Boundary Lyer Theory

CIIAFTER VIII
Gencral propertiee of the boundary-layer equatione
12cforc: passing to t.lw ca.lcr~l:ll~ion of furtl~cr cxarnplcs of bountlary-layer llow
in t.ha next, chnpt,rr, we prol)os': first, t.o tlisc~lss some grncral propertics of the bound-
:~ry-l:rycr t:quatiorls. 111 tloing so wo shall ronfinc our atttlention to steady, twotlimension:tl,
ant1 ir~c-o~r~~)rt~ssiI)l(~ l)o~~n(lar~ I:ly~rs.
Alt,hougl~ t.Iw ~)o~~ntl;trj--l:i.yt~r rcl~~:ttions have hen simplified to a great axtmt.,
as coml)arctl \vit.l~ t,hr Navir~.-St.oltcs rclr~at,ions. thoy arc still so tlifficult from t'he
matJrrn~at.ical point of vicw tht. not vdry marly gcncml ~t~atcn~rnts :rbout tlicrn
ran I,c matle. 'I'o I~c:gin wit.ll, it. is import-antf to not.ice that t,he Navier-Slolrcs
aqlla.t,iotls :trc or t,I~t? rllipt.ic. typa wit,h rrspcct to tllc c:oordin:~l,cs, whcrms Pranrltl's
l~o~~t~tI:~.ry-l;t~~t~r cq~~:~,I~iot~s :tro p:ir;th~lic, It, is :L cowo~~~~t~rwr or lhc sin~plifying
nss~rrnpt~iol~s in Imuntl:rry-layer t,hcory that tho prcssuro can be assumeti constant
in R clirrction n.t right :~nglcs to the hountlwy Inycr, whereas along tho wall the
1wess11rc can be rcprdetl as bcing "imprcsscd" I)g the external flow so that it bccwncs
a givrn f~lnc:I.ion. The rcsr~lt~ing omission of t,hc arlnntion of motion porpcntlicul:ir
t.o the tlirccliott of flow can be i~~tcrprctctl physically I)y stat,ing that a fluitl
~);trt.ic.la in tha l)our~tl:~ry Iaycr has zcro mass, and sulTcrs no frictional drag, as far
;rs it,s motlion in t.11r t.mnsvcrsc ttirecLion is conccrnrcl. It is, tl~crcforc, clear t,ha.t8
with sr~t:lr f~lnrl;trnrt~t,al cl~angcs introtl~~ccd int,o the cqtlat,ions of n~ot~ion we mnsb
nnt.ic.ipatc t.ll:~t, tllrir solut,ions will exhibit certain rn:ltI~cmatical singnlarities,
nn(1 t.ll:lt, :tgrrrrnrnt I,c:t,wrcr~ ol)scrved :t11(1 ~illt:ulat,ed phrrlon~ona cannot always
'I'ho assu~npt~ions which warc rnatlc irt tho tlcrivation of t,hc tmuritlary-layer
rq~tntions are s:~tisfictl with an increasing tlcgrce of accuracy as the Itaynolds number
ir~c:rrnses.
,,
l hils hountl:~ry-layer thcory can bc regardcd as a process of nsymplolic
i~itrgmtiol~ of t,llr Nn.vicr-Bt,olrrs rqnn.t,ions at wry In.rgc Itcynoltls nurnl~c~~s*. 'rhis
sl.:~trmrnt, Irntls 11s now to R tlisc~ission of the yclnt,iortship bet.wcen t(11c Itcynoltls
nirmhcr and t.he chn.rnctt~ris(.ics of a t~~indary 1h.yer on our individrlal body-under
consitlcrat,ion. It, will 1)a reanllctl t,hat in thctlcrivat~ion of the boundary-layer equations
--- - - - - --
t C/. Set-R. Vllf nncl IXj.
* 'I'llo srg,~n~rnt, t~ont,ninctl in tl~in nwtion wan nlronrly tlinc~ls.st?d in Sec. Vllf on high-order
II~)~~~OX~III:~~~OIIR.
'I'IIv itll~l~lilir~~tio~~ is givm I~c.rr for t.lw mltc of Iwl.l.rr ~l~~tlrrnt~it~tlil~g.
a. Drpel~denrc of the rhnmcteristicn of n. boundary lnyer on the llry1101dn IIIIOIIIPT 151
tlinler~sionlcss quantities were used; all velocities were referred to the free-stream
velocity IT,,, all lengths having been retfuced with thr aid of n cl~aractcristic length
of thr botly, 11. 1)cnoting all tlirnensionless magnitutfes I I a ~ prime, thus v/fJm, =u',
. . . , x/L = z', . . . , wc obtain the following equations for the steady, two-tlimrnsionnl
CRSR :
scc nlso cqs. (7.10) t,o (7.12). Itere R dcl~otcs t.lro ltcynolds nurnbrr for~ntyl wit.11 t,)lc
nit1 of 1.11~ rcfcrencc qunntitics
It is seen from eqns. (8.1) and (8.2) that, the boundary-layer solution dcpcnds on
ow parameter, the Iteynolds number R, if the shape of the botly, and, hcnc:c, t,hc
potential motion U1(x') are given. By the use of a further transformation it is
possible to clirninnt.~ the? Rcynoltls number also from cqns. (8.1) nnd (8.2). If wt: p111.
eqns. (8.1) and (8.2) transform into:
with the boundary conditions: v' = O and v" =O at y" -0 and 71' .= U' at y" =a.
, J , hese equations do not now contain the R.rynolrls numl)cr, so that the solutions
of this system, i. e. the functions u1(z', y") and v" (sf, y"), are also independent of the
Reynolds number. A variation in the Reynolds numbcr cnnscs an nffinc t,rnns-
formation of tho boundary lnynr during which tho ordinn.t,o nntl the vclonily in 1,11(.
transverse dircction arc mult,iplictl by R-'I2. In othcr words, for n given botly tho
tlimcn~ionless velocity components M/U, ant1 (v/U,) . (U, L/V)'/~ n.ro fur~cl.ions
of the dimensionless coortlinates z/L and (?//I,) . ((I, I,/V)'~~; the functions, marc,.
over, do not depend on the Reynolds nun~bcr any longer.
The practical importance of this principle o/ nim.ilat.il?y wifl~ resp-1 lo Ilrynold.~
nirmher consists in thc fact that for a given body shape it suffir:cs to find the solrtt,iot~
to the l~oundary-layer problem only once in terms of the above tlimcnsionless varia1)lcs.