Boundary Lyer Theory

Icror~~ ()I(, phj.sic*:~l ~)oint. of vicw it ml~st I)c conclutlnd that 1.11~ mtio of t,hc height
(IS prot,t.~~sions t,o t,l~r I,o\~~~cl:~r,y-l:~yc~r t,l~ic*lz~~rss sholtltl I)(: the tlct.rrmining factor. In
l);~rli(~~~l:~r,
I,IIv ~~II(-IIOIII~~I~O~I is cxl)rcl.c(l 1.0 rlrl)n11(1 on 1,111: I,l~idrrl~w or t,l~r In.minar
s\~lr-l:lyc.r d,, so I.II:II. k/ii, ln~~st, I)c rrg;~rclvtl :IS an inlport,atlt tlirnonsionlcss ntltnbcr
\,:Ili(.l~ is (*l~:~r:~(*l,~~risli(; of l,hf: ki11(1 of' ro11g~111~:ss. If, is d~ar 12hn,k roltgh~lcss \trill ca11se
IIO i~l(-rt*:~sr in ~I&I:III(~ ill (::ISVS \vh(w: I.II(\ I)I.(~~IIS~OIIS arc: SO s111:~ll (or 1,111~ 1)011rl(l:~ry
I:~yrr is so II~ick) t.llnl, I,llc-y ;~rv all c-orlt;rit~c.~l wilhit~ 1.l1c I;lrr~ic~:rr scrl)l;~jw, i. c. if
k < (St, and t.11~ wnll may I)c cwwidcrrtl I~ytlr;ir~lic:~lly smoot,lt. This is simi1a.r to the
:I,I)s~~IIT or t,l~c ~ I I ~ I I I ~ of ~ I ( ro~~gltn(w
* ~ on rrsist:tncc in lln~g~~~~-l'oisc~~ill~: flow. I:I,sII~~:~)I(-II~,s 011 ro~tgl~ pip(-s 11:l.v~ IWCII
w1.1.ivg1 o111, 1)). .I. Nilr\~r:t~lsc [.4(;J-\-, ~vllo IIS(YI (~irv111:lr 11ips (-ov(~rc~l 011 I,IIc it~si(Ic $IS
t igltt Iy :IS ~)ossil)lv \vit.l~ s:~t~tl of :I t1cfinit.r gr:\i~~ sizt, glc~cvl 011 ($0 l llc w:~ll. 13y c-hoosing
1)il)cs of v:~rying (Ii:~n~r~t.rrs n~ttl Ity (hnging IIIC sim: ofgrnit~, IIC wn.s :10lc 1.0 vary the
rc-l:l( it.(, ro~~gl~ncw f.,/ll liom nbol~t 1l.500 1.0 1/15, 'l'ho rrgl~l:~rit.irs of I)rhaviour tlisc~)v(.r~~l
(111ri11g 1.l1r ( WII~SC of l.I~rsv mr:~s~~rrtn(mts can .bc: corrrl:~t.ctl \vit.l~ l.l~osc hr
stno01 11 pil)~*s ~ Ia, I sitt11)lc tn:~tlnw-.
\\'v sII:III Iwgi11 I)), (l(w.ril)i~~g Nik111x11sr's III(~:LSII~(-I~I(~II~S :III(I w: SII+III t,It(:n SIIO\V
t.11:11. 111r rrl:~I~io~r I)c~l~wrc~~~ t.l~c ~wisl,:~nc~r hrlnrtl;~, nntl I.llc: vrlori1,y tlisl.riI)~ction, \vl~ic.h
\\r I;)IIII(I rilrlirr ill t11r (.:IS(- 01' sn1ool11 oil~vs. van I)c esl.rntlr~tl t.o t,l~r rnsc of ror~gh
l)il~~,s ill :I II:II
IICII W I ~ .
t 111.whnl. Follows wc, sl~nll 11sr t.llr sylnhol ks Lo rlcnotr LIIC grain size ill Nikuratlnc's narltl rorlgll-
wsu, wnrrvit~g I.hc ny1111)ol k for :III$ ol,llrr ltillrl of rollghncsn.
: rl'l~c nu~~~t*rir~:ll ~:~IIII.s of t,? I-*/v q11otw1 11rrr will Iw hivc(1 1:tI.w ~~OIII 1.11~ velocity dist.riht~inl1
law. Tl~cy :IM: v:lIirl OIII~ for rrw~l~~~rssrs ol~k~incd with mtld.
Fig. 20.18. Rosiatance formuln for rough pipes
The size of t,hc roughness is so smdl that, all prottrusio~~s :ire conf.ni~~c:tl \vit,hit~ the
laminar snl)-layer.
Protrusions cxtcntl pnrl.ly out.sitln Ihc I:rminnr sub-l:~ycr ;1n(1 1.11~ atltlit,io~~:~l ~.rsisl:~nty~,
as cornp:irc:d with a srnoot.l~ pipe, is ~n:iitrl~y cl11o t>o lhc limn tlr:ig c:spr.ric.~~c.c-tl Ipy 1.11~
pl'oLrusions in the bo~~rltlary layer.
Vclocity dis~ribution: '['he vc.Ioci(.y gr:uliotlt no:cr :I. rollfill \v:1.11 is Ivss slwp I.Il:rr~
f.l~al. nrar a sn~oot.l~ onc, ns can I)c scrn from Vig. 20.1!), in whicl1 l.I~r vcloc~i1.y ~-:~l.io
71/11 has Gccn plol~ctl against the distance ratio y/ll for a smoolh a~ltl hr scvcwl