Discrete Dynamics in Nature and Society, Vol. 1, pp. 99-110Reprints available directly from the publisherPhotocopying permitted by license only

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Universality of Oscillation Theory Laws. Types andRole of Mathematical Models

POLINA S. LANDA

Department of Physics, Moscow State University, Moscow 119899, Russia

(Received 6 December 1996)

The universality of oscillation theory laws is discussed. It is suggested that all models ofconcrete systems be separated into four categories: models ‘‘portraits" of investigatedsystems, models of the type of "black box", aggregating models, and models of certainphenomena which can occur in real systems. As an example of the model of the fourthtype, the equation of oscillations of a pendulum with a randomly vibrating suspensionaxis is considered for the purpose of clarifying the mechanism of the transition toturbulence.

Keywords." Oscillations, Turbulence, Noise-induced phase transitions

INTRODUCTION

The theory of oscillations is the science that stu-dies oscillatory motions irrespective of their phy-sical nature. By oscillatory motions are meantany limited changes of body state taking place ina long time interval. Then, because these changesare limited, they must necessarily be "hither andthither" [1]. Such a definition of the theory ofoscillations is very common. We know that othersciences study changes of state of bodies too.How does the theory of oscillations differ fromthem? An answer to this question has been givenby Mandelshtam [1]. Contrary to other sciencesfor which the prime interest is in what happensto a body at a given instant, the theory of oscilla-

99

tions concerns "the general character of a processtaken as a whole over a long interval of time".

THE UNIVERSALITY OF OSCILLATIONTHEORY LAWS

In common with any other science the theory ofoscillations has laws of its own. It is essentiallythat these laws are general and not depending onthe concrete type of the system studied. Unfortu-nately, these general laws up till now are notwell-formulated. However, many of the specia-lists in the oscillation theory use, sometimes in-tuitively, these laws in their studies. By using theknowledge of the general laws one can profitably

100 P.S. LANDA

predict different phenomena from diversified areasof science. The discovery of light combinationscattering effect by Mandelshtam [2] is a typicalexample of such a prediction. As for the analogybetween light combination scattering and theusual objects of oscillation theory, Mandelshtamwrote [3], "From the point of view of the theoryof oscillations, wireless telephony and light com-bination scattering are the same. It is modula-tion. Sound--in radio, and atomic oscillations--incombination scattering". Thus, one of the generallaws of the oscillation theory, which implies thatin any nonlinear system modulation is possible,was the basis for Mandelshtam’s prediction. An-other example of the general laws of the oscilla-tion theory is the possibility of synchronizationof any self-oscillatory system by means of thecorresponding external action with different me-chanisms of the synchronization in the cases ofslight and strong actions [4-6]. As a further ex-ample we can point to the possible routes for theappearance of chaos and stochasticity in any self-oscillatory systems [7].The knowledge of the general laws of the oscil-

lation theory is also important in that for studyof any oscillatory phenomena one can use simplemodels and be sure that in concrete more compli-cated systems these phenomena will be also ofthe same character. This fact is used tacitly by allresearchers.

It is evident that the general laws of the oscil-lation theory must manifest themselves identicallyin systems of diversified physical nature. Regard-ing this, Mandelshtam told students in one of hislectures on the theory of oscillations, "All of youknow such systems as a pendulum and an oscilla-tory circuit, and also know that from the oscilla-tory point of view they are similar. Now all thisis trivial, but it is wonderful that this is trivial."These ideas have not yet become fashionable. Inthe paper "L.I. Mandelshtam and the theoryof non-linear oscillations" Andronov [8] wroteof the lectures and seminars of Mandelshtam,

"Lectures and seminars of Mandelshtam havesometimes contained new scientific results whichwere not published. But, perhaps the greatest sig-nificance of these lectures lies in the methodicalinculcating of habits of oscillatory thinking, in thegeneral rise of the oscillatory culture". Unfortu-nately, many even prominent scientists, studyingconcrete problems, are still lacking in "oscillatoryculture". This is the main cause of the fact thatup to the present the scientific works, which areabsolutely erroneous from the point of view ofthe general laws of the oscillation theory, occurfrom time to time.

There is much speculation that the generalityof the laws of the oscillation theory is based onthe similarity of the equations describing oscilla-tory processes in different systems. Every so oftenthis is so indeed. However, the laws mentionedabove (and many other ones) are independent ofthe concrete form of the equations. It is onlysufficient that the equations belong to a certainclass, which in common is moderately broad.

TYPES OF MATHEMATICAL MODELSAND THEIR ROLE

The availability of analogies between oscillatoryprocesses in systems of diversified physical natureis why the theory of oscillations got its subject ofinvestigation, and thereby took the status of anoriginal science. A dynamical system is such asubject [9,10]. A dynamical system is a systemwhose behavior is predetermined by a set of rules(algorithm). In particular, and most frequently,the behavior of a dynamical system is describedby differential, integral or finite-difference equa-tions. Obviously, a dynamical system is a modelof a real system. So we can say that the theoryof oscillations studies abstract models, but notconcrete systems. These models can be conven-tionally separated into four categories: models"portraits" of investigated systems, models of the

In western literature this effect is usually called the Raman effect.

UNIVERSALITY OF OSCILLATION THEORY LAWS 101

type of "black box", aggregating models, andmodels of certain phenomena which can occur inreal systems.The models of the first type are constructed on

the basis of the description, as more detailed aspossible, of all elements of the system studied.Such models, e.g., are studied in physics. An ex-

ample is the Navier-Stokes equations and theirfinite-difference approximations describing fluidmotion with a high degree of accuracy. As an-other example we can point to certain models ofheart activity or breathing [11]. These models, asa rule, are very complicated and cumbersome fordetailed analysis and revealing principal featuresof system’s behavior.When the models of the second type are con-

structed, the system studied is considered as acertain "black box" with given inputs and the out-puts which are able to be measured. Further, amodel, as simpler as possible, is chosen. The freeparameters of this model are determined from theconditions of the minimum, according to a givencriterion, of the difference between the model andthe system outputs for identical inputs. A numberof examples of such models for chemical reactionsis given in [12]. The same approach is concep-tually used for reconstruction a model from ex-

perimental data (see, for example, [13]).The models of the third type are constructed

on the basis of the analysis of aggregate behaviorof individual elements of the system studied. Aclassical example of the models of this type isthe model "prey-predator" by Lotka-Volterra[14,15]. The other interesting examples are themodel of an immune reaction illustrating anoscillatory course of some chronic diseases [16,10]and the model of the economic progress of hu-man society [17,10].

Finally, the models of the fourth type are con-structed for analysis of a certain phenomenon nomatter in what system it occurs. For example,the phenomenon of self-excitation of self-oscillations can be modeled by the van der Polequation; the phenomenon of synchronization ofperiodic self-oscillations can be also modeled by

the van der Pol equation but with an externalforce; oscillatory character certain chemical reac-tions is demonstrated by the models "brussela-tor" and "oregonator"; the transition to chaosvia an infinite period-doubling bifurcation se-quence can be modeled by a logistic map [18];the transition to chaos via intermittency can bemodeled by a map of the form [7]

Xn+l Xn -[-ax,,

where z is an even number, a is the parameter;some processes in continuous media can be mod-eled by coupled map lattices [19]. Similar exam-ples can be given in abundance. It should beparticularly emphasized that the mere possibilityand utility of such models are based on the uni-versality of the oscillation theory laws.

THE TRANSITION TO TURBULENCEAND OSCILLATIONS OF A PENDULUMCAUSED BY RANDOM VIBRATION OFITS SUSPENSION AXIS

In this section we dwell more closely on .a modelof the transition to turbulence in the form of theequation of oscillations of a pendulum with arandomly vibrating suspension axis [20]. In [21,22,10] we hypothesized that turbulent processesin nonclosed fluid flows are not self-oscillationsbut are caused by fluctuations both internal (nat-ural) and external (technical). This hypothesis isbased on the fact that instability of nonclosedlaminar flows is convective. By this is meant thata small disturbance being given at some pointof the flow will not indefinitely increase withtime but will drift downstream. It follows from thisproperty of convectively unstable systems thatthey are not self-oscillatory systems, but onlyamplifiers of disturbances. To make these sys-tems self-oscillatory it is necessary to form feed-back. However, in nonclosed flows such feedbackis absent, i.e. for sufficiently large Reynolds’ num-bers such flows are amplifiers of fluctuations.

102 P.S. LANDA

Because the turbulent processes closely resem-ble self-oscillations in appearance, we furthersuggested that owing to the amplified fluctuationsa certain phase transition can occur in the fluidflow and, as a result, the flow transforms into aqualitatively new state associated with the excita-tion of noise-induced self-oscillations. It can bebelieved that the onset of turbulence, which ischaracterized by presence of large-scale, com-paratively regular, structures against a back-ground of fine-scale random motions, is just sucha transition. To make sure that such a transitionis possible in principle, we studied, both analyti-cally and numerically, a simple model in the formof the equation for oscillations of a pendulumwith a randomly vibrating suspension axis

[23,24].

3.1 Noise-Induced Oscillations of a Pendulumwith a Vibrating Suspension Axis

The equation studied is

b + 2/3(1 + og(2) +w(1 + (t))sin g) 0,

where is the pendulum’s angular deviationfrom the equilibrium position, 2/3(1 + Og2)( isthe value proportional to the moment of thefriction force which is assumed to be nonlinear,w0 is the natural frequency of small pendulum’soscillations, and (t) is the acceleration of thesuspension axis that is a comparatively wide-band random process with nonzero power spec-trum density n(w) at the frequency w= 2w0.

Setting T A cos(w0t + 4)) and solving Eq. (1) ap-proximately by the Krylov-Bogolyubov methodwe obtain the following truncated equations forthe amplitude A and phase 4) of the oscillations[23,24]:

(2)

where

is the parameter characterizing the extent towhich the noise intensity is in excess of its criticalvalue for which the phase transition arises,-7+aw02, 7 is the coefficient of the firstnonlinear term in the expansion of sin , K1(2w0)/2 is the parameter characterizing theintensity of suspension axis’ vibration, (t) is arandom process with zero mean value and theintensity K1, M ( cos2 ), and (2(t), much liketo (l(t), is a random process with zero mean valueand the intensity K2 (n(0) + K1)/4.The Fokker-Planck equation associated with

Eqs. (2) is

Ow(A, )Ot

(3 ) Ow(A,oc O)Wo

s TA2 M

Aw(A, c/)))K,w 02 Kzw OZw(A, 4))q80A2 (AZw(A’ 4))) + 2 0O 2 (3)

Finding from this equation the steady-stateprobability density for the amplitude A, we cancalculate the mean value of the steady-state am-plitude. It is

/wgK1 r(l + 1/2)(A)-- V3-- F07+l)r/

0

for > 0,

for r/< O.

It follows from this that for > 0 the parametricexcitation of pendulum’s oscillations occurs un-der the effect of noise. This manifests itself in thefact that the mean value of the amplitude be-comes different from zero.

Numerical simulation of Eq. (1) showed that,as one would expect from the theoretical results,

UNIVERSALITY OF OSCILLATION THEORY LAWS 103

when the intensity of the suspension axis vibration,characterized by the value of (2v0), is in excessof a certain critical value cr proportional to thefriction factor /3, the excitation of pendulum’soscillations occurs that makes itself evident in thefact that the variance of the pendulum’s angulardeviation becomes nonzero. This phenomenoncan be considered as a noise-induced phase tran-sition and the birth of an induced attractor. Thelatter follows from the fact that the correlationdimension of the attractor constructed from thedata obtained by the numerical simulation withusing Takens’ technique turns out to be finite[23,24]. However this attractor is very noisy. Thisis evidenced by the fact that the embedding di-mension calculated by both well-adapted basistechnique [25] and the method of Broomhead-King [26] is very large.Examples of the noise-induced oscillations and

the dependence of the variance of the pendulum’sangular deviation on the relative power spectrumdensity (20)/cr found by numerical simulationof Ec[. (1) are depicted in Fig. 1. It is seen fromthis figure that close to the excitation thresholdthe pendulum’s oscillations possess the propertyof a peculiar kind of intermittency,2 i.e. over pro-longed periods the pendulum oscillates in theimmediate vicinity of its equilibrium position (so-called "laminar" phases); these slight oscillationsalternate with short strong bursts ("turbulent"phases). Away from the threshold the durationof laminar phases decreases and of turbulent in-creases, and laminar phases ultimately disappear.The variance of the pendulum’s angular devia-tion increases in the process. We emphasize thatturbulence for transient Reynolds numbers exhi-bit also this property [29-31]. It is no chancethat the first theoretical works concerning theintermittency phenomenon were made by the spe-cialists in the field of turbulence [32].

Interestingly enough that in the case of thependulum the intermittency observed is different

,i

0.0E;

0.03

0.04

0.02

0.0t

0t.5; 2 2.S 3 3.; ,q .; S ..;

,()/()

(d)

FIGURE The dependencies q(t) and b(t) for 0v0=l,/3:0.1, c: 100 (a) (2)/cr: 1.01, (b) (2)/cr: 1.2, and (c)(2)/cr:5.6; (d) the dependence of 2 on (2)/cr (thestraight line 2 =0.01151 ((2)/r-1) is shown as a solidline).

from all types of the intermittency described in[27,7,28], although they are similar in their exter-nal manifestations. It is so called "on-off-inter-mittency".3 The term "on-off-intermittency" was

The intermittency is described, for example, in [27,7,28].Author is indebted to S.P. Kuznetsov and A. (enys for calling his attention to this fact.

104 P.S. LANDA

recently introduced by Platt et al. [33], though a

map associated with the similar type of intermit-tent behavior was first considered by Pikovsky[34]. It is essential that this type of intermittencycan occur not only in dynamical systems but instochastic systems as well [35]. In [35] the statisti-cal properties of on-off intermittency were foundfrom the analysis of the map

x,+ a(1 + z,)x, +f(x,),

where zn is either a certain deterministic chaoticprocess or a random process, a is the bifurcationparameter, f(xn) is a nonlinear function. It wasobtained that the mean duration of laminarphase has to be proportional to a-1.

Let us calculate the mean duration of laminarphase using Eqs. (2) and the Fokker-Planckequation (3) associated with them. We shall as-sume that the pendulum oscillates in laminarphase if the oscillation amplitude A is no morethan a certain value e. Then the mean durationof the laminar phase - will be determined by themean duration of random walk of the representa-tive point inside the circle of radius e on theplane o, b. This duration can be calculated (see[7]) using the steady-state solution of Eq. (3) withthe boundary condition

w (A, dP)IA= =0. (4)

Because the value of e is assumed to be small, wecan neglect the term (3/4)/3-A2 in Eq. (3). In so

doing the solution of Eq. (3) with the boundarycondition (4) is

8G0 (e I_2A 2_w (A, 4) co02K1 (1 2 r/)A

where Go is the value of the probability flow

4 rl Aw

across any circumference inside the circle of radiuse. The value Go is determined from the normal-

ization condition by integrating the expression (5)over the circle of radius e. Using this condition andtaking into account that - 2G [36] we obtain

16re(6)- co02K1 r/

It follows from this that for small r/ the meanduration of the laminar phase is inversely propor-tional to 7. This result agrees with [35].

If the intensity of the suspension axis randomvibration is under its threshold value then theexcitation of pendulum’s oscillations can be in-itiated by small additional low-frequency vibra-tion of the suspension axis. The inclusion of thisvibration can be carried out by substitution intoEq. (1) of + a cos COat in place of , where a

and COa are, respectively, the amplitude and fre-quency of the additional vibration of the suspen-sion axis. The results of numerical simulation ofEq. (1) for different values of a are representedin Fig. 2. We see that the excitation of oscilla-tions, as the amplitude a increases, is of a thresh-old character. For COa 0.318, (2CO0)/t%r --0.51the threshold value of a is equal to 1.1. Thedependence of the pendulum’s angular deviationmean square on the difference between the ampli-tude a and its threshold value is found to beclose to linear (see Fig. 3(a)). For a>ar theoscillations excited are not distinguished fromthose excited due to random vibration only. Theoscillation intensity is greater, the larger the a.

By this is meant that the low-frequency vibrationinitiates the noise-induced phase transition andthe birth of the induced attractor. In the casewhen the intensity of the suspension axis randomvibration is in excess of its threshold value addi-tional low-frequency vibration contributes signifi-cantly to the intensification of noise-inducedoscillations. The dependence of 2 1/2

on a fort(2)/tcr(2) 2.23, COa 1.5 is given in Fig. 3(b).

Let us consider now a possibility of suppres-sing noise-induced pendulum’s oscillations byhigh-frequency harmonic action. Numerical simu-lation of Eq. (1) with ( + a cos COat in place of (,

UNIVERSALITY OF OSCILLATION THEORY LAWS 105

FIGURE 2 The dependencies o(t) and qb(t) for wo=l,/3:0.1, c= 100, (2o0)/cr=0.51, w,:0.318 (a) a= 1.12, (b)a-- 1.2, and (c) a-- 1.5.

where COa>20, shows that such suppressiondoes occur. The results of the simulation for suf-ficiently large value of 03a are represented inFig. 4. It is seen from this figure that, for smallamplitudes of the high-frequency action, the ac-tion has little or no effect on the noise-inducedoscillations existing (see Fig. 4(a). As the ampli-tude increases the intensity of the noise-inducedoscillations decreases rapidly and the duration of"laminar" phases increases in the process (seeFig. 4(b-d). When the amplitude is in excess ofa certain critical value (for the case considered itis equal to 42) the oscillations are suppressedentirely. As the amplitude increases further theoscillations arise again, but because the condi-tions of the corresponding parametric resonancecome into play. If the frequency of the additionalaction is not-too-high then the perfect suppres-sion does not occur: as the action amplitudeincreases the intensity of the noise-induced oscil-lations first decreases to a certain minimal value,which is smaller the greater the action frequency,and then increases. True, this minimal value isattained for the larger amplitudes of the action,the higher its frequency. The aforesaid is illu-strated in Fig. 5.We emphasize that the initiation and suppres-

sion ofnoise-induced pendulum’s oscillations occurby means not only of additional parametric ac-tion, but of additional additive action as well.

0.4

I .t t .2 1.3 t .4 I .5 .6 t .7 t .8 t .8 2

(a) (b)

FIGURE 3 The dependencies of o21/e on a for o0--1, /3--0.1, --100 (a) (2w0)/cr:0.51, 0Va’--0.318 (the straight lineqoI/2 0.48 (a 1.1) is shown as a solid line), (b) #(2w0)/t%r 2.23, o,= 1.5.

106 P.S. LANDA

b)

FIGURE 4 The dependencies of :(t) and qb(t) for v0--1,/3--0.1, c---100, (200)/cr---5.6, wa= 19.757 (a) a=5, (b)a 15, (c) a 30, and (d) a--40.

3.2 The Comparison of the Results of the Studyof Pendulum’s Oscillations with a Number ofNumerical and Real Experiments ConcerningTurbulence in Pipes and Free Jets

A number of numerical and real experimentsconcerning turbulence shows that there exist theprofound parallels between turbulent processesand noise-induced pendulum’s oscillations. Theseparallels reinforce our hypothesis about fluctua-tional origin of turbulence.One of the indications that turbulence is not a

self-oscillatory process is a numerical experimentperformed by Nikitin [37] which relate to thesimulation of turbulent flow in pipes of a finitelength. He studied fluid flow in a circular pipe ofradius R with a given velocity at the input cross-section and with the so-called soft boundary con-ditions at the output cross-section; these latterare

0 2U Oq 2 Oq 2 g]

OX 2 OX 2 OX 2

where u is the longitudinal component of the flowvelocity, and /are, respectively, the radial andazimuth components of the vorticity f= rot V,V--{u, v, w} is the vector of the flow velocity incylindrical coordinates x, r, 0. The components ofthe flow velocity at the pipe input cross-sectionwere assigned in the form

( r2)u uo + A Re(u’(r)e-i"’t)cos O,

v- A re(v’(r)e-it)cos O,

w A Re(w’(r)e-i"t)sin O,

where u’(r), v’(r) and w’(r) are the eigenfunctionsfor the Orr-Zommerfeld equation associated withthe given value of the disturbance frequency v, R isthe pipe radius, A is the disturbance amplitude.The disturbance frequency v was chosen equalto 0.36uo/R, and the velocity u0 and the piperadius R corresponded to the Reynolds numberequal to 4000. At initial instant the Poiseuille

UNIVERSALITY OF OSCILLATION THEORY LAWS 107

8.4

8.3

L8.28.1

81.8 18 18

(a) s.4

I18e 18x 18

8.4

8.3

L8.28.1

818e

i"’! ’1

18j-

FIGURE 5 The dependencies ofand (d) Oa= 19.75.

8.4

8.3

on a for w0=l, /3=0.1, c= 100, (2)/cr(2)=5.6 (a) a=3.5, (b) wa=6, (c) o= 11,

velocity profile was given for whole flow., i.e.

v, _{t=0 b/0

When the disturbance amplitude A stands outabove a certain critical value, after a short time,random high-frequency pulsations appear in thelower part of the pipe from a certain value x x0onward. The value x0 depends only slightly onthe distance from the pipe axis r, and it is thesmaller, the greater is the disturbance amplitudeA. The occurrence of the turbulent pulsations isaccompanied by considerable change of the pro-file of the mean flow velocity’s longitudinal com-

ponent: it decreases at the pipe axis and increasesnearby the pipe wall. The instantaneous distribu-tions of the velocity’s longitudinal component insteady regime are shown in Fig. 6 for A/uo =0.04;this figure is taken from [37]. If one diminishesprogressively the amplitude of the periodic dis-turbance A then, from a certain its value onward,the turbulence region drifts downstream and theflow in the pipe becomes laminar. As is known

(see, for example, [38]), the Poiseuille flow in acircular pipe, as differentiated from the Poiseuilleflow in a plane channel, is stable with respect toinfinitesimal disturbances for any Reynolds num-bers. However, in the case of sufficiently largeReynolds numbers such flow is unstable with re-

spect to disturbances having a finite value. If anattractor associated with turbulence would existin the system as the disturbance is absent andunder the effect of the disturbance the phase tra-jectory would only come into the attractive do-main of this attractor, the turbulence should notdisappear when the disturbance resulting in it wastaken away. The fact that the turbulence disap-pears implies that such an attractor is absent.True, it is possible that an attractor comes intoexistence under the effect of an asynchronous dis-turbance in itself (see [39,40]). In this case it hasto disappear when the disturbance ceases. Againstthis possibility are the following facts. Firstly, asis known from the general theory of the asyn-chronous excitation of oscillations in systemswith hard excitation [39,40] such an attractor canappear only within a narrow range of system’s

108 P.S. LANDA

0.4

0.7

0.I0 20 40 60 80 x 100

FIGURE 6 Instantaneous distributions of the steady longitudinal velocity component for A/uo--0.04: (a) close to the pipe axis(for r/R =0.02) and (b) close to the pipe wall (for r/R =0.93).

parameters, whereas the transition to turbulencewas observed by Nikitin over a wide range ofReynolds’ numbers. Secondly, asynchronous exci-tation of self-oscillations, is equally possible bothfor positive and negative mistunings between thefrequencies of the disturbance and self-oscilla-tions, whereas the transition to turbulence wasobserved only for low disturbance’s frequencies.As is seen from the foregoing, such asymmetry isintrinsic just to noise-induced oscillations.

So, the fact that the turbulence disappears,when the disturbance is taken away, implies thatthe turbulent pulsations resulting from the distur-bance are more likely to be noise-induced oscilla-tions than self-oscillations. Therefore it can beconjectured that the turbulence development ob-served for A > Acr is a result of the noise-inducedphase transition bringing into existence an in-duced attractor. The similarity of the characteris-tics of turbulence arising in the pipe consideredabove to the characteristics of turbulence in apipe with periodic boundary conditions [31],when there is feedback and self-oscillations areexcited, counts in favor of this hypothesis. The

superficial similarity is illustrated by Fig. 7 con-structed from Nikitin’s data. If the conjecturemade is correct, the role of the periodic distur-bance for the development of turbulence reducesonly, on the one hand, to ensuring necessary am-plification of small fluctuations, which are alwayspresent even in numerical experiments, and, on

the second hand, to initiating the phase transi-tion, much as is demonstrated with the exampleof the pendulum’s oscillations.The parallels between turbulent processes and

noise-induced pendulum’s oscillations can be alsotraced by the example of the turbulence develop-ment and controlling this development in jetflows. It is known [41-43,22] that large-scale tur-bulence in subsonic jets arises at a certain dis-tance from the nozzle inside the boundary layerwhose thickness increases almost linearly withthis distance. The onset of the turbulence is ac-companied by change of the mean flow velocity.It is also known [44,45,22] that acoustic actionupon a jet in the region of its outflow from thenozzle, depending on the frequency, can eitherinitiate the turbulence development or suppress

UNIVERSALITY OF OSCILLATION THEORY LAWS 109

0.2ca)

0.2b)

0 1 0.1

:::s 8 :::s

-O I -o. 1

9 109 299 399 499 O 109 200 309 409

FIGURE 7 The form of turbulent pulsations in a pipe (a) with periodic boundary conditions and (b) with given harmonicdisturbance at its inlet.

0.660 80 100 120

FIGURE 8 The experimental dependence of e, on the rela-tive amplitude of acoustic pressure /?a measured in decibelsfor St 2.35, x/D 8.

it. The initiation occurs at low frequencies of theaction, whereas the suppression occurs at highfrequencies of the action. Such phenomena wereobserved experimentally by various investigators.The experimental dependence of the relativeroot-mean-square pulsation of the longitudinalcomponent of hydrodynamical velocity eu on theacoustic pressure for St 2.35 is shown in Fig. 8taken from [46]. Here St fD/Uo is the Strou-chal number, fa is the frequency of the acousticaction, D is the nozzle diameter, U0 is the meanflow velocity at the jet axis. We see that theturbulent pulsations first decrease as the ampli-tude of acoustic wave increases and then theyincrease. This experimental dependence closelyresembles one of the dependencies of 21/2 on a

shown in Fig. 5 for a pendulum with an addi-

tional high-frequency vibration of its suspensionaxis when the frequency of the action is not too

high.

CONCLUSIONS

The universality of the oscillation theory lawsmakes it possible to use for the study of compli-cated concrete systems not only models "por-traits" but far more simple models as well. Anexample of such a simple model for the elucida-tion of the character of the turbulent develop-ment is considered. It turned out that the excita-tion of noise-induced oscillations of a pendulumwith a randomly vibrating suspension axis andthe appearance of turbulent flow in pipes of a

finite length and in free subsonic jets have manycommon features, in spite of perfect distinctionbetween the equations describing these processes.

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110 P.S. LANDA

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