1、NASA ttcnnncAr MEMORANDUM NASA TM X-71903 (NASA-TR-X-71903) a THEOEETICAL STUDY OF 176-21 427 THE BCOUSTIC IHPECAICE OF OBIFICES IN THB PRESENCE OF 1 STEADY GRAZING FLOW (NASA) 23 p BC $3.50 CSCL 20D Unclas A THEORETICAL STUDY OF THE ACOUSTIC IMPEDANCE OF ORIFICES IN THE PRESENCE OF A STEADY CRAZING
2、 FLOW by Edward J. Rice Lewis Research Center Cleveland, Ohio 44135 TECHNICAL PAPER to be presented at Ninety-first Meeting of the Acoustical Society of America Washington, D. C. , April 5-9, 1976 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-A THE
3、ORETICAL STUDY OF THE ACOUSTIC IMPEDANCE OF ORIFICES IN THE PRESENCE OF A STEADY GRAZING FLOW by Edward J. Rice ABSTRACT 03 Q) CD act An analysis of the oscillatory fluid flow in the vicinity of a circular orifice c) , /rn 2 nondimensional radius (r */d) dimensional radius, m r at orifice radius (ro
4、 = 9 ) nondimensional time (w t*) time, sec magnitude of radial perturbation velocity near orifice, dsec nondimensional radial perturbatlon velocity (uf/u0) complete dimensional radial velocity (see Eq. (6), m/sec radial component of grazing flow velocity (see Eq. (I), m/sec radial component of pert
5、urbation velocity, m/sec same as u but for steady onfice flow orifice perturbation velocity based or, flow rate and orifice area, m/eec orifice perturbation velocity at vena-contracts, m/sec grazing flow velocity, m/sec 6 cqmponent of grazing flow velocity, dsec Provided by IHSNot for ResaleNo repro
6、duction or networking permitted without license from IHS-,-,-complete dimensional 8-component of velocity (see Eq. g), dsec 0-component of perturbation velocity, dsec cp-component of grazing flow velocity, dsec complete dimensional cp-component of velocity (see Eq . (8), m/sec cp-component of pertur
7、bation velocity, m/sec phase angle between pressure and velocity dimensionless acoustic impedance polar angular coordinate (see Fig. 1) wavelength, m density, kg/m 3 average uniform density, kg/m 3 density perturbation, kg/m 3 azimuthal angular coordinate (see Fig. 1) circular frequency, rad/sec THE
8、ORETICAL DEVELOPMENT In this section first the differential equations will be presented in dimen- sionless form, Next, they will be simplified by a magnitude analysis and finally the solutions will be derived. Differential Equations The geometry of the system considered here is shown in Figure 1. Bo
9、th rectangular and spherical coordinate systems are shown centered at the ori- fice center. The steady flow velocity is shown parallel to the x axis and al- though uniform in the rectangular coordinates, in spherical coordinates the ex- pressions are: Provided by IHSNot for ResaleNo reproduction or
10、networking permitted without license from IHS-,-,- u = -V, sin 6 cos cp (1) The differential equations, for the purpose of brevity, will be given with all of the assumptions inserted. The variables will be considered to be given by the sum of a steady component and a time varying component as, Aster
11、isks are used here to denote complete dimensional quantities for the dependent variables and dimensional quantities for the coordinates. The isentmpic relationship is assumed valid so that The radial coordinate and time will be nondimensionalized as follows. The time varying pressure is normalized b
12、y the peak far-field pressure P1 qJP (12) Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-and thus using Equation (9) The time varying radial velocity is normallzed by the amplitude near the orifice (Uo) thus The perturbation velocity (ut), the compl
13、ete velocity (u*), and two components - - of the steady grazing flow (u, v) are shown in the two-dimensioml sketch of Figure 2. The simplest possible solutiol will be sought for the velocity and then tested to see if it has provided any information or insight into the physics of the grazing flow imp
14、edance. With this in mind, only axisymmetric pertuhations of the velocity will be considered and thus, It should be noted later in the development of the equations that even though the velocity perturbatioil is axisymmetric, the pressure distribution around the uri- fice is asymmetric The viscous te
15、rms in the momentum equations will not be considered. With the above assumptions, the equations of motion can be written as follows: Co. tinuity Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-r Momentum Although considerably simplified from the comp
16、lete equations, Equations (16) to (19) are still much too complex for a simple closed form solution. An order of magnitude analysis on the coefficients must now be made in a manner similar to Reference 15 except that V, must be considered. Only the coeffi- cients need to be consider( d since the var
17、iables themselves are of order unity due to the nondimensionalization. From several references (Refs. 4, 6, 7, 8, 10, and 14 e. g.) the grazing flow resistance can be approximated by Thus Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-since M, is at
18、 most 0.6 or 0.7 for the usual acoustic liner and Mo is much less than this. Also provided the wavelength, (A) is much larger than the orifice didmeter (d) which is almost always the case in a practical acoustic liner. The first term in the continuity Equation (16) is probably the smallest of all wi
19、th the third term be- ing next smallest in magnitude (essentially Mach numbers to the third power when the terms in the square brackets are considered). Also since p is of 2 order unity the term .FC can be dropped where it is added to unity. Thus the problem to be solved is shown to be incompressibl
20、e and the equations have been reduced to. Note that Equation (24) is nonlinear in u and we will restrict the solution tr, the region in which the grazing flow velocity dominates the orifice velocity. An approximate nonlinear solution can probably he found tollowing the meihod of Reference 15 but tha
21、t will not be attempted here. A discussion of the range of validity of the linear solution will bc expanded upon in a later section. With this restriction in mind, Equation (24) can bc written as Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-he bou
22、ndary conditions to be applied involve the excitation pressure im- posed upon the system in the far field and the inevitable separation of the flow at the upstream edge of the orifice for inflow. This separation implies that for sufficiently short orifices (no reattachment) the ambient or back cavit
23、y pressure will be felt at the upstream orifice edge. This separation region is shown in Figure 2. For outflow the orifice boundary condition would be modi- fied and it is thought that the solutions could be made to model outflow. This is not done in this paper and the solutions which follow are int
24、ended for inflow only at this time. The boundary conditions can be expressed as follows Solution to Differential Equaiio The solution to the differential equatios follow in a similar manner to that of eference 15. Equation (23, can be immediately integrated to give, where the negative sign was chose
25、n sinca the velocity is opposite to the radial coordinate for inflow. The function F(t) depends upon t only since u was assumed not to be a function of 8 or q. Using Equation (30), Equation (27) can be written. Equaton (31) can be integrated to obtain, Provided by IHSNot for ResaleNo reproduction or
26、 networking permitted without license from IHS-,-,-Use Equation (28) in Equation (32) as r+ to obtain, E EPRODUCIBILIIY OF THE ORIGINAL PAGE 35 POOR where f(t) can be only a function of time due to the boundary condition of Equa- tion (28). Thus, Equation (32) can be written, In solving for F(t) we
27、will seek solutions of the form, Substituting Equation (35) into Equation (34) and imposing the upstream boundary condition (Eq. (29) at the orifice edge, after considerable arithmetic, A and 6 can be determined such that and Csing Equations (35), (36), ar.d (37) in Equation (30) yields, Provided by
28、 IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-and then Equation (34) will ve, It should be noted here that the velocity and pressure Equations (38) and (39) also satisfy the 0 and cp momentum Equations (25) and (26). ACOUSTIC IMPEDANCE Before calculating acou
29、stic impedance one more point must be made. Usually impedance is based upon the velocity calculated from orifice flow rates and orifice area. Note that the area of the hemisphere at r = ro is twice the orifice area. The value of u calculated at r = ro will thus be doubled when used to calculate impe
30、dance. The acoustic impedance ulll be defined as the far field pertu p s = 0 wiiich of course was the boundary condition reflecting flow separation at this point. The back cavity pressure is felt here. However at the downstream edge (r - ro, 0 -: n/2. cp = a), p, = 2 or twice the far feld pressure.
31、In fact if the pressure is integrated over a hemisphere arour!d the orifice to fina an aver;tge pressure, it is found that this average is euual to the far field diiving pressure. The pressure has just redistributed due to the grazlng flow. The pressure in thc high pressure region of the flow must y
32、et be relieved in passing to the back cavity which requires a further expansion. In oiher words, the average prcssurc at r = rO still has r magnitude 9. The fluid must expand to have zero prcssurc as it passes to the back cavity side of the orifwe. The vclocity will thus increase Provided by IHSNot
33、for ResaleNo reproduction or networking permitted without license from IHS-,-,-to a maximum at the vem-contract-. This will be analyzed using a one- dimensional appmach to find an average vena-contracts velocity, utc, and then a discharge coefticient will be cdculated. The further expansion in the o
34、rifice can be estimated from the Bernoulli equation, where UOus is the dimensional perturbation velocity at r = ro, and the pres- sure drop is from .? to zero in the back cavity. Using Equation (42) tl. sre results, Frm flow continuity considerations (use Eq. (42) to calculate flow rate through the
35、hemispherical area) the full flowing orifice velocity (based on orifice area and flow rate) is, The orifice discharge coefficient is defined here as t le ratio of the full flowing orifice velocity to the vena-contracts velocity and c; .n then be calculated from where 9123 was assumed to be small com
36、pared to unity. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-This result has been found empirically in Reference 14 and will be dis- cussed further in the next section along with pressure measurement comsri- sons using Equation (43). DISCUSSION OF
37、 RESULTS In ths section the previously derived theory will be compared to data where it is available. The acoustic resistance is well established by experi- mental data. The acoustic inductance is less defined since measurements lump both the orifice end effects and within the orifice mars effects i
38、nto one inductive quantity without any way of separating them. Also the steady orifice inflow theory will be compared to experiments. Acoustic Impedance Acoustic resistance. - The real part of Equation (41) (resistance) com- pares well with published experimental results. It is well established that
39、 resistance depends upon the grazing flow Mach number. The coefficient of Moo in Equation (41) is 0.5 which compares favorably with 0.3 from References 4 and 7 for arrays of orifices and with 0.7 from Reference 14 for a single orifice. It is not Jlltended that these theoretical results replace the e
40、mpiricisms since there is much more work to be done on the ;letails of the flow within the orifice, but only to show that the theory appears reasonable. Acoustic reactance. - The theoretical results for orifice reactance a re not so easily compared to data as was resistance. Since the major conclu-
41、sions of this paper concern the reactance, it is best to first put the theory and the actual flow into proper perspective. Figure 2 shows sketches of thc inflow to the orifice for conditions such that the grazing flow is complete1:- dominant as assumed in the present theory. The flow patterns of fig
42、ure 2 have been established for steady orifice flow in Reference 14 and for oscil- latory orifice flow in Reference 16. Th? hemisphere of radius equal to the orifice radius is shown on Figure 2 and the solutions given in this paper are expected to apply outside of this hemisphere. The total inductan
43、ce is determined by adding the inductance provided by the fluid within this hemi- sphere and within the orifice itself to the inductive oriiice end correction given by Equation (41). Going back for a moment to zero pazing flow and low Provided by IHSNot for ResaleNo reproduction or networking permit
44、ted without license from IHS-,-,-pressure amplitudes, the orifice inductive end correction is given by 0.8 5d (Ref. 18) which would give 0.425d at each end of the orifice. Equation (41) accounts f over half (0.25cD of this cornction which is outside of the hemisphere sbowu in Figure 2 with the remai
45、nder wing within this hemi- sphere. Hard conclusions can thus be made about only half of this mass, but it is suspected that the conclusions will apply to the bulk of the attached mass. Experimental data (Refs. 6, 8, and 17) show that the total inductance is re- duced with a grazing flow. Since Equa
46、tio (41) shows the inductive orifice end correction to be independent of grazing flow velocity, then the reduction in in- ductance which is observed must be coming from the fluid within or very close to the orifice. This conclusion is contrasted to the idea in which an orifice slug flow exists with
47、an attached mass end correction which is blown away by the grazing flow. Additional study must be devoted to the flow in the orifice. The orifice flow (with a grazing flow velocity) is not a slug which accommodates more mass flow by an acceleration of the slug. Instead the flow may have very nearly
48、a constant velocity (see Ref. 16) with additional mass flow accommodated by an increased flow area. A sketch of these flow patterns is shown in Figure 3. It may be that a constant velocity, variable area type of flow has essentially no inductance at all. This latter point is currently being studied. Steady Orifice FLW For steady inflow into an orifice some discharge coefficient data are given in Reference 14. At low orifice velocity to grazing flow velocity ratios a fit to that data can be made as This result compares fa
