A sound power measurement technique optimised for low-speed fan tones

Abstract

Air-moving devices such as fans are routinely characterised experimentally in terms of their sound power, and sound quality which depends on both broadband and tonal noise levels. The International Organization for Standardization method 5136, widely used in industry and academia, estimates sound power in third-octave bands radiated into a duct. Since the International Organization for Standardization method is not designed for tone measurements and overlooks the unevenness of the modal power distribution, an extended method has been developed and implemented based on mode decomposition. The ‘two-port’ source model is formulated to include higher-order modes and applied for the first three modes which require six independent stationary measurements on each side or ‘port’. The resulting experimental rig is much shorter than the International Organization for Standardization rig and does not require anechoic terminations. Both methods have been used to characterise the same fan, and the results compared to understand tone measurement shortfall of the International Organization for Standardization rig.

Keywords

Acoustics noise source aerodynamics aeroacoustics fan compressor turbomachinery duct ducted rig measurement sound power tones broadband higher order mode decomposition ISO 5136

Introduction

This article describes the design and implementation of a new experimental rig suitable for acoustical characterisation of a fan unit – a source of flow and sound. The rig is designed to allow for flow measurements linked to both aerodynamic performance and acoustic measurements simultaneously as most of the noise is aerodynamically generated¹ (and therefore a function of operating point). Trends in both tonal and broadband noise components are of interest so the rig must give accurate operating point control, and the acoustic measurement technique should be practical, and give insight into generation mechanisms.

There is a large body of literature on fan noise measurement and characterisation. In terms of gas turbine-related research, Tyler & Sofrin’s seminal paper on compressor tonal rotor-stator interaction noise² links blade/vane numbers to the azimuthal order of the duct modes generated. They validate their theory by azimuthally traversing a microphone at some fixed radius – the microphone being inserted on a cantilever at the open inlet of the rotor. Individual azimuthal modes are identified by effectively inducing a Doppler shift between the traversing microphone and the azimuthal mode of interest which is spinning at a unique rate. In order to speed up the measurement process, a rake of microphones has subsequently been used to measure radial and azimuthal modes simultaneously. A system developed at NASA Glenn in the 1990s³ has been used in a variety of turbofan tests from low-speed concept rigs to full-scale production engines.

There are several potential disadvantages of using an in-flow measurement rake:

The rotation mechanism must be robust and allow measurements which are accurate in terms of spatial location and unaffected by vibration.

Inlet measurements can produce additional noise sources from interaction which may be very prominent in low-speed applications.

Outlet measurements with the pressure transducers facing the oncoming flow are most likely to be contaminated by flow noise.

Anechoic terminations are often required at the duct ends. This eliminates reflections and prevents additional reflection at or transmission through the fan which would be measured along with the waves coming directly from the source.

Furthermore, the difference between gas turbine applications and the small-scale regime of interest in the present work means that this kind of technique is less applicable:

The scale is much smaller so the duct diameter is lower and fewer modes are cut-on.

The frequencies of interest are lower because rotational speeds and velocities are lower.

The lower harmonics of BPF are the most significant.

Amplitudes are much lower so more sensitive transducers are required which are generally larger. Consequently, a measurement rake may have a larger disturbing effect.

The International Organization for Standardization (ISO) has developed standard 5136 for the ‘Determination of sound power radiated into a duct by fans and other air-moving devices’.⁴ This method is widely used in academia⁵ and industry to characterise fans, in particular for HVAC applications. The aeroacoustic rig, used extensively by Dyson, is built to this acoustic standard and the associated standard for measuring overall fan performance.⁶

The set-up, pictured in Figure 1, consists of a long, straight duct with the test case located in the middle. Anechoic terminations at each end serve several purposes. Principally they stop reflections of the sound power emitted by the source which would lead to axial standing waves. They also minimise extraneous noise from the surrounding environment, the inlet orifice plate and the outlet throttle which controls the fan operating point. A single microphone is positioned in the flow on the inlet and outlet sides of the fan. To account for the variation in sound pressure/intensity across the duct section, the standard relies on the theory that there is a radial position which will give sound pressure readings representative of the whole section. A simplified higher-order mode model gives this optimum radial position as midway between the duct axis and wall.⁷ The model assumes the first 10 cut-on modes within each band carry equal power. Since the measurements may be influenced by flow noise, particularly at the outlet where the microphone points into the flow, there are requirements on using flow conditioners and specialised microphone shields depending on the outlet velocity magnitude and direction (i.e. swirl).

The existing ISO standard rig set-up has several advantages and disadvantages. The main advantage of the method is that it allows quick, standardised measurements of overall sound power level based on third-octave bands. However, the suitability of the rig to study the narrow-band tones of interest in this work is to be determined. A narrow-band tone superimposed on a broadband spectrum of frequencies can contribute little to the band power. Tyler & Sofrin’s theory for tonal interaction noise shows that specific azimuthal modes are generated depending on rotor/stator numbers – one expects the excited azimuthal modes to carry more power than the others – so the equal mode power model does not apply. The in-duct method is derived and then validated against⁸ third-octave measurements made according to the (standardised) free-field SPL survey method, ISO 3746.⁹ Narrow-band spectra are rarely analysed and compared in any depth as this is not what the method is designed for. On one occasion the standard authors do present a narrow-band comparison of averaged spectra; significant deviations between the methods are seen at some frequencies around the cut-on of higher-order modes (Figure 21 in Holste and Neise⁸).

The approach in this work is to develop a new and more compact facility designed for measurement of broad and narrow-band noise emitted at the fan openings. Mode decomposition of the underlying sound field isolates reflections in the duct and removes the need for anechoic terminations (which increase the length of the rig and are not effective at all frequencies). On each side of the fan, at least one independent sound field measurement per mode is required to find its amplitude. These measurements are stationary and located on the duct surface which causes minimal disturbance or flow noise contamination. The well-known ‘two-port’ method¹⁰ is extended to include characterisation of the transmissive and reflective properties of the fan, including higher-order modes. This mode-based appraisal of the sound field and source has several advantages over the in-duct approach including:

Passive and active properties of the source are quantified,

Equally accurate for tones and broadband in principle,

No reliance on the assumption of the equal power mode model, and

Comparison of in-duct measurements against a different method.

The next section of this article gives further details on the rig at Dyson. The design of the new rig is then presented in ‘New rig design’ section while details of the fan are given in ‘Test cases’ section. This rig implements the acoustic mode decomposition and two-port methods outlined in Mode decomposition and Two port source analysis with higher-order modes sections, respectively. Comparisons are made between the measurements taken in both rigs to investigate the significance of the points outlined above.

Dyson ISO rig

The main components of the Dyson ISO rig are shown in a schematic in Figure 2.

Air enters the rig via the inlet orifice plate which is geometrically similar to the ISO specification⁶ and the volume flow rate Q is calculated based on the pressure difference across the orifice. The static pressure increase across the fan is measured between axial stations upstream of the fan and downstream of the flow straightener. This is taken to equal to the fan stagnation pressure increase $Δ p 0$ since the areas are the same and the flow is fully developed. At each axial station, the pressure is determined from a pneumatic average of four surface pressure taps equally spaced around the circumference.

The operating point is controlled using a throttle located downstream of the outlet anechoic termination to prevent contamination of the noise measurements (pictured in Figure 3(a)).

The microphones are located some distance upstream and downstream to allow evanescent modes to become attenuated. The microphones used are B&K 4189 ½-inch free-field microphones with a dynamic range of 15–146 dB. Each is fitted with a nose cone shield to minimise flow noise effects on the measurement (Figure 3(b)). According to the standard, a sampling tube would be required for mean flow velocities above 20 m/s⁷ – the level encountered in the current context is around 2 m/s. The strut, which holds the microphone midway between the duct axis and wall, has a streamlined shape to minimise disturbance. The flow straightener (ISO 5801 ‘star’ type⁶) at the outlet of the fan removes swirl which would otherwise affect the stagnation pressure increase and microphone readings. According to the standard, an average must be taken of at least three sound pressure levels at different azimuthal locations to account for circumferential variation. This is achieved by rotating the fan duct section relative to the rig and microphones which remain stationary.

The inner diameter of the rig is 0.14 m and the length is ∼14 m. The considerable length of the rig is mainly due to the required spaces between components and the microphones, and to a lesser extent the length of the anechoic terminations (∼2 m each). The large spacing from the microphones gives undisturbed conditions (i.e. no swirl, low turbulence) which would otherwise swamp the measured acoustic fluctuations.

New rig design

In this section, an overview of the new rig (Figure 4) is presented. Key dimensions of both rigs and the fan are given in ‘Test cases’ section while details of the acoustic methods are shown in the remaining sections of the article.

A bellmouth inlet flow meter is used to measure the flow rate as it produces less noise and unsteadiness compared to an orifice plate. This is made to the ISO standard geometry⁴ with the throat diameter chosen to give a Reynolds number (based on this diameter; given in Table 1) greater than 20,000 at the flow rates of interest. The fan stagnation pressure increase

Δ p 0

, based on surface pressure taps before the fan and after the straightener, is found in the same manner as in ‘Dyson ISO rig’ section. Flow conditioning devices are used downstream of the bellmouth and the fan duct to remove non-uniformity. The operating point of the fan is controlled by using an iris valve at the outlet to throttle the flow, similar to an orifice plate of variable area. The iris valve is connected to a stepper motor to obtain precise throttle settings. Throttle noise, a potential issue, is controlled by placing coarse foam just upstream of the valve to condition the flow.

Table 1.

Test rig and fan details.

	Dyson ISO rig tests	New rig tests
Duct inner diameter (m)	0.14	0.145
Rig length (m)	14	2.5
Flowmeter throat diameter (m)	0.058	0.050
Fan type	Mixed-flow
Impeller tip diameter, D (m)	0.1
Fan RPM	8400
Number of blades (B, V)	9, 22
Fan flow coefficient, $φ = \frac{Q}{Ω D 3}$	0.031
Fan pressure rise, $ψ = \frac{p 0}{ρ Ω 2 D 2}$	0.06
Fan control	Variable power, constant RPM ± 0.1%

Acoustic measurements are made using two arrays of seven flush-mounted microphones arranged on the inlet and outlet sides of the fan. The microphones used are G.R.A.S. 40DD ⅛-inch pressure microphones with a dynamic range of 40–175 dB. Signals are acquired using a GBM Viper multichannel system capable of simultaneous data acquisition, signal conditioning for optimal ADC resolution and anti-aliasing filtering, and sensor power supply. The microphone arrangement is explained in more detail in ‘Test cases’ section.

As described in ‘Two port source analysis with higher-order modes’ section, the transmissive and reflective properties of the fan are determined using external loudspeakers. These are housed in speaker boxes which can be seen in Figure 5, each containing three speakers, which emit sound into the duct surface through perforations.

Test cases

The same fan is integrated into and measured in both rigs, for which the details are summarised in Table 1.

Figure 6 shows the non-dimensional operating curves of the fan measured in both rigs, and the design point at which it operated during the study. The fan was driven directly by an electric motor and controller which kept the rotational speed constant with high accuracy.

Mode decomposition

Broadband mode decomposition techniques based on auto- and cross-spectra between stationary microphones date back to the work of Seybert and Ross.¹¹ In their article, they determine an alternative to the impedance tube method for measuring acoustic impedance. Broadband excitation produces incident and reflected plane-waves; the amplitudes of each are related to the spectra measured at two flush-mounted microphones.

Above a well-defined frequency, the contribution from non-plane-wave modes becomes significant. Mode decomposition of higher-order modes has been performed experimentally for a square-section duct which approximates the thin annulus of a high bypass-ratio engine.¹² For a circular-section duct, Lavrentjev and Abom¹³ include higher-order modes in their analysis of the fan inlet noise spectrum.

The mode decomposition theory is presented next in a new compact form amenable to optimisation of the locations of the measurements. Subsequently, a novel location optimisation procedure is implemented which gives the set-up shown in ‘New rig design’ section (as published by the authors¹⁴).

Within a duct of circular cross-section of radius R_d with no mean flow (for simplicity), the acoustic pressure p satisfies the wave equation

\frac{1}{a_{0}^{2}} \frac{\partial 2 p}{\partial t 2} - \nabla 2 p = 0

(1)

where a₀ is the speed of sound. By considering a solution with separable functions in cylindrical coordinates

(r, x, θ)

and applying appropriate boundary conditions, the acoustic pressure due to perturbations of frequency ω is

p (r, x, θ, t) = C m, n e i (ω t + m θ) J m (z m, n \frac{r}{R d}) (A_{m, n}^{+} e - ik m, n x + A_{m, n}^{-} e ik m, n x)

(2)

where

(m, n)

are integers that represent azimuthal and radial mode number, respectively. Bessel functions of order m are denoted J_m and the zeros of its derivative

z m, n

, and

A_{m, n}^{\pm}

are the wave amplitudes of mode

(m, n)

travelling in the positive, and negative x direction. The axial wave number

k m, n

is given by

k m, n = \sqrt{k_{0}^{2} - \frac{z_{m, n}^{2}}{R_{d}^{2}}} = \sqrt{\frac{ω 2}{a_{0}^{2}} - \frac{z_{m, n}^{2}}{R_{d}^{2}}}

(3)

The normalisation factor $C m, n$ (section 9.2 in Morse¹⁵) is set such that

C_{m, n}^{2} \int \int^J_{m}^{2} (z m, n \frac{r}{R d}) d S = S

(4)

to make the squared amplitude magnitude proportional to the mode power once the function is integrated over the cross-section (when finding the power). This expression can be simplified by performing the integration over a duct cross-section to give

C m, n = | \frac{z m, n}{J m (z m, n) \sqrt{(z_{m, n}^{2} - m 2)}} |

(5)

Assuming a uniform flow in the axial direction gives a similar result except that the axial wave number now depends on M, the mean-flow Mach number, and on the direction of wave propagation, either with (+) or against (−) the mean-flow

k_{m, n}^{\pm} = \frac{\sqrt{k_{0}^{2} - (1 - M 2) \frac{z_{m, n}^{2}}{R_{d}^{2}}} \mp Mk 0}{1 - M 2}

(6)

For frequencies below the cut-on frequency of a given mode, the axial wave number is purely imaginary and the amplitude of the wave decays exponentially along the duct axis. Conversely, above cut-on the wave can propagate without decay.

Taking the Fourier transform in time of equation (2) and summing over all possible modes gives

\overset{\land}{p} (r, x, θ, ω) = \sum_{n = 0}^{\infty} \sum_{m = - \infty}^{\infty} C m, n e im θ J m (z m, n \frac{r}{R d}) (A_{m . n}^{+} (ω) e - ik m, n x + A_{m, n}^{-} (ω) e ik m, n x)

(7)

In the present article, the three modes (0,0) & (±1,0) are measured experimentally. However, the scheme can be extended in a straightforward manner to additional modes. The Mach number of the mean velocity in the duct was very low (∼0.01) so that the acoustic theory for no mean flow is applicable.

From equation (7), the frequency-domain summation over the first three cut-on modes at a measurement location $(R d, x i, θ i)$ on the duct surface is given by

\overset{\land}{p} i (R d, x i, θ i, ω) = \sum_{m = - 1}^{m = 1} C m, 0 e im θ i J m (z m, 0) (A_{m, 0}^{+} (ω) e - ik m, 0 x i + A_{m, 0}^{-} (ω) e ik m, 0 x i)

(8)

For each mode, there are two unknown amplitudes $A_{m, n}^{+}$ and $A_{m, n}^{-}$ . This implies that a minimum of two independent measurements are required, giving two equations, for each mode present. For the modes of interest, there are two unknown amplitudes for each of $m = - 1, 0, 1$ giving a minimum of six independent measurements (N modes implies $2 N$ unknowns). Denoting the six measurements required $\overset{\land}{p} I, \overset{\land}{p} II, \dots, \overset{\land}{p} VI$ , the system of six simultaneous equations can be written in matrix form as

︸ [\overset{\land}{p} I (R d, x I, θ I, ω) : \overset{\land}{p} VI (R d, x VI, θ VI, ω)] p = [C 0, 0 e - ik 0, 0 x I \dots C - 1, 0 e i θ I J - 1 (z - 1, 0) e ik - 1, 0 x I : : C 0, 0 e - ik 0, 0 x VI \dots C - 1, 0 e i θ VI J - 1 (z - 1, 0) e ik - 1, 0 x VI] ︸ M [A_{0, 0}^{+} (ω) : A_{- 1, 0}^{-} (ω)] ︸ A \pm

(9)

This set of simultaneous equations is solved at each frequency ω for the unknown amplitudes

{A \pm} = M - 1 {p}

(10)

Note that equation (10) works only for deterministic signals. For a signal with noise, average statistical quantities should be used. The formulation is therefore recast in terms of quantities to be averaged over a significant measurement period, valid for both periodic and random signals. Dividing equation (9) by a coherent reference pressure (or voltage) signal $\overset{\land}{r}$ and using the inverse of $M$

[\frac{A_{0, 0}^{+}}{\overset{\land}{r}} : \frac{A_{- 1, 0}^{-}}{\overset{\land}{r}}] = M - 1 [\overset{\land}{H} rI : \overset{\land}{H} rVI]

(11)

where for

i = I, II, \dots, VI

\frac{\overset{\land}{p} i}{\overset{\land}{r}} = \frac{\bar{\overset{\land}{p} i \overset{\land}{r} *}}{\bar{\overset{\land}{r} \overset{\land}{r} *}} = \frac{\overset{\land}{S} ir}{\overset{\land}{S} rr} = \overset{\land}{H} ri

(12)

in which

\overset{\land}{S} ir

and

\overset{\land}{S} rr

represent cross- and auto-spectra, respectively. This approach works only when the coherence is good between the pairs of signals.¹⁶

The matrix $M$ is singular or ill-conditioned if the six measurements are not fully independent. The approach of equation (10) differs from that of other authors¹³ as $M$ and $A \pm$ contain terms corresponding to waves travelling in both directions, as opposed to separate matrices for each direction. This makes the decomposition scheme straightforward and amenable for optimisation. Surplus measurements can be used to form additional entries in the p vector and matrix $M$ to give an overdetermined system of equations to improve mode decomposition results.¹⁷ However, the minimum number of measurement locations (in this case six) is optimised here, as published by the authors.¹⁴

In order to optimise the locations of the six pressure measurements over a wide frequency range, an optimisation scheme has been developed in MATLAB. This tests every possible combination of azimuthal angles with six measurement locations equally split between planes at two axial locations $x A, x B$ (for ease of construction). The aim of the optimisation is to find the azimuthal configuration which minimises the condition number $κ [M]$ at the high end of the frequency range of interest (2 kHz). The condition number quantifies how independent the six measurements are; a lower condition number implies a higher independence of the measurement set. During the first phase of the optimisation, the axial spacing between the planes is fixed, that is

x i = {x A, i = I - III x B, i = IV - VI

(13)

In this case, the optimisation process for the azimuthal locations can be represented mathematically as

min θ i \in [0, 2 π] κ [M (x B - x A, θ i, ω)]

(14)

assuming that the ideal axial spacing would be close to the theoretical optimum for the plane-wave only case corresponding to a quarter of the wavelength,¹⁸ that is

x B - x A = λ 2 kHz / 4

. Afterwards, this assumption was relaxed by fixing the ideal azimuthal distribution and then varying the axial spacing between zero and half wavelength spacing

min x B - x A \in [0, \frac{λ 2 kHz}{2}] κ [M (x B - x A, θ i, ω)]

(15)

The ideal azimuthal configuration was found to be equal angular spacing. However, the optimum axial spacing was found to be higher than the quarter wavelength ideal for the plane-wave region (Figure 7(a)). The condition number for this ideal arrangement at lower frequencies is shown in Figure 7(b).

The peak in condition number visible in Figure 7(b) around 1.3 kHz corresponds to the cut-on frequency for the first azimuthal mode and is due to the axial wave number being close to zero. The peak level at this (almost) discrete frequency is low enough to not lead to large errors.

The theory and optimised microphone set-up requires experimental verification. On each side a seventh microphone is positioned at some arbitrary far-field location $(R d, x VII, θ VII)$ . The spectrum is calculable based on the modal amplitudes from the six-microphone array

\overset{\land}{S} VII = | \overset{\land}{p} VII | 2 = | \frac{\overset{\land}{p} VII}{\overset{\land}{r}} | 2 \overset{\land}{S} rr

(16)

where

\overset{\land}{p} VII

is given by equation (8) as a function of location, mode amplitudes and reference signal

\overset{\land}{r}

. If the mode amplitudes are correct this should match the actual measurement at the verification location.

Initial tests carried out using a loudspeaker excited with random noise show excellent agreement at the verification location. The first microphone is used as the reference signal in equation (16). The problem of high flow noise on the outlet side of the fan is alleviated with the flow straightener in place – its effect on the outlet source power is discussed in ‘Effect of flow straightener’ section. Hydrodynamic pressure fluctuations due to bulk flow unsteadiness and the boundary layers are localised phenomena which do not correlate well between each microphone and the reference microphone – thus they are attenuated when taking the cross-correlations in equation (11). A comparison of predicted (equation (16)) and actual measurements on both sides of the production fan is shown in Figure 8. On the inlet side, the agreement (for tones and broadband) is excellent apart from at amplitudes below the lower end of the dynamic range of the microphones (∼40 dB) and this indicates lower limit of the arrangement as a whole at this operating point. Furthermore, the importance of accurately capturing non-plane-wave modes is highlighted as the plane-wave only prediction deviates massively from the measured value above 1.3 kHz. The agreement on the outlet side is even better as the amplitudes are higher.

The first significant tone above the plane-wave frequencies occurs at around 1.7 kHz. This frequency, a harmonic of the rotational speed of 140 Hz, is investigated in detail in ‘Two port source analysis with higher-order modes’ section, including a comparison to the level measured in the Dyson ISO rig in ‘Fan narrow-band comparison’ section.

Two port source analysis with higher-order modes

Any linear source such as a fan within a duct with openings or ‘ports’ can be modelled as a two-port source using a system of equations which relate its input and output states. This situation is illustrated in Figure 9 for a given mode. The so-called ‘scattering matrix’ formulation relates the pressure wave amplitudes on the inlet and outlet sides of the source, and was introduced by Davies.¹⁹ It allows the waves due to the source alone (denoted by superscript ‘s’ in the figure) to be discerned from the total wave amplitudes travelling each way on either side of the source (denoted $A 1 \pm & A 2 \pm$ ). At lower frequencies, only the plane-wave mode can propagate on each side of the source which results in 2 simultaneous equations

where the

S

is the scattering matrix,

A s

is a source vector and the sign convention for the wave directions of travel is shown in Figure 9. This scattering matrix contains four parameters characterising the reflective (ρ) and transmissive (τ) properties of the source duct element. Extension to the first three modes gives 6 simultaneous equations and 36 unknowns in the scattering matrix

[A 1_{0, 0}^{+} A 1_{1, 0}^{+} A 1_{- 1, 0}^{+} A 2_{0, 0}^{+} A 2_{1, 0}^{+} A 2_{- 1, 0}^{+}] ︸ A + = [ρ 1 0, 0 \to 0, 0 \dots τ 21 - 1, 0 \to 0, 0 : : τ 12 0, 0 \to - 1, 0 \dots ρ 2 - 1, 0 \to - 1, 0] ︸ S [A 1_{0, 0}^{-} A 1_{1, 0}^{-} A 1_{- 1, 0}^{-} A 2_{0, 0}^{-} A 2_{1, 0}^{-} A 2_{- 1, 0}^{-}] ︸ A - + [A 1_{0, 0}^{s} A 1_{1, 0}^{s} A 1_{- 1, 0}^{s} A 2_{0, 0}^{s} A 2_{1, 0}^{s} A 2_{- 1, 0}^{s}] ︸ A s

(18)

where the terms have a degree of symmetry about the leading diagonal coefficients (corresponding to a proportion of reflection without a change in the mode number). For example, the coefficient

τ 12 0, 0 \to - 1, 0

corresponds to transmission of a wave from side 1 to 2 in which the mode number changes (from 0 to −1). Dividing through by a coherent reference signal

\overset{\land}{r}

gives a form where the amplitudes follow from mode decomposition on sides 1 and 2 using equation (11), valid for random signals

[\frac{A 1_{0, 0}^{+}}{\overset{\land}{r}} : \frac{A 2_{- 1, 0}^{+}}{\overset{\land}{r}}] = S [\frac{A 1_{0, 0}^{-}}{\overset{\land}{r}} : \frac{A 2_{- 1, 0}^{-}}{\overset{\land}{r}}] + [\frac{A 1_{0, 0}^{s}}{\overset{\land}{r}} : \frac{A 2_{- 1, 0}^{s}}{\overset{\land}{r}}]

(19)

Note that as in equation (11) the vector components of the above equation represent averaged statistical quantities. Using an external source such as a loudspeaker, the scattering matrix $S$ is found experimentally by producing sound fields in the duct of much higher amplitude and uncorrelated with the source vector $A s$ , which becomes effectively zero. Once $S$ is known, the external sources are deactivated and the source vector can be found based on mode decomposition to give $A \pm$ since all other quantities are known in equation (19) – this is done in ‘Fan narrow-band comparison’ section.

The number of sound fields required increases with the number of cut-on modes. The simpler plane-wave frequency two-port source data is shown next (as done previously by other authors^17,20) followed by the extension to higher-order modes for the first tone above plane-wave frequencies (a harmonic of the rotational speed).

Scattering matrix measurement

Plane-wave frequency range

For the plane-wave frequencies, modelled using equation (17), the four unknown parameters which make up the scattering matrix are found with two independent sound fields $I, II$ which each give two equations

[↑ \frac{A +}{\overset{\land}{r}} I ↓ ↑ \frac{A +}{\overset{\land}{r}} II ↓] = [ρ 1 τ 21 τ 12 ρ 2] ︸ S [↑ \frac{A -}{\overset{\land}{r}} I ↓ ↑ \frac{A -}{\overset{\land}{r}} II ↓]

(20)

as each sound field gives a vector of amplitudes. The external loudspeakers are driven using random noise to cover a range of frequencies at high enough amplitude to give high coherence between the microphones and the reference loudspeaker voltage signal. The duct surface was perforated to enable sound to be emitted into the duct without disturbing the flow. On iteration I, only the speakers on the inlet side are activated and a measurement is taken. For iteration

II

, this is repeated but with only the outlet side activated. The scattering matrix is then determined by inverting the matrix to the right of

S

in equation (20).

Figure 10 shows the measured plane-wave reflection and transmission parameters for the production fan. In the absence of flow, the reciprocity principle implies that the transmission coefficients should be equal. Due to the low Mach number of the flow, it is evident from the figure that there are only small deviations from this principle. Transmission through, and reflection at the source clearly occurs and contributes to the measured waves travelling away from the fan. Furthermore, it can be seen that transmission decreases smoothly with increasing frequency to a low level.

Calculation of the source power once the scattering matrix is known is shown in ‘Plane-wave frequency range’ section for these frequencies.

Fan tone frequency above the plane-wave range

When including the first three modes for which there are six unknown modal amplitudes contained in the vectors $A \pm$ (equation (18)), the scattering matrix $S$ contains thirty six parameters in a six-by-six matrix. At least six independent sound fields are required to solve for the matrix $S$ (i.e. $2 N$ sound fields for N modes). By varying the location and phases of the active external speakers, several independent sound fields could be generated in the duct. The uncorrelated parts of the measured pressure signals are suppressed using the loudspeaker driving voltage as the reference signal and averaging over many spectra using Welch’s method.²¹

The use of single frequency sine wave excitation gives a high coherence with all microphones. This is particularly important when measuring a source with very low transmission for which it is difficult to achieve high coherence between the loudspeaker signal and a microphone measuring on the opposite side of the source. The disadvantage of exciting one frequency at a time is that it takes longer to cover a broad range although a frequency step of 5 Hz has given sufficient accuracy/execution time in the past.¹⁷

Figure 11 shows the 90° azimuthal spacing of the array speakers. To preferentially excite the first azimuthal mode, the speakers were excited with a 90° phase shift between the driving signals as summarised in combination 1 in Table 2 along with the two other combinations. Six sound fields were generated by exciting the inlet and outlet arrays in turn.

Table 2.

Phase settings of the speaker excitation voltages to preferentially excite different modes.

	Relative phase of speaker
Combination	$\tilde{V} 1$	$\tilde{V} 2$	$\tilde{V} 3$
1	0	+90	+180
2	0	−90	−180
3	0	0	0

The mode amplitudes from each excitation set form a vector in the matrices of mode amplitudes (from equation (19) with $A s / \overset{\land}{r} = 0$ )

[↑ \frac{A +}{\overset{\land}{r}} I ↓ \dots ↑ \frac{A +}{\overset{\land}{r}} VI ↓] = S [↑ \frac{A -}{\overset{\land}{r}} I ↓ \dots ↑ \frac{A -}{\overset{\land}{r}} VI ↓]

(21)

from which

S

can be found through an inversion as before

.

It is difficult to excite modes independently using the speaker installation described above. This is because of several reasons:

The sound from the speakers enters the duct through perforations. This yields a distributed source and so we do not have excitation at discrete point sources that are 90° out of phase required for excitation of a pure spinning mode.

For simplicity, there is no fourth speaker at the bottom needed to complete the symmetry of the excitation.

This resulted in some level of dependence between sets. This problem is solved by using an additional speaker at a different axial location to give another data set and equation (21) is solved as an overdetermined system.

At the tone frequency of interest, the scattering matrix is determined for the production fan operating at its design point. The magnitudes of the reflection and transmission coefficients in the scattering matrix are given in Table 3. As at plane-wave frequencies, in general the transmission through the fan is low while reflection is relatively high, particularly at the inlet. In each of the four quadrants of the table the largest values are seen on the leading diagonal which corresponds to reflection/transmission without mode number change.

Table 3.

Scattering matrix $S$ data at 1.7 kHz for production fan in datum operation.

	Reflected as:			Transmitted as:
	(0,0)	(1,0)	(−1,0)	(0,0)	(1,0)	(−1,0)
Incoming (0,0) to Side 1	0.75	0.20	0.36	0.03	0.04	0.01
Incoming (1,0) to Side 1	0.12	0.84	0.19	0.01	0.06	0.01
Incoming (−1,0) to Side 1	0.10	0.04	0.91	0.02	0.00	0.05
Incoming (0,0) to Side 2	0.69	0.08	0.14	0.05	0.02	0.01
Incoming (1,0) to Side 2	0.06	0.57	0.04	0.01	0.04	0.02
Incoming (−1,0) to Side 2	0.04	0.13	0.56	0.01	0.01	0.05

Ideally the flow straightener located between the fan outlet and microphone array should not affect the measured fan characteristics. It is expected that the presence of the flow straightener may produce additional noise (as noted in the standard⁴) or have an effect on higher-order modes. This is analysed in the next section.

Calculation of source power once the scattering matrix is known is shown in ‘Fan tone frequency above plane-wave range’ section for the tone at this frequency.

Effect of flow straightener

The acoustic effect of the honeycomb flow straightener (pictured in Figure 5) is investigated by characterising its passive properties in isolation without the fan or flow. Ideally this should have very low reflection and high transmission. Measurements at plane-wave frequency scattering matrix show almost complete transmission and are omitted here. One would expect that any effect of the straightener will be more pronounced when higher-order modes begin to propagate. This is because the honeycomb passages are effectively ducts of much smaller diameter through which only plane-waves will propagate unattenuated at frequencies with a wavelength greater than the diameter. For the tone frequency of interest, the scattering matrix data is shown Table 4.

Table 4.

Scattering matrix $S$ data for the honeycomb flow straightener at 1.7 kHz.

	Reflected as:			Transmitted as:
	(0,0)	(1,0)	(−1,0)	(0,0)	(1,0)	(−1,0)
Incoming (0,0) to Side 1	0.12	0.03	0.23	0.90	0.18	0.06
Incoming (1,0) to Side 1	0.03	0.30	0.04	0.05	0.84	0.02
Incoming (−1,0) to Side 1	0.05	0.01	0.33	0.03	0.03	0.79
Incoming (0,0) to Side 2	0.06	0.11	0.02	0.95	0.02	0.11
Incoming (1,0) to Side 2	0.06	0.40	0.05	0.03	0.83	0.03
Incoming (−1,0) to Side 2	0.01	0.02	0.36	0.04	0.05	0.80

It is clear from Table 4 that azimuthal mode (±1,0) transmission is marginally less complete than that of the plane-wave mode. This is coupled with the higher reflection coefficients for the azimuthal modes relative to plane-wave. By considering the sum of the square of the magnitudes of the reflection coefficients, it can be seen that only a very small proportion of the acoustic power is reflected by the honeycomb. For the plane-wave mode which contributes most to the power, at most 7% is reflected and this would only change the power prediction for this mode on the order of 0.3 dB. For the azimuthal modes, at most 17% is reflected and this would have an effect on the order of 0.8 dB for these modes. Consequently, the outlet side sound power at this frequency is expected to be accurate to around ±1 dB due to the presence of the honeycomb flow straightener.

Fan narrow-band comparison

Once the scattering matrix is known for a given frequency or frequency range, the source vector in equation (19) is calculable. The quantities of interest, such as the magnitudes of the mode amplitudes are found using averaged quantities for example

| A_{0, 0}^{s} | 2 = | \frac{A_{0, 0}^{s}}{\overset{\land}{r}} | 2 \overset{\land}{S} rr

(22)

where

\overset{\land}{S} rr

is the auto spectra of the reference signal. Calculation of the source sound power, taking into account higher-order modes, requires integration of the axial intensity field over a cross-section. The time-averaged axial intensity as a function of the acoustic pressure

\overset{\land}{p}

(from equation (7) with

A_{m . n}^{+} = A_{m . n}^{s}

A_{m . n}^{-} = 0

) and axial acoustic velocity

\overset{\land}{u}

(related to

\overset{\land}{p}

from momentum equation) is given by

\bar{I} x (r, x, θ, ω) = \frac{1}{2} Re (\overset{\land}{p} \overset{\land}{u} *)

(23)

This is integrated over the cross-section area S to find the sound power

P s = \int \int^\bar{I} x d S

(24)

It can be shown that the cross terms in the integration involving products of the amplitudes of different modes do not contribute to sound power. Equation (24) can therefore be written for a given mode as

P_{m . n}^{s} = \frac{Sk m, n}{2 k 0 ρ 0 a 0} | A_{m . n}^{s} | 2

(25)

where the overall net sound power is found by adding the contribution from each mode, that is

P s = \sum m, n P_{m, n}^{s}

Plane-wave frequency range

For the plane-wave frequency range for which $k m, n = k 0$ equation (25) reduces to

P s = \frac{S}{2 ρ 0 a 0} | A s | 2

(26)

with the subscripts ‘0,0’ omitted. This can be rearranged into a formula relating sound power to sound pressure level

PWL = 10 \log (\frac{P s}{P ref}) = 20 \log \frac{| A s | 2}{p ref} + 10 \log S - 10 \log (\frac{ρ 0 a 0}{(ρ a) ref}) = SPL rms + 10 \log S - 10 \log (\frac{ρ 0 a 0}{(ρ a) ref})

(27)

since

P ref

is 1 pW,

p ref

is 20 µPa and

(ρ a) ref = p_{ref}^{2} / P ref

. This is the so-called plane-wave formula relating sound power and pressure applied in the ISO standard method for all frequencies.^4,7 In the anechoic rig, a microphone directly measures pressure fluctuation amplitude

| A s |

equivalent to root mean square sound pressure level

SPL rms

, independent of the transverse location. This suggests that the ISO rig measurements should be exact in terms of both broad and narrow-band of the fan spectra with the new rig.

To account for the slightly different duct areas, sound power is compared instead of sound pressure in Figure 12. The random nature of broadband noise means that use of power spectral density (instead of power spectrum) is appropriate to remove dependence on the frequency bin Δf. The comparison shows that the broadband levels are in very good agreement at most frequencies – the tonal components vary since their levels do depend on Δf. Above 800 Hz for the ISO rig measurements, the amplitude fluctuates by approximately ±2 dB due to some kind of reflection phenomena. This is likely to affect the accuracy of narrow-band measurements.

Fan tone frequency above plane-wave range

A key interest of the present work is to understand how best to measure a tone above the plane-wave frequency. In the standard, the plane-wave formula (equation (27)) is used to find the sound power based on the linear average of at least three $SPL rms$ measurements at different azimuths.

The reference signal (for equation (19)), which is correlated with the rotational speed harmonic tone at 1.7 kHz, is taken from the motor controller tachometer – coherence values are shown in Table 5. As a guide, the ISO standard states that the turbulent pressure fluctuations should be at least 6 dB lower than the acoustic fluctuations in which case the coherence must be greater than 0.64 (for plane-wave frequencies).⁴

Table 5.

Magnitude squared coherence values with 256 spectra averages.

	Average coherence estimate with reference signal across all 6 measurements
Freq.	Inlet	Outlet
1.7 kHz	0.94	0.86

The magnitudes of the mode amplitudes on the inlet and outlet

A_{m . n}^{s}

are found using equation (22) and the power from equation (25). A modal breakdown of the power on the inlet and outlet sides for the tone of interest is given in Table 6. The data shows that the power is not equally distributed between each mode at this frequency, particularly on the outlet side where the plane-wave mode dominates.

Table 6.

Source data for each mode at 1.7 kHz tone from production fan in datum operation.

	Power (dB re. 1 pW)
	Inlet, $P 1_{m . n}^{s}$ ( $A 1_{m . n}^{s}$ )	Outlet, $P 2_{m . n}^{s}$ ( $A 2_{m . n}^{s}$ )
Plane-wave mode (0,0)	24.9 (42.9)	37.3 (55.3)
Azimuthal mode (1,0)	18.2 (38.7)	25.6 (46.1)
Azimuthal mode (−1,0)	20.6 (41.1)	26.9 (47.5)
Total	26.9	37.9 ± 1 dB
SPL from plane-wave formula	44.9	55.9

Equivalent SPL according to the plane-wave formula is calculated with equation (27).

The accuracy of taking SPL measurements at $r / R d = 0.5$ to represent the tone sound power level via the plane-wave formula (equation (27)) but above the plane-wave range is now explored. Based on the decomposed source modes which exist under anechoic conditions, Figures 13 and 14 show the transverse variation of the predicted SPL on the inlet and outlet sides, respectively.

Comparable measurements are performed in the Dyson ISO rig (‘Dyson ISO rig’ section) with the microphones located mid-way between the duct axis and wall in accordance with the standard. The microphones (B&K 4189 ½-inch) have a diameter which is almost 20% of the duct radius and so effectively measure the area-average of sound pressure over its diaphragm.

Table 7 shows variations in the measured SPL at different angles with a peak of up to 8 dB difference between the largest and smallest SPL on the inlet side. These large variations can be linked to the highly position-dependent inlet sound field shown in Figure 13 existing under anechoic conditions. A difference of almost −6 dB is observed between the tone inlet sound power levels measured in each rig. On the outlet side, both Table 7 and Figure 14 agree that the sound field SPL varies less appreciably with traverse position which leads to better agreement between the power levels determined in each rig. This is due to a more dominant plane-wave mode which carries a greater of proportion of the power, as shown in Table 6.

Table 7.

Fan tone at 1.7 kHz measured in Dyson ISO rig for production fan in datum operation.

	Inlet	Outlet
$SPL rms$ @ 0°	38.9	50.0
$SPL rms$ @ 120°	34.0	53.5
$SPL rms$ @ 240°	42.8	50.1
$SPL rms$ average	39.3	51.3
$PWL ISO rig$ from plane-wave formula (equation (27))	21.2	33.2
$PWL ISO rig - PWL new rig$	−5.7	−4.7

Conclusions

A new method and rig has been developed to measure fan broadband and tonal noise which gives deeper insight into the source acoustic characteristics than existing methods. The aeroacoustic rig at Dyson based on the ISO standard is designed to measure overall sound power levels in third-octave bands. A principal interest here is to measure deterministic (tonal) noise accurately. The ISO standard makes assumptions primarily applicable to broadband, random noise and hence assumes that it contributes most to overall sound power. The new scheme does not require these assumptions as the modal structure of the source sound field is directly measured (using a two stage process):

The scattering matrix is determined using (tonal) excitation from external loudspeakers with the fan active. These sources, controlled to preferentially excite different modes, elicit a response from the source duct location. The reference signal in the mode decompositions is the loudspeaker voltage which is correlated with this response. The noise from the fan is uncorrelated with the reference and is effectively zero when solving for the scattering matrix.

The external sources are deactivated and a single set of measurements are taken of the duct sound field. Along with the scattering matrix, the mode decomposition of this data allows the source sound to be determined.

For plane-wave frequencies, the narrow-band sound powers in both rigs are in close agreement as a single radius SPL measurement in the ISO method is representative of the section area sound power. This also demonstrates that the new method in which duct terminations are not anechoic can accurately isolate waves coming directly from the source from any reflection/transmission. With the production fan in datum operating, transmission through the fan is very low for frequencies at or above the blade-passing frequency for both the plane-wave and first azimuthal modes.

The most significant tone produced by the fan in datum operation above the plane-wave frequency range is measured in both rigs. The breakdown of mode power shows that the power is not equally distributed, which is expected since only a random, broadband source is likely to excite all modes indiscriminately. When higher-order modes carry significant power the SPL becomes strongly dependent on transverse location in both rigs, and calculation of this dependence, based on source mode amplitudes, show a complex variation pattern. The narrow-band tone sound power level is underestimated using the ISO rig method. The agreement between the two methods is better on the outlet as the plane-wave mode dominates.

There are several key conclusions from the work in this paper most relevant to later work:

Transmission through the fan in datum operation is very low for the blade-passing frequency which suggests (tonal) noise generated by the guide-vanes will propagate more readily to the outlet side while the impeller will most affect the inlet side noise.

The flow straightener introduces an uncertainty of around ±1 dB in outlet measurements.

The Dyson ISO rig is best suited to measure broadband levels. At frequencies above the plane-wave range, inaccuracy may be large especially in the presence of a harmonic source that generates a specific azimuthal mode.

Footnotes

Acknowledgements

The financial and technical support, and collaboration from Dyson Ltd. has made this research possible, and we are particularly thankful to Ryan Stimpson of the aeroacoustic RDD team. The technical support of John Hazelwood at Cambridge is also greatly appreciated.

Declaration of conflicting interests

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

Funding

This work was supported by Dyson Ltd.

References

Desvard L, Hurault J, Hohamed Zamzam MAB, et al. An investigation of the effect of uneven blade spacing on the tonal noise generated by a mixed flow fan. Proceedings of the Fan 2012 International Conference, Senlis, France, 2012.

Tyler JM and Sofrin TG. Axial flow compressor noise studies. Warrendale, PA: SAE International. SAE Technical Paper 620532, 1962.

Sutliff

. Rotating rake turbofan duct mode measurement system. J Acoust Soc Am 2005; 118: 1864.

ISO. ISO 5136 – Acoustics – Determination of sound power radiated into a duct by fans and other air-moving devices – In-duct method, International Organization for Standardization, Geneva, Switzerland, 2003.

Kameier

Neise

. Rotating blade flow instability as a source of noise in axial turbomachines. J Sound Vib 1997; 203: 833–853.

ISO. ISO 5801 – Industrial fans – performance testing using standardized airways, International Organization for Standardization, Geneva, Switzerland, 2007.

Neise

Arnold

. On sound power determination in flow ducts. J Sound Vib 2001; 244: 481–503.

Holste

Neise

. Experimental comparison of standardized sound power measurement procedures for fans. J Sound Vib 1992; 152: 1–26.

ISO. ISO 3746 – Acoustics – determination of sound power levels and sound energy levels of noise sources using sound pressure – survey method using an enveloping measurement surface over a reflecting plane, International Organization for Standardization, Geneva, Switzerland, 2010.

10.

Lavrentjev

Åbom

Bodén

. A measurement method for determining the source data of acoustic two-port sources. J Sound Vib 1995; 183: 517–531.

11.

Seybert

Ross

. Experimental determination of acoustic properties using a two-microphone random-excitation technique. J Acoust Soc Am 1977; 61: 1362–1370.

12.

Gerhold C, Cabell R and Brown M. Development of an experimental rig for investigation of higher order modes in ducts. 12th AIAA/CEAS Aeroacoustics Conference (27th AIAA Aeroacoustics Conference), 2006.

13.

Lavrentjev

Abom

. Characterization of fluid machines as acoustic multi-port sources. J Sound Vib 1996; 197: 1–16.

14.

Newman

Agarwal

Dowling

. A six sensor method for measuring acoustic properties in ducts. Proc Meet Acoust 2013; 19: 015085.

15.

Morse

. Theoretical acoustics, Princeton, NJ: Princeton University Press, 1968.

16.

Bendat

Piersol

. Engineering applications of correlation and spectral analysis / Julius S. Bendat, Allan G. Piersol, New York: Wiley, 1980.

17.

Holmberg

Åbom

Bodén

. Accurate experimental two-port analysis of flow generated sound. J Sound Vib 2011; 330: 6336–6354.

18.

Bodén

Åbom

. Influence of errors on the two‐microphone method for measuring acoustic properties in ducts. J Acoust Soc Am 1986; 79: 541–549.

19.

Davies

POAL

. Practical flow duct acoustics. J Sound Vib 1988; 124: 91–115.

20.

Åbom

. A note on the experimental determination of acoustical two-port matrices. J Sound Vib 1992; 155: 185–188.

21.

Welch

. The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. Audio Electroacoust IEEE Trans On 1967; 15: 70–73.