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$ 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 in a solidmaterial of the boundary may be excitedby both the radiating and theconvected fields. Thus a theoryof noise generation in turbulent flowsshould be capable of predicting the radiating and theconvected fluid fluctuations, the "acoustical" and the"turbulent1' properties a1 ike. Motivatedby these requirements the "acoustical theory of turbulence" was developedby the author (Kentzer, 1974a, b), and thisreport addresses itself to the task of summarizing and extendingthe previously achieved results to a point where thetheory is closed and complete and ready to be tested on some simplecases. From itsearly conception the theory was basedon the statistics of wave motions in viscouscompressible fluids in order to take advantage ofthe Fourier mode ofanalysis which is in common useboth in the 2 r statisticalturbulence and inthe field of acoustics. An added incentive to pursuethe wave formulationof the theory of turbulence was themeeting and stimulatingconversations with Academician A. A. Dorodnitsyn and Prof. M. Z. E. KrzywobYockion theoccasion of a round-tablediscussion ofthe numerical computations of turbulent flows, held during the 8th Symposium on Advanced Methods and Problems in Fluid Dynamics, Tarda,Poland, 1967. It became apparent atthat time that whatremained to be done was to choose a proper statistical framework in order to obtain kinetic equationsfor the time evolution of the wave distributions, and that such distributions woulddetermine all statisticalproperties of turbulence including its acoustic modes. Sincethe Symposium at Tardathe author continuedexchanging numerous communications, withProf. Krzywobtocki and others, on thesubject of analogies existing between the wave dynamics ofturbulence and wave mechanics of quantum systems. The analogies suggestedthe quantum-like framework forthe theory mainly on thebasis of the availability of provenmathematical methods developedover the yearsfor the purpose of treating quantum problems. Thus thesufficiency ofthe use of quantummethods inturbulence was recognizedearly. A survey of literature revealed many applications of quantummethods to the studyof turbulence and many arguments forthe necessity of quantum-like formulationof the theory. A discussionof these subjects is included inthis report, Chapters It and 111, inorder to illuminatethe background of thegenesis of the present theory. The wavedynamicalformulation ofturbulence, with its orthogonal decomposition of thefluid fluctuations into the vorticity, entropy, and acoustic modes,was found to be a naturaltool for the study of thenoise generationin turbulent flows. With the objectives of deriving expressions 3 forthe sound sources in turbulence,arising from interactions with the mean flow and with the vorticity and entropy waves, and the determination of thepropagation properties of theacoustic mode, the research effort was intensified under NASA Grant NGR 15-005-174, culminatingin the presentreport which contains both the previous work and new results. The philosophyunderlying the concepts that guided the development of.the present theory will be discussedbriefly. We observe first that laminarflows, considered as solutionsof the Navier-Stokes equations, depend continuously onand aredetermined by the parameters contained in thedifferential equations and inthe appropriate boundary and initial conditions. These parameters,which usually are grouped into nondimensional ratios,are macroscopic in nature. Once laminarflows become unstable, theinitial-boundary value problems for the Navier-Stokes equations are notproperly posed because thesolutions cease to depend continuously on theinitial and/orboundary data. These dataare not sufficient for the determinationof a uniquesolution. As soon as a disturbancein a laminar flow becomes sufficiently irregular so that a large number of wave components(eigenmodes) becomes excited,there arises the need for treating a continuous medium as a system with infinitely many degrees of freedom. To describethe behavior of such a system onemust use an infinite number ofgeneralized coordinates. It was thendecided to choose the wave solutions of theNavier-Stokes equations, linearized around the local mean flow,to serve as a completeorthogonal set of basis vectors. Consequently, thecoefficients of the expansion in terms of such orthogonal modes become thecoordinates in the spacespanned by thebasis vectors. Further, thebasis vectors are functions of the instantaneous local mean flowwhich 4 plays the r$le of the space-time-dependent macroscopic parameters.The analogy to a system of harmonic oscillators becomes apparent and suggests the use of traditional statistical methods for treating such systems. In turn, the arguments of Ehrenfest (1911) applied to a system of oscillators in equilibrium with an energy reservoir convinced the author of the necessity of considering the statistical methods of quantum theory. We will not review herethe state of the theoretical knowledgeof turbulent phenomena. A brief history of theoriesof turbulence and a description of modern theories aregiven in the Introduction, pp. 5-19 of the book by Monin and Yaglom (1971) to which the reader should refer for an extensive bibliography of the subject. Modern approaches to the theory of turbulence apply statistical methods to the ensemble of turbulent flows satisfying macroscopically identical external conditions. Theories that are rigorous and free from any ad hoc statistical approxi- mations have their origin in the work of Hopf (1952) who deriveda linear functional differential equation fora characteristic functional of incompressible turbulent fields. Hopf's formulation is closed and complete, but leads to considerable practical difficulties of solving equations in functional derivatives. With numerical solutions of turbulent flow problems in mind, the present theory followsa more tractable space- time formulation. Admittedly, the mathematical rigor is sacrificed in the process and traded for the chance to use more familiar mathematical techniques and for the relative ease of direct applicationsto practical prob 1 ems. The organization of the material presented here is as follows. In 5 Chapter 11 theapparent analogies between turbulencetheories andquantum theoriesare discussed. The use of quantum concepts and known quantum- likeformulations of theories of turbulence are reviewed briefly. Several arguments forthe necessity of allowing for the discontinuous ("quantized") natureof turbulent energy exchange processesare given in Chapter 111. The present wavedynamical theoryis developed in Chapter IV, its quantum-likeinterpretation formalized in Chapter V. The derivationof thepartial differential equations for the characteristic functions and forthe field probabilities is carried out in Chapter VI. Withthe view towardapplications to practical problems, Chapter VI1 givesthe generalizationof the statistical concepts required in the present formulation, and Chapter VIII gives some simpleresults in the form of expressionsfor averages of the squares of turbulent fluctuations which show separatecontributions of the vorticity, entropy, and acoustic modes. Chapter Vlll alsocontains suggestions for the method ofobtaining dis- tributionfunctions approximately. Equationsin this report are numbered consecutivelywithin each chapter,with the number ofthe chapter followed by a period and the number ofthe equation in that chapter. References are listed in an alphabeticalorder by the surname of the first author and are cited in the text by the name(s) of the author(s) with the year of pub1 ication givenin parentheses. Letters of the alphabet are further used to distinguish works of a givenauthor which appeared in the same year. 6 ...[a physical analogy may be defined as] "that partial similarity between the laws of one science and those of another whichmakes each of them illustrate the other." James Clerk Maxwell "On Faraday's Lines of Force,"Trans. Cambr. Phil. SOC. -10 (1855), Sci. Papers, I, p. 155. II.RELATIONS BETWEENTURBULENCE THEORY AND QUANTUMTHEORY Inprevious publications by theauthor (1974a,b,c), denoted hereafter as I, I1 and Ill, a mathematicalformalism of a theoryof turbulence (TT) of compressible,viscous, heat conducting fluids was developedfrom the Navier-Stokes theory. In I,I1 and 111 the author alluded to the association of the developed theory to the quantum mechanics ofsingle particles (QM) and toPlanck's theory of thermal radiation. These allusionsraise the question whether the association (or more properly,the isomorphism of mathematical structures) of the theoryproposed in I and I1 with QM and, possibly,with other physical theoriesis strictly coincidental, or physically meaningful, mathemati- callysignificant, and, in general,useful. Independently of eachother, many researchershave observed in the pastthat there are analogies between fluid mechanics and physical processesstudied by quantum theory.For instance, Madelung(1926) derivedthe equations for isentropic flows of an invisc id gas bysepara- ting real andimaginary parts of theSchroedinger equat ion of one-particle 7 I quantummechanics and thus formulated quantum mechan'ics in the hydrodynamic form. A comprehensivediscussion of thehydrodynamic picture of quantummechanics isgiven by Wilhelm (1970a,b) who derivesquantum-hydrodynamic uncertainty relations and relatesthe minimum uncertaintyproducts to the interior quantum stresses.In quantum-hydrodynamics the quantum stresses.are quadratic in the gradient ofthe logarithm of the position probability. Wilhelm mentions that "a hidden turbulence (excited by the presence of a particle) kicking the particle to and fro in a randommanner could lead to an explanation of the nonlinear quantum force,. ..; the mechanism giving rise to the uncertainty phenomenon in quantumsystems would be similar to that in classicalstochastic systems." Terms analogous to quantum stresses appear naturallyin the partial differential equations for the probability densityof turbulence in the present formulation, Eq. (6.17). The inverseproblem, namely, the association of fluid mechanics with quantum mechanics,allows for the transformation of the fluid-' mechanicalset ofnonlinear conservation equations into a complex scalar wave equationwith nonlinearity appearing only in the expression for the potentialof the pressure forces., Krzywobtocki (1958) used this approach tostudy diabatic flows with heat addition and followed later with a wavemechanical formulationof the theory of turbulence, (1971a,1971b). Whilhelm (1971) inhis formulation of the wavemechanics of compressible fluidspoints out that the transformed complex scalarequation (the Schroedingerequation) leads to a considerablesimplification in the mathematicaldescription of compressible fluids. As an illustrationof a solution of a fluid problemformulated according to the wavemechanical 8 theory he givesthe example of the propagation of sound waves. 'Green (1965) remarks that fluid mechanics has only statistical significance and that.predictions based on equations of fluid mechanics are only confirmed exactly in an experimental ensemble. He further draws the attention to the fact (p. 174) that "in the macroscopic context there are uncertainties which no amount of careful observation and. calculationcan remove. This situation is not fundahentally different from what is known to exist in quantummechanics, where Heisenberg's uncertainty principle frustrates every attempt to predict the result of a single experiment. " Spalding (1972, 1974), in discussing a turbulent transport of a scalar, observes that the gradient approximationfor the turbulent flux relates the turbulent diffusionto local properties of the flow. Such an approximation is inadequate when the length scaleof turbulence is not small in comparison to the distance over which the gradientsof fluid properties vary. In some situations the coefficient relating the flux to the gradient may even change signs. Spalding (1974) remarks that "the situation is similar in this regard to that encountered in radia'tive transfer; for often the 'mean free path of radiation' is of the same order of magnitudeas the dimensions of the apparatus." He then proceeds to model the turbulent transportafter the radiative processes. Mi 1 lsaps (1974) turns to the fundamentals of the quantum theory to propose the extensionof Poincare'ls (1912) proof of the necessity of quantization to the case of hydrodynamical turbulence. Whitham (1965) in his work on waves in inhomogeneous media discovered the prominent r81e played by the adiabatic invariants. The same concepts of adiabatic invariants were used by Ehrenfest (1911) to provide the proof of the 9 necessity of quantization of a system of oscillators. Huggins (1971) gavean interestinginterpretation of theclassical vorticity field, such as the one givenby the curl of theincompressible turbulentvelocity field. First, he observesthat one can represent the dynamics ofthe three-dimensional solenoidal field by a conserved two-dimensional vorticitycurrent. Classically, onecan have a contin- uous vorticity field flowing in space, while for a quantum fluid all vor- ticity is localizedin quantized cores. Huggins,then, proposes thatthe crosssection of thequantized core be treated as a two-dimensional quantum excitation of the classical vorticity field in a way similar to the treatment of phonons considered asquantum excitations of the classical sound field. To establishthis picture, Hugginsproposes that the classical vorticity field be treated as the intensity of the probability field for a quantizedvortex, that the vorticity current be treated as a probabilitycurrent, and thatthe classical hydrodynamicequation for vorticity be treated as a semiclassicalequation for thevortex probability + field.With w = curlof velocity, and I' = circulation around a circuit C, -k the flux of w/I' through the circuit C is the probability that a quantized vortexthreads the circuit. This interpretation allows one todescribe thedynamical behavior of the vorticity field and leads to an explicit hydrodynamic model for how the fluid fluctuations can create a distribution of vortex rings. The most striking example of a mathematicalanalogy between fluid mechanics andquantum mechanics isprovided by thenormal modes of the ocean. Eckart (1961) showed thatthe depth of the ocean, atwhich the VSisila-Bruntfrequency rises to a maximum, defines a thin layer in whichthe oceancan sustaintrapped waves withfrequencies not exceeding 10 thelocal cut-off frequency. The waves aregoverned by an equation formallyidentical to the Schroedinger equation. He furtherpoints out that this situation defines a set of normaltrapped wavemodes for the ocean mathematicallyanalogous to the vibrational quantum states of thediatomic molecule. Eckart's normal modes ofthe ocean are a specialcase of waveguide effectsin stratified fluids. As Tolstoy (1973), p. 124, observes,the general form of a characteristicequation for an internal waveguide in a stratified medium takesthe form of the Bohr-Sommerfeld quantization condition. Edwards and McComb (1969) studiedthe statistical mechanics of a system farfrom equilibrium in which the dominant process is a flow of energythrough the normal modes of the system. They argued thatin the case of arandomly excited fluid turbulence there is a strongmathematical analogybetween the classical (i.e. turbulent) cascade of energy and the quantum field or the many-body problem. On page 4 of their bookon the mechanics of turbulenceMonin and Yaglom (1971) pointout mathematical similarities of statistical theories of turbulence (TT) andquantum fieldtheory (QFT) statingthat "a far more fruitful, perhaps, is the analogybetween the theory of turbulence and quantum field theory, which is connected with the fact that a system of interacting fields is also a nonlinear system with a theoretically infinite number of degrees of freedom. From this follows the similarity of the mathematical techniques used in boththeories. This allows us to hope that the considerable advances in the one will also have a decisive effect on the development ofthe other." Inparticular, they observe, p. 19, thatHopf's (1952) equation for thecharacteristic functional of an 11 incompressibleturbulent field is formally similar to theSchwinger equations of quantum field theory,which are equations for theGreen's function of interacting quantum fields. Methods similar to those of the quantum field theory and the - quantum mechanical many-body problemwere used by Wyld (1961) to formulate thetheory of turbulence in incompressible fluids. A systematicpertur- bation series is shown by Wyld to be in one-to-onecorrespondence with certain diagramsanalogous to Feynman diagrams. The seriesis arranged and partially summed in such a way as to reducethe problem to the solution of threesimultaneous integral equations in three functions, one of which isthe second ordervelocity correlation. Truncation of the integralequations at the lowest nontrivial order yields Chandrasekhar's equation, and truncationat higher order yields the equations cfiscussed by Kra ichnan. Kawasaki (1974), instudying the statistical mechanics of turbulence farfrom equilibrium, points out that the merit of his approach to the solutionof the stochastic equations of turbulence is that it is formulatedin the language of quantum fieldtheory and many-body problems and, therefore,various techniques developed there should be alsoapplic- ableto turbulence. He exploresthis aspect of his approach by developing a non-perturbativeself-consistent scheme toobtain average values of the grossvariables, the time-correlation functions of the fluctuations, the non-equilibriumsteady state distribution functipn, and theresponse functionto a smallexternal disturbance. He findsparticularly helpful theanalogy to the condensed Bose systems. Similarly Ross (1969) develops a quantum-mechanical prescription,together with Feynman diagrams, forcalculating wave spectra,statistical averages, and particle diffusion 12 in a turbulent plasma. He concludes that "the quantum method provides a relatively simple way of deriving and interpreting equations for the time development of the wavespectrum and particle diffusion." Piest (1974) attempts to developa theory of turbulent fluid motion by means of a classical n-particle molecular statistical mechanics (which is a classical limit of a quantum mechanics of .a system of particles), and derives closed system of equations which are nonlinear and nonlocal in space-time. Quantities are defined which resemble the mean fields of density, temperature, and ve1ocity.of turbulent flow. The nonlocal terms contain equilibrium correlation functions which are physical properties of matter, i.e., space-time-dependent counterparts of viscosity and heat conduction coefficients. In a recent publication Gyarmati (1974) showed that the generalization of dissipative fields to complex scalar fields leads toa generalized variational principle for dissipative processes in media with linear constitutive equations, and that if and only if complex state vectors are used the variational formulation is isomorphic with (has the same mathe- matical structure as) the one-particle quantum mechanics. Kentzer (1974~) showed that the use of Fourier modes as complex state vectorsin the representation of statisticalturbulence, combined with the allowancefor nondifferentiability of the phase function, establishes the operator algebra, the uncertainty principle,and the complementarity principle for the statistical theory of turbulence, in analogy to the similar corner-stones of the quantum theory. The classical limit in the quantum mechanical correspondence principle has as its counterpart in turbulence the case of low intensity turbulence in a steady homogeneous mean flow 13 I1 I I1 11111111 1111, 111 I, I I I I, I I, .I I ...... _. .. .. I in whichcase turbulence may be described by the statistics of non-interacting wave packetsthat follow classical Hamiltonian trajectories. Treating a generalstochastic process in the mathematical framework of the quantum theorySantos (1974) shows that, if non- commutativecomplex algebra is used,an operatorequation canbe associatedwith every stochastic equation. The equationsof motion derivedby Santos for the Brownian motion and for a single particle instochastic electrodynamics coincide with the basic ones of quantum mechanics. Thesetwo examples givecredence to thebelief that quantum- likeformulation of stochastic processes, being a more generalthan the classicalformulation, may benecessary forthe description of some random processes.In words of Santos, "the difficulty might be that the mathematical techniques developed to deal with stochastic systems are not suitable for the specific system ..." Turbulence might be just such a system forwhich the combination of classical fluid mechanics and statistics of real probability distribution functions, asopposed to complex wave functions, may be inadequate. Conversely to Santos'objectives, Han'Ekowiak (1975) adapts methods ofclassical random fields,in particular that of Hopf(1952) and of Monin andYaglom (1971), tothe quantum field theory.Specifically, he con- structsn-point functions (moments) describing quantum fieldswith the aid of solutionsof the classical field equations. In the present work, and inprevious publications, the author independently arrived at the same conclusion,namely, that if a stronglyinteracting turbulence is to be treated by a quantum fieldtheory of stronglyinteracting, infin ite 1 y- many bodies,then such a theoryshould be formulated in terms of so 1 ut ions 14 ofthe classical field equations. Wave solutionsof a compressible viscous fluid in a locally-steady and homogeneous flowprovide a complete orthogonalset of solutions which serve as vectorbasis for a formal expansion of the fields. In the work of Santosand Han'Ekowiak we haveexamples of boththe classicalstochastic processes put in the form of quantum theory, and quantum fieldtheory formulated in terms of solutions of classical random fields. Thus the two theories,the classical and quantum, have been used interchangeablyto study physical processes from different points of view. The above discussionof analogies and. similarities that exist between thetheories of turbulence (TT) and variousphysical theories, such as particle quantum mechanics (QM), ,quantum fieldtheory (QFT), or radiative transfer and kinetictheory, raises an importantquestion, namely, whether TT, inthe form of competing theories (e.g.those of Hopf (1952), Wyld (1961), Kawasaki (1974),Piest (1974), Krzywobtocki (1971b)and thepresent theory),are isomorphic to QM, QFT, to an extensionor general ization of these, orto other physical theories. This question may be discussed in thelight of mathematical logic and foundationsof the quantum theory as employed,e.g., byStrauss (1972). As definedby Strauss, a physicaltheory is "a union of mathematical structure and its physical interpretation." The equivalenceof different theoriesthus has dualaspects, viz., mathematical equivalence (isomorphism) and physicalequivalence. In the words ofStrauss, "mathematical isomorphism is not the same as physical equivalence.. ." (p. 94) .. . "isomorphic formalisms can represent different physical theories, e.g., the well known isomorphism between geometrical optics and classical 15 .. . . ._" mechanics which may both be deduced from the same variational principle ... The view that mathematical formalisms have to be isomorphic if they are to representthe same theory is untenable." (p. 94). (Yet). .. "different interpretations of the same formalism lead to inequivalent physicaltheories." (p. 95). Consequently,mathematical isomorphism of TT and QM or QFT does notimply that they are physically equivalent andno suchclaims will be made. On theother hand, variouscompeting theories may be physicallyequivalent in some, butnot necessarily in allthe aspects asthey are not mathematically equivalent. Depending on theirmathematical structures, the various competing theories may be generalizationsor special cases (subsets) of each other. In this work we areprimarily interested in choosing a mathematical structure for the formulation of TT that wi 11 be more general'than a formulationin terms of a classicaltheory of probability basedon stochasticequations in real variables. Yet, we would liketo avoid complexitiesof functional calculus asused in QFT. Thus we searchfor a convenient passage fromthe field equations of the Navier-Stokes theoryto a statisticaltheory of field probabilities. Of greathelp and inspirationin this task are the Intertheory Relationships which serve asexamples, guidelines, and storehouse of existing knowledge. "Not all will agree that Intertheory Relations may become a heuristic instrument for finding new physicaltheories. However, we can extend our studies to relations of second order, vii., to relations between relations. We may have ground for believing that the new theory (T4) looked for will stand in the same (or similar) relations to T3 as T 2 stands to T 1: 16 In fact, this was precisely the heuristic scheme by which Schroedinger obtained his waveequation' : wave mechanics : classicalparticle mechanics 2 wave optics : geometricaloptics." (Strauss, 1972, p. 268). Essentially,this type of reasoningalso inspired de Broglieto propose his famous matter-wavehypothesis by observingthe formal analogy- between Fermat's .principle of optics and Hamilton'sprinciple.of dynamicson one hand, and thewave-particle nature of light on theother. Inthe case of fluid turbulence, we have thefield equations (the Navier-Stokessystem) that determine all geometrical(causal) attributes of a singleinfinitesimal wave, thetypes of waves, their interactions, and wemay averagethe equations and thus obtain partial differentialequations for the average(the mean or expectation)values. Thus, we know, essentially,the quasi-particles (wave packets) and theforces of interaction, and we arefaced with the many-body problem forstrongly interacting random systems of suchquasi -particles. We are aware of similarities and analogiesto many physical theories, suchas, e.g., the many-body classical andquantum mechanics, the quantum field theory,the theory of random systems,etc. We arethus faced with the choiceof the lntertheory Relations, namely, whichtheories (and at whichlevel of application)are in the T4 : T3 = T2 : T relationwith 1 thetheory of turbulence. Since many such levelsof application may be of interest, no singleIntertheory Relation would suffice as a guide 17 forthe development of TT. Expecting many such relationsto exist, one should, it appears,not to attemptto enumerate all or as many as possible of thelntertheory Relations relevant to turbulence,but, instead, one shouldproceed formally developing a TT guidedby Inter- theoryRelations at whatever step or level of application such relations become apparent and helpful. It is theintention of the author to mention now and thenthe particulartheory from which he borrows the mathematical formalism in the hope thatthe readers will be ableto follow his line of reasoning, extend it, orgeneralize it so as to help in further developmentsand improvements. The steps to betaken ingeneralizing classical mechanics to quantum mechanics,namely, a replacement of commuting operators on an algebraof real functions by a non-commutativeoperator algebra and complexfunctions, will be outlined below. These stepsappear to be sufficient,with the help of the correspondence principle, to postulate a system ofequations for the characteristic functions that define the probability distributions in the wavenumber space forthe several modes of wave propagationincluding the acoustic mode coupled to the vorticity and entropy modes. 18 ..."we cannot at present compare the contents of a nonlinear classical field theory with experience ... At the present time the opinion prevails that a [classical] field theory must first. by "quantization" . be transformed into a statistical theory of field probabilities according to more or less established rules. I see in this method only an attempt to describe relationships of an essentially nonlinear character by linear methods. Albert Einstein "The Meaning of Relativity,'' p. 165, Princeton University Press,New Jersey, 1966. 111. NECESSITY OF QUANTIZATION OF TURBULENCE As is well known, see, e.g., p. 292, Morse E Feshbach (1953), in the limit of large values of actionand energy, the surfaces of constant phase for the wave function become the surfaces of constant action for the corresponding classical system. Thus wave mechanics goes over to geometrical mechanics just as wave optics goes over to geometrical optics for vanishingly small wavelengths. Bohm(1951, p. 264) shows that the classical (deterministic) treatmentis a valid approxi- mation if the spatial gradient of the wavelength X is negligible as compared to unity, that is, if IVXI< where = velocityvector. This condition will be satisfiedfor X -f 0 (shortwavelength 1 imit) or for a uniform flow, Vi = 0. We shouldobserve that the condition on the wavelength may be rewritten as AWAx << 1, or Ax*Ak >> 2rr where k = 2r/X = wavenumber. Inthe latter form of the condition Ax isthe scale of the inhomogeneities of thefluid-mechanical (classical) nature, and we have here a statementof theuncertainty in a simultaneousdetermination of mechanical and wave attributesof motion. Thus classical(deterministic) treatment is valid onlyin the limit asone ofthe scales becomes infinite,that is, as Ax -f a or Ak + (Ax -+ 0). Turbulence, however, is a phenomenon in whichthe energy-containing wavelengths are of the same order of magn i - tude as thescale of the inhomogeneities of the material properti es of turbulence,e.g., the turbulent velocity field. As a consequence, classicalcontinuum mechanics mustbe extended to cope with such a situation. The extensionshould account for the fact that the wavenumber canbe determinedonly within the uncertainty range Ak = 2~/Ax, where Ax is a measure ofthe scale of thefluid inhomogeneities (length scale). Onemay arguethat the Navier-Stokes (NS) theoryof viscous fluid is a valid description of all fluid phenomena on scales much largerthan the molecular mean freepath. Thus, inthe continuumrange there is noneed tointroduce wave representation and thus be restricted bythe limitations on thewavelength. It may be arguedthat the limita- tions on thewavelength arise only in the wave representation andas a consequence ofintroducing the wave concepts. The penaltyfor not using 20 the wave representation is the impossibility, at present and in the foreseeable future,of obtaining global solutions of the NS system of equations, which solutionswould give the finestand minute details of turbulent microstructure togetheror simultaneously with the large scale features of fluid motion. Separation of the mean or averaged motion from the small scale turbulent fluctuations renders the solution of the average motion tractable providedthat certain functions (correla- tions or statistical moments) of the turbulent fluctuationsare known. Attempts at replacing the NS description of the detailsof the turbulent microstructure by an averaged flow with slowlyvarying statistical properties as parameters avoid the necessity of non- classical treatment of the averaged flow only. The statistics of the turbulent microstructure, on the other hand, is characterized by two length, usually separated in magnitude, L = length scale over which statistical properties vary, and X = characteristic length (e.g. correlation length) of the turbulent structure. If L >> X, then the statistics may be determined under local conditions. If the statistical behavior of the turbulent structureis to be determined by the NS theory in regions of volume X3, then, again, we have to consider the motion of eddies of all sizes up to the dimensionX moving through a fluid with irregular (fluctuating) propertiesof all scales up to the dimension X. Analytical solutions of theNS equations describing such motions, even in small volumes of order X3, are not feasible. Numerical studies of comparable situations usually proceed along two directions- computations in the physical space or in the transformed (Fourier) space. We may argue that both formulations are equivalent because ofa one-to-one 21 transformation of thetwo spaces. Inthe Fourier space,however, the limitations on theclassical treatment apply and cannotbe dispensed with. Thus, it isbelieved that similar limitations are present in fluid-mechanicalcalculations in the physical space. It is premature and counter-productive to philosophize on the impossibility or uncertaintyin simultaneous determination of physical variables which areconjugate to each other. It is much easierto give mathematical arguments inFourier representation in terms of wave variables and then translatethe arguments tophysical variables. This procedure necessi- tatesthe association of, e.g., the wavenumber and frequencywith mechanicalquantities, viz., the momentum andenergy, respectively. This is essentially the approachused in the quantummechanics. Inorder not to create an impressionthat quantum effects on the scaleof the quantum of action h (Planck'sconstant) have necessarily any significancein turbulence, we observethat the quantum of action h characterizesthe magnitudes (scales) of phenomena on theatomic particlelevel. In particular, the Einstein-de Broglie relations, - E = 4iw, p = +x, establishthe exact and universalrelations between thescale of mechanicalproperties E (energy) and 'p (momentum) and thescale of wave properties w (frequency)and E (wavenumber) whichhold in applica- tionsto atomic systems. Thus h playsthe r81e of a conversionfactor which changes mechanicalunits to those appropriate in wave representation. Obviously,the scale ratio, which would play the r81e of the quantum of actionin turbulence, cannot be a universalconstant since the scale 22 effectsin turbulence are governed by the non-dimensional Reynolds number. As a consequence, relationsestablishing proportionality between wave and particle attributes, such as thoseof Einstein-de Broglie, are either nonexistent or take a complicatedform of a functional dependence on the boundary conditionsof the hydrodynamical fields. As a point of interest wemay add that many familiar fluid dynamical conceptscould play the r81e of the unit of action (or rather the unit of action density) for the purpose ofconverting the dimensions from those of frequency and wavenumber to thoseof energy and momentum, and viceversa. We observethat the quantities pUL = thenumerator of the Reynolds number, p = dynamic viscosity or the denominator of the Reynolds number, pJi*dz = density x circulation,pJJ?xc*di = density x vorticity x area, pjJJi*(Vxi)dV//i*dz = density x he1 icity x volume/circulation, etc., all have dimensions ofaction per unit volume. The varietyof such quantities,their varied significance, the arbitrariness of the choiceof line, surface or volume integrals,all seem to suggestthe lackof any physicalbasis for establishing a physicalrelation between frequency and wavenumber of waves and their energy and momentum. The work ofEhrenfest (1916) suggeststhat the quantum ofaction is related tothe adiabatic invariants which, in the case of nonlinear dispersive waves in inhomogeneous fluids,are given a prominentr61e by Whitham (1965) - Any physicaltheory of turbulenceshould lead to anagreement with experimentalfacts. At earlystages of the development of a theory it is helpful to check for agreement in some simpleor idealized cases. A state of equilibriumin, say, homogeneous, isotropicturbulence is 23 such a case. One wouldexpect that the energy spectrum should drop off to zero at both zero and infinite frequency,be positive everywhere, andhave a finiteintegral (finite spatial density of energy). Thus thespectrum musthave a maximum. At highfrequencies (vanishingly smallwavelengths) classical statistical behavior is expected, equipartitionshould prevail, and theenergy spectrum should drop off exponentiallywith frequency. The presence ofviscous dissipation at highfrequencies would cause thespectrum to drop off even faster. At this point we notethat, by consideringthe entropy of a system of normal modes and byimposing the condition that the energy spectrum fall off faster than anypower of frequency,Ehrenfest (1911) has shown thatdiscreteness of energy spectrum is a necessary consequence, whence followsthe necessity of quantization of systems of normal modes. It is believedthat the same argument wouldhold in the case of weak turbulencerepresented by an infinite system of Fourier modes. However, Ehrenfest's proof of the necessity of quantization of the energy of a system of oscillators does notgive the magnitude of the spacing of energylevels. Consequently, the spacing of energylevels need not be the same intheories which are not physically equivalent. We may turn now to the case of turbulenceexpressed in terms of Fourier modes. Examination ofthe wave interactionterms, see,e.g., Davidson(1967) or Appendix C of Vedenov (1968),reveals the fact that theRayleigh-Jeans distribution is sufficientfor the existence of an equilibrium. Thus theclassical Rayleigh-Jeans distribution appearsas an equilibrium solution of the turbulent field analyzed classically. It was Ehrenfest (191 1) who first coinedthe term "u1 traviolet catastrophe" to describethe behavior of the classical energy spectrum at high 24 frequencies.Classically, the energy increases as frequencysquared or as a negativefourth power of wavelength.Since it is observed experimentallythat the behavior occurs in turbulent spectra at largewavelengths, and thatthe spectrum drops off approximately (or almost)exponentially as wavelength approaches zero, the question ariseswhether the departure from the classical Rayleigh-Jeans distribution and theapproach tothe Planck's shape of theenergy spectrum is broughtabout either by, (l), appreciable "quantum effects" onscales far removed fromthose governed by the universal constant h, (2), by thepresence ofdissipation (an energysink) at small wavelengths, or, (3), by the absence ofequilibrium. The explanation (3) was proposed in the case of black body radiation byJeans and was vigorouslydefended byhim untilthe publication of Poincare'lsproof (1912) ofthe necessity ofdiscreteness of the energy transfer process. For our purposes we observethat the nonlinear wave interactionprocesses so dominatethe flow of energythrough the spectrum that the approach to equilibrium cannotslow down to a standstill andmust be sufficiently rapid, or atleast non trivial, so as to negateJean's explanation. Should (2) be thecorrect explanation, then the energy would have to increasemonotonely as A" untilthe viscous dissipation would become important. Thus one wouldexpect the energy containing scales to be closeto if notequal to the dissipation scales. This is not the situationobserved in general where theenergy containing scales and thedissipation scales are usually far apart. The separationof scales isespecially dramatic at high Reynolds numbers, e.g. in atmospheric turbulence.This leaves us withthe necessity of a discreteenergy 25 exchange process (the"quantum" effect) as the only plausible explanation. The similarity of general features of the turbulent energy spectrum and Planck's distribution permitsone to assess the behavior ofturbulence spectrum in slowly decaying homogeneous and isotropic turbulence. It is known that during the decay process the energy distributionis slowly and constantly modified, with themaximum value of the spectral energy decreas- ing and shifting toward longer wavelengths. The same is true for Planck's distribution, c1 1 E(X) = - , c2% T" . x5 eC2/X-I The radiation temperature T should be replaced in case of turbulenceby the mean square of the turbulent velocity fluctuations. The maximum of E, Emax, occurs at c2/X = 4.96 and is given by Thus, if a turbulent spectrum is approximated by a Planck's distribution, and if the decaying process is represented by decreasing temperature, then, since h % 1/T, h increases (maximum shifts toward longer max max max wavelengths) and Emax decreases during decay. The total energy density, E = lE(A)dh, integrates to give the Stefan-Boltzmann Law,E %T4 or the Lighthill's ''eights power of velocity law'' of acoustic radiation in turbulence. We conclu de that the typeof energy spectrum one may expect to . .. obtain for equil ibrium turbulence (modeled in terms of normal modes which exchange energy by a discrete process) possesses correct general 26 characteristics, a properbehavior at zero and at infinite wavelengths, and a proper decay behavior. The classicalRayleigh-Jeans spectrum leads to the "ultraviolet catastrophe" and cannotrepresent the energy distributionexcept in the limit oflong waves which is notof interest inturbulence. The most generalargument in favor of a discrete (as compared to a continuous)energy exchange process was givenby Poincark (1912). Millsaps (1974) was the first to observethat Poincare'ls proof applies toturbulent spectra as well as toelectromagnetic radiation. The conditions, as stated by Poincare',are satisfied by turbulentspectra: 1. E(w) -f 0 as w -+ 0, 2. E(w) + 0 as w + 03, 3. 0 5 JmE(w)dw < for all time, 0 5 t 5 9 0 4. The First and Second Laws ofthermodynamics hold. Poincare' has provedthat, if conditions 1 - 4 hold,then it is necessarythat the energy exchange inthe spectrum occur in a discrete ordiscontinuous manner. Further,he gave theexpression for the form ofthe distribution at large times, which form is a generalizationof thePlanck's distribution valid at equilibrium, t = a. Hereagain, the mathematicalargument does notyield the numerical value of the quantum of energy, it even does notrequire that such a measure ofdiscontinuity of energy exchangesbe a constant. As a consequence,one would expect that in the case ofturbulence the energy spectrum would approach the Poincare'ls generalized form sufficiently far away fromboundaries and far downstream fromthe transition point, and that such a distribution would have to be modifiedto account fo'r lack of homogeneity and isotropy. 27 Anotherargument in favor of quant.izationof turbulent energy exchange processesfollows from the statistical mechanics. We observethat the difference between the classical Boltzmann statistics (BS) and eitherthe Bose-Einstein (BES) or Fermi-Diracstatistics (FDS) is dependenton thetreatment of the elements of the statistical system (particles,oscillators, or normal modes) as distinguishable or indis- tinguishableentities. Which statisticsapplies mustbe determined by experiment.In quantummechanics thechoice between BES and FDS is made on thebasis of symmetry orantisymmetry of the wavefunction. If . one assumes thatsimilar particles are distinguishable (or indistinguish- able),then it follows as a consequence of such an assumptionthat BS (oreither BES or FDS) mustbe used. Inclassical physics the identifying or "labeling" of particles in case of similar or equivalent particles depends critically on thecontinuity of their trajectories. In the Fourierrepresentation there arises a possibilityof indeterminate changes (inmultiples of 2~)of the phase ofthe wave.The phase determinesthe trajectories of the wave packets. Thus thedifferentia- bility of the phase impliescontinuity of wave packettrajectories. The assumption ofdifferentiability is not a necessary one. Thus, the distinction between a classical and nonclassicaltreatment lies in the arbitraryassumption of differentiability of the phase. The classical treatment may thusbe readily generalized by relaxingthe conditions of differentiability of the phase, integrability of the trajectories, and distinguishiabilityof particles. This amounts to a generalization ofthe geometrical mechanics of wave packetmotion characterized by continuoustrajectories to wave mechanicswhich admits discontinuities 28 in 'trajectories,in .the number of particles,in their energy,etc. We havementioned earlierthat the classical (geometric) mechanics fails to describethe system accurately when the scale of inhomogeneities of the medium is comparable tothe wavelength considered, as is the case withturbulence. Consequently, It followsthat it is necessary in wave (orFourier) description of turbulence to generalizethe geometric (deterministic)description to the nonclassical (probabfl istic) mode of treatment.This is accomplished by not requiring that the phasebe differentiable.Consequently, the trajectories of wave packets (and their numbers) become discontinuous and similar wave packetscannot be labeled and thus become indistinguishable. The statisticsappropriate toindistinguishable wave packets isthe Bose-Einstein or Fermi-Dirac statistics. The nonlinearinteraction terms, derivable from the Navier- Stokestheory, show that a two-particledistribution function ("wave- function") is given by a productof one-particle distributions, at least atthe level of theNavier-Stokes theory. Thus the"wavefunctions" are symmetric and we make thechoice, subject to eventual experimental veri- fication,of the Bose-Einstein statistics. Later on, indiscussing statistical moments withrespect to the probabilitydistribution in the phasespace, it is mentioned(and may be readily verified) that without a1 lowance for the uncertainty in the phase (discontinuity or indeterminacy of phase) thestatistics reduces to a trivial casewhere allthe correlations are symmetric and moments of variousorders may be put in the form of products of moments of first order. Then all moments wouldreduce tozero in the case ofisotropic turbulencein contradiction with experimental facts. In conclusion, it is not only sufficient, according to Gyarmati's (1974) proof, to treat turbulence asa complex field withcomplex state vectors asa basis, but it is also necessary to allow for discontinuous ("quantized") energy exchange processes in turbulence. A formulation of a statistical theory ofturbulence based on quantum mechanical principles generalizes classical statistics, extendsand uptates it to a higher level. Thus, such an uplation offers hope of the eventual solution of the general problemof turbulent motion. In this work we outline an approach to the quantum-like wave dynamics of turbulence, which approach avoids the difficulties of solving functional partial differential equations. 30 "Seeing that I can find no subject specially useful or pleasing - since the men who have come before me have taken for their own every useful or necessary theme - I must do like onewho, being poor, comes last to the fair, and can find no other way of pro- viding for himself than by taking all the things already seen by other buyers ..." Leonard0 da Vinci, Codice Atlantico, folio 117v Iv. WAVE REPRESENTATION OF TURBULENTMICRO-STRUCTURE We shallconsider a system ofequations governing the flow of a viscous,heat-conducting compressible fluid, + ('U*V)p + PV-G = 0, RT p1 - -ai + (ii*V)i + - Vp + RVT - [V*V; + (V-l)VV.U] = 0, at P -P where p = mass density, ti = velocity vector, T = temperature, 9 = dissipationfunction (rate of dissipation of mechanical energy), y = ratio of specific heats = c /c = specificheat at constant pressure, P v' cP c = specificheat at constant volume, Pr = Prandtl number = c /IC, V - P1 K = heat conductivity, V = viscosity number = 2 + p /p and 2 1' 9 p2 = first andsecond coefficients of viscosity (then the bulk viscosity 2 -4 is p1 + p2 = p1(V - -) 0), R = gas constant = c - cv. 7 3 2 P We shall first separate the flow variables into.the mean and fluctuating components, The definition of theaveraging process will be made specific later. At thispoint it sufficesto say thatthe averages of the f ationsvanish by definition, 7 (4.5) When this system is averaged, we obtainthe Reynoldsequations for the mean (averaged) fluidflow: + v- ( ) + 33 where v = pl/ = kinematic viscosity of the mean flow. When Eqs. (4.6) - (4.8) are subtracted from (4.3)-(4.5), we have the system for the determination of the fluctuations: (4.10) 34 I The curlybrackets are used to collect terms inthe following order: first, termscontaining only linear terms in the fluctuations thatare independent of the derivatives of the mean flow,then products of thefluctuations and thederivatives of the mean flow,terms bilinear inthe fluctuations, and lastlythe terms trilinearin the fluctuations. Equations (4.9)-(4.11) will now be written as a linear system in thefluctuations subject to three kinds of forcing functions (sources), thosethat are linear in the fluctuations and vanish when allderivatives of the mean flowvanish (representing first order interaction with the mean flow), and thosethat are bilinear and trilinear in the fluctuations (representingthree-wave andfour-wave interactions,respectively). This 35 amounts to takingall but the first curly brackets to the right-hand-side: a- where c = (RT)' = isothermal speed of sound. The symbols m f 1, and . i' q. denotemass-like, force-like, and heat-likesources of i-thorder I inthe fluctuations. The varioussources as defined abovehave anits of frequency,sec-l. The expressionsfor the sources may be obtained by referringto Eqs. (4.9)-(4.11),for instance, for the mass-like sources we havefrom Eq. (4.9) : = - [ij'*V + p'V*]/ ml - = [ The formof the system (4.12)-(4.14) is rather arbitrary inasmuchas - theprimitive physical variables p, u, T couldbe replaced by any other setof unknowns, e.g., p, pi, p,where p = pRT, etc.With the wave representationin mind, we observethat, in general, any initialdistur- bance in anyone or all of the primitive variables will be split into several waves, each wave carryingperturbations in the several of the primitivevariables. Thus theamplitudes of the waves, and theratios of theamplitudes of the perturbations in the primitive variables carried by a given wave, become naturalvariables in the wave representa- tion, and notthe primitive variables themselves. In order to simplify the notation, we shall usefrom now on the following: - - - = c, Thischoice of the nondimensionalization symmetrizes the system (4.12)-(4.14) if thelocal reference quantities, p, c, im, are regarded as constant and takeninside the derivatives, e.g., - %I-at - at(-a e' = *at . This wi 11 facilitate the treatment of the P P linear part of the system of equationsunder the assumption that theflow is steady and homogeneous and thereference quantities are constant. The symmetrizedsystem becomes: + cWVT + cVp v(V2'r + 5;JV-U)1 = T1 + + F3 Dt - F2 37 ~a where = "+U*V Is the substantial derivative. Symbol ical ly, wemay -Dt at write where L (u.) is a lineardifferential operator represented by a ij J symmetricmatrix, and li,Bi, Ti arevectors linear, bilinear, and trilinearin the fluctuations u respectively. j' We now expand thevector of the fluctuations u in terms of j complex exponentials: (4.16) where theintegration with respect to the volume in the wavenumber - space(dc = dkldk2dk3) is extendedeither to the cut-off = kmax orto infinity if $, = 0 for = k > k Eq. (4.16) defines u as max' j a Fouriertransform, at t = 0, of $aPuj, lPaj I = 1. Thisis a local representation for, if $, and P depend onspace and time, then (4. 16) aj can nolonger be inverted by taking a Fouriertransform. The subscr ipt a denotes a contribution of the a-mode of wave propagation and u = Cu isthe sum ofcontributions to thefluctuations of the j a aj j-th quantity from all wave modes. If $a, Paj Y ray and wa aretaken as localvalues (independent of and t) then wemay impose thecondition that the form (4.16) solves the 1 inear part of Eq. (4.15) , that is, 38 which islinear and homogeneous in u.. As thecondition of the J existence of non-trivialsolutions of the linear system (4.17), see Kentzer (1974a,b) ,. we obtain the characteristic equation where X = -r + i (6-i - w). Under theassumption that the viscosity number G, which mustbe no smallerthan 4/3 inorder that the bulk -1 “1 < viscosity be non-negative, is equal to Pr , that is Pr = V = 3/4, the above expressionfactors out into the product (X + vk‘)‘ (X + vk2/Pr) [X(X + yvk‘/Pr) + a2k2] = 0. (4.18) By analogy to the characteristic determinant of the theory of characteristicsin the inviscid case, the first doubleroot will be identified with the vorticity mode, the second linear factor with the entropy mode, and thequadratic factor gives rise to the acoustic mode of wave propagation. Setting each factorin Eq. (4.18)separately equal to zero we impose thenecessary and sufficient conditions for the existence of infinitesimal wave-type solutionsfor the locally steady and homogeneous mean flow. Forthese solutions we have a = 1,2 Ta = vk2, wa = i-i;, (vort i c i ty modes) ” a=3 VPr’- Vk2 war = U*k, (entropy mode) (4.19) a = 4,s = ‘yvk2 ra 2~r wa = fi-c 2 ak( 1-K2)* (acoustic modes) 39 where K = yvW(2aPr) = Knudsen number basedon the man wavelength, 27r/k, and a = (yRT)’ = adiabatic speed of sound. We observehere that all modes have the same frequency i*L when K = 1. Thiscondition corresponds to k = 2aPr/(yv) = lo5 cm”, or to wavelengthsapproaching themolecular mean freepath. This value of k will be taken as the cut-offvalue, kma,, beyond whichthe continuum fluid mechanics does not apply andany resultspredicted by theNavier-Stokes theory will loose theirphysical meaning. The vorticity and entropy modesmay be interpreted as standing waves convectedby the mean motion,their frequencies having the standard formfor a Doppler shift, io’;. The acoustic modesmay beviewed as plane waves travelingthrough a moving medium at a reduced speed of sound, a* = a(1-K2)’, inthe directions + ’; and -i. Thisalso implies that an acoustic wave is a quadraticsurface, a convectedexpanding sphere. Usingrelations (4.19), one may returnto the 1 inearsystem (4.17) to determinethe eigenvectors 4aPcrj. This is possibleonly if K < 1. Due to homogeneity of (4.17), the components ofthe eigenvectors may be determinedonly up to a common factor. We may find,therefore, the participationcoefficients, P underthe condition that they be of unit aj magnitude, c P .P* = 1 J aJ aj where theasterisk denotes a complexconjugate.. Further,since the P correspond todistinct eigenvalues of the aj system (4.17), theyare also orthogonal, that is, = 0 for a # 13. (4.20) Jc ’ajPBj 40 I The vectors P aregiven byKentzer (1974a,b) inthe form aj ‘(4.21) where m = 1,2,3, g= -ak{.’K+i (l-K2)’}, k = lEl. The vectors P and P areunit polarization vectors perpendicular 1j 2j tothe wave vector i. Thisindicates that the vorticity mode is If forpractical purposes andease ofcalculations P3j, Pbj, and P 5j areapproximated by their inviscid limits: P = {l, 0, 0, o,-l/m/{y/(y-l)I* 33 P = {ak,-yckm, akm}/{2ya2k2}‘ 4j (4.22) P = {ak, +yckm, ak-/{2ya 22%k 5j then all P arereal and orthonormal, 1 if cx=B 0r.j 0 if cx#B 41 We maynow observethat in the expansion (4.16) the amp1 itudes 9, are the Fourier coefficients of u. withrespect to the state .J vectors as basis. The five-foldinfinite system ofstate vectors (4.23) is complete and may beused torepresent any piecewisecontinuous, square- integrable function of x,t. Since, in a generalnon-stationary inhomogeneous case, 4 ra, and w a arefunctions of i; and of x and t, theexpansion (4.16), while still valid, is not a Fouriertransform and isnot a solutionof Eq. (4.15), we generalizethe results and proceed as fol ows. We firstwri'te the integrandin the expression (4.16) as (4.24) 00 0 where ra, and wa arelocal values, functions ofonly, and The space-time dependence of $Ja,ra, and wa is implicitthrough their dependence on thefunctions of the mean flow. We tryto avoid introducing functionaldifferentiation asdone, e.g., byHopf (1952) and avoid replacingthe relation (4.16) by a functionalanalog of a Fouriertrans- form. It shouldbe observed that, treating the integrand (4.24)above as a product,the linear operator L.. operating on u will give a zero 'J j contribution when operating on thestate vectors (4.23) whilekeeping 42 constant, and a non-zerocontribution will come onlyfrom L operating (la ij on QCr whilekeeping the state vectors constant. But this contribution must equalthe source terms on theright-hand-side of Eq. (4.15). Let L~~ (QJ = - rw; where the random function ui = ui(i,i,t)is, in general, complex. Then = ri + gi + T~ . The new variable is intended to accountindirectly for the u*a inhomogeneity and unsteadiness of the mean flow(for the "slow" variation of $a, ra, and ua with 2< and t) while, at the same time, 0: is to bechosen so as tosatisfy Eq. (4.15). This means thatthe "slow" changes inthe amplitude $ and inthe parameters and wa are due to a To! thesource terms only. For convenience we shallmodify the state vectors (4.23) by adding u; tothe frequency ua,0 and, instead of Eq. (4.151, we obtain because thetime-derivatives of u. appear onlyalong the diagonal of J the symnetric operator L. .. 'J 43 Omittingthe superscript ( )O, we multiplyboth sides of Eq. (4.26) and wesum over j and integratewith respect to El. The result,under thecondition that 4a(0), $,(O) = 0, is I f we define the "1 ocal" mean Value of any function of by then Equation(4.27) is an integralequation for w* because w* appears B B on bothsides of the equation. In particular, w* appears inthe B arguments ofthe exponentials and in 8 (thebilinear interaction j terms and containthe time derivatives of and T). Of greater T2 42 importance is theinfluence of w B on theresonance conditions. We note that a substitution of the expansion for the perturbations into 7 j' Bj' and T. wi 11 resultin the appearance inthe numerator of Eq. (4.27) of ,J terms ofthe general form 44 Forthe acoustic modes, besidesthe Doppler term i-i,we have the intrinsicfrequency, +ak(l-K2)’ tak.Acoustic waves withthe like sign (waves of mode a=4 or a=5) wouldcontribute only if all the vectors ii, El, ill, ill1 areparallel. Only acoustic waves ofopposite sign (e.g. a=4 interactingwith a=5) couldresonate. The additionof themodifying term u* and presence of vorticity and entropy waves, a’ enlargethe resonance possibilities. This observation is important in calculating sound generationin turbulence. Only numerical calculations wouldassess theeffects of neglecting u: inthe interaction terms. Similarconsiderations arise in the calculations of the various mean (average orexpectation values) terms that enter into the Reynoldsequations (4.6)-(4.8). The number of termsthat need evaluation is verylarge due to double and triple summation overthe five modes. Inmodeling of turbulence it will be necessary to reduce the number of suchterms. It is believed,for instance, that the effectof the acoustic wave field on the vorticity mode is negligible. Similarly,at low Mach numbers, entropy mode will have negligibleeffect on vorticity. However, theacoustic field will be determinedprimarily by the vorticity mode and, to a lesserextent, by theentropy mode. Inconclusion we shouldstate that the partial differential equationsfor the mean flowshould be solvedsimultaneously with the equationsfor the fluctuations. The latterare disposed off by introduction of the random variable u*, and theproblem of the fluctu- ations reduces tothe solution for the amplitude function @(G,E,i). The squareof its magnitude plays the r6le of the probability distri- butionfunction in the wavenumber space, which will bedetermined in 46 terms ofthe characte.ristic function, a probabilitydensity in the physical space. Finally, we notethat Eq. (4.26) may be interpreted as C ‘ is theaverage, with respect to the distribution $a, of the differ- -1 entialoperator $a L.. (Ga)Paj. All attemptsat separating or ‘J factoringout this operator failed. It was thendecided to use a heuristic approach, to derive a system of nonlinear partial differential equationsfor the characteristic functions underthe assumption that $a and w areindependent of X and t, and CY thenpostulate that such a system, suitablymodified, is validalso when the dependence on and t is a1 lowed for. The nextchapters developthis approach. "We havedone considerable mountain climbing. Nowwe are in the rarefied atmosphere of theories of excessive beauty andwe are nearing a high plateau onwhich geometry, optics, mechanics, and wave mechanicsmeet oncommon ground. Only concentratedthinking, and a considerable amount of re-creation, will reveal the full beauty of our subject in which the last word has not yet been spoken." [Cornelius Lanczos] "The Variational Principles of Mechanics," p. 228, University of TorontoPress, Toronto, 1949. V . WAVE-PART I CLE DUAL I TY Wave Packets as Quasi -Particles i (i*ii-wt) The complexexponentials, e , (the phase factors) afford thefollowing interpretation, see, e.g., Lighthill (1965), p. 15. Assume thatthe phase s = G*K-wt is differentiableat least twice. Then, regardingtime as a fourthcoordinate, x4, and frequency as minus the fourth wavenumber component, w = -k4, we write s = xmkm,m=1,2,3,4, and we havekm = 'Waxm. From theassumption of differentiability it follows that that is, the wavenumber vectoris irrotational. If we definethe frequency as a givenfunction of i, t, andby 48 F(xrn’ k rn) w(xi.,ki,t) + k4 0, i = 1,2,3, then If wenow introduce a parameter T such that 4 dkndxmakn aF then “ ”” axmd-r d.r lTFl axn ’ With ~4,Eq. (5.1) givesdt/d-r = 1, or -r = t+const., and wemay write Eq. (5.1) and (5.2)as dx aw dk i=- i=”,aw j = 1,2,3 dt ax ’ dt . (5.3) j axJ This is a Hamiltoniancanonical form of the equations of motion of a wave packethaving a wavenumber k and frequency w and locatedat j x attime t. Thisset of canonicalequations forms the basis of geometri- j cal mechanics. Lighthill (1965) observesthat J. L. Synge pointedout a more generalprinciple from which Eqs. (5.3) follow, namely, thatthe motionof a wave packet is such as to make theintegral stationaryalong the path in space-time betweentwo fixedpoints. 49 The connect ion between geometrical mechanics' and wave ion mot is a subject of an inspiring book by Synge (1954). To display this connection in the present case, we need only to introduce a conversion factor, h = energy x time = action, and write for the phase factor -[x*- - 2l~i hi - hw 2l~i - - h 2l~ -21T tl T(X*P Ht) e =e - where, using the Einstein-de Broglie relations,we have set - hi momentum, = = p=z= H -21Thw the Hami 1 tonian (energy). With these substitutions, the variational principle of Synge becomes -" 21T 6/ dt = 0 h L dx where 4j = -& = velocity, L = the Lagrangian function. As a consequence of the various interpretations givenabove, one may look at integrals of the form is a as a sum of contributions to the average valueof e due to infinitely many plane waves with number densityin wavenumber space aa, or due to a system of wave packets that, in absence of interactions, follow Hamiltonian trajectories,or as a sum of contributions of a system of quasi-particles that obey Hamiltonian equationsof motion. 50 Thiswave-particle duality permits one to look at the waves as particles, and viceversa. Since k = spatialfrequency (number of waves perunit distance j alongthe x -coordinate), and w = temporalfrequency (number of waves j perunit time), then the Synge principleis the principle of conservation of the number of waves or of the number of wave packets. Inpresence of wave interactions, not only the wave amplitude changes (this may be interpreted as a change in the number densityof wave packets),but also waves arecreated (excited) or annihilated (de-excited) so that the number of waves at a given location and at a giventime having frequency w and wavenumber may change due to wave resonances in whichthe frequency and wavenumber (butnot the number of wave packets)are preserved, i .e., These relations,multiplied by h/27r, representconservation of momentum and energy of the quasi-particles. The groupvelocities U and thegeneralized forces F actingon aj aj the wave packetsare given by theHamiltonian equations (5.3). Using Eqs. (5.3) we obtain dx i=dt U = Uj + caa(k./k)(l-2K2)(1-K2)-+ aj J 51 where cCY= 0 for CY = 1,2,3, and c4 = 1, c5 = -1, and U = velocity j of the mean flow. The group velocities of the acoustic waves, CY = 4,5, change discontinuouslyduring an interaction, while those of the vorticity and entropy waves do not. A modificationof the expressions for the fre- quencies, to be introducedlater, will add a random component to all groupvelocities. It will be the random component that will change duringresonant interactions. The Uncertainty Principle In order to allow for wave interactions we must relax the condition ofconservation of the number ofquasi-particles. This will require thatthe phase function change discontinuouslyalong quasi-particle trajectory.Further, the observable physical quantities, which are functionsof the phase factor (andof the state vectors), should remain continuous,that is, remainindependent ofthe random discontinuities inthe phase. Thus we generalizeprevious results to the discontinuous phase, " s = x-k - ut + 2m, n = 21, 22, 23,. .. Conversely, if the phase s is allowed to change discontinuouslyby arbitraryintegral multiples of 2~,then the integrals of the state vectors, e.g.,/$e (s+2an)di, wi 11 remainunaffected andno amount of experimental measurement ofthe functions of such integrals will determinethe actual value of the phase. Thus the phase will remain indeterminate and uncertain,with the uncertainty in the determination of phase being As L- 2~. 52 1 1 Correspondingly, due to thefact that now the phase s isnot 8' differentiable,the phase functionis not single-valued function of space-time,and #k.dx. > 27r , $wdt 2 IT. (5 J J= 4) Thewave number is nolonger irrotational and the ph,ase-space integralssatisfy the inequality //dk.dx. > IT. J J= From Eqs. (5.4) and (5.5) it followsthat AkAx L- 2~, Aw At 1.- 27r. Multiplyingboth sides by h/2~we obtainthe Bohr-Sommerfeld quantization of action conditions familiar from quantum mechanics: $pjdxj = //dp.dx. > h. J J= Equation (5.6) corresponds tothe Heisenberg's form of the uncertaint.y principle, ApAq 2- h, and AEAt 2- h. We note that the mechanical picturerequires introduction of theconversion factor h (Planck's constant) and that such a constantfactor plays only the r6le of a scale factorwhich assigns a particularnumerical value of momentum and energy to a wave packet of a given wavenumber and frequency. We introducedhere the Planck's constant only to show the comn mathematical structure(isomorphism) of wave and particlemotions. We onlyassign a particular numerical value to the uncertainty of phase, viz., As 121~.- Thus, we "quantize"the phase only, anddo notintroduce any scale 53 ~ .9 - . factors,which factors must depend on theboundary conditions in the case of turbulent flow. The uncertaintyof phase,.being a property of the Fourier transform, mustbe carriedthrough the development of the theory. Thus, we adopt the uncertainty of phaseas the uncertaintyprinciple to beincorporated intothe mathematical structure of thetheory of turbulence. The ComplementarityPrinciple We notethat using the Fourier transform one introducesthe products xmkm and wt intothe phase function s. Thus xmand km, or w and t, become conjugate to each other forming "complementary pairs of variables each of whichcan be better defined only at the expense of a correspond- ing loss in the degree of definition of the other." Thesewords are usedby Bohm (1951), p. 160, todefine the quantummechanical principle of complementaritywhich carries over into the turbulence theory in wave (thatis, in Fourier) representation. The Correspondence Principle As theamplitudes @a ofthe waves diminishto zero, one may neglect the higher order terms (the bi 1 inear and tri 1 inear interaction terms), and if the mean flow becomeshomogeneous and steady,then the plain wave solutions satisfy the 1 imiting (1 inear) form of the equations for thefluctuations. In that casethe wave amplitudes 9, remainrigorously constantin space-time and the wave resonances may be neglected. We will referto the small amplitude steady homogeneous case as the "classical 1 imit!' to which a turbulencetheory in wave representation must reduce. As a guideline in the development ofthe theory we shall 54 ~~~ " adoptthe correspondence principle, namely,the. principle that the resultsof the theory should reduce to the classical form of thelinear, smallamplitude wave mechanics of steady homogeneous media. .. Operator Formal i sm i Inthe Fourier analysis of functions of the phase factors , we have e .- - -=aF -iwF, VF = iEF, F = Foei (x-k-ut) at ¶ fromwhich follows the equivalence of operators We may introduce now a characteristicfunction $, which in the "classical" limit is defined as Here we omitthe subscript a and allow w to becomplex ingeneral. We observe the following properties of $: $*c$ = J$*c$di = cJ1$1 2di, c = constant, 55 We see, therefore,that to any dispersionrelation o'= H(G,i,t) there will correspond an operatorequation or a differentialequation of the form whichequation has a formalsolution iHt $=e As long as thedispersion relation is a polynomial in i, theoperator H will be a properspatial differential operator. The introductionof thecharacteristic function has twoimportant advantages. First, a dispersion relation for a given mode of wave propagationdetermines a partialdifferentia? equation for the corresponding characteristic function. Second, thederivatives of the characteristic function determine the averages or moments of functions of wavenumber with respect to the distri- bution function $I*(i)$I(i)E f (E) 2- 0. In turn, the moments of the distributiondetermine the distribution, thus we have here a method of computingthe distribution function f(k) and of calculating all the requiredfunctions of the distribution, in particular, the interaction integrals and thesource functions. The characteristicfunction in the general, inhomogeneous, nonsteady,large amplitude case will beobtained by generalization of the above results for the. 1 imiting, "classical" case. Modi~ -~ f icat ion ofWave Frequency We shall now interpretthe wave packetmotion in termsof the particle mechanics in order to adopta form of the frequency modification term that is consistent with the wave-particle duality. Considera particle motion in which the momentum i = hc/21~ is reckoned relativeto the moving medium. Then = hc/21~+mi isthe momentum relative to the coordinate frame in whichthe fluid carrying the waves has velocity i. The particle mass is m. The energy ofthe particle is then 1 h 1h H = mV = U-k-- -(-)2k2 mV & + -mu22 + -21T + 2m IT + where V is thepotential energy per unit mass. Since H = hw/21~, we have forthe frequency -- h 21~m 1 w = U*k + k2 V). + -h (9'+ The first contributionto the frequency is theDoppler term. The second term is thekinetic energy ofthe particle motion relative to themoving medium.The lastterm is the sum ofthe kinetic energy of the particle as if it weremoving with the velocity of the medium plus thepotential energy. In quantummechanics theclassical limit isobtained by allowing the quantum ofaction h to tend to zero. If we use the same argument 1 here and require that w remain finite, then we must set V = U2. - -2 Thus, we are left with 57 The termproportional to"k2 leads in qua'ntum mechanics to a highly dispersive wave motion.In t.he case of turbulencethis term offers means o.f introducing.turbulent dissipati-on and. dispersion due to a random , motion of the wave packets relative to the mean flow, The frequencies of vort,icity and entropy modes, Eqs. (4.19)., have only one contribution, namely theDoppler term, and thusrepresent a convectedpattern of standing waves. A modificationof frequency for thepurpose ofaccounting for the wave interactions, see p. 43, allows one to model the wave motion because theform of the random function w*(E) introducedthere is arbitrary.Consider a seriesexpansion for w*(ii), = w + w.k + w..k.k. + wijkkikjkk +... 0 ~i IJ I J The constant a. couldbe absorbed into the potential V and, bythe previousargument, terms independent ofshould vanish being inversely proportionalto h. The w.k. termwould introduce a driftvelocity II relativeto the mean velocityof the fluid. Thus the first meaningful term is w k.k.. If w is to be regarded as analogous tothe energy of ij I J quasi-particles, a functionquadratic in mmenta, then such an analogy wouldrequire that we use onlythe diagonal terms of wi,j. Further,the simplestexpression of second orderin ki containing one parametric constant is w* = ( Admittedly, such a drasticapproximation to the wave interactionterms, whichputs allthe information about the complicated interaction processes into one parametricconstant, leaves a lot to be desired. However, even such a simpleapproximation has some redeemingfeatures, namely: 58 1. thewave-particle analogy is preserved; 2. with areaccounted for approximately even in a non-steady, inhomogeneous fl.ow with large fluctuations, and forsmall fluctuations in steady homogeneous flow; 3. thedifferential equation corresponding to the choice of u* is theSchroedinger equation which is a wave equationwith strong dispersion. Thus the varioustypes of waves, in particular the vorticity and entropy waves, will notonly be convected by thefluid but will also be dispersed and spreadout relative to the moving fluid. The dispersionis governed by wave interactions.Both the interaction of waves with the mean flow and wave resonances(higher order effects or wave "collisions")contribute to the dispersion. With ~9complex, there will also be attenuation or amplification due to interact ions. Inconclusion, using the wave-particle duality wemode1 theturbulent motion by allowingthe wave packets to have frequencymodified by a term thatrepresents kinetic energy of the motion relative to the mean flow. The added term has thedesired properties and it depends on the wave inter- act ions. "Shall I refuse my. dinner because I do not fully understand the process of digestion?" 0. Heaviside quoted by T. V. and M. A. Biot, ~tIiematica1'Methods'in Engineering, McGraw-Hill Book Co., New York, 1940. vl.TURBULENTTRANSPORT EQUATIONS DifferentialEquations for theCharacteristic Functions For a given 1 inearsystem I! (I$a) = 0 we have theplane wave " solutions $or= (x*k - wat) with @orand wa independent of space-time. Because thesystem is linear, a weighted sum or an integralover the i-space is also a solutionof L($Ja) = 0 for any distribution @a(i)independent of and t. An equationfor $J may beobtained with local mean valuesof the flow " properties, e.g., the mean velocityU(x,t), as parameters, by substituting a thedifferential operators i- and -iV for w and i, respectively,into at-" thedispersion relation w = w[k;U(x,t)],that is, the differential equationfor I$ takesthe symbolic form 60 Inorder to generalizethese results to unsteady inhomogeneous flows we shall write the characteristic function J, as where we omitthe subscript a. By writing + and w asfunctions of z, L, and t, ratherthan as functionsof and offunctions of and.t, we avoidtreating J, as a functionalof the mean flowproperties. A functionalequation for such a functionalcould be derivedby Hopf's (1952) formalismfor any case 4f fluid flow. Inavoiding the use of functionalcalculus we haveno formal way ofderiving an equationfor the characteristicfunction (6.2) inthe general case.Observing, however, that I) and 9 may be viewed as wave functions in spaceand momentum representations, and thatin the quantum mechanicsthe Schroedinger equation may be derivedformally for the noninteracting (free particle) case as done here in steps 1 eadi ng towards Eq. (6.1 ) , and that the Schroedingerequation for the interacting particle case is postulated to be the same equationwith a properspace-time dependent interaction term, we proceedlikewise and postulate that I) in its generalizedform, Eq. (6.2) continuesto satisfy Eq. (6.1). An additionalheuristic argumentbehind thispostulate is theobservation that, following Madelung's(1926) steps, we may change the wave-function-description of our system of waves into a hydrodynamicdescription of a transport of the probabi 1 ity density I $I2 in a medium of constant translational velocity and constantdensity and pressure. A generalizationof such a transportequation to a transport of the probability density in nonuniform medium is a naturalstep if the effects of wave interactions 61 are taken care of even approximately. Realizingthat the simplest expression for theterm u* modifying thefrequency to account for wave interactions,which expression would lead to dispersive wave motion, is u* = (<-ic)k2, wemay write the dispersion relation for the vorticity waves, a = 1,2, as w = [i*i+ <,k2 - i (v+S,)k2], a = 1,2 where v = molecular kinematic viscosity, 5, = vorticity diffusion coefficient, ca = vorticity dispersion coefficient. Beforewriting the corresponding differential operator, we observethat i, <,, v, and 5, are,in general, functions of and, therefore,the order in which the functions of x and theoperator kt. -iV appear will affect the result; following the practice of quantummechanics, we shall assume, subjectto experimental verifica- tion,that the dispersion relation mustbe first symnetrized and onlythen should be replaced by theoperator -iV. The operator 1"" correspondingto w = [T(U*k+k*U)+io.r,k-ik.(v+S,)k] is Dividing v bythe Prandtl number we have thecorresponding operator for the entropy mode, a = 3: Inthe case ofthe acoustic waves thedispersion relation contains 2 theterm ak[l-(%) '3% whichleads to an improperoperator. Even withthe term approximated by thebinomial expansion or in the inviscid 62 ~~ @, - .4 limit we would need theoperator corresponding to lkl =-[kt+kq+kG] . i1 Sincethe acoustic waves give rise to a quadratic wave surface, we return for the moment to the quadratic term of the characteristic equation (4.18) which must vanish as a conditionfor the existence of acoustic waves. With X = -i[w-i=k-(s-ic)k2J,the quadratic dispersion re1 at ion becomes ~~-2i=k~2(<-ic)k~~+(U,k)~+2i=E(<-ic)k~+[(<-ic)k2]2+i$2~ - ii*i$k2-i32(<-ic)k2-(E)2k4-a2k2 = 0 . We choose to syrnrnet'rize thedispersion-relation (6.5) as follows: - a2k2 = 0. (6.6) The differentialequations for the characteristic functions a are obtained by substitution of the operators i-at and -iV for w and E, respectively. From Eqs. (6.3), (6.4) and (6.6) we have a = 1,2 (vorticity mode) : aJ, -,= -,= at - $i*~$a+v= (UJI,) I+~V*C~,V~,]+V*C(~+S,)VJ,,J, 63 inviscid medium at rest, and with 5, = 0 and constant .7)- (6.9) reduce to (6.10) (6.11) Equations (6.10) and (6.1 1) formally resemble the free-particle Schroedinger equation, and the relativistic Klein-Gordon equation (oftenreferred to as the 64 " "relativistic Schroedinger equation"), + -4a2- {e2 (#2-h2)- (moc2) 2)J,, h where c = speed of light, e = electric charge, mo = rest mass, # and - A = scalar and vectorpotentials. We may observe at'this point that for vorticity and entropy waves 5, plays the r6le of the reducedPlanck's constant h/2a. Th is is a consequence of choosing UI;= (Ca- i5,) k2. In absence of electromagnetic fields the non-relativistic Hamil- tonian is H = .p2/2m, and in the relativistic case we have H = c(p2 +moc2 2)4. Evidently,the acoustic mode leads to the analogy with the relativistic Klein-Gordonequation because the real part of the intrinsic frequency, containsthe square root term in which -(&)2 replaces m2c2. 0 Thus theanalogy would call for an imaginaryrest mass thatvaries inverselywith the square of thewavelength. The modificationof the frequency,the term ck2, behaves non-relativistically.In absence of interactions, 5 = 0, and Eq. (6.1 1) reduces to the wave equationwhich is also a limit of the Klein-Gordonequation for vanishing electro- magneticfield. The Klein-Gordonequation is generallyaccepted as valid for spinlessparticles. The mainobjection to its use is thefact that theequation is of second orderin time and that, as a consequence,it requiresspecification of twoini.tia1 conditions. But this was to be expected because, insteadof two equations for theseparate acoustic 65 modes, we obtained a singleequation for the function which is in an unknown relationto Q4 and J, introducedearlier. If we consider J, to 5 be a two-component vector wave function and followPauli's procedure, see,e.g., pp. 388-398 of Bohm (1951), we could,in principle at least, reducethe problem to two first orderequations for the two components of an acousticvector wave function.There would always remain, of course, the difficulty of interpreting the meaning of the components of such a vector wave function and their relation to the characteristic functions Q4 and $ We proposetherefore to derive an approximate 5' setof first order equations for $4 and $ whichequations retain some 5' ofthe arbitrariness expected for a two-component treatmentby Pauli's procedure,but do notpresent any difficulty in interpreting their mean ing . - " -k If we introduce a unit normal n' so that I El = k-n if n "lip- and neglect K2 as compared to unity, then we have f rom Eq. (4.19) for the two acoustic modes w4 = U*k + ak*n4, = Umk- ak*n w5 5' - so that We need only two orthogonal modes and two linearly 5 = -"4' independentequations for Q4 and J, Such equations may beobtained 5' for anytwo distinctvectors i4 and inparticular, twovectors with 5' opposite directions. Inthe present notat ion the operator corresponding to ai*; is - -ian=V. If the speed of sound, a = a (i,t) , isnot constant, then we shouldsymmetrize thisoperator to -$*[aV( )+V(a )]'. The equations - for $J~and wi 11 thencorrespond to i4= n,and = -n. The Q 5. 5 66 .. .. dispersion relation for theacoustic modes a = 4,5 is 2 w = g k2 i(& 5 + ca(i*n)a fi-E + a - + a )k where c = 1 for a = 4 andca = -1 for a = 5. We symmetrize this a relation as follows The correspondingdifferential equation for $a .is i- --2a c n*[v(a~~~)+a~$~], a = 4,5. Inorder to display the properties of thisequation we rewrite it as follows, - 2-[v-C1 + C~;-V~]$~. (6.12) The left-hand-side of Eq. (6.12) is a total(directional) derivative along a characteristic cone of an inviscid mean flowtaken in the direction of -=ax U+c,an.- Thisdirection, if is arbitrary,corresponds dt to the direction of the group velocity for an acoustic wave with wavenumber i; parallelto n'. The groupvelocity is The first termon the right-hand-side of Eq. (6.12) represents dissipation due tomolecular and turbulent random motions. The second term,as in the case of theSchroedinger equation, gives a turbulent 67 dispersion,=const. Go! being the analog of the Planck's constant h/2a for theacoustic case. The 1-astterm takes the form of theforcing function with the force potential 1 V(;;,t) = - 5 Note thatthe potential V and, therefore,also the force, -VU dependon thedirection of the unit normal n'. Such a force has, therefore, two 1 1 components, -VV*c and paV(G*Va), an isotropic and anisotropic components; 2 respectively. Thus theacoustic modes aresubject to a dispersive force in presence of gradients of the "index of refraction" a/aowhere a. a. is a referencevalue. The dependence of the equation (6.12) on the choice of thc direction ofthe normal n' should haveno effect on thestatistical properties of sound inturbulence, partly, because in computing statisticalfunctions (moments of the distribution) we must sum overthe two acoustic modes, and partly because, withdifferent directions of the normal, the equationsfor I/.J4 and arelinearly independent. Similar situation I/.J 5 arisesin the theory of characteristics of inviscid gas dynamicswhere theparticular choice of the characteristic normal is immaterial as long as the chosen directionsare not parallel to assure linear independenceof the resulting equations. Some comments, regarding justification of.the manner in which Eq. (6.12) was derived,are in order. We notethat the derivation starts withthe assumption that n' is parallelto E. Only afteris replaced by its associatedoperator, -io, thedirection of n' becomes arbitrary because any connectionwith a particular direction of the wavenumber I!! It vectordisappears. We couldde.rive (6.12) underthe assumption that n' isantiparallel to (in the direction opposite to c) and wouldobtain, forthe same givenacoustic mode, theequation (6.12) withthe sign of n' reversed. Thus each mode, a = 4 and a = 5, has tosatisfy Eq. (6.12) withsigns reversed, and only when theequations for and $ are to be $4 5 solvedsimultaneously, the opposite signs have to be used. We conclude that each mode satisfies Eq. (6.12) witheither (+) or (-) signin front of n'. As a consequence, wemay applyto $ successivelythe operator correspondingto Eq. (6.12) twice, first with (+) and thenwith (-) sign inorder to el irninate the normal n'. The result wi 11 be a second order equationof Klein-Gordon type for and an identicalequation for $ 14~ 5' whichequation may notagree with Eq. (6.9) due to different symmetri- zation used. Withconstant U, a, 5, and 5 identicalresults would be obtained as theproduct of the operators corresponding to (+) and (-) signs is identicalto the operator derived from the product of the dispersionrelations (the quadratic dispersion relation), i .e., Thisshould be compared with Eq. (6.5). It isfelt that Eq. (6.12) is notonly correct, but that it could be obtainedby following Pauli's procedure of introducing a two-component wave function and reducingthe Klein-Gordon-typeequation to two equationsfor the two"components" of 69 the acoustic wave function, namely and $ $4 5' It remains tointerpret the two acoustic wave modes. First, we observethat the directions of propagation of the two waves for a given wavenumber areequal and opposite.Physically, the two waves represent, relativeto a planenormal to E, an incident and a reflected wave,The two typesof acoustic waves (or wave packets)are distinguishable by the property of their "parity", and the"conservation of parity'' wouldimply that a given wave cannotreverse its directionof propagation except upon collision(interaction) with other waves orwith boundaries. Using twoseparate acoustic modes in calculations may beviewed as equivalent to theimposition of the radiation condition on the wave packetmicromotion. Further, it may beconcluded that a Klein-Gordon-typeequation cannot describethe propagation of sound correctlyin presence of turbulence with theturbulent scale comparable to the wavelength of sound. DiffusionEquations for the Field Probabilities Following Madelung(1926) we will now putthe partial differential equationsfor the characteristic function $ (whichequations are linear in $) into a nonlinear hydrodynamic form. We observethat Eqs. (6.7), (6.8) and(6.12) arein a common form correspondingto Eq. (6.12) : = 1 ,...,5 where c = 0 for a = 1,2,3, c4=l, c -1, v =v2= v = kinematic viscosity a 5 = v/Pr, v -v -yv/(2~r). v3 4- 5- If the coefficients of the above equationare constant, then the solution is of the form $ = +(~)exp€-rt+i[x-~-w(k)tl). Inthe general case,however, we shallallow tj to have the most generalform of a complex function, namely (6.14) where R and S arearbitrary real functions of space-time to be determined from Eq. (6.13), and S may be multiple-valued.In particular, departing fromMadelung's formalism we shall write the phase S asan integral We shall keep in mind that VS hasan irrotationalpart, VA, and a rotational component Vxi. Thisis a naturalgeneralization of Madelung's formalismthat is consistent with the fact that the phase of thestate vectors, s = i-i-wt hasas its gradient Vs = E which is assumed to be in general a rotationalvector, $ E-dG # 0. Consequently, S is a multiple- valuedfunction but its gradientis not. Differentiating Eq. (6.14) we have [E+iR-] as eiS= [T+alnR i-& as , at at at at at at at VJ, =[VlnR + ii]$. Substituting into Eq. (6.13), dividing by $, and separating real and imaginary parts, we obtain dropping the subscripta + [6+can+2 1 = V*[ (u+S)~]+V*[~V~~R]+~[~(VlnR) 2-V2]. (6.16) Multiplying Eq. (6.15) by 2R2 and introducing the probability density, or "turbulent intensity", P a' we have omitting the subscripta: - [V-~+C~-V~]P. The above equation may be put in the form of a conservation law for the probability density P: . - 1 -ap + v-j = ~P{v= (v+~)[i; (vI~P)~+ at (~i)- v21} where the probability current in Eq. (6.17) is - j = [i + cai + 23i - (v+c)VlnP]P. (6.18) Equation (6.17) is of the form similar toa conservation of chemical species equation,, - + v*[pa(cI + = r; at 9 a’ where = species diffusion velocity, and fi = net rate of production a a of species a. Consequently, we may use the following terminology - v = c an + 2cV -(va+Sa)VlnP = probability diffusion velocity, a a a = net rate of probability production of speciesa. An alternate form of Eq. (6.17) is Since we arenot interested, per se, inthe multiple-valued phase function S and sinceonly its derivatives havephysical significance, we takethe gradient of Eq. (6.16) withthe understanding that VS = i may have a rotational component. The resultis + v*{v[ (u+s)i] + ;v[SvlnPlI - (i.v)ij - ijx(vxi) - ix(vxij). (6.20) Forthe purposes of interpretation we shall add toboth sides of ai ai 1 Eq. (6.20) theterm -at + V[s(U2+V2)] to obtain This may be compared to the momentum equationfor a compressible gas in a movingframe ofreference which, at a giveninstant, is rotating with 1 aCI angularvelocity = -Vx(i+i) andhas an acceleration = 5 - 2 7 -{E 1 +V[p(U2+V2)]} relativeto a Newtonian(inertia1)frame. Such a fictitious,flowing medium, a "probability gas", issubject to the pressureforce per unit mass, - VTI, 74 ""._."_ .. I. ..,,,. ..,.,, .. - and a viscous force per unit mass V-u, where the viscous stress tensoris - 1 u = V[ (v+s)~] + ~v[~vz~P]. We conclude that the "probabi 1 i ty gas" flows relativeto and is convected by an accelerating and rotating medium and, as a consequence, there appear additional fictitious forces acting on the "probabilitygas". We also note that the pressure and viscous forces are solely due to the micromotion (wave motion relative to themean flow), and that the coupling between theseveral modes, a = 1, ...,5, is implicit through the dependence of the diffusionand dispersion coefficients, 5, and 3,, on the interactions among the several modes. The probability density of each mode, P is not conserved but is diffused and dissipated by the a' velocity ? by the random micromotion,and by the molecular viscosity. a' The probability density field P and the probability velocity are a 5a strongly coupled. At this point we may add that nonlinear, diffusion-type equations were proposed for the study of turbulenceby many authors. These phenomenological theories postulate an equation of the general form ap - + (ij + i)-VP = Ve (EVP) + AP, (6.21) at - where P = "turbulent intensity", v = self-diffusion velocity,E = "eddy" viscosity, A = turbulent source function,a function of 6 and P. Equations of type (6.21) were proposed by, e.g., Kolmogorov (1942), Prandtl (1945), Nee & Koviszniy (1969) and are discussed in great detai 1 by Saffman (1970). The present theory derives equations of the form (6.21) with explicit expressions for the functions G, E, and A. Further, 75 ...... - .. ... the present theory couples the probabi 1 ity transport e’quation to the equation for the probabil ity velocity ii which velocity contributes to theprobability transport in a substantial way. Further,the present theoryis developed for the compressible case so as to isolate,the acoustic mode ofpropagation explicitly. The theory may be easily extended tochemically reacting gasesand to ionized gases(plasmas). ..."the logical operations of the calculus of probability cannot be imitated by the averaging operations. The relation between the calculus of probability and the calculus of mean values is not one-one; only the former determines the latter." M. Strauss Modern Physicsand its Philisophy, D. ReidelPubl. Co., Dordrecht,Holland, 1972, p. 188. VII. AVERAGES, MOMENTS, AND CUMULANTS Generalization of Definitions Giventhe characteristic function $(x,t) = I $(x,k,t)exp{i[x*k - w(x,k,t)t]>dc with w complex ingeneral, we seek variousexpressions in which 1$%$=1$1~ playsthe r6le of a probabilitydistribution function in the phase space (;,E). It is assumed thatthe function I/J is known as a solution of a partialdifferential equation, and that $ may be specifiedin terms of its own moments. We define the moments of powers of the components of the wavenumber vector = {k ,k ,k 1 as 123 Mmi".mN = I k6..knNml f(i,k,t) dE "1 ' *" N 1N 77 where m +m mN = = orderof the moment, nl...n indicatesthe 12+...+ K N sequence inwhich.the components kn , N 5 K, arearranged, and j f(i,ii,t) = $*(i,ii,t)$(i,ii,t). The averages or expectations are defined as ml N ml .. .m If all the moments of arbitrary order are known, then the distribution function f(G,c,t) may beconsidered also known and givenin terms of its moments. The distributionfunction f may be calledaccurate to order K if the distribution gives correctly all moments up to and including those oforder K. It will be necessary now to expressthe moments and averages in terms of the characteristic function J, which is assumedknown. The numericalvalues of the moments in the steady homogeneous case aregiven in terms of thespatial derivatives of J,(i,t), theFourier transformof $(E,w). For, at t = 0 and with s = - w(L) t 78 I We observethat for Vln$ to be equal to a pureimaginary number we musthave I$ I = constant. Since ii coresponds to the vector operator V, higherorder products of components of k aregiven by corresponding derivatives of $. Thus We thenmultiply both sides by $* to get m K "1 = (i) $knl k N dk = (ilKMml".m N ... n @*@ N "1"'" N N K Dividing by (i) $*$ we have In the probability theory one usuallydeals with a characteristic function F which is a Fouriertransform of a real distribution function f(k) -> 0, that is and theexpectations and moments of various orders are given by 79 . , ...... "" .. One alsoconsiders cumulants C related to the moments and definedas Pqr K K a 1nF c = (-i) P9 r ax Pqrax ax The cumulantsare polynomials of order K in the various moments of all orders up to and incl ud ing K. Inthe present case these definitions must be generalized to reflect the fact that @ and itsFourier transform $J are complex functions thesquares of whose absolutevalues play the r61e of probability densities, and thefact that, 'in the most generalcase, I/J isnot a single- valuedfunction of space coordinates. Thus theorder of differentiation cannot bechanged withoutaffecting the result's. A generalization to the inhomogeneous time-dependent case follows the practice of a formalgeneralization of the operator relationships. Observe that if @ = @(i,i,t) and w = w(i,K,t), then,evaluating the derivativesat t = 0 (atthe local time), we have = i/[i + Varg@- iVZnl@l]$(X,L,t)eiSdi. According tothe correspondence principle,this expression must reduce tothe "classical" one, thatis, we must recoverthe case of the steady homogeneous wave motion in the 1 imit as @(G,E,t) + $(i), 80 ...... "" .. . . ._ ...... - . - - ...... Lim vl$+ 0 VJ, = i&(E)eiSdk. Thus we postulate that, in the inhomogeneouscase, the role of the wavenumber vector(which is proportional to the momentum ofthe quasi- particles)is taken over by the effective wavenumber vector (E + Vargcj - ivlnlcjl) where Vargcj is a random function arising from'the uncertaintyin the phase s = - W(i,E,t)t k 2m, n = 1,2,3 ,... The imaqinary component, -iVlnlcjl, representsthe attenuation or amp1 ification of the momentum of the quasi-particles in wave interactionswhich result inthe change inthe number densityof the quasi-particles. The sum (E + Vargcj - iVlnlcjl)gives three contributions to VI) due to,respectively, the changes alongthe Hamiltonian trajectories of the quasi-particles, changes due tothe uncertainty of phase (uncertaintyin the number of particles), and changes due to interactions among the waves. Withthe understandingthat in evaluating various integrals of functions of k in the inhomogeneous case we will express them in terms ofthe derivatives of 9, then, because and theeffective wavenumber vectorare in the same relationship to the derivatives of I), the distinction between the two wavenumbers isimmaterial. As a consequence, we postulatethat (7.1) holdsalso in the inhomogeneouscase and, due tothe fact that may varyin space, the r ighthandside of Eq. (7.1) is comp lexin general. Thus we should write as a generalization of Eq. (7.1) : m ml N K 1 either 81 m m K 1 N 1 a or Both of the above equationsgive a realvalue for the average of a real quantity. However, Eq. (7.2) leads to symmetrictensors, e.g. The wave function $ by itself hasno physical meaningand only its amp1 itude squared, R2 = P, has a significance when interpreted as a probabilityintensity. Consequently, the generalization should leave R unaltered so as to rendered P single-valued.This leaves us with a modificationof the phase S as theonly alternative. But the phase may be left arbitrary up to any integralmultiple of 27r, orin general, could berepresented by any multiple-valuedreal function of i and t. That is whywe made the choice of writing S as a path-dependent(multiple-valued) functionof G, thatis, so that,in general, is different from zero and the spatial derivative of S (gradient of S) 82 From thedefinition of the average, Eq. (7.3), - - Differentiatingfirst with respect to xand thenwith respect to P x p,q = 1,2,3, we obtain qy It should be observedthat the differentiation of JI is noncommutative, for is different from zero if Vxi # 0. Applyingformula (7.3) and adoptingthe convention that the order of differentiation is indicated by theordering of the subscripts when read from right to left, we obtain The skew-symmetric part of this tensor is The threenon-zero components of the skew-symmetric partare given by the 1 three components ofthe vector fi = 5 Ox?.The vector fi has units of the -2 inversesquare oflength (cm ).. If we multiplythe phase S by., an appropriatescaling factor h withdimensions of action(energy x time), then hi = momentum, and hfi = angular momentum. The tensorinvariant, A generalexpression for an average of a thirdorder product is 84 Notethat if S weresingle-valued (E = 01, thenthe above third ordertensor would be symmetric with respect to any pair of indices and couldbe expressed in terms of products of lowerorder averages. Thus it is due tothe non-single-valuedness of S = arg J, that the.differentiation of J, is noncommutative, and the differential operators follow the rules ofthe complexnoncommutative algebra. Such an algebra was proposedas a mathematicalstructure which permits a representationof any stochastic process in a quantum-mechanicalframework and permitsinterpretation of quantum mechanicsas a st0chas.t ic process, see Santos(1974) who developed a quantum-likeformalism to deal with generalstochastic systems. The central moments ofvarious orders, defined as moments of various powers ofdeviations of components of wavenumber vector k fromtheir P mean values ck >, P m .m m 1 . . N ml N =J[kn - N3 = M3 - 3M2M’ + 2(M1)3,etc. PP PP P ...... The cumulants will bedefined as successive space derivatives of the 1 owest order aver.age, 1 Forexample, C = - Ambiguityin the Definitions of Moments, Averages andCumulants Some generalremarks, concerning the practical use of moments, averages and cumulants,are in order. First, we shalldiscuss the ambiguitythat arises in the definitions of higherthan the first order correlations and thereasons for making tentatively the choices of the definitions as given on preceding pages. 86 ...... i We will appeal to the mathematical principles asused in quantum mechanics. The basic principle is the fact that the average or moment of a real quantity should be real, and the mathematical method of calculation (the definition)must be such as to assure the reality. We shall start with the simplest caseand return to the earlier observation that in the steady homogeneous case, and that = constant in order that The above implies that, while $*k.$ in view of the fact that e ii-E is a function of 2 and is therefore J canonically conjugate to c. The difficulty arises, however, due to the ambiguity in the particular choiceof symnetrization of higher order powers of k The list below gives partial results for second order j' averages. 87 SYMMETR I C FORM EXPRESSION, FOR Obviously, any combinationof the symmetrized forms may beused to define theaverage value. Another possibility exists in an anti-symmetricordering of terms followedby a multiplication by theimaginary unit i. Thiscorresponds to a generationof a Hermiteanoperator by first obtaining a Hermitean conjugateoperator and thenmultiplying it by i, which is also a Hermiteanconjugate operator. The resultis a Hermiteanoperator and only such will give a realfunction. Thus wemay write The lastexpression corresponds to Eq. (7.3). We alsonotice that integrandor whether we take an anti-symmetric integrand multipl ied by i. Since k isthe derivative of the phase s = i-i;under the integral sign, j and arg @ isthe phase of @,the choice here is obvious.Following this 1 ine of thought, we shallassociate products of components of with derivatives of the argument of @,and we shall usesymmetric ordering for odd powers of and ant i-symmetr icordering for evenpowers. Then Eq. (7.3) may beused for a1 1 ordersof the averages. This choice is somewhat heuristic and requires verification by a referenceto experi- ment. Admittedly,anti-symmetric ordering is not in a common use and is introducedhere to achieve a desired result. All theexpressions for moments, averages and cumulantsare under- stood to beevaluated at a giveninstant of time (t = 0). The time- evolutionof the various correlations should not be studiedin terms of the t ime-dependence of the state vectors $exp { i (x-k - w t) 1, because such a representation of quantum fields in terms ofthe active Schroedinger picture is known to beunstable in time, see, e.g. p. 5 ofDirac (1966). However, thetime-history of the correlations is known if .the space dis- tributionof the wave function $ is known at everyinstant of time. Thus thecorrelations become implicitfunctions of time through the time- dependence of @ governedby the postulated non-linear partial differential equations. The statevectors are used hereonly for manipulative purposes toprovide an instantaneouscoordinate basis for the representation of functions of time at a giveninstant. The averages or expectations of arbitrary functions of L are F(E) = J@*e-isF(L)$eis dE /J@*$di If F(E) is a polynomialin k, thenthe expectation value of F(i) may beexpressed in terms of a correspondingpolynomial of averages of E. 89 If the coefficients of the polynomial depend on;, then each term of the polynomial should.be symmetrized first beforethe wavenumber vector is replaced by a corresponding differential operator. If F(C) is not a polynomial, one is faced with the useof improper operators (e.g., with derivatives of fractional order, etc.). Expressions of the type (7.6) arise in wave interaction terms. An evaluation of such terms, in cases where F(C) is not a polynomial, may be carried out approximately by a suitable choice of an approximation to the probability distribution function f = $*@. If the approximate distribution f(G,c,t) contains a number of arbitrary functions of G and t, such functions may be determined by the requirement that f generates correct valuesof an equal number of moments. Thus, in principle, all interaction integrals of the form (7.6) may be determined with arbitrary, but finite accuracy. 90 "Neither seeking nor avoiding mathematical Gercitations we enter into problems solely with a view to possible usefulness for physical science. Lord Kelvin and Peter Guthrie Tait, "Treatise on Natural Philosophy," Part 11. Cambridge University Press, 1895. VIII. ENERGIES AND DISTRIBUTIONS Moda 1 Energ i es We are now able to write the expressions for the squares of the fluctuations. From Eq. (4.16) we have h2 Due to the fact that ua contain the term (--)ak , the two delta functions will vanish simultaneously only if a = B or at E = 0. Assuming that Cp(0) = 0,-we obtain where the participation coefficients P are given by Eq. (4.21) or, aj 91 I obtainthe squares ofthe fluctuations by applying Eq. (8.1) for a particularvalue of theindex j. Thus, with /f (k)dk =JIawa=Pa, where a P isthe probability density assumedknown as a solution of thepartial a differentialequation (6.19). we haveusing Eq. (4.22) I pl I = p21 ui I 2= p2p- P3 + +p4+p5)],1 Y k2+k2 k2 I w'12' c2 P <=>+ -P <3>+ {2 k2224k24 where the symbol <( )> indicatesthe average with respect to the a-th a 2 distributionfunction, and where c2 = Ri = i/i. The expressions lull J are,actually, sums of moments of IP .I2 withrespect to the distributions aJ f a (E) thatinclude all non-trivial contributions of a1 1 five modes, a = 1, ...,5. Summing up thesquares of the velocity components we obtainfor the turbulentkinetic energy 2 IV'I +1w' Here, it is expected that the contribution of the vort ici tymodes 92 wil-? be dominant, so that the turbulent energy, neglecting acoustic contributions, would become(;/PI (P1+P2). Fluctuations of other physical variables may be expressed in terms of the u' For example, using the perfect gas law, p = pRT, we have j' and = y-l, + 1(P +P )+ [P3+ i(P4+P5)], y32y45 Y The above expression for the expectation value of the square of pressure fluctuations illustratesthe fact that contributions of the several modes of wave propagation are separated outand are given in terms of the probability densities Pa . For instance, the expectation of the square of the pressure carriedby the acoustic modes is 2y-1 -2 21- P (p4+p5) Obviously, this quantity is radiating acoustically, while the contributionof the entropy mode, -. 3(y-1 k2P3, represents the Y effects of the "pseudo-sound" that is convected by the turbulent medium. Thus the separation into radiated and convected contributions amounts to separation of the contributions of the acoustic and non-acoustic modes. Finally, we observe that linear combinations of the diffusion equations for the field probabilities P Eq. would serve as a' (6.19), 93 transport equations for the turbulent kinetic energies. Such a turbulent energy transport equation was proposedby Nee and Kova’szna’y (1969). In the present theory the transport equationsfor Pa are coupled with the equations for the probability density velocity field due to the use of a complex wave function (the characteristic function $). The use of a complex wave function and the allowance for multiple-valuedness of the argument of thewave functions, make present results moregeneral. Of special interest are the spectral distributions of the fluctu- ations. We turn now to the problemof an approximate determination of the wavenumber distribution functions. With the help of the distribution functions one may evaluate expressions such as<,r> which cannot be kl+k% given in terms of derivatives of the characteristic function a’ Approximate Distribution Functions The knowledge of the distributions is required for evaluationof various statistical correlations which enter into the equations for the mean flow, Eqs. (4.6)-(4.8), and for the evaluation of the random func- tions u*a which are given in terms of the interaction integrals, Eq.(4.27). We observe that the mean flow depends on the correlations of second and third order. As a consequence, a distribution function which is fitted to give correctly all first, second, third, ... order moments of the wave- number vector, would be expectedalso to predict accurately various cor- relations up to and including those of the first, second, third, ...order. The procedure for fitting an approximate form of the distribution func- tion f(i,L,t) will be illustrated below. At a given instant (t=const.) the wavenumber distribution function 94 = f,(i,c,t) is a functionof six variables. Ue know that,in the case ofnoninteracting linear oscillators in thermal contact with a reservoir of constant energy, the principle of stationary value of entropy leadsto the Planck's distribution of energy (Bose-Einstein statistics) whichgives the energy spectrum in terms of the wavenumber or frequency. By thecorrespondence principle, infinitesimal fluctuations in a steady homogeneous mean flowshould approach Planck's distribution. Thus the, limiting form of f(L) is known and wemay simplygeneralize it to the nonsteady inhomogeneous caseretaining the general features of the k- dependence of f (i,k,t) and allowing for space-time dependenceand for anisotropy. By argumentsleading tothe derivation of Planck's distribution, e.g., see Bohm (1951)p. 19, one may show thatthe energy dE inthe physical volume V containedin waves ofintrinsic frequency " L w'= w-U.k = a' (k) in the wavenumber range dE centeredat k is at a thermodynamic equilibrium where isthe average energy per particle of the background with which the wavemodes arein thermal equilibrium. This distribution gives the Planck'sfunction for theacoustic waves, w'zak, when w' is linear in k, 2 and a generalizedPlanck's form for w' quadraticin k, w'=yk , for vorticity and entropy waves. The phase space density of energy, u = - -dE , is dc 95 and the phasespace density of the adiabatic invariant is where n = phasespace number densityof the excited states. Thus the adiabaticinvariant U/w' isproportional to the quantum of action h. If we introduce a wavenumber distribution function f4*6 such that U/W' = h*f(i) with a new constant h*, then at equilibrium wemay write For vorticity and entropy waves wemay write in general h*w'/E = f3y(kl-cl) 22+ f32(k2-c2)2+ 333B2(k-c )2 2 with h*, Bi, cibeing functions of spaceand timeto bechosen so as to satisfy the following seven moment conditions: 2 2 P = JfdE, P Carryingout the integrations with the approximate form (8.3) used in equation (8.21, we have 96 2 sin P av P from (8.5) , Subst itution in Eq. (8.6) gives a quadratic in Bi with two roots where 2 Real solutions are possible for b - 4ac 2 0. This places an upper bound on the secondmoments, This example illustratesthe following facts. 1. An approximate distribution function may be expressed in terms of 2 moments such as P, 97 I the probability density function P and the probability velocity Vi, and oftheir gradients. The functions P and Vi aresolutions of the transport equations discussed in the preceding ,.sections. Fitting of the generalized Planck's type distribution functions requires a solution of a simultaneousset of nonlinearalgebra'ic equationsleading to nonunique and, for certain values of the moments, tononexistent solutions. The raleplayed by the quantum of action h is takenover by h* = f3 f3 f3 P/[~IT<(~)] inthe present example. 123 Possible alternatives to the use of Planck's type distributions are suggestedby the following observation, Thus a Planck'stype distribution is a particular power seriesin the 2 -ck Gaussian distribution e . Frankiel and Klebanoff (1973) have shown thatfourth- and six-orderGram-Charlier distributions give excellent approximations tocarefully determined experimental data in turbulent boundarylayers. The Gram-Charlierdistributions of j-thorder (one- dimensional)are defined as where Hj (k) = (-1) jetk2 -(edj -&k2) 9 d kJ and i.(k)is the averagedvalue of H.(k). The Grarn-Char1 ierdistributions J J 98 are of the form of a sum of products of Herrnite polynomials fi. Ck)H. (k) .J J times the Gaussian term. The advantage of Grarn-Charlier distributionis thatthe polynomials i.(k) are given explicitly in terms of the averages J of various powers of k and thetedious algebra involved in fitting a givendistribution is thus avoided. It isinteresting to notethat Hermite expansions in terms of Gaussian variableswere interpreted by Wieneras expansions around the stateof perfect disorder ("white noise"), see,e.g., thediscussions of Wiener-Hermiteexpans ions in Canavan (1970), Crow and Canavan (1970), and Meecham (1970). It appears on thebasis of the experimental data ofFrankiel and Klebanoffthat some almost-Gaussiandistributions may serve as good approximations for the vorticity modes whichcontain most of the turbulent energy. It is further suggested thatfor the acoustic modes, in absence of experimentaldata, the Planck's type distributions beused even though the state of equilibrium is not likely to be approachedunless the turbulentflow is enclosed by the walls of a ductin internal flows. In externalflows the acoustic energy will be radiatedoutwards into the infinite space leading to large fluxes of acoustic energy and to the absence of equilibrium. 99 "For with slight efforts, how should one obtain great results? It is foolish even to desire it." Thomas Jefferson's favorite quotation from EURIPIDES. . IX. SUMMARY Thewave theoryof turbulence formulated here leads to a closed system of nonlinear diffusion-type equations for the probability densities of each separate mode of wave propagation and for the associated probab i lityvelocity fields. The steps inthe der ivationof these equat ions will be summarized briefly. Separatingthe primitive phys icalvariables into the averages with respectto a probability distribut ion and intothe turbulent fluctuations, theNavier-Stokes equations for a viscouscompressible fluid are split into two coupledsystems ofequations for the averaged values, Eqs. (4.6)- (4.8), and forthe fluctuations, Eqs.(4.9)-(4.11). The latter system is putin the form (4.15) inwhich the 1 inearpart of the equations for the fluctuations is equated to termswhich are interpreted as sources forthe fluctuations(or forcing functions) arising from the interactions with the mean flow and interactions among thefluctuations. The linearpart of the system forthe fluctuations is then Fourier- analysedresulting in a five-foldinfinite set of eigen-solutions correspondingto five orthogonal modes of fluid oscillations identified as:two vorticity modes, an entropy mode, and two acoustic modes.The 100 eigensolutions, which represent. infinitesimal waves in a steady homogeneous mean flow at instantaneous, local conditions, are then employed as a complete vector basis for the purposeof forming an integral representation of the solutions. The absolute values of the Fourier amplitudes squared are interpreted as the probability densities in the wavenumber space. The wavenumber vector and the frequency of the Fourier modesare associated with momentumand energy of quasi-particles (wave packets). This association is used to model the wave interaction terms asa stochastic function,u* = (c-ic)k2, quadratic in the wavenumber (momentum) to be added to theexpression for the frequency (energy) of non-inter- acting wave packets. Conditions imposed on the stochastic function ug, Eq. (4.27), are that the average ofu* satisfies the averagedinter- c1 action terms so that each orthogonal mode a is coupled through the interaction terms with the remaining four modes,a = 1, ...,5. Thus ut models the interactions of the fluctuations withthe mean flow and the interactions of the wave-wave type throughthe wave resonance. It is then observed that the interaction terms may be determined if the probability distributions of the orthogonal modes in the wavenumber space areknown. Thus the central problem in the development of the theoryis the determin- ation of the distributions. An operator formalism is introduced in Chapter V so as to associate differential operators with dispersion relations for each orthogonal mode. A characteristic function, that reduces to a Fourier transform of the Fourier amplitudes in the steady homogeneous mean flow,is then sought as a solution of the Schroedinger-type differential equation determinedby the operator formalism from the dispersionrelations. The solution, analogous to a wave functionof the quantum theory, is taken.to have a generalform of a complex functionwith. a multiple-valued phase. Separation of real and imaginaryparts, and differentiation of theimaginary part with respect to space, resultin transport equations for thespatial probability density (for the squares of the absolute values of the amp1 itudes of the characteristicfunction), Eq. (6.17), and vectorequations for the "probabilityvelocities," Eq. (6.20), whichvelocities are shown to equal theaverages of the wavenumber vector with respect to the probability distributionin the wavenumber space. The probabilitytransport equations arenonlinear and of the diffusion type, similar to conservation of chemicalspecies equations inreactive flows, and alsosimilar to the equationsof the quantum theoryin the hydrodynamical form. The latter similarity is clearly apparent because ofthe presence of terms analogous to quantum stressesof the hydrodynamical form of the quantum theory. The transportequations are nonlinear even in the limiting case of infinitesimalfluctuations in a steady homogeneous mean flow,the ''classical" limit of thepresent theory. It is thenpostulated that these transportequations may be generalized to the nonlinear wave motion encountered instrong turbulence in presence of non-steady inhomogeneous mean flow.Further, the transport equations for theprobability density are of the same formas the nonlinear equations for the transport of turbulentintensities proposed in.the past by many researchers. The novelfeature of thepresent theory absent in phenomenological theories isthe strong coupling of the probability density transport equations to theequations for the probability velocity fields, which fields are, in general,rotational. The latterfact is a generalizationof the hydrody- namicalform of the quantum theory in which the. velocity field is 102 irrotational having been obtained by taking a gradient of a single-valued function. Because the spatial probability distributionis obtained by squaring the absolute valueof the characteristic function, the standard definitions of moments, averages,and cumulants of the theory of probabilityhad to be generalized' to the present caseof complex characteristic functions. In Chapter VI1 these definitions are carefully developed with regarddue for the uncertainty principle. The microscopic uncertainty principle arising from the Fourier representation carries over into the macroscopic uncertainty in the form of the multiple-valuednessof the phase of the characteristic function and rotationality of the probability velocity field. Thus the circulation of the probability velocity becomesa macroscopic analog (an expectation value) of the phaseintegrals of the quantum theory, = <4 i*di> . C Many important averages (expectation values of the squares of the fluctuations) are expressible directly in terms of the probability densities of the various orthogonal modes. Thus, conveniently, the contributions of the vorticity, entropy,and acoustic modes are given separately and explicitly in terms of the solutions of the probability density transport equations. Likewise, various moments of arbitrary powers of the wavenumber vectorare expressible in terms of the probabil ity densities and probability velocities. In turn, the moments of a distr i but ion determinethe distribution function, and the centralproblem of the proposedtheory is solved.For practical purposes it suffices to determine thedistributions only approximately. Thus themodeling of turbulence and sound generatedby it wouldinvolve the use of assumed forms of the dis- tributionfunctions which forms should contain enough arbitrary functions to meet a finite number of moment conditions. Two suchforms arediscussed in Chapter VIII, thePlanck's type distributions that maximized the entropy of a system of waves at equilibrium, and Gram-Charlier distri- butionswhich were found successful and accuratein representing extensive experimentaldata. Beforethe partial differential equations of the present theory could be appliedto test cases, onehas to considerin detail the boundary conditionsto be imposed on theprobability densities and probability velocities.Numerical integration of the partial differential equations wouldinvolve a simultaneoussolution of the Reynolds system for the averaged mean flow, Eqs. (4.6)-(4.8), theprobability density transport equations, Eq. (6.17), and theequations for the probability velocities, Eq. (6.20), altogethertwenty-five nonl inear partial differential equationswhich replace the original Navier-Stokes system of five equations fordensity, velocity components, and temperature.This large number of equations may bereduced to only five in the case of a one-dimensional incompressibleturbulent flow, e.g., inpipes and channels.Extensive numericaltesting of various simplified turbulence modelsbased on the equationspresented here will be necessarybefore the present theory could beaccepted as describingthe physical processes in turbulent flows. Extensionsof the theory to chemically reacting flows, to radiative gas- dynamics,and to magnetogasdynamics wouldbe straightforward as it would 104 suffice to introduceadditi.ona1 orthogonal modes. Likewise,the present. theory could he easily reduced to.a quantum-like statistical theory of sound in inhomogeneous media inwhich the ratio of thelength scale of theinhomogeneities to thewavelength of sound is arbitrary. In applications to theacoustics of turbulent noise the present theory offers means ofdetermining, (l),the intensity of pressure fluctuations inthe acoustic modes and, approximately, itsspectral and directional distributions, and, (2), theintensities of fluctuations of arbitrary functionsin any ofthe modes of wave propagationpresent. We observe here that the far field noise outside of a turbulentregion may be cal- culated if theacoustic field at the edge of a turbulentregion is known. The presenttheory is capable ofproviding such information.Secondly, theproblem ofdetermining the noise transmitted from the turbulent boundarylayer through the walls constructed from a solidmaterial requ i res thedetailed knowledge ofturbulent fluctuations not only in the acoust ic mode, butalso in the remaining modes convectedalong the wall. The interaction of turbulence with a solid boundary may be visualized as the excitation of sound waves inthe solid material by the momentum and energy exchange withthe turbulent flow. Thus alsothe fluctuations in the vorticity mode, and to a lesserdegree, inthe entropy mode would be capable of excitingthe sound fieldin the solid wall. The presenttheory provides means of treating suchproblems. X. CONCLUD I NG REMARKS It should be observed that the theory of sound in turbulent flows proposed here is a necessary consequence of the adopted point of view. The point of view held was that sound is just one of the several aspects of wave propagation in turbulence and that the wave representationis therefore both necessary and convenient. The Fourier decomposition into interacting wave fields lead naturally to a quantum-like formulation in terms of complex distribution functions. The formulation imposes the conditions of realityof the expectation values of the fluctuating physical observables and does not require that the turbulent fluctuations tBemselves.be real quantities. Thus the theory was formulated from the start as a statistical theory and the analogy towave mechanics was exploited for the purposeof using a well established mathematical framework as an analytical tool. On the other hand, it becomes clear in retrospect that the question of 'how to transformthe deterministic Navier-Stokes fluid dynamics formally into a statistical fluid mechanics' was answered alreadyin 1926 by Madelung. All that remained to be done was to obtain his hydrodynamical form of quantum mechanics from the equations of the Navier-Stokes theory. This was accomplished here rigorously only for the limiting case of infinitesimal fluctuat ions in steady homogeneous' mean flow. The derivation of the nonlinear Madelung's equations for the hydrodynamic.transport of field probabi lities in the limiting case 106 from the nonlinear field equations of fluid mechanicsprovided the information on how to decompose thephysical variables into interacting waves and alsoprovided the form to which the postulated equations must reduce inthe limiting case. Statistical theories of turbulenceare plagued by the so called closureproblem which arises from the fact that the equations for the lowerorder moments (statisticalcorrelations) contain higher order moments as unknowns. It is necessary to comment on how theclosure problem is avoided inthe present formulation. In order to avoid the closure problem one shouldnot treat the infinitely many moments as unknowns. The vmentsdetermine the distri- butions, and viceversa. If theprobability densities P and their a velocitiesare treated as the dependent variables,then the moments a become known functions of the dependent variables and of their deriv- atives, and only a finite number of dependent variables has to be determinedfrom a closedsystem ofequations. Solutions of theturbulent transport equations must satisfy appro- priate boundary conditions. A discussionof boundary conditions for turbulenttransport equations and theconditions for the occurrence of sharpturbulent-nonturbulent interfaces may be found in the paperby Saf fman (1 970) . Because the governing equations of the theory are of the familiar diffusiontype, existing numerical integration techniques could be used with thecost of computingincreasing only five-fold as compared to the laminarcase. Standard laminar boundary layer computer programs could beadapted to turbulent flow calculations without the need.for an 107 extensiveresearch into computational methods. The theoryas proposed here remains unproven until its predictive capabilities are demonstrated on the basis of some computedexamples. The turbulenttransport equations may be readily simplified by appli- cation of boundarylayer concepts or byreduction to special cases. Forinstance, disregarding all but the two vorticity modes givesthe theoryof incompressible turbulent flows, while the retention of only the two acoustic modes results in a statistical theory of sound of inhomogeneous irrotational(potential) flows of importance in cases where appreciable changes in wavelengths of sound occurover distances of one wavelength. 108 REFERENCES Bohm, D. (1951) "QuantumTheory," Prentice-Hall,Inc., New York, N.Y. Canavan, G. H. (1970) "Some Properties of aLagrangian Wiener-Hermite Expansion,'' J. Fluid Mech., Vol. 41 , Part 2, pp. 405-412. Crow, S. C. and G. H. Canavan (1970) "Relationship Between aWiener- Hermi te Expansionand an Energy Cascade,'' J. Fluid Mech., Vol. 41, Part 2, pp. 387-404. Davidson, R. C. (1967) "The Evolutionof Wave Correlationsin Uniformly Turbulent, WeaklyNon-Linear Systems,'' J. PlasmaPhys., Vol. 1, Part 3, pp. 341-359. Dirac, P. A. M. (1966) "Lectures on Quantum Field Theory," publ.by BelferGraduate School of Science,Yeshiva University, distr.by Academic Press,Inc., New York, N.Y. Eckart, C. (1961) "Internal Waves inthe Ocean," Phys. Fluids,Vol. 4, pp. 791-799. Edwards, S. F. and W. D. McComb (1969) "Statistical MechanicsFar from Equilibrium," J. Phys. A (Gen. Phys.),Ser. 2, Vol. 2, pp. 157-171. Ehrenfest, P. (1911) "Welche Ziige derLichtquantumhypothese spielen in derTheorie d.Warmestrahlung einewesentliche Rolle?" Ann. d. Phys. , Vol . 36, pp. 91-1 18. - (1916) "On Adiabatic Changes ofSystem a in Connection withthe Quantum Theory,"Proc. Amsterdam Acad., Vol. 19, pp. 576-597, alsoin Ann. d.Phys., Vol. 51, pp. 327- 352 (1916) , and Phil. Mag., Vol. 33, pp.500-513 (1917). Frenkiel, F. N. and P. S. Klebanoff(1973) "Probability Distributions and Correlationsin a Turbulent BoundaryLayer," Phys. Fluids, Vol. 16, No. 6, pp.725-737. Green, H. S. (1965) "Fluid Mechanics and itsStatistical Basis," in Recent Advances in Engineering Sciences, Vol. 1 , pp. 171-195, Gordon & BreachSci. Publ., New York, N.Y. Gyarmati, 1. (1974)"Generalization of theGoverning Principle of Dissipative Processes to Complex ScalarFields. Quantum Mechanicsas 'Abstract'Transport Theory," Ann. d. Phys., Vol. 31, pp. 18-32. HaAdkowiak, J. (1975) "Statistical Methods in Quantum Field Theory," J. Math. Phys., Vol . 16, No. 7, pp. 1524-1527. 109 Hopf, E. (1952) "Statistical Hydromechanics and,FunctionalCalculus," J. Rat. Mech. Anal.,Vol. 1, No. 1, pp. 87-123. Huggins, E. R. (1971) "DynamicalTheory and ProbabilityInterpretation ofthe Vorticity Field," Phys. Rev. Letters,Vol. 26, No. 21, pp.1291-1294. Kawasaki, K. (1974) "Contributions to Statistical MechanicsFar from Equilibrium, Ill." Progr.Theor. Phys., Vol. 52, NO. 5, pp. 1527-1538. Kentzer, C. P. (1974a) "AcousticalTheory of Turbulence,"Arch. of Mech., Vol. 26, No. 5, pp.805-816. - (1974b) "AcousticalTheory of Turbulence,"Vol. 1 , pp.128-141 of the Proceedings of the Second Inter- agencySymposium on university Research in Trans- portation Noise, NorthCarolina State University, Raleigh, N. C., June 5-7, 1974. (1974~) "Isomorphism ofStatistical Turbulence and Quantum Theory,''a paper presented at the 50th Annual Meeting ofthe Indiana Academy of Science, DePauw University,Greencastle, Ind., Nov. 1, 1974. Kolmogorov, A. N. (1942)"Equations ofTurbulent Motion of an In- compressibleFluid," Izv. Akad. Nauk SSSR, Ser. fiz., Vol. VI, NO. 1-2, pp. 56-58. Krzywobtocki, M. Z. E. (1958) "OnSome Aspects ofDiabatic Flow and General Interpretation of the Wave MechanicsFundamental Equation,''Acta Phys. Austriaca,Vol. XII, No. 1, pp. 60-69. - (1971a) "Turbulence and Refractivity Changes and Their Sensing Based upon the Wave MechanicsTheory,'' Proc. Symposium onPropagation Limitations in Remote Sensing, NATO, XVll Annual Symposium, ColoradoSprings, Colorado, June21-25, pp. 35-1 to 35-40. - (1971b) "Wave MechanicsTheory of Turbulence,"Fluid Dynamics Transactions,Vol. 6, Part II , pp.365-390. Lighthill, M. J. (1965) "Group Velocity,'' J. Inst. Maths. Applics, Vol. 1 , pp. 1-28. Madelung, E. (1926) "Quantentheoriein Hydrodynamischer Form," Zeitschrift fir Physik, Vol. 40, pp.322-325. Meecham, W. C. (1970) "EquilibriumChacteristics of Nearly Normal Turbulence," J. Fluid Mech., Vol. 41 , Part 1, pp. 179-188. UO Millsaps, K. (1974) "A Thermodynamic Constraint on theEquilibrium Spectrum of Homogeneous IsotropicTurbulence," Mech. Res. Comm. , Vol . 1 , No. 3, pp. 177-178. Monin, A. S. and A. M. Yaglom (1971) "Statistical Fluid Mechanics," Vol. 1, the MlT Press (originallypubl. by Nauka Press, Moscow, 1965, underthe titleStatisticheskaya Gidromekhanika - Mekhanika Turbulentnosti). Morse, P. M. and H. Feshbach (1953) "Methods of Theoretical Physics," .McGraw-Hill BookCo., Inc., New York, N.Y. Nee, V. W. and L. S. G. Kovsszndy (1969) "SimplePhenomenological Theory ofTurbulent Shear Flows,'' Phys. Fluids,Vol. 12, p. 473. Piest, J. (1974)"Molecular Fluid Dynamics and Theory ofTurbulent Motion,''Physica, Vol. 73, pp. 474-494. Poincar;, H. (1912)"Sur lathgorie des quanta,''J. de Physique, VOl. 2, p. 5. .. Prandtl, L. and K. Wieghardt(1945) "Uber ein Formelsystem fir die ausgebildeteTurbu lenz,"Nachr. Akad. Wiss. Gijttingen (Math.Phys. K1 .), Vol. IIA p. 6. Ross, D. W. (1969) "Quantum-Mechan icalInterpretation of Plasma Turbulence," Phys. Fluids,Vol. 12, No. 3, pp.613-626. Saffman, P. G. (1970) "A Model for Inhomogeneous TurbulentFlow,'' Proc. Roy. SOC. Lond., A. Vol . 317, pp. 417-433. Santos, E. (1974) "QuantumlikeFormulation of Stochastic Problems," J. Math.Phys., Vol. 15, No. 11, pp. 1954-1962. Spalding, D. 8. (1972)"Mathematical Models of FreeTurbulent Flows," lnstitutoNazionale di Alta Matematica Symposia Mathematica,Vol. IX, pp. 391-416. - (1974) "TurbulenceModelling: Solved and Unsolved Problems,''Proc. of aMeeting on TurbulentMixing in Non-Reactiveand Reactive Flows, PROJECT SQUID, Purdue University,Lafayette, Ind., May 20-21, 1974, pp. 85- 115, Plenum Press, N.Y., 1975. Strauss, M. (1972) "Modern Physics and Its Philosophy," D. ReidelPubl. Co., Dordrecht, Hol land. Synge, J . L. (1 954) "Geometrical Mechanics and de Broglie Waves," Cambridge UniversityPress. Tolstoy, I. (1973) "Wave Propagation," McGraw-Hill Book CO., Inc., New York, N.Y. 111 1111 Vedenov, A. A. (1 968) "Theory of Turbulent Plasma ," trans . by S . Chomet , American ElsevierPubl. Co., Inc.,'New York, N.Y. Whitham, G. 6. (1965) "A GeneralApproach toLinear and Non-linear Dispersive Waves Usinga Lagrangian,'' J. Fluid Mech., Vol. 22, Part 2, pp. 273-283. Wilhelm, H. E. (1970a)"Hydrodynamic Model of Quantum Mechanics," Phys. Rev. D, Vol. 1 , No. 8, pp.2278-2285. - (1970b) "Formulation of Uncertaintythe Principle Accordingto the Hydrodynamic Model of Quantum Mechanics,"Progr. Theor. Phys., Vol. 43, No. 4, pp. 861 -869. Wyld, H. W. Jr. (1961) "Formulation of theTheory of Turbulence in an IncompressibleFluid," Annals of Physics,Vol. 14, pp. 143-165. 112 NASA-Langley, 1976 CR-2671