Boundary Lyer Theory

734
XXlV. Prrc b11r1111lcnt flows: jot8 RIIC~ wnkcs
Inserting eqn. (24.12) for t.lio rate of increase in width, wc obtain
or
h N (p x cell d)1/2 (two-tlimonsiond wake) . (24.15)
Inserting this vr~lur into cqn. (24.14) we fintl that the rate at which the 'depression'
in the vclocity curve tlccrcascs downstream is rcprcscntcd by
(two-dimensional wake) .
Tn other words, t,lie width of a two-dimrnsional wake increases as ii and thevelocity
tlccreases as 1 /fi .
Circular wakc : Dcnot.ing tlir frontal arca of thc body by A we can write
its drag as D = 4 c, A e M,Z and the momentum, eqn. (24.13), becomes
J - e U, u, h2. ICqunting 11 and J, wc ohtain
Inserting this v:rluc into rqn. (24.12), wr fi~d Illat thc increase in width is given by
or
d 1)
6' ,iz - P CII A
h N (p c, A x)1ln (circular wake) . (24.18)
Tnscrting eqn. (24.18) into (24.17) we fir111 for Lhe clccreasc in the ilcpression in the
vcloc.ity profile tho rxprossion
c,# d
(circular wake) .
- ( )
'I'nblc 24.1. I'owrr Itiws for tho inrrcnso in witl1.h nntl for tho docrcnm in tho ccntm-lino vclocity
in terms of distance z for problclns of free turbulent flow
Fro jet boundary
Two-dimenuionel jet
Circnler jot
'I'wo-dintonsionnl wake
Circulnr wake
width
b
laminar
--
:ent-rc-line velocity
or Ul
Z0 '
1
Z-IIJ i
turbulent
:entrc-line velocity
'J,",,: or 'JI
, .Lhus, . for a circular wakc we find that, Lho wirlLIi of Llic w:dtc incrc~nsos in ~rroport.ion
to x'I3 arid that the velocity decreascs in proportion to x-~IR.
The power-laws for thc width and for the vo1oi:ity in ftllo centre 1i:~vo 111:ori
summarizctl in 'l'able 24.1. Tho corresponding laminar eases which wtm partly
considered in Chaps. IX and XI have been added for completeness.
c. Examples
Tlie prcccding c:stiniatcw givc in Ll~crnsclvcs n vory good i(lo:~ of t,llo OHSI:II~.~:L~
features cncountcred in problems involving free turhulcnt flows. We shall, howover,
now go one stcp f~~rthcr arid shall exaniinc scvcral pRrticl~lar c:ases in muc:li grcatm
detail deducing the complete velocity tlistributiori function from the ccl~~:~tions
of motion. In order to achieve this result it is necessary to draw on ono of tho hypotheses
in eqns. (24.3) to (24.5). The examplcs which hnvc bccn sclcatd Iicro for
consicler:~tion all have tlic common fc:hire that tho velocily profiles wliicli owur
in thcm are aim.ilnr to each othr. 't'liis means that thc velocity profilcs at tlifi'~:rcnt~
distances x can IN made congn~cnt by n, suitsblc choicc of a vclocity and :r width
scalc fnctor.
1. The urnoothing out of a velocity discontineity. As our first cxarnlrlr wc- s1i:~ll
consider tlic problem of the smoothing out of a velocity tliscontinuil.y wltit:h was
first treated by 1,. Prandtl [27]. At time 1 = 0 thcre are two strcams moving at
two different velocities, U1 and U2 respectively, their boundary bcing at y == 0
(Fig. 24.2). As already mentioned, tlic bonndary ncross wliic4i the vcloc4t.y v:trit-s
discontinuously is unstable and the process of turbulent mixing sinoothcu out. the
transition so that it becomes continuous. The width of the zone ovcr which this
continuous transition from velocity U1 to velocity 1J2 takes place incrcnscs with
incrca9ing time. We are hcre concerned with a problem in non-steady parnllcl llow
for which
u = u(?y,l) ; v = 0. (24.20)
Thc convectivc terms in eqn. (24.1) vanish idcntically. Making use of I'randtl's
mixing theory, eqn. (24.3), we can transform eqn. (24.1) In give
Fig. 24.2. The amoothing out of n velocity
discontinuity, after Prandtl [27]; a) Initial
pattern (t =O), b) Pattern at later instant a)