Boundary Lyer Theory

l!Ni IX. Ihnrt solutions of thc .stt*n(ly-stntr 1)onndnry-layer equntioll~ j. 15oundnry layer of second orclcr 1!)7
Pnrnl~nln in n ~ymmetric strclnm: 'r'h sccond-o~dcr l)out~tlary layer on a parabola
in n ~yrnrnrl.rit: st.rrn.111 was rnlcnlnt,cstl I)y M. Van I)yltc (see c~luo ('lhal). VII, 171). 111
t.110 ~~cigl~l)o~~~.l~ootl of st,n.gn:~t,ion, wr II:LVC
111 t.hc rnsr of the pnrnl~ola wc haw nt onr tlisposid a r1111ncricn.l solut,ion of t,hc co111pbt
r Nn.vit:r-Stokrs cc]llnt,ions drrc to It. '1'. I):Lv~s I I I I nnd c:Ln use it. for a tlirc!ck
cval~lnl ion of thr irnpr.ovc:n~rnt mntlr by t.hc sccontl-order t,l~oory. Pignre 0.17 sl~ows
a plot. of the skin-fricl.ior1 cocfficicnt from (9.!)7) at a st,ngnat,ion point, of a parabola.
in t,rrrns of 1.11~ Reynolds nurnhcr forlnrtl wit.11 the radius of cnrvature nt, thc vertex.
It follows I'ron~ cqn. .(9.!)7) Lhat
Cnrvc 2 in Fig. 9.17 is a plot, of this relation, wl~crcas Curve 1 dcpictss the first-order
solut,ion. Curve 3 hi14 IJCCII plo1,tctl with t.hc rcsult.~ of It. 'I?. I)nvis's n~~mcrical solution.
'l'11r ronsiclcrablc i~nlwovcrncnt~ cflkct~ctl by the sccontl-order tl~cory in the lower
rango of Jtcynolds n~lmlms is clcnrly visible. In wtltlition, t,he dingrams give an unslnl)ignous
intlicnt,ion that t,he sccontl-ordrr t.htory allows us to itlent.if.v the range
of vnlitlity of first,-order lhcory. Jf an c~~or of up to 2% is to be tolcmtcd, it follows
tl~at first-ortlcr thcory applies at J