icb

twin many practical aerodynamic applications such as ground trans-portation vehicles, ventilation ducts and jets, etc. A typical flowspeed of automobiles, for example, is in the range of 50100 km/h(or M = 0.040.08) and flows are mostly turbulent and at mod-erately high Reynolds numbers. The computation of low-subsonicturbulent flow noise is, however, a difficult task because the noisesources are highly localized in the turbulent boundary layer nearthe wall or in the wake, while the acoustic wavelengths far exceedthe hydrodynamic length scales. In this case, a direct numericalsimulation (DNS) employing the full compressible NavierStokesequations becomes very difficult and expensive, coping with thefact that a long-time computation is often required to representthe turbulence statistics, i.e. the noise sources.

The sound in bio-medical or biological fluids is also in the rangeof very low Mach numbers. For example, the vocal fold of the hu-man larynx [1], or an insect flapping wings [2] produces the soundby periodically disturbing the flowwith body oscillating at the fre-quency range of 50200 Hz. The associated Mach number can befigured asM = Ub/co = (Lc f )/co, where Ub is the moving speed ofthe body, Lc the distance of travel, f the oscillating frequency of thebody, and co the speed of sound. For a biological body with lengthscale of 12 cm, the associated Mach numbers are in the range ofM = 0.00150.012.

For computing sound of fluids at such low Mach numbers,numerical difficulties lie not only in scale disparity between the

Correspondence to: Korea University, Department of Mechanical Engineering,136-701 Seoul, Republic of Korea.

E-mail address: yjmoon@korea.ac.kr.

method is based on a hydrodynamic/acoustic splitting techniqueproposed by Hardin and Pope [3]. The hydrodynamic flow field issolved by the incompressible NavierStokes equations, while theacoustic field is computed by the perturbed Euler equations withacoustic source obtained from the incompressible NavierStokesequations. The idea of splitting the hydrodynamic part and theacoustically perturbed part from the full compressible flow field is,however, not a straightforward task because of the physics coupledbetween these two fields. It has been found [4,5] that an unstablevortical mode can easily be excited by the non-linear terms inthe perturbed momentum equations when the source terms areimproperly treated; for example, either lack of physical diffusion orlack of grid resolution of the perturbed vorticity ( = u). Herethe prime denotes an instantaneously perturbed quantity from theincompressible state.

To avoid such vortical instability, the linearized perturbedcompressible equations (LPCE) [5] are formulated by eliminatingthe terms related to the generation of the perturbed vorticity.The details of the linearized perturbed compressible equations(LPCE) are reviewed in Section 2. In present formulation, amaterialderivative of the hydrodynamic pressure (DP/Dt) is derivedas an acoustic source in the linearized perturbed compressibleequations. At low Mach numbers, the material derivative of thehydrodynamic pressure (scaled by itself) in the near field of theincompressible flow is found very closely related to the dilatationrate of the compressible counterpart, because the flow speedrelative to the speed of sound is substantially low that any thermaleffect during processes is nearly negligible. The near field of thecompressible flow computed by the direct numerical simulationand that by the present hybrid method with acoustic source areEuropean Journal of Mechan

Contents lists available a

European Journal of

journal homepage: www.e

Sound of fluids at low Mach numbersYoung J. Moon Computational Fluid Dynamics and Acoustics Laboratory, School of Mechanical Engineerin

a r t i c l e i n f o

Article history:Available online 9 February 2013

Keywords:Low subsonic flowTurbulent flow noiseLES/LPCE hybrid method

a b s t r a c t

The sound of fluid at lowMain aerodynamics but also inof O(102) or even less andimportant role inmany aspecis of our present interest andIn this paper, the linearized pacoustic source term,DP/Ddemonstrated by the presen

Cro

1. Introduction

The sound of fluids at low Mach numbers is often encountered0997-7546/$ see front matter Crown Copyright 2013 Published by Elsevier Massodoi:10.1016/j.euromechflu.2013.02.002cs B/Fluids 40 (2013) 5063

t SciVerse ScienceDirect

Mechanics B/Fluids

lsevier.com/locate/ejmflu

g, Korea University, Seoul, 136-701, Republic of Korea

h number is a special research area that poses diverse applications not onlyio-medical or biological fluids. The related Mach numbers are in the ordertherefore the compressibility effects are substantially low but still play ants. A hybridmethod of splitting the hydrodynamic field and the acoustic fieldattention is given to the linearized perturbed compressible equations (LPCE).erturbed compressible equations are reviewed with some discussion on thet . A few selected applications of aerodynamic noise and bio-fluid sound arehybrid method.n Copyright 2013 Published by Elsevier Masson SAS. All rights reserved.

hydrodynamic field and the acoustic field but also in stiffnessof the system of the governing equations. In regard to this, ahybrid approach has been sought as an alternative. This hybridn SAS. All rights reserved.

Y.J. Moon / European Journal of Me

examined in Section 3 with discussion on the process of soundgeneration.

In Sections 4 and 5, a few selected applications of aerodynamicnoise and bio-fluid sound are demonstrated by the present hybridmethod.

2. Computational methodology

2.1. LES/LPCE hybrid formulation

The present LES/LPCE hybrid method is based on a hydro-dynamic/acoustic splitting method [3], in which the total flowvariables are decomposed into the incompressible and perturbedcompressible variables as,

(x, t) = 0 + (x, t)u(x, t) = U(x, t)+ u(x, t)p(x, t) = P(x, t)+ p(x, t).

(1)

The incompressible variables represent hydrodynamic flow field,while acoustic fluctuations and other compressibility effects areresolved by perturbed quantities denoted by ().

The hydrodynamic turbulent flow field is first solved by incom-pressible LES. The filtered incompressible NavierStokes equationsare written as,

Ujxj

= 0 (2)

0Uit

+ 0 xj

(UiUj)

= Pxi

+ 0 xj

Uixj

+ Ujxi

0

xjMij, (3)

where the grid-resolved quantities are denoted by ( ) and the un-known sub-grid tensorMij is modeled as

Mij = UiUj UiUj = 2(Cs)2|S|Sij. (4)Here, is the mean radius of the grid cell (computed as cubic rootof its volume), Sij is the strain-rate tensor.

After a quasi-periodic stage of hydrodynamic field is attained,the perturbed quantities are computed by the linearized perturbedcompressible equations (LPCE) [5]. A set of the linearized perturbedcompressible equations is written as,

t+ (U ) + 0( u) = 0 (5)

u

t+(u U)+ 1

0p = 0 (6)

p

t+ (U )p + P( u)+ (u )P = DP

Dt. (7)

The left hand side of LPCE represents effects of acoustic wavepropagation and refraction in an unsteady, inhomogeneous flow,while the right hand side only contains an acoustic source term,which is projected from the incompressible LES flow solution. Itis interesting to note that for low Mach number flows, the totalchange of the hydrodynamic pressure, DP/Dt is only considered asthe explicit noise source term.

The filtered incompressible NavierStokes equations are solvedby an iterative fractional-step method (Poisson equation for pres-sure), whereas the linearized perturbed compressible equations

are solved in a time-marching fashion. To avoid excessive numeri-cal dissipations and dispersions errors, the governing equations arechanics B/Fluids 40 (2013) 5063 51

spatially discretized with a sixth-order compact finite differencescheme [6] and integrated in time by a four-stage RungeKuttamethod. For example, the first and second derivatives with respectto x are implicitly calculated with a five-point stencil, i.e.

1f i1 + f i + 1f i+1 = a1fi+1 fi1

21x+ b1 fi+2 fi241x (8)

2f i1 + f i + 2f i+1 = a2fi+1 2fi + fi1

1x2

+ b2 fi+2 2fi + fi241x2 , (9)where 1 = 1/3, 2 = 2/11, a1 = 14/9, b1 = 1/9, a2 = 12/11,and b2 = 3/11.

Practically, when using a high order scheme to the stretchedmeshes, numerical instability is encountered due to numericaltruncations or failure of capturing highwave-number phenomena.Thus, a tenth-order spatial filtering (cut-off wave number, k1x 2.9) proposed by Gaitonde et al. [7] is applied every iteration tosuppress the high frequency errors that might be caused by gridnon-uniformity. For the far-field boundary condition, an energytransfer and annihilation (ETA) boundary condition [8] with bufferzone is used for eliminating any reflection of the out-going waves.The ETA boundary condition is easily facilitated with a rapidgrid stretching in a buffer-zone and the spatial filtering which isdamping out waves shorter than grid spacing. So, if a buffer-zonehas grid spacing larger than out-going acoustic wave length, thewave can be successfully absorbed by the ETA boundary condition.

2.2. Linearized perturbed compressible equations (LPCE)

The compressibly perturbed field was originally calculated bythe perturbed compressible equations (PCE), obtained by subtract-ing the incompressible NavierStokes equations from the full com-pressible NavierStokes equations.

The perturbed compressible equations [4] are written as

t+ (u ) + ( u) = 0 (10)

u

t+ (u )u + (u )U + 1

p +

DUDt

= 1f vis (11)

p

t+ (u )p + p( u)+ (u )P

= DPDt

+ ( 1) q (12)where D/Dt = /t + (U ), f vis is the perturbed viscous forcevector, and q represent thermal viscous dissipation and heat fluxvector, respectively. At low Mach numbers, the perturbed viscousforces can be approximated as

f vis,i = 0

xj

uixj

+ uj

xi 2

3ukxk

ij

(13)

by assuming viscosity = 0 (=constant), and and q are ex-pressed as

= ujxk

+ ukxj

23ulxl

jk

ukxk

, (14)

qj = k xj

p

. (15)

Since perturbed variables are residuals of the total variables

with incompressible components subtracted, they represent notonly the acoustic fluctuations but also the other compressibility

52 Y.J. Moon / European Journal of Me

effects such as coupling effects between the hydrodynamic flowand theperturbed field. Oneparticular component of the perturbedvariables related to the consistency of the acoustic solutionis perturbed vorticity ( = u), a non-radiating vorticalcomponent generated in the PCE system. This fluctuating quantitybecomes unstable for various reasons and generates unwantederrors in acoustic calculations [4].

Here, attention is given to identify the terms associated withproduction and diffusion of the perturbed vorticity in its transportprocesses. The perturbed vorticity transport equations, derived bytaking a curl on Eq. (11) with mathematical identities and theincompressible NavierStokes equations employed, are written as

t

+ (u )

= 1[( )u

I-a

+ ( )u I-b

]

I

1[(u )

II-a

+ ( u) II-b

]

II

+ 13( p)

III

+ 1 Fvis

IV

, (16)

where Fvis = (f vis 02U)/.In Eq. (16), one can clearly see that perturbed vorticity is

generated and diffused by source terms on the right hand sidethrough: (i) coupling effects between the hydrodynamic vorticityand the perturbed velocities (terms I and II), (ii) entropy field(term III), and (iii) viscous force (term IV). Term I is related tothe three-dimensional effect of vortex stretching: stretching ofhydrodynamic vorticity by perturbed velocities (term I-a) andstretching of perturbed vorticity by total velocities (term I-b). TermII represents a more direct coupling between the hydrodynamicvorticity and the perturbed velocities. The convective effect ofhydrodynamic vorticity by perturbed velocity is represented byterm II-a, whereas term II-b is related to the dilatation rate effect.Term III is not so important for low Mach number, non thermally-driven flows and term IV only provides physical diffusion tothe perturbed vorticity. In the previous study [5], it was shownthat term II-a is the most dominant source term that generatesperturbed vorticity and term II-b is considered less important atlow Mach numbers.

It is interesting to note that perturbed vorticity is not a radi-ating acoustic quantity but a convecting hydrodynamic vortical.Its physical meaning represents modification of the hydrodynamicvorticity through interactions between the hydrodynamic vortic-ity and the velocity fluctuations. At low Mach numbers, the mag-nitude of the perturbed vorticity is small but, if falsely resolved,it becomes self-excited and grows to affect the acoustic solution.Since term II-a is related to the gradient of hydrodynamic vorticity , perturbed vorticity usually appears at the edge of the hydro-dynamic vorticity and its length scale is similar to (or sometimessmaller than) the hydrodynamic vortical scale. Therefore, acousticgrid resolution must carefully be handled in calculation.

By neglecting the second-order, non-linear terms such as (u )u, the original PCE, Eqs. (10)(12) can be re-written as

t+ (U ) + 0( u) = 0 (17)

u

t+(u U)+ 1

0p= ( u) ( U)

0

DUDt

+ 10

f vis (18)chanics B/Fluids 40 (2013) 5063

p

t+ (U )p + P( u)+ (u )P

= DPDt

+ ( 1) q (19)with mathematical identity, (U )u + (u )U = (u U) +( u) + ( U). Since the left hand side of Eq. (18) does notgenerate any vortical component, only the right hand side termsare responsible for the generation of perturbed vorticity. The firsttwo terms, u and U correspond to the dominant sourceterms (terms I and II) in the perturbed vorticity transport equationsand the last two terms are associated with the entropy and viscouseffects (terms III and IV).

To show the Mach number dependence of each term, the per-turbed momentum and energy equations, Eqs. (18)(19) are com-bined into a convective wave equation, neglecting the viscous andthermal effect terms and then aMach number scaling is conducted.The hydrodynamic variables are scaled by their free stream values:0 ,U U, and P U2. For the perturbed variables,a Mach number expansion approach [9,10] is employed; for exam-ple, u = U + Mu(1) + M2u(2) + M3u(3) + . So, the perturbedvelocity, u Mu(1) and from the linear acoustics, p (c)uand (/c)u. The time is also scaled by l/c, where l is areference length scale and c is the speed of sound.

The resulting convective wave equation is written as

2p

t2O(M)

+ (U ) p

t+Ut

p

O(M2)

P02p +

u

t P + (u ) P

t+ P

t( u)

O(M3)

P {( U)+ ( u)} +

0

DUDt

+(u U)

O(M4)

= t

DPDt

O(M)

. (20)

Each term has the order of c3u(1)/l2 (or c3U/l2) multi-plied by a Mach number to the power denoted in Eq. (20). It isclearly shown that the terms responsible for the generation of per-turbed vorticity (i.e. the right hand side in Eq. (18)) have a Machnumber dependency O(M4), whereas the leading-order termsare O(M). It is also interesting to note that the only explicitacoustic source term, DP/Dt on the right hand side of Eq. (20) hasthe same order as the first term in the convective wave equation,2p/t2.

Now, it is evident that the first two terms on the right handside of Eq. (18) are not so responsible for sound generation at lowMach numbers and thereby one can exclude these to suppressthe generation of perturbed vorticity. The third term related to amomentum correction to the perturbedmass can also be neglectedat low Mach numbers. The last term (perturbed viscous force)is not necessary any more because there is no generation anddiffusion of perturbed vorticity. With the thermal terms neglectedin the perturbed energy equation, a set of linearized perturbedcompressible equations (LPCE) now read Eqs. (5)(7).

Because a curl of the linearized perturbed momentum equa-tions, Eq. (6) yields

t= 0, (21)

acoustic solutions. The derivation of LPCE with detailed discussionon the characteristics of the perturbed vorticity can be found inRef. [5].

3. Acoustic source

The aerodynamic noise at low Mach numbers is often domi-nated by vortex interactions with solid walls. When a vortex in-teracts with the solid body, its strength changes in time and as aconsequence, the circulation in the neighboring fluids is alteredand so are the local streamlines. The time-varying streamlines aredirectly connected to the pressure change in time and space andtherefore a so-called vortex sound is produced.

A generation of dipole tone from a circular cylinder is, for ex-ample, due to an alternating formation of vortex behind the cylin-der and therefore circulation around the cylinder oscillates in time.When vortex is formed at the upper side, a negative circulation

It is clearly noticeable that the rate of change of the dilatationrate acts as a sound source in the near field, and Fig. 3 showsthe generation and propagation of the sound wave generated inthe near field by the snapshots of the two-level contours of thedilatation rate, positive (red) and negative (blue).

The same physics of sound generation can also be representedby the rate of change of the hydrodynamic pressure experiencedby a material element of the fluid (i.e. 1P

DPDt ), which is computed

by solving the incompressible NavierStokes equations for thesame flow condition (ReD = 150). The instantaneous 1P DPDt contoursin the near field during one period of dipole sound generationare shown in Fig. 2. One can note a good resemblance betweenthese two total derivatives of the density and the hydrodynamicpressure, both scaled by itself and plotted with the same contourlevels. It is also clearly noticeable in Fig. 4 that the rate of changeof 1P

DPDt acts as a sound source in the near field. The propagation of

the sound produced by the rate of change of 1PDPDt is well shown by

the instantaneous pressure fluctuation contours, computed by theFig. 2. Instantaneous 1PDPDt contours (incompressible NS eqs., ReD = 150).

the LPCE prevents any further changes (generation, convection,and decaying) of perturbed vorticity in time. In fact, the perturbedvorticity could generate self-excited errors, if is not properlyresolved with the acoustic grid. Hence, the evolution of theperturbed vorticity is pre-suppressed in LPCE, deliberating the factthat the perturbed vorticity has little effects on noise generation,particularly at low Mach numbers. For hybrid methods [5,11], thisis an important property that ensures consistent, grid-independent

cific time interval, the flow field and lift force are at the oppositephase when the vortex is formed at the lower side.

The instantaneous dilatation rate ( 1

DDt ) contours in the near

field during one period of dipole sound generated from the cylinderat ReD = 150 andM = 0.2 is shown in Fig. 1. This is computed bysolving the full compressible NavierStokes equations. The upperand lower four figures represent respectively the processes ofexpansion and compression over the upper surface of the cylinder.Y.J. Moon / European Journal of Me

Fig. 1. Instantaneous 1

DDt contours (comaround the cylinder lowers the stagnation point at the frontal faceof the cylinder with creation of positive lift force. At a certain spe-chanics B/Fluids 40 (2013) 5063 53

pressible NS eqs., ReD = 150,M = 0.2).linearized perturbed compressible equations with acoustic source,DP/Dt .

motions and therefore, pressure or density changes most rapidlyduring a transient development of the flow field.

The locations satisfying the null condition (i.e. 1

DDt = 0,

ux = 0, and vy = 0) are marked in circles on the snapshots ofthe dilatation rate contours in Fig. 3. In the top four figures, onecan note a process of expansion along the trace of the markedcircles on the right, whereas the emission of the compressionwave can be traced along those on the left. The bottom fourfigures also show the formation and emission of the sound waveat opposite phase along the traces of the marked circles. Thesame process of sound generation can also be traced in Fig. 4,which shows the instantaneous pressure fluctuations p perturbedfrom the incompressible hydrodynamic pressure P , computed bythe linearized perturbed compressible equations with acousticsource, DP/Dt acquired from the incompressible NavierStokessolutions. Themarked circles found in compressible flow solutionscan be found in the incompressible flow solutions with conditionsatisfying 1P

DPDt = 0, Ux = 0, and Vy = 0. Along themarked circles,

length, Rec is 1.3 10 and the Mach number,M is 0.06. This freestream Mach number is considerably low, as far as capturing thecompressibility effects are concerned.

For incompressible large eddy simulation, an o-type grid is em-ployed to treat four rounded-corners of the leading and trailingedges. The computational domain is set to r = 10c and a spanwiseextension is chosen as 3% of the plate chord with flow periodicityassumed at the side boundaries. The computational domain con-sists of 65720121 (about 2.8millions) points in x, y, and z andis divided into 32 blocks for parallel computations. A minimal gridsize for x and y is 0.0005c (or1x+min = 1y+min 3), while a uniformgrid spacing of 0.0015c (or1z+ 15) is used in the spanwise di-rection. The computation is conducted with 1t = 1 106 s for400,000 iterations (or 0.4 s).

The boundary layer is triggered approximately at x = 0.2c bythe leading-edge separation bubble and becomes turbulent down-stream towards the trailing-edge of the plate. It was found that thethickness of the boundary layer, is 1.12h at x = 0.2c from thetrailing-edge and the turbulent Reynolds number, Re is approxi-Fig. 4. Instantaneous pressure fluctuation p contours (linearized perturbed compressible eqs. (LPCE) with acoustic source DP/Dt from the incompressible NS eqs.,ReD = 150,M = 0.2).

If one is asked where and when the sound is originated, it canbe estimated at an instant by a point where the dilatation rate andboth the linear strain rates in x and y directions are null. During therepeated process of expansion and compression, the rate of changeof the linear strain rates are coupledwith that of the dilatation rate,via conservation of mass. The condition of null for the dilatationrate and the linear strain rates represents an inflection point, i.e. amaterial point (or line in 3D) that takes only translation in flow

4. Applications on aerodynamic noise

4.1. Trailing-edge noise

This case considers a flow (Uo = 20 m/s) over the flat plate atzero angle of attack (experiment described in Ref. [12]). The platehas a chord length of c = 10 cm with thickness h = 0.03c andspan L = 3c. The Reynolds number of the flow based on the chord

554 Y.J. Moon / European Journal of Me

Fig. 3. Instantaneous 1

DDt contours; positive (red) and negative (blue) (compressible

figure legend, the reader is referred to the web version of this article.)the sound generation process of expansion and compression aswell as their emissions can also be well traced in the figures.chanics B/Fluids 40 (2013) 5063

NS eqs., ReD = 150,M = 0.2). (For interpretation of the references to colour in thismately 230. The iso-surfaces of the second invariant property of thevelocity gradients (Q = 200) clearly show the noise sources near

a/c = 2.5, corresponding to the frequency St = 0.2 atM = 0.06.Besides, the figure shows other high frequency waves being em-anated from the trailing-edge as well as from the shear-layer reat-tachment point. There will also be the waves diffracted at theleading and trailing-edge of the plate, and all of these will con-tribute in part to the far-field noise measured at the microphonelocation.

In order to predict the far-field SPL spectrum, a computationalprocedure described in Ref. [13] is followed. Since the microphoneis located at 20c from the plate, the 2D acoustic field computedby the LPCE for the domain of 10c needs to be extrapolated to20c and also to be corrected for 3D spectral pressure. Finally,the 3D spectral pressure radiated by the simulated span h needs

Fig. 7. Sound pressure level spectrum at r = 20c vertically away from the mid-chord of the plate; computation (blue), experiment (black). (For interpretation ofthe references to colour in this figure legend, the reader is referred to the webversion of this article.)

coherence function of the surface pressure, (z) in the mostdominant noise source region, i.e. the trailing-edge of the plate.The spanwise coherence length of the surface pressure, Lc() isthen calculated by a Gaussian law, (z) = exp{(z/Lc())2}. Thelargest value of Lc() is approximately estimated as 7h at St = 0.2but in most cases, Lc() is below h [13].

The far-field SPL spectrum for the actual span 3c is nowcompared in Fig. 7 with themeasured data of the Ecole Centrale dethe trailing-edge (see Fig. 5(left)), i.e. convecting turbulent eddieswithin the boundary layer and the vortex shedding at the trailing-edge. Fig. 5(right) also shows the wall pressure fluctuations moni-tored along the plate from A to I (A: x = 0.2c , I: the rear face of thetrailing-edge). One can notice the leading-edge separation, convec-tion of turbulent eddies, and the vortex shedding at the trailing-edge in the time history of the wall pressure fluctuations.

The flat plate self-noise is now computed by the linearizedperturbed compressible equations. Fig. 6(left) shows the noisesources near the trailing-edge, i.e. the acoustic source, DP/Dtcomputed by LES interpolated onto the acoustic grid using bilinearinterpolation. The acoustic grid (347 247) with minimal normalspacing at the wall five times larger than that of the hydrodynamicgrid allows the same time step used in LES (i.e. 1t = 1 106 s)for the LPCE computation.

An instantaneous pressure fluctuation field (1p = (P + p) (P + p)) around the plate in Fig. 6(right) clearly shows the ra-diation of the dipole tone generated by the vortex shedding atthe trailing-edge. The acoustic wavelength of the tone is close toY.J. Moon / European Journal of Me

Fig. 5. Instantaneous Q iso-surfaces around the trailing-ed

Fig. 6. Instantaneous DP/Dt contours on the acoustic (top) and hydrodynto be corrected for the total span 100h (or 3c) employed in theexperiment. This procedure requires information on the spanwisechanics B/Fluids 40 (2013) 5063 55

ge (left); wall pressure fluctuations along the plate (right).

mic (bottom) grids (left); instantaneous pressure fluctuation field (right).Lyon [14]. The numerical results are signal-processed by applying ahanning window function with the sampling frequency of 50 kHz,

Fig. 8. Directivity patterns of1prms at r = 20c for different Strouhal numbers.

the block length of 0.04 s, and the number of averages of 10. Theagreement is found excellent, especially for the match of the tonalpeak (peak level deviation is 2.7 dB), its spectral broadening, aswell as the other broadband part. This comparison indicates thatnot only the noise sources but also their turbulence statistics arewell captured by the incompressible LES, while the propagation,scattering, and diffraction of the acoustic waves around the plateare accurately computed by the LPCE.

The directivity patterns at r = 20c are also presented in Fig. 8for various Strouhal numbers (or ratios of the plate chord lengthto the acoustic wavelength). At vortex shedding frequency (St =0.2 or c/ = 0.4), it represents a clear dipole. As the Strouhalnumber increases or the acoustic wavelength becomes shorterthan the chord length, the waves diffracted at the leading andtrailing-edge of the plate are well captured; the directivity patternchanges to a finger-like shape. It is worth noting that the first twoplots of St = 0.2 and 0.4 are consistent with what is expectedfrom analytical modeling based on zero-thickness assumption, asshown for instance in the study of Roger and Moreau [15]. Athigher Strouhal numbers, the directivity pattern departs from theanalytical results, essentially by showing a secondary beamingaround 45 and 30 at St = 1 and 2, respectively. This could beattributed to the plate thickness.

4.2. Porous trailing-edge

with porosity of = 0.25 to a small, selected area of the trailing-edge (2h upstream from the edge, with a plenum inside, seeFig. 9), where the vortex shedding and eddy scattering produce adipole sound. The porous surface has a thickness of = 0.001cand is characterized by non-dimensionalized permeability, K =KU0/c = 1 103 for case 1, 1 102 for case 2, and 1 101for case 3. The porous flow is often modeled by employing the Er-gun equation,

Ps = KUs + 0 CE

K

Us Us (22)where CE denotes a dimensionless Ergun coefficient or Forch-heimer constant, which is dependent on porosity and pore struc-ture. The Forchheimer constant is, however, set to zero in thepresent computation because the pore-level Reynolds numberbased on the permeability and the transpiration velocity averagedin time and space, ReK = UtK 1/2/ turns out to be less than unity.

First, an xt plot of the wall pressure fluctuations is examinedalong the plate and in the wake region. As shown in Fig. 10(left),the solid trailing-edge exhibits distinct, regularly-spaced pressuremarks (1tUo/h 5) near the trailing-edge which correspondsto the vortex shedding frequency at St 0.2. This is obviouslythe noise source for producing the tone. With the porous surface(K = 1102), however, the strips of pressure marks are brokeninto pieces (see Fig. 10(right)) by local blowing and suction of theflow in the plenum.56 Y.J. Moon / European Journal of MeA porous treatment to the same flat plate for reduction oftrailing-edge noise is demonstrated, imposing a porous surfacechanics B/Fluids 40 (2013) 5063The influence of the porous surface is more clearly explainedin Fig. 11 that the correlation length of the pressure fluctuations

Fig. 11. Spatial correlation Rpp contours of wall pressure fluctuations (left); Rpp along the mid-span (right).

is substantially reduced in the streamwise direction, i.e. a signif-icant reduction in the size of the dipole noise source. The spatialcorrelation of the wall pressure fluctuations, Rpp along the mid-span of the plate clearly indicates that the streamwise correla-tion length does not exceed 0.01c , i.e. 1% of the chord length withK = 1 102. The uncorrelated pressure field is resulted fromthe transpiration velocity along the porous surface, non-uniformlydistributed in both the streamwise and the spanwise directions.

As discussed before, the far-field acoustics for the actual spanis directly related to the spanwise coherence length of the noise

the spanwise coherence length is observed at St = 0.21, whileat other frequencies no noticeable difference or even slightly in-creased coherence lengths are found with the porous treatment.Thereby, the tonal noise at St = 0.21 is expected to be much re-duced compared to that of the solid case.

Finally, the PSD spectra of the far-field noise are compared inFig. 13 for different permeabilities. It is found that for K = 1 102, the dipole peak at St = 0.21 is significantly reduced by 13 dB,while there is no significant noise reduction with others. This isdue to the fact that with large permeability (e.g. K = 1 101),Fig. 10. Comparison of xt plot of wall pressure fluctuation at the mid-span near the trailing-edge; solid (left) and porous (right).Fig. 9. Depiction of porous surface (left); schematic of turbulent flow over a flat plate with porous trailing-edge (plenum inside) (right).Y.J. Moon / European Journal of Mesource. Fig. 12 shows the spectrally-decomposed spanwise coher-ence lengths Lc at x = 0.02c. The most prominent reduction ofchanics B/Fluids 40 (2013) 5063 57the fluid flow in porous medium encounters almost negligibleresistance and therefore the plate with porous trailing-edge is

58 Y.J. Moon / European Journal of Me

Fig. 12. Spanwise coherence length of wall pressure fluctuations for solid andporous trailing-edges.

Fig. 13. Comparison of PSD spectra at r = 20c for porous trailing-edges withdifferent permeabilities.

regarded as a plate with a shortened chord. In contrast, with smallpermeability (e.g. K = 1 103), the porous surface behaves likea solid surface so that the wall pressure fluctuations as well as thePSD of the far-field acoustics are almost identical to those for thesolid case. Only within a certain range of permeability (e.g. K =1 102), however, the porous surface provides a mechanism ofnon-uniformly distributed transpiration velocity along the lower,upper, and back-end surfaces and subsequently allows pressurefluctuations to be modified along the porous surface.

5. Applications on bio-fluid sound

5.1. Human larynx

An experimental study has been conducted at the departmentof Phoniatrics and Pediatric Audiology of the University hospitalErlangen, Germany, to measure the vibrating surface of the hu-man hemilarynx. Fig. 14 shows the human hemilarynx with a tube

inserted to blow air from the compressor and the optical mea-surement system: glass plate at the glottal midline, vocal fold andchanics B/Fluids 40 (2013) 5063

brass calibration cube on the left side of the plate, glass prism,and high-speed camera on the right. The surface grid evolutionof the human hemilarynx measured during one period of mo-tion is monotonically interpolated in time and space: raw geom-etry(left); first interpolation(middle); second interpolation(right)(see Fig. 15).

A 2D computational domain of flow and sound is configuredto include the hemilarynx vibrating at a fundamental frequencyof 140 Hz within a vocal track, 28 times of the vocal fold length(L = 12 mm). Fig. 16 shows themoving grids at four different timeintervals in one period of the vocal fold motion at the mid-plane.The first two figures are at the closure phase of the vocal fold,whereas the last two are at the opening. The case considered herecorresponds to a flow rate of 0.0004m3/s with pressure differenceof 2.88 kPa, and in terms of non-dimensional quantities, thecorresponding flow condition is at ReL = 840 andM = 3.1103.

Fig. 17 shows the instantaneous vorticity contours of the flowwithin the vocal track at the same time intervals as the movinggrids. A periodic disturbance introduced by the mucosal wave mo-tion of the hemilarynx continuously makes the gap flow pulsatingat 140 Hz and as a result, the vortices are shed downstream the vo-cal track with a specific scale. The corresponding acoustic field iscomputed with the samemoving grids by the linearized perturbedcompressible equationswith acoustic source acquired from the INSsolutions. As shown in Fig. 18, the top five figures are the instan-taneous acoustic fields at closure of the vocal fold, whereas the re-maining figures are at the opening phase. One can clearly note thatthe soundwaves generated in the human hemilarynx are very sub-tle to the vocal fold motion at each stage.

The computed 2D acoustics monitord at 11L upstream thehemilarynx are compared in Fig. 19(left) with the experimentaldata measured at the University Hospital Erlangen. Consideringthe fact that the computed sound wave is a 2D solution, it depictswell the basic characteristics of the sound produced by the humanhemilarynx. The main vocal sound at fundamental frequency of140 Hz is well resolved by the present 2D model, whereas thehigh frequency waves observed in the experiment is substantiallysimplified as a discrete tone with frequency corresponding to thescale shown in the vorticity pattern. From the present result, it isworth to note that there exits an interesting correlation among thehemilarynx motion, the vortical flow structure within the vocaltrack, and the corresponding acoustic field. A similar correlationcan also be noted in Fig. 18 (right) on the computed sound wavefrom the larynx vibrating with the same fundamental frequency.The difference of the sound wave profile between the hemilarynxand the larynx is attributed by the Coanda effect of the pulsating jetin the gap, which was clearly depicted in the computed flow fieldof the larynx.

A computation of the sound wave at this low Mach number(i.e.M < O(103)) can hardly be accessible with the compressibleNavierStokes equations and for this reason, the present hybridformulation may be considered as a viable tool for computingdiverse bio-medical fluid and sound applications at such lowMachnumbers. The full 3D surface geometry of the human larynx isnow undertaken by the present LES/LPCE hybrid method, andone can expect more complex vortical structures produced by thethree-dimensional surface motions of the human larynx and theirassociated sound waves in wider range of frequency scales, asobserved in the experiment.

5.2. Bumblebee

The unsteady flow and acoustic characteristics of the flappingwing are numerically investigated for a two-dimensional model

of bumblebee at hovering and forward flight conditions. In thisstudy, the time-dependent flow and acoustic fields are computed

nFig. 17. Instantaneous vorticity contours at four different time intervals in oneperiod of the vocal fold motion; top two at closure, bottom two at opening.

for a prescribed flapping wing motion which mimics the real wingkinematics employed by superposition of the pitching and heavingmotions, sometimes referred to as a figure-eight motion [16].

Here, all the specific parameters of the flapping wing are basedon Bombus terrestris, bumblebee [17,18]; chord length (c) is 0.8 cm,wing span (R) which is the distance from base of the wing to tipis 1.7 cm, beat frequency (f ) is 170 Hz, and stroke amplitude ()is 150, defined as the angle swept out by the leading edge fromdorsal reversal (start of downstroke) to ventral reversal (start ofupstroke) in the mean stroke plane.

For a computational domain of circle (extended to r = 500c),the incompressible NavierStokes equations are solved with mov-ing hydrodynamic grid (401 181, see Fig. 20(right)), to computeFig. 16. Moving grids at four different time intervals in one period of the vocal fold motion at the mid-plane.

Fig. 20(left) illustrates the flapping motion of a two-dimensionalellipticwing (chord length c and thickness d = 0.1c). The unsteadymotion is replicated by a wing motion of two strokes (down andup), and each stroke consists of three stages (transverse, tangential,and rotational motions).Fig. 15. Monotonic interpolatioFig. 18. Instantaneous pressure fluctuation contours at ten different time intervalsof the hemilarynx surface grid.Y.J. Moon / European Journal of Mechanics B/Fluids 40 (2013) 5063 59

Fig. 14. Measurement of 3D surface of the human hemilarynx vibrating at the fundamental frequency (left); schematic of optical measurement system (right).in one period of the vocal fold motion; top five at closure, bottom five at opening.

flapping velocity of the wing J = U/(2fR) [18] is also set to0. The Reynolds number based on the maximum translationalvelocity, Rec = Umaxc/air is 8800 andMach numberM = Umax/c0is 0.0485, where c0 is the speed of sound. Fig. 21 shows the timevariations of the drag and lift coefficients of the flapping wing. Itis indicated that the mean drag coefficient (averaged over 10 peri-ods) is nearly 0, while the mean lift coefficient is about 0.66. Fromthe definition of CL = FL/0.50UmaxcR, one can calculate the liftforce on a three-dimensionalwing, employing the properties of thebumblebee [17]. In this study, the maximum translational velocityUmax of the bumblebee is 17 m/s so that the lift force generated bya pair of 3D wing is about 3.11102N. This value is large enoughto support the weight of bumblebee, 8.63 103N, although thethree-dimensional effects neglected in this study could partiallyreduce the lift force.

Now the computed sound fields of the flapping wing in hover-ing motion are presented in Fig. 22. The result indicates that theflapping wing sound is generated by two different basic mecha-nisms. First, a dipole sound is generated by a transverse motionof the wing (two on the left). Due to the fact that the dipole axischanges its direction from downstroke to upstroke, a drag dipoleis generated at wing beat frequency (St = fc/c0 = 0.004), whilethe lift dipole is produced at 2f (i.e. St = 0.008), similar to the

Fig. 21. Time history of drag and lift coefficients (hovering).

corresponding to the lift dipole (St = 0.008) is not present orweakat 0 and 180, while the wing beat frequency (drag dipole) is alsonot present at 90 and 270. At other angles, both the drag and liftdipoles clearly exhibit their peaks. This result is similar to the pre-vious observation by Sueur et al. [19], indicating that thewing beatfrequency ismost dominant in front, whereas the second harmonicis most appreciable at sides.

Another sound source is associated with the vortex edge-Fig. 20. Flapping motion of an elliptic wing (downstroke by hollow, upstroke by filled) (left); moving grids at transverse and tangential motions (right).

the flow of a wing flapping at a non-dimensional beat frequency,St = fc/c0 = 0.004. The sound field is then computed by LPCE onthe moving acoustic grid (251 91), with minimum grid spacingfive times that of the hydrodynamic grid. Note that all the variablesinvestigated here are non-dimensionalized by the speed of soundc0, chord length c , and air density 0.

For hovering case, the flapping angle is set to 0 and the ad-vance ratio defined by the ratio of the flight speed to the mean60 Y.J. Moon / European Journal of Me

Fig. 19. Sound wave profiles at 11L upstream the vocal fold; red (hemilarynx/cominterpretation of the references to colour in this figure legend, the reader is referred todrag and lift coefficients. Hence, the flapping wing sound is direc-tional, as shown in Fig. 23. The sound pressure level (SPL) peakchanics B/Fluids 40 (2013) 5063

pt.), blue (larynx/compt.), black (hemilarynx/exp.), L: vocal fold base length. (Forthe web version of this article.)scattering during tangential motion of the wing. In Fig. 22(twoon the right), one can identify the sound waves (bracketed) at

Fig. 23. Sound pressure level spectra around a hovering insect ( = 0 and = 0; U = 0 m/s) at r = 100c and every 45 position.

150c175c from the center with wavelengths () observed as41c and 48c at t/T = 0.5 and 1, respectively. Considering thewave speed c0 (=340 m/s = 250c/T ), the travel time of thewaves is estimated as 0.60.7 (i.e. 1t/T = 150/250175/250).So, it is figured that these waves were generated during t/T =8/109/10 and 3/104/10 at each stroke. Now, one can notethat the flow fields at t/T = 9/10 and 4/10 clearly exhibit thevortical structures that are responsible for producing the dipolesound during tangential motion of the wing. The vortices in theshear layer emanated from the leading-edge scatter at the trailing-edge of the wing and generate waves radiating perpendicularly tothe wing. It is also found that the frequencies of these waves areclose to St(=fc/c0 = c/) = 1/48 0.02 and 1/41 0.024.These frequencies of dipole tones generated at the trailing-edgeagree fairly well with the theory of shear layer instability [20]: thefrequency of the shear layer breaking-off is calculated as 0.021with

by the chord length at t/T = 9/10, for example. Finally, one cannote in the spectrum that the SPL peaks are multiples of the wingbeat frequency with comparable amplitudes (see Fig. 23). Thisfrequency composition closely resembles the buzz sound of flymeasured by Sueur et al. [19].

In order to mimic the forward flight of bumblebee, all theproperties are kept constant, except the advance ratio J = U/(2fR) and stroke plane angle . The advance ratio is generallyused to determine the free stream velocity U. It ranges from 0(hovering) to 0.6 (fast flight) [21,22], and an intermediate value ofJ = 0.3 corresponding toU = 4.5m/s is considered in this study.The stroke plane angle is then determined by the force balancecondition [21]; the mean thrust must be equal to mean drag fora flight at constant speed. In the presence of free stream velocity(J = 0.3), it is found that the mean drag coefficient is nearly 0 at = 40. At this angle, the thrust force generated by the flappingloading by transverse motion and (third and fourth) vortex edge-scattering during tangential motion.Y.J. Moon / European Journal of Me

Fig. 22. Instantaneous pressure fluctuation contours around the wing in hovering mSt = 0.017ML/ , where ML = 0.044 is a local free stream Machnumber and = 0.035 is the momentum thickness normalizedchanics B/Fluids 40 (2013) 5063 61

otion; flow fields representing the associated sound sources: (first and second) wingwing is almost the same as the drag force caused by the forwardflight of bumblebee. So, the stroke plane angle is determined as

ing and trailing-edge of the wing during transverse motion are notdeveloped as symmetric as for the hovering case and so are the in-duced velocity fields. Therefore, these vortices cannot self-propelaway from the wing but rather remain in the stroke paths. Besides,the ratio between the free streamvelocity and themaximum trans-lational velocity of the wing is close to 0.26 and so the convectioneffect is quite weak. As a result, the vortices drifting around theflapping wing encounter complex wing-vortex interactions. Whencompared with the hovering case, this clear distinction in vorticalflow structure is expected to change the aerodynamic sound char-acteristics for the forward flight case.

The sound fields for the flapping wing in forward flight are in-vestigated by comparing the sound spectra in Fig. 24. Similar to thehovering case, the transverse motion of the dipolar axis results indrag (St = 0.004) and lift dipoles (St = 0.008). It is, however,important to note that the directivity change is not as clear as thatat hovering. The dominant frequency does not vary significantlyand both the drag and lift dipoles exhibit their peaks with compa-rable amplitudes, regardless of directions. One may also note thatthe dipole tones generated at the trailing-edge (St = 0.02 and0.024) are not as distinct as for the hovering case (Fig. 23). These arelargely due to the prominent interactions between the wing andthe vortices, being considered as a discernible difference in acous-tic feature between hovering and forward flight. This indicates thatthe radiation pattern and frequency composition can change withflight conditions and it is expected that these could be used as somebiological functions such as communication, territory defense, andecholocation.

6. Conclusions

noise via porous material at the same low-subsonic, turbulentflow condition. For applications of bio-medical and biological-fluidsound, capacity and future potential of the method has been welldemonstrated by the present study. The flow and sound in bio-medical and biological applications is a unique research fieldwhichrequires a very sophisticated analysis tool to emulate very weakcompressibility effects; for example, sound of blood flows in thecirculatory system or sound of airway flows in the respiratory sys-tem. It is also shown that at lowMach number, the acoustic sourcerepresented by amaterial derivative of the hydrodynamic pressure(i.e.DP/Dt) scaled by itself in the incompressibleNavierStokes so-lution is very closely related to the dilatation rate of the compress-ible counterpart, at any instant, and so is the rate of change of 1P

DPDt

to that of the dilatation rate.

Acknowledgments

The author would like to thank Prof. Roger, M. at the EcoleCentrale de Lyon and Prof. Doellinger, M. at the University HospitalErlangen for providing experimental data for the flat plate and thehuman larynx, respectively, and students, Dr. Seo, J.H., Dr. Bae Y.M.,Mr. Jo, Y.W. for their contributions during graduate study at KoreaUniversity.

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Fig. 24. Sound pressure level spectra around a hovering insect ( = 4

= 40, which is very similar to the real bumblebee [18] for theflight speed at U = 4.5 m/s.

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Sound of fluids at low Mach numbersIntroductionComputational methodologyLES/LPCE hybrid formulationLinearized perturbed compressible equations (LPCE)

Acoustic sourceApplications on aerodynamic noiseTrailing-edge noisePorous trailing-edge

Applications on bio-fluid soundHuman larynxBumblebee

ConclusionsAcknowledgmentsReferences