Boundary Lyer Theory

The vclwil y IL at some point i11 tlrc velocit,y field is proportional to tlte free
strrnm velocity IT, l,he vcloci0y gratlicnt au/ar is proportional to Vld, antl similarly
a2tr/~y2 is proporlional to V/d2. Ilcnce the ratio
Thereforc, tllc condition of ~irnilarit~y is sat,isfictl if the ql~antil~y p V d/p f~as the same
value in bol,l~ flows. The (pntity p V d/p, which, with 11.1~ = v, can also IN wriLt,cn
ns V d/v, is a tlimcnsiotlloss nnrnl)cr \>cen.tlsc it is the mt.io of t,l~c t,wo forces. It is
known as t.110 Ilayitnk1.c ~slr.?ttl)ar, R. Thus t,wo flovs arc similar when the lt:lin
three crqna(.ions :
F : )I -4- 0 : 0 ,
the solution of wlticl~ is
Din~ctlsint~lcss quantities: 'I'hn reasoning followctl in tho precetling drrivi~f ion
of the Rcynoltls numl~er can be e~t~entled to inclndc the casc of diffcre~~t Itrynolrls
numbers in the consitlerat,ion of the velocity ficltl ant1 forccs (normn.l :mtl tangont.i:rl)
for flows wiLh geornetrica.lly sitnilar boundaries. Let thr position of :L point in (.he
space around the gcomctrically similar bodies bc intlica1,cd by thc coortlin:tl.t~s 1, !/,
z; t~llen tho rat,ios z/d, y/d, z/tl arc its tlinicnsiotlless coortlirt:~l,cs. Tl~c vc~loc:il.y c:otltponcnt,s
arc lnatlc dimensionloss by relirrring tllern to the free-stream vch:iI,y V,
thus 711 V, 111 V, w/ V, and lhc normal and st~caring strosscs, p :~ritl t, can bo mn.clcr tlirnct~sionlcss
by reforring thorn t,o Lllc tloubfc of t,llc tlyrtatnic lieatl, i. e. to p V2 t.hus: p/p 1'"
and t/p V2. The previously cn~~nciatcd principle of dynnmical sinlilarit,y can Im c~x1)rt~s-
sod in :Ln alternative form by asserting t ht for the two gcornctricnlly similar sys1,cnls
with equal Reynolds numbers the dirncnsionless quantitics 141 Y, . . ., p/p V2 i~nd
t/e V2 depend only on the dimensionless coortlinatcs x/d, y/d, z/d. If, Ilowcvcr, the
two systems are geometrically, but not dynamically, similar, i. c. if t.lleir Rcynoltls
numbers are different, t,llen the tlimensionless quantit,ies under consideratlion innst,
also depend on the chamctcristic quantities V, d, Q, 14 of the two ~ystcrns. Applying
the principle t,llat physical laws must be independent of the systcn~ of nnit.s, it. fi~llows
that tl~e tlimensionless quarifities u/ V, . . ., p/e V2, T/Q VZ can only depend on a
dimcrlsionless combinatlion of V, d, Q, and 11. which is unique, being the Itcynolds
number R = V d e/p. Thus we are led to the conclusion that for t01c two gcon~cbrically
similar systmns which have different Rcynolds numbers antl which arc bring
compared, the dimensionless quantities of the field of flow can only be funcI.ions of
tlic tthree din~ensionless space coordinates z/d, y/d, z/d and of t.lw Rcynolcls
number R.
The precc(ling dirr~cnsinnal annlysin can bc ~~lilizctl to tu:~ltc an irnport,:r.ttt,
assertion about the t.otal force excrtcd l)y a fluid strealn on an imrncrsotl hotly. 7'11c
force acting on tho bocly is the surface intcgral of all normal and ~llcaring stmsst:s
acting on it. If P denotes the component of the resultant force in any given direction,
it is possil~le
to write a tlirncnsionless forco coefficient of the form P/d2 Q V2, 1~11,
stead of the a,re:b d2 it is cnstomary to clloose a diKcrcnt charactcrist.ic aro:l, A, of
t,he immersed body, e. g. the frontal a.rea exposed by the botly to tile flow tlircct