Infrasonic Far-Field Radiation From a Turbulent Wake

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1 NRL Rqert 7421 Infrasonic Far-Field Radiation From a Turbulent Wake JOHN A. DESANTO PrnpuglIati( Brancth AmoIsxic'. Division June 29, 1972 *4k NAVAL ESEWARCH LAOmRaroRv Wambhgig, D.C. Approvej for public releasn: diributim unlimited. L..., LA....

2 Set tait CIdsLfcation DOCUMENT CONTROL DATA - R & D,$r al flas.,fal-mhon of title., bod. of ab..tartg id,) -..,olat,.,n -.,.t b. entered hen the overall report I, rnsh..fed) 0" "NATING ACTIVITY-, Utpoat. aeafho,) aa.aepoa' SaC.u!AP CLASSIFI(CATON Naval Research Laboratory UNCLASSIFIED Washington, D.C ) -o, I REPORT 'IYLF INFRASONIC FAR-FIELD RADIATION FROM A TURBULENT WAKE 4 OCSCuP TII, C NOTES (ry9p of tepoet and 1noi.v.its dates) Interim report on a continuing problem. AU T.4OR *, (i.i'st noerw, oddfe -nitfaf. lest nlme) John A. DeSanto 6t ROP. OATtr to. TOTAL NO OF POACG ts rio or Ores June 29, * CONTRACT OFF GPA04T 0O SW. Ol NGAOtU'S QIPOPT * iuft IRSI NRL Problem SO1-40 To N. ORojc NRL Report 7421 RF o..0.o,,,it. OnC E,ONT ROM, (Any other fhebet. that ".. be *..,84#d o_ d. 10 OISTRIGUTiON STATEMENT Approved for public release; distribution unlimited. 11 SUPPLEMENTA.v NOTES I* SPONSOING *lt.i 'AM ACT"IVITY Department of the Navy (Office of Naval Research), Arlington, Va AMSTITACT The acoustic far-field pressure and intensity due to a turbulent wake is calculated using Lighthills free-turbulence theory. The radiation intensity is plotted as a function of the Mach number and of the ratio of wake wavelength to radiating wavelength. DD `,N``,.1473 (PAGE ) S/'N 0I0.O07.s6o01 I Seutltv clas sftalom

3 Security CISqsification 14 LINK A LINK a LINK C KEY WORDS HOLI v T 006l9 WT ROLIE WT Wakes Turbulence Extremely low radio freq&'encies Electromagnetic radiation Far-field DD,movs ACK_,PArT, ) U1II Clava icatinn - -

4 / INFRASONIC FAR-FIELD RADIATION FROM A TURBULENT WAKE INTRODUCTION The problem considered in this short report is the calculation of the far-field intensity (or pressure) due to radiation from volume turbulence. The specific question to be answered is: What is this far-field intenpity at extremely low frequencies (infrascund)? In the second section the far-field pressure is calculated using Lighthill's relation for the pressure due to volume turbulence 41,2). To do this requires certain assumptions about the turbulent velocity field, namely, that it be isotropic and have the form of an outgoing spherical wave. The basic result of the second section is that the infrasonic farfield intensity is proportional to M 4 (A/X) 6 where M is the Mach number, A is the turbulent wavelength, and X is the radiated wavelength. For Infrasound (large X) this factor is very small. A plot is given showing the intensity as a function of Al and A/X. Finally, the last section contains a short summary of both assumptions and omissions. THEORY 'The spectral l)ressur p(r) at the field point r (in three dimensions), due to volume turbulence arising from a turbulent velocity field Umn ro) at the point ro (m 1. 2, 3), is given by Lighthill (1,2) as PO P0 c e eirro i~r-.rol p(r) =. monvm B4 (ro)vj (ro) -rodr 0 (1) where 47r f o differentiates on the "zero" coordinate. The Einstein summation convention is assumed here. The density of the medium is p0 and k = 21r/X. Further manipulations of this volume integral yield (1,2) p(r) = P0 a drolv(ro)vn(r)!ikr- r01 (2) In the far field (large r) caklr-rol ir - rol cikr ~.c~oo. rom"o0 (3) r

5 2 JOHN A. DeSANTO where r ro= rro cos 00, and hence, for large r, tieikr '4 p(r) ((aman-rjqmn (4) where Qmn P0?fdrouv(ro)vn(ro)e-d-roc 0 (5) If the turbulence is assumed to be isotropic, VM(r) = Vm(r), (6) and behaves like a spherical wave (with wavenumber K which could have a positive imaginar part), then CiKr vmr(r) = Vm (7) Qmn then can be evaluated as (K 2v/A) Qmn = P0 d1 d ro doo fk 2080 door 2o~ sin O0 Vm(ro) vn (ro)e ijroco 0 P ipovmvu In +_2W (8) 2kK2 l where integral tables (3) have been used for the value of the r 0 integration. Introducing twe adiabatic compressibility Ks, defined by KS = (P0C 2 )-l where c is the velocity of sound in water, Eq. (8) becomes Qm1 Ksc 2 2kK 2 Further approximation of this quadrupole term by its diagonal values yields ( 6 mn is the Kronecker delta) Qmn 2K"mK Imn 1n - (9) 2A/

6 NRL REPORT where M = v/c is the Mach number and v =rn. In addition, in the far field, the term in brackets in Eq. (4) can be written as Using the approximation am leilr( ei(r' dr or rm~nqr) OXm ajxn ar axm 3r 3Xn with E1. (9), and the result 6 mn 6 mn 6nn 3, it is possible to write Eq. (4) as -3i edu 2~ A_ý) 1_+ p(r) K In - (10) p~) 2 kr K, X A and the far-field intensity I as- 21 S9 cii4 f.a 4 + 8() (kr) 2 K4 (\,{\ l (1 L\-2Xj For very low frequencies (infrasound, A large), Eq. (11) can be written 6 _IW 8 tkr) 2 Ks ) (12 Hence, in addition to the usual spherical spreading loss, there is a dependence on the Mach number Al raised to the fourth power, and to the ratio of turbuler.t wavelength to radiated wavelength raised to the sixth power. There is also a fact3r of X in kr. Figure I is a plot of the ratio 1(r)/I 0 versu.s A/A from Eq. (12). The reference level is a comparison source I0 having an output of 0.64xi0" 12 watts/cm 2 (a pressure level of 1 dyne/cm 2 ). The quantity I/1o is plotted in decibels at a range r of 1 naut mi and for an infrasonic wavelength X = 500 ft (or a frequency of 10 Hz). The intensity at a new wavelength X1 can be found from the relation 11 = 1(W/XI). For example, doubling the wavelength (XI = 2), frequency = 5 Hz) decreases each of the 10-Hz %urves by 12 db. Halving the wavelength (20 Hz) increases the curves by the same amount. REMARKS There are several assumptions and omissions in this calculation which should be restated. They are a. Lighthill's turbulence theory for free turbulence is assumed valid, as is the farfield assumption. Effects due to boundary layer turbulence have been omitted.

7 4 JOHN A. DeSANTO +100 M.O"2 r a I NAUT MI. A FT (10 Hz) +60 * o Hz 5Hz 20 Hz -80 z 5 z 20 Hz LOG (A/A) I I I I I A/h Fig. --Relative plot of radiation intensity 1I!0 vs the r2tio of turbulent to radiated wavelength AA for several values of the Mach number Af at a range r of I naut mi. For the corresponding values at I yd add 66 db. b. The turbulence was assumed to be isotropic and to radiate as a spherical wave without decay. c. No attempt was made to treat the time dependence of the pressure or turbulent velocity fields. REFERENCES 1. Lighthill, MJ., Proc. Roy. Soc. A222:1 (1954). 2. Morse, P.M., and Ingard, K.U., "Acoustics I," in "Encyclopedia of Physics." S. Fliigge, ed., Berlin: Springer, Burington, R.S., "Handbook of Mathematical Tables and Formulas" 4th Ed., New York: McGraw-Hill, 1965.

A0A TEXAS UNIV AT AUSTIN DEPT OF CHEMISTRY F/G 7/5 PHOTOASSISTED WATER-GAS SHIFT REACTION ON PLATINIZED T7TANIAI T--ETCIUI

A0A TEXAS UNIV AT AUSTIN DEPT OF CHEMISTRY F/G 7/5 PHOTOASSISTED WATER-GAS SHIFT REACTION ON PLATINIZED T7TANIAI T--ETCIUI AA113 878 TEXAS UNIV AT AUSTIN DEPT OF CHEMISTRY F/G 7/5 PHOTOASSISTED WATER-GAS SHIFT REACTION ON PLATINIZED T7TANIAI T--ETCIUI APR 82 S FANG. 8 CHEN..J M WHITE N14-75-C-922 UNCLASSIFIED NL OFFICE OF