numerical study of wind-tunnel acoustic resonance …€¦ · wind-tunnel acoustic resonance...

NUMERICAL STUDY OF WIND-TUNNEL ACOUSTICRESONANCE INDUCED BY TWO-DIMENSIONAL AIRFOIL

FLOW AT LOW REYNOLDS NUMBER

Tomoaki Ikeda�, Takashi Atobe�

Yasufumi Konishi��, Hiroki Nagai�� and Keisuke Asai��Japan Aerospace Exploration Agency, ��Tohoku University

Keywords: aeroacoustics, low Reynolds number flow, aerodynamic characteristics

Abstract

In the aeroacoustic measurements of a wind-tunnel test, the acoustic resonance should beavoided, associated with the walled test section.The present numerical study of an NACA0012airfoil focuses on how the wall resonance affectsunsteady flow motions via a feedback process, bycomparing with the airfoil placed in a free stream.Tonal frequencies observed in the present simu-lations agree well with our previous wind-tunnelexperiments, represented approximately by thediscrete resonant modes derived through a simplegeometrical relation. More importantly, however,the present results indicate that rather strong wallresonance may alter the hydrodynamic flow mea-surements as well. The acoustic feedback processstimulates the transitional boundary layer on thesuction side, which would increase lift force inthe acoustically resonant channel by suppressingtrailing-edge separation. At a higher angle of at-tack, the increment of lift force becomes moresignificant due to the noticeable size reduction ofa separation bubble, understood via the compar-ison of three-dimensional instantaneous vorticalstructures.

1 Introduction

This paper presents the numerical simulations ofa two-dimensional flow around an NACA0012airfoil, confined within wind-tunnel walls at alow Reynolds number. Trailing-edge noise is of-ten observed at moderate Reynolds number, as

acoustic scattering from a sharp edge when co-herent eddies are shed into wake. The tonalnoise emission is supposedly retained via the for-mation of an acoustic feedback loop, the reso-nance between hydrodynamic instability wavesin boundary layer and acoustic disturbances trav-eling upstream. Specifically, at low Reynoldsnumber, the frequency of tonal noise, or vortexshedding goes down to O�1�, normalized withchord length and uniform-flow velocity. In termsof unsteady aerodynamics, the emitted acousticdisturbance serves as an unsteady component ofaerodynamic force. As shedding frequency low-ers, the unsteady aerodynamic force may becomenon-negligible [3, 4].

In the experiments to measure airfoil tones,open jet facilities are often employed with ane-choic environments [e.g., 2, 11]. In closed wind-tunnel environments, however, solid walls shouldbe acoustically lined to avoid unwanted reso-nance within a test section [e.g., 7, 13]. Other-wise, a closed channel induces tones of acous-tic resonance, whose frequency is determinedmostly by the section hight at low Mach num-ber, less influenced by flow velocity [7, 10, 12].Our previous experiment of an NACA0012 air-foil, conducted in the Mars Wind Tunnel [8] alsoshows fixed tones, nearly independent of flow ve-locity. The fixed tonal noise did not vanish com-pletely even after an acoustic absorber was in-stalled on the test section. In the study, one ofthe primary purposes was to attain experimen-tal evidence of an acoustic feedback loop at low

1

T. IKEDA, T. ATOBE, Y. KONISHI, H. NAGAI AND K. ASAI

Reynolds number, which turned out to be difficultunder the influence of wall resonance.

The present paper discusses how the wall res-onance affects the mechanism of airfoil tone gen-eration. For this purpose, numerical simulationsof two-dimensional NACA0012 airfoil are con-ducted with a rigid-wall boundary condition ap-plied on the test-section walls. Under the influ-ence of wall resonance, unsteady motions are al-tered from those observed in a uniform flow with-out external walls. We focus on how the acous-tic disturbance due to wall resonance is fed backinto the airfoil unsteadiness. In addition, it is ofinterest how the acoustic disturbance, amplifiedby the resonance in a wind tunnel, may alter thehydrodynamic state of an airfoil flow. The de-velopment of vortical motions in a suction-sideboundary layer, prompted by an acoustic feed-back loop, has a great effect on the formation of alaminar separation bubble [5]. The presence of aseparation bubble at the leading edge can signif-icantly improve the aerodynamic performance ofan airfoil. The time-averaged flow field is com-pared with the case solved in a uniform flow.

2 Reference Experiment

We briefly summarize our previous experimentalwork [8] for the comparison with the present nu-merical study. The unsteadiness of low Reynoldsnumber flows around an NACA0012 airfoil wasexamined in the Mars Wind Tunnel (MWT), thedecompression experimental rig operated in avacuum chamber. A wind-tunnel test can be con-ducted at reduced pressure, so that relatively highMach number flows, up to approximately M �0�8, can be achieved at low Reynolds number, ina range between 103 and 105 [1]. Geometricallytwo-dimensional test models can be placed in awalled test section. The spanwise width is 100[mm], while the height is slightly diverged to-ward downstream to relax the effect of boundary-layer development on the channel walls. Themean height of the test section is 160 [mm].

In the experiment [8], high-frequency pres-sure transducers were mounted on both sides ofthe NACA0012 airfoil at 90% chord, with 50[mm] chord length, to measure the pressure fluc-

tuations associated with vortex shedding. On theother hand, time-averaged pressure distributionswere obtained by using the pressure-sensitivepaint on the airfoil. The wind-tunnel tests wereoperated at chord-based Reynolds numbers Re �1�1�104 and 4�7�104, at M � 0�2. In addition,Mach number dependence was further investi-gated as a follow-up study, by increasing Machnumber up to M � 0�4. We will reference theexperimental case at Re � 1�1� 104 to comparewith the present numerical results.

3 Numerical Approach

Compressible Navier-Stokes equations aresolved on multi-block structured numerical grid.Employed numerical schemes are summarizedin our previous study [4]. Two-dimensionalgeometrical configurations are determined tomimic the MWT experiment [8]. An NACA0012airfoil, with chord length L, is placed in themiddle of a two-dimensional channel. Thetest-section height H is given as 3�2L. We definex and y axes in the streamwise and wall normaldirections, respectively. In three dimensionalcases, z axis will be added in the spanwisedirection with periodic boundary condition. Theorigin of the 2D x-y coordinates is defined atthe trailing edge of the airfoil of chord lengthL. However, regardless of angle of attack α, themid-chord location is fixed at the half heightof the test section. In addition, the local X -Ycoordinates are introduced in the chordwise andnormal directions. Their origin comes to theleading edge of the airfoil.

The numerical domain is defined in the rangeof �10 � x�L � 15, while the test section ofthe actual wind tunnel corresponds to the region�4�5 � x�L � 3�5. In the experimental setup,a contraction region is introduced in the up-stream section, while a diffuser section is addeddownstream with ejectors. In the present nu-merical study, characteristic inflow/outflow con-ditions are applied at both ends of the do-main, with sponge layers to damp unwantedacoustic reflections. Inflow density and veloc-ity are defined as ρ∞ and U∞, and employed fornon-dimensionalization, as well as characteris-

2

Wind-Tunnel Acoustic Resonance Induced by 2D Airfoil Flow at Low Reynolds Numbery�

L

x�L

Fig. 1 Subset of numerical grid for anNACA0012 airfoil at the angle of attack α � 5°,settled in a wind tunnel. Each color indicates ablock of structured grid, coarsened to the factorof four for visualization clarity.

tic length L. A free-slip condition is imposedon both the channel walls to avoid the develop-ment of boundary layers on them. The wallsserve as an acoustical reflector, in addition to aflow blockage. The entire domain is decomposedinto 7 numerical blocks. The surface of the air-foil is discretized by 600 cells; more than 70%of the cells are clustered on the suction side toresolve the vortical motions that may develop onthe boundary layer. Total number of numericalcells is 1�7�105. Present grid resolution is deter-mined by reference to the two-dimensional con-vergence study of [5]. Numerical grid is shown inFig. 1. The Reynolds number based on the chordlength L and inflow mean velocity U∞, is 10�000.Inflow Mach number is altered from 0�1 to 0�4 toexamine the dependence of wall resonant modes.

In the reference experiment [8], resonanttones were observed in the range 3° � α � 7°with one degree increments for α at M � 0�2.Whereas no tones were detected on the pressuretransducers at less than α � 3°, presumably dueto the non-negligible level of background noisein the MWT, rather broadband spectra were ob-served at α � 7°. In the present numerical study,we focus on the tonal noise specifically betweenα � 3° and 7°. In the parametric study on Mand α, mostly two-dimensional calculations areperformed. However, by increasing α, three-dimensional motions should develop, which af-fects aerodynamic force estimation. In the

NACA0012 case at the present Reynolds num-ber, lift force is overpredicted in two-dimensionalcalculations at α � 7° [3]. Therefore, we con-duct three-dimensional calculations for the casesat α � 7°. In the present wind-tunnel configura-tion, two Mach numbers are compared: M � 0�2and 0�3 in three dimensions. Spanwise lengthL is discretized with 128 cells, which leads to2�2�107 numerical cells in total in three dimen-sions. On the spanwise grid convergence, wealso conducted the case of doubled resolution,256 cells for the spanwise domain length L, atM � 0�2 and α � 7°. Only minimal differencewas observed in resonant frequency and statis-tics. We adopt the case of 128 grid cells for datapresentation and discussion. For comparison, wereference our previous numerical study [3]. TheNACA0012 airfoil was placed in a free stream atM � 0�2 and Re � 10�000, solved by an equiva-lent numerical approach. Three-dimensional re-sults are also available for α � 7° and 8° as thefree-stream case.

4 Frequency Selection in a Wind Tunnel

4.1 Discrete Resonant Mode

Runyan and Watkins [12] derived the discretemodes of resonant frequency f , induced by a linedipole located in the middle of a hard-walled two-dimensional channel:

f �c∞�

1�M2

H

�m� 1

2

�(1)

where c∞ is speed of sound, M � U∞�c∞, H ischannel height, and m is positive integer. Eq. (1)can be considered as the resonant relation of air-foil tones in a wind tunnel.

Firstly, the frequency dependence is exam-ined on the angle of attack and Mach number.When an airfoil is placed in a uniform flow, Kar-man vortex shedding is induced by wake instabil-ity at small α. Then, as α increases, an acousticfeedback loop mechanism may arise, which sig-nificantly alters frequency. Fig. 2 compares thevortex shedding frequency of the present numer-ical cases, and the tone frequency observed inthe MWT experiment. At α � 3°, two numeri-cal cases agree fairly well, while no tones were

3


0 2 4 6 80

0.5

1

1.5

2

2.5

3

α [deg.]

Nor

mal

ized

Fre

quen

cy

f

m = 2

Free Stream Hard Wall CFD Hard Wall Exp.

Fig. 2 Primary peak frequencies of v-velocitysampled at 0�1L downstream from the trailingedge for the inflow Mach number M � 0�2 andvarious angles of attack α, compared with the re-sults of free-stream simulations [3] and an MWTexperiment [8].

obtained in the experiment. However, at α � 3°,the shedding frequency of the hard-walled chan-nel case discontinuously reduces nearly to thatobserved in the experiment. For α� 3°, the tonefrequency of the present numerical results is al-most fixed as observed in the experiment, ex-cept the two-dimensional case at α � 6° whereno tones can be recognized. On the other hand,the frequency of the free-stream case keeps de-creasing up to α � 7°. At the angle of attack,the free-stream shedding frequency results in aconsiderable difference from the resonant tone ofEq. (1).

The same discrete nature can be observed atdifferent Mach numbers. In the case M � 0�1, thenormalized shedding frequencies are fixed rela-tively low around 1�5 at α � 3°, which approxi-mately corresponds to the mode m � 1; the dataare not shown. At M � 0�3 and 0�4, the presentresults agree very well with those observed in theexperiment, as shown in Fig. 3, with a few ex-ceptions. In Fig. 3-(a), the tone frequency of thenumerical result drops to the mode of m � 1 atα� 7°, which is inconsistent with the experimen-tal observation. Indeed, this lower mode does oc-cur in the experiment at α � 8°. Also in Fig. 3-(b), the secondary mode at α � 5° in the numer-

(a)

3 4 5 6 70

0.5

1

1.5

2

2.5

3

α [deg.]

Nor

mal

ized

Fre

quen

cy

f

m = 3

m = 2

m = 1

Hard Wall CFD Hard Wall Exp.

(b)

3 4 5 6 70

0.5

1

1.5

2

2.5

3

α [deg.]

Nor

mal

ized

Fre

quen

cy

f

m = 2

m = 3

m = 4

Hard Wall CFD Hard Wall Exp.

Fig. 3 Primary peak frequencies in the wake:(a) M � 0�3; (b) M � 0�4. In (b) at α �5°, both primary and its sub-harmonic (or sec-ondary) frequencies are plotted; at α� 7°, a two-dimensional result is shown for M � 0�4. Alsosee the caption of Fig. 2.

4

Wind-Tunnel Acoustic Resonance Induced by 2D Airfoil Flow at Low Reynolds Number

ical result matches the primary frequency in theexperiment. The secondary mode in the numer-ical result is the sub-harmonic of the primary-mode frequency, which is very close to the shed-ding frequency of the free-stream case. Sam-pled in the wake, the primary mode is supposedlymore amplified by the wake instability.

The difference in the frequency selectionmechanism can be recognized in the vortex shed-ding patterns. Fig. 4 compares the three numeri-cal cases at α� 5°. In the free-stream case Fig. 4-(a), the shedding frequency is determined primar-ily by the wake instability, still with a small effectof the feedback mechanism. A regular Karmanvortex street forms in the wake, with only rec-ognizable excitation of an instability wave nearthe trailing edge. On the other hand, other twowind-tunnel cases are apparently affected by thefeedback mechanism associated with the reso-nant mode. Vortical motions arise in the suction-side boundary layer in the middle of the chord,unlike the free-stream case. Vortex shedding pat-terns seem affected by the superposed modes ofdiscrete resonance, other than the primary mode.

The noise scattering from the airfoil confinedin the walled passage seems rather different fromcommon dipole sound emission [e.g., 4], due tothe interference of wall reflection. Fig. 5 presentsinstantaneous sound pressure distributions thatcorrespond to the primary resonant mode m � 1to 4. As can be seen from the figures, pressurefluctuation is antisymmetric on the middle of thechannel, y � 0. The integer m is determined bythe number of wavelengths that exist in the wallnormal directions above and below the airfoil.Due to the Doppler effect, acoustic waves trav-eling in the upstream direction have large ampli-tude with a lattice pattern formed by the interfer-ence, while the acoustic fluctuation in the wakeregion is rather small.

While the two-dimensional pressure fieldsshow relatively clear lattice patterns, assuminga complete coherence in the spanwise direction,the resonant mode of three-dimensional calcu-lations becomes somewhat obscure at a higherangle of attack. Still, the sampled spectra in-dicate that three-dimensional fields should alsobe under the influence of wall resonance, with

(a)

y�

L

x�L(b)

y�

L

x�L(c)

y�

L

x�L

Fig. 4 Instantaneous ωz vorticity distributionsat α � 5°: (a) free stream case at M � 0�2; (b)wind tunnel case at M � 0�2; (c) wind tunnelcase at M � 0�3. Color scale ranges between�10� ωzL�U∞ ��10 from blue to red.

5


(a) m � 1: M � 0�1 & α � 3°

(b) m � 2: M � 0�2 & α � 5°

(c) m � 3: M � 0�4 & α � 4°

(d) m � 4: M � 0�4 & α � 3°

Fig. 5 Instantaneous pressure fluctuations forvarious primary antisymmetric resonant mode mof two-dimensional results. One contour levelof pressure normalized by ρ∞U2

∞ denotes: (a)1�10�2; (b) 3�10�3; (c) 2�10�3; (d) 1�10�3.Total number of contour lines is 50.

(a) m � 2: M � 0�2 & α � 7°

(b) m � 1: M � 0�3 & α � 7°

Fig. 6 Instantaneous pressure fluctuations ofthree-dimensional results on an x-y plane. Onecontour level denotes: (a) 8�10�4; (b) 4�10�3.Also see the caption of Fig. 5.

tonal modes shown in Figs. 2 and 3 at α � 7°.Fig. 6 presents instantaneous x-y views of three-dimensional cases. Although the instantaneousviews are affected by a broadband feature inthe present cases, antisymmetric modes could bebarely identified in the vertical direction acrossthe airfoil.

4.2 Symmetric Resonant Mode

In addition to the resonance anti-symmetric onthe middle of the test section, symmetric modesarise in the numerical simulation. The presenceof a symmetric mode may contradict the dipolesound generation due to Karman vortex shed-ding. However, it is observed clearly at a lowangle of attack, when an apparent anti-symmetricresonant mode does not occur. The discrete sym-metric mode can be written as:

f �c∞�

1�M2

Hn (2)

where n is positive integer.Fig. 7 is an example of sound pressure distri-

butions at α � 2°. The acoustic pressure fluc-tuation is nearly symmetric on y � 0. As inFig. 2, the shedding frequency at this angle of at-tack coincides with that of the free stream case.

6


Fig. 7 An example of symmetric resonantmodes: instantaneous pressure fluctuations atα � 2° and M � 0�2. One contour level denotes3�10�4. Also see the caption of Fig. 5.

Therefore, the wall resonance of this symmetricmode does not seem to have direct relevance tothe vortex shedding, assumingly independent of afeedback mechanism. When velocity fluctuationsare sampled in the wake just behind the trail-ing edge, the frequency of vortex shedding f �2�5 is an only distinguishable peak. Any sym-metric modes are almost unrecognizable. How-ever, when sampled in the acoustic region apartfrom the wake, symmetric modes present domi-nant peaks as shown in Fig. 8. As expressed inEq. (2), these modes are composed of the basictone f � 1�53 at n � 1, and its higher harmon-ics. On the other hand, a significant peak can beseen at the antisymmetric mode m � 1 in Eq. (1):f � 0�766. This corresponds to the subharmonicof the basic tone of symmetric modes. There-fore, we may assume that the presence of thislowest antisymmetric mode, relevant to antisym-metric vortex shedding, could activate its higherharmonic modes as symmetric resonance.

In Fig. 8, the vortex shedding frequency f �2�5 also shows a sharp peak, but its spectral levelis relatively small. We should also notice thatthere is another sharp peak at f � 2�1, which co-incides with the resonant frequency at α � 3°,but not exactly with the mode m � 2 in Eq. (1):f � 2�3. Supposedly, this resonant frequencyf � 2�1 is not solely determined by the reso-nance relation of Eq. (1), compared with othertones that agree very precisely with the predictedmodes as in Fig. 8. Rather, it is selected viaan acoustic feedback process where the instabil-ity of a suction-side boundary layer has a domi-nant effect. An acoustic feedback loop also deter-

0 1 2 3 4 5 610

−14

10−12

10−10

10−8

10−6

10−4

PSD

Normalized Frequency f

Symmetric Antisymmetric

Fig. 8 Power spectral density of v-velocity sam-pled at 0�5L above the trailing edge at α� 2° andM � 0�2.

mines discrete resonant modes [2]. Unstable, suf-ficiently amplified frequencies close to the wallresonant mode would be selected in the feedbackmechanism.

5 Aerodynamics under the Influence of WallResonance

It is of great interest how the acoustic resonancemay affect the aerodynamic measurements of alow Reynolds number flow in the closed test sec-tion. In the numerical study [5], it was shownthat the onset of an acoustic resonance couldimprove aerodynamic performance by forminga laminar separation bubble behind the leadingedge at Re � 10�000. The formation of a sepa-ration bubble at the forward part of an airfoil de-velops a transitional boundary layer that preventsbulk separation, initiated by the acoustic distur-bances fed back at the leading edge.

In the present study, lift forces of NACA0012are compared among the numerical results ofthe free-stream case, and the hard-walled wind-tunnel cases at M � 0�2 and 0�3, in addition to awind-tunnel experiment conducted at fairly lowMach number in a common laboratory environ-ment [9], as shown in Fig. 9. In the free-streamsimulations, three-dimensionality becomes sig-nificant at α �� 7, where two-dimensional calcu-lations overestimate lift. Here, three-dimensional

7


0 2 4 6 8

0

0.2

0.4

0.6

0.8

α [deg.]

Cl

Free Stream ( M=0.2 ) Hard Wall WT ( M=0.2 ) Hard Wall WT ( M=0.3 ) Low Mach Experiment

Fig. 9 The comparison of time-averaged lift co-efficients between three numerical cases and anexperimental result of a low Mach case [9].

results are presented both for α � 7° and 8°. Thefree-stream calculations agree well with the lowMach experiment at α� 6°. However, in spite ofthe three-dimensionality considered in the simu-lations, appreciable differences exist at α � 7°.Moreover, the lift in the wind-tunnel computa-tion becomes even higher than that of the free-stream case, in a considerable range of angle ofattack, also depending on Mach number. This canbe regarded as the modification of hydrodynamicstates via strong acoustic resonance.

In the referenced experiment [9], the air-foil of chord length 75 [mm] was used to attainRe � 10�000 in a ground environment. Thus,the flow Mach number is estimated to be M �0�006. Since the acoustic component of veloc-ity fluctuation has the Mach number dependenceof M3�2 in two-dimensional acoustic scattering,the magnitude of velocity disturbance fed backinto the boundary layer would be about 0�5% ofthe present numerical cases at M � 0�2. Thedifference should have a great effect on the de-velopment of hydrodynamic unsteadiness in thesuction-side boundary layer. Similarly, in thepresent numerical cases, a higher Mach numberflow often results in strong acoustic resonance,as observed in the difference of M � 0�2 and 0�3in Fig. 9.

The difference in aerodynamic characteristicsis investigated in more details. Pressure distribu-

tions on the suction side are compared in Fig. 10,among the free-stream case, the wind-tunnel sim-ulation, and the MWT experiment at M � 0�2, forα � 3°, 5°, and 7°. As the resultant lift forcesagree at α � 3° & 5°, the two numerical casesshow almost no discernible difference, regard-less of the flow-passage walls. Besides, they alsocoincide well with the experimental profiles ob-tained through the graphical image of pressure-sensitive paint [8]. However, at α � 7°, an ap-preciable deviation arises in the pressure distribu-tion, while all the cases present a plateau, or flatprofile behind steep adverse pressure gradient atthe leading edge, and a following pressure recov-ery region toward the trailing edge. The plateauoften denotes the presence of an adjacent separa-tion bubble. As the separated flow reattaches inthe middle of the chord, a relatively steep pres-sure recovery would occur behind the reattach-ment. This resembles the “separation ramp” con-cept [6], which is considered to be a favorablefeature for better aerodynamic performance. InFig. 10, the profile of the wind-tunnel simulationindicates a less pressure recovery at the leading-edge separation, and an earlier occurrence of aseparation ramp, compared with the free-streamcase. However, the experimental result is some-what between these two numerical results: thepressure recovery at the leading-edge separationagrees with the free-stream case, while the sep-aration ramp at the trailing edge better coincideswith the wind-tunnel computation.

In Fig. 11, the separation bubbles are drawnon the time-averaged velocity field of the threenumerical cases that were compared in the liftcurve diagram Fig. 9: the free-stream case atM � 0�2, and the hard-wall wind-tunnel compu-tations at M � 0�2 and 0�3. At α � 5°, the free-stream and wind-tunnel cases almost perfectlymatch at M � 0�2, similarly to the pressure dis-tributions Fig. 10. They exhibit a typical sepa-ration bubble of a trailing-edge stall, elongatedin the streamwise direction. However, the wind-tunnel case at M � 0�3 reduces the size of a sepa-ration bubble, although the separation location isalmost identical with the other two cases. Usu-ally, a trailing-edge stall causes a fairly flat pres-sure profile toward the trailing edge. The aero-

8


0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

1.5

2

X/L

−C

p

α = 7.0 α = 5.0 α = 3.0

Fig. 10 Time-averaged pressure distributions onthe suction side of the airfoil at M � 0�2: —,wind-tunnel case; - - -, free-stream case; Æ, MWTexperiment.

(a)

0 0.2 0.4 0.6 0.8 1

−0.1

0

0.1

0.2

0.3

X / L

Y /

L

Free Stream ( M=0.2 ) Hard Wall WT ( M=0.2 ) Hard Wall WT ( M=0.3 )

(b)

0 0.2 0.4 0.6 0.8 1

−0.1

0

0.1

0.2

0.3

X / L

Y /

L


Fig. 11 The comparison of separation bubblesof time-averaged velocity fields between the freestream case, and the hard-wall wind tunnel casesat M � 0�2 and 0�3 at: (a) α � 5°; (b) α � 7°.

dynamic performance is lowered if the separa-tion location shifts to the leading edge at a higherangle of attack, without any reattachment in themiddle of the chord [5]. The acoustic resonanceenforces the suction-side vortical motions, short-ening the trailing-edge separation bubble, as alsounderstood from Fig. 4.

On the other hand, as shown in Fig. 11-(b)at α � 7°, the size of separation bubbles variesamong these three-dimensional cases. Whilethe separation locations coincide well just down-stream of 10% chord, the acoustic resonance ap-parently affects the mixing strength of the sep-aration shear layer. Still, none of the presentcases causes reattachment upstream of the trail-ing edge. The wind-tunnel case at M � 0�2 ex-hibits the smallest separation bubble, while thehigher Mach number case M � 0�3 settles inthe middle, unlike the cases at α � 5° shownin Fig. 11-(a). Although not shown in Fig. 10,the upper-surface pressure profiles of the hard-walled wind-tunnel cases are very similar be-tween M � 0�2 and 0�3, in spite of a recogniz-able difference in separation bubbles. Presum-ably, this difference is caused by the stabilityof the shear layer that depends on a base-flowMach number. As will be shown bellow, thesuction-side boundary layer exhibits significantthree-dimensional motions, developed in adversepressure gradient. Nonetheless, we should notethat the separation ramp behavior, observed inFig. 10, is not necessarily the evidence of reat-tachment in the middle of the chord, unlike thenumerical study of thinner cambered airfoils [5].

Finally, the three-dimensional motions areexamined via visualization at α � 7°. Fig. 12shows the vortex visualization of the Q-criterioncolored with streamwise vorticity; nominallytwo-dimensional spanwise vortical motions canbe recognized by green. Behind the leadingedge, the separation shear layer is first recog-nized, elongated in the streamwise direction, es-pecially in the free-stream case. Then, it developsinto two-dimensional instability waves. How-ever, right after the spanwise vortices roll up,they are broken up into small longitudinal eddies,showing a transitional boundary layer, still sus-taining large spanwise motions. In the present

9


(a)

(b)

(c)

Fig. 12 The comparison of three-dimensionalnumerical results at α � 7°, via instantaneousQ-vortex isosurfaces at Q � 10�U∞�L�2, coloredby streamwise vorticity ωxL�U∞ ranging between�1 and �1 from blue to red: (a) free stream caseat M � 0�2; (b) wind tunnel case at M � 0�2; (c)wind tunnel case at M � 0�3.

0 0.5 1 1.5 20

0.1

0.2

0.3

max(w

′ rms)/U∞

X /L


Fig. 13 The rms fluctuation of w-velocity ofthree-dimensional cases at α � 7°, above the air-foil (0 � X�L � 1) and in the wake (1 � X�L).Local maxima at each X location is shown.

results, the wind-tunnel case at M � 0�2 seemsto present the most upstream transition of vorti-cal motions, as in Fig. 12-(b). In spite of strongeracoustic resonance, the higher Mach number caseslightly delays the transition at M � 0�3 shownin Fig. 12-(c). The free-stream case sustains arelatively large laminar region behind the lead-ing edge as in Fig. 12-(a), but also supposedlyaffected by the acoustic feedback effect associ-ated with a trailing-edge noise, when comparedwith the extremely lower Mach-number case inFig. 9, conducted in a common laboratory envi-ronment [9].

Spanwise velocity fluctuation is indicative ofthe extent of transitional boundary layer devel-opment. Fig. 13 compares the chordwise growthof rms spanwise velocity obtained in the three-dimensional results at α � 7°. While the free-stream case shows the least w fluctuation downto 80% chord, both the wind-tunnel cases presentapparently larger growth rates at the forward partof the chord, due to the acoustic resonance. Thehard-wall wind tunnel case at M � 0�2 attains thelargest rms value, about 12% larger than the free-stream case. However, the maximum value of thehard-wall case at M � 0�3 is just comparable withthe free stream case. This is presumably the sta-bilization effect of a higher Mach number flow.The instability of a shear layer would be low-

10


ered by increasing Mach number. Still, the acous-tic disturbance is large enough to reduce the sizeof the separation bubble comparing to the free-stream case, by stimulating transitional motionsin the suction-side boundary layer.

6 Conclusions

The wind-tunnel wall resonance was investi-gated numerically, associated with airfoil tonesof dipole nature at Re � 10�000. Confined ina hard-walled passage, two-dimensional airfoilflows solved on a compressible Navier-Stokescode successfully reproduced discrete tones thatapproximately satisfies the resonant modes sug-gested by Runyan and Watkins [12], clearly af-fected by the wall resonance, compared to thefree-stream case. However, the discontinuouschange of tonal modes on angles of attack in-dicates that the tonal frequency would be deter-mined via the instability of suction-side bound-ary layer and the acoustic feedback mechanism,as originally suggested by [2], not solely the ge-ometrical relation of resonance [12]. The presentresults agree very well with the tonal modes ob-served in our previous experimental study [8].

The onset of wall-resonant modes affects notonly the observed aerodynamic sound, but alsothe aerodynamic characteristics as time-averagedquantities. In the present case, external wallresonance amplifies the hydrodynamic instabil-ity waves that may develop into significant vor-tical motions in the suction-side boundary layer.The transitional boundary layer driven by the res-onance reduces the size of separation bubbles,which eventually increases the lift. At relativelyhigher angle of attack, α� 7°, transitional behav-ior is already observed in the free-stream result,with spanwise vortices breaking-up into longitu-dinal eddies. However, the wind-tunnel calcu-lation accelerates the transition, diminishing thelaminar region behind the leading edge. The re-duced size of the separation bubble also signif-icantly alters the surface pressure profile with aseparation ramp, as if the aerodynamic perfor-mance of an airfoil were improved.

7 Contact Author Email Address

The corresponding author can be contacted [email protected] (Tomoaki Ikeda).

References

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[6] M. D. Maughmer and D. M. Somers. De-sign and experimental results for a high-altitude, long-endurance airfoil. J. Aircraft,26(2):148–153, 1989.

[7] E. C. Nash, M. V. Lowson, andA. McAlpine. Boundary-layer insta-bility noise on aerofoils. J. Fluid Mech.,382:27–61, 1999.

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[9] T. Ohtake, Y. Nakae, and Y. Moto-hashi. Nonlinearity of the aerody-namic characteristics of NACA0012 aero-foil at low Reynolds numbers. J. JapanSoc. Aeron. Space Sci., 55(644):29–35,2007.

[10] R. Parker. Resonance effects in wake shed-ding from parallel plates: calculation of res-onant frequencies. J. Sound Vib., 5(2):330–343, 1967.

[11] R. W. Paterson, P. G. Vogt, M. R. Fink, andC. L. Munch. Vortex noise of isolated air-foils. J. Aircraft, 10(5):296–302, 1973.

[12] H. L. Runyan and C. E. Watkins. Consider-ations on the effect of wind-tunnel walls onoscillating air forces for two-dimensionalsubsonic compressible flow. Technical Note2552, NACA Langley Aeronautical Labora-tory, Langley Field, VA, 1951.

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Acknowledgements

This work was supported by Grant-in-Aid forScientific Research (C) from Japan Society forthe Promotion of Science (Grant No. 25420139).Computational resources were provided byJAXA Supercomputer System.

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