Boundary Lyer Theory

568 XVIII. F~~ndamentnls of turbulent flow
,I
1 his is cxplnint\d 1)y the reqnir~mcnt~ of cont,inuity accortling to which, as we know,
t,l~e r:~,l,c. ol' Ilow through any cross-scction ronains constant in time. The integral of
1.11~ c:orrrl:~tion function It, that is, thc quai~t~ity
IS I.llc sc:c.ol~tl vrloc.ilty, 11~' in eqm (14.1:1). is n~wsurcd at. tht: same 1ora.t.ion
l)111 :II. a (~iIl'(w111, ~IIS~.:III~< of I,imn (71~' at, inst,ant tl and 11.; at instant t2 = tl -1- 0, we
obtain the so-called autocorrelalion funcliott. The provision of spacc-t,imc corrcl t .1011s, '
that is, of observations of two velocity components, oath measured at a tlilrrrrnt
location in space and at a different instant in tirnc, allows us to gain a gootl tlr:tl
of insight. As an example, we reproduce in Fig. 18.8 snch space-time corr~lnt~ions
Dist.anco from wall:
y/d = 0.24
T5orrrtdary-layer t.hickncss:
8 - 16.8 nini
Fig. 18.8. Space-time correlations of velocity flnctuatio~~s in Lhr turlnrlcnt Iwu~ltlary lnynr on
e flat plate, as measured by A. J. Favre, J. J . Caviglio nnd 11. .J. lhnns [tfi]
ol)t.nincd by A. J. Favre and his coworkers (161 in tl~c Lnrl~ulcnt bountl:~ry Inyrr on
a fl:l.t plata. 'l'l~e 1nrnpor:tl tlispl;~cctncl~~l,, t,,,, of t.hn m;tsimcr~~~ oI'(-:~(.ll (:Itrvv is inll)osc(I
by 1.h~ passage of turbulent ctlrlics; tho odtlics move wit,ll nn a~~~~rosit~~;~l~c:
vc*lotil.y
which is equal t,o 0.8 U,. 'l'11e tlccrcasc in the maxima is thc rcsult of ;L procrss wl~ich
can be visualized as follows: With the 1)assago of time, tho t,url)ulc~~l, ntltlirs losr t,l~cir
intlividnality t,hrougll mixing with the surroutlding I,urbnlc~~t, flt~icl. Concurrc:nt,ly,
ncw etltlics c:ont,inttously spring into being.
An albrnativc tlcscript,ion of the st,nicturc of turbulcricc is obt,:~inctl wltc~~ a
fr~q~~e,zcy analysis of the motion is provided instcad of a correlation fu~irt~ion. 'l'l~is
leads 11s t,o the concrpt of the speclr.w~rr of a t.rrrl)~~lcnl, st,rc:~ni. f,ct I, clcnot.r thc -- Srcquency
a.7~~tl F(n) (171, t.11~ fr:wtional con1,rnt of tlw root-t~~r:~~~-s(~~~ar(:
v:~lur, IL'~, of
t.11~ longil.utlinal Iluct,uat,ion wl~icl~ bclongs t.o 1 .l~ frcqucnvy ir~l~c~rvnl fro111 11, - t.o 11. I ,111.
'I'IIO fnnct,ion F(n), which rcprcsent,~ - t,l~c tlcllsi1,y of 1.11~ tlislxil)r~l.io~~ ol' IL'VII ?I, is
Icnown as t,l~c .spechd rlistrih~~tio,a of 71,'" l3y tlcfi~~itiorr, wr: must, I~avct