Boundary Lyer Theory

\\+ere j = 0 (plane flow) or j = 1 (flow with axial symmetry). The boundary condit,ions
arc 16 -- I) = 0 at y = 0 n.ntl 11. =- 11 (n.) at ?/ - 0. For turbulent flows u-and v
a.rc t.he n~pprop~.ink lnrnn vclocit,ics n,tltl ~1 rrprescnt.~ a suitably defined cdtly viscosit,y,
scc for inst,nncc A. M. 0. Smit.11 n.nd 'l'. Cc1)coi [Dl]. I'or Inminer flows €1 .= 0.
7'11~ l,rnl~sft,~.r~~al,iot~ of rqt~s. (9.50) ancl (9.57) to din~en~iot~l~s~ vnria1)Ies incorporaLes
11otJ1 t11(, l$ln.sir~s ant1 lhc Mn~lglrr l.rnnsfot~tnn.titr~~ st^ nlso II. (:orrl.lcr I:13, 341) nntl
is tldined n.s ft)llows :
Tllc cont.it1ui1.y equn.t.io~~ is sat.isfictl I)y the st,rcarn function
nntl E, is t,llc rdtly visvosit,y from ecln. (39.2). The s~~l)script,s tlrnotx- part.inl ctifkrctit.ia.Iion,
anrl the qunnt,it,y
5
77~0; /=O; / =On.r~tlq=cm; I,-- I. (9.63)
Fi~~it~e-tlifl'c?rcncc: cqnnt,ions of sccontl order can I>(: SOIVC~ (by mnt,rix inversion
ront.inrs) rn11r11 nlorc cfficiont.ly trllan t.llirtl (or higher) ordcr equations. It is of intcr-
?st,. t,hct~eforo, t,o rrclurc equations (9.01) to sccond ordcr. To this end the variable
I;' -= /, is int,~ducrtl and eqn. (9.01) is rewritken as
INF'v],l I /FII --F~)-T-~((FE'~--~~F',~]. (9.64)
'J'llis rqua(iot~ now conlains two unknown functions, f and 1'1, hut tllrse ale related
by thr sirnplr rxprcwion
In the absence of srlctior~ or t)lowing the boundnry eonditions nrr
This strip is completely covered by a grid with lines drawn parallel to t,he ( and
coordinates as illustro.t,ctl in Fig. 9.16. Tho stq sizr A[ rcl)rcsc~~t.s t.11~ tlist,nnrc.
bet,wcer~ t,wo snr.crssivc grid lincs 5 = const,a~~t; it is prost~~nctl i.o I)c stn:~ll IIII~, is
ot,hcrwise rrnspccifictl. 'I'hc corresponding step sizes in t.hc q-tlirrctiou nrr spc:c.ilictt
t,o vary in geometric progression. The rnl,io Octwecn t,wo s~~ccrssive grid lincs, TI,
and qn+l, is denoted by I< = I -1 k where 1 kJ varies from 0 t,n 0.05 in l.ypical cases.
Each notlal point is itlcntified by a dou1)lc intlcx m, ?L which tlclinrs it.s posit,ion
Fin, 7, according tso
111 writing t>he Anitc-tlilTerence quol.icnt,s it is corivcnlent t,o int.rocluce t,he moan of
two successive Aq-values
In the step-by-step calculations the solution is considcretl known at 5,n ancl ell
preceding grid lines, and the variables F nntl / are sougl~t at. [,,,, 1.
Fig. 9.16. \'ari.zble stcp size finit.c-rliffrronce grid
for th rnlcr~lnt,ion of laminnr and turbuletlt'
Iiounrlnry Inycrs
x knon.~~ vnlucs,
O r~nknnwn rnll~rs