Contains reports on two research projects.U.S. Navy (Office of Naval Research) under Contract N00014-67-A-0204-001

Galaitsis, A. G.

Price, R. H.

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IV. PHYSICAL ACOUSTICSAcademic and Research StaffProf. K. U. IngardDr. C. KrischerGraduate StudentsA. G. Galaitsis P. A. MontgomeryG. F. Mazenko R. M. SpitzbergA. STUDIES OF SUPERSONIC JETSOur study of supersonic jets continues using a modified version of the coronaprobe described in a previous report. 1 The following modifications have been incor-porated: A current collector has been placed around the central discharge wire.This makes the discharge more stable and less dependent on the flow velocityand the geometry of the surrounding system. A 10-M2 resistor has been placedin series with the central discharge wire, thereby decreasing the discharge cur-rent and prolonging probe life. The probe current is now measured as the volt-age drop across a 1000-~2 resistor between the current collector and ground.In Fig. IV-1 a longitudinal trace of the probe output as a function of distancealong the center line of the jet is shown. This curve represents the densityvariation along the jet, and the shock cell structure of the jet is clearly notice-able.As well as this steady-state response, we have also investigated the spec-trum of the flctuations in the probe current. Under certain operating conditionsof the jets this spectrum contains lines which undoubtedly correspond to shockcell oscillations. Of particular interest is the amplitude variation of these oscil-lations with distance along the jet. A typical example of such a variation isshown in Fig. IV-2, which corresponds to an oscillation frequency of 8. 8 Hz.It is interesting to note that the maximum amplitude is in the third cell, andno detectable oscillations occur in the first and second cells.While engaged in this work we have also studied some of the characteris-tics of the probe itself. For example, the variation of the probe output voltagewith density for some different probe bias voltages is shown in Fig. IV-3. Self-oscillation of the probe will occur under certain conditions. As can be seenfrom Fig. IV-4, the frequency of these oscillations depends on both operatingThis work was supported by the U.S. Navy (Office of Naval Research) underContract N00014-67-A-0204-0019.QPR No. 101 Ao600D0c 40200 5 10 15 20 25DISTANCE FROM JET NOZZEL (cm)Fig. IV- 1. Probe output voltage measured along the axis of the jet as a function ofdistance from the jet nozzle for a pressure ratio of 18. 6 and a probebias of 900 V.0.40.3E0.2Z0.1Fig. IV-2.8.7 kHz8.8 kHz5 10 15 20DISTANCE FROM JET NOZZEL (cm)The rms jet noise along the jet axis as measured by the fluctuations ofthe probe output voltage. The bandwidth of the spectrum analyzer is ±6 Hzand the probe bias is 900 V. The curves have not been corrected forchanges in the small-signal gain of the probe, which are due to changes ofthe average gas density. Note that the higher frequencies appear closerto the jet nozzle in each shock cell.QPR No. 101* 1000 VO 900 VA 800 VA 700 VI I I I I I IPRESSURE (Torr)Fig. IV-3. Probe output voltage as a function of ambientpressure under low flow-rate conditions, forvarious probe bias voltages. These curvesare used to calibrate the probe for measuringthe gas density in the supersonic argon jet.Fig. IV-4.PRESSURE (Torr)Probe self-oscillation frequency as afunction of pressure for various probebias voltages.60E400O020CI I(IV. PHYSICAL ACOUSTICS)voltage and gas density. These oscillations do not appear to be present whenthe probe operates in the jet. A more detailed analysis of this matter will bepresented in a future report.R. H. PriceReferences1. R. H. Price and U. Ingard, "Techniques for Measuring Density Fluctuations inJets," Quarterly Progress Report No. 100, January 15, 1971, pp. 70-72.B. NONLINEAR TRANSMISSION OF SOUND THROUGHAN ORIFICEExperimental data have already been reported on nonlinear transmission of soundthrough a sharp-edged orifice. At high enough levels of sound (i. e. , >130 dB) sep-aration of flow occurs and the average velocity u at the orifice becomes a non-linear function of Ap (the pressure drop across the orifice) 3PAp 2 ufu, (1)2C 2where p is the density of the air, and C 1 is the orifice coefficient. If A D and A °are the duct and orifice cross section, respectively, then (1) can be written in termsof the average duct velocity U.2 c ( juU. (2)Ap - 2 A o U U. (2)1To be thorough, we include in the right-hand side of (2) a resistive, as well as a reac-tive, term to account for viscous and inertial effects in the vicinity of the orifice.The equation of motion then becomesAp = Q U U - pcOU + dU for x = 0, (3)whereQ = p/2C (AD/Ao ' t m (AD/Ao),4with C the speed of sound, t the effective thickness of the orifice, and 0 amQPR No. 101(IV. PHYSICAL ACOUSTICS)resistive coefficient that can be expressed in terms of the geometry of the orificeand the coefficient of viscosity.As a result of the nonlinearity the transmitted wave will contain a line spectrumrepresenting the fundamental and overtones of the driving frequency. In a previousattempt to explain the experimentally obtained spectrum 5 the tube was assumed tobe infinite in both directions from the plate. The theoretically obtained spectrumthen contained only odd harmonics of the frequency of the incident wave. Althoughthese harmonics were indeed predominant in the experimental data, even harmonicsdid occur. In the following analysis we shall re-examine this matter, using a morerealistic model of the experimental arrangement.In the experiment1 the orifice is at x = 0, while a highimpedance piston locatedat x = -L moves according tou = u cos wt,p Othereby generating a wave that is partially transmitted through the orifice. In thesteady state the incident, reflected, and transmitted waves have the formPI (x, t)PR(X, t) =PT(x,t)n=1I cos n + A sin nn nRn cos n + U sin nTn cos n + V sin nand the corresponding velocities areUI (x, t)UR(x, t)UT(x, t))n= 1I cos n + An sin n-R cos nl - U sin niT cos n + V sin n'T cos n4 + V sin n4)n n(6)where = kx - wt, i = kx + wt. The coefficients of (5) will be obtained by substituting(5) and (6) in (3), as well as in the two boundary conditions that express continuity ofvelocity:UI(-L, t) + UR(-L,t) = u (7)(8)U (0, t) + UR(0, t) = UT( 0 , t) - U.The calculations are straightforward and yieldQPR No. 101 c(IV. PHYSICAL ACOUSTICS)A +U =Vn n nI -R =T(In-Rn)cn + (Un-An )s n = pcuo 5n(10)(I +R )s + (U +A )c = 0n nnl n nnlIn+ Rn + (-1)Tn +q V = BA - U + (0-1)V - q T = Bn n n ninn-1k [1- 6n][Tj .T-V . V j ] +j=1n-[1-6n] [TnjVj+TjV nj ] +2Bj=12B ij= 1[Tn+j T+V n+Vj]n+ n+] ]](11)[Tn+j Vj- j n+j],c - cos nKL,nq nKt (AD/A),qn mDos = sin nKLnQB-2 Z2'2p cThis system of equations has been simplified further, and computer work is beingcarried out to obtain approximate values of the coefficients for a given set of param-eters. The results will be forthcoming.A. G. GalaitsisReferences1. U. Ingard and H. Ising, J. Acoust. Soc. Am. 42, 6-17 (1967).2. U. Ingard and S. Labate, J. Acoust. Soc. Am. 22, 211-218 (1950).3. S. Whitaker, Introduction to Fluid Mechanics (Prentice-Hall, Inc. , Englewood Cliffs,N. J. , 1968).4. U. Ingard, J. Acoust. Soc. Am. 25, 1037-1061 (1953).5. U. Ingard, J. Acoust. Soc. Am. 48, 32-33 (1970).QPR No. 101(9)where

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