NASA CONTRACTOR REPORT WAVE THEORY OF TURBULENCE IN COMPRESSIBLE MEDIA Cxeslaw P. Kentzer Prepared by PUKDUEUNIVERSITY West Lafayette, Ind. 47907 for LangleyResearch Center NATIONALAERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. MAY 1976 I TECH LIBRARY KAFB. NM ___ - 1. Report No. 2. GovernmentAccession No. 3. Recipient's Catalog No. NASA CR- 2671 ~~~ ~~ 4. Title and Subtitle , 5. ReportDate May 1976 WAVETHEORY OF TURBULENCE IN COMPRESSIBLEMEDIA 6. Performing Organization Code ~ 7. Author(s) 8. Performing Organization Report No. Czeslaw P. Kentzer Kep.ort No. 75-2 ~~ ~ 10. Work Unit No. ~~ . ~~ 9. Performing Organization Name and Address Purdue University 11. Contractor Grant No. School of Aeronautics and Astronautics West Lafayette, IN 47907 NGR 'I 5-005- 174 13. Type of Repon andPeriod Covered 2. SponsoringAgency Name and Address ContractorReport NationalAeronautics and Space Administration Washington, D. C. 20546 14. SponsoringAgency Code 5. SupplementaryNoteS Technicalmonitor - S. Paul Pao, NoiseControl Branch, Langley Research Center. FinalReport .*- - ~. .. ~- 6. Abstract The generationof sound inturbulent flows cannot be determined without a solutionof the problemof turbulence itself since both soundand turbulence are manifestations of the same phenomenon of random fluidfluctuations andbecause sound and turbulence are strongly coupled. In practical appl cations,one is usually interested either in the far-field noise away fromturbulent flow regions or inthe turbulent noise transmitted through solid boundaries in contact with turbulent flows. In the first case, it sufficesto determine the acoustic mode ofenergy propagation at the edge of theturbu- lentregion, and inthe secondcase, both pressure and momentum fluctuations,whether radiating acousticallyor being convected by a turbulentflow past an elasticsolid durface, are of interest. Thus, a theoryof noise generation in turbulent flows should becapable of predicting the radiating and theconvected fluid fluctuations alike. Motivated by these requirements the "acoustical theory ofturbulence" was developed by theauthor. The research effort was intensifiedunder NASA Grant NGR 15-005-174, culninatingin the present report which'contains both the initial work and new results. The statisticalframework adopted is a quantum-like wave dynamicalformulation in terms of comple distributionfunctions. This formulation results in nonlinear diffusion-type transport equations for the probability densities of the five modes of wave propagation:two vorticity modes, oneentropy mode andtwo acoustic modes. Thissystem of nonlinearequations is closed and complete. The technique ofanalysis in this report is chosen such that direct applications to practical problems can be obtainedwth relative ease. " " -~ . .- 7. Key-Words(Suggested by Authoris) ) 18. Distribution Statement Coherent/lncoherentradiation, generation of sound, Wave theoryin turbulence Unclassified - Unlimited . " I Subject Category 71 19. SecurityClassif. (of this report) 1 20. SecurityClassif. (of this page) 21. NO. of Pages I 22. Price. 118 Unclassified . ~~ ~~~ ~ Unclassified $5.25 1 . - ." .. - . " 'For Sale by the National Technical Information Service, Springfield, Virginia 22161 . TABLE OF CONTENTS Page LIST OF SYMBOLS V 1. INTRODUCTION 1 11. RELATIONSBETWEEN TURBULENCE THEORY AND QUANTUMTHEORY 7 111. NECESSITY OF QUANTIZATIONTURBULENCE OF 19 IV. WAVE REPRESENTATION OF TURBULENTMICRO-STRUCTURE 31 W AVE -PARTICLE DUALITY V. WAVE-PARTICLE 48 Wave Packets as Quasi-Particles 48 The UncertaintyPrinciple 51 The Complementar.ity Principle 53 The Correspondence Principle 53 Operator Forma 1 i sm 55 Modification of Wave Frequency 57 VI. TURBULENTTRANSPORT EQUATIONS 60 DifferentialEquations for the Characteristic Function 60 DiffusionEquations theforField Probabilities 70 VII. AVERAGES, MOMENTS, AND CUMULANTS 77 Generalization of Definitions 77 Ambiguityin the Definitions 86 v111. ENERGIES AND DISTRIBUTIONS 91 Modal Energies 91 ApproximateDistribution Functions 94 IX. SUMMARY 100 X. CONCLUDING REMARKS 106 REFERENCES 109 LIST OF SYMBOLS a = (yRT) 4 _- adiabatic speed of sound A - scalarpotential of the probabi 1 ityvelocity i - B - vectorpotential of the probability velocity i vectorfunction bilinear in the fluctuations defined by Eq.(4.15) 'i - c = 0, for a=1,2,3; = 1 for a=4; = -1 for a=5 a c c - specificheats at constantpressure andvolume, respectively P' v ml* * *m C - cumulants ofthe wavenumber probabilitydistributions "1"'" N E - energy fl, F2, F3 -. force-1ike sources, defined by Eq.(4.13) f(G,i,t) = $*@ - probabilitydistribution in the phasespace h - Planck'sconstant H - Hami 1 tonianfunction li - vectorfunction linear in fluctuations, defined byEq.(4.15) - k - wavenumber vector K = yvk/(2aPr) - ratioof molecular mean freepath to the wavelength L - Lagrangianfunction L..(uj) - linearpart of the Navier-Stokes differential operator IJ il,h2 - mass-likesources, defined by Eq.(4.12) ml M - ordinary moments ofthe wavenumber distribution "1 N - n - unit normal vector V ml N - central moments ofthe wavenumber distribution "1"'" N p - pressure - p - quasi-particle momentum vector Pr = C~U/K - Prandtl number = spatialprobability density of the a-th mode pa R2 - P - participationcoefficients of the j-th unknown inthe aj . a-th mode of propagation 41 9 42' 43 - heat-likesources, defined by Eq(4.14) R - gas constant,also amplitude of the characteristic function $ s = (imE-Ut) phase function ofthe Fourier components a a - S = J(VA + VX~)*di - phase funct ion of the characteristic function $ t-time T- absol Ute temperature - T- mean or averageabso lutetemperature vectorfunction tril inearin the fluctuations, def ined by Eq. (4.15) Ti- - u- velocity vector - u' - fluctuating component of the velocity vector u- x-component of the velocity fluctuations u- vector of unknowns, j = 1,. ,5 - j u- mean or average velocity vector u- group velocity of the a-th mode aj u- phasespace energydensity v-- y-component of the velocity fluctuations V = 2+p /p viscosity number 21 - V= VS - probabilityvelocity v - volume vi V - potentialenergy ofquasi-particles a = 1,..,5 - denotesthe a-th mode of wave propagation y = c /c - ratioof specific heats PV attenuationfactor, imaginary part of complexfrequency ra - V - vectordifferential operator realpart of the random function w$ <a - <(z) - Riemann zetafunction K - heatconductivity coefficient X - wavelength 1-1 - coefficientof dynamic viscosity pl, p2 - first and second coefficientsof viscosity - imaginarypart of the random function w* Ea- a p - mass density of the mean flow p' - fluctuationsin mass density 4 - wave amplitude;also the dissipation function in Chapter IV $J = Re characteristicfunction of the a-th mode, Eq.(6.2) a is - - circularfrequency of the a-th mode, Eq.(4.19) Oa 2 w$ = (ca-iCa) k random functionmodifying the frequency w - a Special Symbols ( );? - indicates complexconjugate <> indicatesaverage with respect to the probability distribu- a - tion of the a-th mode of wave propagation (-) - indicates mean value, sum of averageswith respect to all modes of wave propagation vii "It seems to me that the test of 'm we or do we not understand a particular subject in physics?' is 'Can we make a mechanical model of it?'. 'I Lord Kelvin Quoted in P.Duhem, "The Aim and Structure of Scientific Theory," Princeton University Press, p. 71, 1954. I. INTRODUCTION The generationof sound inturbulent flows cannot bedetermined without a solutionof the problem of turbulenceitself since both sound and turbulenceare manifestations of the same phenomenon of random fluid fluctuations and becausesound and turbulenceare strongly coupled. As a matter of fact, carefully worded definitions are needed to distinguish the sound proper(a coherent or incoherent acoustic radiation) from a "pseudo-sound" orthe non-radiating pressure and other fluctuations convectedby the fluid and diffusingthrough it. Solvingthe problem of turbulentnoise in terms of the properties of turbulent flows rather than as a particularaspect of such flowsentails another difficulty. Tur- bulence is amendable onlyto a statisticaldescription. Thus only - " statistical properties of sound generatedby turbulence may be inferred from itsstatistics. C-nsequently, it may be concludedthat real practical advantages inthe analysis of turbulent noise lie in, l', a common theorythat would treat soundand turbulencesimultaneously as two different manifestations of the same random phenomenon, 29, thatthe theory bebased on the statistical methods inthe form commonly used in the fields of acoustics and turbulence alike permitting a convenien'trepresentation ofthe "acoustical" and "turbulent"functions, and, 3', thatthe theory be ableto separate those aspects of the problem that are referred to as "acoustical"from those that are traditionally associated with the purely "turbulent"motion, eventhough this distinction is not clear anddepends on theparticular definition employed. Inpractical applications one is usuallyinterested either in the far field noise away fromturbulent flow regionsor in the turbulent noise transmitted through sol id boundaries 'in contactwith turbulent flows. In the first case, it sufficesto determine theacoustic mode of propagation of energy at the edge of the turbulent region, because onlythe acoustic mode is capable of radiating far away from its source.In the secondcase, bothpressure and momentum fluctu- ations,whether radiating acoustically or being convected by a turbulent flowpast an elastic solid surface, are of interest becausesound waves