Boundary Lyer Theory

4GO XVI. Origin of turbulence 1 1). Principles of the theory or stnbility of Inrninnr flows 4G1
JIerc c, t1cnot.c~ the vclocit,y of propaption of t,hc wave in the z-dircction (phase
vclooity) wl~crcas c, again dcLertnincs t,ho tlcgrce of damping, or nmplificntion, ticpending
011 it,s sign. I'hc arnplit,~~tle function, 4, of the flnct,tlation is ass~~~nccl to tlcpcntl
on y only ~C(:ILIISC the mcm Ilow tlcpcnds on y alone. I'roni cqn. (16.10) it is possible
to ol)t,ain 1.11r c:ornponcnlss of thr: pcrt,rrrl)nt.ion vcloc:it,y
11,' -
t)l/,
-
i)y
: ,#,'(?/) (.""' PI' , (10.12)
1) -=
Ry1 --
ax
i u +(?I) &my PI) .
Irlt.rotl~tc:ir~g I,l~c:sc- valnrs inh rqns. (16.7) and (10.8), we obtain, after the elirninat.io11
of prrssnrc, l11c following, ordinary, fourth-order, differential cquation for the
n.tnl)litwlc 4(y) :
I
When the mean flow IJ (11) is specified, eqn. (IF. 14) contains four pamrnnt.rrs, n :~n~cl~ a,
R, c, anti c,. Of t.hcse the rtcynoltls nrlml)or of the moau flow is liltcwiso spccill(:d
and, f~~rthcr, the ~avcl~ngth i = 2 n/a of the disturb:m(:c is to bc consitlcrctl given.
In t.his rase the differentid equation (IF.l4), togcthrr with thc Imlntlary c:ontlitiorrs
(16.15), f\rrnish one oigrnf'uncLion +(?I) and one complex cigcnv:~ll~r: c == c, 1 i ci
for (wh pir ol' v:~l~tt.s a, R. I Ivrc cr rt:~~rcsw~l~s I,IIc ~)II:LSC v(hwiI,.y 01- 1.11~ l~rw(:ril~t:~l
t1isturbnncc wltcrcas thc sign of c, tlcLcrnlit~cs wllctllrr bho wave is amptilictl (ci :.O)
or tla.tnpctl (ci < O )t. For c, < 0 the correspontling flow (IJ, R) is stnth for lllc ~ivcn
valur of a, wllcrrns c, > 0 tl~notcs il~stabilit~y. Thc limiling case c, - 0 corrrsponds
1.0 nrnLml (indiKcrmt) dist~rrl~ancrs.
11nm1)cr is the critical Reytrolds nuinher or limit
of Ian~inar flow untlcr cot~sitlcrxLion.
Fig. 16.8. Curves of neutral stabi1it.y for
two-di~nensionnl borlntlary layer wi(.tr
two-dimensional disltlrbrmccs
(a) "non-visrolls" inslal,ilily; in thc rnsc nC vclocity
~~rnillrs or 1yp :t wilh pin1 *d i~~flrxion
PI, thr rllrrc or 1w111rnl slnbility is or typc :,
(1,) "visrow" Instahllity; in I11e raw or vriorit,)
proIllrr nl type 8 uilhotrt poi111 of lnflrxion, llw
rurve ol erutml shbilily is or lypc b
Tl~n nryrnglotm Tor ll~n rrlrvc or rwulral stnbility
a al. R --+ r, arc ohlxincd rroln llw "rrirlionlrar"
slal~ilit?. rrlllalio~i (16.16)
r ,
I he rxprritnrntd rvi(Irncr eonwrnit~g t,ransit,iot~ fro111 l:~n~in:t,r 1-0 I,II~I~~III~II~,
flow rcfrrrrtl 1.0 ~)roviorlsly I(.nds 11s t,o cs~~rcl, that, ;LI, SIII:LII Ib(~,y~~oI~ls I I I I I I I I ~ for ~ ~
wl1ic.11 I:~minar llow is ol~servrtl. all \v:rvrl~ngt.l~s wo~lltl l)rorluc*v ot~lg sl.:rl)lr tlist.llrl):~l~c~s,
wl~rrrn.s :lip I:~r.gt-r ltc!g~~oltls rtlln~l)rrs, li~r wllic~l~ I.II~IIIII~:III, IIow is 011svr\~1~1,
ul~st,:~l)lc tlist~lirbarlccs o~tght, t,o corrcs~~ontl to at, I(ywt, sonto \v:Lv~I~II~~.IIs. IIo\r.(:v(~,
it is nwrssnry t,o rcn~nrk at tallis poitlt t.l~:rt~ t.11~ vrit,ic:n.l I