modal approach for sound generation in corrugated pipes: a...
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POSTER 2017, PRAGUE MAY 23 1
Modal Approach for Sound Generation in CorrugatedPipes: A Phenomenological Model
Viktor HRUSKA1
1Dpt. of Physics, Faculty of Electrical Engineering, Czech Technical University, Technicka 2, 166 27 Praha, Czech Republic
Abstract. The sound generation in a corrugated pipe isstudied based on a phenomenological approach relying onthe feedback mechanism provided by the van der Pol typeof governing equations. Necessary basics of vortex soundtheory are presented. Effects of low Mach number flows andmissing corrugations are simulated as well as different meanflow velocity pulses. The discussion aims for connecting themodel features and parameters to the published experimen-tal behavior of such pipes.
Keywordsvortex sound, corrugated pipes, Van der Pol equation,dynamical systems
1. IntroductionCorrugated pipes are inseparable part of many devices
ranging from simple foot pumps to vast engineering instal-lations containing flexible risers. The decisive structural ad-vantage of the corrugated pipes is local rigidity still allowingglobal flexibility. The acoustical study focuses on the cor-rugation pipes drawback: fluid flowing past the corrugationscauses sound production. The vortex sound source is not of aparticular power per se but the mode locking with pipe reso-nance can result to sound of considerable amplitude causingunwanted noise or even structural problems. [3]
There are multiple ways to study this phenomenon in-cluding simplified acoustical approach as well as costly CFDsimulations. In the following text we make use of a previ-ously proposed phenomenological model relying on model-ing the pressure vortex sources as forced van der Pol oscilla-tors [1].
2. Theory2.1. Vortex Sound Theory Requirements
Since the full description of the flow field in the cor-rugated pipe would unduly encumber the text only a brief
Fig. 1. Simplified schematics of a corrugated pipe.
review of the major features will be given and the reader isreferred to existing literature for further commentary.
Sound is generated or absorbed in corrugated pipesdue to the unstable flow patterns of nonzero vorticity andby the hydrodynamic reactions to such instabilities by thepipe walls (Fig. 1). We begin our quick analysis with theHelmholtz decomposition of the flow velocity field v:
v = ∇ (φ0 + φ′) +∇×A , (1)
where the φ and A are scalar and vector potentials re-spectively. Mean value of the scalar potential is denoted with0, the prime denotes perturbations of the mean scalar poten-tial field.
According to Howe [5, 8] the time-averaged soundpower 〈Pvor〉 generated or absorbed by the airflow in thelow Mach number limit in a control volume V could be ex-pressed as:
〈Pvor〉 = −⟨∫
V
ρ0 (ω × v) · uac dV
⟩, (2)
where ω = ∇× v is the vorticity and uac the acousticvelocity defined as the unsteady part of the potential velocityfield (uac = ∇φ′).
2 V. HRUSKA: MODAL APPROACH FOR SOUND GENERATION IN CORRUGATED PIPES: A PHENOMENOLOGICAL MODEL
Three important features follow from these considera-tions: there is a feedback-loop between the acoustic and con-vective flow fields, the sound should be generated predom-inantly in regions of higher acoustic velocity and the flowcan be acousticaly generative as well as dissipative, basedon the interplay among the flow velocity, the vorticity andthe acoustic velocity. See e.g. [5] for more details. Hencethe use of the van der Pol type of equations known to have astable limit cycle resulting from strengthtening of weak dis-turbances and attenuating the strong ones.
2.2. Model Equations
Let U denote a mean flow assumed for simplicity tobe uniform and time independent through a corrugated pipeof length L. The pipe length is assumed to be much biggerthan the pipe cross-section so the system should be deemedone dimensional. The van der Pol type of equation is used tomodel an elementary source Pn located at n-th corrugationat a stable point xn [1]:
d2Pndt2
+AωS
[(PnBUα
)2
− 1
]dPndt
+ω2SPn = C
∂p
∂x
∣∣∣∣x=xn
,
(3)
where A, B, C and α are parameters described below,ωS is the Strouhal angular frequency and p = p(x, t) is theacoustic pressure inside the pipe. Note that the evaluation ofacoustic pressure spatial derivative gives (with proper con-stant) an approximation of the local acoustic velocity.
The acoustic pressure p is governed by a convectivewave equation driven by an external force by unit volumefsrc,x:
(D2
Dt2− c20
∂2
∂x2
)p = −∂fsrc,x
∂x. (4)
We identify U as the unidimensional convective flowvelocity for the wave equation so the operator D2
Dt2 yields∂2
∂t2 +U ∂2
∂x∂t +U2 ∂2
∂x2 . The external force per unit volume iscomposed of the elementary sources contributions. Makinguse of the Dirac delta function δ sampling property it gives:
fsrc,x(x, t) =
N∑n=1
Pn(t)δ(x− xn) , (5)
whereN is the number of corrugations. Combining thelast relations we arrive to the final form of the wave equation:
∂2p
∂t2+U
∂2p
∂x∂t+(U2 − c20
) ∂2p
∂x2=
N∑n=1
Pn(t)∂
∂xδ(x−xn) .
(6)
Considering the time-harmonic solution to the ho-mogenous convective wave equation gives the convectiveHelmholtz equation for the acoustic pressure. For simplic-ity, we consider the ideally reflecting pipe ends so the wavenumber has values of kπL where k is an integer and the spatialmodal shapes Φk(x) are:
Φk(x) = sinkπx
L. (7)
We are aiming to express the acoustic pressure as a sumof orthogonal modes truncated at an appropriately chosenK:
p(x, t) =
K∑k=1
qk(t)Φk(x) . (8)
Using the orthogonality of the modes we arrive at thefollowing equation for modal amplitudes q(t) (see e.g. [6],p. 189 for a detailed decomposition procedure):
qk + 2kζωqk + k2
(ω2 − π2U2
L2
)qk (9)
=2kπc20L2
N∑n=1
Pn(t) coskπxnL
,
where the dots denote time derivatives as usual and thesecond term in the left hand side represents an ad hoc addeddamping (see below). The pressure vortex sources equation(3) becomes:
d2Pndt2
+AωS
[(PnBUα
)2
− 1
]dPndt
+ ω2SPn (10)
=πC
L
K∑k=1
kqk(t) coskπxnL
.
The Eqs. (9), (10) presents a default set for furtherwork. We make the equations nondimensional by introduc-ing a scaled time τ = ωt where ω = πc0
L , the Mach numberM = U
c0and the Strouhal angular frequency:
ωS =1
2πSrU
Λ, (11)
where Sr is the Strouhal number and Λ is a character-istic length (see below).
This procedure yields a new set of coupled ordinarydifferential equations:
POSTER 2017, PRAGUE MAY 23 3
Pn +Aν
(P 2n
B2c2α0 M2α− 1
)Pn + ν2Pn (12)
=CL
πc0
K∑k=1
kqk(τ) coskπxnL
qk + 2kζqk + k2(1−M2
)qk (13)
=2k
π
N∑n=1
Pn(τ) coskπxnL
,
where ν = 2M LΛSr. Note that the coefficient (1−M2)
on the right hand side of the modal amplitudes equation is theonly consequence of using the convective wave equation.
2.3. Meaning of the Coefficients
The proposed model contains various coefficientswhich should be shortly examined in the physical contextnow.
Behavior of the van der Pol oscillator is governed bythe coefficients A and B which controls the strength of non-linear damping and the limit cycle size respectively. Popescuet al. [2] connects the coefficient A with the ratio of bound-ary layer thickness to the pipe diameter. Hence the numericalvalue of the coefficient should be much less than one, sincein fully turbulent flow the feedback mechanism does not lockand therefore the singing phenomenon would not take place.The physical meaning of the parameter B is still missing.Its value is believed to be connected with the resonant effectof the corrugation cavity and its interaction with the cavitytrailing edge.
The coupling coefficient C is reported [2] to be con-nected with a ratio of sounding frequency to the eigenfre-quency of the single corrugation (usually approximated as aHelmholtz resonator). Therefore its value should depend onthe mean flow velocity provided that Strouhal law is a validapproximation. In this article the relation C = 104M2 isused.
The coefficient α presents possible generalization ofthe system enhancing its capabilities to fit on the experimen-tal data. The value of α should be near 2 and this particu-lar value is used throughout this article. The coefficient ζaccounts for acoustic losses, it’s value being small enoughfor the modes to be considered real over the pipe length.The characteristic distance of Strouhal law Λ was originallytaken as the length of a corrugation pitch. Nevertheless, therecent literature shows that extending this distance by the ra-dius of the upstream edge leads to more realistic results [5].For the sake of simplicity we hold the Strouhal number con-stant (Sr = 0.4).
3. Numerical ResultsThe linear stability analysis shows that the origin of
the dynamical system is an unstable focus for any physi-cally meaningful coefficient values (see e.g. [7]). There-fore a limit cycle will be reached whenever any of the initialconditions is nonzero. We hold all the acoustical initial val-ues to be zero and all the pressure source variables to be ofO(10−4
)in a random distribution (variable over the pre-
sented trials). Beside the above mentioned coefficients therest is defined as follows: A = 0.005, B = 0.0001, α = 2,ζ = 0.001. Length of the pipe was 0.4 m with 50 corru-gations and 10 computed acoustic modes. The characteristicdistance Λ was 6 mm, the sound speed 340 m·s−1. The clas-sical Runge-Kutta 4th order scheme was used on Eqs. (12),(13).
The presented data were obtained from eq. (8) (for xat the open end) and subsequently analysed by means of theHilbert transformation to obtain the instantaneous amplitudeand frequency. In order to ease the comprehension the datawere low-pass filtered before the visualization. The refer-ence value for the SPL computation was 2 · 10−5 Pa.
3.1. Velocity sweeps
The first scenario simulates the system subjected toa linear velocity sweep implemented as the growing Machnumber.
Effect of Low Mach Number Flow
Fig. 2 shows data obtained by solving the equationswithout any exception (the green curve, referred to as ”refer-ence run”). The red curve represents the solution without theMach number term in the Eq. (13), i.e. the solution neglect-ing the flow effects. It is apparent that the difference betweenthe simulations becomes mildly noticeable with the growingMach number but the flow effect is very weak, essentiallynegligible in a qualitative point of view.
Effect of Missing Corrugations
The previous commentary and Eqs. (12), (13) impliesthat the sound is mainly generated in the regions of acousticvelocity anti-nodes. Fig. 3 shows a system with two corruga-tions missing at each end of the pipe simulating smooth inletand outlet regions. All the acoustical modes have a velocityantinode at these regions and therefore the loss of power isspread rather equally among them.
In the subsequent case (Fig. 4) only the corrugationscorresponding to the 3rd mode velocity antinodes are re-moved which results in attenuation of this mode.
4 V. HRUSKA: MODAL APPROACH FOR SOUND GENERATION IN CORRUGATED PIPES: A PHENOMENOLOGICAL MODEL
0.03 0.06 0.09 0.12
80100120140160
Mach number
SPL
(dB
)
0.03 0.06 0.09 0.12
1
2
3
4
5
6
7
Mach number
Mod
enu
mbe
r
Fig. 2. Instantaneous amplitudes and frequencies of the systemsubjected to a linear velocity sweep. The green curveshows a reference run, the red curve shows a simula-tion with neglected effects of the mean flow on the wavepropagation.
3.2. Velocity pulses
Simulation of the system subjected to three gaussianshaped velocity pulses is presented on Fig. 5. The dimen-sionless time corresponds to the number of Runge-Kutta al-gorithm steps (referred to as ”RK4 steps” in the figure). Notethe asymmetry of the sounding frequency profile due to themode locking which may result in missing of some of themodes.
4. Discussion & ConclusionsHigher Mach number values are inaccessible without
further careful analysis because the ”non-acoustic” flowscould not be deemed incompressible any more and the rel-atively simple Howe’s formula (2) would not hold. Besidethat, the radiation losses due to the convection of acousticwaves outside the tube are of O(M) but the effects on thepipe eigenfrequencies is of O
(M2)
[4]. Therefore the pre-sented model is valid only in the low Mach number limit. Ithas been proven that in such limit the influence of convectionon the nonlinear dynamical system is negligibely low.
Using the nontrivial velocity build-up (sweep or pulse)is necessary for system to behave realistic. If a set of dif-ferent Mach numbers is examined assuming that their valueis reached immediately, the frequency lock-in plateaus doesnot appear which is inconsistent with experiments.
0.03 0.06 0.09 0.12
80100120140160
Mach number
SPL
(dB
)0.03 0.06 0.09 0.12
1
2
3
4
5
6
7
Mach numberM
ode
num
ber
Fig. 3. Effect of two corrugations missing at each tube end. Thegreen line shows the reference run, the red line the casewith missing corrugations.
Weakness or totall absence of the first acoustic mode(see e.g. Fig. 2) is caused by the lacking feedback. In prac-tice this is assumed to be a result of the flow insufficientreceptivity at lower Reynolds numbers [4].
It is worth to note that the ratio LΛ appearing in the
scaled pressure vortex source eigenfrequency ν = 2M LΛSr
is in order proportional to the number of corrugations N ,precise value being dependent on a specific design of thepipe through the trailing edge radius and corrugations spac-ing. This feature is promising for a possible future fitting thephenomenological model on the measured data.
The effect of corrugations compliance on the soundtransmission could be accounted by introducing the effectivesound speed ceff [4]:
ceff = c0
√ViVp
, (14)
where Vi is the volume of a corrugation segment de-fined by its minimal inner diameter and Vp the volume ofthe whole corrugation segment. It is straightforward thatceff < c0. Nevertheless, a possible transformation c0 → ceffin the wave equations would not qualitatively change the pre-sented system behavior.
Note that the presented SPL corresponds to the acous-tic pressure in the pipe, not to the radiated pressure, whichwould be considerably lower.
There are multiple ways to be followed in a future re-search. The internal pressure amplitudes ask for a nonlin-
POSTER 2017, PRAGUE MAY 23 5
0.03 0.06 0.09 0.12
80100120140160
Mach number
SPL
(dB
)
0.03 0.06 0.09 0.12
1
2
3
4
5
6
7
Mach number
Mod
enu
mbe
r
Fig. 4. Effect of the 3rd mode dominant corrugations removal.The green line shows the reference run, the red line thecase with missing corrugations.
ear finite-amplitude formulation of the resonator describingwave equation. Other possibilities include analytical solu-tion of the proposed set of equations or accounting for theradiation losses.
AcknowledgementsResearch described in the paper was supervised by
Doc. Michal Bednarık, FEE CTU in Prague and supportedby the Grant Agency of the Czech Technical University inPrague, grant No. SGS17/130/OHK3/2T/13.
References[1] DEBUT, V., ANTUNES, J., MOREIRA, M. A phenomenological
Model for Sound Generation in Corrugated Pipes. In ISMA 2007.Barcelona, 2007.
[2] POPESCU, M. JOHANSEN, S. T., SHYY, W. Flow-Induced Acousticsin Corrugated Pipes. Commun. Comput. Phys., 2011, vol. 10, no. 1, p.120-139.
[3] RAJAVEL, B., PRASAD, M. G. Acoustics of Corrugated Pipes: AReview. Applied Mechanics Reviews, 2013, vol. 65.
[4] RUDENKO, O. NAKIBOGLU, G., HOLTEN, A., HIRSCHBERG, A.On whistling of pipes with a corrugated segment: Experiment and the-ory. Journal of Sound and Vibration, 2013, vol. 332.
[5] NAKIBOGLU, G., BELFROID, S., WILLEMS, J., HIRSCHBERG, A.Whistling behavior of periodic systems: Corrugated pipes and multi-ple side branch system. International Journal of Mechanical Sciences,2010, vol. 52.
[6] LIEUWEN, T. C. Unsteady Combustor Physics Cambridge, New York,Melbourne, Madrid, Cape Town, Singapore, So Paulo, Delhi, MexicoCity: Cambridge University Press, 2012.
0 2 4 6
·105
0
0.04
0.08
0.12
RK4 steps
Mac
hnu
mbe
r
0 2 4 6
·105
4070100130160
RK4 steps
SPL
(dB
)
0 2 4 6
·105
1
2
3
4
5
6
RK4 steps
Mod
enu
mbe
r
Fig. 5. System subjected to a flow velocity pulses depicted at thetop. Middle: the instantaneous amplitudes, bottom: theinstantaneous frequencies.
[7] MANNEVILLE, P. Instabilities, chaos and turbulence. 2nd ed. Lon-don: Imperial College Press, 2010.
[8] HOWE, M. S. The influence of vortex shedding on the diffraction ofsound by a perforated screen. Journal of Fluid Mechanics, 1980, vol.97, p. 641–53.
About Authors. . .
Viktor HRUSKA was born in Prague, Czech Republic. Af-ter studies at the Faculty of Mathematics and Physics of theCharles University and the Academy of Performing Arts inPrague he deals with the acoustics of unsteady flows as adoctoral student at the Department of Physics, Faculty ofElectrical Engineering, Czech Technical University.