramp $(ψ)$ was adjusted to 0 $°$ , 8 $°$ and 16° with respect to the flow direction to generate adverse pressure gradient in the test-section. The test-rig also consists of two 350 mm long side plates which, as shown in a recent study,²⁶ are required to maintain the pressure gradient generated by the ramp. The airfoil blade sits on a linear traversing system that consists of two parallel linear Zaber traverse stages that allow electronic adjustment of tip clearance height. A rotary actuator, sandwiched between the blade and the traversing system, is used to electronically adjust the blade geometric angle of attack $(α_{g})$ . A 350 mm diameter turntable with a 3D-printed insert on the bottom ramp provides a slot with a clearance of approximately 0.5 mm through which the blade can freely move in the vertical direction. To minimise the noise generated by flow seeping through this gap a 0.1 mm thick gasket paper sheet was laid around the blade/ramp junction. Note that because the geometry of the blade ramp junction changes with each ramp angle, three different gasket papers were laser-cut to obtain the appropriate sealing profile. The Reynolds numbers in the present work are in the range where the laminar flow separation on the blade generates strong tonal noise that can contaminate both near and far-field pressure fluctuations. Therefore, a 0.4 mm thick zig-zag trip was applied at the 10% chord location on both sides of the blade to force transition to turbulence. The trip was terminated 40 mm below the tip in the spanwise direction to avoid contamination of the tip leakage flow due to the presence of the trip.

Figure 1.

UNSW pressure gradient test rig for tip clearance flow noise measurement.

The undisturbed, incoming flow characteristics for the three ramp angles considered in the present work were established by measuring the mean wall pressure and boundary layer characteristics on the endwall without the blade in the test-section. These inflow conditions and their measurement have already been discussed in Ref. 27, but will be summarised here for completeness. The mean wall pressure was measured using a streamwise row of pressure taps which are shown in a top-view schematic in Figure 2(a). The schematic also shows the location of the blade tip and the closed portion of the test-section. Note that, as highlighted in the schematic, the pressure taps near the blade tip are located on a turntable embedded on the top endplate (see Figure 1) and therefore can be rotated with the airfoil, if required. The coordinate system (shown in Figure 1) that will be used throughout the paper to discuss the near-field measurement results has its origin at the endwall (the top endplate) directly above the blade nose, with $x$ being the streamwise coordinate pointing downstream, $y$ the wall-normal coordinate pointing away from the endwall and $z$ pointing towards the suction-side of the blade. Figure 2(b) shows the mean wall pressure measured on the top endplate as a function of streamwise distance at $U_{\infty}$ = 35 m/s for the three bottom ramp angles ( $ψ$ = 0 $°$ , 8 $°$ and 16 $°$ ). Note that here $x$ = 0 coincides with where the blade leading-edge was eventually placed during the tip clearance flow and noise measurements. As expected, an increase in the bottom ramp angle generates a strong adverse pressure gradient within the enclosed test-section and some distance downstream of where this section ends. However, the open boundaries further downstream reduce this pressure gradient, bringing the flow to a near zero pressure gradient (ZPG) configuration (but with a higher background pressure) near the downstream end of the top endplate. It appears that for both non-zero ramp angles, the presence of a hybrid, open/close type test-section generates two regions of linear pressure gradient. The first region with a stronger pressure gradient (highlighted using superimposed dashed lines) extends from the beginning of the closed section to approximately where the blade leading-edge was later positioned. Beyond this point, a milder pressure gradient due to open side boundaries is observed. Note that the open section in the present work was necessary to allow far-field sound measurements (discussed later) performed using a microphone array placed outside the flow. In this work, we will identify the three different inflow pressure gradient conditions through the Clauser pressure gradient parameter $(β)$ calculated based on the slope of the mean pressure distribution within the first region.

Figure 2.

Pressure gradient test rig flow characteristics. (a) Top-view schematic showing the layout of the pressure taps used to measure the undisturbed (empty test-section) pressure gradient and wall pressure fluctuations. The streamwise mean wall pressure, boundary layer mean velocity profiles, turbulence intensity profiles and wall pressure spectra for the three bottom ramp deflection angles are shown in (b), (c), (d) and (e), respectively. The wall pressure spectra shown were measured at $x / c$ = −0.25 (the most upstream tap on the turntable). The legend for (b) – (e) is shown in (c).

The inflow boundary layer statistics at $U_{\infty}$ = 35 m/s for each pressure gradient configuration were measured using a single-sensor hotwire position approximately 5 mm upstream of the eventual blade nose position (see Ref. 27 for more details). Figure 2(c) and (d) show the boundary layer mean velocity and turbulence profiles, respectively for the three bottom ramp angles. The mean velocity profiles are shown in non-dimensional form in wall units with the friction velocity for each pressure gradient estimated by applying a law-of-the-wall fit to the measured data. The results here show that the inflow boundary layer for the three pressure gradients is self-similar in the log region, while in the wake region of the boundary layer the velocity consistently increases with pressure gradient. The boundary layer turbulence intensities (Figure 2(d)) consistently increase with an increase in pressure gradient, with the largest pressure gradient configuration ( $ψ$ = 16 $°$ ) showing noticeably larger turbulence levels across the entire boundary layer. The behaviour of the mean velocity and turbulence profiles observed here is typical of boundary layers under a pressure gradient.²⁸ Finally, Figure 2(e) shows the undisturbed boundary layer wall pressure spectra for the three pressure gradient configurations measured using the most upstream tap on the turntable (see Figure 2(a) for the tap location). As expected, increasing the pressure gradient results in a consistent increase in the magnitude of the low frequency pressure fluctuations (up to about 2 kHz), while the higher frequencies at this location remain nearly unaffected by the pressure gradient.

The inflow wall pressure and velocity measurements discussed above were used to estimate the Clauser pressure gradient parameter

(β = δ^{*} (d P / d x) / τ_{w})

for each bottom ramp angle. The pressure gradient value

(d P / d x)

was estimated by calculating the approach pressure gradient, i.e. the slope of the two dashed lines in Figure 2(b). The displacement thickness

(δ^{*})

and wall shear stress

(τ_{w})

were estimated from the hotwire mean velocity profiles. The shear stress was estimated by fitting the measured boundary layer profile to the log-law with

κ

= 0.41 and

C^{+}

= 5.0. Table 1 below shows the boundary layer statistics, Reynolds numbers and the

β

values corresponding to each bottom ramp angle used in the present study. Here,

δ_{99}

and

θ

are the boundary layer thickness and momentum thickness, respectively,

H

is the boundary layer shape factor,

u_{τ}

is the shear velocity, and

R e_{δ}

R e_{θ}

and

R e_{τ}

are the Reynolds numbers based on the boundary layer thickness, momentum thickness and shear velocity, respectively. Table 2 lists the tip clearance heights

(h)

considered in the present work. The table also lists the ratio between the clearance height and incoming boundary layer for each pressure gradient configuration, along with the tip clearance height based Reynolds number in the present work. Note that all measurements (near and far-field pressure fluctuations) were performed at a free-stream velocity

(U_{\infty})

of 35 m/s.

Table 1.

Approach boundary layer parameters and Clauser pressure gradient parameter at $U_{\infty}$ = 35 m/s.

Configuration	$δ_{99} (m m)$	$θ (m m)$	$δ^{*} (m m)$	$H$	$u_{τ} (m / s)$	$R e_{δ}$	$R e_{θ}$	$R e_{τ}$	$β$
ZPG	24.0	2.6	3.5	1.32	1.36	53,900	5850	2100	0
$ψ = 8 °$	27.3	3.4	4.6	1.36	1.27	61,200	7550	2200	0.51
$ψ = 16 °$	30.3	4.0	5.5	1.38	1.23	68,000	8850	2400	1.11

Table 2.

Tip clearance heights and Reynolds numbers in the present work.

$h (m m)$	$h / δ (Z P G)$	$h / δ (ψ = 8 °)$	$h / δ (ψ = 16 °)$	$R e_{h}$
1.2	0.05	0.04	0.04	2670
3.2	0.13	0.12	0.10	7120
5.2	0.22	0.19	0.17	11,570
8.2	0.34	0.30	0.27	18,250
10.2	0.42	0.37	0.34	22,700
15.2	0.63	0.56	0.50	33,800
20.4	0.85	0.75	0.67	45,400

The airfoil blade used in the present study is shown in Figure 3(a). The blade has a total span of 645 mm and a chord-length $(c)$ of 212.3 mm. The blade has a tapered geometry with three different spanwise segments as shown. The base and tip segments of the blade have a constant cross-section, with a tapered segment sandwiched in between. The cross-sectional profiles of the tip and the base segments are shown in Figure 3(c). The base segment profile has a smaller camber (2%) and a larger maximum thickness (24%) than the tip segment (4.6% camber and 12% maximum thickness). The tip segment of the blade is further split with a separate 25 mm span machined section (Figure 3(b)) to allow machining of pressure taps on the tip for pressure fluctuation measurements (discussed in the next section). The edges of the blade tip are rounded with a radius of 3.8 mm on both the pressure and the suction sides to simulate a real airfoil blade, as opposed to one with sharp corners. Note that the nose of the rounded portion of the tip does not have a constant radius, instead it varies around the nose to ensure continuity between rounded portion of the tip and the pressure and suction sides of the blade.

Figure 3.

Geometry of the airfoil blade and pressure tap distribution used to measure surface pressure fluctuations on the blade tip and rounded edge of the tip. (a) Spanwise geometry of the tapered blade showing the different blade segments. The 25 mm span tip section used to generate the pressure taps on the flat and rounded surface of the tip is shown in (b). The cross-sectional profiles of the base and tip segments is shown in (c). The layout of the pressure taps on the tip and the rounded edge of the tip is shown in (d).

Surface pressure fluctuation measurements

The unsteady surface pressure on the blade tip was measured on both the tip surface (referred to as the flat surface from hereon) and the rounded edge of the tip towards the suction-side (referred to as the rounded surface from hereon). Figure 3(d) shows the measurement locations on both the surfaces in a top-view projection of the blade tip. On the flat surface, the pressure was measured using 16 taps distributed along the camber line of the tip airfoil section. On the rounded surface, there were 13 measurement locations, each of which is 2 mm downstream of the closest measurement location on the flat surface along the chordline. These measurement locations can also be seen in the 3D rendering of the 25 mm span tip section shown in Figure 3(b).

The surface pressure fluctuations were measured using the remote microphone technique of Ref. 29 and follows a methodology similar to that used previously¹⁸ to measure the pressure fluctuations on a tip with sharp edges. The remote microphone method involves drilling 0.8 mm taps on the measurement surface (shown in Figure 3(b) for the present measurements) and installing a 0.8 mm OD and 0.5 mm ID stainless steel tubing flush with the taps. A vinyl tubing is then attached to this steel tubing to transfer the surface pressure fluctuations to a condenser microphone placed outside the flow-field. The pressure taps used in the present work to measure the pressure fluctuations on both the flat and rounded surfaces of the tip are shown in Figure 3(b). The vinyl tubing that is attached to these pressure taps via the stainless steel tubing is routed to the remote microphones via channels machined inside the blade and a set of holes near the base of the blade (shown at the bottom in Figure 3(a)). Note that throughout the measurements these tubing routing holes remained underneath the bottom ramp, i.e. outside the flow region. The vinyl tubing used in measurements was nominally 750 mm in length. The vinyl tubing was connected to condenser microphones (GRAS 40PH-10 CCP) that sit in a rapid prototyped housing, similar to that used in Refs. 18 and 29.

The remote microphones were calibrated using a comparison calibration method which requires exposing the remote microphone and an unmodified, factory-calibrated reference microphone to the same noise source. The calibration curve is then obtained by dividing the cross-spectrum between the source signal and the remote microphone by the cross-spectrum between the source signal and the reference microphone. The calibrations in the present work were performed using a bottle-shaped hand-held calibrator similar to that used by.³⁰ The calibrator houses the sound-source (a white noise speaker in this case) and a reference microphone (an unmodified GRAS 40PH-10 CCP microphone) inside a bottle shaped cavity and an orifice located at the top of this cavity is then pressed against each pressure tap to obtain smooth calibrations up to about 6 kHz. The frequency range of meaningful pressure fluctuations in the present work is between 100 Hz and 6 kHz, where the lower limit is determined by the background noise in the facility and the upper limit by the noise in the calibrations due to the relatively long pressure tubing (approximately 750 mm) that connects the pressure taps to the remote microphones.

The pressure signal from each remote microphones was acquired at a sampling rate of 65,536 Hz for 32 s. The resulting time-series pressure was Fourier transformed using Welch’s periodogram with a record length of 8192 data points, a Hanning window and a 50% overlap between adjacent records to estimate the auto and cross-spectral densities of surface pressure fluctuations. In this work, the surface pressure fluctuation statistics will be presented as power spectral densities, root mean square (RMS) values and the frequency-domain coherence function between pair of remote microphones. The uncertainties in the estimates of these quantities were estimated using the spectral uncertainty estimation methods outlined in Ref. 31. The dominant uncertainty in the spectral estimation comes from the averaging procedure and depends on the number of records used to obtain the periodogram. In the present work, 511 records were used which translates to an uncertainty of 8.9% in the spectral estimates. The RMS pressure presented in this work was obtained by integrating the autospectra, and assuming that the bias in the spectra is independent of the frequency, the uncertainty in the RMS values is 4.4%. Finally, the uncertainty in the coherence function estimate is a function of coherence value itself, and is therefore frequency dependent. In the present work, the largest uncertainty in the estimates of the coherence function was found to be 0.05 (or 5% coherence).

Far-field sound measurements

Tip clearance flow noise was measured using a 64-microphone spiral phased array placed outside the flow region towards the suction-side of the blade. The geometric centre of the array (the reference observer location in the present work) was located 1.28 m from the blade trailing-edge at zero angle of attack. In the streamwise direction, the array centre was 25 mm downstream of the blade trailing-edge and in the wall-normal direction it was located 0.23 m below the top endplate. The array utilises 64 GRAS 40 PH-10 CCP array microphones which have a flat (to within $\pm$ 2 dB) frequency response between 10 Hz and 20 kHz. The aperture of the array was 0.9 m and its spatial resolution, defined as the minimum separation between two sources for which the beamformed source map produces two distinct peaks, was estimated using the Sparrow limit.³² The spatial resolution depends on the array aperture and the acoustic wavelength, and for the present array, it is approximately 480 mm, 48 mm and 24 mm at 1 kHz, 10 kHz and 20 kHz, respectively.

The array response was acquired simultaneously with the surface pressure microphones for 32 s at a sampling rate of 65,635 Hz using a national instrument PXI data acquisition system. The acoustic pressure time series from the array was Fourier transformed and averaged using Welch’s periodogram with a 50% overlap and a Hanning window to obtain the array cross-spectral matrix (CSM) with a narrowband frequency resolution of 8 Hz. This CSM was then beamformed in 1/ $12^{t h}$ octave-bands using the conventional frequency-domain beamforming algorithm as described in Ref. 33 to reveal the tip clearance noise sources. Figure 4(a)–(c) show the beamformed source map at a single frequency of 4 kHz and clearance height of $h$ = 1.2 mm for all three pressure gradient configurations. The outline of the airfoil blade, the top and bottom endplates and the downstream edge of the closed section of the facility have been superimposed on each source map for spatial reference. Note that the $X - Y$ coordinate system used to present the beamformed sound-source maps here, and elsewhere in the paper, is different from the near-field coordinate system $(x - y - z)$ discussed earlier. The far-field system has its origin at the array’s geometric centre with $X$ being the streamwise coordinate pointing in the downstream direction, $Y$ pointing upwards towards the endwall and $Z$ pointing from the source to the array. The tip clearance noise source in each case is clearly visible towards the top of the image, along with a slightly weaker blade trailing-edge noise from the two-dimensional flow region away from the tip clearance. For the case shown, the effect of pressure gradient on the clearance noise source is not significant and for each case the dominant source is located closer to the trailing-edge of the blade. Besides the clearance and trailing-edge noise, there is also a third source near the leading-edge/bottom wall junction that is visible only for $β$ = 0 and 0.51 (Figure 4(a) and (b)). This source, which is only present in certain cases and is not apparent at frequencies below about 3 kHz is believed to be due to a slight flow leakage between the boundary layer trip placed on the airfoil and the gasket paper which is used to seal the junction. As discussed momentarily, the contribution of this spurious source to the far-field sound can be neglected once the beamformed source maps are deconvolved and spatially integrated to obtain the tip clearance noise spectra.

Figure 4.

Beamformed sound source maps at 4 kHz for the three pressure gradient configurations at $U_{\infty}$ = 35 m/s. (a) ZPG, (b) $β$ = 0.51 and (c) $β$ = 1.1. The deconvolved source maps corresponding to the beamformed source maps in (a), (b) and (c) are shown in (d), (e) and (f), respectively.

The spatial resolution of the beamformed source maps was improved by using a variant of the original DAMAS deconvolution algorithm³³ that works in the spatial frequency domain (wavenumber-domain), and as a result it is significantly faster. This algorithm, referred to here as the DAMAS-2E algorithm, was originally proposed by³⁴ and it is an extension of the DAMAS-2 algorithm by.³⁵ The DAMAS-2 algorithm solves the DAMAS deconvolution problem approximately in the wavenumber domain by assuming a shift-invariant array point spread function (PSF) which is a reasonable assumption if the source region is small compared to source-to-array distance.³⁵ However, this assumption may not be reasonable for extended sources such as those considered in this work. The DAMAS-2E algorithm relaxes this assumption by embedding the DAMAS-2 algorithm inside an outer loop that accounts for the variation in the array PSF. This computational effort required for this outer loop is equivalent to a single iteration of the original DAMAS algorithm and therefore while the DAMAS-2E computation times are slightly longer than the DAMAS-2 algorithm, it is significantly faster than the original DAMAS algorithm. The performance of this algorithm has been shown to be comparable to the DAMAS algorithm in an unrelated tip clearance noise work¹⁷ and also in the case of the measurement of free tip vortex noise from a three-dimensional airfoil.³⁶ Figure 4(d)–(f) show the deconvolved source maps corresponding to the beamformed source maps in the first row, where it can be seen that the DAMAS-2E algorithm can narrow down the different source regions on the blade effectively. Note that in these deconvolved source maps, there are a few ghost sources that appear above the top endplate where there is no flow. This is due to the fact that beamforming utilises a free-field Green’s function which does not account for the reflection of the tip clearance noise sources from the nearby wall. These sources, however, are part of the tip clearance noise problem and therefore will be included in the analysis presented in this paper.

Tip clearance noise source levels were obtained by spatially integrating the deconvolved source maps. The integration allows isolation of the tip clearance noise and exclusion of spurious sources such as the trip/junction source highlighted in Figure 4(a). However, it is not possible to clearly isolate the tip clearance noise near the trailing-edge of the blade with that originating from the two-dimensional flow over the trailing-edge. Furthermore, the three-dimensional flow near the tip also modifies the behaviour of the trailing-edge flow noise itself closer to the tip, rendering a clear separation between the two sources difficult. Nonetheless, to assess the effect of the contamination of the tip clearance noise by the trailing-edge noise, an exercise depicted in Figure 5 was undertaken. Figure 5(a) shows a close-up of the tip clearance noise sources for $h$ = 5.2 mm at 2 kHz. The tip clearance noise and the reflections are clearly visible in this map, along with weaker trailing-edge noise source levels towards the bottom-right. The source levels were integrated over a rectangular region with fixed upstream, downstream and upper boundaries, but the lower boundary was varied between the dash-dot line and the solid line at the bottom. The source levels for each of these integration regions are shown in Figure 5(b) where it can be seen that gradually increasing the lower boundary of the region increases the source levels across the frequency range. Above 1.2 kHz, the levels saturate and the final lower boundary of the integration region (the levels corresponding to which are shown using markers in Figure 5(b)) which was chosen based on visual inspection of many beamforming source maps ensures minimum contamination of tip clearance noise by the trailing-edge noise sources below it. Below 1.2 kHz, the array PSF and the uncertainty in source localisation is quite large which possibly results in the trailing-edge noise being also included as the lower boundary of the integration region is increased. Due to this large uncertainty at lower frequencies, the lower integration region boundary was selected based on a visual inspection of the deconvolved source maps to minimise the inclusion of trailing-edge noise sources. It is also worth noting that, although the absolute integrated levels obtained by spatial integration may be contaminated by trailing-edge noise, or may exclude some portion of the clearance noise below 1.2 kHz, the relative variation in source levels with clearance height and pressure gradient is still meaningful since the same integration region was used for all cases considered in this work.

Figure 5.

Effect of the integration region lower bound on the far-field sound spectra obtained by integrating the deconvolved sound-source maps. A sample deconvolved source map at 2 kHz for $h$ = 5.2 mm and ZPG configuration is shown in (a), along with the integration region bounds used for the analysis of the integrated levels. The integrated source levels obtained by varying the lower integration region bounds is shown in (b), with the source levels corresponding to the final integration region shown with red markers.

Results and discussion

In the following sections we will consider the tip clearance noise and surface pressure fluctuations for both zero and adverse pressure gradient configurations. We will first discuss the far-field sound behaviour for the ZPG configuration, followed by an analysis of the effect of pressure gradient on the far-field sound. Next, the surface pressure fluctuations in ZPG configuration are presented before discussing how the presence of pressure gradient modifies their behaviour. Recall that all measurements were performed at $U_{\infty}$ = 35 m/s for seven clearance heights between 1.2 mm and 20.4 mm (see Table 2). Note that each clearance height considered here, regardless of the pressure gradient configuration, is smaller than the incoming boundary layer height.

Zero pressure gradient tip clearance noise characteristics

Figure 6(a) shows the tip clearance noise spectra under ZPG condition obtained by spatially integrating deconvolved source maps for all seven clearance heights in the present work. The source levels are a strong function of clearance height, particularly at frequencies above about 800 Hz. It is possible that the weaker effect of clearance height on far-field sound below this frequency is a consequence of poor source localisation performance of the array at these frequencies. An increase in clearance height generally raises the noise levels, but the magnitude of this increase varies across the frequency range. At frequencies below about 2 kHz, the source levels show a monotonic increase with clearance height, but between approximately 2 – 4 kHz the two largest clearance heights show a dramatic increase in noise levels. It is possible that this sudden increase is associated with the tip vortex becoming more prominent as the gap becomes larger and the cross-flow across the tip becomes more intense. The spectra for the largest clearance also shows the presence of a knee around 3 kHz which could be associated with tip vortex noise which is known to form a broadband hump in case of the flow over an unbounded tip.³⁷ At frequencies larger than 5 kHz, the source levels are less sensitive to the clearance height for $h >$ 1.2 mm; however, the spectra for the smallest clearance height ( $h$ = 1.2 mm) shows significantly lower levels and a broadband hump centred around 6.9 kHz. It is interesting to note that the acoustic wavelength corresponding to this frequency is approximately 49 mm which is close to the quarter-wavelength associated with the blade chord-length (212.3 mm).

Figure 6.

Tip clearance noise spectra for ZPG configuration. (a) Total tip clearance noise for different clearance heights. The contribution of the different clearance subregions to the total tip clearance noise for $h$ = 1.2 mm, 8.2 mm and 20.4 mm is shown in (b), (c) and (d), respectively. The far-field rms pressure (in dB) in the three subregions as a function of clearance height obtained by integrating the acoustic spectra between 1.2 kHz and 10 kHz is shown in (e), while (f) shows the same result obtained by integrating the spectra between 4 kHz and 10 kHz. The three vertical arrows in (b) – (d) show the frequencies for which beamformed source maps are shown in Figure 7.

We will now consider how the tip clearance noise sources are distributed in the three subregions of the tip clearance—leading-edge region, mid-chord region and the trailing-edge region—and how changing the tip clearance height impacts this distribution. For this analysis, the final integration region shown in Figure 5(a) was subdivided in the streamwise direction into these three subregions and the deconvolved source levels within each subregion were then integrated separately to evaluate their contribution to the overall tip clearance noise. The subdivision was performed such that the downstream boundaries of the leading-edge and mid-chord integration regions were located at $x / c$ = 0.25 and 0.75, respectively. Figure 6(b)–(d) shows the contribution of the sources in the three subregions to the total clearance noise for $h$ = 1.2 mm, 8.2 mm and 20.4 mm, respectively. We first note that these comparisons have a large uncertainty below about 1.2 kHz due to the relatively poor source localisation performance of the array, hence the comparisons are only shown above this frequency. Secondly, note that the spectra for the leading-edge and mid-chord subregions show large fluctuations and missing values at certain frequencies due to low signal-to-noise ratio (SNR) for sources in these regions. For the smallest clearance (Figure 6(b)), the source levels in the leading-edge and trailing-edge subregions show comparable magnitude up to about 6 kHz, past which the leading-edge region of the clearance becomes the dominant contributor to the far-field sound and the trailing-edge source levels drop rapidly. The mid-chord region generally has low SNR, but interestingly, there are visible humps in the spectra associated with this region with prominent peaks around 2.2 kHz and 6.9 kHz where the source levels in this region are comparable, or even greater, than those near the leading and trailing-edge of the clearance. The origin of these peaks in the mid-chord region source levels is not clear, but the fact that they occur at discrete frequencies suggests that they may be related to acoustic interference within the gap. Also, note that it is an increase in the mid-chord region source level which is responsible for the broadband hump in the total tip clearance noise centred around 6.9 kHz for this clearance.

An increase in the clearance height to $h$ = 8.2 mm (Figure 6(c)) changes the distribution of sources across the three clearance subregions dramatically. For this clearance height, the leading-edge region ceases to be an important sound-source region and the noise from the trailing-edge region accounts for almost all of the clearance noise up to about 3 kHz, past which the levels drop slightly due to an increase in the mid-chord region source levels. At frequencies above about 4.5 kHz the contribution from the mid-chord region exceeds that from the trailing-edge region, and the former does not show any prominent humps observed previously for the smallest clearance. Note that, although not shown, a slight increase in the clearance height from 8.2 mm to 10.2 mm leads to a slight reduction in the mid-chord region source levels which further decrease as the clearance height increases to 15.2 mm. Finally, for the largest clearance height considered in the present work ( $h$ = 20.4 mm, Figure 6(d)), the trailing-edge region of the clearance accounts for most of the tip clearance noise radiation, except around 10 kHz where a comparable contribution from the mid-chord region is observed. The subregion analysis presented here shows that changing the tip clearance height not only affects the magnitude and shape of the total tip clearance noise, but it also reorganises the dominant source regions in the streamwise direction. This reorganisation as a function of clearance height can be further understood by considering the far-field rms pressure in the three subregions as a function of clearance height as shown in Figure 6(e). The rms pressure was obtained by integrating the subregion acoustic spectra between 1.2 kHz and 10 kHz for each clearance height. This plot clearly shows that the leading-edge source levels decay rapidly with increase in clearance height, while the sources in the trailing-edge region become more dominant as the clearance height increases. The mid-chord region sources also increase with clearance height up to $h$ = 8.2 mm, past which they show an asymptotic behaviour. However, the subregion spectra considered earlier revealed that the mid-chord region sources become important only above about 4 kHz. Therefore, Figure 6(f) shows the same plot as Figure 6(e), but obtained by integrating the spectra above 4 kHz. This plot clearly shows that at higher frequencies the mid-chord region source levels increase with clearance height reaching a maximum for $h$ = 8.2 mm, before decaying as the clearance height increases further. Also note that for $h \leq$ 10.2 mm, the mid-chord region source levels are comparable to, or in some cases exceed, those in the trailing-edge region.

The movement of the dominant tip clearance noise source region with frequency and clearance height is further illustrated through a series of beamformed source maps for three clearance heights at three frequencies in Figure 7. The frequencies at which the source maps are shown are highlighted using vertical arrows in Figure 6(b)–(d). As shown by the subregion spectra Figure 6(d), for the largest clearance height (last column in Figure 7) the dominant source region is concentrated close to the trailing-edge of the clearance and accounts for most of the tip clearance noise generation. A reduction in clearance height to 8.2 mm (middle column) shows the same behaviour at the lowest frequency (2.2 kHz), but for the two higher frequencies the source in the mid-chord region also becomes important. Finally, for the smallest clearance height of 1.2 mm (first column), the source maps at each frequency have a significantly different appearance than those for larger clearances. Here, each of the three subregions of the clearance are efficient sound radiators at the lowest frequency (Figure 7(a)), while at 4.2 kHz the dominant sources are concentrated at the extremities of the clearance. Finally, at the highest frequency of 6.9 kHz, the trailing-edge region produces less noise than the two subregions further upstream. Overall, the integrated source levels and the beamformed source maps presented here for the ZPG configuration show that the tip clearance noise is a strong function of both the frequency and the clearance height. For large clearances, the dominant source is focussed near the trailing-edge of the clearance—likely due to a weaker blockage provided by the larger gap forcing the strongest cross-flow to occur further downstream. On the other hand, for smaller clearances more noise is produced in the leading-edge and mid-chord regions probably because of a larger pressure difference between the pressure and suction sides of the blade in these regions. It is also worth noting that this behaviour is qualitatively similar to that observed in case of a tip with sharp edges.^17,18

Figure 7.

Beamformed source maps for three clearance heights, $h$ = 1.2 mm (left column), 8.2 mm (middle column) and 20.4 mm (right column), at three frequencies, 2.2 kHz (top row), 4.2 kHz (middle row) and 6.9 kHz (bottom row). The three frequencies at which the maps are shown are highlighted using vertical arrows in Figure 6(b)–(d). The colourbar next to each map shows the sound pressure level (SPL) in dB.

Effect of pressure gradient on tip clearance noise

We now consider the effect of pressure gradient on the tip clearance noise. We will only consider the integrated source levels in this section since the appearance of the beamformed source maps is similar for the three pressure gradients. Figure 8 shows the total tip clearance noise source levels for the three pressure gradients at four different clearance heights. The strongest effect of pressure gradient is observed for the largest clearance (Figure 8(d)), where an increase in pressure gradient lowers the noise levels noticeably across a wide range of frequencies. For smaller clearances, the pressure gradient mostly affects the noise levels in the mid-to-high frequency range, where an increase in the pressure gradient can raise or lower the noise levels, depending on the frequency. For the smallest clearance ( $h$ = 1.2 mm; Figure 8(a)), increasing $β$ to 0.51 has negligible influence on the far-field sound up to about 8 kHz, past which a slight increase in source levels is observed. A further increase in $β$ to 1.11 generates a low-magnitude peak between 5 and 6 kHz and lower levels above 6 kHz. As the tip clearance height increases to $h$ = 5.2 mm and 8.2 mm, the spectra up to 2.5 kHz remain independent of the pressure gradient and a slight increase in source levels for the largest pressure gradient is observed around 3 kHz in both cases. This increase in source levels for $β$ = 1.11 around 3 kHz is reminiscent of the spectral knee observed for $h$ = 20.4 mm under ZPG and it is possible that a substantial increase in pressure gradient leads to the tip vortex associated noise becoming more prominent for smaller clearances. At frequencies higher than those associated with this knee, a slight reduction in noise levels with increasing $β$ is observed for both $h$ = 5.2 mm and 8.2 mm. It is also worth noting that, regardless of the clearance height, the spectra for each pressure gradient also show a spectral dip at 772 Hz which is more prominent for the adverse pressure gradient configurations. The acoustic wavelength associated with this spectral dip is approximately 444 mm which is close to the diameter of the test-section (455 mm). It is possible that this dip is a result of a facility acoustic mode which becomes more prominent as the bottom ramp is tilted to produce the pressure gradient.

Figure 8.

The effect of pressure gradient on the total tip clearance noise source levels for $h$ = (a) 1.2 mm, (b) 5.2 mm, (c) 8.2 mm and (d) 20.4 mm.

The strongest effects of pressure gradient are observed for the largest clearance height of 20.4 mm considered in this work (Figure 8(d)). The spectrum for each pressure gradient at this clearance height shows a prominent spectral knee around 3 kHz which, as suggested earlier, is likely due to the tip vortex associated noise becoming more prominent as the gap becomes larger and the vortex gains strength due to a stronger cross-flow. The magnitude of this knee is nearly independent of pressure gradient up to $β$ = 0.51, but a slight increase in magnitude is observed as the pressure gradient increases to $β$ = 1.11. At frequencies above and below those associated with this knee, a reduction in source levels with increasing $β$ is observed, with a slightly larger reduction observed at lower frequencies. The uniform variation in the source levels with $β$ away from the spectral knee motivated attempts to scale the far-field sound spectra for the largest clearance height considered here. Figure 9(a) and (b) show two different scaling of the spectra based on the incoming boundary layer and clearance height for each pressure gradient. In both instances, the spectral magnitude is scaled on the clearance height to the boundary layer height ratio for each pressure gradient, while two different Strouhal numbers based on the incoming boundary layer thickness and clearance height are considered in Figure 9(a) and (b), respectively. The scaling in (a) with Strouhal number based on the boundary layer thickness collapses the spectra reliably at frequencies below those associated with the spectral knee ( $f δ / U_{\infty} <$ 2). On the other hand, using a clearance height based Strouhal number collapses the spectra at frequencies higher than those associated with the spectral knee ( $f h / U_{\infty} >$ 2). Note that neither scaling works at frequencies associated with the spectral dip mentioned earlier since the source of this dip is likely an acoustic phenomenon, as opposed to a fluid dynamic process. Regardless, the scaling here shows that, at least for clearances where the blade tip is located in the outer parts of the boundary layer, the reduction in the clearance noise level with increasing pressure gradient could be a result of the tip being immersed in a thicker boundary layer. The time-scale of the low-frequency noise scales on the boundary layer thickness, while the higher frequencies scale on the clearance height (or perhaps another constant lengthscale).

Figure 9.

Scaling of the far-field sound radiated by the largest clearance height in the present work ( $h$ = 20.4 mm) under three different pressure gradient conditions. The magnitude is scaled on the clearance height to incoming boundary layer height ratio for each pressure gradient; (a) and (b) show the scaling with Strouhal numbers based on incoming boundary layer thickness and clearance height, respectively.

Lastly, before we move on to discussing the surface pressure fluctuations on the blade tip, it is worth pointing out that although the effect of pressure gradient on the overall tip clearance noise source levels is less prominent for small-to-intermediate clearance heights in this work, the pressure gradient does have a noticeable effect on the source levels in the mid-chord region. To illustrate this, Figure 10 shows the source levels in the mid-chord region (left-column) and the trailing-edge region (right-column) for $h$ = 8.2 mm (top-row) and 20.4 mm (bottom-row). For the largest clearance (bottom-row), because most of the noise is generated by the trailing-edge region, the source levels in this subregion (Figure 10(d)) reflect what we had previously observed in the total tip clearance noise spectra (Figure 8(d)). For the smaller clearance of $h$ = 8.2 mm (top-row), the source levels in the trailing-edge region are a weak function of $β$ , except a small peak between 5 and 6 kHz for the largest pressure gradient. However, in the mid-chord region (Figure 10(a)), a noticeable reduction (up to 4.5 dB) in noise levels with increasing $β$ is observed above about 3.5 kHz. This means that under APG conditions, the sound sources in the mid-chord region are modified significantly, but this fact is not reflected in the total tip clearance noise spectra since those spectra also contain a significant contribution from the trailing-edge region where the sources show a weaker dependence on $β$ .

Figure 10.

Effect of pressure gradient on the far-field sound source levels in the mid-chord region (a, c) and trailing-edge region (b, d) for $h$ = (a, b) 8.2 mm and (c, d) 20.4 mm. The legend is the same for all plots and shown in (d).

Pressure fluctuations on the tip under zero pressure gradient

In this section we will consider the unsteady surface pressure on the flat and rounded surfaces of the blade tip under ZPG condition. Recall from section 2.2 that the surface pressure fluctuations were measured at 16 locations on the flat surface and 13 locations on the rounded surface (see Figure 3(d)). The useful frequency range of the pressure fluctuations is between 100 Hz and 6 kHz and all the spectra presented in the subsequent discussion will be restricted within this range. The following discussion will also consider the pressure fluctuations in form of the root mean square (rms) pressure which were obtained by integrating the pressure spectra between this same frequency range.

Figure 11(a) and (b) show the rms pressure on the flat and rounded surfaces of the tip, respectively as a function of the chordwise measurement location for all seven clearance heights under ZPG condition. The relationship between the surface pressure fluctuations and the clearance height is complex and the behaviour is a strong function of the chordwise measurement location. On the flat surface, the smallest clearance shows weak pressure fluctuations across most of the clearance, except near the extremities of the blade where a sharp rise in pressure magnitude is observed. This is consistent with the far-field sound results (Figure 6) which show that the dominant sound-source is focussed near the leading and trailing-edge of the clearance. As the gap size increases further, the cross-flow intensity across the gap increases resulting in an increase in pressure fluctuations across most of the tip surface, with the largest increase observed in the trailing half of the tip ( $x / c >$ 0.5). Interestingly, unlike the consistent increase in the far-field sound levels with clearance height observed earlier (Figure 6(a)), the pressure fluctuations in the trailing half of the clearance first increase with clearance height up to $h$ = 8.2 mm, before gradually decreasing for larger clearances. Near the leading-edge of the tip, the pressure fluctuations for the two largest clearances are considerably lower than those for smaller clearances which is qualitatively consistent with the trend observed in the far-field which show little noise radiation from this region of the clearance. In contrast, the pressure fluctuations at the most downstream port near the trailing-edge are nearly independent for $h >$ 1.2 mm.

Figure 11.

Surface pressure fluctuations on the blade tip under ZPG configuration. Root mean square (rms) pressure on (a) flat surface and (b) rounded surface as a function of chordwise location for seven clearance heights. The three vertical arrows highlight the locations for which pressure spectra for different clearance heights are shown in (c) – (h). The spectra in the second row (c – e) show the spectra on the flat surface, while the third row (f – h) show the spectra on the rounded surface. The chordwise measurement location for each spectra is shown inside each plot. The legend for each plot is the same and shown in (h).

The behaviour of the pressure fluctuations on the rounded surface (Figure 11(b)) is different than that on the flat surface. The pressure fluctuations on the rounded surface show a stronger variation with clearance height in the leading half of the blade tip, but a weaker dependence on clearance height is observed in the trailing half of the tip. In the leading half, the pressure fluctuations for $h >$ 1.2 mm reach a peak around $x / c$ = 0.3, followed by a monotonic increase with chordwise distance in the trailing half of the blade tip. For the smallest clearance, the pressure fluctuations peak close to the leading-edge of the blade which again is consistent with this region being a strong radiator of noise for this clearance. For larger clearances, the pressure fluctuations in the leading half of the clearance decrease with increasing clearance height which is also consistent with the low acoustic SNR from the leading-edge region for these clearances. The rms pressure for each clearance height also shows a localised peak at $x / c$ = 0.62.

Figure 11(c)–(h) compare the surface pressure spectra for all seven clearance heights at three chordwise locations marked using a vertical arrow in the rms pressure plots (Figure 11(a)–(b)). The middle row here (Figure 11(c)–(e)) shows the spectra on the flat surface, while the bottom row (Figure 11(f)–(h)) shows the spectra on the rounded surface. We first note that the spectra on the two surfaces have a different appearance to one another and those on the flat surface have a more complex shape and dependence on clearance height and chordwise location. Near the leading-edge ( $x / c$ = 0.15), the spectra on the flat surface for the two smallest clearances (Figure 11(c)) show larger low-frequency magnitude, but the higher frequency pressure fluctuations are suppressed, suggesting that the presence of the tip near the wall brings large-scale structures near the wall due to the adverse pressure gradient imposed by the blade. At this location, there is a range of intermediate clearance heights ( $h$ = 8.2 mm and 10.2 mm) where the spectra show large pressure magnitude within the considered frequency range without any significant roll-off present. A further increase in clearance height results in lower pressure fluctuations in this region. Further downstream in the mid-chord region (Figure 11(d)), a larger variation in spectral magnitude with clearance height is observed. Here, the smaller clearances ( $h \leq$ 5.2 mm) show weaker pressure fluctuations across the frequency range, before a sudden rise in broadband pressure fluctuations for $h$ = 8.2 mm and 10.2 mm, showing that the increase in rms pressure for these clearances is associated with excitation of a broad range of frequencies in the flow. It is interesting to note that this increase does not translate to a significant increase in the far-field sound levels from the mid-chord region for these clearances. However, the integrated source levels from the mid-chord region (not shown) do show slightly larger far-field sound levels (2 – 3 dB) above about 4 kHz for these two clearances, suggesting that the flow in the mid-chord region has a greater impact on the mid-to-high frequency noise, an observation which is reminiscent of the behaviour noted previously for a sharp-edged blade tip with a different cross-sectional profile.¹⁸ Finally, in the trailing-edge region (Figure 11(e)), a behaviour similar to that in the mid-chord region is observed, i.e. the pressure fluctuations rise rapidly as the clearance height is increased to 8.2 mm and 10.2 mm and then fall as the clearance height is increased further. There are, however, some notable differences between the spectra here and in the mid-chord region. The spectra for $h$ = 8.2 mm and 10.2 mm show a broadband hump at low frequencies (less than 1 kHz) and a sharp increase in the pressure fluctuation magnitude at frequencies above 1 kHz is observed for $h$ = 3.2 mm. The spectra for the two largest clearances also show a weaker hump around 700 – 800 Hz. These results show that while there are some general features of surface pressure fluctuations on the flat surface which are common in the different regions of the blade tip (e.g., a sharp increase in levels for $h$ = 8.2 mm and 10.2 mm), there are also some notable differences.

Now consider the surface pressure spectra on the rounded surface of the tip at three chordwise locations as shown in Figure 11(f)–(h). Regardless of the measurement location and the clearance height, each spectra has a similar shape—a flat, low-frequency regime, followed by a gentle roll-off at higher frequencies. This roll-off begins at lower frequencies in the two upstream regions ( $x / c$ = 0.17 and 0.62), compared to the trailing-edge region ( $x / c$ = 0.78). In the leading-edge region ( $x / c$ = 0.17), the spectral magnitude decreases with increasing clearance height without affecting the shape of the spectra significantly. This reduction with clearance height is not uniform though and the spectra for $h \leq$ 3.2 mm and 5.2 mm $\leq h \leq$ 10.2 mm are closely grouped together, with the smaller clearances showing considerably larger magnitude across the frequency range which is consistent with them being a strong radiator of leading-edge clearance noise. Further downstream in the mid-chord region (Figure 11(g)), the pressure fluctuations are less sensitive to clearance height, but an increase in pressure levels for the intermediate clearance height range ( $h$ = 8.2 mm and 10.2 mm) seen on the flat surface can also be observed here. Finally, at the most downstream measurement location available on the rounded surface (Figure 11(h)), the pressure fluctuations generally increase with clearance height within the range of heights considered here, but the spectra for $h$ = 15.2 mm shows slightly elevated pressure fluctuations, compared to those for $h$ = 20.4 mm. Also note that, unlike the spectra on the flat surface in this region (Figure 11(e)), no broadband humps are present on the rounded surface. To further understand the differences and similarities between the pressure fluctuations on the two surfaces of the blade tip, we will now compare the pressure statistics on the two surfaces at three chordwise locations for three clearance heights, $h$ = 1.2 mm, 8.2 mm and 20.4 mm.

Figure 12(a) shows a schematic of the blade tip and the three pairs of flat/rounded surface pressure taps which will be used to draw a comparison. Figure 12(b)–(d) show a comparison between the spectra on the two surfaces at three chordwise locations for three clearance heights. At each chordwise location, the spectra on the two surfaces for $h$ = 8.2 mm and 20.4 mm have been scaled by 20 dB and 40 dB, respectively to shown the comparison for all three clearances within the same plot. Note that the chordwise location highlighted above each plot refers to the location of the tap on the flat surface; the closet rounded surface tap is located slightly downstream (0.01 $c$ ) downstream of this location. We first note that at each location, the pressure fluctuations for the smallest clearance (bottom set of curves in each plot) on the rounded surface are considerably stronger than those on the flat surface. This is likely a result of weak cross-flow within this small gap. As the gap height increase to 8.2 mm and flow across the tip intensifies the pressure fluctuations on the flat surface intensify and exceed those on the rounded surface in the leading-edge and mid-chord regions, with the leading-edge region exhibiting significantly large pressure fluctuations due to the lack of a spectral roll-off in the measurable frequency range. In the trailing-edge region ( $x / c$ = 0.82), the pressure fluctuations on the two surfaces are comparable in magnitude across most of the frequency range. A further increase in the clearance height to 20.4 mm lowers the pressure fluctuations on the flat surface in mid-chord and trailing-edge regions ( $x / c$ = 0.6 and 0.82, respectively), but in the leading-edge region they remain considerably larger than those on the rounded surface.

Figure 12.

Comparison of pressure fluctuations on the flat and rounded surface of the blade tip for the three pairs of taps shown in (a). (b), (c) and (d) compare the pressure spectra on the two surfaces for three clearances at three chordwise locations, with the $x / c$ location noted above each plot being the chordwise location of the tap on the flat surface. The legend for (b) – (d) is shown in (d). Note that the spectra on the two surfaces for $h$ = 8.2 mm and 20.4 mm have been scaled by 20 dB and 40 dB for clarity. The coherence between the taps on the flat and rounded surface for the three clearance heights is shown in (e), (f) and (g), with the legend shown in (g).

Figure 12(e)–(g) show the coherence function between the three pairs of sensors on the two surfaces highlighted in Figure 12(a). The pressure fluctuations across the two surfaces for the smallest clearance height remain well correlated up to about 1 – 2 kHz at each chordwise location, with the pairs near the leading and trailing-edge of the clearance exhibiting a stronger coherence. For the two larger clearances ( $h$ = 8.2 mm and 20.4 mm), the coherence across the two surfaces at the two downstream stations (Figure 12(f) and (g)) is largely negligible, but a noticeable correlation up to about 2 kHz is observed close to the leading-edge (Figure 12(e)). Here, the coherence curves for $h$ = 8.2 mm and 20.4 mm show a broadband hump centred around frequencies of approximately 300 Hz and 450 Hz, respectively. Overall, the comparison between the pressure fluctuations on the flat and rounded surfaces presented in Figure 12 shows that for very small clearances, the pressure fluctuations are well correlated across the two surfaces of the tip with stronger pressure fluctuations on the rounded surface. For larger clearances, the pressure fluctuations on the flat surface intensify in the leading-edge and the mid-chord regions, but they are slightly weaker than those on the rounded surface towards the trailing-edge of the tip. An increase in clearance height also decorrelates the pressure fluctuations across the two surfaces.

Effect of pressure gradient on the tip pressure fluctuations

Finally, in this section we consider the effect of pressure gradient on the surface pressure fluctuations on the blade surface. Figure 13(a) and (b) show the rms pressure on the flat and rounded surface, respectively for three clearance heights and three pressure gradients. As before, an offset of 20 dB and 40 dB has been added to the rms curves for $h$ = 8.2 mm and 20.4 mm, respectively to show them on the same plot. On the flat surface (Figure 13(a)), the largest effect of the pressure gradient is observed for $h$ = 8.2 mm, where a noticeable reduction in surface pressure with increasing $β$ is observed across the leading-edge and mid-chord regions. In particular, an increase in the pressure gradient notably lowers the rms pressure in the mid-chord region of the blade tip (between $x / c$ = 0.3 and 0.65). This is consistent with the behaviour of the far-field sound source levels in this region which showed that for this same clearance height an increase in $β$ leads to lower noise levels above about 3.5 kHz (see Figure 10(a)). It is likely that the reduction in both the near and far-field pressure levels in the mid-chord region is a result of a reduction in the velocity of the cross-flow passing through the gap as a streamwise APG is imposed upon the flow. Closer to the trailing-edge, however, the effect of pressure gradient on surface pressure fluctuations is not significant and this probably contributes to the low-frequency far-field sound being less sensitive to the pressure gradient (since the low-frequency sound is generated mostly from the trailing-edge region). If we now consider the surface pressure spectra shown in Figure 13(c)–(e) for three clearances at a single location marked by the vertical arrow in Figure 13(a), we see that the pressure magnitude has a strong inverse relationship with $β$ for $h$ = 8.2 mm, while for $h$ = 1.2 mm and 20.4 mm they are less sensitive to the pressure gradient, although for the latter a noticeable increase in low-frequency pressure fluctuations with pressure gradient can be observed.

Figure 13.

Effect of the adverse pressure gradient on the tip surface pressure fluctuations. The rms pressure on the (a) flat surface and (b) rounded surface as a function of chordwise location for three clearance heights. Note that an offset of 20 dB and 40 dB has been added to the rms curves for $h$ = 8.2 mm and 20.4 mm, respectively to show them on the same plot as the curve for $h$ = 1.2 mm. (c) – (e) show the surface pressure spectra at each pressure gradient for three clearance heights on the flat surface at a single location ( $x / c$ = 0.48, position highlighted using an arrow in (a)). The pressure spectra on the rounded surface at $x / c$ = 0.27 (position highlighted using an arrow in (b)) are shown in (f) – (h). The legend is shown in (h).

Now consider the effect of pressure gradient on the unsteady surface pressure on the rounded surface of the blade as shown in Figure 13(b). Qualitatively, the pressure gradient does not change the behaviour of the rms pressure on this surface, but a reduction in the pressure magnitude with increasing $β$ can be observed across the entire surface for the two smaller clearances. The rms pressure for the largest clearance ( $h$ = 20.4 mm) also shows this reduction for $x / c >$ 0.4, but further upstream an increase in rms pressure with increasing $β$ is observed. Considering Figure 13(f)–(h) which compares the pressure spectra for different $β$ at $x / c$ = 0.27, we note that for the two smaller clearances the reduction in rms pressure with increasing $β$ is driven by pressure fluctuations above approximately 300 Hz, whereas the increase in rms pressure with $β$ for the largest clearance is driven mostly by a noticeable increase in low-frequency pressure fluctuations below about 2 kHz. These results, along with those on the flat surface, show that the effect of pressure gradient on the near-field pressure is a strong function of clearance height and frequency with the largest effect of $β$ observed on the flat surface for intermediate clearance heights in the present work (note that, although not shown, the results for $h$ = 10.2 mm are similar to those for $h$ = 8.2 mm).

We will now consider the effect of pressure gradient on the correlation between the pressure fluctuations on the flat and rounded surfaces of the blade at three chordwise locations. Figure 14 shows coherence between the two surfaces for three clearance heights and at three chordwise stations. Each row in this figure corresponds to the coherence function for a particular clearance height noted to the left, while each column corresponds to a chordwise location noted at the top. Note that the chordwise location shown is that of the sensor on the flat surface and the coherence is calculated between this sensor and the closest sensor on the rounded surface which in each case is located 0.01 $c$ downstream of the sensor on the flat surface. Regardless of the pressure gradient, the most significant correlation between the pressure fluctuations on the two surfaces is observed for the smallest clearance (Figure 14(a)–(c)), while for the two larger clearances only the leading-edge region ( $x / c$ = 0.15) shows statistically meaningful correlation. The effect of the pressure gradient on coherence is mostly quantitative and for all cases with meaningful coherence values ( $>$ 20% coherence), a reduction in coherence magnitude with increasing $β$ is observed.

Figure 14.

Effect of pressure gradient on the coherent pressure fluctuations across the flat and rounded surfaces of the tip. Each column shows the coherence between the three pairs of remote microphones highlighted in Figure 12(a), with the chordwise measurement location of the flat surface microphone noted above each column. Each row shows the coherence function of a specific clearance height noted to the left of that row. The legend is shown in (i).

Finally, Figure 15 shows the effect of pressure gradient on the coherent pressure fluctuations in the leading-edge, mid-chord and trailing-edge regions of the flat surface. The microphone pair for which the coherence was calculated are shown in the schematic in Figure 15(a), with the left, middle and right columns showing the coherence in the leading-edge, mid-chord and trailing-edge regions, respectively. The top, middle and bottom rows in this figure show the coherence for $h$ = 1.2 mm, 8.2 mm and 20.4 mm, respectively. In the leading-edge region, the pressure gradient has a mild effect on the coherence and an increase in clearance height decorrelates the pressure fluctuations across the two locations, a behaviour similar to that observed in Figure 14. Further downstream in the mid-chord region, stronger correlation between is observed for each clearance height when compared to the leading-edge region, with the magnitude of correlation again decreasing as the clearance height increases. In this region, the coherence at lower frequencies for the two smaller clearances (Figure 15(c) and (f)) shows a noticeable effect of pressure gradient with the pressure fluctuations for $h$ = 8.2 mm becoming less coherent as $β$ increases. For the smallest clearance ( $h$ = 1.2 mm), the coherence magnitude decreases as $β$ is raised to 0.51, but then a large increase is observed for the strongest pressure gradient. Finally, in the trailing-edge region, the coherence decreases rapidly as the clearance height increases, with the smallest clearance height showing the same trend as that observed in the mid-chord region, while for $h$ = 8.2 mm, a reduction in coherence magnitude with increasing $β$ is observed. Overall, it appears that on the flat surface pressure fluctuations are more coherent for small clearances and the coherence magnitude at low frequencies is noticeably affected by the imposed APG on the flow.

Figure 15.

Effect of pressure gradient on the coherent pressure fluctuations in the leading-edge, mid-chord and trailing-edge regions on the flat surface. The schematic in (a) shows the three pairs of remote microphones in each region for which the coherence is shown in (b) – (j). Each column shows the coherence in a specific region noted above the column, while each row corresponds to the coherence at a specific clearance height noted to the left of the row. Each plot shows the coherence for all three pressure gradients and the legend is shown in (j).

Conclusion

The present work discusses the behaviour of near and far-field pressure fluctuations in an idealised tip clearance flow (a stationary single blade adjacent to a static wall) under zero and adverse pressure gradient (APG) inflow conditions. A purpose-built test-rig capable of generating adverse pressure gradients in an open-jet facility was designed for the measurements. The test-rig consists of an adjustable bottom ramp which can be deflected to obtain different pressure gradients with a Clauser pressure gradient parameter up to $β$ = 1.11. In the present work, a zero pressure gradient (ZPG) configuration and two adverse pressure gradient configurations with $β$ = 0.51 and 1.11 were considered. The tip clearance flow was generated by placing a single, tapered airfoil blade with a rounded tip section at a geometric angle of attack $(α_{g})$ of 12°. The surface pressure fluctuations were measured on both the top surface of the tip (the tip plane) and the rounded edge of the tip using a remote microphone method. The tip clearance noise was measured using a 64-microphone phased array and the array response was beamformed to extract the tip clearance noise source levels. The measurements were performed for seven tip clearance heights $(h)$ between 1.2 mm and 20.4 mm at a single free-stream velocity of 35 m/s. The Reynolds number based on the clearance height ranged between 2600 and 45,400. Each clearance height considered in the present work was smaller than the incoming boundary layer height for every pressure gradient configuration.

The tip clearance noise under ZPG was found to be a strong function of clearance height, with the levels generally increasing as the gap height increases, but the magnitude of this increase is not uniform across the measurable frequency range (600 Hz – 10 kHz). For the smallest clearance in the present work ( $h$ = 1.2 mm), the noise sources across most of the frequency range were concentrated near the leading and trailing-edges of the clearance. Consistent with this, the pressure fluctuations at the extremities of the clearance were found to be substantially higher than those in the mid-chord region. As the clearance height increases, the contribution of the leading-edge noise sources to the overall clearance noise decays rapidly and the mid-chord and trailing-edge regions of the clearance emerge as more efficient radiators of noise. For clearance heights between 3.2 mm and 10.2 mm, the trailing-edge region of the clearance remains the dominant source region at low frequencies (up to about 3.5 kHz), while at higher frequencies the contribution of the mid-chord region sources becomes comparable, or even larger, than the sources in the trailing-edge region. A further increase in clearance height to 15.2 mm and 20.4 mm weakens the contribution of the mid-chord region sources such that for these clearances the trailing-edge region becomes the dominant contributor to the total clearance noise across most of the measurable frequency range.

The near and far-field pressure fluctuations are both affected notably due to the adverse pressure gradient. The total tip clearance far-field sound for small-to-intermediate clearance heights (up to $h$ = 10.2 mm) decreases slightly (up to 1.5 dB) in the mid-to-high frequency range with an increase in $β$ . This reduction in source levels with $β$ is shown to be a result of weaker source levels and surface pressure fluctuations on the tip in the mid-chord region of the clearance under APG conditions. However, because the total tip clearance noise contains significant contribution from the sources in the trailing-edge region which are only weakly dependent on $β$ , this reduction in the mid-chord region source strength with increasing $β$ is not reflected in the total tip clearance noise spectrum. For the largest clearance height considered in the present work ( $h$ = 20.4 mm), adverse pressure gradient leads to a larger reduction in the total tip clearance noise levels at both low and high frequencies, with the levels gradually decreasing as $β$ increases. The far-field sound magnitude for this clearance can be scaled on the clearance height to boundary layer ratio for each pressure gradient, with low-frequency time scales scaling on the incoming boundary layer thickness for each $β$ and higher frequencies scaling on the tip clearance height. The effect of pressure gradient on the surface pressure fluctuations on the blade tip and the rounded edge of the tip was found to be slightly different. On the blade tip, an increase in $β$ generally reduces the pressure fluctuations across the entire clearance, with the largest reduction occurring for intermediate clearance heights ( $h$ = 8.2 mm and 10.2 mm) in the mid-chord region. On the rounded edge, the fluctuating pressure magnitude also decreases with $β$ for clearance heights up to 10.2 mm, but for larger clearances an increase in magnitude with $β$ is observed in the leading half of the tip. Regardless of the clearance height, the correlation between the surface pressure fluctuations on the tip and the rounded edge of the tip was only mildly affected by the pressure gradient.

Footnotes

Acknowledgements

This publication was made possible by N62909-23-1-2100 from the Office of Naval Research Global. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of the Office of Naval Research Global.

ORCID iD

Manuj Awasthi

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Office of Naval Research Global (N62909-23-1-2100).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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Near and far-field pressure fluctuations in tip leakage flow under an adverse pressure gradient

Abstract

Keywords

Introduction

Experimental setup and instrumentation

UNSW anechoic wind tunnel & pressure gradient tip clearance noise test-rig

Surface pressure fluctuation measurements

Far-field sound measurements

Results and discussion

Zero pressure gradient tip clearance noise characteristics

Effect of pressure gradient on tip clearance noise

Pressure fluctuations on the tip under zero pressure gradient

Effect of pressure gradient on the tip pressure fluctuations

Conclusion

Footnotes

Acknowledgements

ORCID iD

Funding

Declaration of conflicting interests

References