Boundary Lyer Theory

742
XXIV. Frcc t,url)rllcnt flown; jctn atid wnkcn
bllz = 0.441 h, we have 0.44 1 ]/lii /I == : ant1 thus
1 0 = -- = 0.18.
b
The precctling so111th1 const,it~~tcs an approximni.ion for large tli~t~ances X;
rncnsnrcmont~ ~llnw t,llaf. it, is valid for z/c, d > GO. 111 the casc of srnnllor distanccs
it is possible to calculate additionid farms for tho velocity, t,hc terms bcing proportional
to %--' and x-~/~, resprctivcly.
Shearing stress hypo thesin from eqn. (24.5): From eqns. (24.1) and (24.5) we
now obtain
i)ll
ax
a2UI
ayZ
(24.38)
The virtrtal lzirwmatic visrosity is here E,= k1 ul,,, h and, hcnec, constrant nnct
equal to E,, say. Consequently, the tlifferential~equation for ul is identical with that
for a laminar wake, eqn. (9.30), except that thc laminar kinematic viscosity v must
be replaced by F,. Thus we can simply copy the solution which was found in Chap.
IX. Denoting r] = y i- , wo obhin from cqns. (9.31) and (9.34) t.hat
so that finally
The valllc of half tJlc wit1t.h at ldf tho depth is I),/, -.. 1.075 I/F,,//I., c,, rl (.r c,, rl)'ly.
Comparing witall tho prccctline; mcnsurctl vah~e of bIl, it is Ii)ul~(l t.lt:~t, fhc vml)irit::tl
qunntit,y 6, has t.hc vnlne
Eo . .. = 0.0222 .
ar D
The preceding solution shows that the vc1ocit.y distribnt.ion in t,llc wnko c:w be rcprcscnt.ct1
by Gausrt's function. The allarnativc sohrt.ion from cqn. (24.3!)) is scott plol.t.cd
in Ipig. 24.5 ns curve (2). 'rhc tlifkrcncc bctwocn this rtolnI.iol~ it~tfl 1.11:1t in rqn.
(24.37) is vory sln:dl.
\V. Tollmien [53] solvcci the same problcm on Chc lxtsis of voll Ii:irm;in's
tlypothcsis from eqn. (lD.l!)). Tn tho nrigllbourhootl of t,llc point,s of inflrsion in tho
velocity profile, wl~crc Pir/ay2 = 0, it, Itas provctl nc:t!c:ss:try to in:lkc: :~~l~lil,iol~i~l
assumptions. Extcnsivc cxpcrimcnts, which wrrc carried olll, by A. A. 'I'ownscwl 1.541
in tho wako of a cylinder antl which wcrc concerned with t~~rlnllcnt fI~~~t~~:~t,ion
at Reynolds numbers near 8000, showed that at a distancc equal to al,out 160 1.0 180
tliamcters thc trlrl~cllent microstructrlrc is not ynt ftrlly tlevclopetl. I~urt.hcrn~on?,
osoillngrams taken in t,hc strcanl dcmonsLmtc that the flow is f111ly t ~~rl~~~lnnt~ o111.y
aro~~nd the ccnt.re, nnd ~I~~rtuntcs br.twcen laminar and tliri)ulent, in the ~lrighhlrhood,
of the outer boundaries of the wake. Mcnsc~remenk on circ~tlar cylinders at, very
large Reynolds numbers wcrc dcscribcd in Chap. I1 ; cf. 11. I'fcil 126b 1.
Circular 11111lkecn have Ixcn invcst.ignf.ed by Miss I,. M. Swain [41] who Oiwt~I
the calrr~lation on the hypothesis in eqn. (24.3). She obtained thr same rxprrssion
for vclocity as in thc two-din~ensional casc, cqn. (24.37), but thc powcr laws for tllc
width antl for thc ccnt,rc-line vclocity wcrc found to be tlilTercnt, namely b - XI/:'
and ulm,, N X - ~ / ~ as , already shown in Tablc 24.1.
Until recently, it has bccn ncccptctl tht t.hc valociby disLribuLion in :I W:I~C
becomes indcpentlent of the shape of the body far cno11g11 bchind it,, antl is thrrcfore
of a universal form. This belief was put. t,o ~ IIC t.est in a scrirs of ~~p(~ritnrr11,rt
performed by If. Iteichardt ant1 It. J3rmshaus [31] and re1:itctl to wakcs bchintl bodics
of revolnt,ion. Tt turned out that in cach individual casc the vclocity profiles rcmnin
similar at varying distanccs behind the body. Ne~crt~hcless, the profiles behind bluR
bodies (plates, cones with a ratio diamcter/height = 1) tcnd to be fullcr than t,hot;o
behind lender ones (for examplc a cone with a ratio tlian~cter/hcight = 114 to 116).
IJiffcrenccs of t.his kind have not bccn observed in two-dinrcnsional wakcs.