Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

AIAA-97-1644-CP

Numerical Analysis on Self-Excited Tone NoiseRadiated from a Two-Dimensional Airfoil Wing

Shinji NAKASHIMAAdvanced Technology R&D Center, Mitsubishi Electric Corp.

8-1-1 Tsukaguchi-Honmachi, Amagasaki 661 Japanand

Sadao.AKISHITADept. of Mechanical Engineering, Ritsumeikan University

1916 Noji-Cho, Kusatsu, Shiga Pref. 525 Japan

ABSTRACT

This paper describes the numerical analysis on thegeneration mechanism of discrete tone noise radiatedfrom a two-dimensional wing immersed in uniformflow. Linearized approximate equation of the smallperturbation of the boundary layer flow is introducedto the interactive dynamical model between the per-tubed flow and the sound near field emitted from thetrailing edge. The numerical solution of the Greenfunction of the Fourier-Laplace transformation of theperturbed flow variables is derived given the meanvelocity profile of the boundary layer. The solutionestimates well the measured configuration of the per-turbed flow development at the discrete sound fre-quency.

1. INTRODUCTION

From the nineteen-eighties to now on observationsof tone-like noise have been reported on a two-dimensional airfoil wing immersed in a uniform flow,the chord length based Reynolds number of whichranges around 10s at a low Mach number. A kindof self-excited mechanism is suggested on this sound,because ladder-like variation of the sound frequen-cies is accompanied with the flow velocity increase*1J.The flow velocity turbulence measurement of the suc-tion side boundary layer revealed the linear instabil-ity wave in the flow by the authors*2) and Nash andLowson(3). They suggested the coupling between theunstable boundary layer flow and the sound wavegenerated near the trailing edge on the self-excitedmechanism, the schematics of which is stated as fol-lows; the boundary layer flow becomes unstable atthe midway from the leading edge due to the ad-verse pressure gradient on the suction surface of theCopyright © 1997 by the American Institute of Aeronautics andAstronautics, Inc. All rights reserved.

airfoil, then a series of eddies develop downstreamtoward the trailing edge, eddies generate the soundwave on passing the trailing edge, the upstream prop-agated sound wave promotes the rise of the unstableflow in the boundary layer, which means the forma-tion of closed loop of self-excited mechanism. Theauthors conducted the control experiment where thetone noise was decreased by 6dB at maximum byintroducing the trailing edge flap motion synchro-nized with the boundary layer flow at the discretefrequency*4^. It should be noted that the amplitudeof the transverse oscillation of the flap is very small,order of 10~2mm, which eliminates possibility of can-cellation of the total circulation around the airfoilby the flap motion. These experimental result sug-gests the justification of the proposed mechanismsof the sound generation. But the feedback mecha-nism presents no quantitative models that are necces-sary for constructing the control scheme for reducingthe radiated sound pressure level. In order that theactive control for suppressing the self-excited soundfrom a wing may be promising, the interactive modelbetween the sound wave and the unstable flow in theboundary layer should be established. A theoreticalmodel is demanded for examining the active controlsystem.

Crighton discussed the interaction of sound wavewith the shear flow instability*5^. Tarn analysed theexcitation of instability wave in a two-dimensionalshear layer in free space by sound wave*8\ He ap-plied the Green function approach to the excitationof Tollmien-Schlichting waves in low subsonic bound-ary layers by free-stream plane sound waves*7). Thispaper applys the Green function approach to the ex-citation of unstable eddies in the low subsonic bound-ary layer by the sound near field emitted at

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Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

the trailing edge of a two-dimensional wing. Theauthors conduct the finite difference approach forobtaining the Green function of the small pertur-bation equation, while Tarn applied an orthonormalmethod, utilizing the mean velocity profile of theboundary layer flow mesurement. The sound nearfield model employed in this paper was proposed byTakahashi and Kaji(8) and Davis(9). The analysismodel in this paper provides with the generationmechanism of the self-excited tone noise. The resultsof the computation agree well with the results of themesurement by auhtors^ in estimating the develop-ing configuration of the unstable eddies of the tonefrequency in the boundary layers near the trailingedge.

2. BRIEF DESCRPTION OF EXPERI-MENT

in the boundary layer is shown in Fig.4(a) on the suc-tion surface and in Fig.4(b) on the prssure surface,the profiles are similar at 65mm, and 75mm down-stream from the leading edge. These velocity profilesare utilized lator in the analysis of the development ofthe perturbed flow. As shown in Figs. 2 the discretetone of 850Hz dominates the wing-radiating noise.The following analysis is focussed only on the toneand flow fluctuation at 850Hz in the boundary layer.

Fig. 5 presents the development of the flow veloc-ity fluctuation amplitude along the chord length onthe pressure surface. The rise of the fluctuation am-plitude toward the trailing edge agrees well with theprediction detailed below.

Anacholc Room

60

40

20

Wind Tunnel

850 Hz

425 Hz

1700 Hz1275 Hz

1.0 2.Q 3.0FREQUENCY, kHz

4.0 5.0

Fig.2(a) Power spectrum of the noise

10 850 Hz 1700Hz

1.0 2.0 3.0Frequency, kHz

4.0 5.0

Fig. 1 Schematics of experimental setup

An experimental setup where the flow turbulencemeasurement was conducted in the boundary layerflow over the wing surface is shown in Fig. 1. The dis-crete tones were observed radiating from the NACA0012 two-dimensional airfoil wing immersed in theuniform flow. The power spectrum of the radiatednoise at a flow velocity f/o = 15m/s is shown inFig.2(a). We can find a series of discrete tones, thedominant peak of which is at 850Hz. the flow fluctu-ation spectrum measured at 70mm from the leadingedge in the suction side boundary layer is shown inFig.2(b). Two dominant peaks are found at 850Hzand 1700Hz. The development of the displacementthichness of the boundary layer is presented on bothsuction surface and pressure surface in Fig.3. The in-creasing rate of the displacement thichness with thedistance from the leading edge is roughly constant onboth surfaces, the main-stream-wise velocity profile

Fig.2(b) Flow fluctuation spectrum in the boundarylayer (70mm from the leading edge)

1.4

1.2

0.8

0.6

0.2

• »- - Suction Surface-A- -Pressure Surface 'Uo=14.6m/S

f-004"Uo=16.3m/s

Uo=15.8m/s

Uo=15.0m/s&•"'

„ ,Uo=15.0m/s Uo=15.0m/s

^=0.0170rfx

60 65 70 75Distance from Leading Edge X, mm

80

Fig.3 Development of the displacement thickness

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1.20.2 0.4 0.6 0.8 1.0Non-dimensional Flow Velocity U

Fig.4(a) Flow velocity profile in the boundary layeron the suction surfaces; 6

O .Flow, velocityat 65.0mm70.Qmm75;6mm---

~5T 0.4 0.6 0.8 1.0Non-dimensional Flow Velocity U

Fig.4(b) Flow velocity profile in the boundary layeron the pressure surface

0.1

5 g-- Iu- < 0.01 r

c f-.9 'u

X)CO

0.001

Experimental Data

••A

y/ S =0.270.891.78

Calculation Resulty/J-1.0

Calculation Result

Calculation Resulty/S-2.0

Trailing Edge

3. MODELLING

Although the boundary layer flow is on the curvi-linear surface around the wing, we supposed a uni-form parallel flow and the two-dimensional boundarylayer flow on a flat plate for the analysis model. Wedefine the coordinate system as illustrated in Fi.6.The displacement thickness of the boundary layer8b and the velocity profile are assumed locally in-variant along the plate surface. The incident soundwave excites the unsteady flow perturbation in theboundary layer flow. When the perturbed unsteadyflow is unstable, the velocity increases exponentiallydownstream toward the trailing edge. We assume theperturbed flow variables are small compared with thesteady flow variables, even if they become unstableand grow large. The first order small perturbationequation system derived from the compressible vis-cous flow equation on a two-dimensional boundarylayer is written down for the perturbed velocity com-ponents Ud and Vd along x and y— axis, respectivelyand the perturbed pressure pd in the following.

duddt

dt

du dpd

__dx Oy Re\dx* dy

)= 0(la)

= 0 (16)

(lc)

where every variable is non-dimensionalized by thedisplacement thickness of the boundary layer Sband/or the uniform flow velocity Uo, u denotes xcomponent of the mean velocity in the boundarylayer, M does the Mach number of the uniform flowvelocity and Re does the Reynolds number based onthe velocity UQ and the length 8b. The followingboundary condition is imposed on Ud and vj.

• 2 5 - 2 0 - 1 5 - 1 0 - 5 0 5Nondimensional Distance from Trailing Edge. X T / S ,

Fig.5 Development of the velocity fluctuation ampli-tude on the pressure surface

at y =

oo

(2a)

Boundary Layer

Fig. 6 Schematic configuration of the boundaryboundary layer flow and the incident sound wave

The perturbed variables are devided into the vari-ables denoted by the subscript s representing thequantity caused by the incoming sound wave, andthe variables without any subscript representing thequantity excited by the sound waves, as follows.

(3)

Then we have the same equation for the varibleswithout subscript as for the variables with the sub-script d.

du _du du dp 1~ ~ '

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OV _ 0V Op 1 j v v v is , / jt\^7 ~*~ ^^aZ "*" ̂ 17 "" rTI I a«2 "̂ a..2 ) = ^ '^/

dp _dpdt

1(4c)

The boundary condition of the variables are writtenas follows, considering the v, is zero at y = 0 on thewing surface.

.- = o,= u = 0 • oo

(5a)

(56)where the second equation in Eq.(5a) will be detailedbelow.

The authors should note that the same equationis valid for the varibles with the subscript s, whichis suggested from Eq.(3) and Eqs.(l). The soundwave is emitted at the trailing edge, and propagatesthrough the boundary layer flow and in the uniformflow. The displacement thickness of the unsteadyboundary layer of the sound wave flow, uf and v,,is estimated to be 1/^/Re • fi/2, where fi denotesthe nondimensionalized angular frequency fiofo/t/oof fio, and therefore under one-tenth of the thicknessof u and v (10). This fact suggests that the boundarylayer of the sound wave has negligibly small thick-ness. We can assume that the boundary layer flowprofile of the sound wave is confined into the zero-thickness layer on the wing surface. The equation ofthe sound wave variables u, and v, is represented bythe following equation of unsteady nonviscous veloc-ity potential $ in the uniform flow.

-0 (6)

where the following sinusoidal sound wave of thenondimensional angular frequency fi is assumed onut and v,.

-s-ay(7)

This modeling of the sound wave boundary layeryields the non-slip boundary condition on the wingsurface, which is written in Eq.(5a).

4. SOLUTION METHOD

4.1 Sound Wavemnd wave

Main flow U/

L^Trailing Edge ( L ,0)

Fig. 7 Sound pressure field emitted at the trailingedge

The sound wave is emitted at the trailing edge inthis analysis, and it is generated by the developedvortices passing the trailing edge as illustrated inFig.7. The sound field is derived by Takahashi andKaji^8), Davis^9^ and Howe^11^, the velocity potentialof which is written down as follows,

(7,8111(0/2)

where XT denotes the x—coordinate from the trailingedge, r is defined the radial distance of the field pointfrom the trailing edge and & the angle as illustratedin Fig.7. C, is defined as the constant, which is de-termined with the experimental data in the paper.The solution expressed by Eq.(8) is well establishedon the trailing edge noise.

4.2 Instability Flow Analysis

We will find the solution of Eqs.(4) and (5) assum-ing the perturbed non-dimensional boundary layerflow excited by the sinusoidal sound wave of angularfrequency fi, which is expressed below,

U, = u,(x) exp(—t'fif) (9)

As we are applying the Green function approach, theboundary condition at y = 0 for the Green functionof the variables u, v, p is expressed by

vg = 0, ug = 6(x-£) exp(-ifi;£); at y = 0 (10)

where the subscript g denotes the Green function ofthe corresponding variable and &(•) denotes the deltafunction. As we will apply the Fourier-Laplace trans-form relations , the following notations are dennedon the transform and the inverse transform of thefunction f(k, u] and f(x, t) respectively,

/(*,«)=i rJrJ-<f(k, o>) exp i(kx — (lla)

f ( x , t) exp i(— kx + <jjt}dtdx

(116)where k denotes the wavenumber variable and LJdoes the angular frequency variable. T denotes theintegral path extending to infinity in the complexA:— plane. The Fourier-Laplace transforms of theGreen functions of u, v, p satisfy the following equa-tions,

dy Re dy dy(12a)

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Re dy2f k2

-JRe dy

k dy

= 0 (126)

(12c)

The boundary conditions, Eq.(5a) and Eq.(lO), onthe Fourier-Laplace transform of the variables isrewritten as follows,

dy(13o)

"9 — *s — o ; y ~> °° (*36)The equation systems has been transformed to theordinary simultaneous equations for the two vari-ables pg and vg and the resolved expression of ugby pg and vg. The general solution of the Eqs.(12a)and (12b) for pg and vg that satisfy the boundarycondition, Eq.(13b), is written with linear combina-tion of the two solution groups, 0 and x, as follows,

_D~ Xv

(14)

where D and F are the unknown constant tobe determined satisfying the boundary condition,Eq.(13a). The solution for D and F is expressedby

DF

ikex.p(— iy=0

(15)

where A(Ar, us) is defined by the following equation

The solution for u g ( y , k , a j ) is derived with Eq.(12c)as follows by the inspection that F = 0 is valid fromEq.(15) when <t>v(y = 0) = 0 is assumed.

»(0)— dy-fArO.-n) (17)

We should notice the condition of u> and k where thedenominators of Eq.(17) become zero, since the con-ditions yield the eigenvalues of the equation system.The eigenvalues are obtained with the equation forA(k,us) such as,

= 0 (18)

The eigenvalues we are interested in are the instableroots of k, which are denoted as k — k+ and obtained

from Eq.(18) when the imaginary part u>i(> 0) is var-ied continuously in the complex w—plane, while thereal part U>R = fl is fixed. The situation is illustratedin Fig.8, where u>i decreases to zero in the complexu—plane and the corresponding complex k moves inthe same complex plane. The unstable fc is in thelower part of the complex plane (Im(fc) < 0) when wcrosses the real axis'7).

Imag0.2'

-0.2

Time-growing mode

.2 -0.4 "••0.6Real

Spatially growing modeFig.8 Trajectory of o> and the corresponding instablek+ in the complex plane

The physical Green function which pertains insta-ble nature is obtained by the inverse Fourier-Laplacetransform as follows,

= aexp[ik+(x -0 - XU]H(x -Pg

(19)where H(•) denotes the unit step function, and a andug, vg, pg are defined as follows,

a = (20)

• + TJ- dy

(21)Finally the physical perturbation velocity u is ob-tained by the convolution intergral of ug with u, =d®(x, y = 0)/dx as follows,

f°°u= / u,(£)ug(x,y,t;£)dt

J-oorx

= a ut(£)ug ex.p[ik+(x - f) - iJ— oo

(22)

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4.3 Numerical computation technics

The Green functions pg and vg are calculated nu-merically with the finite difference equation technics.Note that Eqs.(I2a) and (12b) are the ordinary dif-ferential equations of y, which means the solution isadequate to any x— coordinate. Suppose that pg andvg are assigned on the node poits of y— coordinate,then pg and vg are transfered to the vector vari-ables composed of the variables allocated at the nodepoints. Eqs.(12a) and (12b) are rewritten in the formof the simultaneous linear equations of pg and vg asfollows,

+ Ask2 + A3k + Ao)pg + (Aik)vg = 0 (23a)

B2k + Bo)vg = 0 (236)where ^4s and Bs are defined as follows containingthe differentiation operator.

where the differentiation operators are transfered tothe finite difference operators and the subscript ymeans the derivative of y. The above equations arewritten to another linear simultaneous equation sys-tems on the variables vectors, vg and pg as follows,

(/ - kB~lA) 9 \ —= o (27)

where / denotes the unit matrix, and A and B aredefined as follows,

r> _ o --BO -

A = O

(28a)

(28.6)

We should note A and B are normal square matrix,respectively. Then, \I—kB~lA\ = 0 is the character-istic equation of k. The equation is an algeraic equa-tion of very high order of A;. We circumvent the diffi-culty of obtaining correct roots of k from the charac-teristic equation by utilizing a recursive calculationtechnics as illustrated in Fig.9. This technics startswith assuming a initial value of k and ends with ob-taining a very small error of recursion. We should

note that the boundary layer ranges over 0 < y < oo,which means some node points in the outer regioncan not be assigned. But this difficuty is overcomedby transfering y— coordinate to 77—coordinate withthe equation,

77 = tanh(ay) (29)where a = 0.2 is assumed in the computation. Thespace of 0 < y < oo is transfered to the space of0 < rj < 1 by the equation.

Given Reynolds numberand frequency.

Given Initial value k1 ?

Substitute k, intoeq.(28.b)

(26) Fig.9 Flow chart of calculation technics of kNow we are obtaining the solution of pg and vg, whichmeans we should obtain the solution of the vectors,<j>v, <pp and x«> XP- The equation systems of 0s andXS are written as follows including the variables atthe boundary.

C v C p 0 0D v D p 0 0

0 0 1 0

.0 0 /3i 02

<Pv Xv0p XP

0wO XvOVpQ XpO

f O Oo o0 1

i -I U

where <pvO, 0pO, XM and Xpo denote the variable onthe boundary y = 0 of <pv, (j>p, x*> and XP, respec-tively. Cs Ds and /3s are defined as follows,

Ot; — £t%K\L-tt

+ ^-jM+M2(0-Gfc) 2

(31)

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Since this equation is a normal simultaneous linearequation, we can obtain the solution of 0s and x&by solving the equation. We should add that forobtaining a in Eq.(20) we need the derivatives of k atk = k+ ,then the solution of Eq.(30) is derived at thedeviated value k from fc+ such as k = k+(l+e), e =±0.005.

5. RESULT OF ANALYSIS

5.1 Mesurement DataWe utilize results of the mesurement as the knowndata in the calculation. Now we introduce the formu-lation of the measurement data shown in section 2.Fig.3 shows the development of the boundary layerthickness along the chord length. The variations ofthe thickness seem linear with the x-coodinate asfound in Fig.3. We assume the starting point of theboundary layer at the point where the linear curvecrosses the x—axis, which means that the boundarylayer starts at x = 43 (mm) on the suction surfaceand it also starts at x = 50(mm) on the pressuresurface. These values are utilized at the integrationin Eq.(22), where -co is substituted by these valuesas the lower limit of the integral.

We assumed the nondimensional mean velocityprofile of the boundary layer as given. The profilesare approximated by the following equations fromthe measurement shown in Figs.4.

u = I - exp(-1.10y3 + 0.440j/2 - 0.0559y)

; at suction side (32a)

u = l-exp(-0.01514j/5-f 0.2014y4-0.7139y3

+0.2398y2 - 0.4854*/) ; at pressure side(32.6)

The solution of the emitted sound field demands theconstant C3 to be determined. The constant is fixedwith the measurement data shown in Fig.5. The fig-ure illustrates the variation of the flow fluctuationalong the chord length on the pressure surface. Asthe amplitude of the flow fluctuation is very smallin the pressure surface, the curve in the figure seemsto represent the amplitude development of the soundfield. Therefore

by a hot wire anemometer, as defined below.

C, = 0.0212 (33)

is fixed as stated above. The calculated curve basedon Eqs.(8) and (33) in the figure agree well with themeasurement.

In order to compare the velocity fluctuations be-tween the measurement and the computation, thevelocity fluctuation / should be expressed as the in-tensity of the fluctuation supposed to be measured

/2 = + + 0062(0! + 02) (34a)

= tan-1 ^ —- (34c)

02 = ten~1 \ —-

5.2 Comparisons between Calculations andMeasurements

Non-dimensional Frequency flFig. 10 Variation of the flow fluctuation amplitudegrowth rate with the angular frequency fi

g 0.6

65.0mm from leading edg«.(R«=900)70mm (Re=l100)

75.0mm (R«=1200)

0.2 0.4 0.6 0.8 t \2Non-dimensional Frequency fl

Fig. 11 Variation of the flow fluctuation phase veloc-ity with the angular frequency ftFigure 10 shows variation of the imaginary part ofk+ with the non-dimensional angular frequency Q.The imaginary part represents the special growthrate of the flow fluctuations. The calculated curvehas the peak at fi ~ 0.5 at the Reynolds numberRe = 500. The magnitude of maximum is about0.3, while the measurement data range from 0.24to 0.34, which means the measurement has smallervalue than the calculation, but the global tendencyis similar in them.

Figure 11 shows variation of the flow fluctuationphase velocity, the real part of k+, with the non-dimensional angular frequency fi. The calculatedcurve has the maximum value of about 0.5, -while

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Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

the measurement ranges from 0.52 to 0.58, a littlelarger than the calculation.

Figure 12 shows the development of the velocityamplitude of the maximum flow fluctuation on thesuction surface along the chord length x. The cal-culated curve based on the mean velocity profile atx = 70(mm) agrees well with the measurement data.The measurement data indicate that the velocity am-plitude does not exceed 10% of the velocity of themean stream, while the calculated curves reach farlarger. This error comes from the correctness limitof the linearized differential equations. Figure 13shows the same development on the pressure sur-face. The calculated result suggest that the calcu-lated curve based on the mean velocity profile atx = 65 ~ 75(mm) is far lower than the curve rep-resenting the sound near field caluculated by Eq.(8),and that the velocity fluctuation on the pressure sideis governed by the sound near field. Figure 14 showsvariation of the phase delay of the maximum flow ve-locity along the chord length x. The calculated curvebased on the mean velocity profile at x — 75(mm)agree well with the measurement data in a range ofx < 65(mm). We can guess that the phase delayfrom a point x — 65 (mm) to the trailing edge is just720deg.

M=0.046,Re=>900(65.0mm from leading edge)

0.000150 55 60 65 70 75

Distance from Leading Edge, x mm

Fig. 12 Development of the flow fluctuation ampli-tude on the suction side

M=0.044,Re=480(75mm)

M=0.044,Re=°400(70mm)

M=0.044,Re=320(65.0mm from leading edge)

55 60 65 70 75Distance from Leading Edge, x mm

800

600

2 400

13 200>l— 0<DQo -200

-400

-600

-800

M=0.046,Re=900(65.0mm from leading edge)

M=0.042.Re=1200(75mm)

50 80

Fig. 13 Development of the flow fluctuation ampli-tude on the pressure side

55 60 65 70 75Distance from Leading Edge, x mm

Fig. 14 Variation of the phase delay of the flow fluc-tuation on the suction side

The distribution of the flow fluctuation velocityamplitude is calculated along y—axis by use of thedisplacement thickness and the mean velocity porfileat x = 70(mm). Fig.l5(a) and (b) show the dis-tribution curves at x = 70(mm) and x = 75(mm),respectively. It is found that the calculated curvesagree well with the measurement data at y > 0.8and that the configuration of the curves are similarto those of the measurement data. Figures 16 (a) and(b) show the variation of the phase delay of the flowfluctuation velocity along y— axis by use of the samedata above. The calculated curves agree well withthe measurement data at y > 0.5.

6. CONCLUSION

The numerical analysis was conducted on the gener-ation mechanism of discrete tone noise. Linearizedapporximated equation of the small perturbation ofthe boundary layer flow is introduced to the inter-active dynamical model between the perturbed flowand the sound near field emitted at the trailing edge.The numerical solution of the Green function of theFourier-Laplace transform of the perturbed flow vari-ables is derived given the the measurement data. Thesolution estimates well the measured configuration ofthe perturbed flow development.

References(1) Paterson,R.W., and et al. "Vortex noise of isolatedairfoils", J. of Aircraft, Vol.10, 1973.(2) Akishita,S., "Tone-like noise from an isolated two di-mensional airfoil", AIAA Paper No.86-1947, 1986(3) Nash,E.C. and Lowson,M.V., "Noise due to bound-ary layer instabilities", Proc. First Joint CEAS/AJAAAeroacoustics Conference, Vol.2, pp.875, 1995.(4) Nakashima,S., Ohtsuta,K. and Akishita,S., "Discretetone reduction control on a two dimensional wing", AIAAPaper No.93-4395.(5) Crighton,D.G., "Acoustics as a branch of fluid me-chanics", J. Fluid Mechanics, Vol.106, pp.261, 1981.(6) Tam,C.K.W., "Excitation of instability waves in a

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Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

two-dimensional shear layer by sound". J. Fluid Mechan-ics, Vol.89, pp.357, 1978.(7) Tarn The excitation of Tollmien-Shlichting wavesin low subsonic boundary layers by free-stream soundwaves", J. Fluid Mechanics, Vol.109, pp.483, 1981.(8) Takahashi,K. and Kaji,S., "Analytical Study onSound Generation by Vorticity Waves Passing throughTrailing and Leading Edges of Semi-Infinite Plates",Proc. DGLR/AIAA 14th Aeroacoustics Conference,Vol.1, pp.476, 1992.(9) Davis.S.S., "Theory of Discrete Vortex Noise", AIAAJournal, Vol.13, pp.375, 1975.(10) Schlichting,H., "Boundary-Layer Theory",Chap.XV, Sixth Ed. McGraw-Hill 1968.(11) Howe.M.S., "The influence of vortex shedding on thegeneration of sound by converted turbulence", J. FluidMechanics, Vol.76, pp.711, 1976.

_ott.

e3

0.015-

-O- Experimental data—— Calculated Result

nimcnsionlcss Distance from Airfoil Surface y.

-O- Experimental data—— Calculated Result

0.5 1 1.5 2 2.5 3 15 < <.5

Dimcnsionlcss Distance from Airfoil Surface y.

(a) Variation at x = TOfrnm)200,—————,————:—————i—————,—————

-O- Experimental data__ Calculated Result

Dimensionlcss Distance from Airfoil Surface y.(b) Variation at x = 75(mm)Fig. 16 Variation of the phase delay of the flow fluc-tuation velocity

(a) Distribution at x = 70(mm)

-O- Experimental data__ Calculated Result

Q ______i™0 O.i i 1J 2 2^5 3 M~ « t.i

Dimcnsionless Distance from Airfoil Surface y.

(b) Distribution at x = 75(mm)Fig. 15 Distribution of the flow fluctuation velocityamplitude

480