Statistical Theories of Turbulence

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•

C. C. LIN

Statistical Theories of Turbulence

rRl1'\.Fl01' 11 r. \\,) l ll\R.\R)

Nll~IBEH

PRINCETON AERONAUTICAL PAPERBACKS

10

PHI'.'\C El"O N :\ERONA UTIC AL PAPERBAC KS

l. LIQUID PRO PELLANT ROCKETS Duvid Altman, Ja mes ~I . Garter, S. S. Pe nne r, l\1urtin Sum merfle ld ~Ligh Te mp•·,·ii .. n. It b recognized that. the details of turbulent flow are so complicated that statistical dcscripl ion must be used. Indeed, only statistical properties of turbuleut nlution arc experimentally reproducible. The purpose of the prc:>ent section is to give a more comprehensive treatment of the su1ti::;tical theory. Current literature on the statistical theory of turbul ence is mainly limiU!d to the treatment of the case of hQmogeneous turbulence' \vithout any es..~ntial mean motion. Su perficially, one might think that there is little to be known about such fluid motions. Actually , the very absence of mean motion allows one to go more deeply into th e inherent nature of the turbulent flow itself. .:'IIany basic concepts have been developed in the study of homogeneous turbulence, and these concepts now gradually 6.nd their way into the study of shear flow. Since there is available an account of the theory of hon1ogeneous turbuJence [JI with a complete di scussion of the rnath ematical background, a somewhat different presentation is adopted in the present ~ectio n . Following the historical order, the isotropic case is taken up first. It i~ hoped that this wi.l l be heI pf ul to th.•c r(;adcrs who wish to gel. 11n idea of the essentials without goin~ thriiugh all the prelirninaril's required in a complete mathematical tn::atrncnt. I o the later parts of this section, other aspects of the s tatisti cal theory aod their applications will be treated.' \Ve have, however, omitted several other approaches to the problem . is Taylor's vorticity scale defined by 1

>.' = -

(av) or' -·

(7-6)

The relation (Eq. 7-5) essentially gives the rate of decrease of kinetic energy. I t was first established by Taylor [15), both theoretically and experimentally. Tbe equations correiiponding to tbe bigber powers of r ";u be discussed in cCJnnection with tbe small scale structure of turbulence (Art. 13).

C,8. The Spectral Theory of I Hotropic Tu rbul ence. The early adoption of statistical correlations for tbe desc ription of isotropic turbulence is at least partly due to tbc fact that they are relatively easy to measure. Another powerful method for dcRcribing a fluctuating field is to analyze it into Fourier components, i.e. to adopt the spectral approach. It is well known that tbc spectral theory and the correlation theory are

( 17 )

C,8 · SPECTRAL TliEORY OF ISOTROPIC TURBULENCE

C · ST:I TISTIC.iL THEORIES OF TURBULENCE

intimately connected with each other by simple mathematical transformations. Physically speaking, however, the two methods of description put. different cmpha11is on tho different. aspects of the same phenomena. The spectral theory is often found to give a clearer description of the basic mechanism of turbulence. Spectral analysis has long been used for the study of electromagnetic waves, such as the radiati on of heat and light. It \Vas first introduced into the study of turbulence by 1'aylor (19(. Taylor made spectrum measurements, behind a grid in a wind tunnel, 'lf the velocity fluctuation as registered by a hot wire fixed in the \\•ind tunnel. This is a fluctuation in time. But Taylor assumed' that "the sequence of changes in u at the fixed point are simply due to the passage of an unchanging pattern of turbulent motion over the point." T he variation is then essentially the same as that in space, and the spectrum he observed corresponds to a onedimensional Fourier analysis of the field of turbulence in the direction of the wind. 'fhe field of turbulence in the wind tunnel is obviously not homogeneous in the direction of the wind. However, in developing the theory, we shall consider a homogeneous field and its Fourier analysis. In isotropic turbulence, the analy~is would he the same in all directions, provided we are always dealing with the component of velocity in the direction chosen for the analysi~. The transverse component in general has a different spectrum whether the turbulence is isotropic or not. In the case of turbul ent motion, we may formulate the Fourier transform relations between the power spectrun\ and the correlation function as follo,vs. If ~F1 (1-....

0

Ol

Q) c 6

xo

........u0

a.

lb 4

~

Vl

x

2

XO b

0

x (x

k

x x

0

100

200

300

(

400

Frequency, hertz Fig. C,8. Experimental veri6ce.tion of the Fourier t ra nsform relation betwee n spe.ce eorrelat1on and ti me spectrum £or turbule nt Ouctuo.tions behind a grid in a. wind tunnel (after Stewart and T ownsend (el?IJ.

-

• lao -at d• aF

= -2•

1a·o •'Fd.

(8-11 )

Exactly as in the case of the correlation theory, one cannot proceed much further with the basic equation (Eq. 8-7) wit hout a more specific knowledge of IV. However, v.·ith the physical interpretation that IV (•, t ) represents the t ransfer of energy among various frequencies, it has been found possible to obtain certain plausible formulas connecting rV(•, t) with F( «, t) and to make reasonable deductions. (Cf. Art. 17.) 17 C,9. S pectral Anal ysis i n One Dimen sion. We shall now develop briefly the one-dimensional spectral analysis of a field of turbulence and derive the Fourier tra nsform relations (Eq. 8-1). I n a ho77!-0gtneous (not necessarily isotropic) field of turbulence, let u (x ) be the velocity at the point x in the direction of the x axis. I t remains finite as z - ± oo. This makes its Fourier analysis more difficult than that of a function which vanishes rapidly at infinity. For such a function, (x), we have the pair of Fourier transform relations

,,

.

This leads to an equation of the form

~ at +iv=

(8-7)

In the above equation, 1-V(K, t) is con nected with the triple correlation function h(r, t) by the following relations:

(8-8)

where a ( - • ) is equal to the com plex conjugate a • (K) for real (x), and ja (•)J2 is a measure of the energy content associated with the wave number or spatial freq uency • · li owevcr, since the velocity fluctu ation u(z) in a homogeneous field of turbulence does not approach zero as x -+ ± oo, we cannot put u (z) in place of (x) in t he above relation. Instead we

( 20) (

21 )

C ·STATISTICAL THEORIES OF TURBULENCE

C,9 · SPECTR..t l, A N ALYSIS IN ONE DIMENSION

first consider a(,,, X) = :;-

fx

i>J1r

- }(

l

and

and then try to adopt a suitable limiting process as X--+ co. In fact, we want to consider first the amplitude not at " but associated with a finite range of values of"· \Ve integrate Eq. 9-1 between" and " + A1t, obtaining

'

1.·+A• ( X )d • a"•

AA(•, .X) =

IC

= ]_ 2.-

f

x u(x) e'"dx{e«A•l• -

-x

x

Here, we may t.9.ke the limit as X -+

oo,

li'R ( E) =

(9-1)

11,(x)e'"dx

1)

i} / _·. F 1(1(" ) · i.e. ·i(•s)d•1d•2d•1 is the energy contained in the range K,, "• + 1. 'f~ obtain the energy per unit mass !F1(•1) lying bet,veen • 1, "• + dK and assoc!ated .with one component of the motion, one must multiply this ex· press1on with the factor ( 1 - K?/ •' )/2 and then integrate for all values oJ 1

co nfildering all ","s " ; tb the same magni t ude "· \\'e have finally

It can be shown that 4>,1 re present!! the cnerp;y density in the wave number space. I n the case of isotropic turhulcnce, (11-3)

Because of the condition of vani11hing divergence of the correlation tensor, we obtain ( 11-4)

( zs )

C ·STATISTICAL THEORIES OF TURB ULENCE

C,12 · LARGE SCALE STRUCTURE OF TURBULENCE

and 4>;; can be expressed in the form 4>1; = \V(«,.)(Kt - K,K;)

+ x.(.:..)xj(iddi;».•, •I) begins with a value 10 and ends with a value 4. u The experimental agreeme~t io this CIUle should be ".':r-cp~d wi~h some re8ervation, since eo little data are available. See (30,SII for deLailed d1scuss1ons.

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C, 15 · THE PROCESS OF DECAY

C · ST.i TISTICAL THEORIES OF TURBULENCE

For large in itial Ilcynolds numbers of turbulence, von I-2 sh ows that the pressure terms P,. must be dropped when the nonlinear effect represented by T., is negligible in the spectral equation (Eq. 11-6), which becomes simply aF,, (Jt

et

-

2 ~~'F, ,

(15-5)

The general solution of this equation is F .,(•-, !) = F., (• • , lo)e- " •' (1- 1,1

(15-8)

From this, we may calculate the correlation tensor by a Fourier transformation. For large values oft - t 0 , only s mall values of Kare important. Thus one may try to expand F •• in powers of •~ and retain only the lowest terms. Following t his method , Batchelor and Proudman [St] found that the longitudinal correlat ion coeffi cie nt f (r, I) is of the form ( Eq. 14- 13) fo r isotropic turbulence and certain very special cases of anisotropic turbulence. P re,;ous to th.is investigation, Batchelor and Townsend [48] compared the e xperimental curve for f(r , l) with the Gaussian curve (Eq. 14-13) and found good agreeme nt. At that time, this agreement was explained by assuming F,,(.. , l ) to be essentially expandible as a Taylor series in ••. S ince this assumption is now found to be not t rue in general, other te ntative explanation8 are suggested by Batchelor and Proudman (SI] . A critical examinat ion of this problem is clearly warranted. Early period of decay. Much experimental information is availaLle during the early part of the decay proceSl!. Recently, Stewart and Townsend (22) summarized their res ults and compared them with fiome of the a bove self-preserving hypotheses. They cautioned agains t the a ssumption of complete self-preservation, but did not include ca se (c) in their discussion, which seems to fit all their experimental findings. In Fig. C,15b, the law of decay observed by Stewart and Townsend ($2) is presented. Although the variation of >.' and u-• both follow the linear law, as they would in the case of complete Aimilarity, the origin of time (or :z axis) must be taken differently for the two straight lines. I t

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C,15 · THE PROCESS OF DECAY

C · ST.iTTSTICAL THEORIES OF TURBULENCE

can easily be seen that the Reynolds number of turbulence R, steadily decreases in the case shown in the figure, contrary to the la\v of decay for coinplete si1nilarity. The earlier experiments of Batchelor and Townsend [SS] also show a definite trend for the decrease of R,. It should be noted that the curves for ~ 2 and 11:- 2 versus time both have fairly large slopes, and it is therefore more difficult to detect any slight deviation

:\ more definite verification of case (c) is provided by their measurement of the spectrum, which is reproduced in Fig. C,15c and C,15d. Fig. C .1 5c gi,·es the one-dimensional spectrum Fi(•) whi ch shows a large departure from !>intilarity at lo"' values of•· Fig. C, L5d shows that, for 200 ~~~-.-~~--..~~~~~~~

x/M

0.4

•

..... -i..=-

150

~

d)l2 JOv dx=u

N

E

' 60 5250 f) 80

~8}10500

60 30 21000

x, cm F ig. C, 15b.

Decay of turbulence behind grids of diff~ring

shapes. (After Stewart a nd Townsend 122).)

........

oi.=...~~~~~

from a straight line. On the other hand, R, should remain constant according to the assumption of complete similarity and is more sensitive for detecLing the departure from such la\VS. The comparison or the intercepts of the straight lines in Fig. C,15b is also a very sensitive method for detecting the same effect. On the bnsis of the discussions of case (c) of Art 14, it may be expected that R, should decrease linearly in t, if uh > 0, i.e. if the energy in the large eddies is smaller than that corresponding to full similarity. The observed decrease of R, indicates that this is indeed the case. ( 40}

0

0. 1

~~--'~~~....__~~-'-~~_,

0.2

0.3

0.4

0.5

0.6

K/K, Fig. C, 15d.

S~ctrum

of

iJu / iJI ffucluntions.

each experiment, all the points for •'F 1(otropie turbulence,'' and s;;~umed that the double correla tion ( E be obtaineta in in this 1nann