discrete tones of isolated airfoils

Discrete tones of isolated airfoils

Christopher K. W. Tam

Department of Mathematics, Florida State University, Tallahassee, Florida 32306 (Received 9 September 1973; revised 7 January 1974)

Recent experimental measurements reveal that discrete tones are emitted by isolated airfoils under certain operating conditions. Arguments are presented that these tones are unrelated to vortex shedding processes as has been suggested. It is proposed that the tones are generated by a self-excited feedback loop of aerodynamic origin. The loop consists of large-scale unstable disturbances in the boundary layer and wake flow and the feedback acoustic waves. Quantitative deductions based on the feedback loop model compare favorably with experimental measurements. It is also found that the proposed model is consistent with the observed characteristic features associated with the discrete tone phenomenon.

Subject Classification: 28.65; 5 0.55.

INTRODUCTION

Noise from isolated airfoils is often thought to be broad band. Sharland 1 suggested that this broad-band noise was generated by vorticity in the wake shed from the airfoil, by the turbulent boundary layer adjacent to the airfoil, and by random disturbances in the incoming flow which caused unsteady fluctuations in the angle of incidence. Recently, several independent experiments revealed a rather surprising observation that discrete tones are emitted from isolated airfoils under certain

operating conditions. Over a wide range of flow Rey- nolds number based on the chord of the airfoil, the total noise power emitted could be dominated by the discrete tones. Clark •' studied the noise from a NACA 65 series airfoil in a laminar flow condition. He found a well-

defined sharp peak in the noise power spectrum, using one-third octave-band measurements. Hersh and Hay- den, 3 in their investigation on the effect of leading edge serra!ion on sound radiated from airfoils, made exten- sive noise power spectrum measurements of a NACA 0012 airfoil. Their results (one-third octave-band mea-

surements) indicated that the total noise power was con- centrated in a narrow band of frequencies. Smith e! al. 4 carried out field measurements of the radiated noise

from sailplanes. The spectrum for a Libelle sailplane at 104 ft/sec from their Fig. 66 shows a strong tone in the 1000-Hz one-third octave band suggesting the possible presence of a discrete tone. More recently, Pat- erson e! al. • performed a well-planned and carefully executed series of noise measurements on a NACA 0012

and a NACA 0018 two-dimensional airfoil. Their experiments include both one-third octave and 10-Hz narrow-

band measurements. The 10-Hz narrow-band results

demonstrate undoubtedly the existence of discrete tones. Further, the frequencies of the tones appear to be well- defined functions of the flow velocity. Their relationship forms an organized pattern in a frequency versus velocity plot. As the velocity increases, a multiplicity of tones can arise. The overall results of the experiments seem to indicate that the tones are generated by an as yet unknown, but highly ordered flow phenomenon. Based on their data, Paterson e! al. believed that discrete tones are generated by a full-size helicopter rotor. Ow- ing to the variation of actual speed along the rotorblades, the discrete tones emitted from different sections of the

blade will have different frequency. As a result, they are often confused as broad-band noise. Patersonetal.

estimated the frequency range of the discrete tones of the tail rotor of a Sikorsky helicopter using a modified NACA 0012 section. They found good agreement with measured values.

In the past, aerodynamic noise with discrete frequencies were often regarded to be the consequence of vortex shedding. This belief was motivated by the well-known Karman vortex street which occurs in a highly organized fashion in the wake of bluff bodies. Following this line of reasoning, Paterson e! al. • suggested that the observed discrete tones were vortex noise. However, they did not elaborate physically how sound waves were produced by the vortex shedding process or the vortex system itself. They argued that one should be able to correlate the discrete tone frequencies by means of a Strou- hal number with the thickness of the boundary layer at the trailing edge of the airfoil as a length scale. This is analogous to the successful Strouhal number correlation of the Karman vortex street. By means of such a correlation, it is possible to give a rough estimate of the tone frequency even though the measured data actually scattered around the correlation formula. Earlier, Clark •' attempted a simple Strouhal number correlation of the peak frequencies of his data. His measured results, like those of Paterson e! al., did not fall on a straight line in a frequency-velocity plot. He concluded that a satisfactory simple correlation of this kind did not exist. Simple as it is, it is believed that Clark's conclusion is correct. To provide further evidence that the observed discrete tones have no relation to vorticity in the wake flow, the observational facts of the various experiments are summarized below. Many of these facts are contradictory to the concept of vortex noise. Indeed there is very little compelling evidence to suggest that the discrete tones are in any way generated by vortices in the wake of an airfoil.

In this paper, a new generation mechanism for the observed discrete tones of isolated airfoils is proposed. It is believed that these tones are the direct consequence of a self-excited feedback loop formed by the acoustic field, the boundary layer, and the wake flow of the airfoil. The physical processes involved are explained in

1173 J. Acoust. Soc. Am., Vol. 55, No. 6, June 1974 Copyright ¸ 1974 by the Acoustical Society of America 1173

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1174 C.K.W. Tam: Discrete tones of airfoils 1174

I0000

5OO0

N

uJ 2000

IOOO

700

ANGLE OF SYM ATTACK, DEG :

_

SECONDARY TON ES o o,,u ' /

// - upper

.• / . lower

I • / I / I I 50 I00 200 500 500

VELOCITY, U, FPS

I , 6 ßANGLE OFATTACK J • j• U=131FPS o

> •o 8 FPS 01: ...... .

o•[

0 ! I I I I I I 1500 1500 1700 1900 2100

FREQUENCY, Hz

FIG. 3. Farfield spectra, NACA 0012 airfoil at 10-Hz band- width showing multiplicity of tones. [Data from Paterson ef al. •]

FIG. 1. Effect of velocity on fairfield tone frequencies, NACA 0012 airfoil. [Data from Paterson et al. 5]

detail in Sec. III. It is shown that the proposed noise generation mechanism based on a self-excited feedback loop is consistent with the observed facts. Quantitative results are derived which compare favorably with experimental data.

I. CHARACTERISTIC FEATURES OF OBSERVED PHENOMENON

The experiments of Clark, •' Hersh and Hayden, 3 and Paterson et al. s reveal many characteristic features of the flow and acoustic fields associated with the emission

of discrete tones from isolated airfoils. These charac-

teristic features are of great importance, for they are the clues by which we can hope to understand the genera-

5OOO

2OOO

N

u_ I000

5OO

ß DATA FROM PATERSON ET AL. o SECONDARY TONES

NACA 00'2 AIR FOIL, 6 ø ANGLE Of ATTACK

--

i I I

50 I00 200 500

VELOCITY, U, FPS

FIG. 2. Effect of velocity on farfield tone frequencies showing fine structure of data points.

tion mechanism of the discrete tones. Below is a brief

description of these special characteristics.

(a) The presence of discrete tones is associated with the presence of laminar boundary layer on the pressure side of the airfoil. Hersh and Hayden 3 performed a trip wire experiment of the boundary layers of both the suction and pressure sides of the airfoil. They found that a trip wire had very little effect on the discrete tone signal when placed on the suction side. However, if the boundary layer on the pressure side was tripped so that it became turbulent, the tone would disappear. The same experiment was repeated by Paterson et al. • with an identical conclusion. Further evidence that laminar

boundary layer on the pressure side of the airfoil is es- sential to the generation of discrete tones is provided by Paterson et al. They found that the transition line in the angle of attack versus velocity plane between tone and no-tone regimes, coincided with the pressure surface transition line measured by McCroskey. 6 Thus, the no-tone regime corresponds to the turbulent regime.

(b) The frequency of the discrete tone increases with increase in flow velocity. Figure i shows the measured data of Paterson et al. • It can be seen that all the data

points lie within a narrow band in the frequency velocity plane. Similar results were obtained by Clark, •' although his frequency measurements are less accurate. Pater- son et al. pointed out that the measured values exhibited a fine scale structure. Figure 2 is a plot of the experimental results at a 6 ø angie of attack. The data show a ladder-type structure. With small increase in velocity, the frequency of the discrete tone exhibits a 0.8-power dependence on velocity. At some velocities, however, more than one discrete tone was apparent. Figure 3 contains several 10-Hz narrow-band measurements

showing the multiplicity of tones as obtained by Paterson et al. Paterson et al. further reported that increasing the flow velocity occasionally caused a sudden jump in frequency from one ladder step (in Fig. 2) to another.. Similar behavior was found when the airfoil was placed at other angles of attack.

(c) The sources of the discrete tones are located very near the trailing edge of the airfoil. This result was found by Paterson et al. • using autocorrelation and cross-

J. Acoust. Soc. Am., Vol. 55, No. 6, June 1974


1175 C.K.W. Tam' Discrete tones of airfoils 1175

o 90 q o

.2 80-

rr 70-

IJJ

z 60- o o r.,,o 50

!

!

o

o / •o

/ o ½P o / •

! o ! E•O

NACA 0012 AIR FOIL

ANGLE OF ATTACK 6 ø

I

IOO

oDATA FROM PATERSON ETAL.

200 :300

VELOCITY, U , FPS

FIG. 4. Effect of velocity on farfield tone amplitudes.

correlation measurements. Moreover, by comparing the cross correlation to the autocorrelation of the dis-

crete tone above and below the airfoil, it was found that there was a phase shift of 180 ø. This indicated that the noise source had the character of an oscillatory dipole.

(d) At small velocity, the sound pressure level of the discrete tone increases with increase in velocity. How- ever, at moderate flow velocity, the amplitude of the tone becomes saturated. Figure 4 shows the experimental results of Paterson et el. ,5 which clearly demonstrate the phenomenon of amplitude saturation.

II. VORTEX NOISE CONCEPT AND ITS INADEQUACY

Paterson e! al. 5 attempted to explain the observed discrete tones as a phenomenon associated with the process of vortex shedding from airfoils. They seemed to believe that the noise measured in the tone regime quali- tatively resembled discrete-frequency vortex shedding noise normally associated with bluff bodies. They then argued that since the presence of a laminar boundary layer on the airfoil pressure surface appeared to be cen- tral to the existence of the tone, a Strouhal number of 0.2 (that associated with bluff body shedding), referenced to twice the laminar boundary layer thickness at the air- [oil trailing edge, should give the relevant nondimension- al frequency scaling law. Thus, we have

S=0.2=2fS/U , (1)

where S is the Strouhal number, f is the tone frequency, 5 is the boundary layer thickness, and U is the flow velocity. The assumption that the length scale was a characteristic dimension of the wake followed Roshko, 7 who postulated that the vortex shedding frequency must de- pend only on wake width and wake velocity. As a first approximation to the airfoil laminar boundary layer thickness at the trailing edge, Paterson et al. used the result obtained for a flat plate (zero pressure gradient)-

8 = 5c/R z/2 , (2)

where c is the width of chord and R is the Reynolds number based on c. Substitution of Eq. 2 in Eq. 1 resulted in the following frequency scaling formula'

(3)

where • is the kinematic viscosity and/4 is a constant equal to 0.02. Equation 3, unfortunately, does not fit the measured data. Recognizing this, Paterson e! al. left the constant/4 as a free parameter. They found that the best general fit to the data was obtained by choosing /4 to be 0.011. This is shown as the full straight line in Fig. 1. However, as can be seen, the measured data actually scattered around the above theoretical line in- stead of lying on top of it. We are inclined to think that the scaling formula does not correlate the data, just as was found by Clark. z Although the simple formula of Paterson et al. can give an order of magnitude estimate to the tone frequency, we believe that the qualitative agreement is accidental and the discrete tone is not vortex noise as they suggested.

After a closer examination of the characteristic fea-

tures of the discrete tone phenomenon as described above in Sec. I, we are further discouraged in referring the tones as vortex noise. First of all, an airfoil is not a bluff body. Its wake does not break off to form vortices as in the case of a bluff body. This is evidenced by pictures taken by Mattingly and Criminale 8 using the hydrogen bubble technique (R • 1.6x104-4.0X104). In- stead, the wake rolls up into a vortex street only as a final stage of development initiated by wake instabilities (see Fig. 5). The formation of vortices takes place at a distance so far downstream of the trailing edge of the airfoil that it is hard to associate them to the noise

source of the discrete tone which is known to be near the

trailing edge. Secondly, it is difficult to see, as ad- mitted by Paterson et al., 5 how their vortex noise concept can explain the fine structure of the measured data as shown in Fig. 2. Nor can the jumps in frequency which occasionally take place be easily understood within the same context. Thirdly, it has never been reported in the literature that a solid streamlined body would generate two vortex systems at different frequencies co- existing with each other. If this is not possible for an airfoil, then the observational fact of multiplicity of discrete tones would imply that these tones are not generated by vortices or their shedding processes. With the above observations and reasonings in mind, we feel that

AIR FOIL ,(

LOCALIZED % NOISE SOURCE

/ '% WAKE --/-• • FLOW

/ /

/

/

/ •ACOU STIC • • WAVE

FIG. 5. Sketch of wake flow pattern behind an airfoil. [From picture taken by Mattingly and Criminale s using hydrogen bubble visualization technique. ]



1176 C.K.W. Tam: Discrete tones of airfoils 1176

there is sufficient justification to abandon the concept of vortex noise of Paterson et al. altogether.

III. SELF-EXCITED FEEDBACK GENERATION MECHANISM

Now we propose that the observed discrete tone is generated by a self-excited feedback loop of aerodynamic origin. The loop consists of an unstable laminarbound- ary layer on the pressure side of the airfoil which, upon merging with the boundary layer on the suction side, forms the near wake flow as shown in Fig. 5. Forced by acoustic waves that impinge on the boundary layer, disturbances are initiated in the wake flow at the sharp trailing edge of the airfoild (point A, Fig. 5). The disturbances, in the form of large-scale unstable waves, grow in amplitude as they propagate downstream along the wake. When these disturbances acquire sufficiently large amplitude, they cause the wake to vibrate laterally as shown near point B in Fig. 5. These lateral vibrations of the wake induce the emission of acoustic waves

in a manner very much like the lateral vibrations of a solid flat plate. The flow visualization pictures of Mat- tingly and Criminale s on the wake of an airfoil indicated that the lateral vibrations of the wake were impercepti- ble near the trailing edge of the airfoil. They became noticeable rather abruptly at some distance downstream. Since acoustic waves are produced by the actual unsteady lateral displacement of the wake, the noise source region in the present case is highly localized. As a first approximation, we will regard it as a point source at B in Fig. 5. The acoustic waves so generated propagate in all directions. Part of the waves will reach the pressure side of the airfoil near the trailing edge where they force the boundary layer to oscillate. In this way, the feedback loop is completed.

According to the feedback model above, disturbances are continuously being initiated at the sharp trailing edge of the airfoil. For this process to be most efficient, it is necessary that the flow be unstable there. We will now use this instability requirement to deduce that the frequency of the discrete tones generated by the feedback loop must lie in a certain restricted band. For this pur- pose, we will approximate the boundary layer on the pressure side of the airfoil by that of a flat plate. The stability characteristics of a flat plate boundary layer (Blasius solution) have been studied extensively by various authors in the past. Figure 6 shows the stability boundary on a dimensionless frequency versus Reynolds number plot calculated by Lin 9 and Shen. •0 The region of instability is enclosed by the neutral stable curve. In this figure, co is the angular frequency, U is the flow velocity, v is the kinematic viscosity, and R• is the Rey- nolds number based on the displacement thickness. Now) from this diagram, it is easily seen that for a given flow velocity and chord width, the frequency of unstable disturbances is restricted to a narrow range bounded by the upper and lower branch of the neutral stable curve. To verify this idea we will use the data of Paterson et al. 5 with v taken to be 1.6x 10 '4 ft•'/sec at 68 øF. By means of the neutral curve of Lin and Shen, it is straightfor- ward to find that the frequency of discrete tones observed by Paterson et al. should be restricted to a band between

400

NEUTRAL STABLE

•X• CURVE

I I I I I I I I I I I

500

200

I00

øo iooo 2000 - R I

FIG. 6. Curve of neutral stability for Blasius flow. [From Lin 9 and Shen. 10]

the two dotted curves in Fig. 1. The upper and lower dotted curves correspond to the upper and lower branch of the stability boundary of Fig. 6. From Fig. 1 it is clear that almost all the experimental points do lie within the frequency band required by stability consideration. The few that are slightly outside the upper dotted curve indicate that if the true boundary-layer neutral stable curve is used (with pressure gradient properly accounted for)• the upper branch would be slightly above that of the Blasius profile. In any case, considering the approximations involved• the agreement between theory and experiment can be regarded as highly favorable. (Note that the bounds imposed on the frequency by instability consideration of the wake flow are less stringent than those of the boundary layer; hence, we need not consider them here. )

Let us now turn to the feedback loop itself. In order to form a loop, the phases of the large-scale disturbances in the wake flow and the feedback acoustic waves

outside the wake must satisfy certain integral relation- ships. From hydrodynamic stability theory (see Lin 9) the disturbances in the 'wak• are known to have the fol-

lowing mathematical representation:

p(x, y, t) = Re {•(y) exp[i(krx - •ot) - k ix]} , (4)

v(x, y, t) = Re {,(y) exp[i(k•x - cot) - kix]} .

Here, p and v are the pressure and velocity associated with the large scale disturbances. x and y are the coor- dinates in the direction of flow and normal to the direc-

tion of flow, respectively, with respect to a Cartesian coordinate system centered at the trailing edge of the airfoil. co is the angular frequency. k r and k i are the real and imaginary part of the wavenumber. Re{ } de- notes the real part. it is also well known that the acoustic waves generated by a line source at B in Fig. 5 can be represented mathematically by

p = Re {•(r)e i(r/a-t)o•}, (5)

where a is the speed of sound and r is the radial distance



1177 C.K.W. Tam' Discrete tones of airfoils 1177

I0000

5OO0

2OOO

IOOO IOO

DATA FROM PATERSON ET AL. / SECONdArY tONE NACA 0012 AIRFOIL, IO ø /"•

ANGLE OF ATTACK /,,,•'•.•.•,

11.8 nU ø'8

/ I I I [ 200 :500 400 500

VELOCITY, U, FPS

FIG. 7. Effect of veIocity on tone frequencies at a 10 ø angie of attack.

with respect to a cylindrical coordinate system whose axis coincides with B, the line source. Let L be the dis- tancebetween A and B in Fig. 5. The phase change acquired by the large-scale disturbances in going from A to B along the wake according to Eq. 4 is Lco/c, where c = co/k r is the phase velocity of the large scale distur- bance. Similarfly, the phase change acquired by the acoustic waves in going from B back to A, according to Eq. 5, is Lco/a. The total phase change in going from A to B and back along the feedback loop is therefore equal to

(l/c + l/a) coL .

Since nothing should change by going around the closed loop once, the uniqueness of solution requires that the total phase change must be equal to an integral multiple of 2v, i.e.,

(1/c + l/a) col = 2•rn , (6)

with n = 1, 2, 3, .... Therefore, the frequency of the discrete tones is given by the expression,

f= (1/c + i/a)L =ng(U' a) . (7) In Eq. 7, the function g(U, a) which depends on the flow velocity U and the angie of attack a, is unknown. How- ever, the formula does stipulate that the frequency of the discrete tones must have a ladder-like structure with

various steps given by various integer values of n. To test the validity of Eq. 7, we will compare it with the data of Paterson et al. s Figure 2 shows the ladder structure obtained at a 6 ø angie of attack. By fitting Eq. 7 to line EE with n set equal to 4, we find

f=6.85 n U ø'8 Hz; U in ft/sec, n = integer . (8)

Now by letting n = 5, 6, and 7, we obtain the parallel dotted lines as shown. These lines seem to match the

experimental points remarkably well. Figure 7 shows the tone frequency data of Paterson et al. measured at a 10 ø angle of attack. By fitting Eq. 7 to line FF with n set equal to 4, we find

f=ll.8nuø'SHz; Uinft/sec, n=integer . (9)

As before, by letting n=2, 3, 4, and 5, we obtain the parallel dotted lines shown. Again, these lines match the measured values remarkably well, indicating that the discrete tones are generated by a feedback mechanism.

Finally, we would like to examine whether our proposed discrete tone generation mechanism is consistent with the observed special features outlined in Sec. I. From the above discussions and comparisons with experiments, it is clear that the self-excited feedback loop is consistent with features (a) and (b). In (b), the multiplicity of tones merely implies that more than one unstable wave modes of the boundary layer and wake flow are excited. In this case, more than one feedback loop are in operation at the same time. Also, according to our model, sound is generated by the lateral vibration of the wake flow. Thus, the nature of the noise source is like that of an oscillating dipole just as described in (c). To explain point (d), it is necessary to understand that the energy which drives the oscillations in the feedback loop and the emission of discrete tones comes pri- marily from instabilities in the wake flow. Recently, Ko, Kubota, and Lees n studied the effect of finite amplitude disturbances in the wake flow of a flat plate. Their results indicated that the amplitude of the finite disturbances would exhibit a saturation phenomenon. In our model, the discrete tones are generatedby the large- scale disturbances in the wake flow. It is, therefore, natural to expect a saturation of the tone amplitude when its energy source, namely, the amplitude of large scale disturbances is saturated. (Note: in Eq. 7 the length L is insensitive to the tone frequency as evidenced by the phenomenon of multiplicity of tones. On the other hand, the phase velocity c of the unstable wave in the wake flow is actually a very weak function of co. How- ever, within the range of flow velocities considered, this dependence is so weak that it can be neglected. )

ACKNOWLEDGMENT

This work was supported by the National Science Foun- dation under grant GK-35790.

lI. J. Sharland, J. Sound Vib. 1, 302 (1964). 2L. T. Clark, Trans. ASME, J. Eng. Power 93, Set. A, 366

(1971).

3A. S. Hersh and R. E. Hayden, NASA Contr. Rep. CR-114370 (1971).

4D. L. Smith, R. P. Paxson, R. D. Talmadge, and E. R. Hotzo, TM-70-3-FDAA, U.S. Air Force Flight Dynamics Lab., Wright-Patterson Air Force Base, Ohio (1970).

5R. W. Paterson, P. G. Vogt, M. R. Fink, and C. L. Munch, J. Aircr. 10, 296 (1973); Rep. K910 867-6 United Aircraft Re- search Lab., East Hartford, Conn. (1971).

6W. J. McCroskey, NASA TN-D-6321 (1971). ?A. Roshko, J. Aeronaut. Sci. 22, 24 (1955). 8G. E. Mattingly and W. O. Criminale, J. Fluid Mech. 51,

233, (1972).

9C. C. Lin, The Theory of Hydrodynamic Stability (Cambridge U.P., Cambridge England, 1955).

løS. F. Shen, J. Aeronaut. Sci. 21, 62 (1954). llD. R. S. Ko, T. Kubota, and L. Lees, Fluid Mech. 40, 315

(1970).



discrete tones of isolated airfoils

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