1、N 00 *o cv = U I FLUID MECHANICAL MODEL OF THE ACOUSTIC IMPEDANCE OF SMALL ORIFICES Alan S. Herslb and Tlbomus Rogers Prepared by HERSH ACOUSTICAL ENGINEERING Chatsworth, Calif. 9 1 3 1 1 for Lewis Research Center NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, 0. C. MAY 1976 Ei Provided b
2、y IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TECH LIBRARY KAFB, NM c 1. Report No. I 2. Government Accession No. I 3. Recipients NASA CR-2682 I 4. Title and Subtitle I 5. Report Date FLUID MECHANICAL MODEL OF THE ACOUSTIC IMPEDANCE OF SMALL ORIFICES 7. Auth
3、or(s) I 7 Performing Organization Report No. Alan S. Hersh and Thomas Rogers I None 10. Work Unit No. 9. Performing Organization Name and Address Hersh Acoustical Engineering 9545 Cozycroft Avenue 11. Contract or Grant No. NAS3-17858 Chatsworth, California 91311 - - 13. Type of Report and Period Cov
4、ered 12. Sponsoring Agency Name and Address Contractor Report National Aeronautics and Space Administration Washington, D. C. 20546 14. Sponsoring Agency Code 1 15. Supplementary Notes Final Report. Project Manager, Edward J. Rice, V/STOL and Noise Division, NASA Lewis Research Center, Cleveland, Oh
5、io 16. Abstract A fluid mechanical model of the acoustic behavior of small orifices is presented which predicts orifice resis- tance and reactance as a function of incident sound pressure level, frequency, and orifice geometry. Agree- ment between predicted and measured values (in both water and air
6、) of orifice impedance is excellent. The model shows the following (1) The acoustic flow in the immediate neighborhood of the orifice (i. e. , in the near field) can be modeled as a locally spherical flow. Within this near field, the flow is, to a first approxi- mation, unsteady and incompressible.
7、(2) At very low sound pressure levels, the orifice viscous resistance is directly related to the effect of boundary-layer displacement along the walls containing the orifice, and the orifice reactance is directly related to the inertia of the oscillating flow in the neighborhood of the orifice. Prev
8、iously, orifice resistance and reactance were modeled by empirical end correction expressions. The model also shows that, at low to moderate sound pressure levels, the resistance can be dominated by weak nonlinear jet-like losses but that the overall impedance can still be constant (i. e., independe
9、nt of incident sound pressure level) providing the orifice resistance is very small relative to the reactance. This is shown to occur when the amplitude of the incident acoustic pressure P is less than p w(D + L) , where w is the sound radian frequency, D and L are the orifice diameter and thickness
10、, respectively, and p is the fluid mean density. (3) When P/p w(D + L) 1, the orifice impedance is dominated by nonlinear jet-like effects. This corresponds to very high sound pressure levels, at which the orifice behaves in a predomi- nately quasi-steady manner. Thus, the model establishes explicit
11、ly the quasi-steady nature of the flow in orifices exposed to intense sound. (4) When P/p w(D + L) = 0 (l), orifice resistance and reactance are roughly equal. 2 7. Key Words (Suggested by Authodr) “ . 18. Distribution Statement Sound absorbers; Acoustics; Nonlinear Unclassified - unlimited acoustic
12、 impedance; Acoustic impedance; STAR Category 71 Orifice flow 19. Security Clanif. (of this report) “1 2Or- also harmonic term where obvious vi Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1. INTRODUCTION Cavity backed orifices are extensively use
13、d in the aircraft industry as acoustic devices to reduce or absorb internally generated jet engine machinery noise. The efficient application of these de- vices depends intimately upon the selection of the “optimum“ impe- dance to maximize the sound absorption. The sound absorption theories of Morse
14、 and Cremer2 for rectangular ducts without flow show that the sound absorption decreases rapidly from its maximum value for off-optimum wall impedance. This sensitivity has also been shown by Rice3 to exist for ducts containing flow. These studies demonstrate clearly the importance of accurately spe
15、cifying the wall impedance in acoustically treated ducts. Despite the extensive use of cavity-backed orifices in industry as devices to absorb undesired sound, their detailed acoustic behavior is not well understood. It has been shown by Ingard and others that the absorption characteristics of these
16、 devices are directly related to their impedance. Thus, most acoustic studies of the behavior of cavity-backed orifices consist of the measurement and prediction of their impedance. The purpose of this report is to present a fluid mechanical model of the behavior of isolated small orifices as a func
17、tion of incident sound pressure level, frequency, and orifice geometry. It is believed that this model will provide the necessary first step in understanding the behavior of cavity-backed orifices. Rayleigh was the first to predict the impedance of orifices by using the concept of lumped elements in
18、 a simple mechanical os- cillator analogy (i.e., the slug-mass model). His model is essentially non-fluid mechanical but gives the actual acoustical impedance char- acteristics for low sound pressure levels when an empirical end cor- rection is added to the slug mass. Rayleighs model was modified fi
19、rst by Sirignano6 and later by Zinn7 by introducing fluid mechanical concepts. To simplify their models, they assumed that the character- istic dimensionsof both the orifice or cavity are very much smaller than the incident acoustic wavelength and, further, that the acoustic flow through the orifice
20、 is one-dimensional, incompressible, quasi- steady, and calorically perfect. Both authors base their models on an integral formulation of the conservation of mass and momentum applied to two control volumes, one being the volume bounded by the orifice inlet and outlet surfaces and the other the cavi
21、ty. To solve these integrals, they used the method of successive approximations with the first order solution corresponding to the linear case of very small sound pressures inci- dent to an orifice. The orifice nonlinear behavior is introduced through the higher order terms and represent only a seco
22、nd order approximation to the (linear) first order solution. Thus their conclusions apply only to weakly nonlinear acoustic pressures and not to the intense sound pressures existing within rocket chambers or jet engines, the intended application of their models. There is a serious deficiency common
23、to both of their models. Sirignano assumes the loss in acoustic energy at the orifice outlet is equal to the jet outlet kinetic energy. Zinn assumes that, at Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-the orifice inlet, the axial inlet flow is z
24、ero but allows a radial inflow to preserve continuity. Both of these assumptions are difficult to understand because they violate their original assump- tions. For example, Zinns assumption that the flow in the orifice is one-dimensional (i.e., au/ax = 0) clearly contradicts his zero inflow and jet-
25、like outflow assumption. Sirignano violates the conservation of momentum by arbitrarily including a momentum term equal to ypu2/2 (see the third term on the RHS of Eqns. lO(a) and 10(b) of his paper). It is interesting to note that these two assump- tions lead directly to a one-half difference in th
26、eir estimate of the orifice nonlinear resistance. Another major deficiency of their models is that to first order (i.e., the so-called linear orifice impedance regime), both models predict the cavity resistance but not the orifice reactance. Despite these criticisms, Sirignano and. Zinn were -the fi
27、rst to assume that the behavior of the nonlinear acoustic flow in the neighborhood of the orifice is quasi-steady and that the concept of a discharge coefficient properly connects the orifice inflow to the outflow. Measurements of the behavior of small isolated orifices by Ingard and Ising and by Th
28、urston et. a1 have provided valuable data and much needed physical insight. These studies are reviewed below because of their importance in the development of the fluid mechani- cal model described in Section 2. Ingard and Ising used the arrangement shown in Figure 1 to study experimentally the acou
29、stic nonlinearity of an isolated orifice. Figure 1 shows an orifice plate mounted at one end of a circular cylinder. The experimental program consisted of taking simultaneous measurements of the acoustic pressure within the cavity and the acoustic velocity in the orifice. The sound pressure level wi
30、thin the cavity was measured with a small condenser microphone. The acoustic velocity in the orifice was measured with a hot-wire probe placed at the center of the orifice. The air within the cavity was excited at a frequency of 150 Hz by means of an electromagnetically driven piston located at the
31、bottom of the cavity. The acoustic nonlinearity is described in terms of the behavior of the orifice impedance. Ingard and Ising defined the orifice impedance as the ratio of pressure within the cavity to the funda- mental harmonic component of the orifice inlet velocity. The funda- mental harmonic
32、component of the orifice velocity was calculated by performing a Fourier decomposition of the measured orifice velocity time-history. The magnitude of the impedance is given by z*= -p:/u: where pr and UT represent the amplitudes of the harmonic cavity pressure and orifice velocity rzspectively. Th:
33、phase angle $1 between the acoustic pressure p1 and velocity u1 was determined 2 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-graphically by comparison of the pressure and velocity traces dis- played simultaneously on an oscilloscope. The results
34、of their study are summarized in Figure 2 in terms of RT the orifice resistance, and X* the orifice reactance. The data plotted against orifice inlet velocity, ma be divided into two regions, one where Phfand the other fiz In the region where R?“x“, gfl and the data shows that, to first order, RL8ui
35、. Ingard and Ising offer the following interpzetation of their data. At low sound pressures (corresponding to Rfi, the measurements show (see Fig. 2) that the orifice resistance is proportional to the ori- fice velocity. The measurements also showed that the orifice reac- tance is very insensitive t
36、o orifice velocity, decreasing at the very highest sound pressure levels measured, to a value roughly one- half the linear value. Ingard and Ising interpreted the orifice resistance data in terms of Bernouillis Law suggesting that the flow behavior through the orifice is quasi-steady. The hot-wire m
37、easure- ments indicated that at these high sound pressure levels, the flow separates at the orifice forming a high velocity jet. Thus during one-half cycle, the flow incident to the jet is irrotational; it is highly rotational (in the form of jetting) after exiting from the orifice. During the other
38、 half of the cycle, the flow pattern is reversed. The loss of one-half of the reactance at these high pres- sure levels was accounted for by assuming that one-half of the end correction is “blown“ away by the exiting jet (in their experiments L?D?fl. The orifice impedance measurements conducted by I
39、ngard and Ising and Thurston, et. a1 provide this relationship directly. 2.1 Approach The analysis starts with the equations describing the conserva- tion of mass and momentum written in spherical coordinates where only a radial inward flow u is assumed. For orifices small compared to the incident s
40、ound wavelength it is logical to assume that the flow approaches the orifice primarily in a spherical manner. The origin of the coordinate system is assumed to be located somewhere in the orifice interior as sketched in Figure 4. Assuming spherical symmetry, the flow field incident to the orifice wi
41、ll be assumed to be independent of the azimuthal angle $I (defined in Figure 4a). The flow field contains a uniform steady-state part and an oscillating acoustic part. A key element of the proposed model is the use of the experimental data of Thurston et. a1 and Ingard and Ising to normalize the equ
42、ations describing the conservation of mass and radial momentum of the oscillating flow field. The experimental data shows that two distinct regimes exist; the regimes are defined by the relationships that exist between the amplitudes of the incident driving acoustic pressure Pand the resulting ampli
43、tude of the acoustic velocity Vin the orifice. For sufficiently low values of p, the data showed that P*d v“ and for sufficiently high values of P: P“-(v”)” where ( )* denotes that the term within the brackets is dimensional. The regime whose P*,V* relationship is characterized by Eqn. (3) is often
44、called the linear regime while the regime characterized by Eqn. (4) is called the nonlinear regime. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-It is clear from dimensional analysis that the proportionality term in Eqn. (3) must have dimensions o
45、f density times velocity while that of Eqn. (4) must be density. The only density term suggested by the physics of the flow is P*, the mean fluid density. Thus, for orifices exposed to intense sound levels, the nonlinear case described above, the relationship characterizing P* and V* is P*= e“ (v“)“
46、 At low sound pressure levels both Ingard and Isings and Thurstons et. al. measurements show that the orifice impedance is dominated by the reactance X (see Figures 2 and 3), where X may be written (see Eqn. 2b) Since1 X* I IR* I (and thus P*/V* = Z*-X* for most practical appli- cations), where R* i
47、s the orifice acoustical resistance, P* is related to V* as follows P*= p*uJ * ( D*+ L*) v“ where D* and L* represent the orifice diameter and thickness respectively. Equation (3a) can also be deduced from a dimensional argument. If viscosity plays a negligible role in affecting orifice reactance (s
48、uggested by the success of Rayleighs slug mass model in predicting orlfice reactance) then the only other avails- ble combination is w*(D*+L*) . Equation (3a) suggests that at low sound pressure levels where P* = Z*V*-X*V* that X*=? * “(D*+ L“) Figure 5 shows that orifice reactance does indeed behav
49、e according to Eqn. (5); the experimental data of Ingard and Ising and Thurston et. al. collapses into a single correlation curve. This agreement is remarkable when one considers the vast differences between these experiments. Ingard and Isings measurements were conducted with an orifice diameter of 0.7 cm. in air exposed to sound fre- quency of 150 Hz while Thurstons et. al. measurements were conducted 6 Provided by IHSNo
