Boundary Lyer Theory

2 (i
TI. O~~tlittr of Imun~lnry-lsyw throry
E~tin~nIin~t of houndnry-lnyer thickllr~s: 'rhc t,l~ickness ofa boundary layor whir11
llas riot sepnrnlrtl can I)(! casily rst,irnnLrtl in thc following way. Whcrcas friction
forccs can be ncglcctctl with rcspoct t.o incrt,ia forccs out,side tho bourltlary Ixynr,
owing to low viscosit,y, thry arc of a comparable order of magnitrldc inside it. 'rhc
inert,ia forcc prr nit volun~ is, as cxplninctl in Scct,ion l e, equal to Q 71 &L/~x. For
a pIat,o of longlh 1 tho gr:ttlinnt arr/a:r is proportional to ll/l, where IJ tlrnotes thr
velocil,y onLsitlv the! I)ountl:wy Inyrr. Ilct~rc Ihc irlnrl,in forcc is of tho ortlcr I, 1J2/1.
On the othcr l~antl the friction forcc per nr~it volurnc is equal to at/@/, wllirll, on tho
assurnpt~ion of lnrninnr flow, is cqunl t,o 11, a21t/i)?/2. The velocity gratliont al~/ay in a
tlirrcLion prrl~rnrliculnr t,o t.l~c wall is of t,lm ordcr Ill6 so that thc friction forcc ])or
~ti)il~ ~olt~tnv is i)~/&y - lI/d2. Proni the cotdit.iorl of equality of the friction :md
inertia forcrs tho following rc.l:ll ion is obhined:
U e UZ
t4 82 - 1
or, solving for I Itr Imuntl;~r~-layrr tlriclcr~rss Ot:
The I~nlnr,ric:nl f:~rt,or wltid~ is, so f:w, st.ill untlct,crn~ined will be drduc:ctl Iatcr
(C!l~:lp. VII) from tho exact solut,ion givcn by [I. 13lasius 141, and it will turn out
t.llnt it is cqrlal 1.0 5, al)proxinlatcly. llrncc for lnmiarrr flow in the bountlary layer
wn hnvo
(2.1 a)
'rho tlinlrt~sionlc~ss 1,our~lnr~-lnyer thirknrss, rcfcrrctf to the length of the plate, 1.
twronles .
wllorr R, clrnotcs tho ltcynoltls nunlber rclatod to the Icngth of the plat.c, 1. Tt is
won from cqn. (2.1) tallat thc boundary-layer thickness is proportional in 4; and
t,o I. If I is ropla.cetl hy the variable tlist~nce z from the leading edge of the plate,
it is seen that d increases proporti~nxt~ely to ii. On tho other hand tho relative
boul~(~ary-Iaycr t,I~ickncss O/i decrems with increasing Reynolds number as I I ~ R
so that in tho limiting case of frictionless flow, with R -+ oo, tllc boundary-layer
t.lrickness vanishes.
We are now in a position to estimate the shearing stress zo on the wall, and
consrq~~ontly, t.hr t,ot,ni drag. According to Newrton's law of friction (1.2) we have
- - --
t A ~~lore rigororts tlrfiniliott of Im~lrtclnry-Iayrr thicknrsn in given st the end of lhia section.
wherc sl~bscrip~ 0 tlenotes the value at the wall, i. e. for y = 0. Witll thc estimate
(au/a~)~ - U/d we obtain 7, - ,u U/d and, inserting the value of d from cqn. (2.11,
we have
We cart now for~n a dirncnsionlrss sl,rcss with rcTrrnrlrc lo I, llz, ns c~xpl:~ittc~cl
in Cltnp. I, ant1 obtain
c,, - =
I'q .
1
- -
The numrric:ll fartor follows from 11 Blasius's cxart solution, atttl is I 328, so tll:~~,
the drag of a ~~lntr in parallrl 1nmin:~r flow 1)rromc.s
Tltc following nt~mrrical rxamplc will serve t,o il11tst~~rt.c: t.hr l)rec:rcling c:st,i~rt:~ t.iolt :
Laminar flow, stipulntctl here, is obt:~it~rtl, as is known r'ronl exprritnctlt,, for Itcynolds
numbers CJllv not cxceccling :d)outt 6 x 10Ql.o 10% lpor 1nrgc.r I