On the Sound Production by Vortex Whistles

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On the Sound Production by Vortex Whistles

Ulf Kristiansen, Muriel Amielh, Daniel Mazzoni

To cite this version:

Ulf Kristiansen, Muriel Amielh, Daniel Mazzoni. On the Sound Production by Vortex Whis- tles. E-FORUM ACUSTICUM 2020 - 9EME EDITION, Dec 2020, Lyon, France. pp.1527-1534,

�10.48465/fa.2020.0775�. �hal-03215259�

ON THE SOUND PRODUCTION IN VORTEX WHISTLES

Ulf Kristiansen ¹ Muriel Amielh ² Daniel Mazzoni ²

1 Acoustics Research Centre, Department of Electronic Systems, Norwegian University of Science and Technology, 7491 Trondheim, Norway

2 Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE UMR 7342, 13384, Marseille, France

ulf.kristiansen@ntnu.no, amielh@irphe.univ-mrs.fr

ABSTRACT

Some first measurements (section 2) agree with earlier findings that the frequency of the produced sound increases linearly with the velocity of the flow into the whistle, and identifies by directivity and acoustic power measurements the sound source to be of the dipole type. Considering the balance equation of the angular momentum in the flow, theory is developed for predicting the generated frequency (section 3). It is shown that the interior friction is of im- portance here. PIV (Particle Image Velocimetry) together with hot wire anemometers and microphones/hydrophones are used to study the flow close to the exit opening (sec- tions 4 and 5). The oscillatory velocities in the exit plane are used to predict the radiated sound pressure levels.

1. INTRODUCTION

The main characteristic of a vortex whistle is that it pro- duces a pure tone sound proportional to the amount of fluid flowing through it. The normal design is such that a flow enters a circular volume tangentially (”I”), spirals up in the core (”C”), and exits with an increased rotation rate through a cylinder of narrower bore (”O”), see Fig- ure 1a. The whistle was first described in the literature by B. Vonnegut in 1954 [1]. He documented the above re- lationship between frequency and flow and found that it works equally well in water as in air. R. Chanaud did fur- ther experimental studies on the whistle (influence of cav- ity dimensions and exit tube length) as well as qualitative flow observations in water [2, 3]. Chanaud regarded the sound source to be a rotating dipole. The flow exits the pipe as a swirling annulus with a rotation corresponding to the frequency of the sound. Vortex whistles have seen practical applications as flow meters [6] and more recently as spirometers [7].

2. CHARACTERISATION OF THE VORTEX WHISTLES

2.1 Geometry of the vortex whistles

Vortex whistles are presently investigated in air and in wa- ter, with a glass and a plexiglas bottle respectively, where

the geometries are slightly different, see Fig. 1 and Tab. 1

Geometry air water

Inlet diameter, D _i = 2R _i 14mm 15mm Outlet diameter, D _o = 2R _o 19mm 20mm Oulet pipe length , L o 65mm 77mm Core diameter, D c = 2R c 54mm 50mm

Core length, L c 105mm 88mm

Table 1. Dimensions of the glass (for air) and plexiglas (for water) vortex whistles.

2.2 Frequency vs flow

Figure 2 shows how the frequency linearly increases vs.

velocity both for use in air and water. For air, the centre- line velocity at the entry was measured using a small Pitot tube and then adjusted to the bulk flow velocity through the exit tube while the whistling frequency was deduced from sound measurements (see 2.3). For water, a flowmeter (KDG Houdec, rotameter 134-18 M12-I) gave the flowrate while a hydrophone (Aquarian Scientific AS-1) was used to determine the whistling frequency.

2.3 Acoustic power

Figure 3 shows the acoustic power generated by the glass

whistle in air as function of the projected rotational veloc-

ity at the rim. This velocity is simply related to frequency

as W _rot−rim = 2πR ₀ f . Note the power of 6 dependency

which is a characteristic of dipole type sources. The whis-

tle was positioned in a reverberant room having an out-

let for pressurised air. The sound pressure was measured

by moving a handheld Norsonic 140 sound level analyser

around in the room (random walk) using an integration

time of 60 seconds and 1/3 octave filtering. The whistling

frequencies, which appear as pure tones, were adjusted (by

controlling the air flow) to be close to the centre frequen-

cies of the 1/3 octave bands. The levels were calibrated

by first measuring the output of a reference sound source

(Bruel & Kjaer Type 4204) using the same measuring pro-

cedure.

(a) (b)

Figure 1. Whisling bottle, (a) Glass vortex whistle used in air, (b) Coordinate systems used for the whistling frequency model.

(a) (b)

Figure 2. Vortex whistling frequency vs. flow rate : experiments and prediction models, (a) Glass vortex whistle used in air, (b) Plexiglas vortex whistle used in water.

2.4 Sound directivity

The directivity of the sound source was measured at two frequencies in an anechoic chamber. The values were ob- tained by a Norsonic sound analyser as the equivalent level (one minute integration time) in the third octave band cov- ering the pure tone. The directivities are seen in figure 4 where the measured dB values are converted to absolute levels in Pascal. The measurement distance was 0.75m, and zero degrees correspond to a position on the axis on the outflow side of the whistle. The plots show mini- mum levels on the whistle axis and maximal values at po- sitions at 90 degrees , i.e. in the plane of the exit opening.

The 400Hz line has been multiplied by a factor of 52.73 ((1500/400) ³ ) to be compared with the 1500Hz one based on the assumption that the pressure increases as the cube of frequency.

2.5 The exit flow

The PIV technique, see section 4, allows us to study the

flow behaviour in the exit region. The flow exits the out-

let pipe as a rotating jet. Figure 5 shows snapshots of the

in-plane velocity at the opening. The frequency is 400Hz,

and a zoom is done to observe the location of the highest

velocities. Figure 6 presents the development of the mean

flow in the near field. The presence of a non-centered pre-

cessing vortex (PVC) at the bottle nozzle induces a non-

axisymmetric distribution of the mean velocity magnitude

different from configurations where two lateral inlets in-

sure the axisymmetry of the mean flow [8]. A central re-

circulation zone (CRZ), which extends in the central part is

related to the high swirling level, since here for a whistling

frequency at 400Hz we estimate W _rot−rim /U _bulk > 2.5 .

Figure 3. Acoustic Power vs. projected rim velocity.

Figure 4. The sound pressure (rms values in Pascal) mea- sured at 1500Hz (blue lines) and 400Hz (yellow lines) at different angles. Zero degree corresponds to measure- ments on the axis on the outflow side. The 400Hz values have been multiplied by a factor of 52.73 ((1500/400) ³ ) to be displayed on the same graph.

3. MODEL FOR WHISTLING FREQUENCY PREDICTION

3.1 Conservation of angular momentum

The objective of this section is to demonstrate how the whistling frequency can be predicted by considering the conservation of the angular momentum. Two successive estimations of the whistling frequency are made. In the first one, friction losses are neglected, while in the second one the friction due to the rotating movement inside the bottle is taken into account. Calculations are hereafter de- veloped using the different coordinate systems defined in figure 1b.

Figure 5. Snapshots of the in-plane velocity (PIV) at the opening: highlighting of the rotating jet velocity W _rot−rim for a whistling frequency at 400Hz, glass bottle in air.

Figure 6. Velocity vectors and magnitude (PIV) showing the mean outflow at 400Hz, glass bottle in air.

The balance of the angular momentum is written as : Z

Ω

−−→ OM ∧ ∂ _t (ρ~ v) dΩ

| {z }

= 0 (stationary) +

Z

∂Ω

ρ −−→

OM ∧ ~ v (~ v · ~ n) dσ

| {z }

Convected moment

= Z

Ω

ρ −−→

OM ∧ ~ g dΩ

| {z }

0

+ Z

∂Ω

−−→ OM ∧

− → f

^s

z }| { σ(~ n) dσ

| {z }

losses by friction

(1)

where Ω is the interior volume of the vortex whistle and

∂Ω the closure of Ω, dΩ and dσ are volume and surface elements, σ the friction tensor and − →

f ^s the force exerted on the wall decomposed in a normal component: the pressure and a tangential component : the friction. In a first step, the friction force is neglected ( − →

f ^s = 0).

With ∂Ω = S _i + S _o , using inlet and outlet sections of the bottle, and by considering the velocity vector ~ v⊥~ n on the wall of the bottle , the balance equation(1) is reduced to:

(1) = ⇒ Z

S

i

+S

o

ρ −−→

OM ∧ ~ v (~ v · ~ n) dσ = 0 (2) where ~ n is the external normal to the domain Ω: on S i

~

n = − e ~ _x and on S _o ~ n = e ~ _z .

(1) = ⇒ Z

S

_i

ρ −−→

OM ∧~ v _i (~ v _i · ~ n) dσ = Z

S

_o

ρ −−→

OM ∧ v ~ _o ( v ~ _o · ~ n) dσ (3) where v ~ _i and v ~ _o are the flow velocities in the inlet sec- tion S _i and oulet section S _o , respectively. The previous equation is projected on e ~ z :

(1) = ⇒ e ~ _z · Z

S

_i

ρ −−→

OM ∧ v ~ _i (~ v _i · e ~ _x ) dσ

| {z }

σ

_i

= e ~ z · Z

S

_o

ρ −−→

OM ∧ v ~ o ( v ~ o · e ~ z ) dσ

| {z }

σ

_o

(4)

3.2 σ i calculus

In the cartesian coordinate system (O ⁰ , ~ e x , ~ e y , ~ e z ) associ- ated to the inlet section S i , the flow velocity is equal to

~

v _i = v _i (y, z) e ~ _x and v ~ _i · e ~ _x = v _i (y, z).

By considering any point M (0, y, z) of the surface S i and the point O (R c , R c − R i , −R i ), the coordinates of the vector −−→

OM are:

−−→ OM





−R c

y + R _i − R _c z + R _i



 and v ~ _i (~ v _i · e ~ _x )





v ² _i (y, z) 0 0



 (5) So that:

−−→ OM ∧ v ~ _i (~ v _i · e ~ _x ) = v ² _i (y, z)





0 (z + R _i ) (R _c − R _i − y)



 (6) We deduce:

σ _i = e ~ _z · Z

S

_i

ρ −−→

OM ∧ v ~ _i (~ v _i · e ~ _x ) dσ

= ρ Z

S

i

v _i ² (y, z)(R c − R i − y)dσ

(7)

By using the variable substitution y = r cos(θ), z = r sin(θ), dσ = rdrdθ, S _i = {r ∈ [0, R _i ], θ ∈ [0, 2π]}.

and by symetry, one considers v i (r, θ) = v i (r) and writes:

σ i = ρ Z R

_i

0

Z 2π 0

v ² _i (r)(R c − R i − r cos(θ))r dθ dr (8)

σ i = ρ Z R

_i

0

v ² _i (r)r [(R c − R i ) θ − r sin(θ))] ^θ=2π _θ=0 dr (9) σ i = 2πρ (R c − R i )

Z R

_i

0

v _i ² (r)r dr (10) For calculation simplification, a constant velocity profile is assumed in the inlet section S i : v i (r) = v i = Q

S _i = Q

πR ² _i = constant = bulk velocity in the section S _i , where

Q is the volume flowrate that supplies the bottle. With such assumption, σ i can be simplified to:

σ _i = 2πρ (R _c − R _i ) Q

πR ² _i 2 r ²

2 ^R

i

0

= ρQ ²

πR ² _i (R _c − R _i ) (11) σ i = ρQ ²

πR ² _i (R c − R i ) (12) 3.3 σ o calculus

In the cylindrical coordinate system (O ⁰⁰ , ~ e _r , ~ e _θ , ~ e _z ) asso- ciated to the outlet section S _o , the flow velocity is equal to v ~ o (r, θ) = v ~ o (r) = rω o e ~ θ + v z (r) e ~ z , where ω o is the angular velocity in the outlet section S o and v z (r) is the normal velocity in the outlet section S _o .

For calculation simplification, a constant velocity profile is assumed in the outlet section S o :

v z (r) = v z = Q S o

= Q

πR ² _o = bulk velocity in section S o

(13) For any point M of S o , the associated vector is written as:

−−−→ O ⁰⁰ M = r ~ e r (14)

The vector −−→

O ⁰⁰ O is written as: −−→

O ⁰⁰ O = −L c e ~ z (15) The vector −−→

OM is written as: −−→

OM = r ~ e r + L c e ~ z (16)

~

v o · e ~ z = v z = Q S o

= Q

πR ² _o (17) Then:

−−→ OM ∧ v ~ o ( v ~ o · e ~ z ) = −rω o L c v z e ~ r − rv _z ² e ~ θ + r ² ω o v z e ~ z

(18) The previous equation is projected on the vector e ~ z , so that:

~

e z · −−→

OM ∧ v ~ o ( v ~ o · e ~ z )

= r ² ω o v z (19) And:

(4) = ⇒ σ _o = e ~ _z · Z

S

_o

ρ −−→

OM ∧ v ~ _o ( v ~ _o · e ~ _z ) dσ

= ρω _o v _z Z 2π

0

dθ Z R

o

0

r ³ dr = ρω _o v _z 2π R ⁴ ₀ 4

(20)

We obtain : σ o = ρQR ² _o ω o

2 (21)

3.4 Whisling frequency prediction without friction If for a first step, we neglect the friction force, then :

(4) = ⇒ σ o = σ i

This equality leads to:

(12, 21) = ⇒ Q ²

πR ² _i (R c − R i ) = ω o QR ² _o

2 (22)

As the whisling frequency is related to the angular ve- locity by: f = ω o /(2π), we obtain:

f = Q

(πR i R o ) ² (R _c − R _i ) (23)

3.5 Whisling frequency prediction with friction In the second and more complete approach, the friction forces at the core and the neck walls are taken into ac- count. In particular, friction forces associated to the an- gular movement are considered. With the hypothesis of a uniform circular movement, the relation between the angu- lar velocity ω c in the core of the bottle and ω o , the angular velocity in the exit section, will be given by the following relation:

ω _c = ω _o R ² _o R ² _c = 2Q

π

R _c − R _i

R ² _i R ² _c (24) In order to estimate the pressure losses using the Moody’s diagram ( [9]), a Reynolds number of reference is needed.

For specific pressure losses (as a sharp enlargement), it is usual to consider a Reynolds number based on the bulk ve- locity v m in the upstream smaller section while the dimen- sion of reference remains the larger downstream radius:

Re c = v m R c

ν (25)

where ν = µ/ρ is the kinematic viscosity of the fluid.

We have now to evaluate the friction term − →

f ^s on the bottle walls. The complementary parameter to be determined in order to estimate the Darcy friction factor ξ for pressure losses using the Moody’s diagram is the height of the ru- gosities of the wall. For the plexiglas bottle the relative ru- gosity is estimated to 0.002, whereas the glass bottle have smooth walls. The Fanning friction factor C F is related to the Darcy friction factor ξ : C F = ξ

4 . It is used to ex- pressed the wall shear stress τ p (friction force by surface unit):

τ p = 1

2 ρv ² _m C F (26) where v _m is usually the bulk velocity in the considered pipe.

In the bottle, on lateral walls, we consider that the contri- bution of the circular movement to friction is:

τ p,θ = 1

2 ρv _m ² C F , (27) by considering :

v m ≡ v θ,c = R c ω c and C F = C F

_c

on surface S c

v m ≡ v θ,n = R o ω o and C F = C F

_n

on surface S n

(28) where S _c and S _n are the lateral surfaces of the core and the neck, respectively .

The balance of the angular momentum when friction on surfaces S _c and S _n is taken into account is deduced from the general equation (1):

(1) = ⇒ Z

S

o

+S

i

ρ −−→

OM ∧~ v (~ v · ~ n) dσ = Z

S

c

+S

n

−−→ OM ∧ − → f ^s dσ

(29) with ~ n = − e ~ _x for S _i , and ~ n = e ~ _z for S _o , thus:

Z

S

o

ρ −−→

OM ∧ v ~ o ( v ~ o · e ~ z ) dσ − Z

S

i

ρ −−→

OM ∧ v ~ i (~ v i · e ~ x ) dσ

= Z

S

_c

+S

_n

−−→ OM ∧ − → f ^s dσ

(30)

A projection on e ~ z is applied :

~ e z ·

Z

S

_o

ρ −−→

OM ∧ v ~ o ( v ~ o · e ~ z ) dσ

| {z }

σ

o

− e ~ z · Z

S

_i

ρ −−→

OM ∧ v ~ i (~ v i · e ~ x ) dσ

| {z }

σ

i

= e ~ z · Z

S

c

−−→ OM ∧ − → f _c ^s dσ

| {z }

σ

c

+ e ~ z · Z

S

n

−−→ OM ∧ − → f _n ^s dσ

| {z }

σ

n

(31)

The two first integrals were previously calculated (See for- mulas (12) and (21)), integrals for σ c et σ n remain to be calculated.

We firstly calculate σ _c . In the cylindrical coordinate system(O, e ~ r , e ~ θ , e ~ z ) the surface force − →

f ^s on a lateral wall of the bottle is written as:

− → f ^s





−p

−τ p,θ

−τ _p,z



 (32)

We deduce the expression of the friction force on the sur- face S c :

(26) = ⇒ − → f _c ^s





−p

− ¹ ₂ ρR ² _c ω ² _c C F

_c

−τ p,z



 (33)

For one local point M on the lateral core wall, the associated vector is expressed as: −−→

OM = R _c e ~ _r + z ~ e _z So that: −−→

OM ∧ − → f _c ^s

= (R c e ~ r + z ~ e z ) ∧ −p ~ e r − ¹ ₂ ρR ² _c ω _c ² C F e ~ θ − τ p,z e ~ z

= ¹ ₂ ρzR ² _c ω _c ² C _F

_c

e ~ _r + (R _c τ _p,z − p z) e ~ _θ − ¹ ₂ ρR ³ _c ω ² _c C _F

_c

e ~ _z −−→

OM ∧ − → f _c ^s

· e ~ _z = − 1

2 ρR ³ _c ω ² _c C _F

_c

(34) The surface element dσ is R _c dθdz, then:

σ c = − Z L

c

0

Z 2π 0

1

2 ρR _c ⁴ ω ² _c C F

_c

dθdz = −ρπL c ω ² _c R _c ⁴ C F

_c

(35) By the same way, σ n is calculated:

σ _n = − Z L

o

0

Z 2π 0

1

2 ρR ⁴ _o ω _o ² C _F

_n

dθdz = −ρπL o ω ² _o R ⁴ _o C _F

_n

(36) Thus, the balance equation of the angular momentum in- cluding viscous losses becomes:

(31) = ⇒ σ o − σ i = −ρπ C F

_c

L c ω _c ² R ⁴ _c + C F

_n

L o ω _o ² R ⁴ _o (37) (24) = ⇒ σ o − σ i = −ρπ C F

_c

L c R ⁴ _o + C F

_n

L o R ⁴ _o

ω _o ²

(38)

(12, 21) = ⇒ ρQR ² _o ω o

2 − ρQ ²

πR ² _i (R c − R i )

= −ρπR ⁴ _o (C _F

_c

L _c + C _F

_n

L _o ) ω ² _o

(39)

Finally ω _o is the positive root of the second order equation:

πR ⁴ _o (C _F

_c

L _c + C _F

_n

L _o ) ω ² _o + QR ² _o

2 ω _o − Q ²

πR ² _i (R _c − R _i ) = 0 (40) The whistling frequency with the wall friction consid- eration is deduced using the relation: f = ω o /(2π) after some iterative calculations in order to corretly evaluate the v _m velocities used in the Reynolds number (25,28) :

f = Q s

1 + 16 (C F

_c

L c + C F

_n

L o ) (R c − R i )

R ² _i − 1

8π ² (C _F

_c

L _c + C _F

_n

L _o ) R _o ²

(41) Figure 2 shows the quite good prediction of the whistling frequency when the friction is included in our model, both in air and in water.

4. DYNAMICS OF THE VELOCITY FIELD IN AIR 4.1 PIV setup

PIV images are acquired by a DANTEC Dynamics system that includes a PIV camera FlowSense EO-4M-32 (2072 x 2072 pixels), the software DynamicsStudio that controls synchronization of image acquisition and a pulsed laser (Fig. 7a). A first computer is dedicated to the PIV control.

The pulsed laser is a QuantaRay laser (SpectraPhysics, 200mJ, 10Hz). A laser sheet generator constituted of sev- eral suited lenses is installed at the end of the optical arm so that a flow field of 30 × 30mm ² is illuminated. PIV images are acquired at a 10Hz rate and the time delay be- tween two associated pulses are adapted in agreement with the flow conditions in the range 1 − 5µs. Three types of fields of view are investigated by PIV in order to describe the three-dimensionnal character of the flow at the exit near field of the vortex whistle (Fig. 7c).

A hot film (55R01, DANTEC Dynamics) is located in the bottle exit section (X = 0mm) at the angle 145 ^◦ and 2.4mm from the external edge of the glass bottle (see Fig. 7b and hollow magenta circles in Fig. 10). This hot film is sensitive to the velocity fluctuations associated to the bottle whistling. The hot film was not calibrated, but its voltage signal is acquired and the spectral analysis pro- vides the whistling frequency for each flow condition. The TTL trigger signal generated by the timer box that con- trols the laser pulses is also acquired (channel 1) simulta- neously with the hot film signal (channel 0) on a A/D Na- tional Instruments (NI 9215) at a 25 600 kHz frequency.

To cover the PIV acquisition time required to acquire 1900 PIV fields at 10Hz, a duration of 200s is necessary for the acquisition of the trigger and hot-film signals ( 5 120 000 samples/ channel). A second computer is dedicated to the A/D acquisitions. For the investigated air flow conditions, the whistling frequencies are in the range 400-1600Hz.

Whereas the 10 Hz rate for PIV is too low to catch the

temporal dynamics associated to the whistling, the hot film signal captures the periodicity of the flow. Fig. 8 shows the camera trigger signal (in red). For each descending front of the TTL signal, a pair of images is acquired from which one PIV snapshot is calculated (Fig. 9).

4.2 Phase averaging method

When observing the hot film signal around one TTL de- scending front of the camera trigger, the sinusoidal shape is obvious and allows identifying an associated phase. The phase is calculated by fitting, locally around the TTL de- scending front, the hot film with a sinus function (Fig. 7c).

This identification of the phase is made for each of the 1900 PIV snapshots (j=1,. . . 1900). Then the PIV snap- shots are sorted according to their phase with a 4 ^◦ step between 0 and 360 ^◦ . One can consider that around 1900/(360/4) ≈ 21 PIV snapshots are associated to each phase class. Indeed we associated to each phase ϕ the av- erage of the n PIV snapshots for which ϕ < ϕ j < ϕ + 4 ^◦ . If we call P IV ϕ

_j

the PIV snapshot associated to the ϕ j

phase, then we calculate the phase average PIV field P IV ϕ

associated to the ϕ phase P IV _ϕ = _n ¹ P

ϕ<ϕ

_j

<ϕ+4

^◦

P IV _ϕ

_j

. As the position of the hot film stayed the same when inves- tigating X = 0mm, Y = 0mm and Z = 0mm PIV fields (Fig. 7c), we can deduce for each phase ϕ the correspond- ing triplet (P IV _ϕ,X=0mm , P IV _{ϕ,Y =0mm} , P IV _ϕ,Z=0mm ).

The interesting quantity for acoustic calculations is U _x ⁰ , the fluctuation of the streamwise velocity U x in the X = 0mm plane. This cannot be obtained directly by 2D PIV. Indeed, the velocity components measured in the X = 0mm plane are U _y and U _z according to the Cartesian coordinate sys- tem described in Figure 7c. In order to estimate the phase average Ux velocity field in the exit section X = 0mm, we use the phase averaged U x profile obtained by measure- ments of the PIV field in the plane Z = 0mm on the line (variable Y , Z = 0mm) for each phase ϕ. Then we build the instantaneous U x (r, θ) cartography in the X = 0mm plane in cylindrical coordinates:

U x (r, θ, X = 0mm) = U x (ϕ, Y, Z = 0mm) (42) By considering that each instantaneous velocity compo- nent V (t) is constituted of a mean component V and a fluctuating component V ⁰ (t), we write in the cylindrical coordinate system (O, r, θ, X) :

U _x (t) = U _x + U _x ⁰ (t) U r (t) = U r + U _r ⁰ (t) U θ (t) = U θ + U _θ ⁰ (t)

(43)

Where t (in the time period domain 0-T) is equivalent to the

phase ϕ (in the phase domain 0 − 360 ^◦ ). An example of

the fluctuating velocity fields (U _x ⁰ , U _r ⁰ , U _θ ⁰ ) reconstructed

by phase averaging , in the exit section at X = 0mm, for

whistling frequency at 400Hz is given for the phase 132 ^◦

in Figure 10.

(a) (b) (c)

Figure 7. PIV setup in air, (a) Camera, laser sheet, hot film probe and vortex whistle, (b) hot film location, side view, (c) investigated planes by PIV.

Figure 8. Synchronized hot film and TTL camera trigger signals.

5. ACOUSTICS PRESSURE CALCULATION WITH RAYLEIGH INTEGRAL

5.1 Fluctuating velocity field

As the rotating velocity field repeats itself after one acous- tic period it was tempting to regard the exit opening as a vibrating membrane with each point having a periodic os- cillation. By analogy with a vibrating membrane, sound radiation can be calculated by the Rayleigh integral [5].

The integral requires the normal direction accelerations and ideally that the source is positioned in a baffle. At the low frequencies in question (long wavelengths com- pared to source diameter) the lack of a baffle reduces the calculated sound levels by 6dB, [4] The oscillating veloc- ity amplitudes were measured by the technique described above and are shown in Figure 11. The velocity field was considered axisymmetric (if time averaged) and harmonic for the calculations.

Figure 9. Phase identification for one acquired PIV snap- shot.

5.2 Rayleigh integral

In the time domain, a Rayleigh integral can in general be written as

p(x, t) = ρ 0

2π Z

S

a _n (t − R/c)/R dS

This relates the pressure at a position x and at time t to the normal acceleration at an element a distance R away at the earlier time t − R/c, where c is the speed of sound.

The pressure was calculated in the exit plane, where the pressure will be a maximum and, at a distance of 1m from the centre of the opening. The symbols are sketched in Figure 12.

The normal (axial) velocity can be approximated as:

U x (t, r s , θ) = U x (r s , θ) + u ⁰ _x (r s )sin(ωt − θ) (44) with u ⁰ _x (r _s ) = √

2U _rms

_x

(r _s ), and the normal acceleration:

a _x (t, r _s , θ) = ωu ⁰ _x (r _s )cos(ωt − θ) (45) Knowing the velocity distribution, the Rayleigh integral takes the form

p(r, t) = ρ 0

2π Z

S

ωu ⁰ _x (r s )cos(ω(t − R

c ) − θ) dS

R (46)

Figure 10. Oscillating velocities reconstructed by phase averaging, in exit section X=0mm, for whistling frequency at 400Hz.

Figure 11. Oscillating velocities at 4 different frequencies.

The integration was replaced by a summation of acceler- ations representing small area segments. The circle was replaced by 10 rings of equal widths where each ring again was divided by 1,3,5,7 etc parts in the θ direction for each quadrant. This ensures sub elements of equal size and close shapes. The acceleration a _x must be a representative (mean) value for a given surface element dS. The magni- tude of the acceleration is calculated as ωu ⁰ _x . Figure 13 presents values measured in an anechoic chamber for dif- ferent 1/3 octave center-frequencies (the stars) and 4 val- ues based on the Rayleigh integral (the circles). The ex- perimental values were obtained by adjusting the frequen- cies to the correct values and then noting the SPLvalues in the corresponding frequency band. The line shows a ’fre- quency cubed’ relationship.

Acknowledgments The contribution of A. Yedro Lozano to experiments during her Master training period is gratefully acknowledged.

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Figure 12. The calculations are done in the exit plane.

Figure 13. Sound pressure levels at 1m distance in a free field. Stars designate measurements, circles from Rayleigh integral.

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