Flow-induced pressure oscillations in shallow cavities

Journal of Sound and Vibration (1971) 18 (4), 545-553

F L O W - I N D U C E D PRESSURE O S C I L L A T I O N S IN SHALLOW CAVITIES~ H. H.

HELLER,

Bolt Beranek and Newman Inc., 50 Moulton Street, Cambridge, Massachusetts 02138, U.S.A. D. G. HOLMES AND E. E. COVERT

Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A. (Received 1 April 1971) In a wind-tunnel study of flow-induced pressure oscillations in shallow cavities (lengthto-depth ratios from 4 to 7), a variable-depth rectangular cavity was exposed to tangential flow over the open surface in the Mach-number range from 0-8 to 3. Flow stagnation pressures varied from 2 to 15 lb/in 2 (13.78 × 103-103-35 x 103 N/m2). Third-octave-band and narrow-band fluctuating-pressure spectra obtained at various locations within the cavity and under the approaching boundary layer yielded information on resonant frequencies and associate pressure-mode shapes, as well as discrete-frequency and broad-band pressure amplitudes. 1. I N T R O D U C T I O N

The characteristics of flow-induced pressure oscillations in open cavities depend strongly on geometric, structural and aerodynamic parameters. The occurrence and intensity o f such oscillations are determined by the balance between the energy available from the free-stream flow and the energy dissipated through various loss mechanisms such as acoustic radiation, viscous losses, and convective mass exchange. The structural response to oscillatory energy m a y be so significant as to damage an aircraft structure. Thus, a practical reason for studying such oscillations is to avoid, if possible, critical design configurations or, at least, to make reasonably accurate predictions o f the magnitudes and spectral distributions. This investigation concentrates on shallow, rigid walled cavities--i.e., those having in the streamwise direction a length-to-depth ratio larger than unity, and being stiff enough for structural deformation to be a secondary effect. The experiments were conducted in the Massachusetts Institute o f Technology Naval Supersonic Wind Tunnel at Mach numbers o f 0.8, 1.5, 2 and 3 with unit Reynolds numbers ranging from 0.5 to 5 million/ft. Plate 1 shows the general arrangement; the test section is 18 in (45.72 cm) wide by 24 in (60.96 cm) high; the cavity was 20 in (50.80 cm) long,§ with continuously variable depth, giving lengthto-depth ratios between 4 and 7. t This research was sponsored by the Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio. ~:Presented at the symposium on "Aerodynamic noise" sponsored by the British Acoustical Society and the Royal Aeronautical Society at Loughborough University of Technology, Loughborough, England, on 14 to 17 September 1970. § The cavity was sized so that at Mach number 1'5 the reflected bow shock just cleared the cavity trailing edge, avoiding interference effects at supersonic speeds. At subsonic speeds, interference effects did appear: one of the cavity resonant frequencies excited an upstream-going travelling wave in the test section, which had a wavelength approximately equal to the cavity length, and a group velocity (energy propagation velocity) close to zero. Experiments on a half-scale cavity confirmed this explanation, and indicated that this "interference" effect was of no practical importance in terms of the general cavity behavior. 36 545

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The cavity was narrow; the width being one-third of the length. This resembles the cavities commonly encountered in aircraft configurations more closely than a short cavity spanning the tunnel. Despite the narrowness of the cavity, the unsteady response, at least, appears to be approximately two-dimensional [1]. The primary objective of the study was to obtain data on the fluctuating pressure within and near the cavity. Data was also obtained for steady-state pressure, cavity temperature and the recirculating velocity of the cavity internal flow near the bottom of the cavity. 2. TEST APPARATUS AND INSTRUMENTATION Fluctuating pressures were measured by nine 0.25 in (6.35 mm) diameter piezoelectric microphones (Bolt, Beranek and Newman Model 370)--one upstream of the cavity under the approaching boundary layer, three along the centerline of the cavity floor, two in opposite corners of the floor, one in the rear bulkhead, and one in each glass side wall. A tenth microphone was mounted within a 0.75 in (19.05 mm) diameter movable rod spanning the length of the cavity (see Plate 1). This microphone could be traversed remotely, and permitted a detailed study of the longitudinal distribution of unsteady pressures in the cavity. Static pressures and recirculating velocities in the cavity were determined by static pressure taps and a Pitot-probe, respectively. Cavity internal temperatures were measured with a thermocouple to obtain information on the internal speed of sound. For each test condition, third-octave-band pressure spectra were obtained. Expected accuracy of the pressure spectrum levels is not less than 4-3 dB. Schlieren photographs taken during each test run provided information on the general flow pattern along the cavity. 3. RECIRCULATION VELOCITIES Woolen tufts placed inside the cavity indicated that the mean flow inside the cavity was primarily a single eddy driven by the external flow [1, 2, 3]. The Pitot-static probe visible in Plate 1 was used to obtain estimates of typical Mach numbers of the recirculating flow. The results, presented in Figure l, show that the recirculation velocity increased with depth. 06

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z./D Figure 1. Recirculation Mach numbers as functions of length-to-depth ratio and free-stream Mach number. (a) M = 0'8, (b) M = 1-5; (c) M = 2"0, (d) M = 3"0. o, Laminar boundary layer, o, turbulent boundary layer.

Plate 1. Experimental cavity in Massachusetts Institute of Technology Naval supersonic wind tunnel. ( Note : half-nozzle block.)

(facing p. 546)

Plate 2. Schlieren p h o t o g r a p h s . (a) L a m i n a r b o u n d a r y layer, M .:: 3, Po = 2 Ib/in 2 (13.87 × 103 N / m 2) 103 N/mS), LID 7.

LID = 7. (b) T u r b u l e n t b o u n d a r y layer, M = 3, P0 = 10 lb/in ~ (68'90

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Recirculation velocities were too low to be measured, when the upstream boundary layer was laminar; in this case the shear stress driving the recirculation was much lower than that for the turbulent case. 4. SPEED OF SOUND WITHIN THE CAVITY The temperatures measured by the thermocouple near the center of the cavity floor are considered representative of the static temperature Tc for the entire cavity. The sound speed in the cavity ac is then ac = a~o[1 + r(2t - 1 / 2 ) M 2 ] 1/2, where a~o is the free-stream static sound speed, ,~is the adiabatic exponent, and r, the recovery factor, equals (To - T ® ) / ( T o - T~); here T® and T o are the static and the stagnation temperature of the free stream flow, respectively. Figure 2 shows the r a n g e of the recovery factor at each Mach number. Thus, the error involved in assuming that the cavity temperature is the free-stream stagnation temperature (recovery factor unity) is negligible at the lower Mach numbers, rising to a few percent at Mach number 3.

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5. FLUCTUATING PRESSURE DATA 5.1. EFFECT OF THE UPSTREAM BOUNDARY LAYER Within the Reynolds number range that could be achieved in the tunnel, fluctuating pressures scaled with the free-stream dynamic pressure, q; Figure 3 shows a typical normalized third-octave-band spectrum. The dynamic pressure varies over a range of 14 dB; yet the normalized spectra agree, in the bands containing the major part of the fluctuating energy, within the accuracy of the measuring system. Exceptions occur at Mach number 3, where at -20

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Figure 3. Typical n o r m a l i z e d fluctuating pressure spectrum. M = 0.8. - - - , Po = 2 l b/ i n 2 (13'78 x 103 N / m 2 ) ; - - - , P0 = 5 lb/in 2 (34"45 × 103 N/m2); . . . . . . , P = 10 lb/in 2 (68-90 × 103 N/m2).

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H . H . HELLER, D. G. HOLMES AND E. E. COVERT

low stagnation pressures the approaching boundary layer was laminar. Normalized pressure spectra in the cavity at Mach number 3, for a laminar [Po = 2 lb/in 2 (13-78 × 103 N/m2)] and a turbulent [P0 = 10 Ib/in 2 (68.90 × 103 N/m2)] approaching boundary layer are shown in Figure 4. At this Mach number, there is no evidence of a resonant response when the boundary layer is turbulent; for the laminar case, however, a strong resonant peak occurs. This behavior is confirmed through Schlieren-optical observations: Plate 2(a) shows a laminar approaching boundary layer and a very unsteady flow--note the curvatures of both the shock waves and of the boundary layer ;t in Plate 2(b) the boundary layer is turbulent, and no gross disturbances of the mean flow are visible. 0 ~

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Figure 4. Comparison of normalized fluctuating pressure spectra at Mach number 3, for laminar and turbulent boundary layers. Curve A: Po = 2 lb/in 2 (13.78 × 103 N/m 2) (laminar boundary layer). Curve B: P0 = 10 lb/in 2 (68"90 × 103 N/m 2) (turbulent boundary layer). Mach number, M = 3.

5.2. RESPONSE FREQUENCIES This and other investigations [6-9] have shown that when intense pressure fluctuations occur, it is at one or more relatively discrete frequencies. Various authors [8, 9] have attempted to predict these available frequencies, using theories similar to those advanced to account for edge tones [10, 11 ]. These models predict available frequencies; one, or more, or none may in any given case be excited. In this investigation, discrete frequency peaks in the spectra were identified through narrow-band analysis. These resonant frequencies were normalized either with the freestream flow speed, U, or with the stagnation sound speed, ao: S] = ( f L ) / ( V ) ,

or

$2 = (fL)/ao,

where f is the measured resonant frequency, and L is the cavity length. $1 is plotted as a function of Mach number in Figure 5(a). No attempt was made to distinguish between different configurations on the basis of length-to-depth ratio or of stagnation pressure at each M a t h number, because values of resonant frequencies are relatively insensitive to these variables, although the occurrence of such discrete frequencies is undoubtedly determined by them. Figure 5(a) also shows data points flora other investigations of shallow cavities. At least four curves can be drawn through all of the data points. On each curve the reduced frequency decreases as the M a t h number increases. Such curves are consistent with the semi-empirical model of Rossiter [8]; the curves in Figure 5(a), in fact, correspond to Rossiter's formula SI

fL m-~ U - 1/kv + M '

-~Pressure fluctuations under these conditions can be as high as 50 % of the mean pressure inside the cavity.

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Figure 5. Non-dimensional resonant frequencies as a function of Mach number. (a) Rossiter's formula; (b) Modification of Rossiter's formula, i , LID = 4; A, L/D = 5"7; o , LID = 7; o, Krishnamurty [5]; I~, Rossiter [8]; 71, Morozov [14]; V, White and McGregor [15] (water table).

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where kv is the vortex convection speed as a fraction of the free-stream flow speed U, M is the free stream Mach number, c~is a constant, and m = 1, 2, 3..... Here c~is chosen equal to 0.25, and kv equal to 0.57. The agreement is good at Mach numbers 0.8 and 1.5, poor at Mach number 2 and worse at Mach number 3. In deriving his formula for Sl, Rossiter assumed the speed of sound in the cavity to be the free-stream speed of sound--i.e, he assumed a cavity recovery factor of zero. At low Mach numbers this assumption introduces only a small error, but at high Mach numbers the error is much greater. In fact, recovery factors measured in the current program were close to unity rather than to zero (see Figure 2). Thus, Rossiter's formula can be improved for the higher Mach number range by assuming that the cavity sound speed is equal to the free-stream stagnation sound speed. The formula for the Strouhal number is then

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Figure 5(b) shows the data points and curves representing S~ for the above values of 0c, kv and m. The modified Rossiter-formula, which greatly improves the correlation at Mach numbers 2 and 3 for data from the present study, appears to be the best available approach to correlating the discrete-frequency response of the cavity. 5.3.

L O N G I T U D I N A L D I S T R I B U T I O N S OF T H E U N S T E A D Y PRESSURES

The traversing microphone was used to observe the rms pressure distribution in resonant frequency bands. For each run, the rms pressure in the most intense band was plotted against

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and lower surveys are incomplete; the identification of the lower survey with an n = 3 mode presupposes another mode near the trailing edge). Note, however, that the frequencies at which these "modes" are observed are below the frequencies of the ideal longitudinal normal modes of the cavities, either with a "closed" surface (no displacements) or an "open" surface (no pressure fluctuations). Note also in Figure 6 that the pressure amplitudes appear to increase toward the trailing edge. The increase in pressure amplitude with distance from the leading edge is confirmed by data from the three microphones on the center line of the cavity floor. The variation between front and rear can be as high as 10 dB; the variations in level are less pronounced for the deeper than for the shallower cavities. [

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Figure 7. Non-dimensional frequency, $2, as a function of Mach number. These observations support the idea that a shallow cavity does not act as a classical acoustic resonator responding in its normal modes. If we suppose the acoustic energy stored in a cavity to be proportional to (amplitude of the osciUations) 2 times (cavity volume) and the energy exchanged per cycle with the free stream to be proportional to (amplitude) 2 times (surface area), then the energy stored divided by the energy exchanged per cycle is proportional to depth. Thus, deep cavities have a high Q-factor and resonate in their normal modes; shallow cavities have a low Q-factor and resonate poorly, if at all--though they may be driven by the free stream at relatively definite frequencies. The mechanism by which the free stream drives a shallow cavity appears to resemble an acoustic source at the trailing edge. Thus, a shallow cavity seems to be driven by an acoustic source close to the trailing edge, and the pressure fluctuation levels fall off as 1/r towards the leading edge. This rate of fall-off is just that computed by elementary theory. Note, however, that the low Q-factor of a shallow cavity may be compensated by a more powerful source effect at the trailing edge. A different aspect is revealed when the frequencies are normalized with respect to a0, the stagnation sound speed of the free stream. (Note that, following section 4, ao is a good

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H . H . HELLER, D. G. HOLMES AND E. E. COVERT

approximation to ac, the sound speed in the cavity.) Figure 7 shows $2 as a function of the Mach number. The numers adjacent to the data points are the orders of the "modes", when, as in Figure 6, they could be identified. The dashed lines are the normal mode frequencies for a "closed" cavity. As the Mach number increases, the frequencies approach the normal mode frequencies of a closed cavity. This is reasonable. Reference [12] indicates that, as the Much number tends to infinity, the pressure exerted by the outer flow on the air in the cavity can be approximated by '

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ah

P = Pc ac M ~ x , where p' is the pressure fluctuation at the interface between outer and inner flows, pc and ac are cavity air density and sound speed, respectively, M is the free-stream Mach number, and Oh/Ox is the slope of the dividing streamline. At lower Mach numbers, this expression becomes much more elaborate [13]. Here, we simply note that as M becomes large, pressure fluctuations become large relative to the interface displacement. The impedance presented by the interface becomes proportionally larger as M increases; the closed-box frequencies are then approached. The formula for S* could be further modified to demonstrate this limit, but it would then become even more empirical in nature. 6. CONCLUDING REMARKS Plumblee et al. [4] proposed that pressure oscillations in the cavity are generated by a linear response to pressure fluctuations in the turbulent boundary layer. However, Plate 2 shows clearly that the laminar boundary layer produces more intense fluctuations despite its own lower noise levels. Krishnamurty's data [5] support this conclusion. From these results, it is evident that boundary layer parameters must be included in a complete theory of the response amplitudes of the cavity. Although there is not enough data available to define precisely what these parameters must be, it has been found (i) that transition from a laminar to a turbulent boundary layer can greatly affect the cavity response, and (ii) that once the upstream boundary layer is turbulent, pressure-fluctuation characteristics in the cavity are insensitive to Reynolds number changes, for the Reynolds number range of these experiments. The proposed modification of Rossiter's formula is effective in predicting available oscillation frequencies for a shallow cavity. This investigation indicates that such semiempirical models which propose an acoustic source at the cavity trailing edge are qualitatively correct.

Determination of which frequencies are actually observed requires that such simple formulae are abandoned, and theoretical approaches such as that of reference [13] attempted. A modification of the theory presented in [13], and applicable to shallow cavities, is called for. The more difficult problem of amplitude prediction needs further study. As indicated by the data at Much number 3, non-linear effects are important. REFERENCES 1. D. J. MAULLand L. F. EAST 1963 Journal of Fluid Mechanics 16, 620-632. Three-dimensional flow in cavities. 2. A. RosnKo 1955 NACA TN-3488. Some measurements of flow in a rectangular cutout. 3. U. B. MEHTAand Z. L. LAGAN1969 NASA CR-1245. Flow in a two-dimensional channel with a rectangular cavity. 4. H. E. PLUMBLEE,J. S. GlaSON and L. W. LASSITER1962 WADD-TR-61-75. A theoretical and experimental investigation of the acoustic response of cavities in an aerodynamic flow.

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5. K. KRISHNAMURTY1955 NACA TN-3487. Acoustic radiation from two-dimensional rectangular cutouts in aerodynamic surfaces. 6. L. F. EAST 1966 Journal of Sound and Vibration 3, 277-287. Aerodynamically induced resonance in rectangular cavities. 7. B. QUINN 1963 American Institute of Aeronautics and Astronautics Student Journal 1, 1-5. Flow in the orifice of a resonant cavity. 8. J. E. ROSSITER 1966 Aeronautical Research Council Reports and Memoranda 3438. Wind-tunnel experiments on the flow over rectangular cavities at subsonic and transonic speeds. 9. B. M. SPEE 1966 AGARD Conference Proceedings No. 4, Separated Flows, Part 2. Wind tunnel experiments on unsteady cavity flow at high subsonic speeds. I0. N. CURLE 1953 Proceedings of the Royal Society, Series A 216, 412424. The mechanics of edgetones. 11. W. L. NYBORG 1954 Journal of the Acoustical Society of America 26, 174-182. Self-maintained oscillations of the jet in a jet-edge system, I. 12. L. E. GARmCK 1957 High Speed Aerodynamics and Jet Propulsion Volume 7: Aerodynamic Components of Aircraft at High Speed: Part F, Nonsteady Wing Characteristics. Princeton, New Jersey: Princeton University Press. 13. E. E. COVERT 1970 American Institute of Aeronautics and Astronautics Journal 8, 2189-2194. An approximate calculation of the onset velocity for cavity oscillations. 14. M. G. MoRozov 1960 News of the Academy of Sciences of the USSR. Mechanics and Machine Building 2, 40--46. Acoustic emission of cavities in supersonic airflow. 15. R. A. WrlrrE and O. W. MCGREGOR 1967 Technical Information Service, American Institute of Aeronautics. Dynamics of resonant two-dimensional cavities in aerodynamic surfaces at transonic and supersonic Mach numbers.