Boundary Lyer Theory

of I,ho resultant forcc parallcl to the unciisturbctl initial vrlority is referred to as t11e
drag I), and the component perpencliculnr to that tlircct.ion is callctl lift, 5. Hencc
the dimensionless cocfficicnts for lift and drag become
L I)
C - nnd C, = - - --- - ,
, - A
(I .Id)
18VSA
if the tlynnn~ic: 11cad 4 Q V2 is SCICC~C~ for rcfcrcrlce instcatl of t,hc tlunnt,ity e V2.
Thus tho argumcnt leads to the conclusion that the tlimcnsionless lift a,nd drag
coefficients for geometrically similar systems, i. c. for geometrically sirnilar bodies
which have t h same orientatmion with respect to the free-8trea.m direction, are
functions of orie variable only, nmnoly the Reynolds numhcr:
c,,=/,(R); CD=/~(R). (1.15)
It is ncccssary to strcss once more that this importmt conclusion from Reyr~olds's
principle of similarity is valid only if the assumptions undcrlying it are satisfied,
i. c. if the forces acting in the flow arc due to friction and inertia only. In the
casc of compressible fluids, whcn elastic forccs arc important, and for motions with
free surfaces, whcn gravitational forccs must be taken into consideration, eqrrs. (1.15)
do not apply. In such cases it is ncccssary to deducc diKerent similarity principles in
which the tlimensionless Froudc numlw F = v/G~ (for gravity and inertia) and
the c1imensionless Mach number M == V/c (for compressible flows) are included.
The importance of the similarit,y principle given in eqns. (1.14) and (1.15) is
very great ns far as the scicnccs of th~orct~icsl and cxpcrimcntal fluid mechanics are
concerned. First, the dimcnsior~lcss cocfficicnts, C,,, C,, and R are irlclependent of
the system of unilm. Secondly, their use leads to a considerable sirnplificntion in
the cxtcnt of expcrimcntal worlc. In most cases it is impossible to tlcterminc the
func(.ions f,(R) and /,(R) throrctic:ally, antl exporimcnt,:~i ~ncthotls must be 11sot1.
S~~pposing tl~ali it is tlcsirccl to tlrtcrrnino thc tlr:~~ cocfficicr~t ITI, for a spot,ilic:tl
s11:q)c of hly, c. g. a sjhcrc, tllcn witl~ot~t the application of Lhc principle of sirni1:~rit.y
it wo111tl hc? ncccssary to carry out drag mcasuremcnt.~ for four indepcntler~t variables,
V, d, Q, and p, antl this would const,itute a trcmondous programme of work. It
follows, however, Lhat t2he drag cocfficicnt for sphcros of diKcrcnt tlinmctors with
different stream vclocitics antl tliffcrcnt fluicls clcpcntls solcly on onc v:~ri:~l)lc, 1.h~
Reynolds r1urn1)cr. Fig. 1.4 rcprcscnts thc dmg cocfficicnt of circular cplintlcrs as
a fi~nct~ion of the Itoynolds number antl shows the exccllcnt agrccment hetwceri
expcrimcnt antl Reynolds's principle of similarity. The cxperimentnl point,s for
the drag cocfficicnt, of circular cylinders of widely differing diameters fall on a single
curve. 'The same applies to points ohtnined for the drag cocfficicnt of spheres plotted
against t,ho Iteynoltle number in Fig. 1.5. The sutltlcn decrease in the value of thc
drag coefficient which occurs near R = 5 x lo5 in the case of circular cylinders and
near R = 3 x 10"n the casc of spheres will be discussed, in n~ore detail, later.
Fig. 1.6 reproduces photographs of the stream$nes about circular cylinders in oil
taken by P. JIomann [7]. They give a good idea of the changes in the ficld of flow
associated with various Reynolds numbers. For small Reynolds numbers the wake
is laminar, but at increming Rcynolds numbers at first very regular vortex patterns,
known as Khrmhn's vortcx &recta, are formed. At sLill higher Reynolds numbers,
not shown here, tho vortex patterns become irregular and turbulent in character.
c. Principle of si~nileril.~; 1110 Ilcynolds nnti Mach numlwxs
2 = V'J
Fig. 1.4. Drag coefficient for circular cylinrlcrn n, a function of tlie Jleynoltls n~~nibcr
4 00
700
C~ roo
80
60
10
70
10
8
G
L
7
I
08
0 6
0 4
0 2
0 1
08
0 a
Fig. 1.5. Drag coefficient for spheres aa n fiulction of tho Reynolds nulnbcr
Curve (1): Stokcs's theory, eqn. (6.10); curve (2): Oseen'a thcory, eqn. (0.13)
17