Boundary Lyer Theory

in cross-flow, tlrprntls only on lhn potat~l,i:d vclo~it~y tlistril)ution. IJ(z). By ront.r:~st,,
whcn an axially symmctrical I)oundary laycr is stutlictl, for rxamplc that on a
rotating 1)ody of rcvolntion, it is found that the contour r(a) of the body entcrs
explicitly into thc corresponding rquations. Tile prcsont scction is clcvotctl lo x
more tlotailctl invcst,ignt,io~~ inl,o thc rolntion 1)ctwc~cn two-tli~nr~~sit)t~:iI nlicl axi:illy
symmntric l~our~rlary Inycrs.
In st,~n.tly flow the Oountlary-layor rqr~:rt.ions for Lwo-tlitncnsiond flow :~ntl fnr
axially symmot~rical flow are given I)y cqns. (7. lo), (7.1 I) ant1 ( I 1.278, h), rospectivrly.
l'hc Int,tcr rcfcr to a curvilincnr systc~n of c:oortlin:~.l.c:s with z tlcnot,ing t,l~r cttrrrnl,
arc: Icr~gt~h nntl y tlrnot,ing {,l~o tlist,nncc from t,hr wall in :L tlirrt-tion normal t,u if..
The rcspcctivc vc1ocit.y components n.rc tlcnotwl I)y IL nntl v, and IJw mn.gnit,ntlcs
wit.11 a bar rcfrr t,o tho two-tlinicnsiond cnsr. Wit.11 these syml)ols, wo Itavo for
tho two-tlinirnsional msc:
for t.hc axially symn~ct~ricnl vnsr
Ilrre ~(z) dcnot.cs tJto (list,ancc of a point, on t11c wa.ll from 1.11~ axis of symmntq.
Thr first eqnnfions of l)ot,l~ systems nro iclrnt.icn1, the tliffrrrncr Ixing onl,y in t.llc
npprnrnncy of t,hr rntlirls ~(n.) in t.11~ rqun.l,ion of conI,innit,g.
It, sc~ms 1.1111s rcasnnnl)lc to inqnire wlteLllrr it, is possil)lo 1.0 intlic.at.c a transformal,ion
wl~icll woultl permitf t.hc nsn of t,l~n solt~t.ions of Lltc two-tlirnrnsionnl cnsc
1.0 tlrrivc solr~f,ions of t,l~c n.xinlly syn~rnrt,ricnl cnsr. Such n gvncml rc1at,ionsI1il)
bctwocn t,wo-dimcnsiond nntl n.xially symmctrical I~ounrlary laycrs Itas bccn cliscovcrctl
by ITT. RTn.nglrr [72]. It rr~lrlccs tho calcnlation of thc hminn.r 11ountla.ry Inyor
for am n.xially s.ymmct,ric.:~l botly t,o tl~nf, on a cylintlricnl I)otly. 'l'he givcn body of
rrvolut,ion is nssocin.18rtl wit.11 n.n itlcnl pot,cnt,inl vclocit.y dist,ril)ntion for n rylint1ric:ll
body, the f~lncl~ior~ Lcing rnsily calcnlatfctl from the conhur ant1 the potcnt,inl vcloci1.y
tlist.ril)tlt,int~ or t.ho botly of rovol~tkn. Mnnglrr's tfmnsl;mnation is also valid for
comprcssil)lt: Imtlntlnry In.ycrs, n.s well ns for tllcrmnl boundnry 1:iycrs in In.tnil~:ir
flow. Wr sh:1.11, I~owcvrr, consitlcr il, here only in rclat,ion to incomprrssi1)lo flow.
According to Manglrr, l.hc cqrin.t.ions whic:l~ t.m.nsform tJle coortlin:~.t.es ant1 111~
velocit.ics of t.hc xxinlly symmct.ricnl pro1)lcm to t,hosc of t.he eqiiivalcnt two-tlimcns-
ional problrm n.rc as follows:
Z
w11orr 1, (Icnotrs a const~arit Iw~gth. Itcn~cnibcring that
it, is rasy to verify tht thc syst,rm of c.q~t:itions (1 1.60) l,mnsforms intm oqns. (1 1 .4!))
by t,hn wc: of LIto sul~sl~itulions (Il .GI).
WIC hountla,ry layer on a 1)otly of revolution r(z) having tho itlml pol,rnt.ial
vclocit.y tlisI,ril~clf.ion IJ(z) nnn l)c cv:dt~al.c:tl by con~pnting t h t,\vo-(litllr~~sio~~:~I
I,ot~n(l:~,ry l:tycr for tt, vcloc:it,.y tli~l,rih~rI,ion o(:?), wltc:ro /J r-: ~ I NZ I ti~~l :I: tire rcl~iI3wl ,
I)y oqns. (1 I .GI). Il:~vi~tg c~alcrtli~t,rxl I.hc voloc.il.,y oornpot~ol~l.~ ii, nntl 6 for l.11~ l.wotlimctl~iottal
I)ortntl:~ry Inycr it is possible tlo tlctcrrnine tho con~poncntn I* nntl IT or
tho n.xinlly symmct.rical bountlnry laycr $1 ith tho nit1 of thc t,mnsforlnnt,ion rquations
(11.51).
Iloncc, from rqn. (11.51), we Itavo
l'hc pol.rnt.ial flow of tllc associatd two-tlimct~sional flow bccornrs
J - ---
U(2) = u, 113 L2 2,
so that 0 ( ~ = ) C 5' , wllcrc (: dcnotc~ a constant. 'rhc associat.~d two-tlitncnsio~la.l
flow bclongs to thc class of w~tlgc flows disoussotl in Sce. 1Xa ant1 is givcn
by I1 = C an', with m = + for the present example. l'rom cqn. (9.7) wc find the
wc~lgc nnglp P = 2 m/(m -1-1) = 4. Thc associntctl two-tlimcnsiond flow is t.ltat
past a wcdgc wit,h an anglc n P .= n/2. '1'11~ fact that nxinlly symmct.rical stagnn,l.ion
flow ran be rcduced to the case of flow past, a wcdgc whosc angle is n/2 wa.s st,nt,etl
in Scc. 1Xa and is now confirmed.
11. Three-clin~ensio~~nl Lountlnry lnyers
IJtttil now wc have restricted o~~rsrlvcs nln~ost cxc:l~lsivcly Lo the consitlcrat,ion
of two-tlimrnsionnI :mi axially sylnmnt.rical prol~lcrns. 1'rol)lcms of t.wo-tlinlct~siot~al
nncl of nsinlly syrntncI,ricnl flow havo this in common l.ltat t,ho prcscril)otl 1)oI,cnt.in.l
flow tlrprntls ot~ly on onr sp~cr: coortlil~:il.o, :l.ntl tho l,wo vc:loc:it,y con~l)c~~ottls ill Ih(:
I~o~tntlnry I:~ycr tlc:pcntl on t8wo space roordin:~tcs
sional 1)orrntl:iry lnycr thc cxhcrnal potcnt.ial llow clcpcntls on two coortlin:~.l.cs in
thc w:~ll srlrfaco and t.ltc llow willtin tllc Imuntlnry laycr posscsscs all tl~rcc vcloci1.y
componrnts which, moreover, tlcpcnd on all three spxco coort1in;~tcs in thc gcncr:~l
cmc. 'l'hc flow abont a disk rot,nl,ing in a fluid at rest (Scc. Vb) and rotntion in thc
nc~ig11l)ourllootl of a fixed wall (Scc. Xla) const,it3utr cxarnl)lrs of t,l~rcn-tlimcnsionnl
I)ol~ntl:~ry I:~yors, rrpnrt from Ijcing cxnol sol~~l~ions of tllc Nlbvior-Stokes cqttn.t,io~~s.
(\:LCII. [II I,IIc cnsc ol' n t ,I~t~(~(:-tli~~~t:t~--