Boundary Lyer Theory

Ilcre v, c, :tntl k tlrnol,c 1.I1t: tlrnsity, sl>rc:ilic Itc::rt, :mi contlucl,ivit.y of Ihc lluitl
rc!spwlivrly; 0 in tltc! tlilli~rcnc:c? I)ct,wco~~ Ihc loonl t,t:llll)onrt,r~rc: ant1 (,hat at :t vory
1;rrgc tlisI.:~t~(:c fron~ t11c I)otly, wl~orc: IJlc: l.c:tnpcr:~l,~~rc:, 7', is c:onsl,ant ant1 cq~~:rl lso
'/, i. c . 0 - - '/ - '/',,,. 'I'llt: vcloc(ity lic:ltl w(z, y) :rt1!1 ~(z, y) in oqn. (4.12) is ;~.ssurnctl
to I)c known. 'L'hc t,ernpnrat~~rc: distribution on the I~ountlarics of Lhc botly tlcfinetl
b?~ '/I,, 3 7', is prrsc:ril)ctl nrttl in the sirnplcst cnsc it is constant wit11 respect Lo
sl)nct: and t.imc 1)111., gcncrnlly spcalting, it varies wit11 both. I'rom the pl~ysioal poinl.
of view cqn. (4.12) roprosents the 11c:rt 1):~lanc:c Ihr an clcn~cnt,ary volut~~e. 'l'hc Icfh-
Ilantl sitlc represents t.11~ qu:mt,it,y of heat, c:xcl~:~~~gotl I)y convcc:tiorl, whereas the
rigl~t-ll:rnd side is the ~~~;~n(.it.v of 11r:lt t:xt:I~:~n~cd 11y con(I~t(:tion. T11c frit:l.ion:~l hcatl
gcneratcd in tile fluid is ncglcetctl. Tf 7',, > T,, tho prol)lom is that of detcrrnining
1.ltc tcmperatl~rc field around a hot body which is cooled. By inspection it is scrn
that cqn. (4.12) is of the same form as eqn. (4.6) for the vorticity w. In fact thcy
hccomc itlentiral if the vorticity is replaced by thc tcmpcraturc tliffercncc and t.hc
kincmatic viscosity v by t,hc ratio kip c known as thc thcrmal diffusivity. 'l'hc bountlary
conclit,ion 0 - 0 at a largc distance from thc body corresponds to the condition
tr, = 0 for the undisturbed p,nrnllcl stmam also at a largc distance from the body.
llcncc we may expect that thc solutions of the two equations, i. e. the dislribntion
of vorticity antl that of tcmpcraturc around the body will be similar in chnrnctcr.
Now, tllc tcmpcratlrre dist,riI)ution around the body may be pcrccivcd intnitivrly,
to a ccrtnin cxlcnt. In the limiting ca.sc of zero velocity (fluid at rosl) the inflncncc of
tilo I~eatccl 11otly will extend ~~niforrnly on all ~itics. With very small velocit,ics tho
fluid arountl the hody will still he affectcd by it in all directions. With incrcnsing
velocity of flow, howcvcr, it is clcarly seen that the rcgion affected by the higher
tempcreturc of the body shrinks more and more into a narrow zone in the immetlint,c
vicini1.y of the body, antl into a tail of hcatcd fluid bchind it., 1Pig. 4.3.
. -
-.__ -- -
--__
Fig. 4.3. Annlogy betfween trnlperntuw
and vorticity di~tribution ill tho neigh-
bol~rhd of R body plnml in a strerrrn
of fl\lid
a), b) IAndCq of rrgion or iw.rrhsrd trmprrsture
n) for ~rnnll vclucitlrs
---------__..______
Ir) fur Inrge vrlucitirn uf flow
'rllc so111l.ion of eqn. (4.1 2) ni~rst-, a.s mcnt,ionkd, be of a chn.ra.cter similar t.o that
for vorticit,y. At snlall velocities (viscous forces hrge compared with inertia forces)
t.hcro is vorticity in ihc whole region of llow around the body. On the other hand
for' largc vcloeitics (V~SCOIIS forccs smnll compnrctl wibh ir~ctl~in forccs), we may
rx-prct, i field of flnw in which vorticity is confined to a small Iaycr along the surfacc
of the I)otly antl in a wake behind thc boily, whereas thc rest of the fcld of flow
c. 'l'ho limiting caw nf vory ~nlnll V~RCOIIR li~rtm 79
remains, practically speaking, free from vortioity (scc Vig. 4.1). 11, is, I.l~ercforc,
to be cxpected that in the limiting case of very small viscons forces, i. e. nt 1;rrgc:
Itcynolds numbers, the solutions of the Nevicr-S(.okcs cq~~:~t~ions arc SO ('otlsti(.~tt.rtl
as 1.0 permit a suldivision of the ficld of flow inl,o an cxtcrn:~I rrgion wlti(*ll is fr~o
from vorticity, and a thin layer near the I)otly togcthcr with a wakc I~(:llit~tl it.. I11
1.11~: first, region tho Ilow mny Im oxpnctctl 1.0 sntisfy OIO ctl~~~rtions of I'ri(:(,iot~ltw
flow, the potcr~t~ial llow theory bcing uscd for ih cvalnation, whcrcas in tllc sc~-otr(l
region vorticity is inherent, and, thcrcfore, the Navicr-Stoltcs cq~iations m~~st. hn
uscd for its evaluation. Viscous forccs are importm~t, i. c. of thc santc ortlcr 91'
mngnitt~tlc n.9 inertia forces, only in tl~c scc:ontl region known :is thc bou~tdrtr?y Irr?yrr.
This concept of a boundary layer was introduced into the scicncc of fluid mechanics
by L. Prantitl at the beginning of thc present ccntury: it has proved t,o he vcry
fruitful. The subdivision of the field of flow into tho frictionless oxtcrnnl Ilow iwtl
the cssentinlly viscous boundary-laycr flow pcrmitkd thc reduction of the rnnt,llcmatical
difficnlties inllorent in the Nnvicr-Stokes cqnatior~s to snch an extent, that
it, lmmne possible to integrate them for a large numbcr of cascs. The tloscript.ion
of t,l~csc methods of integration forms t.hc subject of the boundary-laycr thnory prcscntctl
in the following chapters.
From a nt~mcrical analysis of the available soluteions of the Navicr-Stokc~s
cq~~ations it is also poasiblo to show directly that in tho limiting case of vcry lnrgc
Reynolds numbers there exists a thin boundary laycr in which the influcncc of viscosit,y
is conccntratcd. We shall rcvcrt to this topic in Chnp. V.
The previously discussed limiting case in which viscous forcrs heavily outweigh
inertia force3 ((creeping motion, i. e., very small Reynolds number) results in a considerable
mathematical simplification of the Navier-Stokes equations. By omitting
the inertia terms their order is not rcduced, but they become linear. 'J'hc second
limiting case, when inertia forces outweigh viscous forces (boundary layer, i e. very
large Reynolds numbem) present8 greatrr mathematical difficulties than creeping
motion For, if we simply substitute v = 0 in the Navior-Stokcs equations (3.32),
or in the stream-function equation (4.10), wc thereby suppress the derivatives of Lltr
highest order and with the simpler equation of lowcr order it is impossible to satisfy
sirr~ultancously all boundary conditions of the cornpleto tliKrrcntial equnt~ous.
However, this does not signify that the solutions of sucll an equation, sin~ldificd by
t.he elimination of viscous terms, lose their physical meaning. Moreover, it is possil~lc
to prove that this solut,ion agrees with the &mplete solutionof the full ~ svic~:-~toke~
cq11nt.ions nlmost. everywhere in t,he limiting case of vrry large Reynoltls nrtmb~rs.
Tho exception is confincd to n thin lnycr near the wall - the bountlnry In.yc;r. l'h~ls,
thr complete nolution of t.hc Nnvicr-Stmkcs cqtralions c:nn I)c tl~orrgl~l of nrc t:onrcisI,ing
of two sointions, thc so-cnllctl "outcr" solution which is ohtninctl with the nid of
Eulor's equations of motion, and a so-callcd "inner" or bonndnry-1n.yc.r solnt.ion
which is valid only in the thin layer adjacent to the wall. The "inner" solut,ion
satisfies thc so-called houndary-layer eqmtions which are dctlncctl from tho Navicr-
Stokes equations by ~oortlinat~e stretching nnti pwqsagc to tho limit R + m, n.s will
be shown in Chnp. VII. The outer and inncr solutions must he malchcd t,o ench other
by exploiting the condition that thcrc must exist nn overlapping rrgion in which
bbth s&tions are valid.